(Modern Problems in Condensed Matter Sciences 27, Issue 1) G. LANDWEHR and E.I. RASHBA (Eds.) - Landau Level Spectroscopy-North-Holland (1991)

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IX
L. D. L a n d a u ( 1 9 0 8 - 1 9 6 8 )
Introduction
by
G. Landwehr
and
E.L Rashba
After quantum mechanics was developed in the mid-twenties, the new theoret­
ical concept was subsequently applied to problems of solid state physics. Felix
Bloch was the first to address the problem of free electrons in a periodic
potential in 1928. Two years later L.D. Landau, a young theoretician from
Leningrad, worked out in Cambridge the quantum theory of diamagnetism in
metals (Landau 1930). U p to then it had been tacitly assumed that the magnetic
properties of electrons in metals were determined by their spin and by the electron
binding in atoms. According to a theorem by Bohr and van Leeuwen, based on
classical physics, it was argued that free electrons did not contribute to the
susceptibility because a magnetic field did not change the velocity and
consequently the energy of the electrons. Landau showed that this approach was
inadequate. By solving the Schrodinger equation, incorporating a magnetic field
by a vector potential in a gauge which we now call the Landau gauge, he showed
that the motion of the electron perpendicular to the magnetic field is quantized.
The quantization is coupled with a change in the density of states and results in a
non-zero diamagnetic susceptibility.
Landau performed the calculation for a range of magnetic fields and
temperatures in which the difference in energy between two subsequent
magnetic sub-bands hw is small compared with the thermal energy /cT, with
ω = eB/m (h = Planck's constant/271, ω = cyclotron frequency, Β = magnetic
field, m = electron mass). This condition is not satisfied at low temperatures and
high magnetic fields. Landau noted that under these circumstances no linear
dependence of the magnetic moment on Β could be expected and that a strong
periodic variation in Β should occur. He concluded that it should hardly be
possible to observe the periodic effects experimentally, because they would be
XI
xii
G. Landwehr
and E.I.
Rashba
averaged out due to inhomogeneities of the magnetic field. This estimate was
based on the electron rest mass. Referring to Bloch's theoretical work (Bloch
1928) Landau noted that his calculations should, in principle, also be valid for
crystal electrons although the quantitative application of the results should not
be possible.
At the same time at the University of Leiden, the susceptibility and the
magnetoresistance of bismuth single crystals were investigated at temperatures
between 14 and 20 K. De Haas and van Alphen studied the susceptibilities in
magnetic fields up to 15 kG and found an oscillatory behaviour in the high-field
range (De Haas and van Alphen 1930). In the same year, Shubnikov and de
Haas found an oscillatory magnetoresistance (Shubnikov and de Haas 1930).
The new experimental findings showed that Landau's estimates about the
observability of the periodic variations in the susceptibility had been too
pessimistic. It was Rudolf Peierls, then working with a Rockefeller fellowship as
a guest of Enrico Fermi in Rome, who made the first detailed calculations of the
oscillatory susceptibility, which we now call the de Haas-van Alphen effect
(Peierls 1933). Due to the quantum effects in the regime hco > kT (high magnetic
fields, low temperatures) it is not possible to extend the rather general method
used by Landau for the calculations into the low-field range. Therefore, a model
calculation was performed assuming that the number of electrons was so small
that only the lowest quantized energy bands were occupied. In order to empha­
size the essential features of the quantization caused by a magnetic field, Peierls
discussed a two-dimensional model for Τ = 0, which he considered as physically
meaningless, but instructive. It is interesting to note that nowadays we have
been able to realize semiconductor heterostructures, which really behave like
two-dimensional systems.
Peierls showed that oscillatory behaviour of the susceptibility, which is
periodic in l/B, is expected not only in two dimensions, but also for threedimensional systems. In order to allow a comparison with the experiments by
de Haas and van Alphen, he performed rather tedious calculations for finite
temperatures. It turned out that there was qualitative agreement between theory
and experiment. Peierls recognized that bismuth has a particular band structure
with a very small but anisotropic effective mass. The carrier concentration he
estimated was about two orders of magnitude too small. However, in sub­
sequent years the band structure of bismuth was studied in some detail and
reasonable agreement between theory and experiment was obtained (Mott and
Jones 1936).
Further work by Shoenberg (1939) on the de Haas-van Alphen effect revealed
a wealth of detailed information about the electronic band structure of bismuth
close to the conduction-band edges. The experiments by Shoenberg were very
successful, because they were performed at liquid-helium temperatures, whereas
the original measurements had been done with liquid hydrogen.
Whereas a theory was at hand for the interpretation of the oscillatory
Introduction
xiii
susceptibility data, a theory for the Shubnikov-de Haas effect became available
only in 1959 (Adams and Holstein 1959). For a long period it seemed that the
observed quantum effects in bismuth were a speciality of this semimetal. The
theory of the de Haas-van Alphen effect by Peierls, however, clearly indicated
that Landau quantization and its consequences should be most readily
observable in solids with small carrier concentration and mass. This means
that semiconductors were in principle good candidates to observe the quantum
effects. On the other hand, the carrier concentration must be so high that the
electron gas is degenerate. In order to achieve this, the semiconductor has to
be heavily doped, which usually results in a mobility which is so low that the
conditions for Landau quantization are not satisfied. It turned out, however,
that the conditions for the observation of the de Haas-van Alphen effect in
pure metals were more favourable. Following the pioneering work of Shoenberg, the Fermi surface for a large number of metals was determined in great
detail by de Haas-van Alphen measurements. An excellent review of the field
has been given in a book by Shoenberg (1984).
For the above-mentioned reasons the number of experiments in which
Landau quantization is important was limited for a long time. However, after
the second world war a quite remarkable development began. In the thirties,
semiconductor physics was in its infancy, and it was very difficult to produce
crystals with a sufficient quality and purity. The carrier mobility was often low
and the transport data were not reproducible. This changed after 1945 when
germanium single crystals of high purity became available. Optical methods
were applied in order to obtain information about the electronic band structure.
Magnetic fields were applied more and more frequently in conjunction with low
temperatures. Whereas in the thirties only few laboratories in the world had
liquid helium available, the situation had changed entirely after helium liquifiers
could commercially be obtained immediately after the war. Certainly the
invention of the transistor in 1947 speeded up the development.
As the starting point of the rapid growth of Landau-level spectroscopy one
can consider the cyclotron resonance experiments in the early fifties. After
theoretical suggestions by Dorfmann (1951) and Dingle (1951), and a specific
proposal for an experiment by Shockley (1953), the first resonance data for
germanium, employing microwaves, were independently obtained by Dresselhaus et al. (1953) and by Lax et al. (1954). These experiments revealed significant
details of the structure of both the conduction and the valence bands of
germanium. Whereas the experiments for electrons in germanium could be
explained on semi-classical grounds, it was necessary to base the interpretation
of the hole cyclotron-resonance data on a full quantum mechanical treatment of
the Landau quantization.
Although magneto-optics in solids is nothing new - it dates back to
M. Faraday - something qualitatively different was involved when lowtemperature magneto-optical studies in high magnetic fields were performed in
xiv
G. Landwehr and E.L Rashba
the fifties. Due to Landau quantization the electronic bands split into magnetic
sub-bands, which show up very distinctly in optical data because the density of
states of the sub-bands shows sharp peaks under favourable experimental
conditions. Contrary to Shubnikov-de Haas experiments, it is possible to
investigate very pure samples. It turned out that especially the interband optical
absorption in high magnetic fields was a method which could reveal a wealth of
band-structure information. Work along these lines began independently in the
USSR and in the USA in the fifties. Gross et al. (1954) studied the influence of a
magnetic field on the exciton spectrum of C u 2 0 , and Gross et al. (1957)
observed magneto-optical oscillations in this material. Burstein et al. (1957)
found an oscillatory magnetoabsorption in InSb, and oscillations in Ge were
observed by Zwerdling and Lax (1957). N o excitonic effects were resolved in
the InSb-data, which were interpreted entirely on the basis of Landau transitions between the valence and conduction bands. It has become clear, however,
that it was necessary to include the Coulomb interaction between electrons and
holes in the interpretation. It has been shown theoretically (Elliot and Loudon
1959) that in a magnetic field the transitions between exciton states are more
favourable than those between Landau levels. For small effective masses and
high magnetic fields, both the excitonic and the Landau-transitions depend
linearly on the field, so that it is difficult to distinguish them on the basis of the
experimental data.
It became obvious that the interpretation of the magneto-optical data
required considerable theoretical efforts. Part of the problem is the complicated
valence-band structure of the elemental and compound semiconductors, for
which the top valence bands are degenerate at k = 0 and strongly warped.
Application of a magnetic field results in a complicated Landau-level scheme for
the light and heavy holes. The Kohn-Luttinger k '/i-scheme applied by Pidgeon
and Brown (1966) allows detailed analysis of interband magneto-optical data
neglecting many-body effects. For a complete analysis it is necessary, however,
to provide excitonic corrections. Unfortunately it is rather difficult and tedious
to treat excitons in high magnetic fields theoretically, because the binding
energy depends in a complicated fashion on the magnetic field.
In the sixties, semiconductor physics developed very rapidly. The methods to
calculate band structures theoretically were improved very quickly. There was a
very close interaction between theory and experiment, especially in the field of
magneto-optics. Since the precision of the band-structure calculations was
considerably less than the accuracy of the experimental data, the band
calculations were usually fitted. Therefore, it was highly desirable to have
experimental data with high precision available.
The rapid progress of semiconductor physics, to which magneto-optics
contributed considerably, was also related to advances in the generation of high
magnetic fields. Up to the fifties, the magnetic fields used for solid state
investigations were usually produced by iron-core electromagnets which al-
Introduction
xv
lowed field strengths of the order of 2 T, when having reasonable homogeneities.
In the early sixties, dedicated high magnetic-field facilities were built in several
countries which extended the useful field range to 15 or 20 T. In the late sixties,
rather inexpensive superconducting coils became commercially available which
allowed the generation of magnetic fields in the 10 Τ range. This speeded up the
development of magneto-optics considerably, because it became possible to
perform sophisticated experiments in one's own laboratory.
The scientific activities in the field of magneto-optics and magnetotransport
lead to a very large number of publications. Review papers on particular
subfields appeared in due course. To our knowledge, there has been no attempt,
however, of a comprehensive review of the whole field of Landau-level
spectroscopy. In the beginning it was thought that it might be appropriate to
limit the effort to optical magnetospectroscopy. During the discussions, it
became more and more clear, however, that transport effects, which give
information about Landau levels, should be included. Actually, one may
consider transport experiments in which oscillatory structure shows up as
spectroscopy at frequency zero!
Looking at the program of the recent biannual international conferences on
the physics of semiconductors, one realizes that a considerable amount of the
activity in semiconductor physics and especially in magneto-optics is dedicated
to the investigation of two-dimensional systems. The new technology of
molecular beam epitaxy (MBE) has allowed the production of semiconductor
heterostructures in which electrons or holes are confined to narrow potential
wells with a thickness of the order of 100 A or less. This results in boundary
quantization and discrete electric sub-bands. Since the charge carriers are free to
move parallel to the interface of a heterostructure, their motion can be fully
quantized by the application of a strong magnetic field perpendicular to the
interface. The conditions for Landau quantization can frequently be met at low
temperatures and magnetic fields of the order of 10 T, because it is possible to
produce heterostructures with very high carrier mobilities. Landau quantization
shows up in two dimensional structures of high quality in a very pronounced
way, because the density of states consists of a series of sharp spikes. A very
spectacular manifestation of Landau quantization is, of course, the quantum
Hall effect (von Klitzing et al.).
It was decided to include magnetic-field dependent two dimensional effects
only to a limited extent in these volumes. The field of 2 D physics has grown so
rapidly in the last decade and is still developing so fast, that it seemed too early
for a comprehensive review of the whole field. This decision seems appropriate
with respect to the conferences which have been held during the last few years
and which have been appropriately recorded in the form of proceedings. Several
international conferences have been held in the recent past which were dedicated
to semiconductor superlattices and heterostructures. A substantial part of the
papers employed high magnetic fields and Landau-level spectroscopy for
xvi
G. Landwehr and E.L Rashba
analysis. At the last two international conferences T h e Application of High
Magnetic Fields in Semiconductor Physics' in Wurzburg 1986 and 1988 about
two thirds of the invited and contributed papers were concerned with two
dimensional systems (Landwehr 1987, 1989). Also, the conferences of the series
'Electronic Properties of 2D-systems' were devoted to a significant extent to
Landau-level spectroscopy in its widest sense.
There is an extensive literature on the quantum Hall effect available. Recent
reviews were given by Rashba and Timofeev (1986) and by Aoki (1987). A book
on the integral and fractional quantum Hall effects has been edited by Prange
and Girvin (1987), the fractional quantum Hall effect has been covered in a book
by Chakraborty and Pietilainen (1988). It was felt, however, that there was need
for a review of magneto-optics in two dimensional systems.
In the following, the content of this book will be commented by going through
the separate chapters. These have been organized in five groups: Intraband
effects, interband effects, two-dimensional systems, transport effects and various
topics. It was not always possible to make a clear-cut classification according to
this scheme, some chapters concern two subjects or even more. Due to the large
extent of the subjects covered, it was necessary to split the book into two
volumes.
Intraband effects
The first chapter written by E. Otsuka is about cyclotron resonance. It was
mentioned already in the beginning, that with cyclotron resonance investig­
ations the area of modern magneto-optics started. The experiments on n-type
germanium for the first time revealed the multi-valley structure of the conduc­
tion band in a unique fashion and allowed the precise determination of the
effective-mass parameters. Also, the existence of heavy as well as light holes was
demonstrated. It turned out that due to the complex valence-band structure, the
Landau-level structure is rather complicated. To give a recent example for a
two-dimensional system, the Landau levels for GaAs-(GaAl)As, a p-type
heterostructure are shown in fig. 1 (Bangert and Landwehr 1986). The system is
especially interesting, because due to the two-dimensional nature of the hole gas
and due to the special kind of doping employed, the hole mobility is about
100000 c m 2/ V s in the mK range. Consequently, high-quality experimental
cyclotron resonance data can be obtained with a laser spectrometer operating
with submillimeter waves. It is obvious that there is only a faint resemblance
to the simple Landau fan diagram £ = (n + ^)ftco, where η is the Landau
quantum number. The light- and heavy-hole states are strongly mixed at high
magnetic fields. The Landau-level scheme leads to a multi-line cyclotron reso­
nance spectrum and it is no longer appropriate to assign effective masses. The
Landau-level scheme was obtained by solving self-consistently the Schrodinger
and the Poisson equation in the Hartree approximation. In order to obtain
agreement between theory and experiment it was necessary to take the influence
Introduction
xvii
Fig. 1. Landau levels for a p-type GaAs-(GaAl)As heterostructure. Full lines: anisotropy of the
volume bands included; dashed lines: axial approximation; Heavy line: Fermi energy for a hole
concentration of 2.3 χ 1 0 11 c m - 2. (After Bangert and Landwehr 1986.)
of the magnetic field on the electric sub-bands explicitly into account by
incorporating the vector potential in the Schrodinger equation. It should also
be pointed out that it is necessary to calculate dipole matrix elements between
the Landau levels in order to explain the cyclotron resonance spectrum. These
remarks have the purpose of indicating that modern cyclotron-resonance
experiments usually need a thorough theoretical interpretation. This is also the
tenor of the first two chapters on cyclotron resonance by Otsuka and by
Levinson.
Actually, cyclotron resonance is a high-frequency transport experiment with
fundamentally all the complications which characterize transport effects. This is
no serious drawback if one is interested mainly in the effective masses of n-type
materials, which are essentially determined by the peak of a resonance line. The
transport aspect shows up in the linewidth of cyclotron absorption. The line
broadening is influenced by scattering processes of the charge carriers. There is
the electron-phonon interaction and the scattering of electrons by impurities. As
explained in the review by Otsuka one can distinguish between neutral and
ionized-impurity scattering. Also, the scattering by line defects as dislocations
can be studied by cyclotron resonance.
In chapter 1 recent results of cyclotron resonance in III-V and II-VI
semiconductors are reviewed as well as data on materials like HgTe and Te.
That something new may be learnt by studying germanium and silicon with
modern instrumentation can be seen in the last part of the chapter.
xviii
G. Landwehr and E.I. Rashba
Whereas the contribution by Otsuka is mainly experimentally oriented the
emphasis of the chapter by Levinson is oriented on theory. Therefore, chapter 2
supplements chapter 1 very well. Not only single phonon-assisted transitions are
considered but also multiphonon processes. Also, impurity transitions are
treated which, in many cases, are difficult to distinguish from Landau transitions. Whenever the cyclotron energy is equal to the energy of a longitudinal
optical phonon, a level crossing occurs and the cyclotron resonance line can
split into two components. Even if they cannot be resolved, the line-shape can
strongly be influenced. The review shows how these phenomena can be
described theoretically.
It has been theoretically predicted that in semiconductors which have an ionic
part in the binding, polaron effects show up. An electron moving through the
lattice excites virtual phonons which can surround it like a 'cloud'. The electrons
plus the attached phonons are called a polaron. Because of the interaction
between electron and lattice the polaron mass is larger than the bare effective
electron mass. For weak coupling and interaction with optical phonons a
theory based on perturbation theory and the Frohlich coupling constant was
worked out by Larsen. It is difficult to describe the interaction in the case of
resonance, when the perturbation theory fails. In his chapter Larsen discusses
recent results on the polaron interaction, especially in III-V compounds. Even
if the polaron coupling constant is relatively small, the splitting of the cyclotron
line may be readily observed.
The spin of Bloch electrons in semiconductors usually interacts much
stronger with oscillatory electric fields than with ac magnetic fields, provided
spin-orbit interaction is present. As a result, transitions between different spin
states of a particular Landau level are excited predominantly by the electric-field
component of an electromagnetic wave. The intensity of the combined resonance can be an order of magnitude larger than the usual paramagnetic
resonance. The intensity and the line form of the combined resonance are
sensitive to the details of the spin-orbit interaction. The combined resonance
can occur with band electrons as well as with electrons bound to impurities.
However, the combined resonance for bound electrons is usually less pronounced than that for band electrons. Even under conditions when the scattering of
electrons is rather strong, spin-flip transitions have to be considered. Electricdipole resonances are treated - mainly from a theoretical point of view - in
chapter 4 by Rashba and Sheka. In their chapter on combined resonance
the authors treat the subject in a comprehensive fashion. Emphasis is on the
theoretical aspects, although the experimental results are reviewed and commented on, too.
A separate chapter by Hafele has been devoted to spin-flip Raman scattering.
The experimental exploration of this phenomenon has become a useful tool to
obtain band-structure information on semiconductors. The first work dates
back to 1966 when Wolff treated the scattering of free carriers in a magnetic
Introduction
xix
field. Employing the effective-mass theory he predicted a Raman process
involving two Landau levels with a difference in quantum number of two. The
spin effects were taken into account for the first time by Yafet (1966). The spinflip Raman process is only possible in the presence of spin-orbit interaction.
Raman scattering has become an interesting tool, because the Raman scattering
cross section can be rather large. There is also a strong resonant enhancement of
the cross section when the energy of the incident photons is close to the allowed
intermediate states. Because of the large scattering cross sections and the small
linewidths in InSb it was possible to observe stimulated Raman scattering (Patel
and Shaw 1970). It turned out that it was especially advantageous to employ
C 0 2- l a s e r radiation because its photon energy almost matches the energy gap
of InSb at helium temperatures. This allowed the operation of a continuous
Raman spin-flip laser with low threshold.
The chapter deals also with the results which have been obtained during the
last years in diluted magnetic semiconductors. Not only spin-flip Raman
scattering from free electrons and holes has been observed, but also from carriers
bound to impurities. Also, spin-flip scattering from bound magnetic polarons
has been investigated.
In semiconductors with high carrier concentration plasma effects occur. They
are modified by application of a strong magnetic field. In the past, there have
been numerous studies on magnetoplasma effects in single- and multivalley
semiconductors. The analysis of the data has frequently been based on the
Drude model, reviews have been given by Palik and Furdyna (1970), Pidgeon
(1980), Zawadzki (1974), and by Grosse (1979).
In the chapter written by Bauer, emphasis has been on narrow-gap materials.
These are very interesting subjects for studying magnetoplasma effects. In most
cases, the effective masses are low and the free-carrier concentration is so high
that the Fermi energy, the plasmon energies, and the phonon energies are
comparable. Also, the non-parabolicity has usually to be taken into account. In
such cases careful analysis is necessary and complete information on the
dielectric function is required. It is in many cases no longer adequate to model
infrared active phonons by Lorentz-oscillators, a linear-response theory for the
susceptibility is better suited. This kind of treatment is required especially for
combined resonances and spin-flip transitions which cannot be handled on
classical grounds.
Due to progress in Fourier and far-infrared laser spectroscopy, high quality
data on magnetoplasma effects have become available. Instead of reviewing the
whole field, emphasis has been put on the discussion of IV-VI lead compounds,
which, due to their multivalley band structure, their narrow gaps and their high
carrier mobilities, can be considered as model substances. Many of the
complications which can arise in the data analysis are very well exposed.
XX
G. Landwehr and E.L Rashba
Interband effects
It is well known, that studies of magneto-optical interband effects have been a
major tool to obtain information on the band structure of semiconductors. The
pioneering work of Lax and coworkers (see, e.g., Lax and Mavroides 1967) is
well known. The theoretical interpretation was based on transitions of free
carriers between Landau levels. Due to the peaks of the density of states in
high magnetic fields, the optical absorption shows a pronounced oscillatory
behavior. In the early days, the interpretation of the experimental data was
entirely based on transitions between Landau states. In order to achieve an
appropriate description of the data, it was necessary to incorporate the complex­
ities of the valence band structure properly. Pidgeon and Brown (1966) were
the first to take the conduction band and the three valence bands - including
the spin parameters - into account. By solving an 8 χ 8 matrix they obtained
Landau levels which were fitted to the experimental data by adjusting the band
parameters in the computer calculations. Subsequently, the method was applied
to many other semiconductors.
In the meantime, it has become clear that in certain cases the analysis entirely
based on Landau level transitions is not adequate. In semiconductors of high
purity and good structural perfection it is necessary to take excitonic effects into
account. The Coulomb interaction between electrons and holes can result in
bound states, which have small binding energy because of the relatively small
electron and hole masses and large dielectric constants. Excitonic absorption
showed up in the absorption spectra of the semiconductor C u 2 0 which has a
rather large band gap (Gross et al. 1957). It was theoretically shown by Elliott
and Loudon (1959) that due to large excitonic density of states and large
transition probabilities, the oscillatory magneto-absorption spectrum can be
dominated by excitons. In germanium, with its relatively small energy gap, the
bound states arising from Coulomb interaction were verified at a rather early
stage. However, in the prototype narrow-gap semiconductor InSb excitonic
absorption could not be identified for a long time. Improved methods for
crystal growth have allowed, in the meantime, the preparation of samples with
high homogeneity and purity, which made the observation of the WannierMott exciton in this material possible (Kanskaya et al. 1979). The analysis of
the data including the many-body effects lead to a set of band parameters
which is consistent with the intraband data.
In magneto-optical interband studies on samples with a quality which does
not allow the formation of excitons, the data analysis on the basis of Landau
level transitions is still justified. Actually, high electron concentration can result
in screening of the Coulomb interaction between photo-excited electrons and
holes, so that no bound states can exist. Also, potential fluctuations due to
acceptor and donor impurities can prevent the formation of excitons. It should
be realized that the excitonic binding energies of narrow-gap semiconductors
can be extremely small, for example, that of InSb is only 0.5 meV. Therefore, in
Introduction
xxi
order to observe the exciton effects, the requirements for crystal perfection can
be extremely high. On the experimental side, complications can arise from
surface accumulation and inversion layers. Especially on thin samples, such
layers, which can for instance be caused by oxide coverage, can modify or
suppress exciton formation due to the strong surface electric field. Also, sample
strain caused by substrates can show up in the optical spectra and mask the
excitonic effects.
If the excitonic interaction has to be taken into account in the interpretation
of magneto-optical spectra, it can usually be done with sufficient accuracy.
Excitons in high magnetic fields can be treated theoretically analogously to
hydrogen atoms. However, the complications due to the complex valence-band
structure have to be considered properly.
In the chapter by Seysyan and Zakharchenya, emphasis is on excitonic effects.
In their contribution, it is clearly demonstrated that the careful analysis of
diamagnetic excitonic spectra allows to obtain a wealth of detailed band
structure information. The review is comprehensive and gives all the information, which is necessary for a up to date analysis of magneto-optical data.
The subsequent chapter on interband magneto-optics was written by
Pidgeon. It complements the chapter by Seysyan and Zakharchenya very well.
Exciton effects were omitted deliberately, as they did not show up in the
experimental data discussed. Since the field of interband magneto-optics is a
large one, the review has been restricted to narrow-gap semiconductors. Due to
their small effective masses and high carrier mobilities, they are especially
suitable for magneto-optical studies. It is shown that the method by Pidgeon
and Brown (1966) allows quantitative description of the magneto-absorption in
InSb crystals, in which the excitonic interaction is not important. The diamond
lattice approximation can be applied in this case because the inversionasymmetry splitting is so small that it may be neglected.
The chapter consists of a theoretical and an experimental part. In the latter,
preferentially zinc blende semiconductors and semimetals are reviewed. Interband magneto-optics has played an important role in elucidating the band
structure of the zero-gap materials HgTe and HgSe. Also, the investigation of
H g 1_ xC d J CT e has heavily relied on magneto-optics. There is considerable
practical interest in this material, because it allows the production of efficient
photodetectors for the infrared. The review of recent results on the lead salts
supplements the contribution by Bauer on intraband effects in these volumes.
Magneto-optical spectra are influenced by electric fields perpendicular to the
magnetic field. The chapter by Aronov and Pikus deals with this subject. The
magneto-optical transitions in crossed electric and magnetic fields deviate in a
measurable way from those made in the absence of an electric field. Due to the
lifting of degeneracies, additional transitions become allowed. The difference in
energy (in a model which neglects Coulomb interaction between electrons and
holes) between conduction and valence band contains a term which is propor-
xxii
G. Landwehr
and E.L
Rashba
tional to E2/H2
and the sum of the electron and hole masses.
After the treatment of the light absorption in crossed fields for isotropic
parabolic and nonparabolic energy-wavevector relations, degenerate bands are
discussed. Also, indirect transitions and dispersion effects are reviewed. Then
the absorption in crossed fields is treated, including excitonic effects. Also, the
Franz-Keldysh effect in a magnetic field which is parallel to the electric one is
discussed.
From the chapter by Aronov and Pikus, one can conclude that the theory of
optical transitions in crossed (and parallel) electric and magnetic fields is well
established. Two-dimensional electronic systems, which can be realized at the
surface of semiconductors or at the interface of semiconductor heterostructures,
allow to establish high electric fields in the two-dimensional layers. Since it has
been possible to produce structures with high electron mobilities, it is not
surprising that they have been a preferred object for studies of the influence of
crossed electric and magnetic fields on the optical absorption.
It has been mentioned already in the beginning, that Faraday rotation in
solids was discovered about 150 years ago, when solid state physics as a subfield
of physics did not yet exist. Modern work on Faraday rotation can no longer
rely on quasi-classical interpretations. When in a Faraday rotation experiment
the wavelength is decreased and approaches the energy gap, anomalous
behavior is observed, i.e., oscillatory effects occur at energies above the gap.
From experimental data, information may be obtained about effective masses
and about g-factors. Also, excitonic effects can be important for the Faraday
rotation for quantum energies close to the energy gap. It turns out, however,
that the interpretation of the non-oscillatory part of the Faraday rotation is
rather involved.
The chapter on Faraday rotation by Kessler and Metzdorf is a comprehensive
review of our present state of knowledge of the effect. As a basis for the
interpretation by microscopic parameters, the phenomenological theory is given
in all details. By' comparing magneto-absorption and magneto-conductivity
with the interband Faraday rotation, it is demonstrated that the dispersive
effects are a very useful tool to obtain band-structure information. Also,
experimental results of interband magneto-optical rotation and ellipticity are
reviewed, with special emphasis on germanium and silicon. In addition, the
modern experimental techniques are discussed which allow measurement of the
Faraday rotation with a very high accuracy.
The final chapter in the part on interband transitions has been written by
Zawadzki. It is essentially of a theoretical nature. A coupled band k ·/> theory
allows to treat interband and intraband magneto-optical transitions (including
spin-flip transitions) in the presence of a magnetic field, on the same footing.
Two cases are considered which apply to the model substances InSb and PbTe.
The InSb case is representative for semiconductors with the conduction band
minima and the valence band maxima at the Γ-point, and the lead salt stand for
Introduction
xxiii
multi-valley semiconductors with the band extrema at the L-point. Magnetic
energies and transition probabilities are derived for the two cases, using velocity
operators in matrix form. The description concentrates on unifying principles
and not on details. Some examples for recent intraband magneto-optical
experimental data are given.
Two-dimensional
systems
It has been mentioned already that the investigation of the properties of
two-dimensional electronic systems in semiconductor heterostructures plays
a prominent role in current solid state physics. An extensive literature on
two-dimensional systems has become available during the last years. Therefore,
two-dimensional systems are treated in this volume only to a limited extent.
This concerns especially the integral and fractional quantum Hall effect.
Since magneto-optical properties are an important tool to probe the
electronic properties of semiconductor heterostructures, it was decided to
include a not too short chapter on this subject. In the contribution
by Petrou and McCombe, an up to date review of important developments has
been given. The review concentrates on work published after 1981 since earlier
developments have been covered by previous reviews. The projected length of
this chapter dictated that a selection of the subjects to be covered had to be
made. Nevertheless, the review contains the theoretical background necessary
for the understanding of the experimental results. Also, a discussion of the
relevant magneto-optical techniques is given.
The chapter begins with a discussion of the various heterostructures and the
classification of superlattices. Both lattice matched and strained systems are
covered. The electronic structure of two-dimensional systems is discussed in the
effective-mass approximation. The modifications of the density of states in a
magnetic field are given as well as the selection rules for intra- and interband
states.
In the experimental section, the techniques used in the different spectral
ranges - visible, near- and far-infrared spectroscopy - are discussed. In the part
on recent results, the spectroscopy of free-carrier states in confined systems is
treated, with special emphasis of the valence band structure of type one
heterostructures. Also, the anomalies which can show up in cyclotron spectra
are discussed.
During the last years, considerable attention has been paid to the effects of
confinement on shallow impurities in GaAs-(GaAl)As heterostructures. The
control of the density and the location of shallow impurities in heterostructures
is of technological importance for devices like high electron-mobility transistors
and quantum-well lasers. Since most of the experimental information concerning impurity states in heterostructures has been obtained by magnetospectroscopy, this area is rather thoroughly covered. The theoretical found-
xxiv
G. Landwehr and E.I. Rashba
ations for impurity states in high magnetic fields are separately treated for
donors and acceptors, as well as the experimental results.
The last section is devoted to the electron-phonon interaction in confined
structures. Special emphasis is on Raman scattering, which is the most suitable
technique for the study of the rather strong effects caused by the layering on the
phonon spectrum. In addition, experimental studies of magneto-polarons are
briefly discussed.
The chapter by Petrou and McCombe clearly demonstrates that in the last
decade substantial progress has been made in the understanding of the
properties of confined semiconductor systems. Although the selection of the
topics was subjective and mainly based on the authors' own experience, it gives a
very good description of the state of the art.
Collective excitations in solids like plasma oscillations have been extensively
studied in the past. When the dimensionality of a system changes from three to
two, qualitative changes in the plasmon dispersion relations occur. It is well
known that a magnetic field changes the properties of two-dimensional
collective excitations significantly. The contribution by Volkov and Mikhailov is
devoted to the properties of two-dimensional electron systems in high magnetic
fields which were studied recently. The article discusses the basis for the description of bulk and two-dimensional plasmons. Special emphasis is on the theory
of edge magnetoplasmons in two-dimensional electronic systems, both from
the classical and quantum side. N o t only semiconductor heterostructures are
treated, but also electrons on the surface and liquid helium, which can be
considered as a model system which can be described with quasi-classical
methods.
A separate section is concerned with high-frequency methods to study the
quantum Hall effect. This is of special interest because the influence of metallic
contacts can be separated this way. The theory of the quantum Faraday effect is
reviewed and helicons in superlattices are treated.
The study of edge magnetoplasmons and of the quantum Faraday effect
cannot only give information on the electronic structure in the quantum Hall
effect regime but also on the frequency dependence of the magneto-conductivity
tensor.
The chapter by Pankratov and Volkov has a special character which needs
some comment.
Band inversion is well-known from the alloy system Hg1 _ xC d xT e , where the
electron and light-hole bands change their sequence at a particular composition.
This makes special 'band inverting' heterojunctions possible, in principle. They
may be generated by a spatial variation of one of the components. A somewhat
exotic property is the predicted existence of massless interface states which have
a linear dispersion law and which are not spin degenerate. The question whether
such states could be stable in the presence of fluctuations in the composition is
addressed, and the answer is that they should be observable. The authors come
Introduction
xxv
to the conclusion that the predicted properties are dominated by the band
general symmetry properties connected with the inversion caused by the variation of the composition.
So far, states with zero mass have not been verified experimentally. It is
explained that oscillatory effects caused by magnetic fields perpendicular to the
interface should be a suitable tool to prove the existence of these states and
that Shubnikov-de Haas oscillations of different frequency should exist. Also,
the different spacing of Landau levels should allow, in principle, identification
of the interface states.
The chapter, although to a certain extent speculative, was included in order to
stimulate crystal growers to produce suitable heterojunctions and experimentalists to look for the neutrino-like particles predicted.
Although this volume contains a chapter on the magnetophonon effect, which
arises from transitions between Landau levels due to absorption or emission of
optical phonons, it was felt that a section on the magnetophonon effect in twodimensional systems was appropriate. The chapter by Nicholas deals with
special features which have been studied lately in heterojunctions and superlattices. The recent experimental, as well as the theoretical, activities have been
thoroughly reviewed. Most of the theories are based on the assumption that
optical bulk phonons are dominating the magnetophonon spectra. There are
indications, however, that interface phonon modes play a role, too. Studies of the
two-dimensional magnetophonon effect have contributed substantially to our
knowledge of the electron-phonon interaction. However, it is not entirely clear
how electron concentration dependent screening phenomena can modify the
magnetophonon interaction.
Magneto-transport
Oscillatory magneto-transport effects have been observed as early as 1930, when
L. Shubnikov and W.J. de Haas studied the resistance of bismuth single crystals
as a function of a magnetic field at liquid hydrogen temperatures. For more than
25 years it was believed that one was dealing with special properties of bismuth
and that the Shubnikov-de Haas effect had to be considered as something like a
curiosity. This changed in 1956, when an oscillatory magnetoresistance was
observed in InSb by Frederikse and Hosier. Since then, the Shubnikov-de Haas
effect has been observed in a large number of semiconductors. Today, the effect
is employed extensively to gain band-structure information on elemental and
compound semiconductors as well as on certain metals and alloys.
Shubnikov-de Haas oscillations have contributed especially to the acquisition
of information on two-dimensional systems.
The chapter written by Seiler concentrates on the experimental aspects of the
Shubnikov-de Haas effect, the theoretical side is reviewed in a subsequent
chapter written by Hajdu. Emphasis is on semiconductors; metals, semimetals
xxvi
G. Landwehr and E.I. Rashba
and bismuth alloys are excluded. Shubnikov-de Haas investigations on
two-dimensional systems have not been covered, because they are treated
in another volume of this series (Vol. 30). However, within this framework the
review by Seiler is comprehensive and up to date. Without doubt it will be very
useful because there has been no extensive compilation of work on the
Shubnikov-de Haas effect until now.
Experimental investigations on the Shubnikov-de Haas effect are a standard
technique these days. However, the interpretation of the data requires a detailed
theory. It took a rather long time before quantitative formulae became available
which allowed the deduction of effective masses and the Landau-level broadening from the experimental measurements. The quantum transport theory in high
magnetic fields is a rather challenging problem from the conceptional and
technical point of view. Due to the modification of the density of states by strong
magnetic fields, it is not possible to employ perturbation theory. In the chapter
on the Shubnikov-de Haas effect by Hajdu, a review of the foundations of the
theory is given. The contribution is mainly introductory in character, technical
details and properties related to special band structures have been deliberately
omitted. The discussion of two-dimensional systems is brief, due to the beforementioned reasons. In addition to the magneto-transport effects thermomagnetic effects are discussed which can also be influenced significantly by
Landau quantization.
The subject of transport magneto-impurity effects has been treated by GantMakher and Zverev. This phenomenon concerns the resonant interaction of free
carriers with shallow impurities. It can give rise to structure in the magnetoresistance under non-equilibrium conditions, which can be realized by
heating of the free carriers, by an electric field or by photo-excitation. In general,
inelastic interaction between free carriers and impurities is involved. Whenever
the energy difference between two impurity levels, which may be tuned by a
magnetic field, is equal to a characteristic energy (e.g., the cyclotron energy),
resonant interaction can be observed. The magnitude of the effects is usually
small so that frequently double differentiation techniques are employed. It
seemed appropriate to include a chapter on the transport magneto-impurity
effects, because substantial progress has been made in the last 10 years. After the
classification of possible impurity transitions, the conditions for resonant,
inelastic scattering are treated. Examples for different mechanisms are given:
impact ionisation and Auger recombination as well as the decay of excitons at
ionized impurities. Another possible origin of magneto-impurity oscillations is
the resonant capture of electrons coupled with phonon emissions.
The chapter stresses possible explanations of the effect, the presentation of
experimental results has been restricted to a few model cases. The last section
is devoted to the inversion of impurity resonances, which can occur whenever
a system is relatively far away from its thermal equilibrium. These aspects have
a close relation to the area of hot electrons in semiconductors.
Introduction
xxvii
The discrete nature of the density of states caused by Landau quantization is
reflected in the optical absorption spectra. It also can show up in emission
spectra under certain conditions. Radiative recombination between Landau
levels - which is usually dubbed as Landau emission - can occur after
population of an upper Landau level by an applied electric field. The field
strengths necessary for the observation of Landau emission are only a few volts
per centimeter. Power levels of the emitted radiation of the order 10" 8 W can be
obtained, and linewidths of a few c m - 1.
In the chapter by Gornik, the recent work on Landau emission has been
reviewed. This technique has become a useful tool to explore the electronic
properties of semiconductors. The most extensive work has been done on n-type
InSb and η-type GaAs. Again, the work is closely related to the properties of hot
electrons in these materials in high magnetic fields. It has been possible to obtain
useful band-structure information and insight in polaron properties. In the
recent past, the Landau emission technique was applied to two-dimensional
electronic systems like silicon MOSFETs and GaAs-(GaAl)As heterostructures.
The important results which have been obtained in these investigations are
reviewed.
Although semiconductor lasers based on stimulated Landau emission have
been discussed already in the sixties, it took almost 20 years to realize them.
After detailed studies on streaming hot carriers in crossed electric and magnetic
fields, a far infrared laser was eventually operative in 1983. It is based on lighthole transitions in p-type germanium. This is a remarkable achievement,
because it allows to tune far infrared radiation in a spectral range which is
usually somewhat difficult.
The magnetophonon effect is an oscillatory magnetotransport effect different from the Shubnikov-de Haas effect - which was predicted 2 years
before it was experimentally observed. It arises when the cyclotron energy or a
multiple of it is equal to the energy of optical phonons. It can only be observed
under equilibrium conditions at such high temperatures that optical phonons
are excited. In order to observe the magnetophonon effect at low temperatures,
an electron gas has to be sufficiently heated, so that optical phonons are
emitted. The magnetophonon effect can be observed in slightly doped, nondegenerate semiconductors, contrary to the Shubnikov-de Haas effect, which
requires degenerate material.
In the review by Firsov, Gurevich, Parfeniev and Tsidil'kovskii the develop­
ment of the last 20 years has been comprehensively reviewed. The theory of
the electron-phonon interaction and its influence on the magnetoresistance is
treated in detail. During the last decade, numerous experiments on the magne­
tophonon effect have been performed which yielded interesting results on the
electron-phonon interaction and the band structure of a relatively large number
of semiconductors. Also, the spin magnetophonon resonance is discussed which
is related to the usual magnetophoton effect in a fashion, similar to the way in
xxviii
G. Landwehr and E.L Rashba
which paramagnetic resonance and cyclotron resonance are connected. It has
been mentioned already that the magnetophonon effect in two-dimensional
systems has been covered separately in these volumes.
Various topics
The de Haas-van Alphen effect was one of the first manifestations of Landau
quantization. Although there was no need to review the work on this subject
thoroughly, since the recent book by Shoenberg (1984) gives an excellent survey,
it was felt that a chapter involving the de Haas-van Alphen effect should not be
missing from these volumes. By looking at the contents of this book it becomes
obvious that the main objects of Landau-level spectroscopy in its widest sense
are semiconductors and semimetals. Magnetic measurements on oscillatory
effects in semiconductors are difficult to perform, because of the small carrier
concentration in these materials. However, susceptibility measurements on
metals yield a wealth of information on the topology of the Fermi surface. In
addition to this, it is possible to deduce knowledge on imperfections of materials
like small-angle grain boundaries and point impurities. In the chapter by
Alekseevskii, Kaganov and Nizhankovskii, only selected topics are treated. The
purpose of this contribution is to point out that due to the progress on both the
theoretical and the experimental side it is possible to extract interesting
information which is normally not considered when oscillatory phenomena are
analysed.
The chapter by Zawadzki on impurities in semiconductors in high magnetic
fields is related to many contributions of these volumes. Hydrogen-like, shallow
donors are rather well understood. It turns out, however, that a calculation of
the energy levels of impurities in high magnetic fields is challenging. Even for
spherical energy bands it is not possible to obtain analytical solutions of the
Schrodinger equation. Acceptors cause special problems, because of the complicated structure of the valence bands of elemental and compound
semiconductors.
In high magnetic fields, the wavefunctions are modified. This can cause drastic
changes of the carrier concentration at low temperatures, the so-called magnetic
freeze-out. Also, the magnetoresistance in the hopping range is significantly
influenced by changes in the overlap of wave functions of adjacent impurities.
During the recent past, theoretical work on the magnetoimpurity problem has
been motivated by similarities to the properties of magneto-excitons.
The contribution by Zawadzki concentrates on the theoretical treatment of
hydrogen like donors for isotropic, parabolic and non-degenerate bands in high
magnetic fields. Emphasis is on approximations valid in the very high field
range. Shallow acceptors are also treated, but to lesser extent. Experimental
results are reviewed in order to demonstrate the gross features of the problem.
Finally, magneto-donors in two-dimensional systems are briefly touched.
Introduction
χχιχ
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Zawadzki, W , 1974, Adv. Phys. 23, 435.
Zwerdling, S., and B. Lax, 1957, Phys. Rev. 106, 51.
CHAPTER 1
Cyclotron Resonance
EIZO OTSUKA
College of General Education
Osaka University
Toyonaka, Osaka 560
Japan
Landau Level
© Elsevier Science Publishers B.V., 1991
Spectroscopy
Edited by
G. Landwehr and E.I. Rashba
Contents
1. Introduction
3
2. Cyclotron resonance in Si and Ge - as a transport experiment
3
2.1. Electron scattering by phonons
5
2.2. Electron scattering by neutral impurities
8
2.3. Electron scattering by ionized impurities
14
2.4. Electron scattering by dislocations
19
2.5. Electron scattering by excitons
22
3. Cyclotron resonance as a kinetics experiment
23
3.1. Carrier kinetics in InSb
24
3.1.1. Electric field excitation
25
3.1.2. Spin temperature in optical excitations
29
3.2. Carrier kinetics in GaAs
34
4. Cyclotron resonance in the quantum limit
35
4.1. Electron scattering in GaAs
35
4.1.1. Carrier-carrier scattering
37
4.1.2. Effects of phonon scattering
39
4.1.3. Neutral impurity scattering
41
4.2. Ionized impurity scattering in InSb
44
5. Cyclotron resonance in I I I - V and I I - V I compounds
5.1. Employment of very high magnetic
45
fields
5.2. Transport analysis in I I I - V compounds
45
47
5.2.1. Electron cyclotron resonance in GaSb
47
5.2.2. Electron cyclotron resonance in InP
48
5.3. Cyclotron resonance in chalcogenide materials
51
5.3.1. Cyclotron resonance in ZnSe and ZnTe
51
5.3.2. Cyclotron resonance in CdTe, CdS and CdSe
56
6. Cyclotron resonance in most challenging materials
60
6.1. Ionic crystals: alkali, thallium and silver halides; C u 2 0 and H g l 2
60
6.2. Anthracene and organic materials
62
6.3. Materials with peculiar band structures: HgTe, Te and G a P
7. Germanium and silicon revisited
64
67
7.1. Earlier accurate measurements in the millimeter wave region
67
7.2. Transport measurements of Ge in the far-infrared
70
8. Concluding remarks
72
1.
Introduction
The possibility of cyclotron resonance in solids was proposed by Shockley in
1953. The purification of semiconducting materials, like germanium and
silicon, had made it possible to replace vacuum tubes by transistors. As a natural
consequence one could expect cyclotron resonance to become feasible in Ge and
Si. Pioneering experiments were indeed carried out with no delay. Those
experiments established convincing evidence for nonspherical constant-energy
surfaces. The use of microwaves in the early days shifted to far-infrared lasers in
due course.
The primary raison d'etre of cyclotron resonance remains, or at least remained
for a long time, as a tool for the most direct determination of carrier effective
masses. This was evident after infrared lasers became available. However, the
speedy tabulation of effective-mass values for carriers in various materials
apparently discouraged the cyclotron resonance experts in carrying on further
work, since the materials that could be explored with respect to their carrier
effective masses were almost exhausted. The end of one aspect of cyclotron
resonance was just the beginning of another one, however. Applications in the
area of carrier transport and carrier kinetics were developed, making use of
linewidth and intensity analysis of cyclotron resonance. Frequently, the result of
the analysis varied from sample to sample. Cyclotron resonance thus offered
valuable potential as an established technique for characterizing semiconductor
materials. The role of cyclotron resonance, moreover, extended far beyond the
limits of material characterization. New topics in solid state physics were closely
connected to cyclotron resonance measurements. Many of these have already
been reviewed elsewhere (McCombe and Wagner 1975a, b, Pidgeon 1980,
Otsuka 1980, Ohyama and Otsuka 1983). In this article, emphasis will be put on
transport and kinetic studies of compound semiconductors. Basic features found
in elemental semiconductors will be summarized first, since they cannot entirely
be omitted in order to obtain a full understanding of compound semi­
conductors. Topics will be limited to those in bulk materials. Cyclotron
resonance related phenomena in metals will be entirely omitted.
2. Cyclotron
resonance
in Si and Ge - as a transport
experiment
When cyclotron resonance is achieved by an alternating electric field E =
E0 exp icot on a material with D C conductivity σ 0 , the power absorption is given
by
Ρ~α)Ε20σ0/[(ωΤω<)2τ2
+ \1
(1)
where c o c is the cyclotron frequency and τ is the relaxation time of the relevant
free carriers. The double signs ( + ) correspond to two circular polarizations of
3
Ε. Otsuka
4
the radiowaves with frequency ω. The primary condition for the clear emergence
of a cyclotron resonance peak is ω 0τ > 1. Assuming that this condition is met,
one can derive the relaxation time τ in terms of the halfwidth, that is
1/τ ^ Δω
(2)
where ω 1 /2 is the deviation in frequency from resonance, that gives the halfwidth
of the absorption peak. In terms of magnetic field, that is the quantity actually
scanned in the experiment, the relation (2) can be written as
1/τ = (ΔΒ,/Β Γ)ω,
(3)
where ABT is the halfwidth of the resonance line and Br is the resonance field.
Since the cyclotron frequency and hence the resonance field are fixed, all one has
to do is to measure the halfwidth ABr in order to obtain the inverse relaxation
time, l/τ, or the collision frequency of the relevant carrier. Strictly speaking, the
expression (3) should be preceded by a numerical factor, close to unity, if one
takes the energy dependence of τ into account. Its neglect has, however, no
practical consequences in transport; this will justify the use of eq. (3). By
measuring the linewidth of cyclotron resonance, one is able to derive scattering
coefficients for various scattering centers in semiconductors.
The first pioneering work on Si and Ge (Lax et al. 1954, Dresselhaus et al.
1955, Dexter et al. 1956) was so complete that later publications could not add
much. However, the precision measurements of Levinger and Frankl (1961)
established the most accurate carrier effective masses contained in handbooks.
Intricate spectroscopic aspects of the valence bands in Ge were carefully studied
by Hensel and Suzuki (1974). Transport studies making use of millimeter wave
cyclotron resonance were initiated by Bagguley et al. (1961) and somewhat later
by Kawamura et al. The latter group, in particular, performed a series of
experiments on hot-carrier transport (Kawamura et al. 1962), carrier-carrier
interaction (Kawamura et al. 1964) and electron-phonon as well as
electron-impurity scattering (Fukai et al. 1964). Later findings by the present
author and his group made some amendments to the pioneering data of
Kawamura et al. necessary (Otsuka 1986). But their basic idea to apply the
method to transport problems is surpassed. Especially renowned is their
treatment of the carrier-carrier interaction. Kawamura et al. were the first to
point out the contribution of the carrier-carrier interaction to the broadening of
the resonance line. Their intuitive semiclassical argument predicted a shift in
cyclotron frequency of
A(o =
{\/2hm*coc)(Vxx+Vyy)
(4)
due to the Coulomb or screened Coulomb potential V caused by the presence of
a free carrier. In eq. (4), Vxx and Vyy are second derivatives of V and m* the
effective mass of the carriers. The derivation of (4) was based on classical
transport theory, but is in agreement with the results of more sophisticated
quantum mechanical calculations.
Cyclotron resonance
5
The carrier-carrier scattering is essentially, or at least almost, equivalent to
the carrier-ionized impurity scattering. The early predictions of cyclotron
resonance linewidths frequently show up as references in modern cyclotron
resonance works which make use of far-infrared radiation; in this case, electron
scattering by ionized impurities is very important. We shall discuss a few
examples later.
2.1. Electron scattering by phonons
The important phonon modes that show up in millimeter wave cyclotron
resonance in Si and Ge are acoustical ones causing deformation potential
scattering. Piezoelectric phonons are absent in these nonpolar materials. Optical
phonons can become dominant only at high temperatures, where millimeter
wave cyclotron resonance cannot be observed.
According to the theory of deformation potential scattering (Bardeen and
Shockley 1950), the scattering relaxation time of electrons by longitudinal
acoustic phonons can be written as
TL = TOL(kBT)-1s-^2
(5)
with
ToL = (ftW£?)(2m*)-/.
(6)
3 2
Here ε is the electronic energy, E the deformation potential constant, u the
sound velocity and ρ the density of the material. Gold et al. (1956) calculated
numerically the linewidth of the cyclotron resonance caused only by acoustical
phonon scattering; namely,
l
Δ Β / Β , ~ 1 . 2 5 / ω βτ 0
(7)
To = T 0 L ( f c BT ) - 3 / .2
(8)
with
Introducing the relevant material parameters for Ge and Si, one obtains
1/τ 0 = 3.6 χ 10 8 T 3
/2
s"1
for Ge
(9)
1/τ 0 = 2.6 χ 10 8 T 3
/2
s"1
for Si.
(10)
and
These values of 1/τ 0 are indeed in good agreement with those obtained in
cyclotron resonance experiments by Bagguley et al. (1961,1962a, b) and Hensel
(1963). They are also in fair agreement with drift mobility experiments. In order
to compare with drift mobility, however, one has to take account of the relation
<t>d
r ti =
f
< ^ T L> / < r 2 >
= (4Pyft)rOL(kBT)-3>2.
(11)
(12)
Ε. Otsuka
6
The relation between τ 0 and AB given by Gold et al. is simple. It can be further
simplified if one puts
ΑΒ/Βτ=1/ω0τ'0.
(Ία)
This can be considered as a redefinition of the relaxation time. One obtains then,
in place of (9) and (10),
1/τ'0 = 4.5 χ 10 8 T 3
/2
s"1
for Ge
(13)
and
1/τ'0 = 3.25 χ 10 8 T 3
/2
s"1
for Si.
(14)
This set of 1/τ'0 values has been derived for spherical energy surfaces. Actually,
the conduction bands of Ge and Si are multivalleyed. They have ellipsoidal
constant energy surfaces. It can then be expected that τ 0 or τ'0 is a tensor having
its principal axes along the axes of the ellipsoid. Let us write their components as
τ 0 1 and τ 0 !| , or as τ ' 01 and τ' 0 )| . The meaning of the suffices should be clear.
According to Herring and Vogy (1956), one can write
1/TOJ. = 1.25 Α(ζ±Ξ2
+ η^ΞάΞη + C±S2u)(kBT)>'2
(15)
and
I/To,, = 1.25 Α(ξηΞ2ά + »j ι, S d S u + C„ E2u)(kaT)3'2,
(16)
where
A = 3(2m2ml)1/2/4nh*c1
(17)
c i = i ( 2 c 1 2 + 4 c 44 + 3 c n) .
(18)
and
The parameters ξ9 η and ζ can be found in the paper by Herring and Vogt. The
combined elastic stiffness coefficient c1 is found to be 1.532 χ 1 0 12 dyn c m " 2 for
Ge and 1.906 χ 1 0 12 dyn c m - 2 for Si. Ξά and Su are the dilatational and shear
deformation potentials, respectively. From (15) and (16) we have
τ 0| | Ao± = τ'οΐΐΑΌι = (t±D2
+ 1±D + ζ±)/(ξ „ D2 + iy„ D + C„),
(19)
where
D = EJSU.
(20)
It is known that - 1 < D < 0 for both Ge and for Si. For Ge, applying a magnetic
field along <111>, one obtains a relaxation time τχ that is given by
1 Α ι = 1/τ 0χ.
(21)
Then, applying the magnetic field perpendicular to <111>, another relaxation
Cyclotron resonance
1
time τ2, that is given by
l/T2 = (i)(l/To|i + l/Toi)
(22)
can be obtained. Measuring the linewidth at two different geometries, one can
obtain the anisotropy Κ = ΐ 0 | / τ 0 1 both for Ge and Si. Different authors give
different values of Κ (Bagguley et al. 1962b, Ito et al. 1964, Murase et al. 1970).
These differences are considered to arise partially from the dependence of Κ on
frequency, or, to be more exact, on kB T/ha>.
Murase et al. carried out a systematic measurement of this parameter in their
experiment to determine the deformation potentials of Ge and Si. What they did
was to apply a uniform uniaxial stress along < 111 > for Ge and along <100> for
Si to produce a set of up- and down-shifted valleys in the conduction band.
Then, after measuring the population ratio of electrons between up- and downvalleys, they derived the shear deformation potential constant Eu. In the process
of finding the population ratio, a precision measurement of the linewidth at
different geometries had to be made which gave the anisotropy Κ factor. The
derivation of D, and hence Sd, was straightforward from (19) with the help of the
Herring-Vogt parameters ξ, η and ζ. Under the condition kBT/ha>c = 2.5 (at
4.2 Κ and 35.3 GHz), that has been called 'classical', Murase et al. obtained the
set of parameters given in table 1. The determination of deformation potentials,
anisotropy in phonon scattering etc. has been one of the results of precise
cyclotron resonance measurements.
Taking the existence of anisotropy into account, one can present the
temperature dependence like that given by (13)—(14) or (15)—(16). From now
on, if not otherwise stated, we shall take 1/τ' 01 for phonon scattering. Impurity
scattering is measured also in the same geometry, when the 1/τ' 01 contribution is
subtracted from the total observed linewidth. In 3 5 - 7 0 GHz measurements,
1 / τ' 01 = 4.8 χ 10 8 T 3
/2
s"1
for Ge
(23)
1 / τ ' 01 = 3.0 χ 10 8 T 3
/2
s"1
for Si
(24)
and
should hold in the classical regime below 20 K.
One peculiar observation has been that there exists a slight frequency
Table 1
Deformation potentials and associated electron scattering anisotropy constants in Ge and Si derived
from cyclotron resonance at 4.2 Κ (from Murase et al. 1970)
Κ =
Ge
Si
τ0/τ0
2.02 + 0.05
1.40 ± 0 . 1 5
S d( e V )
£ u( e V )
-12.3 ±0.5
-6.0 ±0.8
19.3 ± 0 . 7
9.0 ± 0.4
Ε. Otsuka
8
dependence in the numerical coefficient for 1/τ' 0 1. For Ge, 1/τ' 01 = 4 . 4 χ
1
08 T 3
/ 2 s- i
fr o
22.2 GHz
and 3.8 χ 1 0 8 T 3
/2
s " 1 was found for 9.16 GHz
(Otsuka et al. 1966a). This feature is shown in fig. 1. A similar frequency
dependence has been observed for Si by other authors (Hensel 1963; Ito 1967).
Because one is in the classical region these results are hard to explain. Since the
frequency dependence is not very large, we shall take the 3 5 - 7 0 GHz values of
1/τ' 01 as the standard classical phonon scattering rate.
2.2. Electron scattering by neutral
impurities
As far as impurity scattering problems are concerned, Si and Ge are materials
that can offer the most reliable data for scattering cross-sections, at least in the
TEMPERATURE
( Κ)
Fig. 1. Temperature dependence of the inverse electron transverse relaxation time 1/τ ± in pure Ge
for three microwave frequencies in the classical regime. The coefficient a is apparently larger for
higher frequencies (from Otsuka et al. 1966a).
Cyclotron resonance
9
classical regime. It is possible to dope these substances, with only one kind of
impurity. This is almost impossible in other semiconductors. In most cases,
free-carrier cyclotron resonance is performed under or after intrinsic photoexcitation. At liquid helium temperatures, all impurities are neutralized by
photo-induced carriers. The resonance linewidth is caused by phonon and
neutral impurity scattering. Linewidth measurements were carried out in the
geometry of B||<111> for Ge and £||<100> for Si. The contribution from
phonon scattering [eqs(23) or (24)] can then be readily subtracted. This
procedure is indicated in fig. 2.
So far as the classical regime (hcoc <^ kB T) is concerned, it has been confirmed
for Sb in Ge that no frequency dependence exists between 9 and 35 GHz for the
impurity scattering (Otsuka et al. 1966a). In addition, anisotropy does not exist
in impurity scattering for As, Sb and In in Ge and Li in Si (Ohyama et al. 1970).
These features are different for phonon scattering, which shows a frequency
dependence and also a definite anisotropy in the classical regime. In the
quantum limit, however, the situation changes entirely. Even for impurity
scattering, we have both a frequency dependence and anisotropy. The frequency
dependence can also be expressed as a magnetic field dependence. This changes
the size of the cyclotron orbit. A field dependence means an influence of the ratio
between the cyclotron orbit and the force range of the scattering center. The
anisotropy, on the other hand, arises from the degree of polarization of the
impurity spin system. Systematic studies of spin-polarized electron scattering
by impurities were performed by Ohyama et al. (1970), using 70 GHz in a
He 3-cooled cryogenic system. We shall discuss the result later. For the moment,
we summarize essential features of impurity scattering studies in the classical
regime.
Τ (Κ)
Fig. 2. Inverse relaxation time τ " 1 of electrons in G e derived from the linewidth of cyclotron
resonance. Plotted against temperature, τ - 1 tends to fall on a T 3 2/ line. This is due to the
1
. The
upward deviation from the T 3 2/ line occurs
contribution from acoustic phonon scattering,
· 1
at low temperatures on account of the contribution from neutral impurity scatterings
Ε. Otsuka
10
Scattering by isoelectronic neutral impurities were treated in the early days
(Otsuka et al. 1965). The scattering cross-sections were found to be so small, less
than 7 χ 1 0 " 17 c m " 2 for Si in Ge and less than 3 χ 1 0 " 1 6 c m " 2 for Sn in Ge at
4.2 K, that no special treatment for isoelectric impurities is necessary. They can
become important only if bound excitons become involved as in the case of Ν in
GaR
For electron scattering by neutralized shallow impurities, the so-called
Erginsoy formula (Erginsoy 1950) is generally referred to. This gives a scattering
rate by neutral donors like
l / T ND = 2 0 f t a DN D/ m e,
(25)
where a D is the effective Bohr radius of the donor, ΝΌ the concentration of
donors and m e the effective mass of the electrons. To begin with, expression (25)
was frequently used without distinguishing between electron-donor and
electron-acceptor scattering. In other words, it was originally believed that eq.
(25) accounted for general electron-neutral-impurity scattering. The electronneutralized donor scattering can be simulated, however, within the framework
of the effective-mass approximation, by the electron-hydrogen atom ( e " - H )
scattering, whereas the electron-neutralized-acceptor scattering is equivalent to
the positron-hydrogen atom ( e + - H ) scattering. The former simulation, in the
simplest treatment using the lowest order term in the partial-wave expansion,
leads to the Erginsoy expression (25). The latter one, on the other hand, after the
same approximation used in the Erginsoy treatment, yields an expression
l A NA = C r - 3 ' 2 a - 3 \ - 2 J V A
(26)
with
a = 12.5 + ft2/2me/cB Ta\
(26a)
and
C = 3.4
ft(fc2/2me/cB)3/2/me,
(26ft)
where the suffices involving A refer to acceptors. 1 / τ ΝΑ is proportional to iV A and
the total coefficient of proportionality, C T ~ 3 / a2 ~ 3 / a2 A" 2 , is a slowly decreasing
function of temperature. One can rewrite the expressions (26), in a similar
fashion as Erginsoy's relation,
l/TNA
= C*(T)haANJme.
(27)
One finds that C*(0) = 3.4 and C*(4.2) ~ 2. The dependence of the cross-section
σ on the wavenumber k of the incident particle is given both for e " - H and for
e + - H scatterings in fig. 3. The nearly constant slope for the e " - H case gives
a prefactor of 20 for the Erginsoy case. The slope for the e + - H case is
/c-dependent, particularly for large values of k. For thermal electrons at 4.2 K, one
may expect that the e ~ - N A (electron-neutralized-acceptor) scattering cross-
Cyclotron resonance
11
kaB
Fig. 3. Difference in scattering cross-section between electron-hydrogen atom ( e " - H ) and
positron-hydrogen atom ( e +- H ) collisions. The abscissa gives the wavenumber of the incident
particle (electron or positron). The ( e ~ - H ) line is almost linear and its slope gives a numerical factor
of 20 preceding the Erginsoy formula. The difference is enhanced at high values of kaB (from Otsuka
et al. 1966b).
section is crudely one tenth of the e " - N D (electron-neutralized-donor)
scattering cross-section for the same Bohr radius. As we shall see below, this
simulation is found to be extremely good for group III and group V impurities
in Ge (Otsuka et al. 1966a, b, c, Otsuka 1986). A certain deviation occurs for
the same impurities in Si, however (Otsuka et al. 1968a, b, Otsuka 1986). For
the group V donor Ρ in Si, the deviation from the prediction of Erginsoy is that
the experimental value is larger by nearly a factor of two. Conversely, for the
group III acceptors Β and Ga in Si, the experimental values are smaller than
the theoretical prediction by nearly a factor of two. Thus a factor of 40 difference
exists between e ~ - N D and e ~ - N A scattering cross-sections in Si. This is signi­
ficant. Basically, the observed large difference in scattering cross-sections
between donors and acceptors should also be valid in D C transport. However,
one has to be very careful when dealing with electron transport in p-type Si. If
the material is compensated by several per cent of donors, the main scatterers
12
Ε. Otsuka
of electrons will then be the minority impurities, the donors, and not the accep­
tors. Inverse relaxation times due to shallow impurities (both donors and accep­
tors) in Ge and Si, obtained at 4.2 K, are plotted in figs. 4 and 5, respectively,
against impurity concentration. For impurities in Ge, as seen in fig. 4, both
predictions - Erginsoy's for e ~ - N D and ours for e ~ - N A - are surprisingly
good for explaining the observed line widths of electron cyclotron resonance.
The same is not true, as seen in fig. 5, for impurities in Si. In the author's opinion
the agreement depends on the degree of justification of the effective-mass
approximation.
Basically, the same arguments should also be applicable to donors and
acceptors in compound semiconductors. In many cases of compound semi­
conductors, one must further take into account that the Bohr radius in an
acceptor is smaller than that in a donor, so that the difference in scattering crosssection between e " - N D and e " - N A is generally enhanced.
The argument so far has concerned shallow donors and acceptors - typically,
group V and III impurities in Si or Ge. These are either hydrogenic or
antihydrogenic. Recently, special interest has arisen in deep impurities, say Zn,
Be etc. They cannot be regarded as hydrogenic. Electron scattering by deep
10 K
10 15
IMPURITY CONCENTRATION (cm' 3)
Fig. 4. Inverse relaxation time of electrons at 4.2 Κ due to neutral donors (Sb) and acceptors (Ga
and In) in Ge. Cyclotron resonance measurements are made at 35 GHz. For donors, Erginsoy's
predictions [expression (25)] are straight lines. For acceptors, our prediction is given by expression
(26). The Bohr radii are taken appropriately into account (from Otsuka et al. 1966b).
Cyclotron resonance
IMPURITY
13
CONCENTRATION (cm"3)
Fig. 5. The same as in fig. 4 but for impurities in Si. The deviations from theoretical lines, upwards
for donors ( P and Li) and downwards for acceptors (B and Ga), are considered to be due to the
limited validity of the effective-mass approximation in Si (from Otsuka et al. 1968b).
impurities was treated long ago (Otsuka et al. 1966c, 1968a, b, Murase and
Otsuka 1968). A dramatically stress-dependent cyclotron resonance was ob­
served for Zn-doped Ge. A neutral Zn impurity in Ge acts like an anti-He atom
for an incident electron. Reversing the signs of the charges involved, one may
simulate the electron scattering with the e + - H e collision. Fortunately there
exists a theoretical treatment of this collision (Kestner et al. 1965). Then, within
the range of the s-wave approximation as before, one obtains
l/xZn = 025ha$NzJme,
2
2
(28)
This simulation explains the observed cyclotron resonance
where a$ = Kh /mee .
linewidth for electrons in Zn-doped Ge surprisingly well in the high-stress limit.
The nice agreement between theory and experiment obtained at 1.5 Κ is shown
in fig. 6. In the absence of stress, the cyclotron resonance line is broadened.
Photoluminescence experiments indicate that the broadening is connected with
electron-bound-exciton collisions (Nakata and Otsuka 1984, 1986, Sauer and
Weber 1984, Otsuka et al. 1986).
A summary of available data for scattering cross-sections or scattering
coefficients of electrons by neutral impurities in Ge and Si is given in tables 2
and 3.
Ε. Otsuka
14
Τ
1—I M i l l
1
1
1 I M i l l
I
I
Π
Fig. 6. Inverse relaxation time of electrons due to neutral Zn acceptors in Ge is plotted against the
Zn concentration. The 35 G H z cyclotron resonance linewidths are measured at a uniaxial highstress limit to eliminate the contribution from scattering by bound excitons. The straight line along
the experimental data is drawn after the e +- H e scattering model expressed by eq. (28) (from Murase
and Otsuka 1968).
2.3. Electron scattering by ionized
impurities
It is rather difficult, in Ge or Si, to see a contribution from ionized impurity
scatterings to the cyclotron resonance linewidth. If one illuminates samples with
intrinsic light at low temperatures in order to obtain free carriers, the impurities
are promptly and stably neutralized. If one raises the temperature to such a
degree that one obtains extrinsic free carriers and thermally ionized impurities,
the contributions from lattice vibrations to the linewidth become tremendous.
The first attempt to see the ionized impurity scattering was made by Sekido et
al. (1964) in Ge (47.6 GHz). To avoid neutralization of impurities as much as
possible, the intrinsic photoillumination was minimized. To ionize impurities in
thermal equilibrium, as much compensation as possible was introduced, Sb as
donor and Cu as compensating acceptor. The temperature was only moderately
raised above 10 Κ to encourage thermal ionization and to hold the contribution
Table 2
Neutral impurities in Ge (from Otsuka 1986). Electron scattering rates per impurity (scattering coefficients) and scattering cross-sections are tabulated for
various impurities in Ge at two temperatures, 4.2 and 1.5 K. The electron velocities are taken to be thermal velocities; that is 2.95 χ 1 0 6 cm s ~ 1 and 1.76
χ 1 0 6 cm s ~ 1 at 4.2 Κ and 1.5 K, respectively. Only for S b + is another thermal velocity 2.57 χ 1 0 6 cm s ~ 1 at 3.2 Κ utilized. Effective Bohr radii are given
only for group III and group V impurities. Tabulated values of νσ and σ for Zn and Ni are obtained at the high-stress limit. Cyclotron resonance
frequencies are 3 5 - 7 0 G H z
As
Sb
Sb+
Ga
In
Zn
Ni
Cu
thermal
Au
Sn
Si
Classification
donor
donor
ionized donor
acceptor
acceptor
acceptor
acceptor
acceptor
acceptor
amphoteric
isoelectronic
isoelectronic
Bohr radius
A
36
47
41
40
( W ) 4. 2 K
( c m 3s _ )1
3.8 χ 1 0 _ 5
4.9 χ 1 0 " 5
3 χ 1 0 " 4( 3 . 2
3.4 χ 1 0 - 6
2.7 χ 1 0 - 6
9.1 χ 1 0 " 7
3.2 χ 1 0 " 7
7.0 χ 10" 7
4.0 χ 1 0 - 6
- 1 χ 10"4
<10"9
< 1 0 " 10
Μ 1.5 κ
( c m 3s _ )1
3.8 χ 1 0 " 5
4.9 χ 1 0 " 5
K)
4.2 χ 1 0 " 6
3.3 χ 1 0 " 6
9.8 χ 1 0 " 7
3.5 χ 1 0 " 7
1.0 χ 1 0 - 6
6.4 χ 1 0 ~ 6
- 1 χ 10"4
<1(T9
σ4 . 3 Κ
σ1.5Κ
( c m 2)
( c m 2)
1.3 χ 1 0 " 11
2.2 χ 1 0 " 11
1.7 χ 1 0 " 11
2.8 χ 1 0 " 11
1.2 χ ΙΟ­ 10 (3.2 K)
Ι.15 χ Κ Γ 12
2.4 χ 1 0 " 12
9.2 χ 1 0 " 13
1.9 χ 1 0 - 12
3.1 χ 1 0 " 13
5.6 χ 1 0 " 13
1.1 χ 1 0 " 13
2.0 χ 1 0 " 13
2.4 χ 1 0 " 13
5.7 χ 1 0 " 13
1.4 χ 1 0 ~ 12
3.6 χ 1 0 ' 12
- 3 . 4 χ 1 0 " 11
- 5 . 7 χ 1 0 " 11
< 3 χ 1 0 " 16
< 6 x 1 0 " 16
< 7 x 1 0 " 17
Cyclotron resonance
Impurity
15
16
Table 3
Neutral impurities in Si (from Otsuka 1986). Electron scattering rates per impurity (scattering coefficients) and scattering cross-sections are tabulated
for some shallow donors and acceptors in Si. Values for Β at 2.2 Κ are derived at the high-stress limit. Thermal velocities of electron are taken to be
Ό4.2κ — 2.41 χ 1 0 6 cm s " 1, Ό 2. 2Κ = 1-74 χ 1 0 6 cm s ~ 1 and v l 5K = 1.44 χ 1 0 6 cm s ~ 1. Cyclotron resonance frequencies are the same as for Ge
Ρ
Li
Β
Ga
Classification
donor
donor
acceptor
acceptor
Bohr radius
(A)
(W>4.3K
(WT)l.5K
( c m 3 s _ )1
( c m 3 s _ )1
( c m 2)
( c m 2)
13
18
2.8 χ 1 0 " 5
1.3 χ 1 0 " 5
2.8 χ 1 0 " 5
1.3 χ 1 0 " 5
M 2 . 2 CI
( c m 3 s " 1)
7.7 χ 1 0 " 7
1.2 χ Κ Γ 11
5.4 χ 1 0 " 12
1.9 χ 1 0 - 1
9.0 χ 1 0 " 12
13
9
7.7 χ 1 0 " 7
5.3 χ 10" 7
σ
4.2Κ
σ
1.5Κ
σ
2.2Κ
3.2 χ 1 0 " 13
2.2 χ Η Γ 1 3
( c m 2)
4.4 χ 1 0 ' 13
Ε. Otsuka
Impurity
Cyclotron resonance
17
from phonon scattering low. Sekido et al. (1964) concluded that ionized impurity
scattering data, obtained at 11 Κ and for the ionized impurity concentration
range of 8 χ 1 0 12 - 4 χ 1 0 1 4c m ~ 3 , are in good agreement with the theoretical
prediction by Con well and Weisskopf (1950); namely,
1/ Tl = ln3/2e4Nl/Sm1J2K2(2kET)3/2^
In (1 + βΤ2)
(29)
with
jS = (3fcfc B/^ 2N I 1 /)32.
(29a)
Here the suffix I stands for the ionized impurity. The pioneering work by Sekido
et al. (1964), however, contained some shortcomings that could not be avoided
at that time. First the contribution from neutral impurities, both from Sb
(donor) and from Cu (acceptor), was analyzed using Erginsoy's formula. Second,
repulsive and attractive scattering centers were not separately treated. And
third, the contribution from phonon scattering, which had to be subtracted,
was larger at 10 Κ than at or below 4.2 K.
A more specific experiment was made by Otsuka et al. (1973) with uncom­
pensated Sb-doped Ge samples. The ionization of Sb was achieved by an H 2 0
laser beam (119 μπι) at and below 4.2 K. Cyclotron resonance of the lasergenerated electrons was performed at 35 GHz. For the unionized portion of Sb
donors, Erginsoy's relation could safely be used. Complications due to the
mixture with acceptors could be avoided. The contribution from lattice
scattering was much smaller than in the case of Sekido et al. (1964), so that it
could more accurately be subtracted from the total linewidth. As a result, a
considerable downward deviation of the inverse relaxation time from the
Con well-Weisskopf prediction was confirmed.
In discussing the linewidth against ionized impurity concentration, one has to
take into account the range of the force and compare it with the cyclotron orbit,
like the comparisons made in Kawamura et al.'s work on the carrier-carrier
interaction (Kawamura et al. 1964) or Miyake (1965). Both 1/T,OCN, and
1/T,OCN, 1 2/ relations have been observed. The deviation from the
Con w e l l - Weisskopf theory is not surprising if one takes into account the effect
of a magnetic field. Miyake predicts that for the l/τ, oc N, region, the proportion­
ality coefficient is determined by φ ζ | σ ( ι ; ζ ) > , where vz is the velocity component
of the electrons along the magnetic field and σ(νζ) is their associated scattering
cross-section. According to the energy conservation law, one has
> X
2
= ±mev2z + (N - N')fteoc,
(30)
where Ν and N' are the Landau quantum numbers before and after collision,
respectively, and similarly the velocity components vz and v'z. The change in
vz and hence kz (wavenumber) is thus quantized. The smallest change in kz, or
Akz = 2ll2/l = 21/2/(ch/eB)il2
is then to replace the cut-off length in the
18
Ε. Otsuka
Conwell-Weisskopf theory, if 1/^/2 is smaller than 1/2N113 (Dubinskaya 1969).
To our surprise, this replacement accounts for the observed linewidth very well
for ΛΓ, > 5 χ 1 0 13 c m " 3. Moreover, the modified Conwell-Weisskopf formula
predicts l/τ, almost independent of temperature between 2.0 and 4.2 K, in
agreement with the experimental observations. These features are shown in
figs 7 and 8.
The deviation of the cyclotron mobility from the drift- or low-field Hall
mobility becomes more conspicuous in cyclotron resonance experiments on
compound semiconductors performed at shorter wavelenths. One example is
CdTe by Mears and Stradling (1969), another InP by Chamberlain et al. (1971)
and the third InSb by Matsuda and Otsuka (1979a, b).
Nj
(cm" )
Fig. 7. The inverse relaxation time τ," 1 of electrons due to ionized impurities ( S b +) in Ge is plotted
against the S b + concentration, JV,. Data are taken at 35 G H z under a far-infrared (119 μπι) laser
excitation. The full curves are the Conwell-Weisskopf and Brooks-Herring predictions. The broken
line gives a modified Conwell-Weisskopf prediction with a new cut-off length (from Otsuka et al.
1973a).
Cyclotron resonance
19
T(K)
Fig. 8. The inverse relaxation time τ,~ 1 of electrons due to ionized impurities in Ge is nearly
temperature independent between 2.0 and 4.2 K. The full and broken curves are the same as in fig. 7,
with the ionized impurity concentration being fixed at 8 χ 1 0 13 c m - 3 (from Otsuka et al. 1973a).
2.4. Electron scattering by
dislocations
The effects of dislocations on cyclotron resonance were investigated by Otsuka
and Yamaguchi (1967), using ultra-pure Ge crystals. The method of introducing
90° and 60° dislocations is shown in fig. 9. The theory of D C mobility had been
developed by Dexter and Seitz (1952). A considerable discrepancy was found
between the theoretically predicted D C mobility and the cyclotron mobility.
Dexter and Seitz introduced a potential of the form
V{r9 θ) = -{Εφ/2π)1(\-
2v)/(l - v)]sin θ/r
(31)
that accounts for scattering by the static deformation potential due to an edgetype dislocation. In eq. (31), El is the deformation potential constant, b the
magnitude of the Burgers vector, ν the Poisson ratio, r the distance from the
dislocation line and θ the two-dimensional polar angle from the slip direction.
Obviously, eq. (31) describes a somewhat Coulomb-like long-range potential.
Then an argument similar to that already presented in dealing with ionized
impurity scattering is used to interpret the linewidth of cyclotron resonance. The
point one has to observe is that eq. (31) is two dimensional and anisotropic. The
Ε. Otsuka
20
α) 60° dislocation
b) 90° dislocation
<ni>
MicrovvcM?
field
<n
D-axs
<2II>
<ooi>
Microwave
field
Fig. 9. The method of introducing 60° or 90° dislocations by bending a Ge crystal. The directions of
microwaves and the magnetic fields in cyclotron resonance with respect to the cut-out specimen are
indicated (from Otsuka and Yamaguchi 1967).
experimentally derived inverse relaxation time l / r d due to dislocation scattering
is plotted in fig. 10 against the dislocation density. Measurements have been
made at 35 GHz both for 90° and for 60° dislocations. Numerically, the results
in fig. 10 indicate that
1/τ ά = 2.0 χ 10 6 N\12 s " 1
for 90° dislocations
(32)
1 / Td = 1.7 χ 1 0 6 N \ 12 s " 1
for 60° dislocations,
(33)
and
where Nd is counted in c m " 2. These results are independent of temperature
between 1.5 and 4.2 K. The theoretical prediction by Dexter and Seitz, on the
other hand, yields
l / T d = ( 3 ^ 3 2 ) ( £ 2b 2/ / c BT / i ) [ ( l - 2 v ) / ( l - v ) ] 2 N d .
(34)
In numerical form, it becomes
1 / Td = 6 . 6 x l O ^ T ^ s " 1 .
(35)
This is considerably larger, at liquid He temperatures, than the value of l / r d
derived from cyclotron resonance. The temperature and concentration de­
pendences are also different.
The linewidth argument of the cyclotron resonance in this case starts from the
shift in frequency
/±co =
{\/2me(Dch)(Vxx+Vyy)
(36)
Cyclotron resonance
10
21
10
τ—I
I
I I I
Ge
35
GHz
dislocation
•
J
6 0 ° dislocation
I
I
I
I
1 1 » '
cm
DISLOCATION DENSITY
(cm 2)
Fig. 10. Inverse relaxation time of electrons due to dislocations τ& 1in Ge against the concentration
of dislocations. Effect of 60° dislocations is less than that of 90° dislocations due to mixing of the
screw component, τ ^ 1 is nearly proportional to (dislocation d e n s i t y ) 1 27 (from Otsuka and
Yamaguchi 1967).
as given by Kawamura et al. (1964) in the carrier-carrier scattering. If one
introduces the static deformation scattering potential (31) in eq. (36) and takes
an average, an appropriate expression for l/xd is derived as follows:
l A d = ( ^ d / 3 ^ 1 / (2 | £ 1 | b / m e W )c[ ( l - 2 v ) / ( l - v ) ] / 2 / ? c 2.
(37)
Here Rc is a length parameter, introduced to prevent the averaging integral from
diverging. Physically, Rc is interpreted to be a distance from the dislocation line,
where the carrier becomes trapped by the dislocation, subsequently to disappear
in recombination. By putting Rc = ηΐ, where / = (ch/eB)1/2 and η is a numerical
factor; η is found to be 4.0 for 60° dislocations and 3.7 for 90° dislocations
in order to fit the experimental value of \/τά. The difference in η between 60°
and 90° dislocations in considered to arise from the mixing of the screw nature
in the 60° dislocation. The effective Burgers vector size of a 60° dislocation is
b sin 60° = 0.866 b. If this reduction in b is taken into account, one arrives at a
new value of η, 3.7, also for 60° dislocations.
In the derivation of eq. (37), the temperature does not show up. This agrees
with experimental observations but not with the Dexter-Seitz prediction.
The effect of temperature shows up, however, in the signal intensity. The
signal intensity varies with temperature almost as T~1/2. This is connected with
carrier recombination kinetics. The higher the temperature, the faster the
Ε. Otsuka
22
average electron is expected to arrive at a dislocation line, where the recombi­
nation is supposed to take place.
2.5. Electron scattering by excitons
Sometimes excitons in semiconductors have an influence on cyclotron res­
onance experiments. An exciton can, in a sense, be regarded as a kind of
impurity - a very shallow donor or acceptor. Electron-exciton scattering then
has a chance to make a contribution to the cyclotron resonance linewidth. An
optimum condition to observe this can be achieved in Ge. Electron-exciton
scattering shows up on top of the electron-carrier scattering contribution to the
linewidth. This was confirmed by time-resolved experiments which were
employed for the first time in cyclotron resonance by Yoshihara (1971), Ohyama
et al. (1971). An extra line broadening in the cyclotron resonance due to
electron-exciton collisions is shown in fig. 11. The effect is clearly seen at 4.2 Κ
but becomes less pronounced at lower temperatures because of the
electron-hole drop formation. Another somewhat different experimental ap­
proach has also been made for Si (Ohyama et al. 1973). The electron-exciton
scattering is a typical three-particle interaction. It was treated theoretically
(b)
ζ
ο
ζ
<
LU
Q
χ
< 10,10
1
cr
LU
LU
DC
cr
LU
<
ο
cr
cr
LU
>
Ζ
10 a
20 40 60
80
DELAY TIME ( p
100
s
)
20 40
60
80
DELAY TIME ( μβ )
Fig. 11. (a) Inverse relaxation time and carrier density η (open circles) at 4.2 Κ against delay time in
a strongly photoexcited time-resolved cyclotron resonance of Ge. l/τ, is the total linewidth and 1/τι
the contribution from lattice (phonon) scattering. The difference (solid circles) gives the sum of
electron-exciton and electron-carrier scattering. The presence of a kink in the l/τ, — l/τ, line
enables one to separate out the contribution from the electron-exciton scattering 1/τ β .χ On the lefthand side of the kink, electron-exciton collisions, while on the right-hand side, electron-carrier
collisions are dominant, (b) By lowering the temperature to 2.9 K, the kink becomes less pronounced
on account of the exciton condensation into electron-hole drops (from Yoshihara 1971).
Cyclotron resonance
23
by Matsuda et al. (1975). Electron-exciton scattering is more like
electron-neutral-donor scattering than electron-neutral-acceptor scattering.
This is because of the presence of electron-electron exchange. By the same
argument, hole-exciton scattering is considered more like hole-neutralacceptor scattering than like hole-neutral-donor scattering. In other words,
excitons behave like neutral donors in electron-exciton scattering and like
neutral acceptors in hole-exciton scattering. Thus the electron scattering crosssection by an exciton can very roughly be approximated by Erginsoy's relation
with an appropriate excitonic Bohr radius.
Electron scattering by bound exciton(s) is also detected by cyclotron
resonance. As already mentioned earlier, an extraordinary line broadening due
to scattering by bound excitons was seen for Ge:Zn, and similarly for Si:B
systems (Otsuka 1986; for original data see Murase and Otsuka 1968; Otsuka et
al. 1968a, b). The presence and importance of bound excitons (or bound doubleexciton complexes) was confirmed for the Ge:Zn system by photoluminescence
(Nakata and Otsuka 1984; Sauer and Weber 1984). Application of uniaxial
stress results in line narrowing and signal enhancement of cyclotron resonance.
Correspondence of the line narrowing to the release of bound excitons has been
shown and discussed by Otsuka et al. (1986) for the Ge:Zn system. Unfortunate­
ly, no quantitative arguments, say for scattering cross-sections etc., are available
for the moment.
Cyclotron resonance of the exciton itself, in a sense similar to 'impurity
cyclotron resonance' widely known for InSb (Kaplan 1969), has also been
observed and very briefly discussed (Fujii et al. 1975).
3. Cyclotron
resonance
as a kinetics
experiment
The expression (1) for power absorption contains σ 0 , the D C conductivity. Since
σ 0 is proportional to the carrier density, one may take the cyclotron resonance
intensity as a monitor for the carrier kinetics. In most semiconductors, free
carriers are practically frozen out at low temperatures. Only intrinsic photoexcitation can introduce free carriers in the bands. The photoexcited free carriers
are of course in nonequilibrium. Within the validity of a linear approximation,
the free-carrier density is proportional to the intensity of photoexcitation. Under
a steady state photoexcitation, the carrier density, and hence the resonance
absorption intensity, will be proportional to the carrier recombination time.
This is clear from the simple rate equation
dn/dt = G - η/ττ = 0,
(38)
where G is the electron-hole pair generation rate, η the free-carrier density, or
n = ne = nh, the suffices e and h standing for electrons and holes, respectively,
and τ Γ the recombination time. In the case of a pulsed photoexcitation, the time
Ε. Otsuka
24
variation of the carrier density after the end of a photopulse is supposed to
follow the relation
d n / d i = - η / τ Γ.
(39)
The transient nature of the photoexcited carriers can be traced by time-resolved
cyclotron resonance. The rate equation (39) simply gives an exponential decay
of the resonance signal intensity with a time constant of τ Γ. In reality, however,
the existence of impurities, free excitons, bound excitons, carrier-spin-flip
interaction etc. largely modifies the simple rate equation (39). Indeed, this is the
main reason why time-resolved cyclotron resonance is so useful to study carrier
kinetics. When we have several resonance lines, both the individual and the
relative intensities of these lines may be traced by time resolution and such a
tracing can lead to the correct kinetics of the carriers, either free or bound. Some
examples of nonequilibrium carrier kinetics will be found in InSb as well as in
GaAs.
3.1. Carrier kinetics in InSb
The high electron mobility of InSb is due to a small effective mass that makes
cyclotron resonance at far-infrared frequencies very easy. Moreover, some new
features arise in the resonance spectra. First, donor levels are so shallow that at
liquid helium temperatures an impurity band arises which overlaps with the
conduction band. The magnetic field behavior of shallow impurities is treated in
a chapter by W. Zawadzki in this volume (chapter 21). Second, neutral donors
show a Zeeman transition, the transition energy of which is much larger than the
ionization energy and very close to the cyclotron energy of conduction electrons.
This second aspect results in the side-by-side coexistence of impurity and
cyclotron transition peaks. The first observation of this was made by Kaplan
(1969). On account of the similarity in appearance of magnetoabsorption, the
impurity Zeeman transition was called 'impurity cyclotron resonance' (ICR).
The relevant Zeeman transition is l s - + 2 p + type and is also observed in other
compound semiconductors, InP, GaAs etc., but is rarely called impurity
cyclotron resonance as in the case of InSb.
The presence of conduction electrons even at 4.2 K, without photoexcitation,
in an η-type InSb makes cyclotron resonance experiment rather easy. Very
many experimental groups have indeed performed cyclotron resonance mea­
surements in this material. They include Oka et al. (1968), Johnson and Dickey
(1970), Apel et al. (1971), Murotani and Nisida (1972), McCombe et al. (1976)
and others. In combination with a superconducting magnet, an η-type InSb
crystal can serve as a detector for far-infrared spectroscopy. That is a
straightforward application of cyclotron resonance in InSb, and the device is
called a Putley detector after its inventor.
Detailed analysis of InSb cyclotron resonance data itself offers new insights.
Cyclotron resonance
25
The availability of high-mobility electrons suggests the study of hot-carrier
effects. This was first done by D C transport (Kotera et al. 1966, Miyazawa and
Ikoma 1967). Electric field excitation of conduction electrons was combined
with cyclotron resonance observations by Kobayashi and Otsuka (1972, 1974)
and later by Matsuda and Otsuka (1976, 1979b). Carrier heating by intrinsic
photoexcitation was achieved by Otsuka et al. (1981) and by Ohyama et al.
(1982). At first sight similar, but on closer examination entirely different, results
were obtained in cyclotron resonance spectra depending on whether electrical
and optical excitation was applied (Otsuka 1983a, b, c). Since the experimental
results show a unique behavior in many respects, we shall spend some time on
their description.
3.1.1. Electric field excitation
Conduction electrons accelerated by electric fields deviate in their distribution
function from thermal equilibrium. This deviation is frequently measured in
terms of'electron temperature'. To define the electron temperature, InSb offers a
convenient system. Owing to its narrow-gap nature, the conduction band is
sufficiently nonparabolic. The separation of Landau levels is no longer equi­
distant. For a far-infrared wavelength of about 100 μιη, a good quality InSb
crystal renders a well-resolved resonance spectrum with successive cyclotron
transitions: 0 + - > 1 + , 0 ~ - » 1 ~ , 1 + - > 2 + , · · , where the numerals give the Landau
quantum numbers and the indices the spin orientations. The intensity of each
transition signal is proportional to the electron population at the initial level
multiplied by the oscillator strength associated with the transition. For
simplicity, let us consider the first two transitions 0 +
1 + and 0" -> 1". The
+
electron population at 0 will be denoted by n 0 + and that at 0" by n0-. Then, if
we write
n0-/n0+=
e x p [ - ( ε 0 - - ε 0 + )/kB T e] ,
'
(40)
an electron temperature 7^ can be defined. Here ε 0 + and ε 0 - give the energies of
the relevant levels. Expression (40) assumes a Boltzmann distribution of
electrons over the Landau levels. In this simplest case, the signal intensities are
directly proportional to n0+ and n0-. In a magnetic field that makes 100 μηι
cyclotron resonance feasible, ( ε 0- - s0+)/kB > 4.2 Κ so that only one cyclotron
transition, 0 + -* 1 + , is observed at 4.2 Κ in the absence of an electric field. In the
work of Matsuda and Otsuka (1979a) the next transition, 0"
1~, starts to
show up only after raising the lattice temperature to 13 K. Going further up to
92 K, the third transition, l + - » 2 + , becomes detectable. These features are
illustrated in fig. 12. Even at 4.2 K, one can observe 0 ~ - > l ~ , l + - » 2 + , ···
transitions by applying an electric field, since the electron population of each
initial state is determined by the electron temperature and not by the lattice
temperature.
By a type of differential method, pulsed electric-field-modulated cyclotron
26
Ε. Otsuka
MAGNETIC FIELD ( kG )
Fig. 12. Thermal equilibrium resonance traces in η-type InSb at various temperatures. At 4.8 K,
1 + ) and 'impurity cyclotron resonance' I (ICR) are
only the lowest cyclotron transition C^O*
visible. On raising the temperature, the signal I disappears on complete ionization of donors, while
the second and third cyclotron transitions C 2( 0 " - • 1~) and C 3 ( l + - > 2 +) start to show up (from
Matsuda and Otsuka 1979a).
resonance (REM-CR), Kobayashi and Otsuka (1974) derived the electron
populations of the 0 + , 0", 1 + and 1" levels at several electric fields. Since the
electron distribution was not Maxwellian, the electron temperature was deter­
mined from the change in population at the lowest two Landau levels, 0 + and
0~. A transverse resistivity measurement (E1B) was also carried out for the
identical sample. The transverse resistivity under a strong electric field, p±(E),
and that under a very weak electric field, pL{Th),
were also measured. By
comparing p±(E) at 4.2 or 1.65 Κ with p±(TL) at the elevated lattice temperature
which gives the same value as pL{E), one can determine Te at a given electric
field. This is the method of Miyazawa (1969). The two electron temperatures
determined by cyclotron resonance and resistivity measurements showed a good
agreement, at 4.2 K, between Ε = 2 V c m " 1 and Ε = 7 V c m " l , where Te
changed from 13 to 30 K. A normalized resistivity p*(E), defined for conduction
Cyclotron resonance
27
electrons, could be derived from Landau level population measurements. It was
concluded that pf(E) oc T e " 3 /2 for 4.2 Κ < Te < 8 K. Contributions from impu­
rity conduction were excluded. Cyclotron resonance could thus reinforce the D C
transport argument.
More delicate aspects of electron temperature in InSb were summarized in the
work of Matsuda and Otsuka (1979b). In the presence of magnetic and electric
fields, three electron temperatures can be defined - two for conduction electrons
and one for impurity electrons. Each is further divided into two cases
corresponding to the geometries of Ε IB and Ε || B. For conduction electrons,
we define 'intersub-band' and 'intrasub-band' electron temperatures. The former
is defined in an approximate form by
n ( N ±) = A 1 e x p [ - e ( i V ± , k z = 0 ) / / c Bn n
]
t e ri
where n(N±) stands for the electron populations at the Landau sub-bands
The pre-exponential factor Αγ is determined by the relation
n^XniJV*),
(41)
N±.
(42)
N±
nc giving the total density of conduction electrons. This temperature is the same
as given before. The relative intensities of cyclotron transitions determine 7^ n t e.r
The intrasub-band temperature, on the other hand, involves the lineshape of the
cyclotron absorption. Within a single sub-band, the electron distribution
function can be written as
fNi(kz)
= A2 cxpi-WN*,
kz) - ε ( Ν ± , kz = Ο ) ] / ^ " " » } ,
(43)
where A2 is determined from the normalization condition
Π(Ν±)=Σ/Ν*(Κ).
(44)
It is assumed that the magnetic field is applied along the z-direction. When the
electron system is heated, the average wavevector component of kz becomes
larger so that the so-called kz broadening should be observed. From the
obtained lineshape, one can derive the intrasub-band electron temperature by
an iterative method with respect to a scattering parameter, first referring to the
equilibrium absorption line and then using the high-field-side halfwidth of the
nonequilibrium absorption line. Details can be consulted in the literature
(Matsuda and Otsuka 1979b). Of all the dramatic features of the hot-electron
resonance, a pair of resonance series are given in fig. 13 corresponding to the
transverse and longitudinal geometries of applied fields in the Voigt configura­
tion. Differences in both elative intensity and lineshape are clearly observable.
The third electron temperature concept is connected with the population of
electrons at the donor level. One can measure the population by the intensity of
the impurity cyclotron resonance. On applying an electric field, electrons at the
Ε. Otsuka
28
donor level are ejected into the conduction band. This can already be seen in
fig. 13. The equilibrium electron population at the donor level is a function of
lattice temperature. By correlating the electron population in an electric field at
4.2 Κ with that in thermal equilibrium at an elevated lattice temperature, one
can derive an electron temperature, again for the two geometries Ε || Β and Ε IB.
This procedure is shown in fig. 14. A crossing of the effect of the electric field is
observed for the two different geometries when the electron temperature is
raised. At low electric fields the electron temperature obtained for El Β is lower
than that obtained for Ε || Β and vice versa. A difference in electron temperature
between the two geometries has thus been observed for three definitions.
Geometric effects are observable in the resistivity and cyclotron emission
intensity measurements as well, which in turn reflect different electron distri­
butions, or electron temperatures in a broader sense. Since the El Β geometry is
considered twofold degenerate, one can take the electron temperature as a kind
of three-dimensional vector, having one longitudinal and two transverse
components. Cyclotron resonance has thus given us the chance to introduce the
new concept of'vectorial temperature' with the help of the electric and magnetic
fields.
Before completing the description of electric field excitation, it should be
emphasized that electron temperature studies can also be made by cyclotron
(a)
MAGNETIC FIELD (kG)
(b)
MAGNETIC FIELD (kG)
Fig. 13. Effect of the electric field on the electron resonance in η-type InSb at two different
geometries: (a), Ε IB and (b) E\\B in the Voigt configuration qlB, where q is the propagation
vector of the radiation. N o t e the difference in lineshape and relative size of the resonances, say at
E= 15.4 V c m " 1 (from Matsuda and Otsuka 1979b).
Cyclotron resonance
29
TEMPERATURE ( K )
ELECTRIC FIELD ( V/cm )
Fig. 14. Density of neutral donors in η-type InSb versus lattice temperature under thermal
equilibrium and against electric field for two geometries of the applied fields, relative to the density at
4.2 K. The density is determined by the ICR intensity. The electron temperature corresponding to
each geometry of the field application is derived by connecting the nonequilibrium data to the
thermal equilibrium data (from Matsuda and Otsuka 1979b).
emission as well (Gornik 1972, Kobayashi et al. 1973). A full treatment of that
subject, however, will be made elsewhere in this volume by Gornik, so that we
shall refrain from making any further comments.
3.1.2. Spin temperature in optical excitations
Electric fields are not the sole means of excitation. Optical excitation also allows
the production of hot carriers. High-intensity optical excitation was applied to
GaAs first by Ulbrich (1973) to obtain an optically hot electron system. With
this goal in mind, an optical excitation was employed in cyclotron resonance
experiments on InSb. As stated before, one can observe electron resonance at
4.2 Κ even without intrinsic photoexcitation in η-type InSb. Application of
illumination, nevertheless, produced rather surprising new findings. First, the
appearance of the 0~ -> 1 ~ transition was distinctly observed. Second, the decay
time constant of this transition peak was found to be unexpectedly large. The
process was examined by time-resolved measurements. Intrinsic photoexci­
tation was achieved by a xenon flash lamp, with a pulse width of about 1 μ8 and
at a repetition of 10 Hz. Only more than 10 μ8 after the application of the
excitation pulse, did the resonance spectrum return to its equilibrium state. In
other words, the existence of the 0~
1" absorption signal lasted more than
10 μ8. If we define the electron temperature in the fashion of eq. (40), the above
Ε. Otsuka
30
observation is equivalent to saying that the high-electron temperature caused by
optical excitation does not cool down to the lattice temperature in a time of
about 10 μ8. Such an observation becomes even more significant if one employs
p-type InSb instead of η-type. In a p-type sample, no electron signal, neither
conduction electron cyclotron resonance nor impurity cyclotron resonance,
shows up at 4.2 Κ without intrinsic photoexcitation. Application of photoexcitation makes a dramatic change. Three resonance peaks, ICR, 0 +
1 + (which
we shall denote by C x ) and 0"
1" (which we shall denote by C 2 ) appear after
excitation and remain for a long time. It seems that the C 2 peak disappears first,
in about 10 μ8 after the photopulse, but the two other peaks remain even after
30 μβ. After a lapse of milliseconds, all the signals disappear as expected for
p-type material in thermal equilibrium. This feature is illustrated in fig. 15 in
comparison with the η-type case. The delay time dependence of various signal
intensities as well as the relevant quantities is shown in fig. 16. The coexistence of
ICR throughout with the C1 and C 2 peaks shows that the apparent rise of
electron temperature is not caused by thermal heating of the sample due to
photopulses. In fact, a simple calorimetric calculation, as well as our experience
with Ge, shows a possible rise of temperature due to photopulses which is only
of the order of 0.1 K.
i.D
^
5
2.0
MAGNETIC FIELD ( T)
MAGNETIC FIELD ( Τ )
Fig. 15. (a) Time-resolved electron signals from η-type InSb after an intrinsic photoexcitation pulse.
The zero time is set at the top of the excitation pulse, (b) The same from a p-type InSb sample (from
Ohyama et al. 1982).
1
Cyclotron resonance
31
10
Sample C
42 Κ
:
in
100
•
A
ICFUC 1 +C 2
ICR
0 C,
AC2
10
ο C2/C!
1
0
1
1
I
1
1
1
10
DELAY
I
I
I
I
1
I
20
T I M E ( με )
>
\
30
Fig. 16. Time variation of various quantities obtained from fig. 15b (from Ohyama et al. 1982).
From the time dependence of the intensities of the C x and C 2 lines, one can
confirm the relation
d(Te-TL)/dt=-(Te-TL)/Tc.
(45)
Here T c is a time constant that corresponds to Newton's cooling law. For a
heavily compensated ρ-,type InSb sample, ND = 1.0 χ 1 0 1 4 c m " 3 and JVA = 1.1
x l 0 1 4c m " 3 , we obtain τ ε = 6.5μ8. This value has little temperature de­
pendence between 1.7 and 4.2 K. Apparently, the magnitude of T c obtained
seems a little bit too large in view of the existing theory of hot-carrier relaxation
(Shockley 1951), indeed by two orders of magnitude. One should recall,
however, that the electron temperature in the relation (40) can also be regarded
as a spin temperature, since the two Landau levels involved correspond to
different spin directions. The concept of spin temperature seems to be more
justifiable in the present case than that of the orbital electron temperature. There
are two main reasons for this justification. First, cyclotron transitions at higher
Landau levels such as 1 + ->2 + are not observed. Second, the lineshape of the
0+
1 + transition does not show any indication of the hot-carrier distribution
Ε. Otsuka
32
within the 0 + sub-band. In other words, the absorption curve is almost
Lorentzian, showing very little /c z-broadening. The intrasub-band electron
temperature is thus practically equal to TL.
This does not mean that conduction electrons in InSb can never become
optically hot. In continuous wave (CW) photoexcitation experiments with a
tungsten lamp, one can indeed produce a population in the 1 + level in n-type
InSb, which results in cyclotron emission (Otsuka et al. 1981, Ohyama et al.
1982). This is evidence for the existence of optically hot electrons. For pulsed
photoexcitation, cyclotron emission is observed only during the pulse appli­
cation. Both in CW and pulsed excitations, the emission signals are weaker than
in the case of electric field excitation.
This situation can be interpreted. On photoexcitation, electrons first populate
at least several Landau sub-bands. Then, a prompt energy relaxation starts
within a time interval of about 10" 8 s. The photoelectrons relax to the 0 + and
0" sub-bands. Those falling into the 0~ sub-bands, however, meet a bottleneck
prior to further relaxation to the lowest 0 + sub-band. This is considered to be
the reason for the persistent observation of Cl and C 2 transition lines. If the
intrinsic excitation light is unpolarized, it is reasonable to assume an equal
number of up and down spins in the conduction electrons at first, so that a
considerable portion of electrons populates the 0~ sub-band after the first
process of energy relaxation. The final spin flip between the 0~ and 0 + levels is
expected to occur during collisions with various scatterers. Elliott (1954) first
discussed the possibility of spin flip due to a purely electrical potential arising
from the implicit spin-orbit coupling. The same mechanism was also considered
by Yafet (1961). Another mechanism, taking account of the lack of inversion
symmetry in I I I - V compounds, was then proposed by D'yakonov and Perel
(1971a, b). A third special mechanism, taking into account the electron-hole
exchange interaction was further suggested by Bir et al. (1975) for the case of a
relatively high density of holes. Actual calculations for InSb were carried out
later, for example, by Boguslawski and Zawadzki (1980). Interaction potentials
that can cause spin flip derive from acoustical phonons, deformation potential
and piezoelectric modes, impurities and by other carriers. Impurities are further
divided into neutral and ionized. It is expected that contributions from phonons
are rather small at low temperatures.
Boguslawski and Zawadzki show that, at temperatures below 20 K, the
dominant scattering mechanism for spin-flip transitions is ionized impurity
scattering. The scattering potential is assumed to be of the screened Coulomb
type, or
V(r)=
-(e2/Kr)exp(-qsr),
where qs is the screening constant or the inverse of the Debye-Huckel
length. For a crystal with a donor concentration of ΝΌ = 1 0 14 c m - 3, the
Debye-Hiickel length becomes about 500 A. One should be aware, however,
Cyclotron resonance
33
that this magnitude of the Debye length is of the same order as the effective Bohr
radius of the electron bound to a donor. For an isolated neutral impurity, one
may as well construct a Hartree potential of the form
V(r) = -(e2/K)(l/r
+ l/a*)exp(-r/a*).
(46)
This is indeed similar in form to a screened Coulomb potential. As a% is of the
same order as the screening radius for an ionized center, it becomes all the more
difficult to distinguish between ionized and neutral donors.
Scattering by other carriers, or electron-electron scattering, may also
contribute to the 0~~->0 + transition. This has been treated by Boguslawski
(1980). We shall leave the electron-hole interaction aside, since no free-hole
resonance is observed in the time-resolved resonance spectra. Since the
interaction is again of the screened Coulomb type, this is nothing but an
extension of the earlier work of Boguslawski and Zawadski (1980) dealing with
ionized impurity scattering. One finds that the two spin relaxation times
and
7i i on (standing for electron-electron scattering and ionized impurity scattering,
respectively) are related as
_7T
ΤΓ
_{4y/2J(s/4)\N£
\
5J(s)
where J(s) = — 1 — (1 + s)exp(s)Ei(
— s) with
s = h2q2/2mekBT.
(48)
In the high-temperature limit, one obtains
Tr/Tr=lA3N+/nc.
(47a)
Supposing that N£ ~ n c , one can expect that the two scattering mechanisms
contribute almost equally to spin flip processes. Thus we have to take both
ionized and neutral impurity scattering into account, together with the contri­
bution from electron-electron scattering, to explain the observed value of Tx.
The density of the scattering centers depends on the excitation intensity. After a
strong excitation with intrinsic light, more impurities will be neutralized and we
shall have more conduction electrons. A numerical fitting procedure with the
excitation intensity as a parameter indicates a relative effectiveness of 0.9:1.0:0.8
for spin flipping, in the order of neutral impurities, ionized impurities and
electrons (Fujii 1985).
So far all the derivations of 7\ have been made for more heavily doped
materials than the one described here. Other works (Nguyen et al. 1976, Pascher
et al. 1976, Brueck and Mooradian 1976, Grisar et al. 1976), deal with impurity
concentrations of N D — N A = 1 0 15 - 1 0 1 6c m " 3 , and give smaller values of 7\,
varying from 1 0 ~ 9 to 1 0 ~ 7 s. Ti is considered to be inversely proportional to the
density of scattering centers. Only with such a low impurity concentration
34
Ε. Otsuka
sample as ours, with NA-ND
= 5.8 χ 1 0 12 cm""3, can one obtain such a large
value of T x as 3.5 μ8. Indeed, by normalizing impurity concentrations to a fixed
value, say ΝΌ — NA = 1 0 1 4 c m " 3, all experiments give comparable values of Tx.
3.2. Carrier kinetics in GaAs
Cyclotron resonance of conduction electrons in GaAs can also be utilized in
kinetics studies of photoexcited carriers, if the donor Zeeman transition is taken
into account. The first approach in this direction was made by Ohyama (1982).
Time-resolved measurements of cyclotron and Zeeman transition signals in
photoexcited GaAs samples show various new features of photoexcited carriers.
Different from the case of η-type InSb, the electron cyclotron resonance signal is
extremely small at 4.2 Κ in the absence of photoexcitation. The donor Zeeman
transition signal, on the other hand, is easily observable, the intensity of which
corresponds to the neutral donor density, ΝΌ — NA. After a flash from a xenon
lamp, electron cyclotron resonance shows up with a strong intensity and then
steadily decays exponentially with a time constant of about 7 μ8. The donor
Zeeman transition, in the meantime, weakens at first but starts to grow again,
making an overshoot in comparison with the original intensity at thermal
equilibrium, and, after a few tens of microseconds, reaches a quasi-equilibrium
stage. The signal intensity corresponding to this stage, is expected to reflect ΝΌ,
the total donor density. Combination of this with the original donor signal
intensity in the dark yields the compensating acceptor density. Thus, the
photoexcitation allows a joint determination of ND and NA. Though the
cyclotron and Zeeman transitions behave differently with time, the amount, ne
+ iVo (where n e is the density of conduction electrons and
that of neutral
donors) remains practically constant for the quasi-equilibrium time range. The
entire system comes back to thermal equilibrium after a long passage of time,
say 2 ms. The process is shown in fig. 17 for a typical sample of GaAs with
ND = 1.5 χ 1 0 15 c m " 3 and iV A = 1.0 χ 1 0 15 c m " 3. The slow decay process of the
neutralized donor system must be restricted by the donor-to-acceptor transfer of
electrons as in the case of G a P (Thomas et al. 1965). Let us assign a transition
probability of the form
W(r) =
Wmaxexp(-2r/aO)
(49)
to the electron transfer, where r is the separation between a neutral donor and a
neutral acceptor, αΌ the effective Bohr radius of the donor electron, and Wmax a
constant. One can then explain the slow decay process of the donor signal by
putting
W m ax = 5 x l 0 7 s ~ 1
(50)
with an appropriate ensemble average of r. The decrease of the donor signal at
the beginning may be due to the loss of the binding energy on account of
Cyclotron resonance
Sample
A
35
172
4.2 Κ
Δ_Δ__Δ
t=oo
JL
10
20
30
40
50
DELAY T I M E ( j i s )
60
Fig. 17. Time variations of densities of conduction electrons (ne) and neutral donors (iVjJ) in an
η-type GaAs crystal after an intrinsic photoexcitation pulse. The neutral donor signal intensity decays
to its thermal equilibrium value after a long lapse of time (i = oo) (from Ohyama 1982).
excessive screening by the photoexcited carriers. All these features are more or
less the same for InSb.
Further carrier kinetics studies in η-type GaAs, covering both impurity and
Landau level electron lifetimes, in a way similar to the photoexcitation
technique described here, have recently been made by Allan et al. (1985).
4. Cyclotron
resonance
in the quantum
limit
4.1. Electron scattering in GaAs
Gallium arsenide stands, almost in every respect, between InSb and Ge (or Si).
The substantial technological potential of GaAs has initiated great efforts
worldwide to improve the quality of these semiconductors. Today one has a
better chance of obtaining high-quality GaAs than high-quality InSb. However,
36
Ε. Otsuka
such a high standard of material control as achieved for Ge or Si is still
unavailable. The carrier effective mass, of electrons in GaAs, is 0.067 m0. This is
indeed between 0.014 m 0 for InSb and 0.2 m0 for Ge. Performing far-infrared
cyclotron resonance experiments in GaAs is not as easy as in InSb but much
easier than in Ge. One advantage not existing in Ge is the nondegeneracy of the
conduction band, with the consequence that the electron mass in GaAs is
isotropic. It is no longer necessary to discuss the complications involving T|( and
τ ± . Thus a linkage of cyclotron resonance with transport studies becomes more
straightforward than in the case of Ge.
The smaller carrier effective mass certainly makes cyclotron resonance easier,
since a given infrared frequency requires a lower magnetic field. However, too
small an effective mass sometimes makes the identification of impurity effects
problematic as we have already experienced in InSb. Indeed we have had
difficulties in distinguishing between neutral and ionized impurity scattering in
InSb because of the very large Bohr radius of the donor electron. The Bohr
radius is inversely proportional to the carrier effective mass. Thanks to a
reasonably large effective mass, one can consider the impurity centers to be
sufficiently isolated for high-quality GaAs. Thus one may approach the problem
of impurity scattering in GaAs in a similar way to Ge or Si, to a large extent.
The electron effective mass in GaAs is thus sufficiently large to define an
isolated neutral donor. At the same time, it is sufficiently small to make quantum
limit studies possible, since the requirement that the de Broglie wavelength
should be much larger than cyclotron radius ftcoc > kB Τ is readily fulfilled for
far-infrared wavelengths near 100 μιη at 4.2 K. In fact, ftcoc/kB = 83.3 K, for
example, for a wavelength of 172 μιη (1744 GHz). Cyclotron resonance of
conduction electrons is observable at 4.2 Κ only under or after intrinsic
photoexcitation. In the absence of photoexcitation, the sole absorption signal is
the Zeeman transition of donor impurities.
Typical resonances traces, time resolved, are shown in fig. 18 for three samples
of GaAs, for a single FIR wavelength of 220 μιη. All the signals were obtained
20 μ8 after the end of the excitation photopulse, or, in laboratory language, with
a delay time of 20 μ8. The line appearing near 10 kG ( I T ) is the Zeeman
transition of the donor-bound electrons. This corresponds to ICR in InSb. As
mentioned in the last section, the Zeeman transition plays a role supplementary
to the cyclotron resonance which appears at a higher magnetic field near 33 kG.
Since we are concerned with the cyclotron resonance itself, let us leave the
Zeeman transition for the moment.
All the traces in fig. 18 for the three samples of GaAs are obtained at 4.2 Κ
and at td (delay time) = 20 μβ. In spite of these common experimental con­
ditions, the resonance traces obtained differ considerably from sample to
sample. The contribution from the phonon scattering cannot vary for different
samples. So any difference in resonance behavior, linewidth in particular, should
be attributed to the difference in impurity content. For the three samples shown
Cyclotron resonance
37
GaAs-1
λ=220μπι
Τ\ 4· 2Κ
t d= 2 0 p s
CR.
1S-2R1
'
J_
10
20
30
40
20
30
40
GaAs-3
λ=220μη
T=A.2K 1 S^2P.1
t d= 2 0 Hs
0
10
GaAs-5
λ=220μηη
4.2 Κ
t d= 2 0 Ms
10
20
MAGNETIC
FIELD
30
( kG )
Fig. 18. Absorption traces obtained at 4.2 Κ for three η-type GaAs crystals. Both cyclotron
resonance (CR) and the donor Zeeman transition ( l s - ^ 2 p +) are shown for a far-infrared
wavelength of 220 μπι. All the data are taken in time resolution at a delay time of 20 μ8 after the
excitation photopulse. Lines are more broadened for samples with more doping. Impurity
concentrations for the three samples are given in the text (from Kobori 1986).
in fig. 18, the following impurity concentrations have been determined: GaAs-1:
ND = 1.5x 1 0 1 5c m " 3 a n d i V A = 5.0x 1 0 1 4 c m - 3; GaAs-2: N D = 1 . 5 x 1 0 1 5c m " 3
and i V A = 1 . 0 x 1 0 1 5c m " 3 ; GaAs-3: ND = 5.5 χ 1 0 15 c m - 3 and J V A= 1 . 5 x
1 0 1 5c m ~ 3 . This information is hard to determine from transport experi­
ments.
4.1.1. Carrier-carrier
scattering
If one changes i d, the delay time, the linewidth of the cyclotron resonance also
changes in accordance. For smaller i d, the linewidth is broader. That is due to
the carrier-carrier interaction. Variation of td corresponds to changing the
carrier density. The absolute value of the carrier (electron) density can be
derived from the intensity of cyclotron resonance (Ohyama 1982). In the course
Ε. Otsuka
38
of increasing t d, the resonance linewidth approaches a constant value. Then the
carrier-carrier interaction is no longer contributing to the linewidth. This
feature is shown in fig. 19. Writing the total linewidth in terms of the relaxation
time τ, we obtain, for a series of time-resolved cyclotron resonance measure­
ments, the relation
\/x = a + bn\
(51)
where a and b are constants, η is the electron density and s is an exponent close
to unity. The first term, or the constant a, is evidently reflecting neutral impurity
scattering. We shall rewrite the second term as 1/T cc to identify the contribution
of carrier-carrier scattering. The closeness of 5 to unity means that one is dealing
with the low-concentration region (Arora and Spector 1979, Prasad 1982). The
value s = j is expected for the high-concentration region (Fujita and Lodder
1976, Prasad 1982). For λ = 220μη\ (1364 GHz) and at 4.2 K, one obtains
1/T cc = 2.8 and 3.1 χ 1 0 ~ 4 n s _ 1 for GaAs-1 and GaAs-3, respectively, with
η expressed in c m " 3. For λ = 172 μπι (1744 GHz) and at 4.2 K, one obtains
1/T cc = 2.4 and 3.0 χ 1 0 " 4 n s - 1 for the same samples. Theory predicts 1/T CC
= 6.3 χ 1 0 ~ 4 n s - 1 and 1/T cc = 4.9 χ 1 0 ~ 4 n s " 1 for 220 μπι and 172 μπι, respec­
tively, at 4.2 Κ for low carrier density. One should note that Kawamura et al.
(1964), in their classical treatment, predict 1/T C COCH 1 /2 for low carrier con­
centration and 1/T CC OC η for high carrier concentration. This is in contrast to the
quantum limit treatments. The high concentration case in the classical treatment
* 1 0 12
λ=172μΓη
Τ =4.2 Κ
• GaAs -1
- (Erginsoy)
10'k11
• ocne
10'>10L|13 J
10
I 1*1 1 MM
,14
I
I I 1 I II
10'
ELECTRON DENSITY
10,15
(cm* 3)
Fig. 19. Separation of carrier-carrier and neutral impurity scattering in an η-type GaAs crystal. The
linear dependence part of the inverse relaxation time on the electron density comes from the
carrier-carrier scattering. The electron density is varied with the time resolution. The horizontal
component is believed to arise from the neutral impurity scattering. Erginsoy's prediction in D C
transport corresponding to the donor density in this sample is also indicated (from Kobori 1986).
Cyclotron resonance
39
is similar to the D C transport case. The reason is that one can expect the
presence of another electron within the cyclotron orbit of a particular electron.
Then the localization effect that characterizes the cyclotron motion is weakened
and practically replaced by an ordinary plane-wave scattering problem. In the
far-infrared cyclotron resonance of GaAs at 4.2 K, the quantum limit conditions
hold. The absence of the η 1 /2 dependence of 1/T c c, predicted for the low
concentration case in the Kawamura formulation, is not surprising. On the
other hand, the η 1 /2 dependence expected for the high-concentration case in the
quantum limit theory, has not been observed so far.
The first term on the right-hand side of eq. (51) is primarily determined by the
contribution from impurity scattering at low temperatures, say at 4.2 K. To be
more exact, it is neutral impurity scattering that determines the magnitude of a.
The effects of phonon scattering can show up only at elevated temperatures.
Genuine contributions from impurities and phonons to the linewidth should be
looked for at a delay time for which the carrier-carrier scattering makes a
negligible contribution.
4.1.2. Effects of phonon scattering
By changing the temperature, one can separate the contributions to the
linewidth from various types of scattering. Possible contributions from phonons
are: (i) acoustical piezoelectric scattering, (ii) acoustical deformation potential
scattering and (iii) polar optical phonon scattering. The importance of neutral
impurity scattering depends on the sample. For GaAs-1, the purest of all
samples investigated by us, the contribution from phonons becomes visible
above 10 K. Of the three kinds of the phonon scattering mentioned above, no
quantum limit treatment for the polar optical phonon scattering is available.
This does not matter, since the polar optical phonon scattering becomes
effective only at high temperatures, where the condition fta>c > kB Τ is no longer
satisfied. We shall, accordingly, employ the classical theory for this type of
scattering in order to make a comparison with the experimental results.
Expressed in analytical and numerical form, the relevant scatttering equation
becomes
1/Tpo = 2occuLO[exp(ha)LO/kBΤ)
- 1]
= 7.2 χ 1 0 1 2[ e x p ( 4 3 2 / T ) - l ] " 1 s
(52)
(52a)
where α is the polaron coupling constant and c o LO is the longitudinal optical
phonon frequency, and Τ is measured in K. Quantum limit calculations for the
two kinds of acoustical phonon scattering, piezoelectric and deformation
potential, are available. The quantum theory for piezoelectric phonon scattering
developed by Saitoh and Kawabata (1967), however, contains no dependence
on c o c. Numerically, it gives surprisingly a prediction nearly identical with the
classical calculation given by Meyer and Polder (1953). The classical theory
Ε. Otsuka
40
yields
3n1/2m*1/2e2K2(kBT)112
1
τ ρζ
2 5 / f2 c 2,c
= 3 . 9 x \09T1/2s-\
(53a)
where X is the piezoelectric coupling constant. The prefactor in (53a) changes to
3.7 χ 1 0 9 in the Saitoh-Kawabata treatment. It thus becomes unimportant to
distinguish between the classical and quantum treatments for the piezoelectric
scattering. Accordingly, we shall refer to the older work for the moment.
The final mechanism, acoustical deformation potential scattering, is the
subject of some controversy. Quantum treatments have been presented by Arora
and Spector (1979) as well as by Suzuki and Dunn (1982). The two theories have
almost the same analytical form. The former authors give
_L =
(54)
^
τ ΟΡ
^.OItO^C/cbT
1
^ 3 m * 3 / 2 E 2 ( f c BT ) 3 / 2
with
τ£ρ
2 3 / 72 r 1' 2f t c 1
*
'
Here El is the deformation potential constant and c x is the longitudinal elastic
constant which can be expressed, in terms of the elastic stiffness constants c l 7, as
( 3 c n + 2 c 1 2 + 4 c 4 4) / 5 . The expression Ι/τ^ρ stands for the zero-field scattering
as treated originally by Bardeen and Shockley (1950). Using expression (54), all
we have to know is the exact value of E1. Unfortunately, this is not quite well
established. Values of - 7 . 0 e V (Stillman et al. 1970), - 1 1 . 5 e V (Rode and
Knight 1971) and - 15.7 eV (Pfeffer et al. 1984) have been published. None of
them, however, yields satisfactory agreement with the experimental observa­
tions. A large deviation of the experimental points from the prediction of Arora
and Spector is shown in fig. 20, for λ = 172 μπι, where Εί = —7.0 eV has been
adopted. Changing to Ex = —15.7 eV makes the fitting even worse. Replace­
ment by Suzuki and Dunn's value gives no improvement.
At present, no satisfactory explanation has been given yet for the discrepancy
between theory and experiment. Taking the experimental result as correct, one
has to look for a new analytical approach. For the experimental precision, on
the other hand, it is important to single out as much as possible the acoustical
deformation potential phonon scattering in the quantum limit. As we shall see at
the end of this chapter, such an isolation is realizable in Ge, where we
empirically obtain 1/T D PO C T. Assuming that the same relation holds in GaAs, it
is possible to try a new fit. A much better fitting than before is indeed obtained if
we put
1 / τ Ο Ρ= 1 . 9 χ I O ^ s "
1
(56)
Cyclotron resonance
41
1013 -
Ε
λ=172μηη
- · GaAs-1
Combined —• / /
-
•/ /
)g10,12
. 7 / /
-(Erginsoy)
-Neutral Impurity
ώιο 1 ~~
LU
>
1
·
/ //Acoustic
· * / / Deformation
·* y//7
R B t i e2 i n
χ
<
ω
α:
LU
/
·/ / /
Ρ
<
/
·
• SI /1
• yS / / Ι
t i a l l
s
/
i^nezoelectric
~ Neutral impurity
S
-(Experimental)
/
γ
/Pblar Optical ( 3 )
ι ι r ι ιΑ πι
ι ι ι ι 11 n
10>10
1
10
10 3
102
TEMPERATURE
( K)
Fig. 20. The temperature dependence of the inverse relaxation time of electrons in GaAs is
compared between experimental data and theoretical predictions at a wavelength of 172 μπι.
Phonon-type scattering comprises: (1) acoustic piezoelectric scattering, (2) acoustic deformation
scattering are indicated by broken lines, experimental
and theoretical
(Erginsoy)
(from Kobori 1986).
for λ = 220 μπι. The results for GaAs-3 are shown in fig. 21. A similarly nice fit is
also available for GaAs-1, using eq. (56). For A = 172 μπι, the numerical factor in
eq. (56) has to be changed to 3.0 χ 10 9. For enhanced accuracy perhaps a more
careful treatment of other scattering contributions will be required. Some more
discussions will be presented later in association with the experimental obser­
vations for Ge.
4.1.3. Neutral impurity scattering
In fig. 20, the horizontal lines indicated either by 'Neutral Impurity', (Erginsoy)
or (Experimental), become the final subject of our discussion. The correspond­
ing lines are denoted by ND(C.L.) and ND(Q.L.) in fig. 21. Evidently, there
exists a quantum effect in the neutral impurity scattering. The so-called Erginsoy
formula, that was successful in explaining the neutral donor scattering in Ge at
microwave frequencies, gives in figs 20 and 21 a value nearly one order of
magnitude larger than the experimental observation. Thus it becomes important
to examine the magnetic field dependence of the neutral (donor) impurity
scattering. The results are shown in fig. 22 for two samples, GaAs-1 and GaAs-3.
It seems that a downward deviation from Erginsoy's prediction starts well below
10 kG, and, for the quantum region, there exists a field dependence like B~1,2. A
parallel investigation of the concentration dependence at a given field shows
Ε. Otsuka
42
_
1 0 1 3F
-
|ιο
I
I I I I ιιιι
1 1 1 1 1 III
I
Λ = 2 2 0 μηη
DP(C.L)5
/ 1 /-
η-GaAs-3
T S ( J C/ K B= 6 5 . 3 K
Combined—• //
-
/
V /
O/ /
°/ /
°/ / /
=
/ /
'
DP(Q.L.)I
/
/
''y
Ο / //
-
12
11 I 1 1 lit
.-NDJC.U
©
<
Χ
<
—ι
LU
* 1 0 11
LU
CO
X'
ND(Q.L)
κ
,10
10'
10^
1
/«—po
-
y
1 1 l/UTf
/l
1 1 1 III III
10z
10
TEMPERATURE
1 1 1 1 1 III
10-
( K)
Fig. 21. Improved fitting between theory and experiment is obtained for the temperature
dependence of the inverse relaxation time of electrons in GaAs-3 by introducing an empirical linear
temperature dependence of the acoustic deformation potential scattering in the quantum limit, as
denoted by D P (QL). The classical-limit Bardeen-Shockley prediction is drawn by a broken line,
denoted by D P (CL). Polar optical (PO) scattering and acoustic piezoelectric (PZ) scattering are the
same as in fig. 20. Horizontal lines denoted by N D (CL) and N D (QL) are the classical (Erginsoy)
and quantum limit (experimental) contributions, respectively, from neutral donors in this sample
(from Kobori 1986).
linear behavior. A feature common with the Erginsoy prediction is that very
little temperature dependence is observed for every FIR frequency. Combining
all these influences on the neutral donor scattering, it may be justified to write:
1 / τ ΝΟ = (1.1-1.2) χ I O - ^ d B - ^ T ^ s "
1
(57)
for the quantum region of GaAs, where Β is in gauss and ND is in c m " 3 .
A large difference in cross-section between electron-donor and
electron-acceptor scattering has been observed in Ge and Si. The same effects
also exist in GaAs. However, a difference has so far only been observed in the
quantum limit regime. Three p-type GaAs crystals, A, Β and C, with net
impurity concentrations of NA — ΝΌ = 3 χ 1 0 1 4, 6 χ Ι Ο 15 and 6.5 χ 1 0 1 6 c m " 3,
respectively, were the subject of a 172 μηι cyclotron resonance experiment at
4.2 Κ (Kobori et al. 1987). The electron resonance trace of the crystal Β is shown
in fig. 23, in comparison with an η-type GaAs sample having a donor
Cyclotron resonance
* 1 0 12
43
=
GaAs-3_ (Erginsoy )
UJ
GaAs-1
(Erginsoy)
B
UJ
T = A.2K
• GaAs-1
ο GaAs-3
1 1 010
ι 11 11
I
I
2
I I I MM
10
10 2
MAGNETIC FIELD
J
I l l I III
10 3
(kG)
Fig. 22. Magnetic field dependence of the inverse relaxation time due to neutral impurities obtained
for two GaAs samples at a fixed temperature of 4.2 K. The tendency of decreasing inverse relaxation
time with magnetic field is rather similar for the two samples, though their impurity concentrations
are different. The horizontal lines denoted by 'Erginsoy' give the field-independent classical
predictions (from Ohyama et al. 1986).
λ=172μΓη
η - GaAs
N dr5.5xl0 1 5crrf 3
Ζ
UJ
λ=172μη
p-GaAs-B
N a=6.0x10 1 5crrf :
_L
30
35
AO
MAGNETIC
A5
FIELD
_1_
50
Λ-χ.
55
(kG).
Fig. 23. Difference in electron cyclotron resonance linewidth between n- and p-type GaAs crystals,
having the same order of donor or acceptor concentrations (from Kobori et al. 1987).
44
Ε. Otsuka
concentration of 5.5 χ 1 0 15 c m " 3. In these samples, compensation has been kept
as small as possible, so that the observed large difference in linewidth is
primarily considered to reflect the difference between the electron-donor and
electron-acceptor scattering. An average inverse relaxation time of 1/τ ΝΑ ~ 0.8
χ 10"6 NA s " 1 has been obtained from the linewidth of the three samples. The
prefactor is nearly by a factor of 50 smaller than that for the donor scattering at
the same wavelength. The main acceptor dopant is Be. Its energy level lies
28 meV above the valence band edge. The effective Bohr radius will then be
about several times smaller than that of the donor which has a typical binding
energy of 5 meV. If this small value of the Bohr radius is combined with the
difference between e ~ - H - and e +- H - t y p e scattering, which has already been
discussed in a previous section, the observed difference in the scattering
coefficient by a factor as large as 50 is not surprising. It seems worth mentioning
that the large difference in the electron scattering cross-section between donors
and acceptors observed in the classical regime also exists in the quantum limit.
In other words, both e ~ - H - and e +- H - t y p e scattering have a reduced scattering
cross-section in the quantum limit, apparently described by the same factor.
4.2. Ionized impurity scattering in InSb
The dominant contribution from ionized impurities to the electron scattering in
the quantum limit shows up very clearly in the far-infrared cyclotron resonance
of InSb (Matsuda and Otsuka 1979a). As mentioned already several times,
electron resonance in this material can be observed without band gap photo­
excitation. Conduction electrons are available from donors in the case of n-type
materials and leave donors ionized, except in very high magnetic fields at low
temperatures. The cyclotron resonance linewidth has been measured between
4.2 and 160 Κ for three samples and for three far-infrared wavelengths: 84, 119
and 172 μπι (3571, 2521 and 1744 GHz). General features are, that the inverse
relaxation time is definitely smaller than that predicted by D C conductivity
data. The shorter the FIR wavelength, the smaller the inverse relaxation time;
for the temperature range where the ionized impurity scattering is dominant, the
observed inverse relaxation time is practically independent of temperature.
Matsuda and Otsuka compare their experimental data with the theoretical
formulae derived by Kawamura et al. (1964) and Fujita and Lodder (1976).
Recently, van Royen et al. (1984) made another comparison between theory and
experiment from the theoretical side. They proposed a new approach starting
from the Kubo formula and obtained good agreement with the experiments. The
lowest-order Born approximation is employed to account for the joint contri­
bution from ionized donors and acceptors. Experimental data for a typical
sample with an impurity concentration of 5.5 χ 1 0 1 4c m ~ 3 are presented in
fig. 24 for the three wavelengths mentioned before. The deviation from the D C
data ( ω = 0) is quite obvious. The theoretical prediction, together with one for
Cyclotron resonance
J
I
1
I
I
I
I
10
TEMPERATURE
45
I I
I
I
I
I
I
100
I 1
I
(K)
Fig. 24. Inverse relaxation time of electrons in η-type InSb mainly due to ionized impurities is
plotted against temperature (Matsuda and Otsuka 1979a). Experimental points are obtained for
three far-infrared wavelengths. The full curve indicated by ω = 0 is a result of D C transport
measurements. The broken curves are the predictions of van Royen et al. (1984) calculated for a
wavelength of 84 μπι. At temperatures higher than 60 K, the onset of the optical phonon scattering is
obvious (from Otsuka 1986).
LO phonon scattering, is given by the broken curves. Typically for 84 μπι, we
obtain l/τ, = 1.6 χ 1 0 11 s " 1 at 18-35 K. This corresponds to a scattering
coefficient of 1/τ,Ν, = 2.9 χ 1 0 " 4 s _ 1 c m 3. It is of interest that this value is
almost in exact agreement with the one derived for S b + in Ge at 3.2 Κ and
35 GHz (Otsuka et al. 1973a).
5. Cyclotron
resonance
in III-V
and 11-VI
compounds
5.1. Employment of very high magnetic fields
It is needless to say that the primary requirement for the observation of
cyclotron resonance is that ω0τ > 1 holds. An equivalent formulation of this
condition is μΒ
1 where μ is the carrier mobility. Residual impurities
sometimes limit the low-temperature mobility significantly. In order to obtain
cyclotron resonance in such materials at low temperatures or at higher
temperatures, the use of high magnetic fields in conjunction with high frequen­
cies is necessary. Continuous fields up to 25 Τ have been produced by watercooled resistive magnets, and fields up to 35 Τ by hybrid magnets, a combination
of resistive and superconducting coils (Landwehr 1980). Still higher magnetic
46
Ε. Otsuka
fields have been generated with pulsed coils. This is a sort of challenge in high
technology, since one has to fight the destructive magnetic forces, called the
Maxwell stresses, which rise with the square of the field. In fact, even a most
deliberately designed stress-reduced coil is subject to break at a field of 6 0 - 7 0 Τ
(Miura 1984). Any attempt to perform cyclotron resonance in the so-called
megagauss (100T) region, accordingly, has to be accompanied by the destruc­
tion not only of a coil but also of a precious sample. Only if sufficient
information is obtainable in a single condenser discharge, does it seem justified
to perform such an experiment.
Those who performed such investigations, however, contributed much to the
clarification of the band properties of certain semiconductors. Some of the
experiments were carried out in the nondestructive region. Suzuki and Miura
(1975) demonstrated with λ = 119 μπι that thermally released holes in p-type Ge
displayed quantum cyclotron resonance spectra at 77 Κ similar to those
observed at 1.2 Κ with λ = 5.57 mm (Hensel 1962). Larger cyclotron masses
than those observed at longer wavelengths were found. Since the band gap
varies with magnetic field, changes in the effective mass could be expected and
were indeed confirmed experimentally for the first time in Ge. The same authors
also carried out quantum cyclotron resonance measurements for holes in p-type
GaSb. This material had previously been investigated by Stradling (1966) at a
longer wavelength of 2 mm. Suzuki and Miura found a close resemblance of hole
spectra in GaSb with those in Ge. This resemblance made them confident to
determine the Luttinger parameters yl9 y 2, y3 and κ (Luttinger 1956) for the
valence band of this material. The parameters found are comparable with the
classical values derived by Stradling.
Electron cyclotron resonance was even easier to study and to analyze in the
megagauss range. Small cyclotron masses of electrons in InSb, GaAs and in Ge
enabled Miura et al. (1976) to employ C 0 2 laser wavelengths, 9.5 to 10.8 μπι, in
combination with magnetic fields between 50 and 100T. The ambient temper­
ature was 300 K. Spin splitting in the lowest cyclotron transition was observed
only in InSb. The room temperature cyclotron masses were slightly different
from those found at low temperatures. An interesting discovery was the
dependence of the relaxation time τ on magnetic field as seen in Ge. For one
sample, τ was found, in units of 1 0 " 1 3 s, to be 1.1 at 96 T, 4.1 at 7.8 T, and 1.7 at
zero field as obtained from D C measurements. One thus finds that τ goes
through a maximum at medium fields. In GaAs, only a decrease of τ in the
presence of high magnetic fields was measured. The τ-value obtained from the
D C mobility was 2.5 χ 1 0 " 1 3 s, which was larger than 1.0 χ 1 0 " 1 3 s, the value
obtained with a field of 80 T. This is in contrast with the results obtained at low
temperatures for InSb (Matsuda and Otsuka 1979a, b). The magnetic field
dependence of the relaxation time, however, depends on the nature of scattering.
At low temperatures, impurity scattering is mainly responsible for τ in most
cases. At high temperatures - as employed by Miura et al. - lattice scattering is
Cyclotron resonance
47
more important than impurity scattering. Thus the field dependence of τ should
not be the same in all cases.
Generally, electron cyclotron resonance can yield much more detailed
information about scattering processes than hole resonance. The reason is that
the classical hole resonance has an inhomogeneous broadening due to overlap­
ping of different quantum transitions. If one can achieve an isolated quantum line
for holes, one may of course discuss the hole scattering as well from its linewidth
aspects.
It has indeed been a remarkable development in cyclotron resonance that one
can discuss the linewidth even of room temperature data. However, cyclotron
resonance experiments in pulsed magnetic fields have several shortcomings: they
cannot be repeated. Many times even in the nondestructive case, the repetition
cycle is small. Time-resolved measurements thus become prohibitive. Moreover,
carrier kinetics studies involving cyclotron resonance are beyond the range of
pulsed magnetic fields. With all these drawbacks, cyclotron resonance in
extremely high magnetic fields remains a pioneering spearhead in Landau level
spectroscopy. More experimental results obtained in pulsed magnetic fields will
be discussed later in this section.
5.2. Transport analysis in III-V
compounds
5.2.1. Electron cyclotron resonance in GaSb
While the hole cyclotron resonance in GaSb has actively been explored
(Stradling 1966, Suzuki and Miura 1975), very few studies on electron resonance
have been reported. This is partly because the currently available purest,
as-grown crystals are always ρ type. However, one can have electronic resonance
even in p-type materials after intrinsic carrier excitation. The first and, to the
author's knowledge, perhaps the only data of electron cyclotron resonance in
GaSb were obtained by Hill and Schwerdtfeger (1974). These authors il­
luminated a GaSb crystal, having a large acceptor concentration of 1 - 2
χ 1 0 1 7c m ~ 3 with intrinsic light, and carried out measurements in the milli­
meter wave ( 3 2 - 3 7 GHz) range. The main acceptor level was located at
3 2 - 3 7 meV above the top of the valence band. Secondary acceptors, with a
concentration of about 1 χ 1 0 16 c m " 3 , had an energy level of 11 or 24 meV. Due
to the low value of ω 0τ, the resonance absorption signal was rather broad.
However, these authors succeeded in deriving a band mass parameter of
0.0396 m 0 as well as a polaron mass of 0.0412 m 0, corresponding to an average
electronic energy of 15meV. From the observed linewidth and the known
acceptor concentration NA (to be more exact, NA — ND), the electron scattering
rate by neutral impurities was also derived. The authors were aware of the
difference between electron-donor and electron-acceptor scatterings. They put
l/t\
=
caAhNA/me
(58)
48
Ε. Otsuka
- a modified Erginsoy relation - where aA was the acceptor Bohr radius, 1/τ Α the
scattering rate by acceptors and c a constant that could depend on electron
energy. Erginsoy's relation corresponds to c = 20. This value of c is valid only for
a hydrogenic neutral donor. The empirical value obtained by Hill and
Schwerdtfeger was c ~ 0.1. It was smaller than predicted by Otsuka et al. (1964,
1966a, b, c), as well as by Blagosklonshegya et al. (1969). These theoretical
predictions, however, were derived for electrons in thermal equilibrium. Hill and
Schwerdtfeger obtained their cyclotron resonance signals in strong microwave
electric fields. In other words, the electrons were hot. Calculations of atomic
scatterings, e ~ - H and e + - H , show that at higher energies of the incident
particle the difference between the two scattering mechanisms becomes larger.
This enhancement effect was further discussed by Otsuka (1981, 1983b). The
ratio of scattering cross-section e ~ - H to e + - H could indeed become 10 2 to 10 3
for kaB = 0.5-0.6, where k is the wavenumber of the incident particle (electron or
positron). The photoelectrons in GaSb heated by microwave electric fields had
without doubt wave numbers in this range.
5.2.2. Electron cyclotron resonance in In?
The first cyclotron resonance in InP was reported by Chamberlain et al. (1971).
The measurement was made by the cross-modulation technique at a FIR
wavelength of 337 μπι. Two high-quality samples were employed: one with iV D
- N A = 4 x 1 0 1 5c m - 3 and μ(77 Κ) = 33000 c m 2 V - 1 s " 1, and the other with
MD-NA
= 2 χ 1 0 15 c m " 3 and μ(77 Κ) = 6 0 0 0 0 c m 2 V " 1 s " 1. The observed
effective mass changed slightly with temperature: 0.0815 m 0 at 10 Κ to 0.0829 m 0
at 42 Κ for the sample with higher mobility. After a polaron correction, the band
mass was deduced as 0.081 m 0 at 10 Κ and 0.0819 m 0 at 42 K. These values were
obtained at weak applied electric fields. The effective mass apparently shifted to
a higher value with an increase of the electric field. The authors ascribed this
phenomenon to the nonparabolicity of the conduction band. An interesting
observation was that the resonance line narrowed continuously as the temper­
ature was lowered for the same sample as cited above. The ω 0τ value was 9.0 at
42 Κ and 27.5 at 10 K. The electron mobility obtained from D C measurements,
on the other hand, showed a maximum at 77 Κ and decreased by a factor of
three, when the tempeature was lowered to 10 K. Such a conflict between D C
and cyclotron mobilities, however, is always observed as one deals with ionized
impurity scattering (Mears and Stradling 1969, Apel and Poehler 1970, Otsuka
et al. 1973a, Matsuda and Otsuka 1979a). Without doubt the observed linewidth
arises from ionized impurity scattering.
A similar and complementary experiment was carried out for n-InP at shorter
FIR wavelengths (Ohyama et al. 1983). A different type of modulation method employed by Kobayashi and Otsuka (1974) - was employed to see an exchange
of electrons between donors and the conduction band. Application of electric
fields, 0 to 200 V c m - 1, resulted in a decrease of the donor electron con-
Cyclotron resonance
49
centration and an increase of the conduction electron concentration. The donor
electron concentration monitored by the l s - » 2 p + Zeeman transition and the
conduction electron density by cyclotron resonance. Typical traces of difference
signals at 4.2 Κ are presented in fig. 25.
The cyclotron mass of electrons determined at 119, 172 and 220 μπι agrees
with each other within the experimental error and yields (0.0817 ± 0.0004) m 0.
This is compatible with m = 0.0815 m0 obtained by Chamberlain et al. at 10 Κ
and at a longer wavelength.
In the experiment by Ohyama et al. cyclotron resonance by photoexcited
electrons has also been explored. Typical traces obtained at 4.2 Κ are shown in
ρ-'
Δη
/
<
c
f
< 5
a
r
·-
•V
-
ι
ι
o-ΔΝ0
1
ι
ι
ι
1
i- ,
I
180 V/cm
<
Ζ
Ο
Ι­
Ο.
cr
ο
CO
<
2
A
6
MAGNETIC FIELD ( Τ )
8
10
Fig. 25. Pulsed electric field modulation signals of the electron cyclotron resonance (CR) and donor
Zeeman transition ( l s - * 2 p + )1 for an η-type InP crystal. As the electric field is intensified, the
density of neutral donors decreases so that the downward difference signal of the Zeeman line
deepens. The cyclotron resonance signal is enhanced, on the other hand, since the density of
conduction electrons is increased because of the impact ionization of neutral donors. The inset
shows the relative change in far-infrared absorption as a function of applied electric field. For
comparison, a cyclotron resonance trace for electrons in a pure η-type GaAs crystal is shown at the
bottom (from Ohyama et al. 1983).
50
Ε. Otsuka
fig. 26 for the wavelength of 220 μπι, and at various delay times after the end of
the photoexcitation pulse made by xenon flash lamp. Without photoexcitation
no cyclotron absorption is observed, since very few electrons are present in the
conduction band at 4.2 K. One may note that no Zeeman absorption is
observable in fig. 26, but a kind of trough towards zero field. This is because of
the crossover between cyclotron and Zeeman energies that occurs near the
wavelength of 220 μπι. A corresponding disappearance of the Zeeman transition
in GaAs occurs, for example, for Λ = 433 and 513 μπι, but not for 220 μπι.
Analyzing their Zeeman transition data in InP according to the variational
calculation by Larsen (1968), Ohyama et al. derive a donor binding energy of
7.6 meV.
The sequence of resonance traces presented in fig. 26 is a demonstration of
multichannel time resolution (Nakata and Otsuka 1982). As many as sixteen
traces with different delay times can be obtained in a single scan of the magnetic
field. The delay time dependence of the cyclotron resonance signal intensity
yields the lifetime of photoexcited electrons in the conduction band. From fig. 26,
MAGNETIC
FIELD ( Τ )
Fig. 26. Time-resolved cyclotron resonance traces of photoexcited electrons in the same η-type InP
sample as in fig. 25. The Zeeman transition is not visible at the employed far-infrared wavelength of
220 μπι but a trough near zero field (from Ohyama et al. 1983).
Cyclotron resonance
51
one obtains an electron lifetime as long as 5.3 μ8. This appears a little too long
in view of the theory of electron capture by charged impurities or holes.
Nevertheless, lifetimes of conduction electrons of similar length are also
observed in InSb and GaAs. It is understood that the joint action of the slow
donor-to-acceptor recombination and neutralization as well as screening of
ionized impurities by photoexcited carriers is again playing an essential role in
prolonging the electron life in the conduction band.
The cyclotron resonance linewidth changes with delay time, reflecting the
existence of carrier-carrier interaction. The inverse relaxation time obtained in
the delay time independent region is 4.1 χ 1 0 11 s " 1. This is considered primarily
to arise from neutral impurity scattering. The donor concentration derived by
the Ohyama method (Ohyama 1982) is 3 χ 1 0 15 c m " 3. If this value of ND is
introduced in the classical Erginsoy formula, with the use of a D = 79 A, one
obtains l/τ (electron-neutral-donors) = 6.7 χ 1 0 11 s " 1. This is larger than the
experimental observation. As pointed out in the treatment of GaAs, the
Erginsoy relation is not valid in the quantum limit, or in the presence of a strong
magnetic field. The discrepancy mentioned here, however, seems rather small in
view of the quantum limit treatment. There remains a possibility that the actual
donor concentration, probably including donors with deep levels, is larger than
3 χ 1 0 15 c m " 3 , as derived by the standard Ohyama method that accounts only
for the shallow donors.
5.3. Cyclotron resonance in chalcogenide
materials
5.3.1. Cyclotron resonance in ZnSe and ZnTe
Zince selenide is a material with potential for a blue light emitting diode.
However, fabrication of a device has not been achieved up to now, because of the
difficulty in growing doped p-type material. The presence of native defects
always makes as grown crystals n-type.
A surprisingly sharp cyclotron resonance signal as shown in fig. 27 for ZnSe
was first obtained by Ohyama et al. (1984) with the help of a far-infrared laser.
The obtained effective electron mass was 0.145 m 0. It is in good agreement with
the value 0.147 m0 that Holscher et al. (1985) derived in their two-photon
magnetoabsorption measurements. The mass value cited is the polaron mass. If
one takes a polaron coupling constant of α = 0.432 (Rode 1970), a bare band
mass of m* = 0.135 m 0 is obtained. This is the zeroth-order polaron correction.
By introducing a self-consistent correction in the light of the new experimental
data, one arrives at a modified value of the coupling constant, namely, α = 0.39.
The high ω0τ value enables one to observe a distinct resonance peak even with
millimeter waves (35 GHz). In fact, detailed studies of the temperature de­
pendence of the transport properties have been carried out at 35 GHz (Ohyama
et al. 1985). The resonance linewidth can be fitted crudely in the 1.5-50 Κ
temperature range by scattering contributions from neutral impurities (donors),
Ε. Otsuka
52
ZnSe
f = 1 3 6 4 GHz
( λ = 220pm )
Τ= 4.2 Κ
Β // <11 0>
c
3
m*=0.U5m„
<
Ζ
Ο
»—·
Ι­
Ο.
cr
ο
ω
ω
<
149γτϊ λ
0
1
2
3
4
5
6
7
MAGNETIC FIELD ( Τ )
8
Fig. 27. Sharp cyclotron transition of photoexcited carriers in a high-purity ZnSe crystal (from
Ohyama et al. 1984).
acoustical piezoelectric phonons, acoustical deformation potential phonons and
polar optical phonons. These contributions can be written as
1
1
20haONDme9
e2(mckBT)l/2K2
16(2π) 1 / &
2 2
1
TDP
(59)
'
3m3J2C2(kBT)3/2
(Sny2h*pcs
'
(60)
(61)
and
2αω,t o
LPO
(62)
respectively, where Κ is the piezoelectric coupling constant, κ the static dielectric
constant, Ct the deformation potential constant, c s the longitudinal sound
Cyclotron resonance
53
velocity, ρ the density of crystal and a > LO the longitudinal optical frequency.
Putting α = 0.432, fa»LO/fcB = 3 6 0 K (Aven et al. 1961), C ! = 4 e V , pc s2 = l.l
χ 1 0 1 2d y n c m ~ 2 , ,c = 8.1 (Aven and Segall 1963), X = 0.0437 (Mahan 1972),
a D = 3 3 A and ND = 4.0 χ 1 0 1 4 c m - 3, one can see in fig. 28 an essential
agreement between the results of eqs (59)-(62) and the experimental observa­
tion of resonance linewidth.
A small nonparabolicity in the conduction band arises due to the polaron
effect. The Landau levels are no longer equidistant in energy. In the 35 GHz
measurement, the observed electron resonance signal is a result of closely
overlapping cyclotron transitions. Elevation of temperature, either lattice or
electronic, results in a shift of the apparent resonance peak position towards
higher magnetic field. This is due to an overall electron population transfer to
higher Landau levels. In the far-infrared measurement, a second peak with an
apparent cyclotron mass of 0.149m 0 is observed, with considerably reduced
intensity, adjacent to the main peak that gives a mass of 0.145 m 0 (fig. 27).
Qualitatively, the emergence of adjacent peaks can be explained by Bajaj's
calculation (Bajaj 1970), which actually applied to CdTe, predicts m* = 0.148 m 0
for the second peak and 0.153 m 0 for the third peak in ZnSe. Splitting of the
conduction electron cyclotron resonance due to the polaron effect has also been
observed and discussed for GaAs (Lindemann et al. 1983, Ohyama 1983).
2
5
10
50 100
TEMPERATURE ( Κ )
Fig. 28. Temperature dependence of the inverse relaxation time of conduction electrons in ZnSe
obtained at 35 GHz. Indicating numericals show the contributions from (1) acoustical piezoelectric
phonon scatterings, (2) acoustical deformation potential phonon scatterings, (3) polar optical
phonon scattering and (4) neutral donor scatterings. The dashed line shows the combination of the
above four scattering contributions (Ohyama et al. 1988).
Ε. Otsuka
54
An unexpected feature of cyclotron resonance in ZnSe is the two-dimensional
character of the resonance of the electron system accumulated at grain
boundaries of twinned crystals (Ohyama et al. 1986). It is almost inevitable to
have such grain boundaries inside a sizable ingot of a real crystal. The better the
crystal, the more the tendency of electrons to be accumulated at the boundary is
developed. As one rotates the magnet, the resonance field varies with the angle
of rotation. The lowest field is equal to the resonance field of the bulk electrons
while the highest one diverges when a tilt angle of 90° is approached. This
feature is shown in fig. 29. It should be noted that the isotropic resonance of bulk
electrons coexists with the resonance of the 2 D (two-dimensional) electrons
when the magnetic field is oriented perpendicular to the [111] direction. The 2 D
signal is strongly enhanced as one treats the grown pure ZnSe crystal in the melt
of Zn metal. The 2 D electron resonance can be observed at liquid helium
temperatures, even without intrinsic photoexcitation. The 3D electron res­
onance can be observed only after elevating the temperature, say to 18 K. The
3D electrons are those that have spilled out of the 2 D channel at the twin crystal
boundaries owing to thermal excitation. One should be reminded here that the
existence of a 2 D hole system adjacent to the grain boundary in η-type Ge
bicrystals has been studied recently (Uchida and Landwehr 1983, Landwehr and
Uchida 1986). At the time of writing, unfortunately, the work on grain
ZnSe
35 GHz
20K _
Bin (110)
2
ο
Ε
<
cr
Q
UJ
if)
if)
<
Ζ
LU
ο
cr
ιο
<
_l
Ο
>-
2
υ
>oo—
"Ό-ΟΟΟ—OOCO<
9 0 β 60°
<ΪΪ2>
30°
0° -30° -60°
<111>
Fig. 29. Angular dependence of two resonances in ZnSe. The isotropic resonance is from threedimensional (3D) bulk electrons, while the highly anisotropic resonance is from the two-dimensional
(2D) electron system accommodated within twin crystal boundaries (from Ohyama et al. 1986).
Cyclotron resonance
55
boundaries in ZnSe had to be terminated because the technical reproducibility
of samples was not sufficiently under control.
Another novelity in ZnSe is the appearance of a hole resonance (Ohyama et
al. 1987a). Only in good-quality crystals does an almost isotropic signal show
up, corresponding to m h = (1.04 ± 0.04) m 0. The cyclotron mobility derived from
the observed linewidth was 1.7 χ 1 0 4 c m 2 V " 1 s ~ 1 at 4.2 K. As a consequence of
relatively high effective mass, the observation is possible only at microwave
frequencies. Obviously, the observed signal is caused by heavy holes. N o
anisotropy is observed. The reason for the absence of anisotropy is not clear. A
light-hole signal has not yet been detected.
Zinc telluride is a compound noted for its native p-type character. This is in
contrast with the native η-type ZnSe. Cyclotron resonance of holes thermally
excited from acceptors was observed by Stradling (1968). To observe electron
cyclotron resonance, Clerjaud et al. (1979) used intrinsic photoexcitation. They
extended the electron paramagnetic resonance technique (9.2 and 35 GHz) to
cyclotron resonance and took the differential form of the absorption signal. The
obtained mass was m e = (0.122 ± 0.002) m 0. This, of course, corresponds to the
polaron mass. The peak-to-peak distance in the magnetic field scale was taken
as the linewidth and its temperature variation was investigated. The 9.2 GHz
measurement yielded ω0τ = 3.5 at 3.5 K, corresponding to a cyclotron mobility
of 8.5 χ 10 5 c m 2 V - 1 s " 1. The temperature dependence of τ, and hence the
cylotron mobility, is given in fig. 30. Above 10 K, the variation of τ obeys a
TCK)
Fig. 30. Scattering relaxation time of electrons in a p-type ZnTe crystal, obtained from a 9.2 G H z
cyclotron resonance linewidth measurement, is plotted against temperature. The corresponding
cyclotron mobility is also scaled on the right-hand side (from Clerjaud et al. 1979).
56
Ε. Otsuka
Τ " 3 /2 law beautifully, reflecting the dominance of the acoustic deformation
potential phonon scattering. Deviation of τ from the T " 3 /2 line shows up only
below 10 Κ corresponding to the presence of impurities with a concentration of
NA — ΝΌ~ 1 0 15 c m " 3. All the impurities are considered neutralized by intrinsic
carriers produced by a mercury lamp.
The value of the cyclotron mobility obtained at 3.5 Κ is somewhat higher than
that obtained in high-purity ZnSe at the same temperature; that is, 2.6
χ 10 5 c m 2 V ~ 1 s " 1. The estimated impurity concentrations are nearly the same
but for ZnSe one can estimate ΝΌ — NA~ 1 0 15 c m " 3. The impurity scattering
contribution is thus mainly due to donors in ZnSe, and predominantly due to
acceptors in ZnTe. From what is known about group IV elements in I I I - V
compounds, this difference suggests that the electron cyclotron resonance in
ZnTe should show up easier than that in ZnSe, since the electron-acceptor
scattering is less important than the electron-donor scattering.
Some basic differences can be seen between ZnSe and ZnTe in their cyclotron
resonance behavior, although the apparent chemical as well as the physical
nature of these two compounds are similar. In ZnTe, contributions from polar
optical phonon scattering, as well as acoustic piezoelectric phonon scattering,
are negligible below 60 Κ in comparison with acoustic deformation potential
phonon scattering. In ZnSe, however, only the last mechanism can almost be
neglected in a wide temperature range from 1.5 to 50 K. A substantial
anisotropy of the hole resonance has been reported in ZnTe while an apparent
isotropy in ZnSe has already been mentioned above. The most recent work by
Ohyama et al. (1988) on a ultra-high quality ZnSe crystal, in which the hole
resonance has been detected, however, indicates a certain contribution from
acoustic deformation potential phonon scattering, a small contribution from
impurity scattering and a good agreement with polar optical phonon scattering,
as seen in fig. 28.
53.2. Cyclotron resonance in CdTe, CdS and CdSe
Cadmium chalcogenides have a long history of cyclotron resonance investi­
gations. Cadmium sulphide and selenide crystallize in the wurzite lattice, and
CdTe in the zinc blende. Generally speaking, the zinc blende crystal is the more
favourable for cyclotron resonance work. Indeed more data are available for
CdTe than for the other two compounds. These compounds are noted for the
polaron effects which show up rather clearly in cyclotron resonance.
Cyclotron resonance observation in CdTe goes back to the experiment by
Kanazawa and Brown (1964) performed at a wavelength of 4 mm at 4.2 K. They
obtained a cyclotron resonance signal under intrinsic photoexcitation. An
isotropic effective electron mass of 0.096 m 0, was reported. These authors were
careful enough to examine the photosensitivity against wavelength spectrum of
the resonance. The sensitivity was found to peak strongly in the vicinity of the
direct exciton transition. Moreover, a qualitative comparison was made between
Cyclotron resonance
57
drift mobility
(crr^/tolt sec)
20
AO
temperature
60
( κ)
Fig. 31. Discrepancy between the electron mobility as observed for CdTe between Hall data (broken
curve) and cyclotron resonance linewidth data (open circles are from standard cyclotron resonance;
open triangles from cross-modulation at 2 mm and full triangles from cross-modulation at 1 mm)
(from Mears and Stradling 1969).
the cyclotron mobility obtained in the dark with the Hall mobility extrapolated
to 4.2 K. The cyclotron mobility, 3 χ 1 0 4 c m 2 V " 1 s - 1, was somewhat larger
than the extrapolated Hall mobility, of about 1 χ 1 0 4 c m 2 V " 1 s - 1, but the
authors took the order of magnitude agreement as evidence that the resonance
signal was due to electrons.
Mears and Stradling (1969) performed another experiment on CdTe at the
shorter wavelengths of 1 and 2 mm. This was done in order to confirm
consistency with a separate magnetophonon measurement (Mears et al. 1968).
Carrier excitation was achieved by thermal excitation between 17 and 63 K.
Below 17 K, they employed a cross-modulation technique (Kaplan 1965). The
electron effective mass was confirmed to be 0.0963 m 0 at 14 K.
Most instructive in the experimental results of Mears and Stradling is the
contrast with the Hall effect data. The relevant feature is shown in fig. 31. The
cyclotron mobility steadily rises with decreasing temperature, nearly as T ~ 3 / ,2
while the Hall mobility has a peak at 28 K, below which it decreases. The highest
cyclotron mobility of 2 χ 10 5 c m 2 V ~ 1 s ~ 1 was measured at 11 K. The observed
discrepancy between the mobilities derived from the Hall and cyclotron data is
attributed to the long-range nature of ionized impurity scattering, which also
58
Ε. Otsuka
shows up in InSb (Apel and Poehler 1970, Matsuda and Otsuka 1979b), InP
(Chamberlain et al. 1971) and Ge (Otsuka et al. 1973a).
Cyclotron resonance measurements of CdTe were extended to far-infrared
frequencies by the MIT group (Waldman et al. 1969, Litton et al. 1976).
Waldman et al. observed a magnetic field dependence of the electron cyclotron
mass, which they attributed to the polaron effect. Starting from Frohlich's
Hamiltonian, these authors performed a variational calculation to find the
transition energy between the lowest two Landau levels which was in agreement
with the experimental observations. The coupling constant was chosen as 0.3 or
0.4. This work was the first experimental test of the large-polaron theory.
Somewhat later, Bajaj (1970) did a simpler calculation in terms of Onsager's
theory to explain the same experimental data. He took α = 0.4. Litton et al.
subsequently made a crucial test of the polaron theory, using seven far-infrared
wavelengths ranging from 78.4 to 337 μπι. They deduced a coupling constant of
0.40 + 0.03.
So far no valence band parameters for CdTe have been derived by cyclotron
resonance. They are only available from the excitation spectrum of acceptors
(Svob et al. 1978).
Cadmium sulphide and selenide are also substances of interest from the
viewpoint of polaron studies. The polaron coupling constant α for CdS is 0.6 and
that for CdSe, 0.45. These are not as large as those for alkali halides, but large
enough to cause the so-called polaron pinning. The materials are also noted to
be piezoelectric. The onset of the acoustopiezoelectric polaron coupling is
expected, and was actually observed for CdS (Nagasaka 1977).
The first CdS cyclotron resonance observations were reported by Sawamoto
(1963) and by Baer and Dexter (1964) using millimeter waves. These authors'
experiences were somewhat distressing because of the poor reproducibility of a
resonance signal. They found, nevertheless, very similar values for the electron
effective mass. Sawamoto gave 0.17m 0, while Baer and Dexter found 0.171 m 0
for β He-axis and 0.162m 0 for £_Lc-axis. Experiments were later repeated in the
far-infrared range independently by Button et al. (1970) and by Narita et al.
(1970). Both of these groups observed, in addition to the qualitative confirma­
tion of the cyclotron mass reported earlier, splitting of the resonance line when
the temperature was varied. The splitting was first interpreted to be due to the
piezoelectric polaron effect, but was later reinterpreted as an interference effect
(Cronburg and Lax 1971). This incorrect interpretation was indeed an unfor­
tunate event. But almost the same error (Otsuka et al. 1973b, 1974) and its
correction (Otsuka et al. 1978) were repeated somewhat later for the 'excitonic
polaron' in Ge. In CdS the electron density was changed by a variation of
temperature, while in Ge it was varied by optical excitation. Otherwise the
interference mechanism was the same. These misinterpretations emphasize the
importance of employing wedge-shaped samples in carrying out cyclotron
resonance experiments with laser radiation to avoid such interference effects.
Cyclotron resonance
59
A convincing answer to the piezoelectric polaron problem in CdS was given
by Nagasaka only in 1977 (Nagasaka 1977). The dependence of the cyclotron
resonance frequency on the magnetic field is shown in fig. 32. Three straight lines
give the tentative bare band masses, 0.174m 0, 0.182m 0 and 0.188 m 0. The
observed shift in photon energy from the bare mass line was essentially
explained by Miyake's theory (Miyake 1968), which predicts — Am*/wi*oc
Τ2
/3Β-1
Cadmium selenide was explored by laser cyclotron resonance by Miura et al.
(1979b), as well as CdS, in megagauss experiments at temperatures between 130
and 300 K. The laser wavelengths employed were 28 and 16.9 μιη. The best fit of
the bare-mass values were found to be 0.165 m 0 for CdS and 0.116 m 0 for CdSe.
Miura et al. make a remark that the band mass obtained at high fields is larger
than that obtained at low fields (corresponding to a wavelength of 119 μιη) both
for CdS and CdSe. They ascribe the difference to the piezoelectric polaron effect.
Again Miyake's theory is consulted. The authors state that Miyake's prediction
explains the field dependence but not the temperature effect sufficiently well. It is
true that the double-polaron contribution from LO phonons and acoustic
piezoelectric phonons causes complications.
90
α
n-CdS
Η//β
80
BE
^ 7 0
_
δ
50
(Τ
UJ
Ο 0 0
°26ΟΟ
Ζ 40
UJ
Ο 30
Ο
Χ
°" 2 0
ο 19 Κ
α
38 Κ
10
/
J
I
20
40
1
60
80
MAGNETIC
ί ­
I
.
100
120
140
FIELD
(|<0Θ)
1
160
180
Fig. 32. Magnetic field dependence of the photon energy corresponding to the electron cyclotron
resonance in CdS at 19 and 38 K. The deviation of the experimental data from the straight lines
shows the piezoelectric polaron effect. The three straight lines a, b and c correspond to effective
masses of 0 . 1 7 4 m 0, 0 . 1 8 2 m 0 and 0 . 1 8 8 m 0, respectively (from Nagasaka 1977).
Ε. Otsuka
60
6. Cyclotron
resonance
in the most challenging
materials
6.1. Ionic crystals: alkali, thallium and silver halides; Cu20
and
Hgl2
Ambitious physicists dared to try cyclotron resonance experiments in ionic
crystals. The first trial at 70 GHz was made for AgBr by Ascarelli and Brown
(1962). It was combined with the systematic transport studies, as well as the
optical studies, on silver halides made by Brown's group. Carefully extended
cyclotron resonance measurements on silver halides were later made at 35 GHz
by Tamura and Masumi (1971, 1973) as well as by Hirano and Masumi (1987).
These authors gave m e = 0.41 m 0 for AgCl and 0.29 m 0 for AgBr. In AgBr, even
a twofold hole cyclotron resonance was observed to give m h t= 1.71 m 0 and
m hl = 0.79 m 0.
Apart from a simple carrier mass determination, cyclotron resonance in silver
halides, especially that in AgBr, offers a striking example of polaron dynamics.
Komiyama and Masumi (1978) have carried out a unique high-power cyclotron
resonance experiment for AgBr, using magnetrons in the frequency range
3 5 - 5 0 GHz, to find tremendous peak shifts and line broadening of the electron
resonance. The peak shift amounts to 100% towards higher magnetic fields and
the linewidth is broadened by a factor of more than ten. The power dependence
of the peak shift and the linewidth are given in fig. 33. These features are
1
]
1
τ
1
AgBr 35.0 GHz
ο ZR-3 4.2K
A C-tS7 4.2K
•
ZR-3
17K
Timur* t> Mtsumi
at 1.7K And 346Hi
I
(A)
1
/
!
/
SjJf (B)
A
ΐ
0.1
-80
-70
-60
-50
-40
-30 - 2 0
-10
1
-80
1
1
-A2db
(- SOV/cm)
1
1
1
1
1
70 -60 -50 -40 -30 -20 -10
INPUT MICROWAVE POWER (db)
0
INPUT MICROWAVE POWER (db)
Fig. 33. Microwave power dependence of (a) peak shift and (b) linewidth of the 35 G H z electron
cyclotron resonance in AgBr. The measurement is made at two lattice temperatures and for two
samples. The earlier measurement by Tamura and Masumi is also shown by a broken curve. The full
straight line in (b) is a theoretical prediction considering LO phonon emissions. The regional
partition [ A ] and [ B ] indicates the low-power range, where acoustic phonon emissions are
dominant and the high-power range, where LP phonon emissions are dominant (from Komiyama
and Masumi 1978).
Cyclotron resonance
61
suggestive of not only the polaron nonparabolicity at the bottom of the
conduction band but of the possible existence of the so-called streaming motion
of hot carriers after emitting LO phonons (Conwell 1967, Kurosawa and Maeda
1971). This idea led to the later discovery of population inversion of hot
electrons in AgBr and AgCl (Komiyama et al. 1979, Komiyama 1982), and
eventually to the population inversion and tunable far-infrared laser oscillation,
that was first achieved in p-type Ge (Andronov et al. 1984, Komiyama et al.
1985, Komiyama 1986).
The peak shift observed by Komiyama and Masumi is ascribable to the
sequential excitation of electrons on the Landau ladder levels, reflecting the high
electron temperature and nonparabolicity of the conduction band due to
polaron effects. Such a peak shift can also be realized by a rise of the lattice
temperature (Tamura 1972, Hirano and Masumi 1987). The same, or at least
similar, effects have also been found in CdTe (Waldman et al. 1969), in GaAs
(Lindemann et al. 1983, Ohyama 1983) and in ZnSe (Ohyama et al. 1987b).
Another highlight of the polaron dynamics in AgBr was found by Tsukioka
and Masumi (1974, 1980), who observed time-resolved electron cyclotron
resonance in AgBr after a strong intrinsic photopulse excitation by either a N 2
laser, dye laser or xenon flash lamp. From the observed plasma shift and the
linewidth they conclude that an electron density as high as 2 χ 1 0 13 c m - 3 is
created, interacting with a dense exciton gas, probably as dense as 1 0 15 c m - 3.
The polaron-exciton interaction can be treated like electron-neutral-donor
scattering and shows, in time resolution, a close resemblance to the
electron-exciton interaction observed in Ge (Ohyama et al. 1971). The timeresolved measurement shows that a time constant of 10 ns is required for the line
to narrow, making a good correspondence with an independent time-resolved
luminescence experiment (Baba and Masumi 1987).
In alkali halides, it is much more difficult to create long-lived free carriers than
in silver halides, so that a standard cyclotron resonance measurement is
practically impossible. An alternative method, called. cross-modulation, was
used by Mikkor et al. (1965, 1967) on KBr containing F centers. Crossmodulation is essentially a combination of pulsed photoconductivity measure­
ment and cyclotron resonance. As one sweeps the magnetic field, the photo­
conductivity of electrons released from F centers shows a peak or a dip,
depending on the relative geometry of the applied electric and magnetic fields,
on account of the onset of cylotron resonance. Thus one can find the magnetic
field for resonance, and hence the carrier effective mass. An extended application
of such a technique is available even for the prototype semiconductor Ge
(Gershenzon et al. 1968a, b). With the help of this cross-modulation, Hodby et
al. (1967, 1968, 1974, 1976), as well as Hodby himself (Hodby 1969), later
obtained carrier effective masses for other alkali halides: KC1, KBr, KI, KC1 and
Rbl; thallium halides, T1C1 and TIBr; silver halides, AgCl and AgBr, and two
more substances, C u z O and H g l 2 (Bloch et al. 1978). As an example for a very
62
Ε. Otsuka
difficult experiment, a resonance trace obtained for KBr is shown in fig. 34. The
values obtained for the carrier effective masses in the above materials are given
in table 4. One should note that all the effective masses obtained by cyclotron
resonance are polaron masses. In order to derive the band mass, one has to
make a correction taking the polaron coupling constant α into account. In
I-VII compounds, values of α are much larger than those in I I I - V or I I - V I
compounds. Indeed the largest is 4.09 for RbCl and even the smallest is 1.53 for
AgBr. We shall not enter into proper discussions of the polaron problem in this
chapter but merely present the results of cyclotron resonance. Appropriate
review articles dealing with polarons are available elsewhere (Hodby 1972,
Masumi 1981, 1984) and also in this volume by Larsen, which is particularly
connected with cyclotron resonance. We shall only state here that a further
crucial test of the large-polaron theory has been conducted by Hodby et al.
(1987) for AgBr and AgCl, using frequencies of 137 and 525 GHz.
6.2. Anthracene and organic
materials
Of all the nonmetallic materials investigated by cylotron resonance, perhaps the
most exotic is anthracene - an organic molecular crystal. Burland (1974) first
reported results obtained with low-frequency microwaves of 3.50 GHz. In
general, effective masses of carriers are heavier in organic materials than in
inorganic ones. So a compromise in frequency is inevitable in view of the limited
Fig. 34. Cyclotron resonance signal at 4.2 Κ from photoelectrons in a KBr crystal containing F
centers. The trace is obtained in cross-modulation at 140 G H z (from Hodby et al. 1967).
63
Cyclotron resonance
Table 4
Carrier effective (polaron) masses in ionic crystals obtained from cyclotron resonance (by courtesy
of J.W. Hodby)
Material
KC1
KBr
KI
RbCl
Rbl
AgCl
AgBr
Carrier
e
e
e
e
e
e
e
h (111 ellipsoids)
T1C1
e
h (100 ellipsoid)
TIBr
e
h (100 ellipsoid)
C u 20
e
h
h
Hgl2
e (ellipsoid)
h (ellipsoid)
mp
o I na r(in
o
m 0)
0.922 ± 0 . 0 4
0.700 ± 0 . 0 3
0.536 ± 0 . 0 3
1.03 ± 0 . 1 0
0.72 ± 0 . 0 7
0.411 ± 0 . 0 2
0.2897 ± 0 . 0 0 4
{ 1.71 ± 0 . 1 5 ( m , )
10.79 ± 0 . 0 5 (m t)
0.551 ± 0 . 0 3
{0.58 ± 0 . 0 3 (m,)
10.98 ± 0 . 0 4 ( m t)
0.525 ± 0 . 0 3
J0.55 ± 0 . 0 3 (w,)
{0.74 ± 0 . 0 3 (m t)
0.99 ± 0 . 0 3
0.58 ± 0 . 0 3
0.69 ± 0 . 0 4
{0.31 ± 0 . 0 3 (HI ||c)
10.37 ± 0 . 0 2 ^ * 0 )
{2.06 ± 0 . 0 5 (m|| c)
11.03 ± 0 . 1 0 (m±c)
[1]
[1]
[1]
[1]
[1]
[2]
[3]
[4]
[4]
[5]
[5]
[5]
[5]
[5]
[5]
[6]
[6]
[6]
[7]
[7]
[7]
[7]
[ 1 ] Hodby (1971), [ 2 ] Hodby et al. (1987), [ 3 ] Hodby et al. (1974), [ 4 ] Tamura and Masumi
(1973), [ 5 ] Hodby (1972), [ 6 ] Hodby et al. (1976), [ 7 ] Bloch et al. (1978).
magnetic fields. Carriers are expected to have an anisotropic mass. In his
experiment, Burland set the magnetic field perpendicular to the so-called a-b
plane, and created holes by photoinjection from Ag paste with the help of a
mercury lamp. He observed only one effective mass of 11 m 0. From the linewidth
observed, he derived a carrier relaxation time of 4 χ 1 0 " 1 0 s at 2 K. He found
that the observability of signals depended critically on the sample quality. The
resonance signal entirely disappeared above 30 K.
The experimental details of Burland and Konzelmann (1977) reported later,
indicated the difficulty in dealing with such an organic material. The technical
obstacles are substantial. The first attempt to perform cyclotron resonance
experiments on anthracene should be highly appreciated. In fact magnetooptical resonance in general should not be confined to inorganic
semiconductors.
In fig. 35, the first detection of hole cyclotron resonance in anthracene
obtained at 2 Κ is reproduced.
64
Ε. Otsuka
Η (kilo-oersteds)
Fig. 35. (Top) Cyclotron resonance traces for holes in two different anthracene crystals. The upper
curve is obtained at 2.30 G H z while the lower one is at 3.50 GHz. Both measurements have been
carried out at 2 Κ with the magnetic field perpendicular to the a-b plane. Resonance maxima
indicated by dotted lines correspond to a cyclotron mass of 11 m 0. (Bottom) Theoretical fitting lines
for the above. The full line is for m h = 11 m 0, τ = 7 χ 1 0 " 11 s and ω ε/ 2 π = 2.30 GHz. The broken
curve is for m h = 11 m 0, τ = 4 χ 1 0 " 10 s and ω0/2π = 3.50 GHz. The vertical scale is the ratio of the
microwave to the D C conductivity (from Burland 1974).
6.3. Materials with peculiar band structures - HgTe, Te and GaP
There are semiconductors for which it is difficult to obtain cyclotron resonance
of free carriers. The difficulty can arise from the crystalline or from the band
structure of the material or from both. We shall summarize here some data for
such crystals, for which the difficulties have been overcome.
We shall mention some results on HgTe, Te and GaP. These are materials
with peculiar band structures. Mercury telluride is known as a zero-gap
semiconductor, with a crossover of the Γ 6 and Γ 8 bands. In the presence of a
strong magnetic field, a complicated admixture of states occurs in the degenerate
Γ 8 energy band. As a result, systematic energy shifts, upwards for a certain
group of Landau levels and downwards for another group, occur with increasing
Cyclotron resonance
65
intensity of the applied magnetic field. An energy gap can then arise, and
similarity with an ordinary finite-gap semiconductor is expected. In magnetooptical absorption measurements various transitions, both intraband and
interband, and also impurity associated ones can be recorded. Uchida and
Tanaka (1976) have investigated the magneto-optical transition in HgTe by farinfrared radiation (3 meV < hco < 14 meV) at liquid helium temperatures.
Various peaks, with large linewidths and which frequently overlap with each
other, were observed in the magnetic field range 0 - 6 T. They included intraband
transitions which may be classified as well as cyclotron resonance. Noting the
correspondence of the Γ 8 band of HgTe with the Γ 8 band of Ge, Uchida and
Tanaka tried to make use of the Luttinger effective-mass Hamiltonian and
derived a new set of Luttinger parameters which are slightly different from those
derived earlier (Groves et al. 1967, Guldner et al. 1973). The authors stated that
a reinterpretation of transition lines explained the difference. They also derived
the effective masses for three directions of magnetic field: m e = 0.031, 0.032 and
0.032, in units of m 0, for <100>, <110> and <111>, respectively, while m h = 0.37,
0.45 and 0.48 for the same field directions. Several transitions associated with
impurities and transitions considered theoretically to be forbidden are also
observed. For details and, in particular, for the different Luttinger parameters,
the reader is advised to consult the original papers.
Gallium phosphide has for long remained a controversial material despite its
common use in light emitting devices. The ambiguity has been caused partly by
the low carrier mobility in this material which hampered active studies,
especially ones involving cyclotron resonance. A more essential factor that
makes cyclotron resonance observation quite difficult, however, is the camel's
back structure in the conduction band (Lawaetz 1975).
Electron cyclotron resonance in G a P was first observed by Leotin et al.
(1975), at a wavelength of 337 μπι of an H C N laser, employing pulsed magnetic
fields. Since the minimum of the conduction band in this material occurs close to
the X point, the energy contours were expected to be ellipsoidal, like those of Si.
These authors obtained m t = 0.25m 0 and K = m1/mt = 20. Somewhat later,
Suzuki and Miura (1976) made another experiment at 119 μπι. They obtained a
similar value of m t, 0.254 m 0, but the anisotropy factor Κ was 7.9. Such a large
discrepancy in Κ between the two experiments could not be reconciled. The Κ
values, however, were not derived from the observation of a second cyclotron
resonance peak that should appear in the geometry of Β || <100>, corresponding
to ( m t m , ) 1 / .2 The long awaited second peak finally appeared for 337 μπι
radiation only at 37 Τ (Kido et al. 1981), to make Κ even larger than 21.3, which
was later confirmed to be 28. Further experiments, extended to megagauss fields
(Miura et al. 1983a), confirmed the appearance of the second peak also for the
wavelength of 119 μπι, but with a K-value of 19, while the mt value remained
0.25 m 0. One thus finds that the K-value is strongly dependent on wavelength, or
magnetic field. This puzzling behavior can be understood only by postulating a
66
Ε. Otsuka
camel's back structure for the conduction band. In fact, the twofold degeneracy
at the X point is lifted as a consequence of the lack of inversion symmetry in the
case of III-V compound semiconductors. Landau levels for such a conduction
band have indeed been calculated by Miura et al. (1983b). The levels N * , having
different spins, are degenerate at low magnetic fields but split at high fields.
Selection rules for possible transitions will be changed to yield such a transition
as 0"
1 + . Miura et al. assign this transition to the second peak. In this way
they explain the puzzling field dependence of the anisotropy factor K.
The band parameters of the valence band have been theoretically estimated
for various compound semiconductors, both for I I I - V and I I - V I compounds
(Cardona 1963). To the best of the author's knowledge, no cyclotron resonance
work has so far been reported for holes in GaP.
A similar initially somewhat confusing story involving a camel's back
structure exists in Te. The relevant energy band in this case is the valence band.
Since an undoped crystal of Te is always p-type at low temperatures,
experimental investigations have been focused on the valence band. The first
microwave frequency experiments (Mendum and Dexter 1964, Picard and
Carter 1966) yielded a single cyclotron resonance line for the magnetic field
oriented both parallel and perpendicular to the trigonal c-axis, indicating
ellipsoidal constant-energy surfaces. When the experiments were extended to the
submillimeter range, a multiline spectrum was observed for the configuration
B±C (Couder 1969, Button et al. 1969).
A camel back structure for the upper valence band was first proposed by
Betbeder-Matibet and Hulin (1969). Subsequently the problem was treated
theoretically more rigourously by Doi et al. (1970), and by Weiler (1970)
employing the k · ρ method.
A study of the temperature dependence of the position, shape and width of
the submillimeter cyclotron resonance absorption was performed by von
Ortenberg et al. (1972). The data were compared with the predictions of several
theoretical models. The best agreement is obtained with the valence band
parameters as proposed by Weiler (1970).
This high-feld behavior, like that of GaP, is related to a camel's back band
structure.
Electron cyclotron resonance in Te was first observed above 200 Κ (Button et
al. 1969), where an undoped crystal changes from p-type to η-type. Miura et al.
(1979a, b) carried out measurements in the megagauss range at three wave­
lengths, 10.6,16.9 and 28.0 μιη. Due to the destructive nature of the experiments,
the data were quite limited. Yet the authors derived m ± = 0.186m 0 and
my = 0.085 m 0 for 16.9 μπι and m ± = 0.167 m 0 and
= 0.079 ra0 for 28.0 μπι,
where the suffices indicate the magnetic field either perpendicular or parallel
to the c-axis. These values can be compared with the low-field magnetoabsorption data obtained by Shinno et al. (1973), who gave m 1 = 0.104m 0 and
m|| = 0.070m 0. In principle, the observation of the electron resonance should not
Cyclotron resonance
67
require such high temperatures as employed by Button et al. and by Miura et al.
However, attempts to observe electron resonance at liquid helium temperatures
in p-type material under photoexcitation have failed so far (Nisida and von
Ortenberg 1980). On the other hand, it has been possible to observe electron
cyclotron resonance at 4.2 Κ in p-type tellurium inversion layers (Silbermann
and von Ortenberg, unpublished) realized in a field effect configuration.
7. Germanium
and silicon
revisited
7.1. Earlier accurate measurements in the millimeter wave region
We have seen at an early stage of this chapter that cyclotron resonance in Ge or
Si can still have its significance from the viewpoint of transport studies. The
same idea has been taken by Gershenzon and his group (Gershenzon et al.
1969). The usefulness of cyclotron resonance can also be extended to kinetics
studies and is not limited, of course, to elemental semiconductors. Of all the
materials, however, Ge seems to be especially suited for precision experiments.
The reason will be self-evident. The physical parameters of this substance are
known very well and the material control has been achieved to the highest
degree. As a result, electron scattering cross-sections for a particular species of
impurities are for example, best known for Ge, and somewhat less well known
for Si and practically nothing is available for other materials. It is also possible
to grow a single crystal of Ge essentially free from impurities as far as cyclotron
resonance measurements are concerned. With these facts in mind, several
precision measurements have been carried out that are possible only for Ge.
Some of these have already been described but some topics have remained
untouched. One is the kinetics connected with the existence of electron-hole
drops (Hensel and Phillips 1972, 1974, Ohyama et al. 1974, Fujii et al. 1985).
Monitoring the electron cyclotron resonance signal intensity in time resolution
has offered a new and independent approach to the properties of electron-hole
drops, not easily accessible by luminescence studies. Meanwhile, the suppression
of electron-hole drop formation by means of uniaxial stress has enabled
Kawabata et al. (1977) to observe cyclotron resonance of charged excitonic
complexes, or so-called trions, in Si with a frequency of 49 GHz and at 1.7 K.
Another unique utilization of cyclotron resonance is found in impurityassisted intervalley electron scattering studies (Murase and Otsuka 1969).
Application of uniaxial stress along an appropriate crystallographic axis causes
up- and down-shifted valleys in the conduction band. Electrons transferred from
upper valleys to the lower ones due to impurity scattering are a kind of 'hot
electrons' and contribute to the linewidth of the down-shifted valley resonance.
Such a line broadening can be observable only under optimum values of the
scattering as well as the recombination times with simultaneous use of precision
Ε. Otsuka
68
superheterodyne equipment. The electron intervalley scattering rates associated
with Ga and In in Ge, and Ρ in Si, have been derived at 4.2 K, to be
approximately 3 to 5% of each corresponding intra valley impurity scattering
rate.
The third topic is associated with spin-dependent quantum transport
(Ohyama et al. 1970). This involves spin-polarized electron scattering by neutral
3
impurities in Ge and Si. Superheterodyne equipment for 71 GHz and a H
e
cryostat have been employed for the measurements. Actually cyclotron res­
onance had never been carried out below 1 Κ before. The lowest temperature
under a feeble intrinsic photoexcitation, was about 0.5 K. The electron-neutraldonor collision can be classified, theoretically, into singlet and triplet scattering
because of the presence of two electrons in the system. A very clear distinction
between these two types of collision has been extracted from a precision
linewidth measurement, making use of the emergence of electron resonances at
X109
S
-
Ge/sb
Ν = 5.9 X l 0 1 3c m " 3
α*= 47 A
—
-
o
\
_
3.0
-
—
Λ
_
:
-
Ο
ρ
x---=J(==-^
*<
V ί
—
_ /
Ο
~0
-
2.5
Calculation
/
/
1
/
/
/ /
/ /
1
0.5
1
1 1 1 11
1
TEMPERATURE
1
2
1
3
1
4
1
5
K
Fig. 36. Inverse relaxation time of electrons due to neutral Sb in Ge depends on the geometry
(magnetic field strength) at very low temperatures. Onset of the spin-polarized scattering effect is
illustrated for a sample with a Sb concentration of 5.9 χ 1 0 13 c m " 3 (Ohyama et al. 1970).
69
Cyclotron resonance
different magnetic fields, corresponding to 1/τι = 1/τ1 and 1/τ 2 = ( ι ) ( 1 / τ ± +
1/Τ||). A typical difference feature between the two geometries obtained, for
example, from Sb-doped Ge is shown in fig. 36. The broken lines are theoretical
predictions based on the assumption that, for conduction electrons, their
recombination time is much longer than the spin flip time. The contribution
from lattice scattering, to be given below, has been subtracted off for each
geometry. A variety of 1/τ1 and 1/τ 2 values has been obtained for several
impurities, both donors and acceptors, in Ge as well as in Si.
Resonance linewidth measurements for undoped Ge and Si, that account for
contributions from scattering by acoustic deformation potential phonons, have
also been made down to a temperature of 0.5 K. A strong upward deviation
from the T 3 /2 dependence in the classical regime has been observed, and a
qualitative reference to the theoretical predictions of Meyer (1962) and Ito et al.
(1966) has been made. The results obtained are illustrated in figs 37 and 38 for
Ge and Si, respectively.
All these findings, obtained with millimeter waves as described above, indicate
that cyclotron resonance in Ge and Si may not yet be regarded as complete. On
the contrary, these prototype semiconductor materials have a chance of offering
new openings in research, not necessarily confined to semiconductor physics. Of
ao
f = 71 GHz
Pure-Ge
MXy
•
1/Γ2
/
^4.0
/20
ο
•
ζ
Ρ
Q
/
1.0
-
05
0T
-^itoetdl.
1 — L . .I-
0-5
'
I I I!
1
1
• •
3
4
5
TEMPEFWURE ( Κ )
Fig. 37. The inverse relaxation time of electrons in ultrapure Ge is measured at very low
temperatures. The difference between I / t ^ and 1/τ 2 is inherited from the classical regime. The
deviation from the T 3 2/ line reflects the onset of the quantum limit. Qualitative comparisons with
old theories are indicated (from Ohyama et al. 1970).
Ε. Otsuka
70
Χ10 9
40
Pure Si
f = 71 GHz
ο
ι/τ,
ο
~Q2
0.1
0.5
2
3
A 5
TEMPERATURE ( Κ )
Fig. 38. The same as in fig. 37 but obtained for pure Si (from Ohyama et al. 1970).
particular interest will be the electron scattering problem in the quantum limit.
Even in the millimeter wave range, rather drastic effects of quantum transport
show up. Naturally one is tempted to make further experiments in the farinfrared. Indeed some exploratory experiments have been initiated. It seems
appropriate to introduce such work at the end of this chapter as a possible
example of new precision Landau level spectroscopy.
7.2. Transport measurements of Ge in the far-infrared
It is true that a number of cyclotron resonance experiments have been carried
out in the quantum limit. But it should be pointed out that the high-field
(megagauss) experiments were not always favorable under quantum limit
conditions, since the temperatures were usually high. The cyclotron resonance
studies in GaAs described in this article were made under rather optimal
conditions for the quantitative investigation of various electron scattering
mechanisms. The quantum limit condition could essentially be satisfied, since
the temperature was varied down to the liquid helium range, in a convenient
combination with steady magnetic fields generated by a superconducting
magnet. Of special interest was the behavior of the phonon scattering, and the
acoustic deformation potential scattering in particular. The modern theories of
Arora and Spector (1979) and of Suzuki and Dunn (1982) did not yield a good
agreement with the experimental data.
71
Cyclotron resonance
In order to avoid the involvement of scattering by acoustic piezoelectric
phonons or impurities, an ultrapure Ge sample has been subjected to farinfrared cyclotron resonance, extending the earlier work by Fink and Braunstein
(1974). Measurements were made, in time resolution, of the linewidth corre­
sponding to 1/τ ± of the electron cyclotron transition. Contributions from
carrier-carrier scattering were carefully excluded. The experimental results
obtained at λ = 119 μιη (2521 GHz), for which fta>c//cB = 121 K, is demonstrated
in fig. 39. Above 10 K, no contribution from impurity scattering is visible. The
contribution from nonpolar optical phonon (optical deformation potential)
scattering becomes appreciable only above 300 K. Piezoelectric phonons do not
exist. The acoustic deformation potential phonons, accordingly, are the only
ones to contribute to the electron scattering. As shown in fig. 39, the experi­
mental points lie along a guide line giving a T-dependence. Above 100 Κ some
deviations are discernable. They are considered to be a transition to the classical
regime that gives a T 3 /2 dependence. A similar situation holds also for longer
FIR wavelengths of 172-513 μπι, where deviation from the Τ dependence
10,13E
1 I I I 111
Ί — I
Pure
UJ
2
^
c
1
1 I Μ
I III
1 I 1 1 I4J
1—I
Ge
/ k
B
= l21K
1
Σ 10 1Z
F
ζ
ο
Acoustic
Deformation
Rrtential
-
Quantum Limit-/ 0
/ ι
>
ζ
~*y
Classical Limit-/
ι ι Optical
Inter vail ey-w /<-Deformation
/ /
Potential
"
ι
I
/
10,10
10 L
ι
ι ι ι mi/
i
I I I Mil/
10
10'
TEMPERATURE
ί
I
I
I
Mill
10 J
( K )
Fig. 39. Inverse relaxation time of electrons in an ultrapure G e crystal is plotted against temperature
for a far-infrared wavelength of 119 μπι. The contribution to cyclotron resonance linewidth is
considered to come primarily from acoustic deformation potential phonon scattering. The deviation
from the classical prediction is obvious. The intermediate quantum limit behavior shows a very close
Γ-dependence. Only a slight deviation from the T-dependence starts above 100 K, where the electron
system becomes more or less classical (from Kobori, Ohyama and Otsuka, unpublished).
Ε. Otsuka
72
occurs at lower temperatures (Otsuka 1989). One is thus forced to recognize the
Τ dependence as the intermediate quantum limit scattering by acoustical
deformation potential phonons. This is different either from Arora and Spector
or from Suzuki and Dunn. An old intuitive calculation that makes a reasonable
agreement with experiment is included in Meyer's work (1962), though critical
views have been raised by later authors against his approach. For details and for
further issues of quantum limit cyclotron resonance, Kobori's work (1989) or its
concise version (Kobori et al. 1990) should be consulted.
8. Concluding
remarks
Cyclotron resonance, first observed in Ge and Si, has been applied to a wide
range of materials. Relevant wavelengths, magnetic fields and ambient temper­
atures can be varied considerably these days. The application of cyclotron
resonance is no longer restricted to the determination of the carrier effective
mass.
After a fairly complete survey over various materials, extending to III-V,
I I - V I and even to I-VII compounds, Ge is discussed again, for a genuine study
of carrier scattering in the quantum limit. This represents a cyclic development
in research. Cyclotron resonance studies have, however, not been limited to bulk
materials. As is described in the chapter by McCombe and Petrou twodimensional electron systems have been investigated in detail as well. One
further point should not be forgotten. As pointed out in the treatment of InSb,
for example, the time variation of cyclotron resonance signals can be used to
study carrier kinetics, including spin-dependent phenomena. In the energy
relaxation process of carriers, the presence of impurities, either donors or
acceptors, shallow or deep, plays a vital role. A too specialized view of cyclotron
resonance research, accordingly, is not desirable. A combination with Zeeman
transitions or with photoluminescence, for instance, frequently reveals unex­
pected new aspects. The employment of high magnetic fields further allows the
concept of 'impurity cyclotron resonance' and similarly of 'exciton cyclotron
resonance'.
Despite a large amount of work in cyclotron resonance in the last few decades,
there still exist considerable gaps in our knowledge, both theoretical and
experimental. This holds especially for intermetallic compound alloys, organic
semiconductors, noncrystalline materials, and two-dimensional systems.
Acknowledgements
The writer is grateful to those, both authors and publishers, who are cited in this
chapter and who have given him permission to reproduce figures or tables from
Cyclotron resonance
73
their original articles. He is also greatly indebted to many of his co-workers,
whose names are given in the references, mostly in association with the author
for carrying out experiments that contribute to this article. The tedious work of
preparing the typeset manuscript, as well as redrawing the figures, has been
undertaken by Hiroko Matsumura, whose effort can never be appreciated
enough.
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CHAPTER 2
Phonon-assisted Cyclotron Resonance
Y.B. LEVINSON
Institute of Problems of Microelectronics
Technology and Superpure
Materials
USSR Academy of Sciences
142432 Chernogolovka, Moscow District,
USSR
Landau Level
© Elsevier Science Publishers B.V., 1991
Spectroscopy
Edited by
G. Landwehr and E.I. Rashba
Contents
1. Introduction
81
2. Absorption coefficient for phonon-assisted transitions
81
3. Isotropic parabolic band
83
3.1. A nondegenerate electron gas
84
3.2. A degenerate electron gas
85
4. Spin and band nonparabolicity
86
5. Experimental technique
88
6. PACR peaks (experiment)
88
7. Broadening of PACR lines
93
7.1. Collisional broadening
93
7.2. Broadening due to the optical phonon dispersion
95
7.3. Inhomogeneous broadening
95
8. Fine structure of PACR lines
95
8.1. Pinning (level crossing)
95
8.2. Bound states
97
9. Multiphonon processes
98
9.1. Multiphonon-assisted cyclotron resonance
99
9.2. Multiphonon pinning
101
10. Many-valley semiconductors
103
11. Two-dimensional systems
104
12. Impurity transitions
13. Effect of a d.c. electric
105
field
105
14. Phonon-assisted cyclotron resonance and magnetophonon resonance
106
References
107
7.
Introduction
Transitions of an electron between two Landau levels in an electromagnetic field
may be accompanied by phonon absorption or emission. If one long-wavelength
optical phonon is absorbed or emitted during this process, then the correspond­
ing absorption coefficient of the electromagnetic wave K(v) reaches its max­
imum when the distance between two Landau levels differs from the photon
energy hv by the energy of a phonon, that is, when
hv = ε Γ — ε, ± hco0.
(1.1)
Here et and ε Γ are the energies of the initial and final Landau levels, ha)0 is the
energy of the optical phonon with momentum q = 0, the formula with the upper
sign corresponds to phonon emission and with the lower one to phonon
absorption. This effect is called phonon-assisted cyclotron resonance (PACR). It
was predicted by Bass and Levinson (1965), Klinger (1961) and by Uritskii and
Shuster (1965).
PACR was, for the first time, observed in InSb by McCombe et al. (1967)
evidently by chance. Enck et al. (1969) were the first to have performed
detailed experimental studies of PACR. They generalized as well the PACR
theory for the case of a degenerate electron gas.
2 . Absorption
coefficient for phonon-assisted
The absorption coefficient for
level / to the Landau level /'
perturbation theory (Bass and
transition matrix element with
Μ
^=Σ
γ^ι
transitions
a phonon-assisted transition from the Landau
can be calculated in the second order of the
Levinson 1965, Enck et al. 1969). The effective
the absorption of a photon hv is
+Σ
γ~η
·
Here i is the initial state with N(v) photons with the frequency v, Nq phonons
with the wave vector q and the electron is found to be in the state lkzkx*. Also, f is
the final state with iV(v) — 1 photons, Nq± 1 phonons and an electron in the
l'k'zk'x state. Hamiltonian H\^] describes one-phonon electron-lattice interaction
(see Gantmakher and Levinson 1987) and Hamiltonian HR describes the
interaction between the electron and the electromagnetic field. States v' and v"
are intermediate. The number of phonons in the states v' and i is the same, as HR
is diagonal in the phonon quantum numbers. The number of photons in the
states v' and f is the same, as
is diagonal in the photon quantum numbers.
T h e wave function of the electrons are chosen in the Landau gauge with H\\z.
Y.B. Levinson
82
Therefore, the summation over v' reduces to the summation with respect to the
This concerns the summation over v" as well. The
electron states \kzkx.
quantities Ei9 E{ and Ey entering the denominators of the matrix element (2.1)
are the energies of appropriate states,
Ex — E v, = ε — ε +
ftv,
E{ — Ey>. = ε' — ε — hv,
(2.2)
where ε, ε' and ε are the energies of the electron in the initial, final and
intermediate states.
The absorption coefficient for the transition / -> Γ is expressed in terms of the
transition matrix element (2.1) as follows,
x d(hv + ha>q + s - ε')/(ε)[1 - / ( ε ' ) ] .
(2.3)
Here c'(v) = c/y/ic(v) is the velocity of the electromagnetic wave of frequency ν in
the semiconductor, κ(ν) is the dielectric permeability, L 3 is the normalization
volume, the factor in the front takes into account the induced photon emission.
The summation over the final states f means the summation over the wave
vectors q and polarizations of the emitted or absorbed phonon as well as over
different states of the electron in the /' band, that is, over k'zk'x. The summation
over the initial states i implies that over electron states in the / band, that is, over
kzkx. Finally,/(ε) is the Fermi function.
If the optical phonon dispersion is ignored, that is, if it is assumed that hcoq
= ή ω 0, the summation over q, kx and k'x in (2.3) affects the matrix element only.
The summation over kz and k'z is equivalent to that over the energies ε and ε'.
Therefore,
K(v)~
J d f i g I ( e ) J d e ' g r ( e ' ) | A # e^ |
χ S(hv + hco0 + s-ε')/(ε)[1
2
-/(ε')],
(2.4)
where g^s) is the density of states in the Landau / band, and
| Μ ε_ ε, | 2 = X
| M ^ f | 2.
(2.5)
The density of states gt(e) has a root singularity at the bottom of the band (at
s = sh that is, when kz = 0),
a i e W - e , ) - 1' 2" * , - 1.
(2.6)
The expression for K(v) in the form of (2.4) makes it possible to explain the
mechanisms responsible for the PACR. The root singularity (2.6) is integrable,
because the total number of electron states in the Landau band is finite. But if
the resonance condition (1.1) is fulfilled, the root singularities of both bands
Phonon-assisted cyclotron resonance
83
overlap in the integration over ε and ε' in (2.4), and results in the logarithmic
divergence of the integral. Hence, it appears that when the frequency ν
approaches the resonance frequency (1.1) the absorption coefficient K(v) grows
infinitely. This logarithmic singularity of the absorption coefficient is PACR.
As seen from the above, PACR originates during the transition of an electron
from the bottom of one Landau band kz = 0 to the bottom of the other k'z = 0.
This distinguishes PACR from cyclotron resonance (CR) in which transitions
lkz-+l'kz with all kz contributing to the resonance.
3. Isotropic
parabolic
band
Substituting explicit expressions for perturbation matrix elements HR and H{^]
into (2.3), one can calculate the absorption coefficient. By now, calculations of
this kind have been made only for the simplest model, in which the electron
band and the matrix element of the electron-phonon interaction are treated as
isotropic ones. Even in this model the calculations are cumbersome, and we only
give results after preliminary discussion of some important properties of
perturbation matrix elements HR and H{^\
The theory generally employs the so-called dipole approximation, that is, the
wavelength of radiation responsible for the transitions is assumed to be large
compared to the lengths typical of the electron wave function, namely, to the
and the wavelength of movement of the
magnetic length aH = (hc/eH)1/2
electron along the magnetic field / t ~ / c z - 1. In a nondegenerate electron gas,
the typical value of hkz is the thermal momentum ( 2 m T ) 1 / ,2 and in a degenerate
gas, it is the Fermi momentum (2meFl)1,29
where the energy sFl = sF — sl is the
Fermi energy reckoned from the bottom of the corresponding Landau band.
In the isotropic band in the dipole approximation, the matrix elements HR
obey selection rules in /, that is, if the electric field Ε of the wave is polarized
parallel to the constant magnetic field / / , then Al = 0, but |Δ/| = 1 if E1H.
From this follows a well-known fact that the cyclotron absorption is
observable only in a transverse polarization and only for transitions between
adjacent Landau levels. The matrix elements H£] obey no selection rules in /,
therefore, phonon-assisted /-»Γ transitions may occur between any pair of levels
and at any polarization.
But the absorption in longitudinal polarization differs substantially from that
in transverse polarization. This occurs due to the fact that at E\\H the
perturbation matrix element HR is diagonal in kz and proportional to kz. That is
why for E\\H there is
| Μ ε ^ | 2^ ( ε - ε ζ ) + ( ε ' - ε Γ) ,
(3.1)
i.e., the matrix element is small for transitions in which the initial and final states
of the electron are localized in the vicinity of the band bottom. As a result, the
84
Υ.Β. Levinson
longitudinal polarization turns out to be free of PACR, i.e., the absorption
coefficient has no maximum at the resonance (1.1).
3.1. A nondegenerate electron gas
For a nondegenerate electron gas, the absorption coefficient in the transverse
polarization for the one-optical phonon-assisted transition / Γ is found to be
(Bass and Levinson 1965)
Κ^Λν)
= « r « l t 4 T ( 1 - e-^T)A(v)(N0
+ i ±
i)fc*(0lr
(3.2)
KHKT
Here the formulae with the upper and lower signs correspond to absorption and
emission of a phonon, respectively; nt is the electron concentration in the / band,
hkH = (2mha)H)1/2,
hkT = (2mT) 1( 2 are the magnetic and thermal momenta,
N0 = (eha)°/T-\)~1
(3.3)
is the optical phonon occupation number, and
1
ν
1{ν + ωΗ)2+(ν-ωΗ)2
(3.4)
where ωΗ = eH/mc is the cyclotron frequency. The light frequency detuning Ahv
enters the function
Φ(ξ) = ^έ^0(\ξ\).
(3.5)
Here J f 0 is the Bessel function, and ξ = Ahv/IT, where
Ahv = hv — (εν — £j + hco0)
(3.6)
is the resonance detuning.
Factor bw is dependent on the electron-phonon interaction,
K={
fl,
PO,
| ( ω Η/ ω 0 ) ( / + / ' + 1 ) ,
DO.
(3.7)
Here PO and D O designate the polarization and deformation mechanisms of
this interaction (Gantmakher and Levinson 1987). Then a R = e2 jhcy/κ^ή is the
dimensionless electron-radiation coupling constant, i.e. the fine-structure con­
stant e2/hc, where e2 -» e2JK(y) and c C/^/K^V). The electron-lattice dimensionless coupling constant is
aL = f-l/2w0.
(3.8)
Here τ is the nominal time of the interaction with optical phonons, determined
so that the probability of spontaneous emission of a phonon by an electron with
85
Phonon-assisted cyclotron resonance
the energy ε, close to the emission threshold hw0, is found to be (at Η = 0)
1
1/
ε
V'2
In the case of PO interaction a L is the Frohlich coupling constant. Formula (3.2)
gives a correct expression for K(v) only in the vicinity of the resonance when
Ahv <ξ ha>H. At great detunings this formula provides an estimate of the order of
magnitude of K(v).
Using asymptotic expressions of the function J T 0 one can easily find
ν ^ Ι η Κ Γ 1,
\ξ\<1
(^/2)\ξ\-ι'Λ*-Μ
\ξ\>\.
(3.10)
As a result, the function K(v) in the region \Ahv\ <ζ Τ has a symmetric
logarithmic singularity X(v) ~ l n \Ahv\ near the resonance point. But being far
from the resonance at the distance |Aftv| ^> Τ the resonance peak displays a
certain disturbance in symmetry: at Ahv > 0 the absorption coefficient decreases
according to the root law, K(v) ~ |Aftv|" 1 / ,2 and at Ahv<0 one observes an
exponential decrease in addition to the root dependence, K(v) ~ \Ahv\~1/2
χ exp( — |ΔΛν|/Τ). This is quite evident since in case of photon energy deficit
(Ahv < 0) the initial state is at the tail of the Maxwellian distribution in the /
band.
3.2. A degenerate electron gas
Calculations for a degenerate electron gas were made only for T= 0 (Enck et al.
1969, Bakanas 1970). At T = 0 only transitions with phonon emission occur, and
the positions of initial and final Landau levels must satisfy the conditions
sF — (hv — hco0) < εζ < ε Ρ,
ε Ρ < ε Γ < ε Ρ + (hv — δ ω 0) ,
(3.11)
where ε Ρ is the Fermi level.
The absorption coefficient is found to be
Κ^ν(ν)
=
kHkFl
(3.12)
ακ^-^Α(ν)^νΨ(ξ).
Here hkFl = [ 2 m ^ P — ε ^ ) ] 1 /2 is the Fermi momentum in the / band, and
Ψ(ζ) = 2π In
1+(1+2£)1
cl/2
/2
Ahv
ξ =2(ε Ρ - ε,)
(3.13)
Note, that from (3.11) follows ξ > —\. As in (3.2) formula (3.12) is exact only at
Ahv <ζ hioH. At great detunings it can be considered as an estimate of the order of
magnitude of K(v).
Let us now discuss the PACR peak shape in a degenerate gas. The most
86
Y.B.
Levinson
important properties of the function Ψ (ξ) are as follows:
Ψ(ξ)=<
(π\η\ξ\-\
r-
\ξ\<1;
(3.14)
The following two cases should be distinguished.
(1) Nonresonance magnetic fields when the Fermi level is located far from the
initial Landau level, i.e. ε Ρ - ε ζ ~ & ω Η . Then in the vicinity of PACR, from
|ΔΛν| < hojH it follows that \ξ\ < 1, so that the absorption peak is symmetric, K(v)
~1η|Δδν|.
(2) Resonance magnetic fields, when the Fermi level is close to the initial
Landau level, that is, ε Ρ — ε, <^ ηωΗ. In this case the shape of the peak resembles
that in a nondegenerate gas. The only difference lies in the fact that here ε Ρ — ε,
acts as T. Indeed, at \Ahv\ <ζε¥ — ει the peak is symmetric: K(v) ~ ln|Aftv| and at
\Ahv\$>eF — ε ζ this symmetry is disturbed, that is, if Ahv>09 we have K(v)
~ (Ahv) ~1/2 and if Ahv < 0 then K(v) = 0.
When Τ = 0 in the transitions with emission of a phonon ha>0 the electron
energy grows by hν — ha>0. Due to the Pauli principle the initial energies ε of the
electron fall within the range between ε Ρ — (hv — ha>0) and ε ρ , and the final ε' lie
in the range between ε Ρ and ε Ρ + (hv — ha>0). These ranges vary with the light
frequency ν and/or the magnetic field H. Along with this the numbers of the
initial and final levels / and Γ satisfying the conditions in (3.11) change as well.
The number of active levels / and /' change at such fields Η when either one of
the Landau levels coincides with the Fermi level or it coincides with one of the
range boundaries ε Ρ ± (hv — hcc>0). It is easily understood that at these critical
fields Η the absorption (at a fixed frequency) is to have specific features
(Bakanas and Levinson 1970, Bakanas 1970). We will not discuss it in great
detail here because the spectral features in question have not been experimen­
tally identified so far.
4, Spin and band
nonparabolicity
In sections 2 and 3, the spin splitting of the Landau levels was ignored. Actually,
for each spin orientation σ = + 1 and σ = — 1 there exists a series of Landau
levels which we will denote by ε1σ (or ε * ) . In the simplest model the Landau
levels are given by
ε1σ = ha>H(l + i ) - i<xg/*BH,
(4.1)
2
where μ Β = eh/2m0c = 0.58 χ 1 0 " meV/kG is the Bohr magneton and g is the gfactor of the electron. In this model, the transitions discussed in sections 2 and 3
constitute transitions with no change in spin orientation, the resonance
frequency being independent of spin orientation.
Phonon-assisted cyclotron resonance
87
Formula (4.1) describes satisfactorily the energy levels in the A 3 B 5 semi­
conductor conduction band only in the energy range ε <^ εν A, where e g is the
band gap and A is the spin-orbit splitting. For InSb, with e g = 236.7 meV and A
= 810 meV, noticeable deviations from the simple formula (4.1) are observable
even at ε ~ 50 meV. Agreeable data for the energy spectrum are obtained using
the Kane model by which the energy levels ε1σ are obtained from the solution of
the following equation (Johnson and Dickey 1970):
ε1σ
1 + * a((/ + i)hoH
— 2 gS
a= ^
-
\
,
ft-
- bfrgpkH)
* +
J ' ' - 1 J,
(4-2)
Here the cyclotron frequency and the g-factor describe an electron at the band
bottom. One can easily realize that if ε1σ <ξ ε 8, A, then a = b=\ and (4.2) reduces
to (4.1).
In the Kane model the spin-orbit interaction plays a significant role. That is
energies of transitions with no change in the spin orientation σ are dependent on
<7, that is,
fir - ζε + Φ νε
- Γε
·
( 4·3)
It means that the PACR peaks with different spin orientations are split.
Moreover, spin-flip transitions appear (further on called spin-flip PACR) at
resonance frequencies:
hv = Bp — ε[~ ± Λω 0,
εΖΤ — ε+ ± hw0.
(4.4)
It is seen from the expression for the transition matrix element (2.1) that spin-flip
may occur either in the electron-phonon or electron-photon interaction
process.
The first case has been discussed by Matulis (1967) as well as by Margulis and
Margulis (1982). Matulis assumes in his work that spin-flip is caused by the
lattice-induced modulations of the spin-orbit interaction forming the band
structure. With this spin-flip mechanism the logarithmic singularity in the
absorption coefficient K(v) is obtained only for electron-TO-phonon inter­
action (and not for electron-LO-phonon interaction as in PACR without spinflip). LO-phonon-assisted transitions with spin-flip are also possible, but they
are weak at small kz and, as a result, do not lead to a maximum in the
absorption coefficient K(v).
Since InSb-type crystals have no centre of symmetry the state with k Φ 0 splits
into two states with different spin orientations to the k direction. This splitting
along with lattice scattering altering k, brings about a so-called precessional
spin-flip mechanism (see Gantmakher and Levinson 1987). Spin-flip PACR
based on this spin-flip mechanism has been studied by Margulis and Margulis.
Y.B. Levinson
88
In this case spin-flip PACR is due to LO-phonons, and its intensity depends on
the orientation of the magnetic field relative to crystallographic axes.
PACR with spin-flip due to interaction with photons may be considered
as LO-phonon-assisted combined resonance (Rashba and Sheka, chapter 4
of this volume). Calculations for this spin-flip PACR have been performed by
Zawadzki et al. (1978). A significant difference between this and the case when
spin-flip is due to interaction with phonons resides in the fact that the
logarithmic singularity in the absorption coefficient K(v) ~ln|Aftv| can be ob­
served only in the longitudinal polarization E\\H. At E1H the singularity is
weaker, K(v) ~ \Ahv\ 1η|ΔΛν|. Such a singularity results in no absorption peak.
5 . Experimental
technique*
The most direct method for PACR experimental studies involves measuring the
absorption coefficient K(v) in a magnetic field Η (it is common to fix the energy
of the light quanta hv and to measure the absorption coefficient K(v) as a
function of a magnetic field H). This method suffers, however, a low sensitivity
and is used in samples with electron concentration η as high as 1 0 15 c m " 3 when
the absorption coefficient Κ ^ 1 c m " 1 . At lower carrier concentration the socalled cross-modulation method is applicable. In this case the sample is placed
in a magnetic field H, and the sample magnetoresistance R(H) is measured in a
weak d.c. electrical field. The resistance R(H) changes when the sample is
illuminated since the electrons are heated by the absorbed light, and their
mobility is affected. The change in resistance is maximum at the utmost
absorption, i.e., when there occurs a resonance, for instance, the cyclotron
resonance. The cross-modulation technique permits measuring small absorp­
tion coefficients down to X ^ 1 0 ~ 2 c m " 1 . One should note here that the
interpretation of the cross-modulation data presents some problem compared to
the interpretation of the absorption data, for changes in resistance are due to
rather complex and sometimes obscure processes of hot carrier scattering. What
is more, the d.c. electric field may introduce additional difficulties (Ryzhii 1973).
6. PACR
peaks
(experiment)
The overwhelming majority of PACR experiments have been performed in nInSb. For the first time in this material PACR was observed (evidently, by
chance) by McCombe et al. (1967)**. The absorption peak at ν = ω Η + ω ί 0
*For experimental details see the review by Ivanov-Omskii et al. (1978).
**As mentioned in this work, PACR was observed by Fan and Marfaing at the same time, but their
results had never been published.
Phonon-assisted cyclotron resonance
89
corresponding to the 0 +
1 + transition was detected in the range of magnetic
fields from 40 to 90 kG for hv quanta from 425 c m " 1 (53 meV) until 650 c m " 1
(81 meV), respectively. As seen from fig. 1, this absorption peak only occurs in
the transverse polarization E1H. N o absorption is detected if E\\H at the
frequency ωΗ + O > L .O
At hv = 500 c m " 1 (62 meV) the absorption in the resonance field Η = 54.5 kG
comes to 10% when the sample is 9.25 mm thick. As a result, the absorption
coefficient Κ % 0.1 c m " 1 . Let us compare this value with the theoretical one
from section 3. In experimental conditions (n = 2 x 1 0 1 4c m _ 1, Γ = 6 Κ , Η
= 54.5 kG), only the lowest level 0 + is populated, the electron gas being
nondegenerate: ( ε Ρ — ε£ )/T< — 4. Substituting the parameters of n-InSb,
namely, hcoLO = 284 Κ = 24.5 meV, m/m0 = 0.014, τ = 0.7 ps, into (3.2) for the
transition 0 + -+ 1 + we obtain that in the vicinity of the resonance
K(v) « 0.05 l n - ^ - c m " *.
|Δην|
(6.1)
Here the detuning Ahv should be replaced by the resonance smearing (see
section 7). This smearing is not well-defined, but in any case the logarithm is of
40
60
Θ0
100
H(kC)
Fig. 1. The absorption versus magnetic field Η at fcv = 6 2 m e V in n-InSb (n = 2 x
Γ = 6 Κ ) (from McCombe et al. 1967).
1 0 1 c4 m - 3,
90
Y.B. Levinson
the order of unity, hence the formula (6.1) yields K~ 0.1 c m - 1, which agrees
with the measurement results.
The first experiment with the aim to study PACR was carried out by Enck et
al. (1969). In InSb samples with n » 1 0 1 6c m " 3 at T= 13 K, they observed
absorption peaks identified as 0 + - W ' + transitions (/' = 1-7) with one LOphonon emitted. One of those absorption peaks 0 + - > 2 + is shown in fig. 2.
It is seen from this figure that in the longitudinal polarization no absorption
peak is observed. But unlike the results presented in fig. 1, the absorption
coefficient is almost the same in both the polarizations. It means, that PACR is
not responsible for the main absorption mechanism; i.e. the contribution of LOphonon-assisted transitions is small compared to the absorption due to acoustic
phonon-assisted and/or impurity-assisted transitions. Figure 3 exhibits the
absorption coefficient at the PACR peak as a function of the number of the final
Fig. 2. Absorption near the transition 0 + - • 2 + + LO (from Enck et al. 1969).
k ( c m - )1
-2
10 1— ' — 1 — 1 — ' — '
0-2
0-3
0-4 0-5
1
0-6
Fig. 3. Absorption coefficient at the PACR peak maximum for transitions 0 -* /' in InSb (from Enck
et al. 1969).
Phonon-assisted cyclotron resonance
91
Landau level /'. The solid curve is the result of calculations by the formulae from
section 3.2 with the smearing of levels assumed to be 3 meV, that corresponds to
the experimental resolution.
The transitions 0 + - > / ' + LO with V > 1 in InSb (PACR harmonics) were
experimentally identified by Johnson and Dickey (1970), Dennis et al. (1972),
Ivanov-Omskii et al. (1973), Ivanov-Omskii and Shereghii (1974a). Positions of
the absorption peaks are commonly correlated with the calculations of the
Landau levels from formula (4.2). An example of such correlation is shown in
fig. 4 demonstrating a good agreement between the theory and experiment.
In latter experiments high magnetic fields are employed, up to 200 kG, and a
C 0 2 laser with hv = 117-130 meV as a radiation source. All this along with the
use of cross-modulation technique increases significantly the sensitivity (rather
small absorption coefficients, down to 1 0 " 2 c m " 1 , are measured here) and the
resolution of the absorption peak positions in the magnetic field scan. Thus,
Goodwin and Seiler (1983) detected PACR peaks 0 + - > / ' + with /' up to 23. In
the experiments with large quanta hv, PACR peaks were observed in both the
transverse, and longitudinal polarizations though in the last case they were
weaker Wachernig et al. 1977, Grisar et al. 1978, Goodwin and Seiler 1983).
This could be accounted for by the fact that at large electron energies the simple
spherical model of the conduction band does not work and the kz selection rules
for the matrix elements HR are broken. But, on the other hand, the positions of
PACR peaks for large quanta hv agree with those calculated when the same
band parameters are used as for smaller hv (Goodwin and Seiler 1983).
tiiKmeV)
10
20
90
40
50
Η (kG)
Fig. 4. Positions of PACR peaks and PACR harmonic peaks versus magnetic field Η (from IvanovOmskii and Shereghii 1974a). Initial and final Landau levels are given near the curves (n-InSb,
η = 1 . 7 χ 1 0 1 c6 m ~ 3, μ = 4χ 1 0 3 c m 2/ V s, Τ = 4.2 Κ).
92
Υ.Β. Levinson
Besides, with the use of a C 0 2 laser and strong magnetic fields PACR for light
holes in n-InSb was resolved (Grisar 1978 and Grisar et al. 1978).
At T= hcoLO, the LO-phonon occupation numbers (3.3) are small (in InSb at
Τ = 40 Κ we have nLO ^ 1 0 " 3) . That is why until very recently PACR lines with
phonon absorption have not been detected. Nevertheless, Leshko and Shereghii
(1987) managed, applying strong fields H, up to 400 kG, to elevate the
measuring temperature up to 7 = 7 7 - 1 6 0 Κ and to resolve the transitions
0+
1 + and 0 + - > 2 + assisted by LO-phonon absorption.
In n-InSb, there has been observed the spin-flip PACR as well. The line 0 +
-> 1 ~ with LO-phonon emission was experimentally identified by Weiler et al.
(1974), Zawadski et al. (1978), Grisar et al. (1978). This line is observable only in
the longitudinal polarization E\\H (see fig. 5); the latter means that the line
conforms to the combined LO-phonon-assisted resonance.
In the work by Morita et al. (1980a) the cross-modulation method was
employed to identify three-phonon-assisted spin-flip transitions 0" -> /' + + 3LO
and 0"->/'" + 3LO for ΐ = 3 - 6 (see section 9). It is surprising that all the
transitions are from the excited state 0" and not from the ground state 0 + . The
authors assume that light heating of electrons (see section 5) is responsible for
depletion of the ground level 0 + and occupation of the excited level 0".
Studies of PACR in other materials were casual. In H g x _ xC d xT e (x = 0.2, ε 8
= 64 meV) the ωΗ + coLO line was identified (McCombe et al. 1970). In CdS, the
CR peak in the field H(= 85 kG || c is localized at hv = 5.2 meV, that corresponds
Hll<100>
EIH
120
1 6 0 HfkG)
140
13
3
Fig. 5. Spin-flip PACR in n-InSb (n = 8 χ 1 0 c m " , T = 8 K , μ = 4 χ 1 0 5 c m 2/ V s) at hv =
111 meV. The upper pair of curves and the lower pair of curves correspond to different orientations
of the d.c. electric field Ε (from Zawadzki et al. 1978).
Phonon-assisted cyclotron resonance
93
to the known electron mass m = 0.19. But that peak has a distinctly pronounced
shoulder which can be treated as an unresolved PACR peak at hv = 10.5 meV
with the assistance of a phonon ha>0 = 5.2 meV (Nagasaka et al. 1973, 1977).
7. Broadening
of PACR
lines
Since PACR is caused by the density-of-states singularity at the bottom of the
Landau level, it may only be observed in the absence of broadening of the
Landau levels. The two following reasons of broadening are commonly
discussed, namely, temperature and collisions. As the absorption coefficient
K(v) singularity responsible for PACR is quite independent of the energy
distribution/^) properties, the possibilities for PACR observation are definitely
unaffected by the temperature broadening of the Landau levels. Besides, PACR
insensitivity to the type of the distribution / ( ε ) implies that this resonance is
observable in an electron gas of any degree of degeneracy.
7.1. Collisional
broadening
Collisions of an electron at the Landau level / with impurities and phonons lead
to an uncertainty in the state energy of an order of rt = h/xh where xx is the
electron scattering time in the Landau band /. The collisional broadening F,
of the Landau levels is, in its turn, responsible for smearing of the singularity
(2.6) in the density of states, i.e., the unlimited growth of gx(z) at ε-+ζχ is cut
off at ε — z{ ~
It is evident that the condition to observe PACR in the / - • / '
transition is
εν-ει>Γι
+ Γν = Γ.
(7.1)
This condition can be written as ωΗτ > 1, from whence it is seen that it is the
condition for the magnetic field to be a strong nonquantizing one. Collisional
broadening of the Landau levels can be evaluated from the mobility. For n-InSb,
this estimation yields the scattering time of τ = 1-10 ps, that corresponds to the
broadening Γ = 10-1 Κ « 1-0.1 meV. In n-InSb, for electrons ha)H = hcoLO
= 280 Κ = 24.5 meV at Η « 35 kG, so that the condition (7.1) is well fulfilled
when Η ^ 10 kG. It is a common practice to run experiments at
20 K; in
this case the magnetic field proves to be a quantizing one: ha>HP T.
The above smearing of the singularity of the density of states gi(e) brings
about smearing of singularity of the absorption coefficient K(v). But since the
latter is very weak (logarithmic), the shape of the PACR line is weakly
dependent on the value of τ, and τ Γ. In particular, the magnitude of X(v) in the
absorption maximum is weakly dependent on τ. This greatly discriminates
PACR from CR, where, in the absence of scattering, there is a delta-like
absorption singularity, that is, K(v) ~ δ(ν — ωΗ), and the shape of the CR line is,
therefore, completely determined by scattering.
94
Υ. Β.
Levinson
The above-said concerning the PACR line is experimentally confirmed. It was
demonstrated by Shereghii and Ivanov-Omskii (1980) that the half-width of the
0 + - > 1 + + L O line in n-InSb, which is AHxlkG
at T = 4 . 2 K , is in fact,
independent of the sample quality (defined by the mobility at 77 K). Moreover,
this width is by two orders of magnitude greater than the CR line width (in
samples with similar parameters) which is dictated by the collisional broadening
l/T.
Morita et al. (1980b), point to a rather strong dependence of the PACR line
width on the neutral donor density. But the PACR lines were detected there in
the photoconductivity signal that makes the interpretation of their widths more
complicated (see section 5).
Nevertheless, there is a case when the collisional broadening may strongly
affect the shape of the PACR line (Morita et al. 1975). It is seen from the results
of section 3.2 that at ε Ρ - ε ζ- > 0 the logarithmic singularity K(v) transforms to
the root singularity, the latter being much stronger. That is why the collisional
broadening will have a strong influence on the PACR line shape if Γ ε Ρ — ε,.
Introducing the scattering phenomenologically one should substitute
Ψ(ξ) - Re Ψ{ξ),
ξ = (Ahv + ίΓ)/2(ε Ρ - ε,).
(7.2)
At Γ^>ε Ρ — ε,, the function Ψ(ξ) can be simplified, assuming \ξ\ > 1. Thus we
obtain
K(v) = a R a L ^ ^ , 4 ( v ) { Afev + C(Afev) +
(Ahv)2 + r2
2
r] l
2
1
/
2 1
/
2
J
'
This absorption coefficient reaches its maximum at Ahv =
n
*
]
and max
K(v) ~
\ijf.
Collisional broadening due to optical phonon emission is of particular
interest. Consider, for instance, in what way the / = 1 level broadening depends
on the magnetic field H. The contribution of optical phonons to this broadening
Γ\° has a maximum in such a field Hc when ε1 — ε 0 = ha>LO (see section 8.1). At
Η < Hc an electron at the bottom of the band Ζ = 1 is unable to emit an optical
phonon and, therefore, Γ\° = 0. At Η = Hc, phonon emission transfers the
electron to the bottom of the / = 0 band where the density of states is high. As Η
grows the density of final states decreases and, as a result, Γ\Ό diminishes as well.
The nonmonotonic change of Γ\°(Η) with a maximum at Η = Hc is well
illustrated by the dependence of the CR line width on Η (see the review by
Levinson and Rashba 1973). A nonmonotonic change of this kind should
presumably be typical for the PACR line width also. It is evident that maxima of
the broadening Γ\° will occur at ε 2 — ει = fta>LO and at ε 2 — ε 0 = ha>LQ. It is the
same with higher Landau levels. It is quite possible that the nonmonotonic
dependence of the PACR line width on Η at a fixed hv observed by Morita et al.
(1975) is due to this circumstance.
1
Phonon-assisted cyclotron resonance
7.2. Broadening due to the optical phonon
95
dispersion
The optical phonon dispersion is specified by the following relation
ω, = ω 0 ( 1 - ί 2 / 9 ο ) ,
(7.4)
where the momentum q0 is of the order of the Brillouin zone dimensions, that is,
q0 ~ co 0/s, where s is the sound velocity. An electron in a quantizing magnetic
field interacts with phonons the momentum of which is q^kH.
Therefore,
broadening of the PACR line due to optical phonon dispersion is
δν ~ co0g2/<?o - ω 4/ίο - ™s 2.
0
(7.5)
2
For most semiconductors, ms ^ 0 . 1 K % 0 . 0 1 meV, that is, the broadening (7.5)
is much less than the collisional broadening.
7.3. Inhomogeneous
broadening
When the Landau levels are equidistant, that is, ε Γ — ε^ = hcoHAl, where Δ/ = /'
— /, the contribution to absorption in the vicinity of a certain resonance
frequency may yield transitions starting from different initial levels / and ending
at levels Γ = I + Δ/ with a fixed Δ/. But if the electron band is nonparabolic and,
as a result, the Landau levels are to some extent nonequidistant, the energies of /
- • / ' transitions with different / at a fixed Δ/ are different. This may cause
'inhomogeneous' broadening of the PACR peak. In fact no inhomogeneous
broadening is detected since only one or two initial levels are commonly
occupied, and the band nonparabolicity is great, so that different transitions
with a fixed Δ/ manifest themselves as individual separated lines.
8. Fine structure
of PACR
lines
There are two groups of phenomena which are responsible for the fine structure
of the PACR line, namely, pinning (level crossing) and electron-optical phonon
bound states (see the review by Levinson and Rashba 1973).
8.1. Pinning (level
crossing)
The pinning phenomenon is important in 'resonance' magnetic fields, when the
distance between two Landau levels is equal to the optical phonon energy.
Consider the PACR transition 0->Z with the emission of one LO phonon
occurring at the frequency hv = ε* — ε 0 4- hcoL0. Pinning of this transition take
place at two series of magnetic fields, namely, at fields H'k when ε ζ+ Λ — ε^ = ha>LO
(pinning of the first type) and at fields Hk' when ει — ει _fc = ha>LO (pinning of the
second type).
In the field H'k the final state of the electron-phonon system (that is, an
electron at the level / plus a phonon) is degenerate in energy with a zero-phonon
96
Y.B.
Levinson
state, when the electron is found to be at the level / + k. Those two states are
mixed and pushed apart with the result that the PACR line becomes split into
two components. The arising splitting may be treated differently. At Η Φ H'k,
there are two independent widely spaced lines, namely, the PACR line at the
frequency hv = ε^ — ε 0 + hcoLO and the CR harmonic at the frequency hv = sl+k
— ε 0. If the electron band is near to the isotropic one, then the second transition
is almost forbidden and its oscillator strength is small. As Η approaches Hk, the
lines draw closer together and the oscillator strength from the strong PACR line
is pumped to the weak CR harmonic line. When Η = Hk, these lines super­
impose, and the oscillator strengths of both lines become of the same order of
magnitude.
In the field H'k\ the final state of the electron-phonon system is degenerate
with a two-phonon state: an electron at the level (/ — k) plus two phonons. In
this case the strong PACR line at the frequency hv = st — ε 0 + ha>LO and the weak
two-phonon PACR line (the 0->(l — k) transition with the emission of two
phonons) are brought closer together (the oscillator strength of the two-phonon
PACR line is of second order in the small electron-phonon coupling constant,
see section 9.1).
Mixing of two degenerate states (single-phonon and zero-phonon ones at
pinning of the first type or single-phonon and two-phonon states at pinning of
the second type) produces two levels spaced at about ocl,3ha)LO (Levinson and
Rashba 1973). Transitions to these levels are of the same order of the oscillator
strength. That is why in the fields Hk and Hk the PACR line must split into two
of the approximately same intensity spaced at about
oil/3hcoL0.
A typical pinning situation is qualitatively shown in fig. 6. Curve 1 gives the
PACR line position versus the magnetic field H, and curve 2 gives the position of
that weak line with which the PACR line coincides in the pinning field Hk.
Interaction of the two transitions in the fields close to Hk is responsible for the
appearance of two strong lines. The region of fields ΔΗ is determined by the
Fig. 6. Pinning of two levels (see text). The heavier the curve the stronger is the corresponding
transition.
Phonon-assisted cyclotron resonance
97
condition that the distance between curves 1 and 2 does not exceed 'the pinning
energy' (xl/3ha)LO by the order of magnitude.
In n-InSb,
— SQ = hcoLO at Η % 35 kG and ε^ —
= ha>LO at Η « 41 kG.
Pinning of PACR in those fields were detected by Ivanov-Omskii and Shereghii
(1974a). As seen in fig. 4, near the field H = 3 5 k G for the 0 + - > 1 + + LO
transition one observes an 'irregularity' in the absorption peak position
dependence on H. At greater spectral resolution (of the order of 0.7 meV) it
becomes clear that this irregularity is caused by the fact that at each Η in a small
field region near 35 kG the absorption peak splits into two. A similar splitting
can be seen for the 0 + - > 2 + + LO transition near the field Η = 41 kG.
Though interpreting these pinnings of the second type seems natural there are
a number of problems to be discussed. Indeed, the Landau levels in n-InSb in the
energy range ε ~ ha>LO are almost equidistant. In the field Η = 35 kG the
difference ( ε 2 — ε^) — (ε^ — ε£) ~ 3 meV, i.e., of the same order as the splitting
al/3ha>LO « 2 meV. Nevertheless, in case of an equidistant spectrum the pinning
becomes more complicated since the pinning fields of the first and second types
of pinning coincide, the fields are given by the equation kcoH = O > l o. In this case
one may expect splitting of the PACR line into more than two components
(Gijazov and Korovin 1974, 1975a, 1976). A complex pinning in the range of
magnetic fields, where hv « 2hwLO and hcoLO % hojH was studied experimentally
and theoretically by Devreese et al. (1978), but no well-defined identification of
the lines has been achieved.
8.2. Bound states
The calculation of the shape of the PACR line in sections 2 and 3 are based on
the assumption that the phonon outgoing in the process of photon absorption
and the excited electron do not interact in the final state f. Indeed, the phonon
may be born in the state where it is bound with the electron. There is a direct
similarity to the well-known interband light absorption, when the electron and
hole born in the process of photon absorption may be either free or bound to
form an exciton. The electron-phonon interaction in the final state modifies the
shape of the PACR line in the same way as the Coulomb interaction does the
absorption edge.
It is shown in Levinson's paper (1970) (see as well the review by Levinson and
Rashba 1973), that an electron in the / = 0 band and a dispersionless optical
phonon form a bound state η with its energy below the phonon emission
threshold, i.e. at ε = ε 0 + ha>0 — Wn, where Wn is the binding energy in the η state.
At ωΗ^ω0,
the binding energy Ν Κ η~ α £ δ ω 0. The main contribution to the
bound state is from electron states in the vicinity of the bottom of the band.
The fine structure of the PACR line for the 0 -> 1 + LO transition is dictated
by bound states of a phonon and an electron in the / = 1 band. The energies of
these bound states are close to
hco0. Strictly, those bound states are
Υ. Β. Levinson
98
KM
Κω 0 + Κω,
Ή
Ι\
I
ι\
ι \
\
w
μ­
Fig. 7. The fine structure of the PACR line in a degenerate electron gas due to electron-phonon
bound states (see text).
quasistationary as the electron states of the / = 1 band, participating in their
formation, may decay emitting an optical phonon and transferring the electron
to the / = 0 band. If ει — ε0 < Λω 0, however, the states in the vicinity of the / = 1
band bottom which gives the main contribution to the bound state are
nondecaying and the width Γ of the bound state is, therefore, small. It is
demonstrated in the work by Bakanas et al. (1973) that at εχ — ε0 < hco0 and ωΗ
~ ω 0 we obtain Γ ~ α£Λω 0, that is, the bound state width is small compared to
the binding energy W~oclhco0.
Most well-pronounced bound states are to manifest themselves in PACR for a
degenerate gas where the singularity of the absorption coefficient K(v) trans­
forms from logarithmic to a root one (see section 7.1). The shape of the PACR
line for this case was calculated by Bakanas et al. (1973). It is given in fig. 7. The
inclusion of bound states results in the decrease of the absorption K(v) above the
threshold and the appearing of the sequence of peaks with the width Γ below the
threshold.
The fine structure of the PACR line has not been detected so far. It is of no
surprise since in n-InSb at Η = 40 kG we have W ~ 0.01 meV which is likely to
be beyond the experimental resolution. But in materials with a stronger
electron-phonon interaction, bound states must be pronounced. Thus, in nCdTe at Η = 200 kG the binding energy is quite large, W ~ 3 meV.
9 . Multiphonon
processes
Multiphonon processes have a double effect on PACR. Firstly, the transition
between Landau levels in the light quantum absorption may be accompanied by
the emission or absorption of two or more phonons, not of a single phonon.
Phonon-assisted cyclotron resonance
99
Secondly, pinning of Landau levels (see section 8.1) may be due to two or more
phonons.
9.1. Multiphonon-assisted
cyclotron
resonance
It is evident that the radiation-induced transition between Landau levels may be
followed by the emission of more than one phonon. It is seen, for example, from
the matrix element (2.1), where the Hamiltonian
can be substituted by the
electron-two-phonon interaction Hamiltonian f/J 2) (see Gantmakher and
Levinson 1987). Then the states i and f will differ in occupation numbers of two
phonon modes q' and q".
The matrix element Mj_>f, containing the product
(9.1)
<f|flL>|v'><v'|H|i>
2
R
can be diagrammed infig.8a, where the vertex with two wavy lines q' and q" is
and the vertex with a dashed line ν is H .
Matrix elements Mt^f connecting states differing in occupation numbers of
two phonon modes can also be obtained starting with the Hamiltonian of an
electron-one-phonon interaction when turning to a higher order of the
perturbation theory (the third order). These matrix elements involve products of
the type
R
< f | H L 1 )| v ' X v ' | i i L 1 )| v , r> < v 1 H R| f > .
(9.2)
Q
b
c
Fig. 8. Diagrams for processes when one photon ν is absorbed and two ( q \ q") or three (q, q', q")
phonons are absorbed or emitted.
Y.B.
100
Levinson
They are represented by the diagram in fig. 8b. In the same manner one can
readily construct diagrams and write matrix elements M j _ f , describing tran­
sitions with the assistance of three or more phonons. For instance, fig. 8c shows
one of the diagrams which is responsible for a three-phonon-assisted transition.
It will be easily understood from qualitative considerations which phonons
and phonon combinations may contribute to absorption K(v) and which will
lead to PACR peaks.
Electron states entering into the matrix elements of HL are characterized by
lengths aH and 2n/kZ9 which are much greater than the lattice parameter a0. An
electron only interacts, therefore, with long-wavelength perturbations of the
lattice potential. It implies that the sum of phonon momenta appearing in each
vertex of HL must be small, much less than the Brillouin zone size b0 ~ 2π/α0.
For the matrix element in fig. 8c it means, for example,
\9\<b09
W+
tf\<b0.
Two-phonon-assisted transitions with the assistance of phonons of momenta q'
and q" are possible if \q\ \q"\ < b0 or if \q' + q"\ <ζ b0, i.e., q' « —q".
Not any combination of phonons with a total momentum equal to zero will
result in an absorption peak. An absorption peak appears only in the case when
the total phonon energy hQ corresponds to a singularity of the multiphonon
density of states which is defined as follows
p(Q)=
d 3 9 l \ά\2...δ{Ω-ω^-ω^-
...)d(q1+q2+
...).
(9.3)
Generally speaking, phonons qi9 q2, ... in this equation correspond to different
phonon branches s 1? s 2 , . . . . If the density ρ(Ω) has a sharp peak at a certain Ω*
one can observe a PACR peak at the energy
hv = εν — ει + ΗΩ*.
(9.4)
Two-phonon-assisted transitions are of the greatest interest. Two-phonon
density of states is
(9.5)
It may display a singularity in two cases: (i) at Ω* frequencies, which are
combinations of frequencies of long-wavelength optical phonons, (ii) at Ω*
frequencies constituting combinations of short-wavelength phonons at the
Brillouin zone edge. In other words, PACR with two-phonon emission occurs
during emission of two long-wavelength optical phonons or of two shortwavelength phonons with momenta lying at opposite points of the Brillouin
zone.
A theoretical calculation of PACR with emission of two TA-phonons at points
X or L of the Brillouin zone was made by Mazur (1979). As the electron-two-
Phonon-assisted cyclotron resonance
101
phonon interaction was chosen the macrofield created by two short-wavelength
phonons with almost opposite momenta (Levinson and Rashba 1974).
The presence of two-phonon PACR is demonstrated in fig. 9 (Ivanov-Omskii
and Shereghii 1974a). The positions of the absorption maxima is in agreement
with the dependence hv = εχ — ε0 + hQ*, where hQ* is the energy of the following
two-phonon combinations:
24.4 meV = LO(T),
31.6 meV = 2LA(L),
37.7 meV = TO(L) + LA(L)
(9.6)
or
37.7 meV = LO(X) + LA(X),
48.8 meV = 2LO(T).
Here in the brackets are given the points in the centre or at the edge of the
Brillouin zone to which phonons are related.
PACR peaks for transitions with two LO(T)-phonon emission were also
registered by Morita et al. (1975). Experiments performed by Goodwin et al.
(1980) have provided convincing evidences of the 0 + - > / + transitions (where
/ = 6-16) with three LO(T) phonon emission.
9.2. Multiphonon pinning
The number of possible pinning situations increases if multiphonon processes
are accounted for. Firstly, there are possible pinnings similar to those considered
in section 8.1 when the distance between Landau levels coincides with the energy
of a multiphonon combination hQ*. Consider, for example, the PACR 0 - » /
transition with LO-phonon emission corresponding to the phonon energy
Fig. 9. Peaks of two-phonon-assisted cyclotron resonance (from Ivanov-Omskii and Shereghii
1974a).
Y.B. Levinson
102
hv = — ε 0 + ftcoL0. Let the field Η be such that ει — ε0 = hQ*9 where hQ* = hcoA
+ Λω Β is the energy of the two-phonon combination of phonons A and B. Then
the final state of the PACR transition, i.e. an electron at the I level plus an LO
phonon, is degenerate in the energy with a three-phonon state, i.e., an electron at
the Ζ = 0 level plus three phonons LO, A, B. These two states differ in occupation
numbers of two phonons A and Β and are mixed in the first order due to the
electron-two-phonon interaction H(^] (if phonons A and Β are long-wavelength
ones the mixing may be caused by the electron-one-phonon interaction
in
the second order of the perturbation theory). As a result of mixing and repulsion
of the two degenerate states the PACR line for absorption of the photon with the
energy
hv « εζ — ε 0 + ha>LO « hcoA + hcc>B + ha)LO
(9.7)
splits into two components. The distance between these components is deter­
mined by the coupling constant of the electron-two-phonon interaction a AB
(Gijazov and Korovin 1975b) or by the product of coupling constants of the
electron-one-phonon interaction α Αα Β. In terms of this pinning the experimental
results are interpreted for the PACR transitions 0 + - > 2 + + LO and 0 + - > 3 +
+ LO (Ivanov-Omskii and Shereghii 1974b, and Ivanov-Omskii et al. 1975).
Figure 10 demonstrates a part of the pinning structure of the last of these lines
taken with high resolution. The following combinations play the role of AB
phonon combinations,
(1)
48.8meV = 2LO(T),
(3)
45.6meV =
(5)
43.6meV = 2TO(L).
2TO(r),
(2)
47.8 meV = LO(T) + ΤΟ(Γ),
(4)
44.4meV = 2TO(X),
Another type of pinning for the 0
occurs in such a field Η when
(9.8)
/ transition with LO-phonon emission
(9.9)
hv « ε{ — ε0 + h(D^Q « hwA + hcuB.
In this case the final state, an electron on the I level plus an LO-phonon, is
degenerate with a two-phonon state, an electron on the / = 0 level plus two
phonons A and B. In such a situation the mixing states differ in occupation
numbers of three phonons and the splitting is, therefore, dictated by the product
of the coupling constants a L Oa AB or a L Oa Aa B. In terms of pinning of this kind
the experimental results of Ngai and Johnson (1972) are interpreted for the
transition 0 + - > l + + LO in InSb, where the part of the AB combination is
played by the combination 2ΤΟ(Γ).
It is surprising that the experimental value of the pinning splitting is weakly
influenced by the number of phonons taking part in pinning, that is, by the order
of electron-phonon interaction responsible for splitting. It follows from the
experiments carried out by Ivanov-Omskii et al. (1975) that the splitting of the
two-phonon-assisted pinning in n-InSb comes to 0.4 meV. If this magnitude is
Phonon-assisted cyclotron resonance
103
tvifmeV)
Fig. 10. The pinning structure of the PACR transition 0 + - » 3 + + LO in n-InSb ( w = 1 . 5
χ 1 0 1 c6 m ~ 3, μ = 1.1 χ 1 0 5c m 2/ V s , T = 4 . 2 K ) . The pinning is due to two optical phonons at
the Brillouin zone center or at its edge. Numbers (1-5) correspond to two-phonon combinations
(9.8), see text (from Ivanov-Omskii et al. 1975).
3
given as α 2 / ΛΩ*
and whence the constant of electron-two-phonon interaction
OL2 is found then we obtain a 2 = 0.7 χ 1 0 " 3.
This value is only one thirtieth of the constant of the electron-one-LOphonon interaction in n-InSb ocx = 2 χ 1 0 " 2, while according to the theoretical
estimation (see Gantmakher and Levinson 1987), α 2/ α 1 ^ (m 0/M) ~ 1 0 " 2 - 1 0 " 3 ,
where m 0 is the mass of a free electron and Μ is the atomic mass. This estimate is
also supported by experiments on electron heating and magnetophonon
resonance in InSb for the case of interaction with a pair of phonons TA(X).
It is still unclear as to why two-phonon combinations manifesting themselves
in various processes are different. Compare, for instance, the combinations (9.8)
appearing in the pinning of the line 0 +
3 + + LO and those (9.6) in absorption
+
+
for the transitions 0 - > l . The combination 48.8 meV = 2LO(T) is the only
common one for pinning and absorption.
10. Many-valley
semiconductors
PACR in many-valley semiconductors can be of two types, intravalley and
intervalley.
Transitions at intravalley PACR are to occur between Landau levels of the
same valley assisted by a long-wavelength optical phonon. In the anisotropic
band, the orbit of an electron in the r-space lies out of the plane perpendicular to
the magnetic field H. Transitions, therefore, may take place at any polarization
Ε relative to H.
104
Y.B. Levinson
At many-valley PACR, transitions assisted by a short-wavelength intervalley
phonon occur between Landau levels of different valleys (Bakanas et al. 1969). If
an electron leaves the level Zx of valley 1 for the level l2 in valley 2, the resonance
condition is
hv = ε\2) - είί> ± hco12 = Ηωφ(12 + ±) - hoo^i^ + ±) ± hco
(10.1)
2)
where ή ω 1 2 is the energy of the intervalley phonon, and ω ^ ' are the cyclotron
frequencies in valleys 1 and 2. The frequencies may be different if the field Η is
orientated differently with respect to valleys 1 and 2.
Absorption coefficient calculations for intervalley PACR in a nondegenerate
semiconductor have been made by Margulis et al. (1977).
PACR was examined in PbTe, a many-valley semiconductor (Saleh and Fan
1972). Only intra valley transitions were detected due, evidently, to the fact that
intervalley transitions are forbidden. N o precise identification of the intravalley
transitions was made.
Saleh and Fan (1972) observed also some specific features of the absorption,
the positions of which in the magnetic field scale were independent of the photon
energy hv. The magnitudes of the fields H, corresponding to these features were
found to be larger in samples with higher electron concentration n. Moreover,
no such features manifested themselves in single-valley semiconductors InSb
and InAs, as well as in PbTe if all the valleys were oriented in the same way with
respect to H. In quantizing fields H, the bottom of the spectrum in each valley
goes up by hwH/2. If valleys are orientated differently with respect to / / t h e shifts
of valley bottoms are different, which causes carrier redistribution between valleys.
In a degenerate gas, the redistribution effect is particularly pronounced in such
fields Η when Η changes the bottom of the valley (that is, the lowest Landau
level in this valley) passes through the Fermi level. Electron redistribution
between valleys must produce characteristic properties of absorption, due to the
same reason as redistribution between Landau bands in the same valley (see
section 3.2) does.
11. Two-dimensional
systems
In a thin film (also in an inversion layer or a quantum well) the movement of an
electron in the ζ direction perpendicular to the film plane is quantized. If the
magnetic field H\\z is applied, the movement in the film plane will be quantized
as well. In this situation the energy levels of the electron are as follows,
znl = en + ha>H(l + \ \
(11.1)
where εη is the quantized energy for movement along z. In a film or quantum
well we have
en =
(h2n2/2md2)n2.
(11.2)
Phonon-assisted cyclotron resonance
105
It is evident that in such a situation transitions with emission of an optical
phonon hco0 must produce delta-like peaks at frequencies hv = εηΎ - εη1 + ha>0.
Bass and Bakanas (1976) calculated the absorption coefficient in a nondegenerate electron gas in a semiconducting film for the DO-interaction with optical
phonons. They showed that delta-like peaks appear at any polarization of the
a.c. field E. Korovin and Eshpulatov (1981) considered the case when movement
along ζ is quantized by the MIS-structure field, electrons are degenerated
( T = 0 ) and the electron-phonon interaction mechanism is PO.
12. Impurity
transitions
In n-InSb at Η = 0 the impurity band is merged into the conduction band, while
in strong fields H, donor levels split off, however, from the conduction band (at
Η = 100 kG the donor ionization energy comes to Aef = 4 meV). That is why
one should distinguish PACR transitions starting from the band level 0 + or
from the impurity level (000) + below the bottom of the band (for the field
Η = 100 kG the densities of band and impurity electrons becomes equal at
T^IOK).
It is common practice in experimental work not to discuss the question as to
whether the lines detected are associated with transitions either in the band or
on the impurities. Meanwhile in the field Η = 100 kG the difference in energy
between the transitions 0 +
1 + + LO and ( 0 0 0 ) + - > ( 1 1 0 ) + + LO amounts to
about 1 meV, that is about 2 kG on the magnetic field scale.
The experiments performed by Wachernig et al. (1977) and McCombe et al.
(1969) are exceptions. In these works the impurity lines were discriminated from
the band lines through the temperature dependence of the line intensities. The
authors of the first paper have detected in n-InSb impurity lines of CR
harmonics but no impurity lines of PACR harmonics have been resolved down
to Γ = 8 . 5 K. But in the second work above, the transition ( 0 0 0 ) + ->(110) +
+ LO is distinctly identified in Se-doped n-InSb ( n D = 8 χ 1 0 1 4c m " 3 ) at Τ
= 4.5 Κ in the field Η = 76 kG. This transition disappears as the temperature
increases and at Η « 79 kG there appears the transition 0 + -> 1 + + LO. This
experiment shows as well that TO-phonon-assisted PACR lines are absent (one
can estimate the electron-TO-phonon coupling constant: a T O/ a LO < 0.04 from
absorption measurement sensitivity).
13. Effect of a direct current electric
field
A d.c. field Ε can be applied to the sample in cross-modulation experiments (see
section 5) or it occurs in the vicinity of the sample boundary due to band
bending.
106
Y.B. Levinson
If there is no current through the sample the effect of the d.c. field Ε reduces to
the change in the energy spectrum of the electron. The field Ε removes the
degeneracy of electron states in fcz, that smears the singularity of the density of
states (2.6) and broadens the PACR line. The broadening can be estimated in the
following manner (Bakanas 1976). When absorbing a phonon the centre of the
electron orbit shifts by Ay0 ~ a^q^ where qL is the phonon momentum
component transverse to H. Since the electron interacts mainly with phonons
with qL ~ a„1, we have Ay0 ~ aH. This shift of the orbit causes a change of the
electron potential energy of the order eEaH, that yields the estimate of the
magnitude of the PACR line broadening.
If the field Ε generates a current then it can as well be responsible for the
heating of electrons. N o theoretical studies have been performed on the
influence of heating on PACR.
The experiments have demonstrated that the d.c. field Ε may have a profound
effect on PACR lines. For example, Morita et al. (1980a) and Morita and
Mikoshiba (1980) showed that in some samples of n-InSb in a weak field Ε
only 0 - • Ζ + 3LO lines were detected, while increasing the field Ε produced lines
0 - / + LO.
14. Phonon-assisted
resonance
cyclotron
resonance
and
magnetophonon
PACR and the so-called magnetophonon resonance (MPR) have a lot in
common (Gurevitch et al. this volume, chapter 20; see also the reviews by
Parfenev et al. (1974); Harper et al. (1973)). The MPR are features of the d.c.
magnetoresistance arising in such fields Η when the energy of an optical phonon
coincides with the distance between two Landau levels, that is,
&v — st = ha)0.
(14.1)
Under this resonance condition the scattering of an electron by optical phonons
drastically increases since transitions may occur when both the initial and final
states of the electron lie in the vicinity of a Landau band bottom. It is the
enhancement in scattering that causes the appearance of specific features of
magnetoresistance. The MPR condition (14.1) can be considered as a special
case of the PACR condition (1.1) at ft ν = 0.
Both MPR and PACR can be detected at low temperatures only, T< ha>0, for
at higher temperatures the collisional broadening of the Landau levels is great
(the case of very high magnetic fields is an exception). At low temperatures the
MPR intensity is proportional to the small factor exp( — hco0/T). Indeed, if
scattering is assisted by absorption of a phonon Ηω0 (i.e. scattering goes from the
lower level I to the higher level /'), then this factor appears as the phonon
occupation number N0. If scattering is assisted by emission of a phonon hco0 (i.e.
Phonon-assisted cyclotron resonance
107
it goes from the higher level /' to the lower level /), this factor is due to the small
number of equilibrium electrons at the upper level /'. The PACR intensity
assisted by emission of a phonon hco0 contains no such small factor, as the
energy required to generate a phonon is borrowed from the photon.
The M P R conditions (14.1) coincide with the pinning conditions (see section
8.1). That is why the detection of the PACR pinning by the cross-modulation
method presents difficulties. The features of the magnetoresistance R(H) in the
field Η corresponding to the pinning condition is due to both the resonance
behaviour of light absorption and of electron scattering.
References
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Bakanas, R.K.,
Bakanas, R.K.,
Bakanas, R.K.,
Bakanas, R.K.,
Bass, F.G., and
Bass, F.G., and
1970, Fiz. Tverd. Tela 12, 3408.
1976, Lit. Fiz. Sb. 16, 565.
and Y.B. Levinson, 1970, Zh. Eksp. Teor. Fiz. Pis'ma Red. 1 1 , 1 4 1 .
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Y.B. Levinson and A.Y. Matulis, 1973, Zh. Eksp. Teor. Fiz. 6 4 , 1 0 6 5 .
R.K. Bakanas, 1976, Fiz. Tverd. Tela 18, 2672.
Y.B. Levinson, 1965, Zh. Eksp. Teor. Fiz. 4 9 , 914.
Dennis, R.B., C R . Pidgeon, S.D. Smith, B.S. Wherrett and R.A. Wood, 1972, Proc. Roy. Soc. A
3 3 1 , 203.
Devreese, J.T, J. D e Sitter, E.J. Johnson and K.L. Ngai, 1978, Phys. Rev. Β 17, 3207.
Enck, R.C., A.S. Saleh and H.Y. Fan, 1969, Phys. Rev. 182, 790.
Gantmakher, V.F, and Y.B. Levinson, 1987, Carrier scattering in metals and semiconductors (NorthHolland, Amsterdam).
Gijazov, I., and L.I. Korovin, 1974, Fiz. Tverd. Tela 16, 2693.
Gijazov, L, and L.I. Korovin, 1975a, Fiz. Tverd. Tela 17, 3708.
Gijazov, I., and L.I. Korovin, 1975b, Fiz. Tverd. Tela 17, 2902.
Gijazov, I., and L.I. Korovin, 1976, Fiz. Tverd. Tela 18, 1134.
Goodwin, M.W., and D.G. Seiler, 1983, Phys. Rev. Β 2 7 , 3451.
Goodwin, M.W., D.G. Seiler and D.H. Kobe, 1980, Solid State Commun. 3 3 , 489.
Grisar, R., 1978, Solid State Commun. 2 5 , 1 0 2 3 .
Grisar, R., H. Wachernig, G. Bauer, J. Wlasak, J. Kowalski and W. Zawadzki, 1978, Phys. Rev. Β
18, 4355.
Harper, P.G., J.W. Hodby and R.A. Stradling, 1973, Rep. Progr. Phys. 3 6 , 1.
Ivanov-Omskii, V.I., and E.M. Shereghii, 1974a, Fiz. Tverd. Tela 1 6 , 3 3 7 9 (Sov. Phys-Solid State 16,
2189).
Ivanov-Omskii, V.I., and E.M. Shereghii, 1974b, Fiz. Tverd. Tela 16, 238 (Sov. Phys.-Solid State 16,
153).
Ivanov-Omskii, V.I., B.T Kolomietz and E.M. Shereghii, 1973, Zh. Eksp. Teor. Fiz. Pis'ma Red.
18, 337.
Ivanov-Omskii, V.I., R.V. Parfen'ev and E.M. Shereghii, 1975, Fiz. Tverd. Tela 17, 3661.
Ivanov-Omskii, V.I., L.I. Korovin and E.M. Shereghii, 1978, Phys. Status Solidi Β 9 0 , 11.
Johnson, E.J., and D.H. Dickey, 1970, Phys. Rev. Β 1, 2676.
Klinger, M.I., 1961, Fiz. Tverd. Tela 3 , 1342, 1355.
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Levinson, Y.B., 1970, Zh. Eksp. Teor. Fiz. Pis'ma Red. 12, 496.
Levinson, Y.B., and E.I. Rashba, 1973, Rep. Progr. Phys. 3 6 , 1499.
Levinson, Y.B., and E.I. Rashba, 1974, Zh. Eksp. Teor. Fiz. Pis'ma Red. 2 0 , 63.
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CHAPTER 3
Polaron Effects in
Cyclotron Resonance
DAVID M. LARSEN
University of Lowell
College of Pure and Applied Science
Department of Physics
One University Avenue
Lowell, MA 01854, USA
Landau Level
© Elsevier Science Publishers B.V., 1991
Spectroscopy
Edited by
G. Landwehr and E.L Rashba
Contents
1. Introduction
Ill
2. Weak coupling, bulk semiconductors
112
3. Intermediate coupling, bulk ionic crystals
120
4.
Polarons confined at a heterojunction
References
122
128
L
Introduction
An electron placed in the conduction band of a polar insulator or semi­
conductor surrounds itself with induced lattice-polarization charge. The particle
consisting of the electron with its surrounding lattice-polarization charge is
called a polaron. Recently, the term 'magnetopolaron' has been gaining currency
for referring to a polaron in an applied magnetic field.
Landau-level energies of magnetopolarons are shifted relative to those
predicted by band theory for electrons in a rigid lattice. These energy shifts,
called polaron effects, are most clearly evident in optical experiments and
especially in cyclotron resonance.
This review is, in essence, an annotated guide to the literature on polaron
effects in cyclotron resonance. Since the present chapter is intended, at least in
part, for researchers or would-be researchers in polaron magneto-optics, an
effort has been made to point out unsolved problems in the field.
Intense interest in the magnetopolaron began with the discovery of a
characteristic discontinuity in the η = 1 Landau-level energies of magneto­
polarons when the cyclotron frequency, c o c, given in a simple parabolic band by
ojc = eB/mc,
(1)
where m is the electron band mass, becomes equal to c o LO the frequency of longwavelength longitudinal optical (LO) phonons of the host crystal (Johnson and
Larsen 1966). This discontinuity results in a cyclotron-resonance spectrum
indicated schematically in fig. 1, where the unperturbed cyclotron-resonance
Fig. 1. Schematic plot showing the polaron anticrossing. Solid lines represent the perturbed n =
0 η = 1 cyclotron transition, dashed lines, the unperturbed energies measured relative to the
η = 0, pz = 0, zero-phonon Landau level. The horizontal dashed line is the energy of the η = 0,
p2 = 0 one-phonon state. The sloping dashed line is the unperturbed cyclotron-resonance energy.
D.M. Larsen
112
curve, indicated by the sloping straight dashed line given by eq. (1), is seen to
break into two branches. Different names have been attached to this phenom­
enon; it is called the polaron pinning effect, the polaron resonance, or the
polaron anticrossing. We shall discuss it in more detail in the next section.
Polaron physics has had a long and active history prior to the discovery of the
pinning effect; a fine review of this early work will be found in Kuper and
Whitfield (1963). A very good review which includes discussion of magnetooptical work prior to 1973 is given in Levinson and Rashba (1973). Additional
useful reviews are contained in Larsen (1970), Kaplan and Ngai (1973) and
Devreese (1972).
In cyclotron-resonance experiments one measures the frequency, ω Μ, of the
transition between adjacent Landau levels in the presence of a measured
magnetic field, B. One can then define a cyclotron mass m c, in analogy with
eq.(l),
m c = eB/ca)M.
(2)
In general, mc and m are not the same; m c is dependent upon Β and is generally
larger than m. When polaron effects are not present, m c depends upon Β and
exceeds m because of band nonparabolicity (BNP); m c approaches m a s B - ^ O .
When polaron effects are present, but B N P is absent, m c again is ^-dependent; in
the limit £ - > 0 , m c approaches m*(m*>m), the polaron mass, which will be
defined in section 2. In the polaron resonance region, m c is observed to become
double-valued or to suffer a sudden change when plotted as a function of B.
We shall discuss polaron effects in three categories. Sections 2 and 3 discuss
polaron effects in bulk semiconductors for weak or moderately strong
electron-lattice couplings, respectively. Section 4 contains a review of work on
magnetopolarons either confined to interfaces between two semiconductors
(heterojunctions) or bound in quantum wells.
2. Weak coupling,
bulk
semiconductors
Our starting point is the Frohlich Hamiltonian for an electron in a magnetic
field (Kuper and Whitfield 1963, Devreese 1972), which has the form,
Η — HQ +
^_ph,
where e is the magnitude of the electron charge, bk+ creates an LO phonon of
wave vector k, Ve-ph is the electron-LO phonon interaction, and A is a vector
potential obeying V χ A = B. It is usual to take A either in the Landau gauge,
A = (0, Bx, 0), or the symmetrical gauge, A = j( — By, Bx, 0). It is convenient to
Polaron effects in cyclotron resonance
113
introduce natural units of energy, ha>LO, and length, r 0, the polaron radius,
defined by
r 0 = (ft/2mo> L O) 1 / .2
(4)
In these units we can rewrite Η of eq. (3) in the Landau gauge as,
Η = (Ρχ — hX2y)2 + ρ2 + ρ2ζ + Σ Κ
k
+ Σ*CP( ~
v
x
e
ik
* r)K
h
+ exp(iA · r)bk\
1/2
λ2 = ω0/ω^
(5)
where Ω is the crystal volume (in units of r%) and α is the dimensionless Frohlich
coupling constant, given by
« = WiT— (— - - \ 2 / r 0 ,
(6)
where
and ε 0 are, respectively, the high-frequency and static dielectric
constants of the crystal. For weakly polar crystals like many of the narrow-gap
compound semiconductors,
and ε 0 are not very different. Also, r0 tends to be
large in these materials because m is small. As a result, α is small (a « 0.02 in
InSb and 0.06 in GaAs).
Major assumptions made in deriving eq. (3) are that
(i) the crystal lattice may be replaced by a dielectric continuum;
(ii) lattice polarization waves are supported in the medium; these waves
propagate with a frequency c o LO which is independent of wavelength;
(iii) the conduction band is parabolic, i.e., when K e_ ph = 0 the cyclotron
frequency obeys eq. (1) for magnetic-field strengths considered.
Assumption (i) is expected to be correct if r 0 is much larger than the lattice
constant. Assumption (ii) should be valid for crystals with a LO phonon
spectrum which is flat for a range of wave vectors between 0 and several times
l / r 0. Assumption (iii) is justified if the relevant band gap (usually the gap
between the conduction band and the nearest valence band) sufficiently exceeds
both hcoLO and the excitation energies (measured relative to the bottom of the
conduction band) of the electrons being studied. The use of a one-electron
Hamiltonian like that of eq. (3) is justified when the density of electrons is
sufficiently low. This last condition is almost always satisfied in experiments on
bulk materials since sharp cyclotron-resonance lines, required for accurate
polaron studies, are only observed in crystals of high purity and at temperatures
too low to excite a significant population of electrons across the band gap.
From eq. (5) it is clear that if α is small, so is Vc_ph. For many materials of
DM. Larsen
114
interest α <| 1; in these cases K e_ ph can be treated as a perturbation on the
eigenstates of H0. At low temperatures, defined by
kT <ζ hcoLO
( f t a > LO is typically 2 0 - 4 0 meV) we can make the zero-temperature approxima­
tion and use zero-phonon eigenstates of H0 as the initial unperturbed states.
The most general eigenstate of H0 has the form,
where χ η
Ρζ
is a Landau-level wave function with energy,
ϊ)λ2+ρ2ζ,
(η +
and | 0 > is the vacuum state for LO phonons. The energy of the wave function
described by eq. ( 7 ) is,
(n + i ) A 2 + p 2 + JV.
(8)
From the structure of eq. ( 7 ) and of Ve_ph in eq. ( 3 ) , it is clear that all odd orders
of perturbation theory vanish; thus the lowest nonvanishing correction to eq. ( 8 )
is given by second-order perturbation theory.
To calculate, for example, the perturbed energy of the cyclotron resonance
transition from η = 0 , pz = 0 to η = 1, pz = 0 , which would well approximate the
observed transition energy in the weak-coupling, low-temperature, lowelectron-density limit, one would simply calculate the difference of the perturbed
energies of those states.
Denoting matrix elements by (Larsen 1 9 6 4 ) ,
MnJ(k)
=
(χ,Α^-Ίχ,,ο^Μ;,
one finds that the perturbed energy Ef] corresponding to the η = j , pz = 0 , zerophonon Landau level is given by,
^ ' » - "
+
^ - ? . T ( . ^ J '
M
J G ! R . - . /
>
where ε 7 = (j + i ) A 2 for Rayleigh-Schrodinger perturbation theory (RSPT) and
Sj = Εψ{λ2)
for Wigner-Brillouin perturbation theory (WBPT).
General arguments (not restricted to perturbation theory) indicate (Larsen
1 9 6 9 ) that if, for zero magnetic field, the polaron energy, Epol, can be expanded
as
£ P o i ( P 2 ) = EPOI(0) + ^ P
2
+ 0(p%
(10)
in the limit ρ - • 0 , where ρ is the total momentum of the electron-phonon system,
115
Polaron effects in cyclotron resonance
then in the limit
(n + i ) A 2 ^ 0 ,
p z 2^ 0 ,
with p2 of the same order of smallness as ηλ2, we have for a nonzero magnetic
field
= £ p o (l 0 ) + ^ [ ( n +
Εη(λ2)
iU
2
+ p 2] + 0 { [ ( n
+i
U
]
}
2
2
(11)
where £ , , ( Λ 2) denotes the energy of the nth Landau level with total system
z-momentum pz along the field. The replacement
ρ
2 _
+
η(
+1
μ2
+
2ρ)
is exactly correct to order (η + ^ ) λ 2 + ρ 2 as we have just indicated, but it is not
correct to order
or higher. Thus,
Εη(λ2)*Ερο1[(η
+ ϊ)λ2 + ρ21
in general (Larsen 1964, 1969, 1972, Das Sarma 1984, Larsen 1984a).
In the weak coupling limit (a->0) one can show directly from eq. (9), taking
> l 2- > 0 , that
£ Ρο ΐ ( 0 ) = - α ,
4 = 1 - i a.
m*
(12)
For jλ2 < 1, one can recast eq. (9) in the form of a one-dimensional integral
convenient for computer evaluation (Larsen 1966a, Lindemann et al. 1983).
Results for the η = 0 η = 1 Landau transition in RSPT are given by the solid
line in fig. 2.
Examination of the energy denominator of eq. (9) shows that the right-hand
side of eq. (9) diverges in RSPT when
(j-n)k2^\,
(n = 0 , l , . . . , ; - l ) .
(13)
N o solution of eq. (13) exists for the ground state (7 = 0), which is therefore
divergence-free. However, for the j=l
Landau level (which is of great experi­
mental importance) a single solution exists at η = 0 and λ2 = 1. The correspond­
ing divergence is unphysical and reflects a limitation of RSPT. It arises as the
result of the crossing of the j = 1, pz = 0 zero-phonon initial unperturbed state
with the continuum edge of j = 0, pz = kz one-phonon unperturbed states - a
level crossing which produces the polaron resonance phenomenon described
earlier.
If one diagonalizes H, given by eq. (3), in the subspace of states spanned by the
above-mentioned unperturbed states, one obtains the eigenvalue equation,
116
DM.
Larsen
3
α
Li_
O
CE
LU
Ζ
>
υ
ο
hz
ο
(Τ
(Τ
ο
ο
ο
<
_ι
ο
ο.
0.2
0.4
0.6
0.8
1.0
U N P E R T U R B E D CYCLOTRON
F R E Q U E N C Y / a > L0
Fig. 2. Polaron correction to the energy of the η = 0 - • η = 1 Landau transition for a parabolic band
(solid curve) and for a nonparabolic band in a crystal with InSb band structure with fundamental
gap, EG, equal to \0ha>LO (dots and crosses). The abscissa for the solid line is λ1. D o t s and crosses
represent spin-up and spin-down cyclotron-resonance transitions, respectively (Larsen 1987a).
the right-hand side of which is just the η = 0 term in the WBPT form of eq. (9).
Solutions of eq. (14) [and therefore also of eq. (9) in WBPT form] for Ex <
1 + \λ2 give a lower branch with the qualitative features indicated in fig. 1. As
Ex approaches 1 + \λ2 from below, the corresponding wave function becomes
more and more strongly admixed with η = 0 one-phonon components. As this
happens, the transition strength for the η = 0 - » η = 1 cyclotron resonance
becomes weaker and weaker on this branch. [A more refined calculation
(Larsen 1966b) shows that \λ2 in the energy denominator in eq. (14) should be
replaced by the polaron ground state energy, £ ( 0 P()^ 2) ]
Polaron effects in cyclotron resonance
117
Calculation of the upper branch (E1 > 1 + | i 2 ) , which rapidly gains oscillator
strength with increasing λ2 near λ2 = 1, is more difficult for, in distinction to the
lower branch, energy conserving LO phonon emission transitions can occur
from electrons excited to the upper branch. Experimentally, one sees a jump in
the absorption linewidth as one moves from the lower to the upper branch
(Dickey et al. 1967, Summers et al. 1968).
To understand properly the upper branch absorption one must extend the
simple theory embodied in eqs(9) or (14) and should actually calculate the
absorption profile (Korovin and Pavlov 1967, Harper 1967, Nakayama 1969,
Vigneron et al. 1978). The results indicate that eq. (14) or the WBPT form of
eq. (9), when integrals over k are understood as principal-value integrals, quite
accurately locates absorption maxima on the upper branch for λ2 > 1
(Nakayama 1969), but is qualitatively incorrect for λ2 < 1.
Although low temperatures are employed in the free-electron cyclotronresonance experiments (in order to assure sharp resonance lines), the temper­
ature must not be so low that substantially all of the electrons freeze out on
donor impurities. Thus, one must take into account the fact that the n = 0
electrons contributing to the cyclotron resonance have a thermal distribution of
z-momenta [not just zero z-momentum as assumed in eqs(9) and (14)]. It is
found theoretically that the broadening of the absorption on the upper branch
near λ2 = 1 is sensitive to this thermal distribution (Nakayama 1969).
The foregoing discussion has focussed on discontinuities in the cyclotronresonance spectrum induced by electron-LO phonon interaction. Similar
discontinuities have been observed in the spectrum of electrons bound to donor
ions (and holes bound to acceptors), where the electronic levels are discrete. For
such cases the theory is in some respects simpler than for electrons in Landau
levels. The analog of eq. (14) for impurities is obtained by removing the term fc2
from the energy denominator and replacing \λ2 and \λ2 by the unperturbed
impurity ground state energy and excited state energy, respectively. In addition,
not just LO phonons but any phonon species with a spectrum which is
effectively wave-vector independent and which interacts with electrons can
generally produce discontinuities of the types discussed above, especially in
impurity spectra. We shall return to this point later.
The narrow-gap semiconductors are interesting materials for experimental
polaron studies (even though their α-values are low) because they have small
conduction-band masses; thus the level-crossing condition λ2 « 1 requires only
moderately strong magnetic fields. However, the small band gaps in these
substances produce not only small effective masses but also strongly nonparabolic conduction bands, so the Frohlich Hamiltonian of eq. (3), which assumes
a parabolic conduction band, is not strictly applicable. A more general version
of the Frohlich Hamiltonian can be derived which contains the band nonpara­
bolicity, from which polaron Landau-level energies can be calculated (Larsen
1987a) without great difficulty. For the η =
= 1 cyclotron transition, it is
118
DM. Larsen
found that energy corrections derived as functions of λ2 from eq. (3) are
approximately correct if λ2 is replaced by X 2 , the unperturbed nonparabolic
η = 0 to η = 1 transition energy divided by ftcoL0. A comparison of spin-up and
spin-down cyclotron-resonance transition energies, calculated from the gen­
eralized Frohlich Hamiltonian containing B N P and plotted against Γ 2 , and
transition energies from the Frohlich Hamiltonian of eq. (3) plotted against λ2 is
shown in fig. 2. At small values of λ2, however, an even better approximation
than the one just described is obtained by assuming that the polaron and
nonparabolic corrections are additive (Larsen 1972).
Undoubtedly, InSb, with its sharp resonance lines and low conduction-band
mass, has been the material on which the most intensive experimental studies of
polaron effects in magnetic fields have been undertaken (Dickey et al. 1967,
Summers et al. 1968, Dickey and Larsen 1968, McCombe and Kaplan 1968,
Kaplan and Wallis 1968, Koteles and Datars 1976). Both cyclotron resonance
and combined resonance (in which a cyclotron transition occurs combined with
a spin flip) have been used to study the level crossing. In addition to free-carrier
cyclotron-resonance transitions, impurity cyclotron and combined-resonance
level crossings have been observed. These latter resonances occur between
discrete energy levels of electrons which are very weakly bound to donor
impurities (the donor binding energy in InSb in zero field is approximately
0.7 meV). Since the binding is so weak, by far the strongest optical transition at
the experimental fields of interest ( « 3 T) is close in energy to the free-electron
cyclotron transition. The upper level reached in the transition from the donor
ground state displays, as a function of magnetic field, the expected anticrossing
behavior when its unperturbed energy crosses the energy of a lower-lying donor
level plus ftcoL0. Reviews in Levinson and Rashba (1973), Larsen (1970), Kaplan
and Ngai (1973), discuss polaron anticrossing experiments in weak-coupling
bulk semiconductors.
Although there is good qualitative agreement between theory and experiment,
some puzzles remain. Nakayama (1969) finds satisfactory agreement between
theory and experiment for the splitting between the upper and lower branches at
the level crossing using α = 0.02, as deduced from eq. (6) and the dielectric
constants of InSb (Kartheuser 1972). However, the predicted line widths of the
upper branch at fields above the level crossing are too small - a value of
0.03-0.04 for α is required to obtain satisfactory agreement with the observed
linewidth. Koteles and Datars (1976) studied the level crossing of the n = 2
Landau level with the n = 0 level plus one LO phonon by observing the
transition η = 1 η = 2, where the η = 1 level was thermally populated
( T = 48 K). Using a zero-temperature theory to interpret measurements of the
observed field dependence of the absorption maxima with α as a fitting
parameter, Koteles found a best fit at α = 0.04. It would be premature to
conclude from the foregoing that there is a problem with eq. (6) in view of the
smallness of α in InSb and the oversimplifications implicit in the theoretical
models.
Polaron effects in cyclotron resonance
119
Impurity combined resonance experiments in InSb (which have the advantage
that Reststrahl absorption does not obscure the phonon resonance region)
appeared to yield an anomalously small phonon energy at which the pinning
occurred [23 meV (Dickey and Larsen 1968) or 23.5 meV (McCombe and
Kaplan 1968) versus the expected LO phonon energy of 24 meV]. However, the
phonon energy is not measured directly and deducing it requires knowledge of
the electron g-factor. This latter, in turn, was inferred from the difference in
energy between combined and direct cyclotron-resonance transitions. Direct
spin-resonance measurements performed subsequently yielded g values smaller
in magnitude than those deduced by the difference method described above, for
reasons which are not understood. The 'new' g values are consistent with a
24 meV LO phonon (McCombe and Wagner 1971).
Absorption and emission transitions between Landau levels up to η = 4 have
been observed in GaAs by heating the electrons with voltage pulses applied to
samples immersed in liquid He (Lindemann et al. 1983). The analysis of
transition energies observed is based on the assumption that all transitions
occur between levels with z-momentum close to zero. Using α as an adjustable
parameter, a best fit was found at α = 0.08 compared to a value of 0.06 deduced
from eq. (6). Again, it is not clear how much significance to attach to the
difference between these values. Similar experiments have been reported in CdTe
and InP (Knap et al. 1985).
Perturbations on the cyclotron-resonance frequencies near polaron level
crossings have been observed in HgCdTe (Kinch and Buss 1971, McCombe and
Wagner 1972) and in InAs (Litton et al. 1969, Harper et al. 1970). The value of α
calculated from eq. (6) (a = 0.05) was reported to be in agreement with the value
deduced from the cyclotron resonance data in InAs.
In addition to the simple anticrossing discussed above between the η = 1 zerophonon level and the η = 0 one-phonon level, which occurs near λ2 = 1, more
complicated crossings also occur in the same range of fields. Thus, e.g., in a
parabolic band the η = 2 zero-phonon level, the η = 1 one-phonon, and η = 0
two-phonon level all cross at λ2 = 1 at two LO phonon energies above the η = 0
zero-phonon level. By diagonalizing the Frohlich Hamiltonian in the subspace
of these degenerate states, one can obtain a good semiquantitative description of
the mixed levels. Such a level crossing has been observed and analyzed in
impurity cyclotron resonance in InSb (Devreese et al. 1978).
Level crossing experiments in InSb suggest that donor electrons can couple
significantly not only to LO phonons, but also to excitations of energies A
(Kaplan et al. 1978) and A + ha>LO (Kaplan and Wallis 1968) where A is
approximately 35 c m - 1, the energy of TA phonons near the L-point. Electrons
in shallow donor states associated with the Γ-point in InSb normally would
couple only with phonons with wave vectors near the zone center. This would
rule out a significant coupling to TA(L) phonons. However, the Is state of a
certain donor, called the A donor (believed to be oxygen), although very
shallow, has been shown to contain a significant admixture of L-point wave
120
DM. Larsen
function. Wasilewski et al. (1983) have suggested that the donor ρ states (which
should be purely Γ-like) could couple to the admixed L-component of the
ground state via a TA(L) phonon (Wasilewski et al. 1983). It remains to be seen
whether this explanation can account for the relative magnitudes of the
observed splittings of the upper and lower branch at the Δ + ftcoLO and Δ
anticrossings in Kaplan and Wallis (1968) and Kaplan et al. (1978), respectively.
Experiments on certain weak cyclotron-resonance lines in InSb have seemed
to indicate level-crossing anomalies mediated by electron-two TO phonon
deformation potential coupling (Ngai and Johnson 1972, Ngai 1974). However,
no such effect was observed on the strong impurity cyclotron-resonance line
(Kaplan and Ngai 1974). Thus, the status of this effect remains unclear.
3. Intermediate
coupling,
bulk ionic
crystals
In the weak coupling materials (a ^ 0 . 1 ) discussed in the previous section,
polaron effects are hard to separate from band effects except at the anticrossings.
This is because, away from the anticrossings, energy shifts due to electron-LO
phonon interaction are only of order a. In materials in which α is not small,
polaron shifts can be clearly observed in cyclotron resonance at cyclotron
frequencies well below C O l o. That is fortunate, since materials with larger values
of α also happen to have larger band masses, making it difficult to produce fields
strong enough to achieve the anticrossing condition λ2 = 1.
Since second-order perturbation theory loses accuracy as α increases, one
would like to have a better theory to interpret cyclotron-resonance data at the
larger values of α encountered in the II-VI semiconductors and in polar
insulators. It appears very difficult to extend the perturbation calculation to
higher orders in the three-dimensional case, although the fourth-order correc­
tion for the two-dimensional polaron cyclotron resonance problem has been
evaluated (Larsen 1986, 1987b).
Two theories with the desirable property that in the limit α 0 they approach
the RSPT correction to the cyclotron resonance energy, have been put forward.
The first theory is the generalized Haga theory (GHT), so called because it is a
generalization of a method originally proposed by Haga (1955) for determining
the polaron effective mass, m*/m, in eq. (10). In this method one introduces the
unitary transformation (Lee et al. 1953)
where fk = — v f c/(l + /c 2), and diagonalizes V~lHU, where Ή is given by eq. (4),
in the subspace of unperturbed wave functions having either zero or one phonon
(Larsen 1972, 1974a). Eigenvalues are obtained separately for the ground state
Polaron effects in cyclotron resonance
121
(n = 0) and for the excited states (n = 1, 2,...) for pz = 0, where pz is the total
momentum of the polaron along the magnetic field. Perturbed cyclotron
frequencies of interest are calculated simply as differences of successive
eigenvalues.
The G H T has been extended to include the case of cyclotron resonance in an
anisotropic conduction band with a magnetic field in an arbitrary direction
(Larsen 1974b).
An entirely different approach combining the Kubo formula for the
frequency-dependent conductivity and a generalization of the Feynman polaron
model to the case of nonzero magnetic field (Peeters and Devreese 1982) has
been introduced more recently (Peeters and Devreese 1986). It is natural to refer
to this method as the generalized Feynman theory (GFT) because of its close
connection to the Feynman polaron theory (Feynman 1955).
The G F T has been employed to calculate absorptions for radiation of
frequencies both above and below coLO (Peeters and Devreese 1986).
Extensive series of cyclotron-resonance measurements have been carried out
in CdTe (Waldman et al. 1969, Litton et al. 1976) and the silver halides AgBr and
AgCl (Hodby et al. 1987). In all three of these materials α is large enough for
polaron corrections to the cyclotron-resonance frequency to be large compared
to band nonparabolic corrections, even for λ2 well below 1.
For CdTe using α as an adjustable parameter, one finds that the data can be fit
satisfactorily only for α near 0.40 (Litton et al. 1976). The difference between the
G H T and G F T is not very big for this material. It is evident from a comparison
of the two theories that the G F T predicts a slightly larger polaron nonparaboli­
city than does the G H T (Peeters and Devreese 1984). The value of α derived from
eq.(6) is 0.28 + 0.06 for CdTe (Waldman et al. 1969), which seems to be
significantly smaller than the value 0.40 required for a best fit to the cyclotron
data. The origin of this discrepancy remains unclear.
In the silver halides, which have large bandgaps and are relatively strongly
polar, the shift in cyclotron frequency due to electron-LO phonon interaction
dominates the nonparabolic band effects. Since α is large in these materials
( a « 1.5 in AgBr and 1.8 in AgCl) and narrow resonances are observed, the
experiments pose a strong test of theory. Excellent agreement with the AgBr
data is found using the GFT. The agreement with the G H T is much less good,
the G H T predicting a distinctly smaller nonparabolicity than is evident either
from the experimental data or from the GFT. (Differences in resonance
frequencies predicted by the G H T and G F T grow strongly with increasing
values of a.) For AgCl, where the resonances are much less sharp than in AgBr,
both theories give an adequate fit to the data (Hodby et al. 1987).
It would be of interest to extend the measurements in the silver halides to still
higher magnetic fields and to investigate polaron nonparabolicity in more
strongly polar crystals for which cyclotron resonance has already been observed
in 'weak' magnetic fields (Hodby et al. 1967, Hodby 1971, Hodby et al. 1972).
DM. Larsen
122
4. Polarons
confined at a
heterojunction
The observation of sharp cyclotron-resonance lines associated with electrons
trapped at the interface between two polar semiconductors (especially in the
GaAs-AlGaAs system) has recently kindled both experimental and theoretical
interest in magnetopolarons in such systems. (When describing a heterojunction
we use the convention that the electrons of interest are confined in the firstnamed semiconductor.)
Assuming that the electrons are bound to the interface ζ = 0 by a potential
U(z) and that the magnetic field is applied in the z-direction, one might think
that the requisite polaron energy levels could be found by diagonalizing
Η + I/(z),
(15)
where Η is the bulk Hamiltonian given by eqs. (3) or (5). The unperturbed
electronic states of eq. (15) are more complicated than those of eq. (3) since the
presence of U(z) changes the simple plane-wave state e l f cz of H0 in eq. (3) to more
complicated sub-band states satisfying,
(16)
The perturbation expressions analogous to eq. (9) can be evaluated conve­
niently in closed form as integrals in the special case that U(z) is a harmonic
oscillator potential (Larsen 1984b). However, a useful approximate method for
more realistic potentials for which 'exact' numerical evaluation is impractical
has been described (Larsen 1985a).
Applying this method to the 'triangular potential' model, in which it is
assumed that the electrons are attracted to an infinitely repulsive step interface
by a uniform electric field in the z-direction, one obtains in RSPT curves very
close to those shown in fig. 3. (The curves in fig. 3 are actually calculated for a
harmonic oscillator confinement.) These curves indicate that as the electric field
gets stronger and stronger, the polaron effects increase until the infinitely strong
field limit (the two-dimensional limit) is reached. Thus, confinement increases
the expected polaron effects relative to the bulk (unconfined) case.
In the case that only the lowest sub-band ( 5 = 0) is occupied it is convenient, if
not always accurate, to ignore the higher sub-bands for the purpose of obtaining
the eigenvalues of eq.(15). The accuracy of this ansatz, called the quasi-twodimensional approximation (Q2DA), depends upon η2, the ratio of the energy
difference between the lowest two sub-band unperturbed levels to the LO
phonon energy. In the polaron units this is written as
,2
—
s = 1 -E,
The Q2DA is most useful for large η2 (η2 ^ 1) or, equivalently, strongly confined
polarons.
Polaron effects in cyclotron resonance
0.2
0.4
2
0£
123
0.8
Fig. 3. Polaron corrections (RSPT) as functions of λ2 for various degrees of confinement (Larsen
1984b). The curves with higher values of η2 represent more tightly confined electrons since η2 is the
).
difference in energy between the lowest lying pair of sub-band levels (in units of fauLO
A still simpler, but more extreme approximation is the two-dimensional
approximation, where the Q2DA is adopted and it is further assumed that,
<<t>0(z)\eik
"\φ0(ζ)>
=
exp[i(/c xx +
kyyft
(17)
or φο(ζ) = δ(ζ) (infinitely small confinement distance).
A brief discussion of the accuracy of these approximations can be found in
Larsen (1988).
Since typical values of η2 in experimental heterojunctions lie in the range 0.5
to 2, it is clear from fig. 3 that the two-dimensional approximation strongly
overestimates polaron effects in these systems. The two-dimensional approxi­
mation applies when the electron motion is restricted to z-values which are
much smaller than the polaron radius, given by eq. (4).
124
DM. Larsen
Despite this shortcoming the two-dimensional model remains interesting
theoretically because of its relative mathematical simplicity. Thus, the twodimensional version of eq. (9) can be evaluated explicitly in terms of Γ functions
(Larsen 1984a, Peeters and Devreese 1985). More important, fourth-order
perturbation theory is computationally tractable and corrections of order a 2
have been obtained for both the η = 0 Landau level (Larsen 1986) and the η =
0 - > n = l cyclotron-resonance energy (Larsen 1987b). In these calculations
summations over a multitude of Landau-level intermediate states are replaced
by evaluation of an expectation value of products of exponentials of harmonicoscillator raising and lowering operators.
Since neither the G H T or G F T discussed earlier are exact to order a 2 in the
limit a->0, it is of interest to compare them in that limit in the two-dimensional
case where the exact a 2 term is known for the cyclotron energy. This has been
done (Larsen 1987b, Peeters et al. 1986), and the comparison shows that, at least
for λ2 ^ 0.7, the G F T is more accurate in two dimensions as a - > 0 than is the
G H T (see fig. 4). Interestingly, with reference to our earlier discussion of the
AgBr data, the G H T is found to underestimate the polaron nonparabolicity,
whereas the G F T slightly overestimates it.
The Feynman path integral variational theory is perhaps the most successful
theory available for calculating the polaron ground-state energy as a function of
α at zero magnetic field (Feynman 1955). There is a long-standing theoretical
challenge to harness the power of the Feynman method for calculating the
ground state of a polaron in a magnetic field. The problem is that it is difficult to
maintain the variational nature of the energy obtained (i.e., to be sure that the
energy calculated remains an upper bound to the true energy), while at the same
time obtaining a very accurate energy. Sheka et al. (1976) have treated this
problem in a way which guarantees the variational property of the Feynman
method, but their energies are not very accurate for small a; these energies do
not approach the second-order perturbation result as a - > 0 (Sheka et al. 1976).
Peeters and Devreese (1982) have given a treatment for three dimensions which
was later extended to the two-dimensional case (Wu et al. 1985). Energies in
this theory do approach second-order perturbation theory as α 0 and, in fact,
turn out to be very accurate in that limit if λ2 is not too large. However, it has
been shown that the theory is not variational (Larsen 1985b) since for λ2 greater
than about 1.5, it gives energies below the exact energy in the limit a - > 0
(Broderix et al. 1987).
In addition to the usual anticrossing at λ2 = 1, second-order perturbation
theory based on the Hamiltonian of eq. (15) predicts anticrossings at λ2 = 1 + Es
— E0 for 5 > 0 . Higher-order perturbation theory predicts (Larsen 1987b)
anticrossings at λ2 = n + Es — E0 for integers η and s satisfying n> 1, s ^ O .
Obviously, the only anticrossings accounted for in the quasi-two-dimensional
approximation are those for which s = 0. Thus, this approximation is not
reliable when λ2 >> 1 + Εγ - E0.
Polaron effects in cyclotron resonance
125
Fig. 4. Comparison in the limit α - • 0 of contributions in order a 2 to the polaron correction for the
two-dimensional polaron in the G H T (boxes), for the G F T (crosses), and exact (solid curve). Both
the G H T and G F T are exact to order a.
In bulk semiconductors, discussed earlier, sharp cyclotron-resonance lines are
seen only for low impurity concentrations. Hence, the electron densities are
small and the electrons obey Boltzmann statistics. However, at heterojunctions
the electrons can be well removed from the donors which contribute them to the
junction and sharp cyclotron lines can be observed at electron densities high
enough for the electron gas to be degenerate. In this circumstance the simple
one-polaron picture, which was so useful for understanding bulk cyclotron
resonance, can be expected to break down. We shall examine some of the ways
this happens.
Electronic levels which are already occupied can not be admixed, in a
perturbation calculation, e.g., with a given initial unperturbed state. In this
connection, it is convenient to define the Landau-level filling factor v, a
dimensionless quantity measuring the areal density of electrons at the interface
DM.
126
Larsen
ρ, in units of the areal electron density p M which is the density just sufficient to
completely fill all the degenerate states associated with a single Landau level and
given spin projection. Thus, ν is defined by,
ν =
PIPM,
majc
PM
=
2nh '
Consider now the two-dimensional analog of eq. (14),
2
) — f I22 = — Y'. 2 , 2 V > '2 f o j a( A
£<1 2 D>(A
17
V \λ + 1 - E\'
(18)
where Σ' means summation over all unoccupied η = 0 states. If ν ^ 1 and the
electrons are all in the spin ground state, then Σ' Α = (1 — ν) Σ*. This indicates
how occupation of η = 0 intermediate states can reduce the strength of the
resonant polaron effect. For ν ^ 1 all η = 0 levels with ground state spin are
occupied and Σ'* = 0.
The foregoing argument suggests that nonzero density effects tend to
counteract the confinement effect in heterojunctions so that the actual size of the
polaron effect depends upon both η2 and v.
If one considers processes in which a virtual phonon emitted by one electron
is absorbed by another, one finds a more complicated v-dependent polaron
effect than outlined above. Although such processes appear formally in fourth
and higher-order perturbation theory, their contribution to the polaron shift is
of the same order of α as the right-hand side of eq. (18) because of the high
degeneracy of the Landau levels (Larsen 1984b). A satisfactory treatment of this
many-polaron problem, even neglecting electron-electron Coulomb repulsion,
is lacking.
Another finite-density effect which has to be accounted for is screening of the
electron-LO phonon interaction by the electron gas (Das Sarma and Mason
1985, Wu et al. 1986). Calculations of the effect of screening on the polaron
corrections in strong magnetic fields in G a A s - A l x G a 1 _ x A s hetero-structures
have been reported recently (Wu et al. 1987). The screening, of course, tends to
lower the effective electron-LO phonon coupling strength, thereby reducing
polaron effects.
Several systematic experimental-theoretical studies have been carried out on
cyclotron resonance in GaAs-Al^-Ga^jAs heterojunctions in order to inves­
tigate polaron and band nonparabolic effects (Seidenbusch et al. 1984, Sigg et al.
1985, Horst et al. 1985, Hopkins et al. 1986). At the time of this writing, the
studies continue, and much remains to be understood. Unfortunately, no
experiments showing the level-crossing discontinuity have yet appeared; they
are made difficult partly by the requirement of magnetic fields of 22 Τ or more.
Thus, polaron effects, which are small, have to be separated from other effects
occurring at 'weak' magnetic fields. This can be a tricky procedure, requiring, for
example, an accurate knowledge of the band nonparabolicity in the
heterojunction.
Polaron effects in cyclotron resonance
127
Detailed attempts have been made to fit the experimental cyclotronresonance data. For example, the data obtained in Horst et al. (1985) are
remarkably well fit in Zawadski (1985) by a theory which includes band
nonparabolicity, but neglects interaction of the electron in the η = 1 Landau
level and the s = 0 sub-band level with all other levels except the η = 0, s = 0 onephonon states. Screening and occupation effects are neglected and only one
adjustable parameter is employed, the electric field strength associated with the
triangular potential which binds the electrons to the interface. Given the
simplicity of the model employed, one wonders whether the excellent fit is
fortuitous.
An attempt to fit a wide body of cyclotron data is described in Broderix et al.
(1987). The only adjustable parameter employed is the band mass; screening, the
occupation effect and band nonparabolicity are all taken into account, but the
calculation is carried out in the quasi-two-dimensional approximation. Excel­
lent agreement with the low electron-density (n e = 1.4 χ 1 0 11 c m " 2) data of
Hopkins et al. (1986) is obtained, but for n e = 3.4 χ 1 0 11 c m " 2 , agreement is
satisfactory only at high fields. Likewise, the data considered in Zawadski (1985)
(n e = 4.1 χ 1 0 11 c m " 2) are well-fit only at high fields by this approach.
One should note, however, that the condition for validity of the quasi-twodimensional approximation is not well satisfied, at least in the data by Hopkins
et al. (1986), where the energy separation of the lowest two sub-bands is less than
0.5ha>LO. Thus we expect that the theory of Wu et al. (1988) underestimates the
polaron shift by approximately 20% (Larsen 1988). It appears that a rather
large value of α (α = 0.07) was employed in Wu et al. (1988). Perhaps the quality
of the fit in that work could be preserved by correcting the Q2DA, and using
a smaller α value which would still be consistent with eq. (6) as well, given the
experimental uncertainty in the values of e0 and
in GaAs.
A picture, quite different from the one described above emerges from
cyclotron resonance studies on GalnAs-InP and GalnAs-AlInAs heterojunctions. In these compounds, the cyclotron mass defined by eq. (2) when plotted
against cyclotron energy has discontinuities at energies very close to the two
bulk TO phonon energies in the GalnAs (Nicholas et al. 1985). Those
discontinuities are characteristic of anticrossings. Discontinuities are not found
at any other energies; in particular, they are not observed at the two LO phonon
energies of GalnAs. Magnetophonon measurements in both GalnAs-AlInAs
(Brummell et al. 1983) and GalnAs-InP (Portal et al. 1984) heterojunctions
show interaction with LO phonons but not, apparently, with TO phonons.
[Arguments have been put forward that magnetopolaron oscillations in
GaAs-AlGaAs heterojunctions, originally believed to be due to bulk LO
phonons in GaAs, may actually be associated with the GaAs-like LO phonon in
AlGaAs (Brummell et al. 1987).]
It is possible that this discrepancy between the cyclotron-resonance measure­
ments and the magnetophonon results is connected with the fact that the latter
results are obtained at much higher temperatures than the former. Thus,
128
DM. Larsen
whereas the electron-LO phonon interaction may be strongly screened out at
low temperatures (at which the cyclotron-resonance experiments were carried
out), it may be much less strongly screened at the higher temperatures of the
magnetophonon resonance experiments (Brummell et al. 1987). However, a
quantitative explanation of the data along this line is not yet available.
Cyclotron resonance in an InSb inversion layer in a M O S structure on p-InSb
revealed an anticrossing near the LO phonon energy (Horst et al. 1983).
However, the small (% 1 meV) separation of the LO and TO phonon energies in
InSb makes it difficult to determine which phonon is responsible.
Complicated anticrossing behavior has been reported in n - H g 0 8C d 0 2T e
accumulation layers (Razeghi et al. 1981).
In 150 A quantum wells in GalnAs-InP (Portal et al. 1983, Cheng et al. 1982)
and GalnAs-AlInAs (Singleton et al. 1986), both cyclotron and magneto­
phonon resonance experiments suggest interaction with LO phonons.
Measurements of transmission in a 200 A InAs-GaSb quantum-well
structure as a function of magnetic field reveal anticrossing behavior in the
cyclotron resonance spectrum near the GaSb TO phonon energy (Ziesmann et
al. 1987). However, after accounting for the resonant dielectric functions of InAs
and GaSb at their TO frequencies, the authors find no evidence for polaron
effects involving either LO or TO phonons (Ziesmann et al. 1987).
It is evident that a clear picture of electron-optical phonon interaction effects
in heterojunctions and quantum wells has yet to emerge. N o doubt further work
will clear up the confusion which now reigns in this area of research.
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CHAPTER 4
Electric-Dipole Spin Resonances
E.I. RASHBA
L.D. Landau Institute for Theoretical Physics of the USSR Academy of Sciences
117940 GSP-1 Moscow, ui Kosygina 2, USSR
and
V.I. SHEKA
Institute of Semiconductors
of the Academy of Sciences of the Ukranian SSR
252028 Kiev, prospekt Nauki 45, USSR
Landau Level
© Elsevier Science Publishers B.V., 1991
Spectroscopy
Edited by
G. Landwehr and E.L Rashba
Contents
1. Introduction
133
2. Basic formalism of the theory
137
3. COR theory in the Zeeman limit
140
4. Angular indicatrices and selection rules
144
5. Three-dimensional spectrum with linear terms in the dispersion law
150
6. Inversion asymmetry mechanism for the η-type InSb band
157
7. EDSR and EPR interference
161
8. COR for semiconductors with inversion centre
164
9. COR for narrow-gap and zero-gap semiconductors
167
10. COR on shallow local centres
172
11. Two-dimensional systems: heterojunctions and M O S structures
178
12. One-dimensional systems: dislocations
181
13. Shape of the EDSR band
184
14. EDSR induced by lattice imperfections
188
15. Conclusion
191
Addendum A. Transformation of the reference system and of the Hamiltonian
193
Addendum B. Kane model
195
List of abbreviations
202
References
202
7.
Introduction
Resonance phenomena, which offer a powerful tool of studying intricate
properties of condensed matter, have for a long time been divided into electric
and magnetic resonances. Electron cyclotron resonance (CR), for instance,
belongs to the class of electric resonances, whereas electron paramagnetic
resonance (EPR) belongs to the class of magnetic resonance phenomena.
Electric resonances are excited by the electric vector of an electromagnetic wave,
and electrons experience a change in their orbital state but not in their spin
state.
Naturally, the classification of the motion in terms of the coordinate and spin
degrees of freedom, which in fact provides the grounds for such a simplified
description, is possible only in the absence of spin-orbit (SO) interaction. The
subject of this paper is an electron resonance of a more general type, excited due
to SO interaction. The characteristic features of this resonance, called combined
resonance (COR), are: (i) the electric mechanism of its excitation, and (ii) change
of the spin quantum state, when the quantum numbers corresponding to the
orbital motion either remain unchanged or are changing. In the former case the
resonance occurs at the spin frequency of an electron and is called electric-dipole
spin resonance (EDSR), or electric-dipole-excited electron-spin resonance
(EDE-ESR). In the latter case it occurs at combinational frequencies, that is,
linear combinations of orbital and spin frequencies. We shall call this the
electric-dipole combinational frequency resonance. If the mechanism of its
excitation is not specified, or if the emphasis is on the frequency of the resonance
rather than on the mechanism of its excitation, we shall use the terms spin
resonance (SR) or combinational frequency resonance (CFR).
At present COR, first predicted by Rashba (1960, 1961), is being experimentally discovered and studied for various crystals with different types of
symmetry. It has been observed in 3D systems, i.e. in bulk, in 2 D systems (on
heterojunctions and inversion layers), in I D systems (on dislocations) and in 0 D
systems (on impurity centres). N o w COR is regularly used to investigate the
band structure of semiconductors. The extensive use of the method is accounted
for by: (i) the relatively high intensity of COR (which may exceed the EPR
intensity by several orders of magnitude), (ii) the presence, as a rule, of several
COR bands in the spectrum, and (Hi) the fairly specific angular dependence of
their intensity.
N o w it is necessary to clarify, first, what the source is of the high COR
intensity, and, second, why for more than 15 years after the discovery of EPR by
Zavoisky (1945) and its extensive experimental investigation, COR was not
observed. It is convenient to start this discussion by taking band electrons as an
example.
In the absence of SO interaction, an electron put in a constant homogeneous
magnetic field H, performs two independent motions associated with orbital and
134
E.L Rashba and V.I. Sheka
spin degrees of freedom. The first motion is cyclotron rotation with a frequency
a)c = eH/m*c,
(1.1)
where m* is the effective mass, and e and c are universal constants. A
characteristic spatial scale corresponding to this motion is the magnetic length
r e = (cft/etf) 1 / .2
(1.2)
Accordingly, the minimal electric-dipole moment corresponding to a transition
between neighbouring quantum states under the influence of an a.c. electric field
is pc % erH. The frequency cos of spin transitions is determined by the equation
hcos = |#|μ ΒΗ, where g is the g-factor of an electron in a crystal and μ Β = eh/2m0c
is the Bohr magneton, and m 0 is the mass of the electron in a vacuum. Putting
g ~ 2, we can estimate the magnetic-dipole moment of the transition displayed in
EPR as μ 8~eX. Here X is the Compton electron wavelength: X&4 χ 1 0 " 1 1 cm.
Therefore, if the values of Ε and Η (the amplitudes of the electric and magnetic
fields, respectively) are comparable, the ratio of the CR and EPR intensities is
of the order of / CR/^EPR ~ ( ri/A) 2- For typical values of / / , rH ~ 1 0 " 5 - 1 0 " 6 cm.
This allows us to estimate the value of the ratio of the intensities; typically,
WEPR-IO
1 0.
So, CR is many orders of magnitude stronger than EPR. This property is
inherent in all electric resonances. For instance, for paraelectric resonance, rH
must be replaced by the characteristic atomic quantity, namely, the Bohr radius
rB = h2/m0e2
= 0.5 χ 1 0 ~ 8 cm, and therefore the ratio of the intensities is
^PERAEPR ~ ( Γ Β Α ) 2 ~ 10 4· Since electric resonances are much stronger than
magnetic resonances, one can expect that even weak SO interaction leading to
the coupling of orbital and spin motions will cause intensive electric excitation
of SR. Besides, for band electrons the coupling of orbital and spin motions
makes it possible for the combinational frequencies ω = ncoc ± ωΒ (where η is an
integer) to appear in the spectrum. The intensity of the transition at these
frequencies, i.e., the intensity of the electric-dipole CFR, is generally of the same
order of magnitude as the EDSR intensity. Jointly they form the COR
spectrum*. Usually it is convenient to observe the COR spectrum in cyclotronresonance inactive (CRI) polarizations, since there is no strong CR background.
N o w let us clarify why COR may be absent or, more exactly, very weak. Let us
start with a free electron in a vacuum. The Thomas SO interaction energy is
3^ο = (μΒ/2ηι0€)σ(Εχρ).
(1.3)
*Very often only the electric-dipole C F R bands are ascribed to COR. However, we shall use the
term COR in the sense defined above, in conformity with the original work (Rashba 1960) and with
subsequent reviews (Rashba 1964a, 1979). Thus by COR we understand the entire family of electricdipole spin resonances.
Electric-dipole spin resonances
135
Here σ are Pauli matrices, £ i s the electric field and ρ is the momentum operator.
If Ε is regarded as an a.c. electric field Ε with the frequency ω 5 and ν = p/m0 is
taken as the velocity of the electron, then Ji?so = \μΒσ(Ε χ v/c). Comparing this
expression with the Zeeman energy μΒΗ, and assuming E = H, we see that
JEDSR/JEPRM^/C) 2- I n t h e nonrelativistic limit (v/c)2 <ζ 1 and EDSR is much
weaker than EPR. This result is absolutely clear since up to the T h o m a s 1/2' the
energy J^so coincides with the Zeeman energy in the effective magnetic field He{{
= (v/c)Ε which acts on the electron in the reference system where the electron
rests. Therefore the SO interaction is required to be sufficiently strong. In
crystals, SO interaction is strong due to the fact that the field Ε in ( 1 . 3 ) is a static
electric field of the crystal lattice which is particularly strong near nuclei and the
operator ρ acts not on the smooth functions of the effective mass approximation
(EMA) but on the Bloch functions:
ΨηΜ = unk(r)Qxp(ikr\
(1.4)
the periodic factor unk(r) rapidly varies near nuclei. As a result, the SO
interaction becomes stronger with increasing charges of the nuclei of the atoms
constituting the crystal. In typical semiconductors the SO splitting of the valence
band is A ~ 0.1-1 eV and it may compete with the forbidden gap width EG. Thus
the g-factor and other parameters of the electron are strongly renormalized as
compared to the parameters of an electron in a vacuum. For example, the gfactor may change substantially and may have an anomalous sign (Yafet 1 9 6 3 ) .
As a result, the spin of the electron somehow transforms into its 'quasispin'.
Then due to the difference between μ* = gμB/2 and μΒ the EPR intensity varies:
at larger values of |g| it may be much higher than for an electron in vacuum. But
the COR intensity is determined not by the renormalized value of the g-factor
but by specific terms in the EMA Hamiltonian which simultaneously involve
the Pauli matrices (quasispin) and the quasimomentum operator k (orbit). The
structure of these terms and, consequently, the COR intensity is determined by
the symmetry of the crystal. This problem is discussed in section 2 and
subsequent sections. On the whole, the higher the symmetry of the group G* of
the wave vector in the point of Ac-space corresponding to the band extremum,
and the larger the £ G , the lower the COR intensity. Naturally, the intensity
decreases with a decreasing charge of the nucleus.
Despite the aforementioned restrictions the COR intensity for band carriers in
many crystals is so high that it is impossible to observe EPR against the
background of the COR intensity. However it significantly decreases in cases
where electrons become bound in donor states. It follows from the Kramers
theorem (see section 10) that in this case the intensity involves the factor
(hcoj^)2,
where &x is the ionization potential of a donor. This factor can be very
small if the field Η is weak.
Sticking to the subject of this volume, we shall consider only COR of band
carriers in the Landau levels and also electrons bound to shallow impurity
136
E.L Rashba and V.I. Sheka
centres where Sx is comparable with hcoc and ha>s. But it should be noted that
EDSR is possible also for low-symmetry deep centres. For them, the EDSR
intensity is determined not by the band spectrum of the semiconductor but by
the structure of the electron shell of an impurity ion and by the local symmetry
of the crystal field. On the whole, it is noticeably lower than for band electrons
and for large-radius centres. EDSR for small-radius centres was predicted by
Bloembergen (1961) and experimentally observed by Ludwig and Ham (1962).
The survey by Roitsin (1971) and two monographs by Mims (1976) and by
Glinchuk et al. (1981) are devoted to electric effects in the radiospectroscopy of
deep centres.
The foregoing arguments shed some light upon why EDSR was not observed
and identified experimentally until the conditions of its high intensity had been
found theoretically.
The COR mechanism for free carriers we have discussed is totally due to the
SO interaction entering in the Hamiltonian for a free carrier in a perfect crystal.
According to this approach the presence of impurities or defects, which cause
binding of carriers in shallow levels, diminishes the COR intensity. The theory
based on this concept was developed in the early sixties and preceded the
experiment: its results were summarized in a survey by Rashba (1964a). The
diverse experimental data obtained since then are in agreement with the theory,
and have permitted a number of new parameters of the energy spectrum of
carriers to be found. Later theoretical works were aimed at describing the
experimental results quantitatively within the framework of the original
concept.
However, there is one other line of thinking in COR physics. Originating from
experimental results rather than from theoretical ideas, its essence is the
existence of specific COR mechanisms, caused by defects or impurities. As a
result, in materials with a high concentration of imperfections, COR may prove
to be considerably stronger than in high-quality samples. The paper by Bell
(1962) on EDSR in strongly doped η-type InSb was the first to point to the
existence of such mechanisms, and the problem was first recognized and
formulated by Mel'nikov and Rashba (1971). For the moment, the problem
remains somewhat obscure. That is why there is no doubt that future work on
COR theory must be concentrated on this problem. One can expect the problem
to attract the attention of experimentalists since it opens up new possibilities for
studying disordered systems.
In terms of macroscopic electrodynamics, COR belongs to magnetoelectric
phenomena, first reported by Curie (1894) and reviewed by O'Dell (1970) for
magnetic materials. From the viewpoint of the microscopic mechanism the most
significant feature of COR is the strong coupling of electron spins to the a.c.
electric field in a broad class of crystals. By now different manifestations of this
coupling have been found. This coupling, in particular, is responsible for spin-flip
Raman scattering, discussed in chapter 5 by Hafele.
Electric-dipole spin resonances
2. Basic formalism
of the
137
theory
The COR theory is based on the theory of the band spectrum of an electron in
crystals and on the effective mass approximation. These concepts have been
thoroughly developed and they were reviewed, for instance, in the paper by
Blount (1962) and in the book by Bir and Pikus (1972), which we recommend to
the reader.
In practical COR calculations for specific semiconductors one should proceed
from the band structure of the semiconductor determined by: (i) symmetry
properties and, (ii) by numerical values of the energy spectrum parameters. For
example, for semiconductors with a narrow forbidden gap of the InSb-type it is
often convenient to make use of a multiband Kane model (1957). However, (i) to
elucidate the principal mechanisms of the COR phenomenon and, (ii) to do it
from the same point of view conformably to different systems, it will be more
convenient to use a two-branch (i.e. one-band) model wherever possible. By this
term we understand two branches of the energy spectrum differing only in the
spin state of an electron (or a hole). These branches of the spectrum in crystals
with the inversion centre merge into one band in the entire Ar-space (Elliott 1954)
and in crystals without the inversion centre they stick together in a highsymmetry point and split in its vicinity. Numerical parameters of the twobranch model can be expressed via parameters of a more general model
(Addendum B).
In the framework of the two-branch model, the most general approach to
describe COR in a semiconductor, subjected to external fields (electric and
magnetic), is as follows. The Hamiltonian
for an electron and the operator f
of its coordinate can be derived by means of the method of invariants (Bir and
Pikus 1972) which employs only general symmetry requirements:
JT = J T 0 +
+-*ίο>
*~ = Σ°Μ>
(2.1)
(2·2)
j
/==i|7 t
(2.3)
j
where σ, are Pauli matrices and fc is the operator of the magnetic field
quasimomentum
K= -\Vr-(elch)A(r\
(2.4)
A(r) and q>(r) are the vector and scalar potentials, respectively. The functions ft
and Xt are polynomials over powers of £ j9 being the Cartesian coordinates of the
fc vector. These polynomials are such that
and F possess the necessary
transformation properties with respect to G f c, i.e., the group of k, the wave vector
near which an expansion in powers of kj is performed.
must be a scalar
E.I. Rashba and V.I. Sheka
138
quantity and r is a vector quantity with respect to spatial transformations. Both
and r must be real operators, i.e., must retain their sign at time reversal
t-+ -t. The functions ft and Xt include both symmetrized and antisymmetrized
combinations of k}. In virtue of the commutation condition
[M,'] = ^
" r ,
(2-5)
where jj' and / ' constitute cyclic permutations (e.g., if / = 2 o r ; = y, then / = 3,
/ ' = 1, or / = z,j" = x), the antisymmetrized terms can be expressed via Hs. The
cp(r) potential can be regarded as, for instance, the one created by impurities. It is
assumed that this potential is smooth. In the higher EMA order alongside cp(r)
there emerges a gradient Ε = — 7φ in Η. The corresponding term in
is
analogous to the SO interaction (1.3) for a Dirac electron (eigenfunctions of the
operator
are two-component spinors).
If the a.c. electric field Ε exciting resonance transitions is described by the
vector-potential A, the interaction Hamiltonian is
j ? e = -(e/c)vA,
(2.6)
where the velocity operator is determined by a commutator
v = \?l*,f]-
(2-7)
From eqs. (2. l)-(2.3) and (2.7) it follows that
v = h-lVktf
+ Q{fi\
(2.8)
where Ω(£) is the polynomial over R. Note that the X0)
operators become
important only when the terms of the order fc4 or higher are taken into account
in//*).
A complicated structure of the Ρ operators in eq. (2.3) results from projecting
the multiband Hamiltonian of the kp approximation (Luttinger and Kohn 1955)
onto the conduction band (or valence band). Similar terms also exist in the
Dirac problem. The Dirac Hamiltonian may be treated as a multiband
Hamiltonian, simultaneously incorporating dynamics of differently charged
particles (electrons and positrons, or, in terms of the solid state theory, electrons
and holes). From this point of view interband matrix elements must correlate as
cp = ch/c => PR and the 'forbidden gap' as 2m0c2 =>EG.ln the 1/c 2 approximation
r = r+
h
_Λ „2(σ
4m 0c
,
P)=
x >iV
k
ΛΡ2
+ -^2-(<r x £)·
(2.9)
For semiconductors the coefficient entering in the SO term, is much larger than
the appropriate coefficient in a vacuum, as has been pointed out in section 1.
The interaction of an electron with an electromagnetic wave can be described
Electric-dipole spin resonances
139
not only by the vector but also by the scalar potential φ(Ρ) = — eEF. Then the
COR intensity is expressed via matrix elements of the f operator. Due to the
relation
<f|«|i> = i<» f I<f|f|i>
(2.10)
( ω π is the transition frequency), straightforwardly following from (2.7), the
results obtained by either method coincide. It is worth stressing that the matrix
elements ( f l ^ i ) depend not only on X} but also on J ^ G . This is because the wave
functions of the i and f states are also J^ 0-dependent, this dependence being
quite relevant (Rashba and Sheka 1961a, c).
To compare the EDSR and EPR intensities, it is necessary to calculate matrix
elements of the interaction responsible for EPR. They are determined by the
magnetic component of the electromagnetic field. The corresponding pertur­
bation operator equals
&m = H* VHJe = curl Ά · VHtf.
(2.11)
Differentiation in (2.11) should be performed only with respect to / / e n t e r i n g
explicitly in
but not with respect to Η entering through the vector-potential
A(H), since the appropriate terms are already taken into account in (2.6).
Experiments typically use two types of mutual orientation of the unit vector e
of the electric field of an electromagnetic wave, of its wave vector q and of the
constant magnetic field H: the Faraday geometry (q\\H,e±H) with two circular
polarizations of e (transverse resonance), and the longitudinal Voigt geometry
(qlH,e\\H)
(longitudinal resonance). This choice of polarizations is also handy
for constructing the theory. Therefore, apart from the original reference system
A, associated with the crystallographic axes, it is useful to introduce another
reference system A', associated with the magnetic field in such a manner that
Ζ || Η (Addendum A). In the A system the Cartesian basis is employed and the
vectors r, k and ν are denoted by lowercase letters and their coordinates are
numbered by Latin indices (i,j= 1, 2, 3). In the A' system the vectors are
denoted by capital letters. Their components are chosen in the circular basis
V=(V_,VZ,V+)=(V,9V0,V1),
(2.12)
and similarly R and K. In (2.12)
V±=(Vx±iVY)/y/2.
(2.13)
In the circular basis the coordinates are numbered by Greek indices α, β = Τ, 0, 1
or - 1 , 0 , 1.
The direction of the Z-axis will be chosen from the condition eHz > 0 with the
sign of the charge e taken into account. According to the conditions (2.5) Kz is a
c-number and the other components obey the commutation rule
[/£_, K+ ] = eH/ch = k2H,
kH = r~H Κ
(2.14)
E.L Rashba and V.l. Sheka
140
If we single out the dimensional factor kH from K, the result can be represented
by step-up and step-down operators a+ and a as
R=kHa,
α = ( α , £ , α + ),
aa+-a
a=\,
+
ξ = ^γΚζ.
(2.15)
In the circular basis the commutators of Κ and R are written down as
p e e, f y ] = - i S e, .
(2.16)
So far we have dealt with purely technical aspects of the problem relevant to
the formalism of calculations. In conclusion to this section we shall make an
attempt to discuss the problem in physical terms. This will enable us to
understand qualitatively certain COR mechanisms. Of course, such consider­
ations are no substitute for a consistent analysis of the Hamiltonian (2.2) or for
a more sophisticated Hamiltonian describing the multiband model.
There is a qualitative distinction between band structures of crystals with an
inversion centre and crystals without one. In this section it has already been
noted that in crystals without an inversion centre the spectrum is degenerate
only at certain points of the /r-space, but in the vicinity of these points the
degeneracy is lifted and the spectrum splits into two branches corresponding to
different spin states of the electron. This splitting is due to the Hamiltonian (2.2)
where/) are linear or cubic in k (cf. sections 5 and 6). Since such terms in J^ G are
inherent in crystals without an inversion centre, the COR excitation mechanism
caused by them is termed the inversion asymmetry mechanism. As a rule, it is
fairly efficient.
In crystals with an inversion centre,/) oc H. This is indispensable for ensuring
twofold degeneracy of bands for all k. But the presence of the Η factor
diminishes J^ G, and it may have an observable value only if the forbidden gap
EG entering in the denominator of f} is narrow. Under these conditions the
region, where the dependence of/} on k is quadratic, will be very narrow; this is why
the COR mechanism associated with a small value of EG is often termed the
nonparabolicity mechanism (see section 8). Sometimes under these conditions a
major role is played by the large value of the Xj(k) functions, a possibility made
clear from (2.9) (see also section 9).
Above we have covered the two mechanisms which can be most clearly
specified. But in realistic situations, especially when one is dealing with
degenerate valence bands, to distinguish and interpret individually the contri­
butions of different perturbations (in particular, of those responsible for
warping) is practically impossible.
3. COR theory in the Zeeman
limit
An exact analytical solution of the problem for an electron in a homogeneous
magnetic field Η can be derived only for a few specific cases even for the two-
141
Electric-dipole spin resonances
branch model. Yet, the most interesting situation occurs when the Zeeman
splitting dominates over SO splitting (Zeeman limit). It can be studied in the
general form at cp(r) = 0. In this case an expansion is performed in the parameter
r(^)^<^
s
2o
>
1
/
/ 2^
m
i « ni
,
(3.1)
where c o m ni = min { c o c, ω 8 , ncoc — ω δ } , η is an integer. Here Κ is a characteristic
value of the quasimomentum; for instance, for a band electron it is determined
by formula (3.4). The criterion (3.1) means that the mean energy of SO
interaction is small compared to the spacing between magnetic quantization
levels. Depending on the power / of the quasimomentum tc entering in J^so, the
criterion (3.1) is fulfilled in strong (/ = 1) or weak (/ ^ 3) fields.
Represent the polynomials /0) as
/ ^ ) = ΊΙΣ1ΊΙ,...ΙΛΛ···^
(·)
3
2
where 3t is the SO coupling constant. The matrix elements F\$ ~ 1 are
determined by the symmetry a system possesses. In the systems studied the
number of factors / ^ 4. Keeping in mind (3.2), one gets
y ( £ ) ^ , < F > ' / 2/ f c o W
(3.3)
where (fc2 > can be estimated as
<ft 2P/2m*> ^ m a x t y , T,h(oc},
(3.4)
where η is the Fermi energy, and Τ is the temperature.
In the field of an electromagnetic wave with the vector potential A =
A0e cos{qr — cot} the total perturbation (see (2.6) and (2.11)) is
St = #
e
+ Stm =
--M)
c
= -A0
Re|^(w) + i ^ * ( *
x
f)]exp[ifor -
ωί)]|.
(3.5)
The operators u differ from ν because they include the paramagnetic contri­
bution (2.11) responsible for EPR.
As will be shown in section 7, taking into account both terms constituting St
and their interference may prove to be important (in particular, in η-type InSb)
but at the first stage we shall retain only the term «^, because it is usually
considerably larger than the second. Calculation of matrix elements can be
conveniently performed using eigenfunctions of J0Q, therefore in
one should
eliminate J^so by the unitary transformation
J f ^ e V e - t
(3.6)
in the leading order of the perturbation theory can be treated as
142
E.I. Rashba and V.I. Sheka
nondiagonal*. All the operators (r, v, etc.) are transformed in a similar manner.
In the ^ - l i n e a r approximation the t operator is proportional to J^ G and
^ £ )
+ [ f , J f o ] = 0,
f
t = -i
(3.7a)
00
Jo
Jf s o(^(t))dt,
K{t) = exp(LT 0i)£ e x p ( - U * V ) .
(3.7b)
(3.7c)
Transforming (2.7) analogously to (3.6) and expanding the transformed (2.7) in
T, we get
(3.8a)
where
r = QfrQ~f « r + [ f , r ] = r + r s o.
(3.8b)
In the two-branch approximation under a quadratic dispersion law 3tf0 equals
^o = ^
+ k/*B(<T")-
(3.9)
Here we confine ourselves to the spherically symmetric Hamiltonian 3tf0,
although the theory can also be developed for an anisotropic mass and g-factor
(Rashba and Sheka 1961c). An isotropic Hamiltonian is attractive also because
the transformation properties of r and ν do not depend on the specific form of
the Hamiltonian under the transformation A - • A ' . That is why angular
diagrams of matrix elements are universal. For example, they do not alter when
nonparabolicity of bands is taken into account, or when an electron is bound to
a spherically symmetric impurity centre, etc.
Symmetry properties manifest themselves most explicitly if we take the A'
system and choose the circular basis in it (section 2). Rotation of the vectors
from the A reference system to the A' reference system is realized by the matrices
B, whose explicit form is given in Addendum A:
Q = BR=kHBa,
f=BR.
(3.10)
The vectors k and r are defined by the Cartesian coordinates in A, whereas the
vectors Κ and R are defined by the circular coordinates in A' (section 2). To
diagonalize the Zeeman energy (σΗ) it is necessary simultaneously with the
transformation Β of the Η vector to perform unitary transformation of Pauli
*The generalization of transformation (3.6) for the Hamiltonian involving the coupling of an electron
to an electromagnetic wave has been done by Kalashnikov (1974), who formulated the COR theory
with explicit gauge invariance.
143
Electric-dipole spin resonances
matrices. This transformation matrix S ( 0 , φ) belongs to the D 1 /2 rotation group
representation. Here θ and φ are the polar and azimuthal angles, respectively.
This transformation results in
Sa.S' = Σ Β ι Λ ,
1
« = (1, 2, 3),
α = (Τ, 0,1).
(3.11)
α
The circular components σα of the σ vector are defined according to the general
rule (2.12). It is natural that £, Η and σ are equally transformed at rotations.
In the new reference system in conformity with (2.15)
H0 = Ηω0{α+α
+ i + ±β*σζ + Κ)2.
(3.12)
The eigenfunctions of H0 are oscillator functions φΝ and their eigenvalues equal
^
= ^ c( N +
β*=gμBH/hωc
+ i { 2) ,
i + W *
= gm*/2m0,
<x=±l,
N = 0,1,....
(3.13)
Here ftcoc is the cyclotron energy (1.1) and hcos = \β*\Ηω0 is the energy of a purely
spin transition.
Let us make use of the equality
£ a(r) = £ a exp(iafau ci).
The matrix element of t
(3.14)
on spin functions is
β
fl
α | Τ | Τ > = < 5
'^τ Σ
α
β
":"y*.,-
<·>
3 5ΐ
Commutation of Γ with Λ τ in (3.8b) reduces to the differentiation with respect
to Kf due to (2.16). Passing from matrix elements of the coordinate to those of
the velocity, for the COR matrix element we get
2 1/ 2
< J V ' | | K t| A f T > = ^ , - r
jv' — Ν
— B*
η Af-iV—ρ
Σ
P
—τ 7· {/ } { }α
8
ΛΓ ^ - ( Κ α ι. . . Κ α ι) Ν ).
(3.16)
Similar formulas have been applied to η-type InSn and η-type CdS crystals
(Rashba and Sheka 1961a, c).
In conclusion let us roughly estimate the characteristic scale of matrix
elements of the Ρ and ν operators, responsible for COR. For this purpose it is con­
venient to start with formula (3.8b) for i*so and to use (3.1), (3.7b) and (3.8a),
bearing in mind that the characteristic value of f is
As a result, for COR
matrix elements we obtain
'COR
'y(E)r>
vCOR~rCORa>s.
(3.17)
144
E.I. Rashba and V.I. Sheka
In sections 5-9 we shall consider COR for free carriers mainly in those
systems for which we have reliable experimental data. However, in the next
section we shall formulate preliminarily a general approach to define the
selection rules and to calculate the angular dependence of COR intensities.
4. Angular
indicatrices
and selection
rules
The COR intensity corresponding to each individual transition can be cal­
culated on the basis of the scheme described in section 3. Then we can find the
angular indicatrix of the transition, i.e., the dependence of the intensity of the
transition on the orientation of Η and e. Nevertheless, independent calculation
for each transition, in a number of cases rather time-consuming, is not necessary.
Actually there is usually a relatively small number of different types of angular
indicatrices. The correspondence of a transition to a certain definite indicatrix
Ω(Η, e) is determined by symmetry properties; later in this section we shall
discuss the principles according to which transitions can be classified.
For interpreting experimental data, it is very important and also nontrivial to
examine the problems concerning the selection rules for different transitions and
also the problems of their absolute intensities. These subjects have been
discussed in numerous papers; for example, mainly applicable to narrow-gap
semiconductors of a sphalerite lattice, a most detailed analysis in terms of the
generalized Kane model (1957) has been carried out by Zawadzki and Wlasak
(1976), Weiler et al. (1978), Braun and Rossler (1985) and Wlasak (1986).
Both of the above-formulated problems are closely related to each other, so
that to solve them it is possible to develop a general approach. This approach
uses the notion of angular quasimomentum (AQM), which will be introduced in
this section. Treatment of the problem in terms of this notion is pretty general
and fairly practical. It implies that the high-symmetry Hamiltonian 3tf0 of the
zero approximation, describing an electron in a magnetic field by means of a
certain multiband model, makes it possible to introduce an AQM quantum
number m. Then all transitions m^rn with a given value of Am = m' — m have
identical indicatrices, determined exclusively by the form of the perturbation
operator W and by the polarization of radiation. In virtue of the universality of
the Ω functions, they can be calculated in an explicit form for simple models (e.g.,
for the 2 χ 2 instead of the 8 χ 8 scheme). The problem of the selection rules is
also solved in a general form. For each
' and each individual polarization
there is a maximal ( A m ) m ax such that at \Am\ > ( A m ) m ax the transition is
forbidden for any arbitrary orientation of H. At \Am\ ^ ( A m ) m ax the transition is
forbidden only in the zeroes of the corresponding Ω function.
The only problem which cannot be solved in a general form concerns
numerical values of the coefficients at the functions Ω. These coefficients must be
Electric-dipole spin resonances
145
expressed via matrix elements of Jf7 (e.g., via the constants entering in the
generalized Kane Hamiltonian), and this can be done only by straightforward
calculation, which is sometimes rather cumbersome.
The simplest way to define the selection rules which have been exploited in
some early works is to calculate the contributions of J^so into the velocity
operator vso = i [ ^ D , r]/ft and subsequently calculate matrix elements of vso by
eigenfunctions of the operator Jf0. It is obvious that the number of operator
factors in vso will always be smaller by unity than in J^so. The structure of i ? so
must be compared with the structure of the operator v, determined by formulas
(3.8) incorporating the perturbation of eigenfunctions of J^0, produced by J^so
(Rashba and Sheka 1961a,c). According to (3.8) operator structures of ν and
r s o coincide and their matrix elements differ by the factor equalling the transition
frequency (with an accuracy up to i). Therefore the problem is reduced to
comparing the operator structure of t? so with the operator structure of ν or of r s o.
Since (3.14) holds for a simple band, in this case matrix elements of Τ and Jfso
differ only by the denominator in (3·. 15), then i ? so and r s o have similar operator
structures. Yet, in a more general form of 3tf0, terms in r s o emerge with a larger
number of operator factors than in v so and owing to these terms, the originally
forbidden transitions become allowed.
Let us start with the symmetry arguments leading to the classification of
indicatrices Ω(Η, e). Consider the most interesting case of cubic crystals with a
direct gap at k = 0. From the total Hamiltonian J f 7 of the system, single out the
zero approximation Hamiltonian
describing a spherically symmetric system
in a magnetic field. In all other aspects, it is arbitrary. In particular, it may have
arbitrary dimensionality; for instance, for a 'quasi-Ge' band structure it may be
an 8 χ 8 Hamiltonian. All other terms in
will be treated as perturbation
'.
It includes the anisotropic part of the
operator. Both 3tf and 3tf0 depend on
coordinates exclusively through the operators K+ and K _ . The operator J f 0 is
constructed as a spherical invariant (and J f as a cubic invariant), involving the
product of basis matrices, multiplied by the product of the operators
Since
J^0 is spherically symmetric, its multicomponent eigenfunctions Ψ„ can be
classified according to the angular momentum m. The operators K+ and K _ ,
acting in the A' system, respectively, raise and lower the projection m of the
angular momentum of the ΨΜ function by unity (i.e., K^m e {Ψ,η+α}). On the
other hand, when the commutation properties (2.14) are taken into account, the
Ka operators can be regarded as step-up and step-down operators for an
auxiliary oscillator, changing the value of its quantum number Ν by α (cf.
(2.15)). This allows us to establish the correlation between the angular
momentum m and the quantum number Ν of the auxiliary oscillator by
representing certain components of the multicomponent wave function Ψη in
terms of Landau oscillator eigenfunctions. For instance, in the Kane model the
wave function corresponding to the projection m (m is a half-integer) of the total
146
ΕΛ. Rashba and V.L Sheka
angular momentum onto Η is written as
ClUm+1/2
^3^m-3/2
(4.1)
^5Ym+
1/2
^ 6 ^ m + 3/2
ClUm- 1/2
C ^ m * 1/2
(Addendum Β). The arrangement of bands in the Kane model is illustrated in
fig. 1. This chart clarifies the principle according to which the components of Ψη
Fig. 1. Arrangement of bands in direct-gap cubic semiconductors, described by the Kane model
(1957). EG is the width of the forbidden band, and A is the SO splitting. On the top is the conduction
band; in the middle the valence band, consisting of light hole and heavy hole bands; at the bottom is
the split-off band. The figure applies to the A U Bi v- t y p e of semiconductors. The weak splitting of the
bands (emphasizing their twofold degeneracy) is due to the absence of an inversion centre. On the lefthand side, the splitting is neglected; on the right-hand side, the splitting is shown but its magnitude is
exaggerated.
Electric-dipole spin resonances
147
are constructed; the two upper components correspond to the conduction band
(spin 1/2), the next four components correspond to the valence band (spin 3/2),
and the two lower components correspond to the split-off band (spin 1/2).
Although in the preceding paragraph m is referred to as the angular
momentum of the ΨΜ function, strictly speaking this is not so and m should
really be called the angular quasimomentum (AQM). A genuine angular
momentum is determined by the action of the operator j z on the wave function;
Jz being the projection of the total angular momentum onto the direction of H.
Having only operators K+ and X _ , one cannot construct the operator J z ,
which is evident if the simplest Hamiltonian (3.9) is taken as an example. That is
why it is impossible to define the respective quantum number. Nevertheless, to
find angular diagrams it is not necessary to correlate m with the genuine
momentum. The functions Ω(Η, e) are completely determined (within the scope
of the approximation stipulated at the end of this section) by a change in the m
number which coincides with a change of the genuine angular momentum. It is
noteworthy that the angular momentum can be introduced only in the axial
gauge of A. Nevertheless m retains the meaning of AQM in an arbitrary gauge,
which is clear from its close relationship with the quantum number N.
In the Kane model a set of eight functions Ψ„ of the type (4.1), differing from
each other by numerical coefficients Ch corresponds to each value of m. For
particles (electrons, holes) positioned at the edge of the band, the coefficients Ch
corresponding to this band, are large. This property makes it possible to single
out the functions describing electron and hole states from the complete set. Both
transitions inside the set (Am = 0) and transitions between different sets
(Am φ 0) are possible. Among these transitions there are intraband and interband transitions. For all the systems investigated so far, the form of indicatrices
is universal in the sense that it is exclusively Am-dependent. However,
matrix elements determining the indicatrices were found in the first order in ^ f s o;
involving only the terms containing some of the lowest powers of k. The explicit
meaning of this restriction depends on an individual form of the Hamiltonian.
For the Kane model the appropriate analysis is carried out in Addendum B. For
a more general five-band (fourteen-branch) Hermann-Weisbuch model (1977),
which describes GaAs and a number of other A m B v compounds very well, the
AQM notion retains its validity for a spherically symmetric approximation (it is
broken due to interaction of two p-bands) (Rossler 1984, Zawadzki et al. 1985).
There are grounds for hoping that the universality of the Ω(Η, e) functions will
turn out to be a general property, although no proof of this has yet been found.
Above we have discussed transitions between quantum states of band carriers.
But a similar problem also arises for carriers bound in local centres (Sheka and
Zaslavskaya 1969). Here the angular indicatrix of the transition is also
determined completely by the value of Am. It is natural that in a given case m is
the projection of the genuine angular momentum. Actually the impurity centre
Hamiltonian includes the r-dependent potential energy operator. Thus compo-
148
E.I. Rashba and V.I. Sheka
nents of the operator r are added to the K+ operators, so the total number of the
operators of our theory increases. As a result, the operator Jz is also an operator
of this theory.
Now consider modifications which have to be introduced into the selection
rules due to the second term of (3.8b). For this purpose, consider the 'quasi-Ge'
valence band; but in order to treat the problem analytically, impose specific
restrictions upon coefficients of the Hamiltonian.
(i) Assume that the band is spherically symmetric, i.e., the Luttinger
parameters obey the condition y 2 = 73 (Luttinger 1956). Then from the very
beginning it is convenient to work in the A' system, where Z\\H (section 2).
(ii) Assume that Kz = 0.
(Hi) Assume that the numerical value of the g-factor is such that
j ^ o = aA(R) + bB(R),
a = (h2/2m0)(yi
b = -(h2/2m0)y2,
+ 5y 2/2),
(4.2)
where
A(R) = R2 +
2k2lJz,
B(R) = (JRΫ - i tr{(JR)2} - 2k2HSz.
(4.3)
The operator A(R) commutates with B(R) and with (JR). In the A' system the
eigenfunctions Wm are the four middle lines of (4.1).
As was stated at the beginning of this section, our strategy should be to
compare operator structures of » so and r s o. Since Jfso in formula (3.7b) depends
on the operator R(t), let us start by calculating it. Transform the operator
e x p ( i ^ 0i ) entering in (3.7c)
exp(iJf0t)
+ iBC~1 sin(bCt)}.
= exp(iaAt){cos(bCt)
(4.4)
The operator, introduced here,
C(R) = {\_A(R)Y +
lk y< ,
2
2
H
(4.5)
is proportional to a diagonal 4 x 4 matrix in the Ψηχ basis. At the derivation of
(4.4) we have used commutativity of A and Β as well as the identity B2 = C 2 ,
valid at Kz = 0. From (3.7c) and (4.4) it follows that Ka(t) have the following
structure:
Ka(t) = K^(R2,
t) + VB(R\ KM (i?, t).
2)
2
a
(4.6)
We shall not need the explicit form of the φ functions. According to (3.8b) the
next step is to calculate the commutator [Τ, r ] . Since Jtso is a polynomial over
Ka, calculation of [T, r ] is based on calculation of the commutators [Ka(t), r ] .
From (4.6) and (2.16) it follows that
[ j e e( i ) ,
Λ Τ] =
t) +
KXu \R , t).
2
2
(4.7)
Electric-dipole spin resonances
149
Formula (4.7) enables us, using (3.7b) and (3.8b), to calculate r s o. Moreover, to
follow through this procedure to the end is not necessary. The basic result is
already contained in (4.7). Analyzing this formula, one should bear in mind that
Κ2 = ΚβΚβ, entering in the arguments of χ(Λ1) and χ{2) does not alter m at the
action upon the Ψη functions. Therefore K2 can be ignored. That is why only the
operators entering in (4.7) as coefficients of the χ functions are important. In
fact, the first of them almost coincides with the commutator [Ka, Rx~\ (see (2.16)),
which is employed in the course of the calculation of vso. It is this coefficient that
is responsible for the reduction in the number of operator factors in vso by unity
in comparison to those in
In contrast, the second term in (4.7) raises the
number of operator factors by unity. Due to this, the effect of the operators vso
and r is essentially different. The raise in the number of factors gives rise to the
appearance of extra bands. Since there is no small parameter in the Hamiltonian
(4.2), the intensities of these extra bands and the other bands are comparable.
Calculation for InSb of the effect of inversion asymmetry (Addendum B) and of
warping (Sheka and Zaslavskaya 1969) shows that the indicatrices of the
'original' bands, generated by the vso operator, are not altered due to the second
term in (4.7). The indicatrices of the extra bands are contained in the set of the
indicatrices for the 'original' bands.
This scheme is handy for finding out general properties of the Vx operators
and the selection rules. Concrete calculations can be conveniently performed if
the explicit form of all operators in the basis of the 3tf0 operator eigenfunctions is
used in (3.7).
To determine the indicatrix, corresponding to a given change in AQM Am
and to a given polarization τ (formula (2.12)), let us make use of (B.23). The
explicit relation between the
and Ψ$+α functions in (B.23) is irrelevant, only
the difference between the subscripts being of importance. Therefore it stands to
reason to make this formula more abstract by introducing 'step operators': stepup ( a > 0 ) , step-down ( a < 0 ) and step-zero (a = 0) operators acting on m.
Symbolically they are defined as
4.ΨΜ=>ΨΜ+α.
(4.8)
N o w (B.22) can be rewritten in the operator form
(4.9)
a
the superscript / is defined by formula (3.2). An analogue of (4.9) for the velocity,
expressed via the step operators *>a, defined similarly to (4.8), has the form
(4.10)
α
Formulas (4.9) and (4.10) contain the rule for determining the angular
150
E.I. Rashba and V.I. Sheka
indicatrices of the transitions, induced by small (and, as a rule, not spherically
symmetric) perturbation
':
(i) the operator W should be expanded in the step operators in comformity
with (4.8) in the A' system (Z\\H). This expansion determines the coefficients
(ii) the expansion of the velocity in the step operators is determined by
formula (4.9) where the coefficients are 3t^\
(iii) the AQM values m and m' should be ascribed to the initial and final states,
respectively;
(iv) the transition m->m' in the polarization τ is described by a matrix
element, proportional to ^ _ α ( 0 , φ) with a = m! — m — τ and the intensity of the
transition is determined by the indicatrix
Ω^(θ,φ) = Ω^(θ,φ)
=
\0ι\(θ,Φ)\2,
α = rri — m — τ.
(4.11)
The Ω(ί)(θ, φ) functions are invariants of a group of the crystalline class and,
consequently, their number is infinite. The rank of invariants grows with
increasing power / in the expansion of 2/f' in k. The choice of the set of the samerank invariants should be made separately for a chosen
The Ω functions,
given in sections 5 and 6, may serve as an example.
5 . Three-dimensional
dispersion law
spectrum
with linear terms in the
It is convenient to start the application of the COR theory to concrete
dispersion laws from the case when Jfso is linear in k, i.e., when the expansion of
fi(k) in (2.2) starts with k. Of particular interest is the Hamiltonian describing
carriers in the vicinity of the k = 0 point in noncentrosymmetric crystals having
a symmetry axis of the order no lower than the third. Examples of this are
wurtzite-type crystals (Rashba and Sheka 1959, Casella 1960, Balkanski and
Cloizeaux 1960) as well as sphalerite-type crystals uniaxially strained in the
symmetric crystallographic directions (Bir and Pikus 1972). To construct the
EMA Hamiltonian it is important to bear in mind that carriers are described by
the spinor rotation group representation D 1 /2 corresponding to the angular
momentum J = 1/2. For this representation Pauli matrices are transformed as
components of the pseudovector σ, odd with respect to time reversal. Writing
down 3tf as a sum of invariants, we get
<?ι — rJiQ -ή- « ^ T s ,o
(5.1)
(5.2)
3%0 = δ1(σ χ R)c,
(5.3)
Electric-dipole spin resonances
151
c is a unit vector directed along the symmetry axis. For simplicity it is assumed
in (5.2) that m* and the g-factor are isotropic. This allows us to discuss the
results in terms of the general approach developed in section 4. Besides, this
model is realistic for strained A n iB v crystals and also for a number of hexagonal
A n B v , crystals.
At Η = 0 the energy spectrum has two branches
S±(k)
h2k2/2m*±Slk1,
=
kL = {k2x + k2yyi2,
ζ ||c.
(5.4)
The minimum of energy Smln is achieved on a circumference (on the loop of
extrema) of the radius k0 = m*d1/h2,
Αχ=ηι*δ2/2Η2.
£min=-*u
(5.5)
The velocity operator u, describing all resonance transitions, equals
u = hk/m* + ί i ( c - if') χ φ .
(5.6)
Here
q' = Η2β*φνη*δί9
q' ~ i W O M a . u . ,
(5.7)
where aB is the Bohr radius, and the index a.u. means that the quantity it is
attached to is expressed in atomic units. In (5.6) the first term describes CR, the
second term COR, and the third one EPR. At q\\c the second and third terms
differ from each other only by a numerical factor, therefore all parameters (e.g.,
width and shape of bands) coincide in COR and EPR and their intensities are
related to each other as
W/CR
= (<?') 2.
(5.8)
This ratio increases with increasing q, i.e., with the increasing frequency at which
the resonance is observed. Maximal frequencies at which measurements are
carried out now correspond to wavelengths λ « 100 μιη, i.e., q « 2 χ 1 0 3 c m " 1 in
a crystal. At the values ^ ~ ( 1 0 ~ 2 - 1 0 ~ 3 ) a.u., typical of A „ B VI crystals
(Romestain et al. 1977, Dobrowolska et al. 1982, Ivchenko and Sel'kin 1979,
Pevtsov and Sel'kin 1983) q[ ^ 1 0 " 3. Consequently, COR must dominate over
EPR.
At H\\c the problem is solved exactly (Rashba 1960,1961). The eigenfunctions
are
ψ =
1 m
^ΝΙΨΝ-1
N = mH-i
(5.9)
Here CNl are the coefficients and φΝ are the eigenfunctions of the Landau
152
E.I. Rashba and V.I. Sheka
oscillator. For all Ν > 0 there are two solutions with the energies:
1/2 ^
2u2
+
2m*
h
(5.10)
The index σ = ± 1 numbers the branches of the spectrum. For Ν = 0 there is
only one solution with C01 = 0 and the energy
ft2fc22/2m*.
* 0 ( * J = ito*>c(l - β*) +
(5.11)
At the Zeeman limit, S = σ/2 acquires the meaning of a spin quantum number.
In this case (5.10) is simplified, and after the levels are renumbered, the spectrum
((5.10) and (5.11)) at 0 < jS* < 1 is written as
<$NS(K)
= (N + \)hcoc ± 5Λω 8 + h2k2J2m*.
(5.12)
This spectrum is depicted in fig. 2, which also gives a scheme of transitions in
CRA polarization. This scheme illustrates the universality of the selection rule
Am = 1 for all types of resonances. At moderate magnetic fields, when ha>c ~ hcos
~ Δί9 the arrangement of the levels is much more sophisticated. Nevertheless it
is possible to check that the selection rule Am = 1 also holds for this case.
Using the explicit form of the coefficients Cm one can calculate intensities for
all bands. Then an interesting peculiarity shows up; at the Zeeman limit the
intensity of the CFR band, corresponding to the CFR transition and depicted in
fig. 2 by the dashed line, vanishes. At Ν ~ 1 it is by the factor ~(Al/hcoc) weaker
than the intensity of the EDSR band. This disappearance of the CFR band,
allowed by the selection rules, is at first glance in contradiction with the general
assertions of section 4 and Addendum B. However, this seeming contradiction
is accounted for by the fact that these statements were made for a spherically
symmetric Hamiltonian Jif of a general form (in the case under consideration in
N=2
I
I
ζ—• v
N=l
1
1
I
N=0
<
m = 3,
J,
! .
1
ί
CR
.
COR CFR
Fig. 2. Arrangement of electron energy levels, their classification in terms of the Ν and m quantum
numbers, and quantum transitions allowed in the Faraday geometry (CRA polarization), H\\c. The
figure applies to the case when a>c > co s, g > 0. The short arrows on the right-hand side of the figure
indicate spin orientations.
Electric-dipole spin resonances
153
the H\\c geometry axial symmetry is sufficient). At the same time 3tfL from (5.2)
possesses a specific property: it is quadratic in ic. If we take into account the
nonparabolicity of ^f0, i.e., introduce into it an additional term ensuring
anharmonicity
Jf a =
fooa(F//c2)2,
(5.13)
then the 'accidentally' forbidden transition becomes allowed. The same is true if
we take into consideration the anharmonic correction to 3tfso. The calculation is
analogous to the one given in section 4 for the valence band. It shows that in the
presence of 3tfa there emerges a new contribution to the velocity operator
ensuring nonzero intensity of the transition N\ ->(N -f 2 ) | .
Thus, any violation of the harmonicity of the zero approximation Hamil­
tonian by incorporating (5.3) or (5.13) into it, allows the transition 2ω0 — ω5.
Then the matrix element of the velocity acquires a small factor of the order
Δ Jhcoc or a>Jo)c, respectively.
In the Zeeman limit (section 3) it is possible to find the angular dependence of
matrix elements of the velocity (5.6) in a tilted field H, for instance, employing
(3.16). In this case only F ( 2 1=) 1 and Ff)=
- 1 are nonzero, while Vt do not
involve the Ka operators and therefore are diagonal relative to the Landau
quantum number. Bearing in mind the properties of the Β matrix (Addendum
A), we have (6='cH)
< N | | K +| N i > = - i 2 1 / 2 ^ T ^ r c o s 0,
<JVt|K_|JVj> = 0,
<JVT|K z|iVj>= - y s i n f l
(5.15)
(Rashba and Sheka 1961c). Thus, only EDSR is allowed and CFR is absent.
Equation (5.15) has recently been rederived by La Rocca et al. (1988a, b).
It is of interest to compare this result with the general selection rules
formulated in section 4 and to compare the angular indicatrices. We shall do so
to demonstrate the potential of the general method, using a simple model. For
the case under study, results can also be obtained in a different way; but in
complicated cases (e.g., degenerate valence band) to describe the angular
indicatrices by straightforward calculation is an extremely cumbersome pro­
cedure. At the same time the procedure discussed in section 4 directly yields
angular dependences, and the problem is only to bring them into proper
correlation with electronic transitions.
The SO contribution to the velocity is determined by formula (4.10). To derive
the appropriate expressions for the problem we are investigating, let us start
154
E.L Rashba and V.I. Sheka
with Jfso (5.3), which in the A' system (Addendum A) can be written analogously
to (4.9)
α= - 1
fi^&Pequal
Here
^ o = cos0,
^
^*
1 =
1
=
-i2_
4 = ϊδ^σ χ Κ)Λ = i^K^r
1 / 2
sin0,
(5.17)
- σ α„/£ α,).
(5.18)
7
The subscripts a, a and a" constitute a cyclic permutation from Τ, 0 , 1 , and
properties of ΑΛ as step operators are verified by inspection. The explicit form of
v a +x can be found only provided the explicit form of the J^so operator is used.
Properties of ν Λ τ+ as step operators are dependent on their subscript. Table 1,
relating the polarization of a transient and A Q M change with the angular
indicatrix Ωα = \όα\2 uses these properties.
There is complete agreement between (5.15) and table 1 for transitions
occurring at the frequency ω 8, yet all other transitions illustrated in table 1 are
absent for the Hamiltonian ((5.1)—(5.3)) in the Zeeman limit. From (5.14) it is
clear that incorporation of the anharmonism (5.13) allows the transition
2coc — ω 8 but only in the polarization τ = 1. The transition coc + ω 8 also becomes
allowed. The transition coc — cos becomes allowed but only at τ = 0 , 1 . Angular
indicatrices of these transitions are in agreement with table 1. We should stress
that in contrast to 'extra' transitions in the valence band (section 4), the matrix
elements for both of these 'extra' bands are small over the parameter a)Jcoc.
The example considered illustrates how the generalization of the Hamiltonian
J^0 promotes the appearance in the absorption spectrum of the bands which
should exist according to the general theory but do not exist at a specific form of
(formula (5.2)).
Table 1
Allowed transitions and their angular indicatrices
Am
χ
2
-
Τ
0
1
Ωι
Frequencies
ωε + ω5
1
0
-1
-2
Ωχ
Ω0
Ωι
Ω0
Ωχ
Ωί
Ω,
Ω0
ω8
2wc — ω 8
coc — ω 5
ω 8 — ω0
— 2wc + ω 8
-
-
-
-ω8
The table gives indicatrices Ωτ for all polarizations τ at possible A Q M changes Am = m' — m. At
larger values of \Am\ no transitions occur. The dashes mark forbidden transitions. The bottom lines
give transition frequencies (positive for absorption, negative for emission). The table holds for the
case g > 0.
Electric-dipole spin resonances
155
In principle, the band 3ω0 — ω 8 must also be present since for this band Am
= 2 (i.e. the maximal Am of table 1). However when we choose J^so in the form
(5.3), this band is missing. The reason is that the generalization of J^0 results in
the appearance in the velocity operator of the term, containing a higher power of
k than in ^ f s o, the power being larger only by unity (section 4). From this fact it
follows that the maximal change of Ν (Ν in the Zeeman limit has the meaning of
the Landau quantum number) in the case we're dealing with here equals |AN|
= 2. Therefore the band 3ω ε — ω 8 to which AN = 3 corresponds is forbidden by
virtue of the selection rules with respect to N.
Above we have assumed that m* and the ^-factor are spherically symmetric
(see (5.2)). But in the general case in crystals with the preferred axis c the tensors
of the effective mass and of the g-factor are anisotropic. In this case EDSR
angular diagrams are noticeably more complicated compared to (5.15) (Rashba
and Sheka 1961c).
The most convincing experiments on EDSR, caused by fc-linear terms, were
carried out on η-type InSb samples subjected to uniaxial strain. Bir and Pikus
(1961) were the first to notice the existence of such terms. If we take into account
only the deformation potential C 2 (see table 2, Addendum B), the term in
emerging due to the strain is
(5.19)
A distinctive feature of this Hamiltonian is that it becomes zero when the stress
X is acting along [001], despite the fact that the restrictions imposed by
symmetry alone do not require that J^E = 0 in this case. Kriechbaum et al. (1983)
experimentally discovered that for X||[001] EDSR is very weak, and therefore
they concluded that the most important role is that of the C 2 potential. Their
main measurements were done for X|| [110]. In these conditions J^so is described
by (5.3) with c|| [110] and δ1 o c e x y. These experiments were made in the Faraday
geometry (q\\H) with linearly polarized (or nonpolarized) light. / / | | [ 1 1 2 ] was
tilted with respect to the symmetry axis. In these conditions the transition is
allowed and its intensity is X 2-proportional. This dependence was observed in
experiments at X^ lkbar. A quadratic dependence at small X testifies to the
fact that EDSR excited by fc-linear terms is much stronger than that excited by
k3 terms.
An experiment, somewhat complicated as a result of the technique employed
but equivalent in physical essence, was performed by Jagannath and Aggarwal
(1985). They studied the generation at the frequency ω 3 , using the mixing of two
laser beams with frequencies ωι and ω2 such that ω 3 = ω1 — ω2. The generation
exhibits a strong resonance at ω 3 = ω 8. Its intensity is proportional to the EDSR
intensity. Therefore the three-wave process employed provides an independent
method for studying EDSR. Uniaxially strained η-type InSb crystals with
ΛΊ|//||[111] were used. In this case there also arises the Hamiltonian (5.3) with
Si ccX; Η being oriented along the symmetry axis. The results are collected in
156
E.I. Rashba and VI. Sheka
fig. 3. The band observed at X=0 is ascribed to a magnetic-dipole transition
because in the geometry which was used, an electric-dipole transition is
forbidden for both the inversion asymmetry mechanism (section 6) and
the nonparabolicity mechanism (section 9). At X / O there occurs radiation
with ElH in agreement with (5.15). Its frequency is shifted depending on the
stress, which is accounted for by the g-factor dependence on X, and its intensity
rapidly increases with the stress. As is obvious from fig. 4, the dependence of the
intensity on the stress complies with the law X2 in conformity with the theory.
The situation for p-type InSb subjected to uniaxial stress is much more
intricate due to band degeneracy. The effect of fc-linear terms, induced by the
stress, is partially masked by the fc-linear terms contribution entering in the
valence band Hamiltonian at X= 0. Besides, contributions coming from fc-odd
terms and from the strong nonparabolicity induced by the strain (section 8), are
competing with each other. A thorough analysis of the experimental data
(Ranvaud et al. 1979) pointing to the existence of a few COR bands was made on
the basis of the calculations by Trebin et al. (1979).
Unfortunately there are practically no experimental data on COR caused by
fc-linear terms in uniaxial free crystals. An interesting object for investigation is
\
-
m
\
•j 170 gauss
•
E3IB
Λ
\
\
%
·
Ο
α
d)X=2.24kbar
\
3
\
•\
3
%
52.0
2.
53.0
c)X=l.67kbory
1
IV
\
ί
α) X =0
E 3IIB
s.
•
•I
-
b)X=U2kbar /
i
M§
1\
ι
52.0
/
Λ
\
/ \
\
\
\
% 1
54.0
-· 1
1 V
1
56.0
58.0
60.0
Magnetic Field Β (kG)
>·
1
62.0
64.0
Fig. 3. Far-infrared power as a function of magnetic field (denoted as B) for η-type InSb (n e = 5
geometry. Polarization of the emitted radiation (denoted
χ 1 0 15 c m - 3) at 1.8 Κ in the
as £ 3) is shown, (a) X— 0, the asymmetric shape of the curve is illustrated in the same inset, (b)-(d) The
electric-dipole emission due to fc-linear terms of the Hamiltonian for different values of X(Jagannath
and Aggarwal 1985).
Electric-dipole spin resonances
0
1
2
3
4
157
5
6
CStress X (kbar)]2
Fig. 4. Far-infrared power as a function of X2. The solid line is the least squares fit to the data shown
by dots (Jagannath and Aggarwal 1985).
P b i ^ G e ^ T e crystals experiencing a structural phase transition O h - > C 3 v. In
their cubic modification (at T> Tc) these crystals have a band structure typical
of Pb salts (section 8). Electron and hole bands have four extrema, positioned in
the L points. After the phase transition (T<TC) the symmetry axis is directed
along [111]. As a result, three extrema remain equivalent to each other but are
positioned in very low symmetry points (Gk = C s), whereas the fourth extremum
is on the symmetry axis (Gk = D 3 d) . This situation is reminiscent of Bi (see
section 8) but with the difference that there is no inversion centre in the lattice
having the symmetry C 3 v, and therefore the bands for arbitrary k are not
degenerate. Thus, /c-linear terms appear in the spectrum. EDSR was observed by
Fantner et al. (1980) at χ = 0.01. Calculations by Bangert (1981) show that
EDSR becomes considerably stronger at transition to the rhombohedral phase.
6. Inversion
asymmetry
mechanism for η-type InSb
bands
In crystals with sphalerite symmetry the expansion/(/r) starts with k3. The COR
theory for this case was formulated by Rashba and Sheka (1961a). Here, like
158
E.I. Rashba and V.I. Sheka
in the systems described in section 5, the spinor representation D 1 /2 is acting.
In the reference system formed by the crystallographic axes, apart from the
pseudovector of the T d group constituted by Pauli matrices, there is another
pseudovector whose components are
(6.1)
where
/ and / ' form a cyclic permutation. This notation already implies
noncommatativity of the operators &} (2.5). The electron Hamiltonian equals
J f = h2P/2m*
+ ^μΒ(σΗ)
+
δ3(σκ).
(6.2)
Before calculating the explicit form of the matrix element of the velocity by
means of the formulas from section 3, let us find out, using the rules of section 4,
possible transitions and their intensity indicatrices. The Hamiltonian (6.2) is
approximate and can be derived from the Kane Hamiltonian by projecting it
onto the conduction band (see (B.l), (B.2) and (B.4)). In the two-branch
approximation, eigenvalues of the Hamiltonian J^0 can be chosen as
ψ(+)
Ύ
N+l/2
«/(-)
Ύ
_
-
JV- 1/2 ~
Φν
= |NT>=»|N + i > ,
0
0
s|Ni>=>|N-i>,
(6.3)
Φν
The arrows here mark transformation to the notation in terms of the AQM
formalism.
The notation in terms of the Landau oscillator eigenfunctions φΝ is
convenient since it clearly points to possible transitions between Landau levels.
On the other hand, a general analysis is conducted in terms of A Q M m, which in
accordance with (6.3) can be easily brought into correlation with N:
m= Ν + \
(for spin-up),
m= N —\
(for spin-down).
(6.4)
The probability of a spin-flip transition +
— with a change in the Landau
levels Ν -»N' is determined by the velocity matrix element
<1ΛΠΚτ|ΛΤΤ>=><ΛΓ' ~ ί\Κ\Ν
+ i > = <m'|Kt|m>.
(6.5)
In this form the matrix element of Vx is defined in the basis of the functions with
a given AQM, and therefore the rules of section 4 are applicable to it. In
particular, its angular dependence is determined by the coefficient ^ ( _ i ( 0 , φ)
with a = m' — m — τ = N' — Ν — τ — 1. The intensity indicatrices in conformity
Electric-dipole spin resonances
159
with (4.11) and(B.15) are
ΩΛ = \α.α(θ9φ)\2
= \αα(θ9φ)\2
(6.6)
(the superscript 3 is dropped). They are equal to
Ω0 = 9Ι0,
Ω2 =
Ι2-3Ι0-Ι19
Ωλ=21ΐ9
Q 3 = f ( / 1 + 8 / 0) .
(6.7)
Here 7 0, J x and I2 are cubic harmonics
h2h2h2,
I0 =
h = h2x(h2 - h2z) + h2(h2 - h2x) + h\(h2x 6
x
6
y
h%
6
z
I2 = h + h + h ,
(6.8)
and ^ = HJH. Formulas (6.7) and (6.8) follow from (Β. 11)-(B. 14).
Possible COR transitions in η-type InSb are given in fig. 5. The general
analysis also indicates a possibility of transitions with\Am\ = \AN\ ^ 4. In fact, |τ|
^ 1 and in (B.23) |α| ^ 3. Transitions with |AiV| > 2 are possible only as long as
the dispersion law is nonparabolic (cf. section 5). Therefore in the conduction
band these transitions are weakened (but in the valence band there are no
restrictions for their intensity (cf. section 4)). Thus, for the Hamiltonian (6.2)
spin-flip processes can be accompanied by transitions with AN = 0, ± 1 , ± 2
only. They occur at frequencies ω5 = ω0\β% ω0(1 ±β*) and coc(2 + /?*), if as
usual \β*\ < 1. In contrast to CR and EPR, COR is, as a rule, observable in all
three polarizations. The COR intensity is strongly anisotropic whereas the CR
and EPR anisotropy is weak and is caused by effects not considered above
(warping, etc.).
Let us now calculate the matrix elements of the velocity. If the mean energy of
an electron is of the order ftcoc (3.4) and (m*<53)a u % 1 (this holds for n-type
InSb, see section 7), the condition γ <^ 1 (3.1) is fulfilled even in strong magnetic
fields Η « 5 χ 1 0 4 G. Thus, one can make use of the results of section 3.
The nonzero coefficients F$ equal
^•=1,
F%r=-l.
(6.9)
Now use formula (3.16). The products of this formula have two operator factors
applicable to the case under study and the matrix elements of the velocity are
<N'i\K\m)
=
_~f *_
τ
ΣN Β(τβγ)<
Ν'\αβα7\Ν>.
1) Q (fi
The coefficients Β{τβγ), which are symmetric with respect to all subscripts, are
expressed via ternary products of the elements of the matrix Β (Addendum A),
B{m) being zero. Among the other coefficients there are only four independent
E.I. Rashba and V.I. Sheka
160
m
-\
-%
-%
N=3
N=2
-X
^
N=1
-/
3
2
N=0
1
ΔΠ1
Τ V.
3
2
1
0
-1 -2 -3 -4
1
V2
Q, Q 2Q , Q . a
1
V.
Q 2 Q, Q 0 Qi Q. Q s
^
EPR
Q, Ω, Q, Q„ Q, 0» Q 3
0
-1
Q 2Q ,
CR
Fig. 5. Scheme of COR transitions in η-type InSb for the inversion asymmetry mechanism {g < 0).
Under the transition the respective values of Am and of the intensity indicatrix for all polarizations τ
are given. The transitions with AN = 3 and AN = 4 are forbidden in the Zeeman limit for the
Hamiltonian (6.2). Polarizations in which CR and EPR are excited are also indicated, their
indicatrices being isotropic.
coefficients (Rashba and Sheka 1961a). The fact that there are just four of them
is dictated by formula (B.23). &α(θ, </>), entering in (4.10) as factors at the
corresponding step-operators, are expressed via these coefficients according to
(B.16).
The theory predicted that the EDSR and CFR intensities would exceed that of
EPR (Rashba and Sheka 1961a,b), and this was confirmed experimentally. The
latest data show δ 3 = —56 a.u. (section 7). The EDSR induced by fc3-terms was
first discovered in η-type InSb by Dobrowolska et al. (1983) in the longitudinal
e\\H polarization (fig. 6). The angular dependence of the observed resonance
intensity at a rotation of the specimen around the wave vector q at #||[100] in
agreement with the theory is perfectly described by the function Qx. This
convincingly proves that fc3-terms, i.e., the inversion asymmetry mechanism, are
responsible for the observed EDSR.
161
Electric-dipole spin resonances
1.0 -
#
0.8
,.ϊ
\
I
/
0.6
/
1
I
I
1
1
1
-8i
[oTo]
*
ι
\
1
0.0
i
i
0.4
0.2
\
1
/
-
E
„ο
/>
/'8\
ι
f
|
i
!l
t
I
1
\
X
X
ό
ι
[on]
w
[ooi]
1
[on]
\
[010]
ORIENTATION
Fig. 6. EDSR intensity in η-type InSb as a function of the orientation of Η in the (100) plane. Black
and white dots indicate opposite orientations of the field. The dashed line is the theoretical depen­
dence for this plane (Rashba and Sheka 1961a), normalized to the experimental data for H\\ [ O i l ] .
The data were observed at 4.5 Κ at 118.8 μπι on a sample with nc = 3.6 χ 10 1 cm , 3 4 mm thick
(Dobrowolska et al. 1983).
7. EDSR and EPR
interference
It follows from (3.5) that simultaneous excitation of EDSR and EPR is possible,
but their interference occurs only if certain conditions are satisfied. The
interference term in the absorption spectrum is proportional to the correlation
function < ^ ( 0 ^ ι ( 0 > · Statistical averaging in this function involves in­
tegration in Kz. The integral is nonzero only if jfte and Stm have the same parity
with respect to k. Consequently, interference is possible only if there are terms
which are odd with respect to /c in <?fso(£). The interference is strong if # c and Jf m
do not strongly differ in magnitude. These two conditions are fulfilled in n-type
InSb where
oc fc3 and the EPR intensity due to a large value of the g-factor
( g » —50) is high. The theory was developed by Sheka and Khazan (1985),
Chen et al. (1985b), and Gopalan et al. (1985) on the basis of the work by
Rashba and Sheka (1961a).
The first COR experiments were carried out in the millimeter wavelength
range, and therefore a natural way to distinguish between EDSR and EPR was
to put a specimen into positions where either Ε or Η reach their maximum (Bell
1962, McCombe et al. 1967). In observing the spin resonance in the infrared
wavelength range [λ ~ 100 μπι), as has been done in recent experiments, such
spatial separation is of course impossible, so EPR occurs on the background of
EDSR. This is what creates the possibility of their interference.
162
E.I. Rashba and V.I. Sheka
In the Faraday geometry, in neither of the circular polarizations is there
interference: in one of them EPR is not excited, while in the other the difference
in phases of the matrix elements of Jf e(i) and &m(t) equals π/2. In the
longitudinal Voigt polarization (q±H\\E)
the SR intensity anisotropic part is
proportional to (Sheka and Khazan 1985):
Σ*
I.(9, h) = Ω,(Η) -
Μ*)·
(7-1)
The first term describes EDSR, the second EDSR and EPR interference. The
isotropic term, responsible for EPR, is dropped. A similar result was obtained by
Chen et al. (1985b). In the geometry of fig. 6 the interference term vanishes.
The second term in (7.1) is odd with respect to q and H, therefore Is varies at
separate inversion of both q and H, but remains unaltered at their simultaneous
inversions, which is in essence the effect discovered by Dobrowolska et al.
(1983). It is illustrated in fig. 7. Rotation of the specimen by 180° corresponds to
the reversal of q or H. The dependence of the spectrum on the sign of q, i.e., a
strong spatial dispersion, is at first glance quite unexpected at such a large
wavelength of the light ( ~ 100 μιη). This dispersion is caused by the EDSR and
EPR interference (Dobrowolska et al. 1983). The influence of the interference
upon the angular indicatrix is shown in fig. 8. It is noteworthy that the
1
" V
Q
b'
Λ
41.0
c
42.0
. 41.0
y .4 2 . 0
Β (KG )
Fig. 7. Variation of the EDSR spectrum in η-type InSb at rotation of the sample, observed in the
longitudinal Voigt geometry at 118.8 μιη and 4.5 Κ with ne = 2.3 χ 1 0 14 c m - 3. The sample faces are
in the (110) plane, (a) EDSR for H\\ [ Π 0 ] , q || [110]. (b) The sample has been rotated by 180° about q
relative to (a), (c) The sample has been rotated by 180° about Η relative to (a), (d) It has been rotated
by 180° about q χ Η relative to (a). The sequence (a')-(d') corresponds to configurations (a)-(d),
respectively, but with the magnetic field reversed. In each resonance doublet the higher-field,
stronger line is the free-electron EDSR, and the weaker line is E D S R of donor-bound electrons
(Dobrowolska et al. 1983).
Electric-dipole spin resonances
163
Fig. 8. EDSR intensity as a function of the orientation of Η in the (110) plane for tf||[110]
(longitudinal Voigt geometry). Black and white circles correspond to opposite signs of / / ,
respectively. The solid line is the theoretical angular dependence of EDSR (Rashba and Sheka
1961a). The dashed curves are guides for the eye, connecting experimental points (Dobrowolska
et al. 1983).
interference is strong although the ratio of the EPR intensity to the EDSR
intensity does not exceed 0.02.
It is very important that this interference opens up a unique possibility for
finding not only the magnitude but also the sign of δ3 from SR. To help us
understand the situation it is essential that in the T d group the [111] axis be
polar, i.e., the directions [111] and [TTT] must be physically nonequivalent. The
manifestation of this fact is that the opposite faces of a crystal are not equivalent.
So, for a conventional choice of the direction [111] the face whose external
normal is [111], consists of group III atoms, whereas the opposite face consists
of group V atoms. Therefore these faces exhibit different behaviour when the
sample is etched; this is how they were specified by Dobrowolska et al. (1983).
The choice of a reference system which is consistent with this definition of the
(111) face is unambiguous. It is clear from the arrangement of surfacial atoms on
this face (Gatos and Levine 1960) that if the origin is chosen at a site where a
group III atom is positioned, then one of its nearest group V neighbours lies in
the first octante. Therefore the sign of δ3 has an absolute sense.
First Rashba and Sheka (1961b) estimated δ3 as |<53| % 200 a.u.; McCombe
(1969) found the upper limit for \δ3\ ^ 5 0 a.u., Sheka and Khazan (1985), by
processing the data given in fig. 8, obtained δ3 % — 75 a.u.; Chen et al. (1985b)
and Gopalan et al. (1985), using the whole variety of experimental data, found
that |(53| ?s 56 a.u.; and Cardona et al. (1986a), after performing numerical
calculations, concluded that δ3 = 54 ± 3 a.u. The difference in sign is accounted
164
E.I. Rashba and V.I. Sheka
for by the choice of the opposite reference system. In their following paper,
Cardona et al. (1986b) gave an experimental value of δ3 = 56 ± 3 a.u.
8. COR in semiconductors
with inversion
centre
The COR mechanism considered in sections 5 and 6 is missing in crystals with
an inversion centre. This mechanism is caused by the splitting of bands in the
vicinity of the point k = 0, described by formulas (2.2) and (3.2). In crystals with
an inversion centre, all bands are twofold degenerate in the entire A>space
(Elliott 1954), therefore splitting of the kind which was discussed in sections 5
and 6 is absent. That is why we must take into account other COR mechanisms,
usually induced by higher order terms in k in J^so. Such terms can be constructed
using the method of invariants. Some of these terms can be conveniently
interpreted as the dependence of the g-factor on H.
The COR theory for crystals with the inversion centre was first formulated by
Boiko (1962) for electrons in Si and Ge. In Si the minima of the band are located
on the <001> axes in the general position points. In these points the wave vector
group is Gk = C 4 v. The situation is akin to the one in wurtzite (section 5) in the
sense that EDSR is induced by fc-linear terms in J^so. However, an important
difference is that in Si these terms are //-proportional. Therefore EDSR is much
weaker than in wurtzite-type semiconductors. According to the estimates made
by Boiko (1962), / E D RS ~ / E P .R EDSR must be present in all polarizations. In Ge
the minima are on the boundary of the Brillouine zone in the < 111 > directions
and G* = D 3 d. J^so is linear in Η and quadratic in k. The operators r and ν are
linear in /c. EDSR and the electric-dipole CFR must have comparable intens­
ities, which according to Boiko's estimates (1962) may exceed the EPR intensity
by one order at an electron concentration of ~ 1 0 1 4c m " 3 . With increasing
concentration this ratio must increase. As far as we know, COR has not so far
been observed for band electrons either in Ge or in Si.
The
dependence on k is particularly strong in the presence of narrow gaps
in the spectrum. In this respect it is helpful to study p-type Ge subjected to
uniaxial strain. The strain cancels out the fourfold degeneracy at the top of the
valence band, the gap 2ε'0 in the spectrum being proportional to the stress T. On
the basis of the Hamiltonian derived by Bir and Pikus (1959) the theory was
constructed by Gurgenishvili (1963) for H\\ T\\ [001] and by Hensel (1968) for
#11 T\\ [111]. In the latter case, if ε'0 > η (where η is the Fermi energy of holes)
one can obtain an effective two-branch Hamiltonian J^so by mapping the 4 χ 4
hole Hamiltonian onto a subspace with the angular momentum projection
± 1/2, corresponding to the upper branch of the strained crystal spectrum:
*„αζ¥^%£{£_σ
++Κ
£
o
kH
σ-).
+
(8.1)
Electric-dipole spin resonances
165
Formula (8.1) is written in the reference system, associated with H. It is clear
from this formula that in the longitudinal E\\H polarization there are only
transitions with ω = co c — co s. The experimental data obtained by Hensel (1968)
are exhibited in fig. 9. The solid curves are a result of the exact diagonalization of
the 4 χ 4 Hamiltonian, which was indispensable since the spectrum of holes was
noticeably nonequidistant. This is obvious from the great difference in the
frequencies of transitions of the same type. The high precision of the experi­
mental data and their thorough processing made it possible for the first time to
find the g-factor of holes in Ge.
Schaber and Doezema (1979 a, b) observed EDSR in η-type PbTe in the
Faraday geometry. PbTe is a narrow-gap direct semiconductor with a lattice of
the NaCl-type. The extrema of the bands are located at the L points (on the
boundary of the Brillouine zone). The isoenergetic surfaces are almost ellipsoid­
like and differ only slightly for electrons and holes. It is natural to expect that in
such a system the nonparabolicity mechanism will be dominating in EDSR
(section 9). However, the authors assert that the EDSR intensity observed was
Ge
H0J Kill]
\ \
\
\
\
/
1(h-BRANCΗ
^u^L-—
(0,+)
(1,+)—(2,-)
+
)—-*
-(4,-)
yo,-)—(Ζ,Λ
r
STRAIN PARAMETER, X =
Fig. 9. Positions of COR bands for p-type G e as a function of the dimensionless strain parameter
χ', Γ, H, £ | | [ 1 1 1 ] , the temperature is 1.2K, v = 5 2 G H z . The resonance frequencies are expressed
in terms of the 'effective mass'm*; m is the electron mass in vacuum. Quantum numbers, cor­
responding to the strong strain limit (in contrast to the figure in the original paper, Hensel (1968)),
label energy levels. The '/c H-branch' is interpreted as a transition (0, + ) - > ( l , —) for electrons with
large values of Kz.
166
E.I. Rashba and V.I. Sheka
much higher than implied by the theory. The dominating EDSR mechanism
remains rather obscure.
Of great interest is the mechanism of spin transitions in Bi and Sb. Although
there is an extensive literature on the electronic properties of these materials, the
mechanism of spin transitions in them remains rather vague. The Bi lattice
results from a parent cubic lattice due to a slight trigonal deformation. The
result of this deformation is a semimetallic spectrum with three electron pockets
and one hole pocket (Abrikosov and Fal'kovskii 1962). The electron pockets are
positioned at the low symmetry points of the Brillouine zone and the hole
pocket is located at the high symmetry point. Fermi surfaces of electrons and
holes resemble strongly prolate ellipsoids; the presence of small masses in the
spectrum indicates the existence of narrow gaps. The two-band Cohen-Blount
model (1960) satisfactorily describes certain properties of electrons in Bi. How­
ever, the shape of the Fermi surface shows considerable deviation from this
model (McClure and Choi 1977).
Observations of the resonance at the frequencies ω 8 and ω0 ± cos on electrons
in Bi and Sb were first reported by Smith et al. (1960). However, the assignment
of bands proposed by them was later rejected. A new announcement of the
observation of SR and CFR bands (the frequency coc — ω δ) was made by Burgiel
and Hebel (1965). Due to nonparabolicity, they observed several bands of each
type; their intensities were much lower than the CR intensity.
For electrons in Bi, the following competing mechanisms were discussed:
(i) due to the narrow gap, the nonparabolicity mechanism may be of
importance. This theory was formulated by Wolff (1964) on the basis of the
Cohen-Blount model, in close analogy with the theory of the Dirac electron in a
magnetic field;
(ii) the influence of other adjacent bands is possible (Yafet 1963);
(in) due to its low symmetry, the electron Hamiltonian must involve terms of
the order Hk, which have a lower order in k than the nonparabolicity
mechanism.
As far as we know, there are not yet sufficiently detailed experimental data to
make it possible to find out the dominating COR mechanism for electrons.
Thanks to the study by Verdun and Drew (1976), the situation pertaining to
holes in Bi is now much better understood. They showed that the theory based
on the two-band Hamiltonian (Wolff 1964) cannot adequately describe the
experimental results. The interpretation they proposed is based on the EMA
Hamiltonian involving terms up to /c 4. The coefficients attached to these terms
were found from the calculations made by Golin (1968) and then corrected
within the error-rate admissible for the theory, to get agreement with the
experimental data. This procedure makes it possible to take into account the
real band structure, including several adjacent bands. As a result, Verdun and
Drew (1976) succeeded in describing versatile experimental data on EDSR and
on the electric-dipole CFR in the Faraday and Voigt geometries, including
Electric-dipole spin resonances
167
angular indicatrices of the position and of the intensity of the bands.
McCombe et al. (1974) reported that they observed in the far infrared
spectrum of the η-type B i 0 8 8 S5 b 0 1 51 alloy a band which they assigned as the
electron EDSR with ω 8 > a>c.
9. COR in narrow-gap
and zero-gap
semiconductors
This section is devoted to COR in narrow-gap semiconductors of the InSb type
and in zero-gap semiconductors of the HgTe type. It is natural that the theory
for such semiconductors should be constructed on the basis of the Kane model
(Addendum B). In the framework of this model, by introducing a relatively small
number of parameters, one can describe a wide variety of electron properties of
crystals, including the strong nonparabolicity effect induced by a small value of
EG. It is very important that the Kane model enables one to find Landau levels
with high accuracy in the conduction and valence bands. It is possible to find a
set of parameters entering the Hamiltonian if the theoretical positions of these
levels optimally match the relevant experimental data. The Landau quantiz­
ation theory for the Kane model was developed in papers by Bowers and Yafet
(1959) and Lax et al. (1961). Pidgeon and Brown (1966) and Pidgeon and
Groves (1969) proposed the formulation of the theory which is now used as
standard. The most complete form of the 8 x 8 Hamiltonian, involving fc-linear
and fe-quadratic terms, was proposed by Weiler et al. (1978). Numerical values
of the ten parameters of this Hamiltonian for InSb are contained in the paper by
Littler et al. (1983), where they also carried out comparisons with results
obtained by other workers. The values of most parameters are now definitely
known, although certain parameters still need some improvement (Chen et al.
1985a).
The theory of COR arising due to nonparabolicity mechanism, was put
forward by Sheka (1964) for InSb and later by Kacman and Zawadzki (1976) for
zero-gap semiconductors. In each of these papers the authors employed the
simplest form of the 8 x 8 Hamiltonian which allowed an exact analytical
solution for the problem to be found. In this form, of all the nondiagonal terms
only the P-proportional terms are retained. Numerical calculations of COR
intensities based on a more general form of the 8 x 8 Hamiltonian were initiated
by Bell and Rogers (1966). N o w such calculations have become conventional. It
is noteworthy that in order to find the position of energy levels, which are
experimentally measured with high precision, numerical calculations are indis­
pensable, particularly for the valence band. Even for the conduction band such
calculations are justified if we are dealing with such weak effects as the energy
dependence of the g-factor anisotropy (Ogg 1966, Chen et al. 1985a). However,
for calculating COR intensities, especially for transitions between electron levels,
sufficient accuracy can be achieved in the framework of the two-branch model.
168
E.I. Rashba and V.I. Sheka
By transforming the 8 χ 8 Hamiltonian into a 2 χ 2 Hamiltonian it is possible
not only to simplify the calculations but also to get an adequate idea of the
physical mechanism of the resonance formation.
Transformation of the 8 x 8 Hamiltonian into a 2 χ 2 Hamiltonian is realized
by means of the standard projection procedure (cf. section 3). The k3 term in J^so
with the coefficient δ3 (ΒΑ) in the 2 χ 2 Hamiltonian (6.2) stems from the Pk and
Gk2 terms of the original Hamiltonian. The role of the k3 term in COR was
considered in section 6. The next power term in the expansion containing σ
originates from nondiagonal P-proportional elements of the 8 x 8 matrix. This
term is
(9.1)
where m* and g are determined by formulas (B.l) and (B.2). Since Η oc fc2,, J^so
must be regarded as a quantity of the order of fc4. If we compare formula (B.4)
for <53 with (9.1), it becomes clear that the large factor gm0/m* enters in (9.1).
Therefore although the k3 terms in the EMA formalism are lower powers in
comparison with the fc4 terms, the latter are comparable with them at relatively
small values of k. This means that the fc4 terms are relatively large.
It is important to understand the physical meaning of J^so (9.1). The operator
J^so is diagonal if taken between the eigenfunctions of the operator J^0 (3.9). This
can be regarded as a correction to the g-factor due to nonparabolicity (cf. (B.2)).
Since J^so is diagonal, it cannot excite electric-dipole transitions, and in
particular, it cannot excite COR. The term of the order of ( £ 2 ) 2 , also entering the
Hamiltonian, does not cause spin transitions either. This term can be regarded
as a correction to m* due to nonparabolicity (cf. (B.l)). Thus the nonparabolic­
ity terms in the 2 x 2 Hamiltonian, considered here, do not lead to COR.
Nevertheless, COR does occur but its origin is different. If the projection
operation is performed by means of the Luttinger-Kohn procedure (1955), the
matrix t is determined by formula (3.7a). The perturbation is the terms of the
8 x 8 operator containing P. The transformed operator of the coordinate r is
calculated according to (3.8b)
r = e x p ( f )r exp( - f) « r + [ f , r] + \
(9.2)
Since r can be regarded as a diagonal operator (see the end of Addendum B) and
vso contains only interband terms (table 2), then [T, r] does not contribute to the
2 x 2 operator. Calculation of the second commutator yields
(9.3)
This formula was derived by Yafet (1963). The SO contribution to the velocity
stems from communication of (9.2) with 3tf0. Thus, the COR mechanism, which
Electric-dipole spin resonances
169
is conventionally called a 'nonparabolicity' mechanism, in this case is ensured
not by the nonparabolicity terms in the Hamiltonian but by the SO contribution
to the coordinate operator.
The relation (9.3) is spherically symmetric, and therefore the selection rules
entail from the angular momentum conservation. Since in η-type InSb g < 0,
EDSR is excited in the CRI polarization. For the same reason in η-type InSb in
the longitudinal polarization a transition is excited at the frequency coc + ω 8,
AQM at this transition remaining unaltered.
COR at the frequency coc + ω 8 in η-type InSb was discovered by McCombe et
al. (1967). The experiments were carried out in the range 3 0 0 - 6 5 0 c m - 1,
ensuring the fulfillment of the weak scattering criterion ωτ > 1 (where τ is the
relaxation time). The experimental data collected in fig. 10 show that COR is
observed in agreement with the theory only in the longitudinal Voigt polariz­
ation. In the conditions of the experiments conducted by McCombe et al. (1967)
the nonparabolicity mechanism for the band co c 4- co s is much more efficient than
the inversion asymmetry mechanism. One can check this by using the value of δ3
cited in section 6 and the formulas for the COR intensities for both mechanisms
(Rashba and Sheka 1961a,b, Sheka 1964). Convincing arguments in favour of
the dominating role of the nonparabolicity mechanism are given in the article by
McCombe (1969). These arguments are based on the dependence of the intensity
upon the orientation and magnitude of H. The paper by McCombe et al. (1967)
has been of major importance for COR studies, since the authors were the first
to discover CFR, i.e., spin transitions induced by the a.c. electric field were
observed in conditions satisfying the basic criterion of the theory ωτ > 1 (in an
earlier paper of Bell (1962) EDSR was discovered in conditions where this
100
Η (KG)
Η (KG)
Η (KG)
Fig. 10. Transmission spectra obtained at 500 c m " 1 and 6 Κ for the 9.25 mm thick sample of InSb
with carrier concentration ne = 2 χ 1 0 14 c m - 3 and carrier mobility 5 χ 10 5 c m 2/ V s at 80 Κ. Η is
the magnetic field, and Ε is the a.c. electric field. The doublet structure of the transition reflects the
presence of both free and localized electrons (McCombe et al. 1967).
170
E.I. Rashba and V.I. Sheka
criterion was violated; cf. section 13). In the same paper COR was used for
determining the g-factor for the Ν = 1 level. The energy of a quantum absorbed
in COR is
hojCOR = hcoc + ^BHlg(N
+ 1, H) - g(N9 H)l
(9.4)
Since coc(H) and g(N = 0, H) were known from the data on CR and SR, it
became possible for the first time to experimentally find g(N = 1, if) from CFR.
Since then COR has been systematically used to study g-factors (McCombe
(1969), Appold et al. (1978) in InSb, Pascher (1981) in PbTe, etc.). Some time
later McCombe and Kaplan (1968) observed in the CFR spectrum a distinct
pinning in the region of the resonance c o c = c a L O, and were able to find the
constant of the coupling of electrons to optic phonons (Johnson and Larsen
1966). Observation of the pinning in the CFR spectrum is more reliable than in
the CR spectrum, because the frequency at which the measurements are
performed is remote from the reststrahlen region.
It is clear from (9.3) that matrix elements of EDSR and CFR differ by the
factor KzrH <ζ 1. Consequently, / E DSR/^CFR ~ (h°>c) 1max{f/, Τ} <ζ\. This
inequality was well fulfilled under the actual conditions of the experiment by
McCombe et al. (1967). Therefore, EDSR was weak and was not observed.
Later, McCombe (1969) reported the observation of this transition at Τ = 80 Κ,
where the probability of transition, judging by the afore-given estimate, must be
higher. The temperature dependence of the shape of this band was also
investigated (McCombe and Wagner 1971). Temperature dependence was
observed in the transverse CRI polarization and its origin is ascribed to the
nonparabolicity mechanism.
COR in the valence band is much more difficult to interpret in great detail,
but at the same time is much more informative for finding numerical values of
the parameters. COR in the valence band was studied in p-type InSb by Littler
et al. (1983) and fig. 11 presents the experimental data for the strongest COR
bands. The best fit of the theory with the experiment made it possible to improve
the values of a number of the parameters of the 8 x 8 Hamiltonian. The
difference in position of the resonances at / / | | [ 1 1 1 ] and at / / | | [ 1 0 0 ] testified to
the anisotropy of hole states.
In its electron properties, H g x _ xC d xT e with χ > 0.16 is analogous to InSb (at
χ < 0.16 its band structure is inverted). However, EG is much smaller than in
InSb (EG « 60 meV at χ % 0.2) and due to this the nonparabolicity mechanism is
much stronger. McCombe et al. (1970a) discovered EDSR in this material in the
CRI polarization. The g-factor changed from —200 to — 100 with Η increasing
up to 50 kG, in agreement with the theory (Bowers and Yafet 1959). The
resonance intensity fell with increasing Η according to H'1 in agreement with
the predictions of the theory based on the nonparabolicity mechanism, and its
absolute value was also found in agreement with the theory (Sheka 1964). Some
time later the same team (McCombe et al. 1970b) discovered and studied COR
Electric-dipole spin resonances
171
B(kG)
Fig. 11. COR transition energies, calculated from the 8 x 8 band model (solid lines) and the
observed free-hole transitions (dots). Β is the magnetic field. The numbers correspond to the
(0);+ 2. a+{\)-fc+(0),
3. b~(2)-a+(l);
4. Z>"(3)
following transition assignment: 1. b~{\)-a
- a +( 2 ) ; 5 . f l - ( l ) - f e +( 2 ) ; 6 . f e - ( 4 ) - * a +( 3 ) ; 7. a " ( 2 ) - f c +(3); 8. 6 " ( 5 ) - a +( 4 ) ; 9. a " ( 3 ) - f c +(4); 10.
(5);
+
11. a " ( 4 ) - f c +( 5 ) ; 12. fc"(7)-a+(6); 13. < T ( 5 ) - & + (6); 14. fc"(8)->a+(7); 15. a"(6)
fe-(6)->a
- b +( 7 ) ; 16. &-(9)->a +(8); 17. c T ( 7 ) - > b +( 8 ) ; 18. b " ( 1 0 ) - > f l +( 9 ) ; 19. a _ ( 8 ) - 6 +(9); 20. J T ( l l )
(10);
+
22. 6 ~ ( 1 2 ) - > a +( l l ) . The designations are given as in the article by
- > a +( 1 0 ) ; 21. a~(9)-+6
Weiler et al. (1978) (Littler et al. 1983).
in η-type Hgi .^Cd^Te at the frequency coc + ω 8. This resonance was also excited
by the nonparabolicity mechanism. COR in H g x .^.Cd^Te was also observed by
Golubev and Ivanov-Omskii (1977).
COR studies were also conducted in materials with the sphalerite lattice
which has an inverted band structure. Pastor et al. (1981) investigated pure
samples of HgSe with the electron concentration 4 χ 1 0 16 c m - 3 at T = 4 Κ by
measuring the transmission in the far infrared range. The EDSR band was
discovered both in the longitudinal (E\\H) and transverse (ElH)
polarizations.
This unambiguously points to a considerable contribution of the inversion
asymmetry mechanism. Actually, angular indicatrices are universal, i.e., inde­
pendent of the dimensionality of the Hamiltonian (section 4) and therefore it is
possible to employ the arguments given above for the 2 x 2 Hamiltonian.
Approximately describing electrons by this Hamiltonian, Pastor et al. (1981)
estimated the asymmetry coefficient as <53 » 300 a.u. for the electron band. This
value is 6 times as large as the one in InSb (section 7). COR bands were also
172
E.I. Rashba and V.I. Sheka
observed. The authors think that it is possible to explain the complexity of the
observed bands by assuming that the contribution coming from the inversion
asymmetry mechanism is large. Weiler (1982), after analyzing the experimental
data obtained on HgSe, came to the conclusion that this mechanism is
dominating for HgSe.
H g 1 _ x M n ; cT e with χ = 0.03 was investigated by Witowski et al. (1982). This
material is also a zero-gap semiconductor. EDSR was observed at ElH and
CFR was observed at the frequency co c + ω 8 at E\\ H. Both resonances were used
for studying the dependence ω 8 ( Τ ) in this semimagnetic semiconductor.
Tuchendler et al. (1973) studied properties of HgTe, which is also a zero-gap
semiconductor, in the submillimeter frequency range. The experimental results
were compared with the results of the calculation of the energy spectrum in the
8 x 8 model and its parameters were found on the basis of this analysis. In the
Faraday geometry COR was observed at the frequency a> c + a> s.
In semimagnetic H g 1_ xM n J CT e with x ^ O . l a gap opens ( £ G^ 5 0 m e V ) .
This interesting system has not yet been sufficiently studied but observations of
EDSR and CFR (Stepniewski and Grynberg 1985) as well as CFR involving
spins of Mn ions (Stepniewski 1986) have been reported. (Concerning possible
excitation mechanisms for the latter bands see the end of section 14.)
The energy spectrum of zero-gap semiconductors with the inverted band
structure is modified in the presence of a slight tetragonal deformation. This
situation, according to Bodnar (1978), is inherent in C d 3 A s 2 and is described by
the generalized Kane model. CFR transitions at the frequencies coc ± cos were
observed by Thielemann et al. (1981) in the longitudinal Voigt geometry at H\\c,
c is a unit vector along the tetragonal axis. The COR theory for this band
structure was formulated by Singh and Wallace (1983).
10. COR on shallow local
centres
We shall focus our attention here on large-radius impurity centres while sticking
to our general concept, i.e., establishing a correlation between specific COR
mechanisms and the respective terms of the EMA Hamiltonian. Here, as in the
case of band carriers, EDSR and CFR are possible. In strong fields the orbital
quantum number, changing in CFR, may be both a Landau level number with
which the Coulomb spectrum of the centre is related, and a number of the level
in this spectrum.
Let us start with EDSR. The first problem is whether the electron binding in
the impurity centre affects the EDSR intensity. If it does, the effect should be
especially strong in a weak field when ha>s <^ &l9 Sx is the ionization energy of the
centre. Consider a situation typical of donors. If the band is degenerate with
respect to the spin only, then in the limit Η 0 there is only Kramers degeneracy
in the ground state of the centre. The two spinor functions, belonging to the
Kramers doublet, are Ψ and ΚΨ, where Κ is the time reversal operator:
ΚΨ = ayW*. Writing out the matrix element of the coordinate r corresponding
173
Electric-dipole spin resonances
to the electric-dipole transition between these states, and applying the operator
Κ to this element, in accordance with the well-known rules (Wigner 1959), we
get
(ΚΨ,?Ψ) = (&Ψ,Κ2Ψ)
= -(KrW,W)=
-(ΚΨ,ι·Ψ)
= 0.
(10.1)
Here we have made use of the fact that the operator r is real and Hermitian, and
also of the property K2 = — 1. Consequently, the matrix element vanishes in the
zeroth order in H. In connection with (10.1) it is noteworthy that EDSR on band
electrons, in contrast to the situation dwelt upon here, occurs between the states
which are not Kramers conjugates. In the band, EDSR occurs between the states
with opposite spin orientations but with the same value of the projection of the
vector k onto H. That is why constraints imposed by formula (10.1) are invalid
for EDSR on band electrons.
Using this result, we can get estimates for r C OR and for i ; C OR by analogy with
formula (3.17). Equation (10.1) shows that the transition is forbidden in the
zeroth order, which gives rise to the appearance of the factor ha>J£x in r C O .R The
second small factor emerges when the mixing of levels induced by the SO
interaction is taken into account. This factor is of the order ^fc'/^b where
k~ R~l, and R is the radius of the electron state in the impurity centre. As a
result, we get for the EDSR band (Rashba and Sheka 1964a)
' c o r - (toDjStWiR-'WR,
vCOR ~ r C O aR ; s .
(10.2)
The obtained estimate is correct if / is odd (as is the case for η-type CdS and ntype InSb). If / is even, it is necessary to introduce another small factor, since the
transition is parity forbidden in the lowest order. Comparing (10.2) with (3.17)
and putting in (3.17) k~ r^ 1, we obtain the ratio of the EDSR matrix elements
for bound and free electrons:
(rCOR)bound/(>COR)free - ( t u o j g ^
2
< 1.
(10.3)
The inequality in (10.3) is valid, since in fact I ^ 3. Therefore the binding of
carriers weakens the EDSR excited by the mechanisms induced by the SO
coupling in the electron band.
Naturally, in the opposite limit of strong magnetic fields the difference in the
intensity of the EDSR excitation on free and bound carriers vanishes.
The theory for an arbitrary ratio of ha>s to Sx was formulated in the paper by
Rashba and Sheka (1964a) whose approach we shall follow below. The theory in
this work was developed conformably to semiconductors with the extrema loop.
In this case 3tfso is linear in k (see (5.3)). Since its value is usually small, it stands
to reason to confine ourselves to the Zeeman limit Jtso ^ hcos.
The Hamiltonian J^0 can be represented as
jf0 = jr'Q + jho)cLH
+
&μΒ{σΗ),
174
E.I. Rashba and V.I. Sheka
The eigenvalues of these operators are equal to $„m = £'n\m\ + ftcocm/2 ± hojl
and &'n\m\, respectively. Here LH is the projection of the angular momentum
operator onto the direction of the magnetic field, ρ is the radius-vector in the
plane, perpendicular to / / , and m is the magnetic quantum number. For
simplicity, assume that m* and g are isotropic. In this case to obtain the angular
indicatrices of the longitudinal and transverse resonances, it is possible to make
direct use of the results of sections 4 and 5.
Since J^0 in (10.4) is axially symmetric and permits the introduction of the
angular momentum m, all the results based on the symmetry arguments in
section 5 are valid. Nevertheless, using table 1, one should bear in mind that the
energy spectrum of the impurity centre is richer than the Landau spectrum of the
band electron and that instead of (m' — m)coc±cos
there appear transition
frequencies {$n'm' — £nm)/h- I n the polarization τ the indicatrices ΩΛ (4.11) with
a = m' — m — τ correspond to these transitions, and according to (5.17)
Ωχ o c c o s 20 and Ω0 ocsin 20. Experimental identification of the indicatrices must
help to unambiguously assign the electric-dipole CFR bands. In particular, as in
the case of free electrons, EDSR must be observed in the Faraday geometry in
one of the circular polarizations (depending on the sign of the g-factor) and also
in the longitudinal Voigt polarization.
The matrix element of the spin transition between the 'spin-down' and 'spinup' states of the ground level, calculated in the first order of the perturbation
theory in the parameter Jifso/ha)s, is
< 0 | / C | K m ) < n m | K t| 0 )
|
<Q|K T|wm)<wn|fl t+|0)|
(10.5)
Here \nm} are the eigenfunctions of the operator 3#"0 in the A' system,
corresponding to the quantum numbers n, m (n = 0 is the ground state),
Λωτ = hcos - hcocT/2, and the sign of ω 8 in (10.5) coincides with the sign of the g factor. The angular dependences of the matrix elements in (10.5) are identical to
the ones of band electrons (formula (5.17)).
The presence of poles in (10.5) at
hwx — £'n\m\ — £"c
(10.6)
i.e., at the resonance of the spin transition frequency with the frequency of one of
the allowed orbital transitions in the impurity centre, testifies to a strong
dependence of the spin-flip transition intensity on Η and to the existence of
gigantic intensity resonances in it.
Infinite summation over η in (10.5) makes it difficult to get the results in the
explicit form. Yet, it is possible to obtain them approximately if the sums in
Electric-dipole spin resonances
175
(10.5) are interpreted as second-order corrections for certain auxiliary Hamiltonians. Namely, they are for the transverse (τ = ± 1 ) resonance
and for the longitudinal (τ = 0) resonance
= X'o - t*<»J
+ eRo,
(10.8)
where 3 is the operator of the spatial inversion. In both formulas the last term is
perturbation. One can immediately check that the sum in (10.5) in either case
can be obtained as the coefficient at ε 2 in the expression for the corresponding
eigenvalue of the operator Jf aux . Determination of this coefficient by the
eigenvalue of ^ u x, found by the variational method, is a handy means of
calculating matrix elements (10.5).
Resonance growth of the EDSR intensity, predicted by Rashba and Sheka
(1964a), was experimentally observed by Dobrowolska et al. (1982) for semimagnetic semiconductors. They are a unique object in which the resonance in
the intensity can be observed in the region of relatively weak fields hcoc <ζ £v
Dobrowolska et al. (1982, 1984) used high-quality Cax _ x M n x S e crystals with χ
= 0.1 and 0.2. In these conditions the value of the g-factor is very large {g « 100).
The experimental data are given in fig. 12. In the Faraday geometry, EDSR was
observed in the CRA polarization and hence g > 0. The initial analysis of the
experimental results was made on the basis of the Wolff (Dobrowolska et al.
1984) and Dietl (1983) theories for different versions of the two-level model.
Later, Gopalan et al. (1986) carried out a new analysis of the experimental data.
In fig. 12 the curve found by Gopalan et al. (1986) is plotted as well as the best fit
obtained by us for the Hamiltonian (10.7) with a simple variational function
(containing two exponents). The value we found for δ1 is \δ^ « 1.6 χ 1 0 ~ 3 a.u.
The EDSR mechanism studied above is entirely associated with fc-linear terms
in the dispersion law and is not specific for semimagnetic semiconductors. Yet in
semimagnetic crystals there is a completely different SR mechanism, briefly
described at the end of section 14.
The intensity of transitions at combinational frequencies for weak Η has been
calculated by Edelstein (1983).
The majority of experiments where COR was observed on donors were
carried out in InSb in the conditions ftcoc ^> Sv Dickey and Larsen (1968) and
McCombe and Kaplan (1968) observed that the resonance at the combinational
frequency somewhat shifted with respect to ω0 + ω 8 , due to the effect of the
Coulomb field of the impurity. Another type of transition at combinational
frequencies was observed by Kuchar et al. (1984). Although their proposed
interpretation of the experimental data is rather tentative, there is no doubt that
they observed combinations of ω 8 with frequencies of the transitions within one
Coulomb series. Analogous impurity CFR transitions had previously been
176
E.I. Rashba and V.I. Sheka
4
6
8
10
12
14
PHOTON ENERGY fico (meV)
Fig. 12. Dependence of the EDSR intensity on the photon energy according to Dobrowolska et al.
(1984). Here α is the absorption coefficient in the EDSR peak and Γ is the total resonance width at
half maximum. Measurements were performed for the C d 0 9M n 0 ASe crystal in the CRA polariz­
ation at H\\c. Circles are experimental data (open circles, 4.7 K, black circles, 9.8 K). The solid line is
the best fit with the Gopalan et al. (1986) theory, and the dashed curve shows the best fit with
variational calculation.
reported by Lin-Chung and Henvis (1975) and Grisar et al. (1976) for
frequencies associated with CDC + co s and 2a>c + ω 8 transitions respectively. EDSR
was observed by McCombe and Wagner (1971) and later by other physicists.
The EDSR band on bound electrons, observed by Dobrowolska et al. (1983) is
shown in fig. 7; it is a bit shifted towards weak fields in comparison with the
band corresponding to free electrons.
There are situations when the value of r C OR is much larger than the one
ensuing from (10.2). For instance, the denominator can be much smaller than Sv
Electric-dipole spin resonances
ill
This is possible for a multivalley spectrum and also for COR on excited levels.
Yet the most important case is the case of acceptors in crystals with degenerate
bands (Rashba and Sheka 1964a, b).
By virtue of the degeneracy of hole bands in the Ge- and InSb-type of crystals,
the ground state of the large-radius acceptors is fourfold degenerate. In the
magnetic field it splits into levels with the angular momentum projections
m = ± 1 / 2 , ± 3/2. The levels with the same value of \m\ are Kramers conjugates.
That is why for transitions between the levels with different values of \m\ the
first factor in (10.2) is absent. The second small factor is also missing since the
two-band spectrum corresponding to light and heavy holes implies strong
SO interaction, the spacing between these bands of the spectrum at k ~ R ~1
noticeably exceeding Sx. However, for EDSR to occur, it is necessary to
introduce instead of these two factors a factor which is responsible for the
absence of the central symmetry in the Hamiltonian of the impurity centre. In
the hole Hamiltonian in crystals of the A n iB v type there is a nonrelativistic (and,
consequently, large) term, proportional to J K ( K ) (analogous to the σκ term in
formula (6.2)). The presence of this term enables one to construct the EDSR
theory for acceptors within the framework of the EMA method. Employing the
Kane model (Addendum B) for InSb one can obtain an estimate for the matrix
element r E D RS (Rashba and Sheka 1964b):
PG
REDSR
~
EGEX
O f c 3> ~ i o
3
a.u.
(10.9)
This is larger by three orders of magnitude than the estimate for the EPR
characteristic length. The angular indicatrices of EDSR on acceptors, found by
Rashba and Sheka (1964b), agree with the general rules formulated in section 4.
Observation of SR is simplified in the presence of uniaxial strain, lifting up the
degeneracy of the spectrum (Kohn 1957). In this case r E D RS becomes smaller
compared to (10.9) by the factor (hcos/AE)29 where Δε is the splitting of hole bands
caused by the strain (Bir et al. 1963). Even if this factor is taken into account,
REDSR remains large enough for experimental observation of the resonance.
Much more intricate for the theory is the case of acceptors in Ge and
Si (Bir et al. 1963). Crystals of this type possess the inversion centre and it is
absent only in the site group of the impurity centre. Therefore the value of r E D RS
depends on how the potential changes on the scale of the lattice spacing a, and it
is only possible to roughly estimate the orders of magnitude of EDSR. Since the
site group is a tetrahedron group, the antisymmetric part of the potential can be
modelled as the octupole potential and at a distance R it has the order of
magnitude S^a/R)3. This leads to the estimate:
>EDSR ~
(a/R) 3R
=
(a/R)2a9
(10.10)
i.e., r E D RS <^ a. At the same time, the thus estimated r E S RD exceed λ for Ge and,
particularly, for Si. Bir et al. (1963) obtained an estimate similar to (10.10), but
178
E.I. Rashba and V.I. Sheka
according to their data, due to the presence of the small numerical factors,
r E DRS ~ # for Si and r E
DR
S
<^ λ for Ge.
So far no experimental results on EDSR on large-radius acceptors are
available.
11. Two-dimensional
systems:
MOS
structures
heterojunctions
and
Spin resonance was observed in the inversion η-type layers in the
GaAs-Alj-Ga! _ xA s heterojunctions. In such structures, the normal to the plane
of the junction is along [001]. Under these conditions for a perfectly plane
heterojunction the 2D symmetry group is C 2 v, which is a subgroup of T d. The
symmetry planes are (110) and (1 TO). For the GaAs band structure under these
conditions the Hamiltonian of 2 D electrons is anisotropic and involves two
fc-linear terms,
Ko = < $ ι ( σ Α ~
σ
Α ) + 'Λ
δ σχΚ
~
σΑ
) ·
(
η
Λ)
The presence of the independent constants <5X and δ\ may be considered as the
effect of the 'terminal layer'. The values of these constants are determined by
whether GaAs in the heterojunction is terminated by a layer of Ga atoms or by a
layer of As atoms. The other reason for appearance of two constants will be
clarified in what follows. At present, there are no experimental data that show a
difference in properties of the heterojunction in the [110] and [ Π 0 ] directions,
so, it is reasonable when analyzing the experimental data to confine ourselves to
the isotropic model. Since both invariants entering into eq. (11.1) are unitarily
equivalent, one can set δ\=0,
i.e., take the Hamiltonian of eq. (5.1) with
c II [001].
It is useful to discuss the question of which mechanisms generate δί and δ\.
First, this is the SO interaction in the plane of the heterojunction in a layer
with a width of the order of the lattice spacing. It contributes to both δι and δ\.
To estimate the magnitude of these terms, it is convenient to compare them with
the fc-linear terms in A n B v, compounds. Since in both cases these terms
have a common origin, one can expect that they will have the same order of
magnitude. The only difference is that in the heterojunction the potential is
strongly asymmetric, but the width of the heterojunction amounts to only 10%
of the width of the electron channel and the maximum of the φ function is
probably beyond the plane of the heterojunction. As a result, the real value of
δγ(δ\) can be a factor of about 10 2 smaller than its maximum value, obtained
from first-principles atomic estimates. In crystals of the A „ B VI type with the
wurtzite lattice, deviation of the nearest-neighbour coordination from the
tetrahedral coordination amounts to 1%. However, it is this deviation that gives
Electric-dipole spin resonances
179
rise to the appearance of the linear terms. Therefore, the small numerical factor
has roughly the same order of magnitude as in the former case.
The second mechanism generates only δ[, this contribution comes from the
bulk fc3-terms of eq. (6.2). It can be obtained (Bychkov and Rashba 1985) from
the estimate given by Aronov et al. (1983) for <53,
^ = Kft(2m^)"
<^X
3
2/1
3
G
(11.2)
where a c » 0 . 0 6 for GaAs and <...> denotes the average value over the wave
function of the electron confined in a channel. A reasonable estimate,
<£ 2> ^ ( 4 0 A ) ~ 2, yields δχ « 1 0 " 1 0 eV cm. This contribution exists even for
symmetric wells, and is not very sensitive to the behaviour of the potential near
the interface. The third mechanism is that the fe-odd terms with δ\ = 0 emerge
due to the inhomogeneous electric field in the space-charge layer.
The interference of the second and the third contribution may result in
anisotropy of J4?so. However, it is important only when both contributions have a
comparable magnitude. We shall not consider the case of such an accidental
coincidence. According to Malcher et al. (1986) the second contribution
dominates over the third for electrons in GaAs-Al^Ga^^As heterojunctions.
Stein et al. (1983) observed spin resonance in the G a A s - A l 0 3G a 0 A
7 s
heterostructures with carriers with high mobility (μ > 1 0 5c m 2/ V s ) at Τ ~ 1 K.
The resonance was detected by conductivity modulation of the specimen, its
intensity was high. Bychkov and Rashba (1984) made an assumption that the
observed SR is the EDSR caused by the fc-linear term of eq. (5.3) [cf. the first
term in eq. (11.1)] in the dispersion law, and proposed to find δι from the
dependence of the resonance frequency ν on H, which is shown in fig. 13. An
important peculiarity is that v(H) extrapolated from the region of high Η always
shows a nonzero offset v 0 at Η = 0. It follows from eq. (5.10) that for the Ν = 1
level in the region of strong Η the resonance frequency is
v ( H ) = v 0 + v z( f f ) ,
v 0 » -{6AJnh)
signg,
(11.3)
This linear dependence agrees with the experimental data shown in fig. 13. The
sign of the offset, v 0 is positive, indicates that g < 0 in agreement with other
experimental data. The value of v 0, determined for a non-illuminated specimen
(2) from fig. 13 and eq.(11.3), gives Α1π2.5
χ 1 0 " 6e V and δ1*2χ
1 0 " 3a . u .
Calculating /EDSRMEPR according to eqs.(5.7)-(5.8) we get a value of about 10 7,
i.e., EDSR strongly prevails over EPR. The value obtained for δ1 coincides with
the typical value of this coefficient in the bulk dispersion law for carriers in
hexagonal crystals of the Α π Β ν ι type (see section 5).
Despite the fact that the δ1 found here has a reasonable magnitude, the
approach which led to this result is open to criticism. Lommer et al. (1985) have
shown that in G a A s - A ^ G a ^ ^ A s heterostructures a very important contri-
180
E.I. Rashba and V.I. Sheka
Β [Τ]
>
Fig. 13. Dependence of the SR frequency ν on Η for two samples. Ν is the Landau quantum number.
The carrier concentration Ns was altered by illuminating the sample. The dashed lines represent data
obtained by another method of cooling the samples (Stein et al. 1983).
bution to the dependence ν = v(H) comes from the /c-dependence of the g-factor,
originating from the bulk fc4-nonparabolicity. This contribution alone is
sufficient to describe experimental data satisfactorily, so it is impossible to find
two independent constants from these data. The role of nonparabolicity
enhances with increasing frequency v, and in recent experiments by Dobers et al.
(1988) on quantum wells at ν ~ 60 GHz it played a dominant role. However, the
most important statement (Bychkov and Rashba 1984) that the electro-dipole
mechanism of the excitation of SR dominates in heterojunctions, seems
undeniable, since the magnitude of <5Χ « 1 0 ~ 3 - 1 0 " 2 a.u. has been found after­
wards in independent experiments, e.g., on Shubnikov-de Haas oscillations [on
Si MOS structures (Dorozhkin and Ol'shanetskii 1987), and on A m B v quantum
wells (Luo et al. 1988, Das et al. 1989)]. This statement has been also confirmed
in experiments by Stormer (1988).
Erhardt et al. (1986) investigated absorption spectra of p-layers in
GaAs-AlGaAs heterojunctions in the submillimeter range of the spectrum in
magnetic fields up to 25 T. They observed EDSR and electric-dipole CFR
transitions. There is no adequate interpretation of the observed bands, especially
those which were observed in fields higher than 20 T.
Darr et al. (1976) observed EDSR in the inversion η-layer on the (111) face of
InSb. The resonance intensity considerably enhances with increasing angle θ
Electric-dipole spin resonances
181
between the normal to the surface and H. They supposed that <5X(0 = 0) % 0 but
that this parameter rapidly increases with θ if taking into account a finite value
of the parameter < z 2> / r j , which is usually regarded to be small The reason for
δι(θ = 0) being so small that minor correction terms are dominating, is, so far,
unclear. However, this effect was confirmed by Merkt et al. (1986). These
authors also observed an unusual dependence of the shape of the EDSR line on
electron concentration in a channel. At a low electron concentration ns (ns« 1.6
χ 1 0 11 c m - 2) in the spectrum on the EDSR frequency one can observe a dip
which, with increasing n s, is continuously becoming a peak, distinctly seen
already at ns « 2.8 χ 1 0 11 c m 2. The unusual profile of the EDSR line is ascribed
to the Fano-resonance occurring due to the fact that the EDSR line is observed
on the background of a broad CR band.
12. One-dimensional
systems:
dislocations
Dislocations in crystals are extended defects which may produce the attractive
potential for electrons. This potential localizes electrons in the plane per­
pendicular to the plane of the dislocation, but the motion of electrons along the
dislocation remains free. As a result, one can expect that electrons trapped by
the dislocation will exhibit I D behaviour. The most convincing argument in
favour of the existence of I D energy bands for carriers bound to dislocations is
apparently the discovery of the Ch-line; the Ch-line has been identified as an
EDSR band for I D carriers (Kveder et al. 1986).
The Ch-line was discovered by Kveder et al. (1984) on oriented dislocations in
Si. Annealing led to the reconstruction of dislocations, which resulted in the
disappearance of the original EPR signal corresponding to dangling bonds and
in the appearance of a new SR signal, the Ch-line. The Ch-line is excited by the
electric field Ε parallel to the [ Π 0 ] direction coinciding with the direction of
dislocations, and a slightly anisotropic g-factor, close to g = 2, corresponds to
this line. The measurements were carried out at the frequency ν = 9.5 GHz and
the electric mechanism of the excitation was established by moving a specimen
within the resonator: when the specimen was moved away from the antinode of
£, the signal became 200 times weaker.
The dependence of the EDSR intensity on the orientation of the specimen (at
a fixed reciprocal orientation of Η and E) appears to decisively prove that not
point defects but electrons of the dislocation band are responsible for the Chline. The experimentally observed angular dependence can be accounted for by
assuming the dislocations to have low symmetry, allowing for the invariant
vector perpendicular to the dislocation axis. Let us bring these two directions
into correlation with the unit vectors b and /, respectively. The electron
quasimomentum k can be oriented only along the straight line /. The energy of
the SO interaction and the respective contribution to the velocity can be written
182
E.L Rashba and V.I. Sheka
as
j r eo
(12.1)
= ^ 1( f / ) N * x / ) ) ,
These formulas are analogous to (5.3) and (5.6). The operator vso completely
describes the effect of the SO interaction since in I D systems the operator f
commutates with the spin-independent term in the velocity ν (section 3). The
matrix element describing the EDSR excited by the field E\\e, is proportional to
(12.2)
In deriving (12.2), we have here as elsewhere switched over from the A system
(with the axes x\\b, y\\(b χ / ) , z||/; fig. 14) to the A' system. Using formula (A.5)
for B21 in notations of fig. 14, we get the expression for the angular EDSR
intensity dependence (Kveder et al. 1986):
(12.3)
I = I0 cos; 2% ( l - s i n 20 s i n 2< / > ) .
Formula (12.3) holds if the g-factor is isotropic. Agreement of this dependence
with the experimental data is illustrated by figs. 15a, b. They convincingly testify
to the fact that the model based on the existence of the electron dislocation band
is correct.
The estimate of the lower bound on δί9 following from the experimentally
found ratio / E DSR/^EPR ^ 200 is l^l ^ 203 χ 2nhv ~ 1 0 " 1 4 eV cm. It is by a few
orders smaller than in other cases (sections 5 and 11). The coefficient can be
correlated with the effective force F or with the effective transverse electric field,
£eff = F/e, acting on the electron. A very rough estimate, which makes it possible
to relate F\\b and δΐ9 would be F ~ 5JX2. According to Kveder et al. (1986), who
obtained the estimate in a somewhat different manner, £ e ff ^ ΙΟ7—108 V/cm, i.e.,
Ε
Η
b*l
Fig. 14. Orientation of the d.c. magnetic field H, of the a.c. electric field £ and of the a.c. magnetic
field ft relative to the dislocation axis / and to the invariant vector bll (Kveder et al. 1986).
Electric-dipole spin resonances
183
2 -
1
-
180
Θη
Fig. 15. Dependence of the EDSR intensity on the orientation of the sample: (a) φΗ = 0; (b) φΗ =
π/2. Curves 1 in (a) and (b) correspond to E±H, Ε, Η and / are in the same plane. Curve 2 and 3
correspond to E\\H. The solid line shows the theoretical results and the dots the experimental
results. Curves 1 and 2 are obtained in the linear regime, and curve 3 at higher microwave power
when the resonance is close to saturation (Kveder et al. 1986).
it has an atomic order of magnitude. Therefore it was concluded that the I D
band responsible for the Ch-line lies deeply in the forbidden gap.
A more detailed description of experimental data is given in the paper by
Kveder et al. (1989) while the theory is given in the article by Koshelev et al.
(1988).
Babich et al. (1988) discovered a few new EDSR bands, associated with
dislocations in Si. The EDSR intensity exceeds the EPR intensity by two orders.
The authors ascribe this to paramagnetic centres (1/2 spin) with the symmetry
C s, built in cores of dislocations which are components of the dislocation
dipoles.
184
E.I. Rashba and V.I. Sheka
13. Shape of the EDSR
band
The shape of the resonance curve for band carriers is determined by a number of
factors. These include the scattering of carriers by phonons, the spread of
resonance frequencies for carriers with different quantum numbers, and narrow­
ing due to scattering (this mechanism for EPR in η-type InSb was described by
Sugihara (1975)). Consequently, the theory must be developed to conform to
concrete situations, and no theory of this kind for COR is available. That is why
below we shall discuss only one aspect of this problem, which is important for
understanding the main features of the spectrum and for treating the experi­
mental data.
Figure 16 shows the CR and EDSR spectra in η-type InSb obtained by Bell
(1962). In this work EDSR on free carriers was observed for the first time, and
we shall come back to it in section 14. Here let us note only three things. First,
the electric mechanism of SR excitation was unambiguously proved by moving
within a waveguide a sample whose width amounted to 1/20 of the wavelength:
the spin transition intensity increased when the sample was moved away from
the microwave magnetic field antinode (i.e., the electric field node). Secondly, the
EDSR band is much narrower than the CR band (by 2 orders). And thirdly, the
CR band zero is only slightly shifted with respect to the origin (H = 0). This
φ
-2
I
I
I
ι
-1
0
+1
+2
Magnetic Field (kilogauss)
Fig. 16. CR (broad band) and EDSR (two narrow lines) spectra in η-type InSb. The inset shows the
detailed shape of the spin line. Donor concentration is 9 χ 1 0 1 c3 m " 3. T = 1 . 3 K, frequency
ν = 72 G H z (Bell 1962).
Electric-dipole spin resonances
185
means that ωτ < 1, where τ is the momentum relaxation time, and the Landau
quantization is destroyed by scattering. On the other hand, the SR band is
narrow. Therefore ωτ 5 > 1, where T s is the spin relaxation time.
It is evident that at τ <ζ τ 8 the CR band and all electric-dipole CFR bands will
have the collisional width ~ τ ~ 1 . Under the conditions ωτ < 1 all electric-dipole
CFR bands will fuse with the CR band and will not be observable on its mighty
background. The situation is quite different for the EDSR band. Mel'nikov and
Rashba (1971) showed that the EDSR band consists of a broad band of width
~ τ _ 1 and of a narrow line of width ~ τ 8 - 1 (if only the collisional mechanism of
broadening is taken into account). Distribution of the intensity between them
strongly depends on the dispersion law.
The integral intensity of each of the COR bands is practically independent of
the scattering rate. If ωτ, ωτ 8 ρ 1, all bands may be resolved and can be brought
into correlation with the respective terms in the velocity operator. For each of
the bands the real part of the diagonal components of the conductivity ση can be
written as
ση(ω) oc Re
υJit) = exp(iJf t/h)vj exp( - U f i/fc),
(13.1)
where <...> denotes an average over the ensemble and 2tf is the total
Hamiltonian of the system. The integrated intensity of the absorption in each of
the bands is expressed via matrix elements of the velocity at coinciding times:
Σ
(13-2)
sn<sn.
Here ρ is the density matrix. Therefore as long as the perturbation operator
responsible for the scattering of carriers does not affect the energy spectrum and
the integrated intensity of each band is τ-independent.
matrix elements (vj)n.n,
Overlapping of bands does not bring about any major changes.
With regard to EDSR, the problem is how the overall intensity is distributed
between the broad band and the narrow line. Let us base our consideration on
the strong inequality τ < τ 8. Since the duration of the spin transition is of the
order of τ δ, in the course of the spin transition an electron experiences a good
deal of collisions (of the order xjx > 1), affecting its momentum but not its spin.
We shall call them momentum collisions. From the considerations below it will
ensue that they affect EDSR in a different manner than EPR.
If the g-factor is isotropic, the Hamiltonian for a perfect crystal is
"i/OC
(Pn-Pn')\(Vj)n>n\2-
= je0(R) + &μΕ(σΗ)
- - V(R)A(t).
c
It is convenient to write down the velocity as V= σ+ V-{R)
(13.3)
+ σ_ V+(R)
186
E.L Rashba and V.I. Sheka
+ σζ VZ(K). The Z-axis is directed along H. Since τ 8 ^> τ, the SO interaction is
weak. Therefore one can assume that it is completely incorporated in V(R) by
means of the appropriate canonical transformation (cf. section 3) and is missing
in J^Q. Then at Ά = 0 the eigenstates can be classified according to coordinate
and spin quantum numbers, and it stands to reason that the collision integral
should be divided into two parts, corresponding to momentum and spin
collisions (W and Ws). If the >i-linear correction to the density matrix is written
as p\t) = p(t) — Po> the equation for it in the interaction representation over
is
+ &μηΙ*Η, Pi + &(Ρ') + ^ s ( p ' ) = £tV(t)A(t)9
p 0] .
(13.4)
Here W and Ws are linearized collision integrals. Within the spin line the largest
term is W. Therefore p'(t) should be sought in the condition W(p') = 0. This is
not difficult to do if we bear in mind that the equivalent condition W(p(t)) = 0
corresponds to the equilibrium distribution of electrons in an instantaneous
magnetic field H+ ff(t) with different chemical potentials η^ή and η2(ή for the
two spin orientations. A most general form of p' is
(rji-η)
p'(*91)
(13.5)
=
9//
=
Here £ is the kinetic energy of an electron, £ia(£)
£±ha)J2, η = (ηι + η2)β is
the equilibrium chemical potential, / ( £ ) is the Fermi distribution function and
γ is a complex parameter. This parameter can be found by calculating the trace
of eq. (13.4) over configurational quantum numbers with formula (13.5) and by
taking the fact that W(p') = 0. When the trace is calculated, the term Ws reduces
to a constant, multiplied by γ: thus the spin relaxation time T s naturally appears.
As a result
^l + i(usy
ot
+ L
TS
=
where V_ is the matrix element of the velocity
<y~> =
(13.6)
i2-i/2^<v_yA{t),
ch
t r { r _ [ f l * 2- » i )
tr{/(*2-fl)
-Α^-ηΏ}
-Λ*ι-η)}
V_(R),
(13.7)
The solution of eq. (13.6) permits us to calculate the current and the
conductivity tensor
e2
<K,>j<K-> f
na> ι(ω 8 — ω) + τ 5
,
1
(13.8)
Electric-dipole spin resonances
187
where ηγ and n2 are equilibrium electron concentrations with different spin
orientations.
Formula (13.8) shows that the spin line is described by the Lorentzian of a
small width τ " 1 . An important peculiarity is that the numerator of (13.8)
involves averages of the velocity matrix elements (but not averages of their
squares, cf. (13.2)). These averages in many cases must turn to zero, then the
narrow line must be absent. This is the main difference in the effect of collisional
averaging in EPR and in EDSR. In EPR the matrix element is practically
independent of the configurational quantum numbers and the averaging leads
only to motional narrowing of the band. In contrast, in EDSR the matrix
element, as a rule, strongly depends on the configurational quantum numbers,
and therefore the averaging may greatly reduce the intensity of the SR line.
So, the EDSR spectrum consists of the two overlapping bands of the widths
~ τ " 1 and ~ τ ~ 1 . Of course, it is assumed here that the collisional broadening is
much larger than the inhomogeneous broadening, caused by the dependence of
co s on configurational quantum numbers and weakened as a result of the
motional narrowing. At ωτ ρ 1 one can observe the broad band and the narrow
line on its background. At ωτ ^ 1 the broad band must be practically invisible
on the background of the cyclotron absorption and one can observe only the
narrow line whose intensity at arbitrary ωτ is determined by formula (13.8).
However, this formula holds only at the conventional constraint $τ > ft, where $
is the mean energy of carriers.
To understand in which cases < V_ > Φ 0 and what determines its value, it is
instructive to consider some concrete examples. Two cases are possible where V
is ^-independent. First, the SO interaction can be represented by the fc-linear
terms in the dispersion law (section 5). Secondly, it can be represented by the
same terms but multiplied by Η (section 8). Then the overall intensity is
concentrated in the narrow line (Rashba 1964a, Boiko 1964).
For the SO interaction Hamiltonians of higher orders in k two cases are
possible. If 3tifso is even with respect to k, the velocity is odd with respect to k and
< y± > = 0; hence the SR line is missing. An example of this case is the fc4-terms in
the electron Hamiltonian for InSb. If Jfso is odd with respect to k, then < V± >
Φ 0, its value being dependent on the specific symmetry and on the magnitude of
H. For the fc3-terms in the InSb spectrum (MeFnikov and Rashba 1971)
< F ±> o c < K 2+ X ? - 2 / C l > .
(13.9)
At ftcoc <^ $ the leading terms are cancelled, the matrix element diminishes by
the factor ~ fta>c/<? and the entire absorption almost occurs in the broad band.
At ftcoc~<? the absorptions in the band and in the line have comparable
intensities. Since the angular indicatrix of the matrix elements of V does not
depend on configurational quantum numbers (section 6), the EDSR angular
indicatrices for the line and for the band are the same.
Unfortunately, the EDSR spectrum described in this section, formed by the
188
E.I. Rashba and V.I. Sheka
superposition of two bands of different widths, has not so far been observed
experimentally.
14. EDSR induced by lattice
imperfections
There is experimental evidence of the fact that lattice imperfections induce new
EDSR mechanisms. In this section we shall consider them and also give certain
models of such mechanisms. This interesting aspect of the problem is still the
least developed in the COR theory.
The theory presented in section 6, based on the Hamiltonian J^ 0oc/c 3, does
not describe the experimental data of Bell (1962) for EDSR in η-type InSb under
the conditions ωτ < 1. First, the observed EDSR intensity is unexpectedly high.
Actually, it follows from the experimental results that / E D RS is by many orders
higher than / E PR at E~H. At the same time, according to Dobrowolska et al.
(1983) (cf. section 7), these intensities at ωτ > 1 differ by less than two orders.
But at a strong inequality of the relaxation times (τ 8 Ρ τ) and at a strong
scattering (ωτ < 1), i.e., under the conditions of the experiment carried out by
Bell (1962), the SR line intensity should be additionally suppressed by the factor
~ ηω^/η (section 13). The same conclusion ensues from the too large value of the
ratio /EDSR/^CR ~ 10 ~5 1 0 >4 which could be estimated from fig. 16. In the
second place, the observed absorption was isotropic and the same in both
circular polarizations. The formulas of section 6 yield polarization-dependent,
strongly anisotropic absorption. The theory in section 9 does not account for the
experimental facts either. This controversy points to the fact that in highly
doped crystals (the experiment by Bell was performed under the conditions
ητ ^ h) the role of impurities in COR is modified: their influence is not reduced
to the scattering of carriers, giving rise to level broadening and to the averaging
of transition matrix elements. Inducing transformation of the energy spectrum,
they give rise to the appearance of new COR mechanisms and, consequently,
make a new contribution to the COR oscillator strength, a contribution that
becomes dominant in certain conditions.
Although these experimental data are not yet properly understood, we can
nevertheless mention certain mechanisms which may, in principle, be re­
sponsible for the observed effect. For example, the impurity potential V(r)
generates an 'anomalous' velocity (Blount 1962) which in η-type InSb has the
structure νοζσ xVV(r). Furthermore, impurity centres produce the strain,
calling forth new terms in the dispersion law: in η-type InSn they are linear in k
(cf. section 5).
The second experiment displaying a new EDSR mechanism concerns not the
conduction electrons but the electrons bound in As donors in Ge. Gershenzon et
al. (1970) discovered that in compensated samples the intensity and the width of
the SR spectrum rapidly increases at decreasing T. By moving a sample within a
Electric-dipole spin resonances
189
waveguide it was proved that the new spectrum is an EDSR spectrum. The data
are given in fig. 17. The intensity of the EDSR spectrum is so high that when the
sample is moved towards the antinode of the field H, EDSR is about 70 times
more prevalent than EPR due to the finite size of the sample (T = 1.7 K). Such a
high intensity of EDSR on impurities was all the more unexpected because
EDSR was not observed in experiments on band electrons in Ge, and the
theoretical estimates (Boiko 1962) predicted a relatively low EDSR intensity
(section 8). For bound electrons one can expect EDSR to be weakened by the
factor (hcoJS'i)2 ~ 1 0 " 5 (section 10).
The treatment of these data proposed by Mel'nikov and Rashba (1971) is
grounded on the following basic facts: (i) the Ge spectrum is a multivalley
spectrum; (ii) parameters of the valleys are strongly anisotropic; and (iii) the
random electric field E(r) of charged impurities leads to the mixing of wave
functions belonging to different valleys. If we assume that the field Ε is
homogeneous within the centre, then for the spin Hamiltonian of the ground
state of the donor centre in Ge the following expression is valid:
(14.1)
where
Here η numbers the quantum levels of the centre in the one-valley approxi­
mation, Sn is the energy of the levels, ν numbers the valleys, A is the valley-orbit
splitting, d is the dipole moment of the transition, the quantities Pvu have the
meaning of polarizabilities, and Py, P± and g\\, g± are values of the tensorial
components of Ρ and g in the main axes of electron ellipsoids. Formula (14.1)
holds if the Zeeman and Stark energies are small compared to A. It is clear from
(14.1) that the product (g^ — g i ) ( P | | — P±) has the meaning of the SO coupling
constant. If charged impurities are arrayed chaotically, there follows from (14.1)
a formula for the conductivity tensor per donor:
χ3φ(χ)
dx
άΩχΜ,.-Μ^)
χ δ(ω - ω 8 - a)sax2l(eh)2
- Χ^Λ?]^.
(14.3)
190
E.I. Rashba and V.I. Sheka
It is written down in the basis of the crystallographic axes. Here
σ0 =
2E20H2
8lhA2
μϊί^ιι -g±)2(P\\
" P±)2«> tanh(o>/2T),
00
E0 = 4ne(nJ30),2/3
k sin(fex) exp( —fc 3 / 2) dfc,
φ(χ) =
ο
P A.) El
Mjj = hfij + (he — Ihie^Sjp
g=
Ug\\+2g±),
Mj±=2-li2(Mjx±iMjY),
(14.4)
e and h are unit vectors of the electric and magnetic fields, respectively
(integration (14.3) is performed over the orientation of e), and nx is the charged
impurity concentration.
Although the model is very simplified, the conclusions are in good, at least
qualitative, agreement with the experiment. The uniaxial strain, transforming
the spectrum into a one-valley spectrum, does away with the EDSR band
(Gershenzon et al. 1976). The EDSR band has wide wings, decreasing as
\ω — cos\ ~ 3 /4 (cf. fig. 17a). At ω -» ω 8 au logarithmically diverges, therefore on the
background of the broad band, narrow peaks are seen (fig. 17a). The character­
istic width of the curve has the magnitude \ω — cos\ ~ αω 8, i.e., it increases with
increasing frequency. The absorption intensity is a 0-proportional, i.e., it in­
creases with increasing H. The width of the curve depends on the orientation of
H: it is minimal at / / | | [ 0 0 1 ] (according to (14.3) it is even zero). All the three
conclusions are in qualitative agreement with the data of Gershenzon et al.
(1976). Numerical estimates show that /EDSR>^EPR AT \n ^ 1 0 1 5 c m - 3, which
also agrees with the experiment. The decrease in the ratio /EDSR/^EPR with
increasing T, observed in the experiment, is probably accounted for by hopping
conductivity, i.e., the mechanism discarded by the theory.
b
500 G
Fig. 17. SR spectrum of Ge:As. (a) EDSR in a compensated sample, (b) four-component EPR
spectrum in a noncompensated sample. For either sample Nd — Na = 3.2 χ 1 0 15 c m - 3. For the first
sample the compensation factor is Κ = 0.5, Τ = 4.2 Κ, v= 10 M H z (Gershenzon et al. 1970).
Electric-dipole spin resonances
191
Note also that inhomogeneity of the field Η may also give rise to COR, since
the Zeeman Hamiltonian gμB(σH(r))/2
involves both σ and r. In particular, it
has been shown (Pekar and Rashba 1964) that this mechanism may operate in
magnetic materials due to the interaction of an electron spin with the spin and
orbit variables of other electrons. N o w it is becoming clear that a similar
mechanism is efficient in semimagnetic semiconductors and that it is realized via
the exchange interaction of an electron with magnetic ions.
Since this interaction simultaneously involves r and σ, it allows electric-dipole
spin transitions analogously to the SO interaction. Conformably to parameters
of Cdj.^Mn^Se (cf. section 10) the intensities of both processes may compete.
The exchange mechanism has a number of peculiarities. Since the spins of an
electron and of an ion change simultaneously (the flip-flop process), the
transition frequency is shifted by the Zeeman frequency of the ion. For this
reason electric-dipole transition bands may have a doublet structure (SO and
exchange components). Since spins of impurities play the role of a magnetic
field, the electric-dipole transition between components of the Kramers doublet,
forbidden by (10.1), is allowed, and therefore the matrix element of the transition
is nonzero already in the zeroth order in H. The resonance is allowed in all
polarizations and the angular dependence of the intensity is close to isotropic.
These results were obtained by Rubo et al. (1988, 1989).
An allied COR mechanism for semiconducting alloys of inhomogeneous
composition was proposed by Leibler (1978). In this case, in formula (1.3) there
emerges an extra term for ^ ξ 0, containing a quasielectric field, proportional to
the gradient of the composition. This term apparently makes an extra contri­
bution to the COR intensity.
75.
Conclusion
We intended in this survey to elucidate the main problems pertaining to
combined resonance in solids, mainly in semiconductors, and to shed light on
the latest research in this field.
COR was the first phenomenon to reveal the presence of a strong coupling
between an electron spin and an a.c. electric field in crystals. This coupling gives
rise to a number of new phenomena, discovered later: spin-flip Raman scattering
(Yafet 1966, Slusher et al. 1967) (see chapter 5 of this volume by Hafele) spin
resonances of nonlinear susceptibility (Nguyen and Bridges 1972, Brueck and
Mooradian 1973), and other higher-order processes.
For perfect crystals a complete description of COR is achieved on the basis of
the appropriately derived EMA Hamiltonian. General symmetry requirements
and numerical values of the parameters, which are specific for concrete crystals
and vary in wide ranges, completely determine the Hamiltonian. Accordingly,
for different crystals different COR mechanisms may dominate. Moreover, the
192
E.I. Rashba and V.I. Sheka
intensities of different COR bands in one crystal may be controlled by different
mechanisms. The two main mechanisms are: (i) the inversion asymmetry
mechanism inherent in crystals without the inversion centre, and (ii) the
mechanism usually (and not quite adequately) termed as 'nonparabolicity'. At
present, all the most interesting situations (/c-linear and /^-inversion asymmetric
terms, nonparabolicity) have been observed in experiments and the succession
of these mechanisms has been followed as the strain affecting the symmetry of
the crystal was increased.
Thanks to its high intensity and to the considerable amount of bands in its
spectrum, COR is a mighty tool for studying the band structure of semiconductors. Apart from determining the basic parameters of the spectrum, COR
is used for many specific purposes (such as determining the energy dependence
of the g-factor, measuring the SO splitting of bands in the vicinity of the
symmetry points, finding the deformation potentials and constants of the
coupling of electrons to optical phonons, and so on). Experimental studies of
angular indicatrices of the COR band intensities (which for a variety of systems
must exhibit a universal behaviour) should allow to check the reliability of the
assignment of COR bands.
At present, spin-flip Raman scattering, the physical mechanism on which the
operation of tunable lasers is based, is of practical significance. For possible
applications of COR to quantum electronics (Rashba 1964b), heterostructures
with spin injection look quite promising.
Of particular interest is COR in nonperfect crystals. Theoretical predictions
relevant to peculiarities in the behaviour of COR on electrons bound to
impurities, particularly resonance enhancement of the COR intensity, have
lately received convincing confirmation in experiments on semimagnetic semiconductors. The most fascinating aspect of the problem is the search for new
COR mechanisms induced by imperfections, the mechanisms nonexistent in
perfect crystals. Such imperfections may be randomly positioned: impurities as
well as heterojunctions and dislocations. Recent progress made in discovering
EDSR on 2 D and I D electrons in heterojunctions and dislocations is especially
encouraging. These achievements prove that COR is becoming an efficient tool
for studying defects in crystal lattices. COR may also be applicable for the
purpose of determining the magnitude of random electric fields and strains in
disordered crystals.
COR can be regarded from two points of view. First, as a method of
measuring parameters of crystals, and secondly, as a phenomenon which is in
itself an interesting subject of study. As far as the second aspect is concerned, the
most intriguing and impressive results may be expected in COR studies on
nonperfect crystals.
193
Electric-dipole spin resonances
Acknowledgements
We are grateful to all the authors who kindly gave their consent to their figures
being reproduced in this review. We are particularly thankful to Professor R.L.
Aggarval, Dr. J.C. Hensel, Professor D.G. Seiler and Professor J.K. Furdyna for
providing us with the originals of the figures.
Addendum A. Transformation
the
Hamiltonian
of the reference
system
and of
It is convenient to perform the calculation of the quantum level arrangement,
classification of states and determination of transition intensities in the reference
system A', associated with the magnetic field H. In this system the axis Ζ\\H and
classification of wave functions of the spherically symmetric part of the
Hamiltonian can be carried out in terms of the angular momentum projection
m. Therefore it is handy to use circular coodinates in the A' system:
R = (Rj, R0,
Rx),
Rj = 2-l,2(X-iY\
R0 = Z,
Λ 1 = 2 - 1 / (2 Χ + ί 7 ) .
(A.l)
In the A system, associated with the crystallographic axes, we shall use Cartesian
coordinates:
r
= {xj,
Xi = x ,
y,
ζ.
(A.2)
To avoid confusion, the vectors defined by their coordinates in the A system will
be labelled with lower case letters, whereas those defined by the coordinates in
the A' system will be labelled with capital letters.
Transformation of the tensor corresponding to the angular momentum J
from the A system into the A' system is carried out in a standard manner with
the aid of the matrix Sj(0, φ), which belongs to the irreducible representation
D j . Here θ and φ are the polar and azimuthal angles of the vector Η in the A
system.
Spinors are transformed by means of the S 1 / (2 0 , Φ) matrix constituted from
the Cauley-Klein parameters (Landau and Lifshitz 1974):
s1/2(e,
Φ) =
(A.3)
194
E.I. Rashba and V.I. Sheka
To transform the vectors defined in the canonical basis, one can employ the
matrix S x(0, φ) (since its explicit form will not be needed below, we shall not
write it out). However, since vectors in the A and A' systems are defined in two
different bases, namely, Cartesian and circular, it is more convenient to
represent the switchover from r to R by means of the linear transformation
r = BR,
(A.4)
rt = Β ί αΛ α,
with the unitary matrix £?,
2 " 1/2(^2 _
fl=||B,J|
=
y
2
)
i 2 - 1 / (2 y 2 + <52)
21/2γδ
- 2~1/2(sin
βδ-ay
2~1/2((χ2-β2)
ϊ(αγ + βδ)
- i 2 " 1 / (2 a 2 + β2)
(χδ + βγ
-21Ι2αβ
φ + i cos θ cos φ)
sin θ cos φ
2" 1 / (2 c o s φ — i cos θ sin φ)
sin θ sin φ
i 2 " 1 /2 sin 0
cos θ
— 2" 1 / (2 s i n φ — i cos θ cos φ)
2" 1 / (2 c o s φ + i cos θ sin φ)
-i2"1
/2
(A.5)
sine
Cartesian coordinates are designated in (A.4) and henceforth indicated by Latin
subscripts, circular coordinates by Greek subscripts, it is implied that the
summation over α is being performed.
By virtue of the unitarity of B, its columns are orthogonal, and the vectors Bt
= (Bih Bi0, Bn), defined by their components in the A system, are related to each
other as
BjxBr
= iBf.,
(A.6)
the subscripts j , f and / ' constitute a cyclic permutation. This formula helps to
simplify a number of expressions. So, for instance, for the Hamiltonian (6.2)
formula (3.15) involves products of four elements of the Β matrix. Still, it permits
transformation to the form of (6.4) where the coefficients Β(αβγ),
involving
products of only three matrix elements, enter.
The next problem is to transform the EMA Hamiltonian from the A to the A'
system. Let us confine ourselves to the case where the irreducible representation
D, corresponding to the band under study, coincides with the representation D 7
of the rotation group for all elements of the group geGk (for a more general
case, see, Bir and Pikus 1972). If g e Gk is an improper element, it should be
simply replaced by the element gf (I is the inversion operator) of the group of
proper rotations; therefore henceforth we shall not distinguish between g and gl.
Electric-dipole spin resonances
195
Under these conditions, matrices of the angular momentum J (whose rank
coincides with the dimension of the irreducible representation) can be chosen as
basis matrices via which the Hamiltonian
may be written. When the
Hamiltonian #f transforms from A to A', matrices of the momentum J are
transformed as
S J I S J
=1 Β(θ,
φ)1
= Bja.
(A.7)
This formula has a simple meaning: during rotation, components of the
pseudovector J are transformed as components of the vector r. To transform
the Hamiltonian 3tf from A to A', it is required to perform both the
transformation (A.7) and the transformation from k to Κ by analogy with
formula (A.4).
If the g-factor is isotropic, the Zeeman energy is proportional to (JH) and the
transformation (A.7) diagonalizes it. One can confirm this by taking into
account that Bia = B* = (B+)&:
= ΒίαΒίβ3αΗβ
= J\HA = J0H = JZH.
(A.8)
If the g-factor is anisotropic, the Zeeman energy is diagonalized, as, for
instance, in the paper by Rashba and Sheka (1961c).
Similarly, one can check commutation relations in the A' system:
[ K a, Rf-] = (B-%(B-%ikp
Addendum
B: Kane
r , ] = -i(B-%BJp
= -\δαβ.
(A.9)
model
The Kane Hamiltonian (Kane 1957) has proved to be rather efficient for
describing electron properties of cubic InSb-type semiconductors with a narrow
direct forbidden gap. In the Kane model there are three adjacent bands: the
conduction band, valence band (consisting of the light hole and heavy hole
bands) and spin split-off band. The conduction band has s-type symmetry and
the other bands emerge from the splitting of the original p-type band due to the
SO interaction. Eight basis functions φι with k = 0, corresponding to the s- and
p-states (table 2), are taken as the basis in the kp method (Luttinger and Kohn
1955). The choice is made in such a manner that φί and φ2 are transformed over
the D 1 /2 rotation group representation and correspond to the conduction band,
the functions <p 3-<p 6 a er transformed over the D 3 /2 representation (valence
band) and φΊ and φ8 are transformed over the D 1 /2 representation (spin split-off
band). Interaction of these terms is taken into account exactly via matrix
elements <φ ζ|£/>|<Ρί>· As a result, we obtain the 8 χ 8 EM A Hamiltonian which
should be treated as the zero approximation Hamiltonian Jt0. It involves
diagonal terms and nondiagonal terms Pkj (the terms hk2/2m0 on the diagonal
196
Table 2
Kane Hamiltonian (simplified)
0
0
0
0
0
¥73
0
P
73
73
Ψι = - i s T
'
I
Ψι = - i s j
£ΐΨηι+ 1/2
Ψ3 =- - L ( x + i>0T
^3^-3/2
+
0
73/2Ci_
0
0
7 ^
~EG
72/3^7
0
-y372C/c+
0
0
ΨΑ = V ^ z T — +
7 *
0
0
<Ρ5 =
+
pit
-Ciz
-Bo
v/3/2Ci+
-4=(JC - iy)T +
Q^-l/2
^δΨηι+ί/2
7 *
-Eo
7 *
73 *
7 2
i>>)!
0
0
^βΦηι + 3/2
Ψβ =
>/2
0
0
0
0
~EG-Δ
0
0
0
0
0
0
-EG-Δ
Ψί = ^ [ 2 T + ( x + i y ) i ]
_ L [
<Ρ% =
( _ ix
y
_) 4 ] T
€ΐΨπι- 1/2
^8^m+
1/2
7 3
The right-hand side of the table gives a simplified notation of the Kane 8 x 8 Hamiltonian. The bottom of the conduction band is chosen as the origin for
the energy. EG is the width of the forbidden band, and Δ is the SO splitting. The second column on the right-hand side gives the basis functions φι. The
extreme right-hand side column gives the components of the wave function Ψ„, of the spherically symmetric part of the E M A Hamiltonian. In the A'
system m plays the role of A Q M , and the ψ functions depend exclusively on the azimuthal angle, whereas the factors C depend on other variables. The
following designations are used: A = £,· 4- i(G/P)(£rtcr
+ i r i y ) + \(C2IP)zn-Ah
f a dn / ' constitute a cyclic permutation), i ± = (4X ± idy)/y/2 for the
Cartesian
i } and circular £ a, i a coordinates in the A system.
E.I. Rashba and V.I. Sheka
7 3
Electric-dipole spin resonances
197
are dropped as irrelevant). The Hamiltonian Jf0 is spherically symmetric (see
below). It makes it possible to express via the parameter Ρ the value of m* for
electrons on the bottom of the band and the dependence m*(g) (Kane 1957)
_ 1 _
=
2
2;! + 3 ( E G +
h2 3(EG
m*(g)
+ g)(EG +
<f)
A+g)
'
'
}
A similar formula is easily derived for light holes. And if one introduces the field
Η by means of the replacement k -> £, one gets an expression for the g-factor of
electrons (Roth et al. 1959):
σ(/?)
= -
*K
}
4° m
h2 3(EG
-
+ g)(EG + A + g)
p2
(U
'
Ί)
' '
Here the contribution g = 2, corresponding to a free electron in a vacuum, is
omitted. Since Ρ has an atomic order of magnitude Ρ = 10 eV A ~ 1 a.u., in
crystals with a narrow forbidden gap EG ~(0.1-0.3) eV m* <^m0 is small and
\g\ > 1 is large. It is very important that m*(E) and g ( £ ) change a good deal on
the scale g ~ £ G , the spherical symmetry of the spectra of electrons and light
holes being retained with high accuracy in the entire region. This strong
dependence of the spectrum on g is termed nonparabolicity. However heavy
holes cannot be described by the Hamiltonian J f 0 : their effective mass in this
approximation is infinite. 3tf0 has the same form for crystals with the inversion
centre and for crystals without the inversion centre.
Eigenfunctions of
are characterized by the angular quasimomentum m
(section 4). In table 2 Ψη is represented as a column: the subscript of the ψμ
function in each line equals the value AQM, which should be attributed to the
respective component of Ψη.
The next step is to take into account more distant bands. For this purpose it is
necessary to project approximately the total Hamiltonian onto the subspace
{<P/}. In the Luttinger-Kohn formalism such a projection is performed as a
unitary transformation. As a result, new terms emerge in the 8 χ 8 Hamiltonian.
Most of them do not possess spherical symmetry and among them there are
higher order terms in k compared to J^0. In particular, there are terms which
render the mass of heavy holes finite and are responsible for band warping,
reducing their band symmetry from spherical to cubic. Among them there are
terms which enter into the Hamiltonian of a crystal irrespective of whether or
not the crystal possesses inversion symmetry. We shall denote them as
it is
these terms that are mainly responsible for the warping of heavy hole bands (this
is a 'quasi-Ge' spectrum). Alongside these terms, there are also J^as arising due to
inversion asymmetry which are specific for crystals of the A,„B v-type. The most
complete form of the Kane Hamiltonian studied so far is that written out by
Weiler et al. (1978). Table 2 includes only two types of such inversion
asymmetric terms, namely, those which play a major role in COR (Rashba and
Sheka 1961b, Cardona et al. 1986a, 1987). First, there are the /c-linear terms,
198
E.I. Rashba and V.I. Sheka
contained in the central 4 x 4 square and determined by the invariant (Pidgeon
and Groves 1969):
jr = c
4= ( * ( W >
C
Μ) =
J
J
M '
~ *Μ·>
(β·3)
λ/3
here J f are matrices of the J = 3/2 angular momentum. They may prove to be
important for COR in the valence band for low carrier concentrations. Secondly,
there are the fc-quadratic terms, entering in the Hamiltonian with the constant
G. This constant is nonrelativistic, and therefore the corresponding terms are
not small. The constant <53, which determines the COR intensity in the
conduction band, is expressed via G (Rashba and Sheka 1961b):
d3=-4GPA/3EG(EG
+ A).
(B.4)
As a rule, J^as makes a relatively small contribution to the shape of the bands.
From the viewpoint of this article, the terms
and J^as are important
because at Η Φ 0 they allow a lot of transitions which are forbidden for the
Hamiltonian 3tf0. These are CR harmonics, spin-flip transitions and CFR. For
the latter two groups of transitions (i.e., spin transitions) ^ s is of major
importance.
Here we shall make use of the explicit form of the two terms of the
Hamiltonian ^ s (shown in table 2) to illustrate the general property of matrix
elements dealt with in section 4. This property is that the operator tf" = Jfw
+ Jifas, breaking spherical symmetry, consists of the sum of the operators
[
such that their matrix elements obey the relation:
{Ψ^Ψ^^^-ΑΘ,ΦΥ
(B.5)
^ S - m ' a er functions of the angles θ and φ, determining the orientation of //,
universal in the sense that they are independent of the parameters of the
Hamiltonian. The quantum numbers m and rri enter in them only as a difference
m — rri (Sheka and Zaslavskaya 1969).
Division of Jtif' into separate terms
obeying the condition (B.5), is not a
trivial task. So, different matrix elements of the same term 3tf[ may involve
different powers of k, differing from one another even by parity. For instance, in
the simplified Kane model given in table 2, all matrix elements responsible for
the absence of spherical symmetry (of the type of Ck and Gk2; the latter being
contained in £ 3) form one term
£=
One of the methods, which may be
recommended for dividing
' into separate terms
is to project W onto the
2 x 2 subspace corresponding to the conduction band. Then all invariants
entering in
which in projection will generate equivalent terms in the 2 χ 2
Hamiltonian, should be included in one
This procedure simultaneously
allows us to establish the correlation between each
£ and the appropriate term
of (4.9) and thus to find the value of the superscript /, corresponding to ffl[. In
our case ζ = 1 = 3, which ensues from formulas (6.1), (6.2) and (B.4) as well as
Electric-dipole spin resonances
199
from the generalization of (B.4) if the contribution from the invariant Jfc (B.3) is
incorporated (Rashba and Sheka 1961b). Similarly, of the terms of the order k2
one can single out the invariant J ^ , corresponding to the valence band warping
(table 2 does not contain it). In projecting onto the 2 x 2 conduction band
subspace (Ogg 1966) it becomes evident that in this case ζ = 1 = 4; angular
diagrams were found by Sheka and Zaslavskaya (1969). The analysis of the
resultant 2 x 2 Hamiltonians and evaluation of the indicatrices are performed as
in the situations considered in sections 5 and 6 of this chapter.
The basis {φ,} is a joint basis for the three bands, which is why the functions
φι are transformed over the D = D 1 /2 + D 3 /2 + D 1 /2 representation. Accord­
ingly, the matrix S, transforming
from A to A', equals
Φ)
s1/2(e9
0
s3/2(0,
0
S(0, Φ) =
0
0
0
Φ)
0
(B.6)
s 1 / (2 f l , Φ)
1
At the transformation J^=>SJ^S
9 the matrix elements, proportional to the
components of P/c, are transformed as
Ptj^P(B+k)a
= PKa.
(B.7)
The origin of the B+ matrix in (B.7) can be understood if we transform (JH)
using (A.7): (J/c) =>(BJ, £) = (J, B+/c). To get the ultimate result one must bear
in mind that H = BR (cf. (A.4)) and also the unitarity of B. The relation (B.7)
ensures the spherical symmetry of J^0:
jr0(£) = s-ijr0(B-1£)s=
s-1
jr (it)s.
(B.8)
0
The terms of the
in
proportional to G, after the transformation S
and the switchover to ΚαΚβ acquire the coefficients proportional to the products
of three elements of the Β matrix: one element comes from the B+ matrix
similarly to (B.7) and two elements appear at the switchover from £ to K. These
products are grouped into coefficients of the B(afiy)~type entering in (6.10). The
simplest way to find them is to expand the products ΚαΚβ in the operators
s *
±
2=
K
2 ±
,
^
= { K 0, K ±} ,
± 1
J * 0
= {K
+,K_}-2K2.
(B.9)
The curly brackets {...} mark an anticommutator. In terms of these operators
it is convenient to write, analogously to (B.7), the transformation of the terms of
the operator W involving £ s (table 2):
ij=>jta
= k, + (G/P)
Σ
β=-2
ΚβΒα_β^β.
(B.10)
E.I. Rashba and V.I. Sheka
200
Here b 0 1
equal
£
±1
= — 2, whereas b(Xp=
1 in the other cases. The functions &α(θ9 φ)
= - i | sin 2φ sin 0 sin 20,
Ό
(Β. 11)
@ γ = - i 2 " 3 / [2 c o s 2 0 sin 20 - i sin 2φ sin 0(2 c o s 20 - s i n 20 ) ] ,
2
2
J> 2 = cos 2φ cos 20 - i^ sin 2φ cos 0(2 c o s 0 - sin 0),
= 3 χ 2_
?
3
3 / 2
[ s i n 2φ sin 0(1 + c o s 20 ) + i cos 2φ sin 20],
(B.12)
(B.l3)
(B.14)
(B.15)
If we employ the explicit form of the coefficients Β(αβγ)9
Sheka (1961a), it is easy to verify that*
=
—%n),
^2
&\ = — i ^ ( o o o > »
= %oo)>
^3
found by Rashba and
= 2 B ( T T .0 )
(B.16)
At the transformation A-> A', matrix elements of the operator J^c (B.3) acquire
coefficients, including products of four elements of the Β matrix: three of them
stem from transformation of the matrices κ( J) in quite a similar way as in (B.7),
and one stems from the switchover to K. By means of (A.6) and (B.16) they
reduce to Β{αβγ) and & a. The explicit form of matrix elements of J^c in the A'
system is:
j f 3 3 = - ( 3 1 / /2 2 ) C ( ^ T K 1 - ^ 1 K T ) ,
^
C(^ /e + ^
1/2
34
= 2"
2
i
T
X 0 - 2 ^ 0^ Τ ) >
^ 3 5 = Κ ( ^ 5 ^ ι - 2 ^ Κ 0- 5 ^ τΚ τ) ,
^r
36 =
^
tf44=
(3/2) / c( -^3/e0 + ^ K
x2
2
T
-H55
= 3H66 = - 3 / / 3 3,
#56 = #34,
H 4 6=
- H 3 5,
),
(B.17)
= ( j r + ) i 4.
N o w we have come to the key point in the verification of (B.5). So it is
necessary to consider the action of the operators s/a upon separate components
of the Ψ„ functions, treated in terms of the perturbation theory over
' as
eigenfunctions of the spherically symmetric Hamiltonian Jf0. Let us take two
examples.
If Jf0 is a Hamiltonian of a free electron in the field / / , then the lines of Wm are
eigenfunctions of the Landau oscillator (using the Landau gauge). The values of
Ν in different lines correlate with each other in the way shown in table 2. They
differ from m by a half-integer, so in this case A Q M with an accuracy of up to a
half-integer has the meaning of the Landau quantum number. The operators s/a
and Ka transform φΝ with an accuracy up to a numerical factor as
ΚΨΝ=>ΨΝ+«,
ΚαφΝ=>φΝ+α,
(B.18)
*Note that in the paper by Rashba and Sheka (1961a) the subscripts (123) correspond to the
subscripts (TlO) in this review.
Electric-dipole spin resonances
201
i.e., they act as the step-up and step-down operators. N o w we can explicitly
calculate the action of W on an arbitrary eigenvector Ψ(£\ at a given m the index
t = 1 ... 8. Application of table 2 and of formulas (B.10), (B.17) and (B.18) yields
Σ
ΣΓΆ®-*(θ,Φ)Ψ{Άα·
(B.19)
α= - 3 ί'
The numerical coefficients fJjV are Θ- and ^-independent due to the spherical
symmetry of J4f0. Since φΝ with different Ν are orthogonal, it follows from (B.19)
that (B.5) is fulfilled, irrespective of the values of the indices t.
The second example concerns a spherically symmetric impurity centre in the
field Η (Sheka and Zaslavskaya 1969). In this case, it is handy to employ the
axially symmetric gauge in order to use the axial symmetry inherent in the
problem. In this case m can be defined as a genuine angular momentum (section
4). Singling out the azimuthal angle φ one can represent the Zth component of
the Ψ(£ function as
Ψη«,ΐ(Γ>
Φ) = Xmt,l (r> 9)
(B.20)
^(Ψτηΐψ)
where (r, 5, φ) are polar coordinates in the A' system. The action of the
operators srfa and Ka upon φτη(1 reduces, similarly to (B.19), to the replacement
Pmi^Vmi
(B.21)
+ *
and to a complicated modification of the form of the xmtJ functions; the details of
this modification are irrelevant. Acting in the same way as at the derivation of
(B.19), we arrive at
jry£>=
Σ
«
-
.
(
M
)
(
B
.
2
2
)
α= - 3
This formula can be checked by inspection. It is important that all components of
the Ψ„+Λ function contain φ only via the exponential factor, in a similar way to
(B.20). Yet, /x m, is replaced in it by μΜΐ + α in accordance with (B.21). The explicit
form of the r-, 5-dependent factor does not affect the result. It is of importance only
in that, by virtue of the spherical symmetry of the problem, this factor does not
depend on θ and φ. Therefore from (B.22) the result (B.5) ensues. To get this result,
it suffices to use the orthogonality condition at integration over φ in each line.
At the derivation of (B.19) and (B.22), the C- and G-proportional terms in ^ s
have been used above (table 2). However, verification shows that these formulas
are satisfied if all the terms included in the Kane Hamiltonian by Weiler et al.
(1978) are taken into account in
That is why the result (B.5) is largely
general. However, it holds only if lower-order EMA terms, inducing certain
transitions, are taken into account. For instance, if we consider, alongside the
invariant (B.3), the invariant
7CF( J)tcf, this gives rise to the appearance of a
new angular dependence but the respective terms will have small numerical
E.I. Rashba and V.I. Sheka
202
coefficients. It appears that the most important distortions of angular de­
pendences, determined by the functions
occur for holes due to the term J^,.
In the above, the matrix elements of
' have been calculated. However, the
probability of transitions is determined by the velocity operator v. If we work in
the A' system, matrix elements of the operator {VS0)Z = '\\_J^\ Rx]/h (section 4)
differ from (B.5) only by a change in the subscript of the difference (m — rri) =>
(m — rri + τ). The same result is obtained if we calculate total Vx according to
formula (3.8). Note here that in the Kane model the operator β must be different
from /?/ ( / is a unit 8 x 8 matrix) due to the corrections resulting from the
Luttinger-Kane procedure (Luttinger and Kohn 1955) and caused by the
influence of more distant bands (analogous to r so in (3.8b)). Verification shows
that these corrections to the velocity have the same symmetry as the terms
originating from
', and that these corrections are small.
To summarize, one could write down formulas for the coordinate R and the
velocity V, analogously to (B.19) and (B.22); so, an analogue of (B.22) for the
velocity has the form
Vx<Fm =
t
Λ - . ( Μ ) Ρ „ +« +τ ·
(B-23)
<x= - 3
List of
CR
SR
EPR
COR
EDSR
CFR
CRA
abbreviations
cyclotron resonance
spin resonance
electron paramagnetic resonance
combined resonance
electric-dipole spin resonance
combinational frequency resonance
cyclotron-resonance active
CRI
EMA
SO
AQM
a.u.
MOS
cyclotron-resonance inactive
effective mass approximation
spin orbit
angular quasimomentum
atomic units
metal-oxide-semiconductor
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CHAPTER 5
Spin-Flip Raman Scattering
H.G. HAFELE
Physikalisches
Universitat
Am
D-8700
Federal Republic of
Landau Level
© Elsevier Science Publishers B.V.,
1991
Institut
Wurzburg
Hubland
Wurzburg
Germany
Spectroscopy
Edited by
G. Landwehr and E.L Rashba
Contents
1. Introduction
211
2. Theory of SFR scattering
212
2.1. Classical picture
212
2.2. Spontaneous and stimulated SFR scattering
213
2.3. Semiclassical treatment of stimulated Raman scattering
217
2.4. Higher-order processes by Stokes-anti-Stokes coupling
219
3.
Principles of investigation of SFR scattering
219
3.1. General remarks
219
3.2. Spontaneous SFR scattering
220
3.3. Stimulated SFR scattering
221
3.4. Raman gain measurements
222
3.5. Resonant four-wave mixing, CARS-spectroscopy
223
3.6. SFR scattering from coherent spin states
224
3.7. Spin-flip Raman echo
225
3.8. SFR scattering and electric transport
226
4. Application of the SFR laser
227
5. Experimental and theoretical results of nonmagnetic semiconductors
227
5.1. III-V-compound semiconductors
5.1.1. Indium antimonide (InSb)
227
227
5.1.1.1. Origin of SFR spectra
227
5.1.1.2. Cross section and selection rules
228
5.1.1.3. Effective g-factor
232
5.1.1.4. Line shape and line width
236
5.1.1.5. Spin-relaxation times
240
5.1.1.6. SFR scattering in p-material
5.1.2. Indium arsenide (InAs)
5.2. II-VI-compound semiconductors
5.2.1. Mercury cadmium telluride (HgCd)Te
5.2.1.1. Effective g-factor
5.2.1.2. Line shape and line width
5.2.2. Cadmium sulphide (CdS)
243
243
245
245
245
247
248
5.2.2.1. Origin of SFR spectra
248
5.2.2.2. Cross section and selection rules
250
5.2.2.3. Effective g-factor
251
5.2.2.4. Line shape, line width and relaxation times
252
5.2.2.5. Special applications of SFR scattering
254
5.2.3. Zinc telluride (ZnTe)
256
5.2.3.1. Origin of SFR spectra
256
5.2.3.2. Effective g-factor
257
5.2.4. Other II-VI-compound semiconductors (ZnSe, CdSe, CdTe)
259
5.3. IV-VI-compound semiconductors
259
5.3.1. Lead telluride (PbTe) and lead-tin telluride [(PbSn)Te]
260
5.3.2. Lead selenide (PbSe)
261
6. Experimental and theoretical results of diluted magnetic semiconductors ( D M S )
6.1. Cadmium manganese selenide
[(CdMn)Se] and zinc manganese
262
selenide
[(ZnMn)Se]
263
6.1.1. Origin of SFR spectra
266
6.1.2. Cross section and selection rules
266
6.1.3. Effective g-factor
266
6.1.4. Line width
267
6.2. Cadmium manganese telluride [(CdMn)Te]
6.3. Cadmium manganese sulfide [ ( C d M n ) S ]
References
267
269
271
1.
Introduction
Spin-flip Raman scattering (SFR scattering) provides a practical means of
probing the electronic structure of semiconductors and impurities. It is a singleparticle process in which electrons or holes interacting with the radiation change
spin state. The energy of the scattered light is shifted by the value of the spinsplitting energy
AE = hcoSF = \g*\pBB,
(1)
where g* denotes the effective g-factor, μ Β = ehjlmc the Bohr magneton and Β
the magnetic induction.
In 1966 Wolff (1966) treated the scattering of light by mobile carriers in
semiconductors in a magnetic field for the first time. By means of effective-mass
theory he predicted a Raman process involving a transition between two
Landau levels with An = 2. This work was extended by Yafet (1966) and by
Kelley and Wright (1966), using the actual band structure and wave functions of
InSb-type crystals. Here, also spin-reversal transitions with Δη = 0, As = 1 in the
conduction and valence bands via virtual interband transitions were considered.
The SFR process is only possible in the presence of spin-orbit interaction. A
mixing of spin and orbital states is required for spin-flip transitions (SF
transitions) due to electric-type perturbations, as was already pointed out by
Elliott (1954) and by Elliot and Loudon (1963). Detailed calculations of Landau
level Raman scattering in InSb were then performed by Wright et al. (1969). The
Raman cross section in a semiconductor showed to be surprisingly large since
the Thomson cross section is larger than that of a free electron by a factor
(m/m*) 2. Another essential characteristic of the SFR cross section is an energy
denominator which causes strong resonance enhancement as soon as the
incident-photon energy is close to one of the optically allowed intermediate
states.
It was not long before Slusher, Patel and Fleury (1967) reported the
observation of spontaneous SFR scattering from free electrons in InSb and
Thomas and Hopfield (1968) observed the SFR spectrum from donor-bound
electrons and acceptor-bound holes in CdS.
The large scattering efficiency and the small line width in InSb favour the SFR
process for stimulated Raman scattering and this effect could be detected by
Patel and Shaw (1970).
One important improvement in the experiments with InSb was the use of a
CO-Laser by Mooradian et al. (1970). Its photon energy approximately equals
the bandgap energy at the temperature of liquid helium. With the cross section
resonantly enhanced, efficient, continuous operation of the SFR laser was
achieved at threshold powers less than 50 mW (Brueck and Mooradian 1971).
An effort was made in many laboratories to develop this frequency tunable,
intense coherent light source to an instrument for high-resolution spectroscopy.
212
H.G. Hafele
H0
Fig. 1. Modulation of electronic polarizability by a precessing spin (from Geschwind
Romestain 1984).
and
In the last years SFR scattering has enabled the observation of a great variety
of excitations in nonmagnetic and diluted magnetic semiconductors. SFR
scattering from both free electrons and holes, electrons bound to donors and
holes bound to acceptors, as well as scattering from bound magnetic polarons
and scattering due to SF transitions within the Zeeman multiplets of para­
magnetic ions have been investigated. In addition to pure SF transitions also
combined resonances with a>CR = a > C Ry — a > SF were detected.
From these measurements band structure and one-electron parameters such
as effective g-factors and lifetimes were obtained and information about the
dynamics of carriers could be derived.
A few reviews have been published which emphasized the various aspects
differently. The most comprehensive articles, especially on SFR scattering in
InSb are by Smith et al. (1977) and Colles and Pidgeon (1975). Studies of freehole spin-flip and spin-flip of holes bound to acceptors in various p-type
materials have been summarized by Scott (1980a). A review of SFR spec­
troscopy in CdS was given by Geschwind and Romestain (1984).
This chapter gives a survey of the investigations for a wide range of materials
with emphasis on the solid-state aspect. It comprises a general part in which the
principles of investigation of SFR processes are presented and a second part,
which contains the most important SFR features in the various semiconductors.
Scattering from spin-density fluctuations [see Abstreiter et al. (1984) and Klein
(1975)] and with spin-dependent Raman scattering in magnetic semiconductors
[see Guntherodt and Zeyher (1984)] will not be treated.
2. Theory
2.1. Classical
of SFR
scattering
picture
It is very instructive to start with a classical picture of SFR scattering proposed
by Geschwind et al. (1977). A spin center precesses at frequency ω 8 Ρ = g*μBB in
an external magnetic field. With existence of spin-orbit coupling, it will drag
around its electronic charge cloud and therefore, the electronic polarizability is
modulated at the same frequency.
Spin-flip Raman scattering
2.2. Spontaneous and stimulated SFR
213
scattering
Raman scattering is a special case of a two-photon process. Through the
electron-radiation interaction an incident laser photon with frequency a>L and
wave vector kL is absorbed while simultaneously a photon at ω 8, ks is emitted.
The scattering sample is excited from an initial state |i> to a final state |f>. In
SFR scattering this is associated with a change of the spin state. The process is
only possible if both energy and momentum are conserved,
hcoL - hcos = E{hkL-hks
Et = ηωη,
(2)
= hq.
(3)
E{ and Ex are the energies of the two states and their difference is the spin
splitting
AE = E{ - E{ = |g*|/i Bfl =
toSF,
(4)
where co SF is the spin-flip frequency. With coL > cos the process is referred to as
Stokes scattering and with a > s > c o L as anti-Stokes scattering. The scattering
vector q must be taken over the electronic system. The conditions (2) and (3) are
illustrated in fig. 2. As an example Stokes scattering from conduction-band
electrons is chosen. Initial and final states are situated in two different spin subbands of the lowest Landau level. The corresponding scheme for SFR scattering
from holes in the valence band of a p-type semiconductor is drawn in fig. 26.
α
b
Fig. 2. Virtual interband transitions for SFR scattering from conduction band electrons, (a) Energy
diagram, c - conduction band with degenerate electron gas; 0 | , 01 spin-up and spin-down sub-bands
in the η = 0 Landau level; EF - Fermi level; AE - spin splitting; ν - valence band; states have mixed
spin character, (b) Momentum diagram. kL - wave vector of incident laser radiation; ks - wave
vector of scattered Stokes radiation. Momentum transfer q = kL — ks;q = 2\kL\ sin Θ/2.
214
H.G. Hafele
The Hamiltonian for the coupling of electrons to the radiation field is
Ί2
(5)
2m y Pj-~cA(rj)
where p} denotes the momentum of the jth electron, A(r}) is the vector potential
of the electromagnetic field. It is the sum of the vector potentials of the incident
laser beam ( a > L, kL) and the scattered light ( ω 8 , ks). A(rj) is described in terms of
the photon creation and annihiliation operators ax and ax, respectively, for
photons in direction kx, with frequency ωΧ and polarization ex. Then the
interaction is given by
mc2J
x
x[ex-p
\
sV
J
aifS(m<x>
x
x)1/2
exp(i* A · r)ax + e -p exp( -ikx
· r)af],
(6)
where m is the free-electron mass, ε is the dielectric constant and V the volume of
the sample.
This interaction can be considered as a time dependent perturbation and since
the perturbation is linear in A(rj), the calculation has to be carried out to second
order in H'. The known \A\2 term is ignored because it does not contribute to
Raman scattering in the dipole approximation (Yafet 1966). The calculation
results in a Raman transition probability which is proportional to the matrix
elements A{i9 for exciting or de-exciting the sample and for creating and
annihilating photons (see, e.g., Shen 1975).
The SFR scattering turns out to arise from the interband ρ · A matrix elements
between the conduction band and the valence band which is split by spin-orbit
coupling. In this chapter we are mainly interested in these matrix elements,
because they reflect the relevant features of the band structure. This part of the
transition probability involves the scattering amplitude R which was evaluated
as (Yafet 1966, Wright et al. 1969, Smith et al. 1977)
R =
y [ < f | g - g | t > < t l g L- p | i > _
<f|e L^lt><t|e s*>i;|i>|
V[
hcoL-(Et-Ei)
fta>s+
j'
)
where ν is the electron velocity operator in the presence of a magnetic field.
Equation (7) shows the Raman process to occur in two steps. The electron
makes a virtual transition from |i> to an intermediate state |t> followed by a
second virtual transition from |t> to final state |f>. The sum is carried over all
intermediate states |t>. We see that any theoretical treatment of SFR scattering
is concerned with the identification of the initial and final states and the
operative intermediate states. The two terms between the brackets comprise the
same transitions, but differ in the order in which the emission of the Stokes
photon and the absorption of the laser photon takes place. Total energy of
electron and photons is conserved for the whole transition |i>->>|f>, but not for
[
Spin-flip Raman scattering
215
the single step |i> |t> and |t>
|f>. According to the uncertainty principle, this
is allowed for a corresponding short time.
The existence of spontaneous and stimulated Raman scattering arises from
the properties of the photon creation and annihilation operators. For simplicity,
we assume that in the beginning there is only one laser mode and one Stokes
mode with the numbers of quanta nL and ns, respectively. If |a f> and |a f > denote
the states of the radiation field, then the Stokes process is characterized by
l«i> =
"s>
|a f> = |wL - 1, ns + 1>.
and
(8)
As known, the effect of the Boson operators a+ and a on the states is given by
dL\nL}
= V^LI^L- Ο,
a+ |n s> = y/ns+l\ns
(9a)
+ 1 >.
(9b)
Whence
\ < ^ α ^ } \
2
= ή^η8+1)
(10)
Consequently the Raman transition probability
W(IK\An\2nL(ns+l).
(11)
In spontaneous Raman scattering only a very small fraction of the states in the
Stokes mode are occupied, n s<^ 1. Then the rate of Stokes emission is pro­
portional to n L, i.e., the incident laser intensity. If this intensity is high and
sufficient Stokes photons are being produced in the relevant mode, i.e., n s > 1,
then stimulated Raman scattering becomes dominant due to the common action
of Stokes and laser radiation. N o w the scattering probability is proportional to
the product n Ln s of the numbers of photons present in the incident and scattered
mode. By virtue of the factor ns an amplification of the Stokes radiation is
expected. The rate of change of the number of Stokes photons is given by
dnsdt
dns
di
=
TTdx"
,
/ws\-
-
1 mT l 9
=
f1
l i
( 7 ρ
"
a s s
"
=
"
Here the rate per unit time is expressed by the rate per unit length in the
medium. ns is the refractive index at frequency ω 5, Ν is the effective number of
particles in the material system taking part in the process. A loss term is added,
characterized by the total absorption coefficient a s at ω 5. gs is called the Raman
gain. Integration of eq. (12) leads to
"s(*) = «s(0) exp(g s - a s) x.
(13)
By the Raman process the Stokes mode gains power at the expense of the laser
mode. If the Raman gain exceeds the losses in the material, the photon density in
the Stokes mode grows exponentially. Under actual experimental conditions
spontaneous Raman radiation is not emitted in a definite single Stokes mode,
H.G. Hafele
216
but rather in a continuum lying in a certain frequency interval, which is given by
a line shape function #(Δω).
The magnitude of the scattered intensity may be described by the differential
scattering cross section άσ/άΩ. It is defined as the Stokes energy radiated from
all scattering elements per unit time, per unit volume, per unit solid angle, per
unit incident flux in the medium for each direction of polarization (see, e.g.,
Hayes and Loudon 1978, Smith et al. 1977). The cross section for SFR scattering
turns out to be (Yafet 1966, Wright et al. 1969)
(14)
The summation is over all initial states involved in the Raman process. R is the
scattering amplitude of eq. (7), n s and n L denote the refractive indices for the
scattered and incident laser light, respectively. Apart from the matrix elements
in R the cross section is sensitive to the electron population and the pump
frequency. The first is obtained by integrating the right-hand side of eq. (14) to
evaluate the total cross section. It results in (Colles and Pidgeon 1975, Brueck et
al. 1973),
(15)
O "
n,k z
= \F(0)\2S,
(15a)
where fn] and fni denote the Fermi population factors for the spin-up and spindown states, respectively, in the nth Landau level. |F(0)| 2 is the cross section per
electron at kz = 0, i.e., the kz dependence of F is ignored. At low temperatures,
the coefficient S can be approximated by the population difference
SsxN.-N,.
(16)
Transitions are only possible from occupied to empty states. So a transition is
blocked if the upper spin state is occupied. Referring to the situation drawn in
fig. 2a, the Fermi level EF must lie between the lower and the upper spin level.
Wherrett and Harper (1969) and Makarov (1969) pointed out that for this
situation the one-electron transitions are forbidden. An adequate description is
a two-electron process in which first an electron in the valence band is excited in
the (empty) upper spin level and then another electron of the (occupied) lower
spin level goes back into the valence band hole. Blocking by filling of the spin
states leads to oscillations in the number of electrons contributing to the Raman
process. The calculations of Wherrett and Harper (1969) show turning points in
this number which correspond to electron concentrations and magnetic fields at
which the Fermi level crosses the Landau levels. These quantum oscillations are
well manifested in the output power of stimulated SFR scattering in both InSb
and (HgCd)Te.
Spin-flip Raman scattering
217
A further important characteristic of the SFR scattering cross section is the
resonance denominator in the scattering amplitude R of eq. (7). As a conse­
quence, the intensity of scattered light is resonantly enhanced as the incident
photon energy ftcoL approaches the excitation energy of the intermediate state.
In this situation, the cross section is proportional to an enhancement factor of
the form
(17)
where E% is the magnetic-field dependent energy of the band gap.
2.3. Semiclassical treatment of stimulated Raman
scattering
According to eq. (11) stimulated Raman scattering arises from the simultaneous
action of the photons of the laser and the Stokes modes upon the material.
Unlike spontaneous processes, this can conveniently be treated with a semiclass­
ical approach, where the radiation is described by electromagnetic waves. In this
electromagnetic theory, stimulated scattering originates in a coupling of the
waves by a nonlinear electric polarization [see, e.g., Bloembergen (1965) and
Yariv (1975)]. The induced polarization is of third order in the electric field
strength and can be expressed as (Shen 1975)
P*\cos)
S ( j d 3 ,| £ L| 2 + X&\ES\2)EL,
(18a)
(x&|£ l + χ^\Ε \ )Ε
(18b)
^
2
L
5
2
8>
where EL and Es denote the laser and scattered electric field vectors. The thirdorder nonlinear susceptibilities χ ( 3) are in general fourth-rank tensors, but here
they are assumed to be scalar. The terms with χ { 3) and χ 83) are known to modify
the dielectric constants at a> L and ω 8 and will be neglected later on. The
quantities χ } 3) are the Raman susceptibilities at coL and ω 8 . The relevant term is
the first one on the right-hand side of eq. (18b). It represents a polarization
which is proportional to the electric field Es of the existing Stokes wave and to
the power | £ L | 2 of the pump beam. So it oscillates with frequency ω 8 and emits
Stokes radiation.
The Raman susceptibility at frequency ω has the form (Smith et al. 1977)
ά3)(ω) = Α-
Σ
~r-r
W 2-
<1 9>
ηω{ω / ω — ω 8 + inis
As in eq. (14) summation runs over all occupied initial states |i> with nonblocked final states and R is the scattering amplitude of eq. (7). Here a
Lorentzian lineshape is assumed where T s denotes the half width of the line.
In resonance, ω = ω 8 , the susceptibility has a sharp maximum and becomes
H.G. Hafele
218
negative imaginary, so we can write
(20)
To consider the stimulated Raman process, we take the two waves in the form of
plane waves travelling in the x-direction. Then the propagation is governed by
the wave equations
sLd2EL
d2EL
4 π δ 2/ * 3)
=
"ΕΡ" -
4π θ 2
= ^ a ? ^
S - W -
,
1
)
^
- 5 ? ~ -?-W = Ϊ - Ύ Γ = ^ < * W * > .
( 2 1 a
(2ib)
From this, it follows that the nonlinear polarizations couple the two waves and
constitute sources for the fields. The stimulation of Raman scattering arises from
energy transfer from the laser wave to the Stokes wave. In a simplified
calculation, the transfer in the opposite direction is neglected, and one assumes
both that the pump intensity | £ L | 2 remains nearly constant and that the
variation of the amplitude of the Stokes wave with χ is small. With this
approximations eq. (21b) has the solution (Smith et al. 1977)
Es(x) = Es(0) e x p f
\
\EL\2 - %)x,
2J
nsc
(22a)
or
Is(x) = / s( 0 ) exp(g s - a s)x,
(22b)
where Is is the Stokes energy flux and the Raman gain factor g s is given by
,s = ^ |
£
nsc
l
| > .
)
The peak Raman gain gP in the center of the Lorentzian line can be expressed by
the scattering cross section as
16π 2 c2IL άσ
g
p
=
f e ^ d O
=
g
o
/
'
L
)
where J L denotes the incident laser energy flux.
Since the Stokes intensity J s is proportional to the number ns of photons in the
Stokes modes, eq. (22b) is equivalent to eq. (13). The exponential increase of the
Stokes intensity starts as soon as the gain exceeds the loss, or according to
eq. (23) as the incident laser intensity exceeds a certain threshold value. It is
worth to note that in stimulated Raman scattering the phase-matching con­
dition of eq. (3) is automatically fulfilled since the phase of the laser radiation
drops out.
( 2 3
Spin-flip Raman scattering
2.4. Higher-order processes by Stokes-anti-Stokes
219
coupling
Besides intense stimulated Stokes radiation also higher order Stokes and antiStokes waves with frequencies coL ± η ω 5Ρ are observed (n = 1, 2 , . . . ) . These are
generated by the parametric coupling between the various light waves. When the
Stokes field Es at frequency ω δ has grown up, it beats with the pump radiation at
co L and drives the spin-flip transition at the difference frequency coL — cos = a> S F.
From this, the laser beam is scattered coherently generating intense anti-Stokes
light at c o a = coL + c o S .F This coupling can be described by a nonlinear polariz­
ation source term with the ith Cartesian component
P?HvJ = n%lELjELjE$l.
(25)
In a similar way, second and higher order Stokes as well as higher order antiStokes fields are generated by the successively induced third-order polarizations
(Patel 1971, Wherrett 1972, Yuen et al. 1974b). For example, waves at ω 2 8 =
c o L - 2a>SF originate in the coupling by a polarization, the ith component of which
is given by
P?L(o>2S)
= X{ilijkiE^ESkEl.
(26)
Adding the corresponding wave equation to the system of eqs (21), in the smallsignal approximation one gets solutions in the form of eq. (31) (Shen 1975).
3. Principles
of investigation
of SFR
scattering
3.1. General remarks
Independent of the special techniques, the equipment which is usually employed
to study SFR scattering exhibits common features. The samples are inserted in
low-temperature optical cryostats and are subjected to a steady-state magnetic
field. Excitation of the spectra is always provided by a laser. If possible, the laser
wave length is selected close to the absorption-band edge of the semiconductor
to take advantage of the resonance enhancement. Most experiments were
performed with an A r + - or K r +- i o n laser, a dye laser, H e - N e laser, Co laser
or C 0 2 laser.
The scattered light is collected and analyzed by a monochromator. If needed,
double or triple grating monochromators are used to reduce stray light. In the
visible region, the scattered light is detected by means of photomultipliers
incorporated in photon counting systems. In the IR region, detection is made by
photoconductive or photovoltaic detectors in connection with lock-in amplifiers
or box-car integrators.
A versatile arrangement for SFR experiments at the extremes of both
magnetic field and temperature has been demonstrated very recently. Fiber
optic cables pass the pump laser light to the sample and the collected scattered
H.G. Hafele
220
light to the spectrometer. This system enabled the observation of SFR spectra of
(CdMn)Se, mounted in a liquid 3He-cryostat and placed into the bore of a 30 Τ
hybrid magnet (Isaacs and Heiman 1987).
3.2. Spontaneous SFR
scattering
By far, most investigations on SFR scattering are concerned with spontaneous
scattering. The scattering configurations must be chosen with respect to the
transmission properties of the sample and the polarization selection rules. Four
different arrangements are drawn in fig. 3(a-d). About the different geometries,
the following remarks can be made:
In the arrangement shown in fig. 3a, the scattered light is observed in forward
direction, with both beams propagating perpendicular to the direction of the
external field Β (Voigt configuration). This collinear geometry is often called
q · Β = 0 geometry, because the scattering wave vector q = kL — ks is normal to
the field. The electric vector of the plane polarized pump radiation has the
direction of the magnetic field, whereas the polarization of the Raman radiation
is orthogonal to it. In the usual notation, this scattering geometry is described by
the form y(z, x)y. The first and the last letter denotes the direction of the incident
Fig. 3. Geometries for SFR scattering. The magnetic field Β is chosen parallel to the z-direction. (a)
Forward (collinear) scattering geometry, (b) Right-angle geometry with the scattered light along B.
(c) Right-angle geometry with the scattered light perpendicular to B. (d) Backscattering geometry.
Spin-flip Raman scattering
221
and scattered light, and the letters in the brackets refer to the direction of
polarization of the incident and scattered light, respectively (Hayes and Loudon
1978).
In the arrangement shown in fig. 3b, the scattered light is collected at right
angles to the incoming light and propagates parallel to the magnetic field. Right
and left circular polarizations of photons which propagate along the magnetic
field are denoted by σ+ and <r_, respectively. In these two geometries (a) and (b),
measurements have been performed on e.g., InSb (Brueck et al. 1973).
In the right-angle geometry of fig. 3c scattered light is collected at 90° from
the incoming beam and the magnetic field. For linearly polarized light the
configurations are y(x, z)x and y(z, y)x. SF scattering in this geometry was
reported for ZnTe (Hollis and Scott 1977) and (CdMn)Se (Nawrocki et al.
1981).
A suitable arrangement for crystals in the transparent or opaque region is
presented in fig. 3d. The light scattered from the illuminated face of the sample is
collected in backward direction (configuration y (x, z)y). Raman experiments on
CdS (Thomas and Hopfield 1968) and ZnTe (Oka and Cardona 1981) were
performed using such a back scattering geometry.
A presentation of various geometries for spontaneous Raman scattering in the
presence of a magnetic field can be found in a study of the diluted magnetic
semiconductor (CdMn)Te (Petrou et al. 1983).
3.3. Stimulated SFR
scattering
Corresponding to eq. (22b), stimulated SFR scattering is generated if the Raman
gain is bigger than the absorptive losses in the medium. The conventional
scattering geometry is that of fig. 3a, i.e., the stimulated Stokes radiation is
collinear with the pump beam and perpendicular to the magnetic field. This
q · Β = 0 geometry maximizes the gain length for the stimulated scattering and
provides generally much narrower line widths as in a q · Β Φ 0. The scattering
sample constitutes an optical resonator of length I and reflectivity R if the faces
of the crystal are polished parallel. R may be the natural reflectivity of the
material or may be that of coatings. Then the condition for Raman laser
oscillation is given by
«exp(&-Os)/>l.
(27)
In the first experiments on stimulated SFR scattering with InSb, low pressure,
Q-switched C 0 2 lasers were employed (Patel and Shaw 1970, 1971, Allwood et
al. 1970, Irslinger et al. 1971). High output power up to 1 kW is possible by
means of a high power C 0 2 TEA pump laser with a large InSb crystal of 2 cm
length (Aggarwal et al. 1971). Pumping near resonance by using a CO laser
enabled continuous operation with conversion efficiencies in excess of 50% and
SRF laser power of 1 W in the 5 μιη range (Mooradian et al. 1970, Brueck and
H.G. Hafele
222
Mooradian 1971). In other experiments, even conversion efficiencies as high as
80% were observed and the spectrum could be extended up to the fourth Stokes
order (de Silet and Patel 1973).
The large amount of experimental and theoretical work done on stimulated
SFR scattering in InSb is fully described in the reviews of Smith et al. (1977) and
of Colles and Pidgeon (1975).
The number of semiconductors in which stimulated SFR scattering has been
observed is, so far, not very large. The strong resonant cross section together
with the narrow line width enables the SFR scattering in CdS to become
stimulated at excitation powers higher than 3 M W / c m 2 (Scott and Damen 1972,
Scott et al. 1972). Additionally, this stimulated SFR scattering has been achieved
in InAs by near-resonant pumping with a HF-TEA laser (Eng et al. 1974).
Measurements of stimulated SFR scattering were also reported on H g x _j.Cdj.Te
(Sattler et al. 1974) and n - P b ^ S n ^ T e (Yasuda and Shirafuji 1980). Here, the
band gap energy can be adjusted to the C 0 2 laser wavelength by selecting the
proper value of the concentration parameter x. Stimulation of the SFR
scattering also takes place in semimagnetic semiconductors, which exhibit giant
spin effects. Laser action was demonstrated in (HgMn)Te (Geyer and Fan 1980)
and in (CdMn)Se (Heiman 1982).
3.4. Raman gain
measurements
For precise measurements of the SFR gain in InSb, amplifier methods have been
employed (de Silets and Patel 1973, Brueck and Mooradian 1973, Pascher et al.
1976b). In these gain experiments two CO laser beams are collinearly superposed
in the sample. The intense pump at c o L and the weak probe at ω 8 propagate
normal to Β with polarization orthogonal to one another (EL1ES).
In or near
resonance, i.e., if the condition
ω _ - ω 5 = g*pBB/h,
(28)
is satisfied, the probe wave is amplified in the sample and gains power according
to eq. (22) by energy transfer from the pump wave. Measuring the intensity 7 S(0)
Linewidth = 3G <^> ZOO MHz
I
I
I
1780
I
I
1
I
I
1785
MAGNETIC
I
I
I
I
I
1790
I
I
I
I
I
I
I
1795
FIELD (G)
Fig. 4. Gain of SFR scattering in InSb. n = 8 x 1 0 1 c4 m ~ 3, coL - cos = 4.17 c m " \ T=2K
Brueck and Mooradian 1973).
(from
Spin-flip Raman scattering
223
and Is(l) at the front and the end surfaces of the sample, respectively, allows
calculation of absolute values of the Raman gain factor gs.
Detecting the transmitted power at ω 8 as a function of magnetic field in the
small-signal regime gives the line shape directly. In general, the gain factor g s has
the same shape as the spontaneous SFR line. Therefore, this technique bypasses
the need for spectrometers in measuring spontaneous scattering line widths. The
resolution is only limited by the frequency instabilities between the two lasers.
As will be seen later, the actual SFR line widths of a few hundred MHz are
beyond the resolving capability of spectrometers. From the well-known laser
frequencies and the measured magnetic field at the line center the effective
g-value can be calculated. Figure 4 shows the amplification of the test laser
intensity in a InSb sample due to Stokes gain as a function of magnetic field.
3.5. Resonant four-wave mixing,
CARS-spectroscopy
As already introduced, the Raman susceptibility χ ( 3) is a fourth-rank tensor
parametrized by four frequencies and polarizations. So it permits the mixing of
three independent fields of known frequency, polarization, amplitude and phase.
As a very useful spectroscopic method to measure Raman spectra, a special case
of optical four-wave mixing, the coherent anti-Stokes Raman scattering (CARS)
was developed (Eesley 1981, Maier 1976, Pascher 1984). Similar to the gain
measurement, two laser beams at the pump and signal frequencies coL and ω 8 are
superimposed in the medium. The difference of the frequencies a>L — ω 8 is chosen
to be close to the Raman-active SF transition at ω 8 Ρ. An energy-level diagram
for coherent Raman mixing is presented in fig. 29.
By the mixing process a nonlinear polarization is induced in the material
PfL
=
XuJii -<*>a> co L, co L, -a)s)Ej(x,
OJl)EJ(X,
χ exp i[(2kLx - kSx)x - (2a)L - ω 8) ί ) ] .
<y L)£f(x, - ω 8 )
(29)
It oscillates at the anti-Stokes frequency
(DA =
(DL
+
ω 8Ρ =
2(DL
-
ω 8,
(30)
and acts as a source to amplify anti-Stokes waves. This four-wave mixing
method is advantageous in comparison with direct Raman measurements, as
the signal frequency itself can be chosen away from spectral ranges with
fluorescence or other excitations. Especially it is not influenced by phonon
or plasmon effects. To calculate the CARS intensity the polarization term of
eq. (29) is substituted into the wave equation for the anti-Stokes field £ a , set up
in the same way as and in addition to eqs (21a, b). With the usual slowly varying
wave approximation it results in (Eesley 1981)
(31)
H.G. Hafele
224
where / is the length of the sample, ΔΑ: = 2kL — ks — ka and χ is a vector of unit
length in x-direction. If phasematching is achieved by a non-collinear geometry,
the anti-Stokes intensity increases quadratically with the length Z.
It is important to note that 7 a is proportional to the square of the nonlinear
susceptibility. As shown by Eesley (1981), the third-order nonlinear suscepti­
bility is a sum of 24 different photon-matter interaction terms, four of them
containing Raman resonances. If one of these resonances dominates the others,
the expression can be simplified and the susceptibility may be divided in a
complex resonant and a real nonresonant contribution. Separating real and
imaginary parts we get,
(32)
(33)
The interference between resonant and nonresonant contributions may lead to
complicated line shapes of the CARS spectra, as was analyzed by Pascher
(1984). It turns out that by the CARS technique, line shapes and effective
g-factors can be derived well, in as much as some experimental parameters are
carefully taken into account.
Four-wave mixing was used for studying SF transitions in InSb (Nguyen et al.
1976, Pascher et al. 1980), (HgCd)Te (Bridges et al. 1979, Pascher 1983), PbSe
(Pascher et al. 1983b) and (PbMn)Te (Pascher et al. 1987).
3.6. SFR scattering from coherent spin states
The generation of anti-Stokes SFR radiation by four-wave mixing can also be
viewed in a manner usually applied to electron-spin resonance. It is the adequate
description of non-equilibrium situations and was first introduced for measure­
ments in CdS by Romestain et al. (1974) and in InSb by Nguyen et al. (1976). In
the nonlinear Raman process, the incoming waves at a>L and ω 8 drive the spin
precession at the difference frequency ω 8 Ρ = a>L — ω 5. By this, a transverse mag­
netization is produced. Further coherent scattering of radiation at frequen­
cies coL and a>s from the coherent-spin states results in anti-Stokes and secondStokes light at frequencies
ω 3 = co L + ω 8 Ρ = 2o)L - ω 8,
(30)
ω 2 8 = ws - wSF = 2cos - coL.
(34)
It has been shown that the Raman interaction in cubic symmetry can be
brought in the form of an effective spin-flip Hamiltonian (Yafet 1966, Brown and
Wolff 1972, Romestain et al. 1974, Hu et al. 1976)
ΗrsF = ι « σ · (EL χ Es) e x p [ - i((coL - a>s)f - (* L - * s ) · r)] + c.c,
(35)
where EL and Es are the pump and Stokes fields, respectively, σ are the Pauli
Spin-flip Raman scattering
225
matrices and α is related to the spontaneous differential Raman cross section.
From this, it follows that the vector product
a ( £ L χ Es) e x p [ - i ( a > L - ω 8) ί ]
is equivalent to a linearly polarized effective transverse r.f. magnetic field H1
with frequency coL — ω 8 = ω 8 Ρ. It has the strength (Nguyen et al. 1976)
H\ = -τ
ELES,
η ms Q)Lcos
where m s = 2m/\g\ is the spin mass and F is the resonant enhancement factor for
the Raman interaction.
The solution of the Bloch equation yields a transverse magnetization due to
the resonantly driven spins (Yariv 1975)
.
tow
,
Here y denotes the gyromagnetic ratio, T2 the transverse spin relaxation time, Tx
the spin-lattice relaxation time and σζ0 the equilibrium longitudinal magneti­
zation. The power for the second Stokes excitation generated by Raman
scattering from the precessing spins has been calculated on the basis of a plane
wave approximation (Romestain et al. 1974) and is given by
n
f
» -
f
lddli
» 4 d f l f
. ,
n M
„ 2 ( 1 — cos Akl)
(Afe/)
2
. „
'
)
where άσ/άΩ is the SFR cross section, A 28 the free space second-Stokes
wavelength, n e ff the effective density of scattering electrons, ε the dielectric
constant, and / the sample thickness. Equation (37) in connection with eq. (36)
allows the determination of the relaxation times Tx and T2 if the powers P 2 8a n^
Ps are absolutely measured (Nguyen et al. 1976).
Another kind of four-wave mixing experiment, called time-delayed CARS,
circumvents the difficult measurement of absolute powers to get the relaxation
time T2 (Pascher et al. 1980, Pascher 1983a). Spin precession is coherently
excited by stimulated SFR scattering of a first strong laser pulse at frequency c o L
and subsequently decays freely. The decay is monitored by coherent forward
Stokes (or anti-Stokes) Raman scattering of a second weak probe laser pulse at
ω 8, which is properly delayed with respect to the first pulse. As long as the
precessing spins maintain coherence, a signal at ω 2 8 = ω 8 - ω 8 Ρ can be ob­
served and the dependence of its intensity on the delay time yields the dephasing
time T 2.
3.7. Spin-flip Raman echo
As seen in section 3.6, coherence can be induced between two spin states by
stimulated SFR scattering or by two-wave mixing. After a suitable short and
(
226
H.G. Hafele
intense exciting pulse, normal precession proceeds in the absence of an external
electromagnetic field. However, since the definite phase relationship existing
between the spin states is lost, the magnetization falls to zero within a time T2.
As known, this dephasing time is composed of two constituent parts, T5°m and
T 2n h. The first part, T 2 o m, characterizes stochastic interactions with the
surroundings, by which the phase of the electron-spin precession is randomly
and irreversible interrupted. The second one, T f2n h, corresponds to the phase
decay which is caused by slightly different resonance frequencies co SF of different
electrons, leading to a line width Δω£ρ. Anticipating the discussion in sections
5.1.1.4 and 5.2.2.4, this inhomogeneity may arise from band nonparabolicity,
Doppler effect (diffusion), local strains, dislocations, and so on. The concept of
echo experiments is based on the fact that the phase decay due to these
inhomogeneous line-broadening effects is reversible (Hartmann 1968, Shoe­
maker 1978).
The first Raman echo reported involved SFR scattering from bound donor
states in CdS (Hu et al. 1976, Geschwind et al. 1977, Hu et al. 1978, Geschwind
and Romestain 1984). In this experiment, the sample is irradiated by a certain
sequence of short laser pulses. The first pulse of an argon laser causes a nutation
with a 'tipping' angle (Hu et al. 1978)
(38)
where Δί is the duration of the pulse. Immediately after the end of this π/2-pulse,
all spins are precessing in phase and the resultant macroscopic transverse
magnetization σ τ is a maximum. Later on, the coherence will be destroyed due
to the distribution of the angular frequencies of different spins inside the
inhomogeneously broadened spontaneous spin-flip line. After dephasing, a
second pulse is applied at time τ with intensity and duration chosen to yield
θ = π, i.e., to reverse the relative order of the precessing spins. This results in a
rephasing at time 2τ. A dye laser probe pulse set at 2τ is coherently scattered by
the coherent spin system and generates the Raman echo. By measuring the echo
amplitude as a function of τ the dephasing time T 2 o m is obtained (see section
5.2.2.4).
3.8. SFR scattering and electric
transport
Stimulated SFR scattering can create strong deviations from the thermal spin
population. It turned out that under SF-resonance conditions the electrical
conductivity in n-InSb is changed. Although it is not directly spin-dependent,
carriers are subject to energy dependent scattering mechanisms, which cause
them to have different mobilities in different spin sublevels. Also, the effective
mass is energy dependent, due to the nonparabolicity of the bands.
The behaviour of the electrical conductivity of a n-InSb sample, accompany­
ing SFR-laser action was first observed by means of a pulsed C 0 2 laser (Grisar
Spin-flip Raman scattering
227
et al. 1976, Grisar and Wachernig 1977). To study the mechanism which brings
about the modification of the electrical conductivity under spin-flip resonance
conditions, detailed magnetoresistance measurements have been performed by
Skok and Studenikin (1983), Studenikin and Skok (1986), and Pascher et al.
(1982b). By means of a sensitive magnetic-field modulation technique the SFR
gain at ω 8 and the photoconductive response from the specimen was recorded
simultaneously (see section 5.1.1.5).
4. Application
of the SFR
laser
The observation of intense stimulated SFR scattering in n-InSb samples led to
the development of the SFR laser. This oscillator is a tunable, coherent infrared
light source of high power with narrow line width. An essential feature of
stimulated SFR scattering is the narrowing of the spontaneous emission line. As
with any laser, the line width of a SFR laser is determined jointly by the
spontaneous linewidth, the cavity line width and the emitted Stokes power.
Under different experimental conditions line widths of SFR-laser radiation from
1 MHz to a few 100 MHz are quoted in the literature. With highly sophisticated
equipment Patel (1974) succeeded in demonstrating a line width of less than
1 Hz.
These properties make the SFR laser a useful tool for some applications in
spectroscopy. Especially absorption spectra of molecular gases can conveniently
be recorded with high spectral resolution, which can hardly be achieved by
means of conventional spectrometer arrangements. These aspects of application
are treated in the reviews of Smith et al. (1977), Colles and Pidgeon (1975) and in
papers of Pidgeon (1972) and Hafele (1974). The potential of the InSb-SFR laser
for high-resolution IR spectroscopy has been demonstrated, e.g., in the experi­
ments of Fait et al. (1977), Mozolowski et al. (1979) and Haj-Abdallah et al.
(1985). Unfortunately, the expectations have, so far, not been realized in a lot of
practical applications.
5 . Experimental
and theoretical
semiconductors
5.1. 111-V-compound
results of
nonmagnetic
semiconductors
5.1.1. Indium antimonide
(InSb)
5.1.1.1. Origin of SFR spectra.
The far most numerous and detailed experi­
mental and theoretical studies on SFR scattering have concerned InSb. This
material exhibits the largest Raman scattering cross section yet observed and
the spontaneous line width is extremely narrow. First we consider the situation
H.G. Hafele
228
of an η-type InSb crystal with a full valence band and the lower spin level of the
conduction band Landau state η = 0 (|c, 0 | » populated by electrons. Provided
that the upper spin state |c, 0J,> is empty, an electron from the valence band can
be excited to the upper spin level with the absorption of an incident laser
photon. In the second step, an electron from the lower spin level relaxes to the
valence band with the emission of a Stokes photon. So, for the SFR process the
initial and final states are the conduction band spin-up and spin-down states,
respectively, as illustrated in fig. 2. It is obvious that due to the large interband
matrix elements and the energy denominator in the scattering amplitude R, the
essential contributions arise from intermediate states in the light-hole and
heavy-hole valence bands near the Γ-point. As the light waves do not couple to
the electron spin directly, it is the mixed-spin character of these states which
gives rise to a non-zero value of R.
5.1.1.2. Cross section and selection rules. The various studies on the structure
of the conduction and valence bands in InSb in a magnetic field vary in the
degree in which the three main features, i.e., non parabolicity, finite momentum
kz and higher band interactions are accounted for, reviews are given, e.g., by
Pidgeon (1980) and Zawadzki (1973). For new developments we refer to Weiler
et al. (1978), Weiler (1979), Trebin et al. (1979), Braun and Rossler (1985). For
the calculation of SFR scattering in InSb several authors used the model of
Pidgeon and Brown (1966) (P.B.). It takes into account the strong coupling
between the conduction and valence bands in narrow-gap semiconductors
which introduces a high degree of nonparabolicity. It also takes into account the
most important effects of warping or band anisotropy. In this approach, the
interaction between the conduction band Γ 6, heavy- and light-hole valence band
Γ 8 and split-off* band Γ 7 (three-band model) is treated exactly, while the
influence of higher bands are included in the effective-mass equation to order k2.
The energy-band structure of InSb near the Γ-point is represented in fig. 5.
In the paper of Dennis et al. (1972) it is shown that the P.B. model is adequate
for a correct description of the SFR process in InSb. Following this treatment,
the set of eight coupled equations for the conduction/valence band system has a
solution, where the single-electron wave functions ψ" of the conduction, heavyhole, light-hole and split-off bands (j = 1 to 8, including spins) are expanded as
linear combinations of products of Bloch functions with harmonic oscillator
functions φ„,
Ψ1=
Σ
<"/,o</v,
(39)
where / is to be summed over the six valence and two conduction band states,
the expansion coefficients a"$ depend on the magnetic field and the labels ri are
associated with the / in a fixed relation. The functions uft0 are the band-edge
basis functions for zero magnetic field in the (J, nij) representation. It is the
Spin-flip Raman scattering
229
Elk)
EUO
B*0
Fig. 5. Schematic representation of the energy band structure of InSb near the Γ-point. (a)
Conduction and valence bands without magnetic field, (b) Spin-split Landau levels in a magnetic
field (not to scale).
m i x e d - s p i n nature of the functions u 5 0 t o w 8 0 w h i c h a l l o w s for S F R scattering.
F o r kz = 0, the w a v e functions divide t o a g o o d a p p r o x i m a t i o n i n t o a n a-set ( /
= 1, 3, 5, 7) a n d a b-set ( / = 2 , 4 , 6 , 8) a n d o n e arrives at
^> =
4
ΐ
"
ΐ
θ
,Φ
η + <3"3,θΦπ-1 +455,θΦπ+1 + < 7 " 7 , Ο 0 π
Μ
+1
(40a)
,
and
*
A
b
,j = 422,(A + <6"6,0</>π-1
W
+
4
U
4
0
,<
>
/n
+
l
M
+ «7,8«8.O0n-
( 4
1·
Eigenfunctions b e l o n g i n g t o the a-set a n d t h o s e of the b-set describe c o m p l e ­
m e n t a r y spin configurations.
Further simplifications are m a d e in the u n c o u p l e d - b a n d m o d e l of Luttinger
a n d K o h n ( 1 9 5 5 ) , w h e r e the light- a n d h e a v y - h o l e b a n d s are formed by
*A"
a,
=
433,0</>n-l
W
Φ% = α16η6ί0φη.1
( 4 * 1)
+ < 5" 5 , Ο 0 π + 1 >
+ αη]Αη^0φη+ι.
(41b)
W i t h this a s s u m p t i o n Wright et al. ( 1 9 6 9 ) g a v e a c o n v e n i e n t description of the
S F R scattering a n d o b t a i n e d the cross s e c t i o n
d* _
(*y ( ω* z!_ Υ 8 (
2
άΩ '{mc )
2
Egm J 9\ω
V
2
- ω)
(42
)
'
where £ g = fta>g a n d P = - i / m < S | p 2 | Z > is the interband m o m e n t u m matrix
element. F r o m this it follows that the S F R p r o c e s s r e m a i n s finite for a v a n i s h i n g
m a g n e t i c field. Since in this p a r a b o l i c - b a n d a p p r o x i m a t i o n the c o n d u c t i o n -
}
{
H.G. Hqfele
230
ms = y
to
nn5= - y
τι
6R
Eg
Fig. 6. Transitions comprising the SFR process (not to scale). Contributions arise from three
valence- and one conduction-band intermediate state. 0 | and OJ, denote the spin-up and spin-down
sub-bands in the η = 0 Landau level (from Dennis et al. 1972).
band effective mass ra* ~ 3 / 4 £ g / P 2 , the cross section is roughly proportional to
the ratio (m/m*) 2 multiplied by the resonance factor.
Using the more precise wave functions of eq. (40) for evaluating the
transition matrix elements, Dennis et al. (1972) found that for π-input polariz­
ation there are essentially three intermediate states in the valence bands and one
in the conduction band. The two-stage transition scheme is drawn in fig. 6. The
cross section has its maximum value at zero field. The contribution which uses
the conduction-band state |c, 1|> as the intermediate state becomes only
significant at magnetic fields above 100 kG, where the intermixing of
conduction- and valence-band states becomes strong. Wherrett and Wolland
(1974) calculated the spontaneous cross section for the near-resonance
condition. From this paper, the intermediate levels with the relative magnitudes
of the matrix element products and polarizations are taken, and reproduced in
Table 1.
The dramatic resonance enhancement, which is due to the energy denomi­
nators in the SFR cross section, has been established in all experiments. In a
detailed study the near-resonance behaviour of the spontaneous scattering has
Table 1
Spin-flip transition φ"+1 -• φη2: |c, n] > -*· |c, n[ >
Intermediate states
n = 0|hh, 01>
n ^ 0 | h h , lh, s , n + 1|>
l|lh, s , n - 1|>
n > 2 | l h , n - 1T>
n ^ 0 | c , n + U>
Ol|c,n-lT>
n ^ 0 | l h , n|>
n ^ l | h h , «T>
n ^ 2 | h h , lh, n|>
Relative magnitude of
Polarization for which
matrix element product
resonance can occur
1
1
1
1
Epha)c0/El
Ephwc0/El
- )
(+,z)
(z, - )
(*,
(*,
- )
- )
(+,z)
(+,*)
£ P£ F o / £ g
EpEF0/El
EPEF0/El
(2, - )
Spin-flip Raman scattering
231
been investigated as a function of input-photon energy and magnetic field by
Brueck et al. (1973). Figure 7 shows the results along with the theoretical
resonance curve.
INCIDENT PHOTON WAVELENGTH
6)
nl
200
60
59
58
57
56
5.5
(/xm)
54
'.3
52
1
I
I
I
I
I
I
205
210
2)5
220
225
230
235
INCIDENT
PHOTON ENERGY
l_J
240
(meV)
Fig. 7. Resonance enhancement of spontaneous SFR scattering as the input-photon energy is
varied, η = 1 χ 1 0 16 c m - 3, Β = 40 kG, and T « 30 Κ (from Brueck et al. 1973).
H.G. Hafele
232
There is g o o d qualitative a g r e e m e n t b e t w e e n t h e o r y a n d experiment, with the
e x c e p t i o n of s o m e structure m e a s u r e d for i n p u t - p h o t o n energies a r o u n d
235 meV. T h e s u g g e s t i o n that this m i g h t arise from a s h a l l o w a c c e p t o r level a s
the intermediate state h a s b e e n confirmed later o n ( W a l u k i e w i c z et al. 1979).
T h e structure results from a n interference of a c c e p t o r a n d valence b a n d
intermediate states.
U n i a x i a l stress lifts the d e g e n e r a c y of h e a v y - a n d light-hole b a n d at k = 0 a n d
provides a d d i t i o n a l information, w h i c h p r o v e d helpful in identifying optical
spectra. A quantitative description of v a l e n c e b a n d s a n d of L a n d a u levels in
uniaxially stressed I n S b w a s given b y Trebin et al. ( 1 9 7 9 ) . T h e t h e o r y is b a s e d o n
a k'p H a m i l t o n i a n formed b y t h e m e t h o d of invariants. Stress interaction
modifies the w a v e f u n c t i o n s , especially t h o s e of the valence b a n d s , w h i c h
contribute t o the R a m a n p r o c e s s a s intermediate states. T h i s c a u s e s a remar­
kable e n h a n c e m e n t o f the S F R cross s e c t i o n ( a n d line width) w i t h uniaxial stress
(Wolfstadter et al. 1988).
The polarization selection rules are included in the formulas (7 a n d 14) for the
S F R cross section. T h e y are d e t e r m i n e d b y the o r t h o n o r m a l i t y of the h a r m o n i c oscillator-function parts of the w a v e functions. U s i n g the s y m b o l s of section 3.2,
the p o l a r i z a t i o n s are ( ζ , σ _ ) a n d ( σ + , ζ ) . A s w a s p o i n t e d o u t b y Yafet ( 1 9 6 6 ) , the
sense of circular p o l a r i z a t i o n is d e t e r m i n e d b y t h e sign of the electron g-factor.
S o the sign c a n b e d e t e r m i n e d in principle b y a p o l a r i z a t i o n analysis. It s h o u l d
be n o t e d that t h e ± -selection rule a l s o implies a linear p o l a r i z a t i o n per­
pendicular t o the field, if a p h o t o n is p r o p a g a t i n g perpendicular t o the field. In
a n y case, o n e p h o t o n m u s t be polarized a l o n g t h e field a n d o n e polarized at
right angles t o t h e field.
Accurate m e a s u r e m e n t s o n s p o n t a n e o u s S F R scattering performed b y Brueck
et al. (1973) w i t h a variety of g e o m e t r i e s s h o w quite g o o d a g r e e m e n t b e t w e e n
theory a n d experiment. A p p l i c a t i o n o f the different theoretical m o d e l s w i t h
different a p p r o x i m a t i o n s yields t h e s a m e result. Effects of crystal o r i e n t a t i o n o n
the polarization selection rules estimated from m o r e detailed models, have been
f o u n d t o b e small (Brueck et al. 1973).
5.1.1.3. Effective g-factor.
mined by
8* =
μΒΒ
A c c o r d i n g t o e q s (2) a n d (4) the g-factor is deter­
'^·
(43)
F o r calculating the experimental value of g* o n e m a y use μ Β = 9.2741
x l O ~ 2 1e r g / G a n d the energy relations 1 c m " 1 = 1.98648 χ 1 0 " 1 6 erg o r
1 H z £ 6.62619 χ 1 0 " 2 7 erg. T h e effective g-factor o f I n S b is a b o u t - 5 0 a n d
implies a large R a m a n shift. T h i s high a b s o l u t e value originates from the s t r o n g
s p i n - o r b i t c o u p l i n g a n d the small energy g a p . Principally, a n effective g-factor,
different from the free-electron L a n d e factor of a b o u t t w o , arises in the effective-
Spin-flip Raman scattering
233
mass theory as an anti-symmetric contribution to the energy by the spin-orbit
mixing of the valence band functions, g* determines the paramagnetic splitting
and corresponds to the projection of the total angular momentum on the
magnetic field direction. In the scope of the three-band model, the band-edge
effective g-factor is (Pidgeon 1980)
where A denotes the spin-orbit energy and Ρ the interband matrix element. Due
to nonparabolicity (increase of effective mass) the paramagnetic spin splitting
decreases with energy and, therefore, drops with higher magnetic fields. The
calculation gives
'•"{'-fe-'js^s}
(45
>
where
m c = m c 0[ l + 2(EC - £ , ) / £ , ] ,
(46)
and Ec is to be counted from the bottom of the conduction band (Zawadzki
1973). It has been pointed out by several authors (Cardona 1963, Ogg 1966,
Hermann and Weisbuch 1977) that a proper description of the conduction band
g-factor - and of course of the effective mass - requires including the interaction
of the r 6c conduction band with the p-antibonding conduction band T 7c + r 8 c.
Recently, Braun and Rossler (1985) presented a systematic formulation of the
conduction band Hamiltonian in the presence of a magnetic field by an in­
variant expansion. The coefficients of this expansion are expressed in terms of
the band parameters of an extended Kane model.
Experimental values of the g-factor have been derived from the frequency shift
of Stokes and anti-Stokes radiation as a function of magnetic field. Figure 8
shows the tuning characteristics of spontaneous Stokes scattering as a function
of magnetic field strength in the range 2 0 - 8 0 kG. This figure has been taken
from the paper of Brueck et al. (1973). The solid curve is the kz = 0 electron spinflip energy calculated by Johnson and Dickey (1970). The other curve in fig. 8
presents the effective g-factor as obtained from the tuning curve. It decreases (in
absolute value) from 47 at low fields to 32 at 80 kG.
To investigate the dependence of the g-factor in n-InSb on the magnetic field
in the low-field region, Raman gain measurements have been performed by
Pascher et al. (1978) and by Vdovin and Skok (1986) with similar results (fig. 9).
This method enables determination of g-values, with an accuracy comparable to
the accuracy of from ESR measurements (Isaacson 1968). Extrapolation to zero
field yields the g-factor g 0 at the conduction-band edge [ g 0 = —51.55 ± 0 . 0 2
(Vdovin and Skok 1986), g 0 = - 5 1 . 0 1 (Pascher et al. 1978)].
The slope dg/dB obtained experimentally turns out to be (2.06 + 0.02) T " 1 .
H.G. Hafele
234
MAGNETIC FIELD (kG)
Fig. 8. Tuning characteristic of SFR scattering in InSb as a function of the magnetic field, η = 0.96
χ 1 0 16 c m - 3. - O - Frequency shift, χ effective g-factor as calculated from the frequency shift (from
Brueck et al. 1973).
5
7F 75
o\
0
!
I
0.2
L_J
0Λ
I
0.6
I
I
0.8
H(T)
L_
7.0
Fig. 9. The g-factor dependence on magnetic field. The vertical dashed lines show the points of the
superquantum limit. Δ : ne = 8 χ 1 0 13 c m " 3, 0 : n e = 8 x 1 0 14 c m " 3, A : ne = 1.6 χ 1 0 15 c m " 3, · :
ESR data (from Vdovin and Skok 1986).
If this dependence is calculated from eq. (45), the value 2.0 T " 1 is obtained for
kz = 0. Using the data set of Littler et al. (1983), Braun and Rossler (1985) find
the band-edge value g° = —51.3 and a slope g\ = 2 T - 1. A comparison of the
SFR experiments of Vdovin and Skok (1986) with the results of the Bowers and
Yafet model (1959) shows a similar good agreement.
Recently, SFR scattering in n-InSb under uniaxial stress up to 400 Μ Pa was
investigated with the stress applied parallel to the [100] direction and per­
pendicular to the magnetic field, # | | [ 0 0 1 ] . Both SFR gain measurements and
Spin-flip Raman scattering
235
four-wave mixing experiments were carried out to obtain effective g-values and
line widths (Wolfstadter 1984, Wolfstadter et al. 1988). The g-factor of the
conduction band turned out to be stress independent. In uniaxial stress up to
300 MPa the absolute value of the g-factor decreases by less than 0.5%
compared to zero stress. This is predicted by theory, and is in accordance with
FIR measurements by Kriechbaum et al. (1983).
The theoretical models predict an anisotropy of the effective g-factor in InSb.
This anisotropy was recently confirmed in FIR magnetotransmission measure­
ments using electric-dipole-excited spin-resonance (EDSR) (Chen et al. 1985).
The variation of the g-factor as a function of the angle between the direction of Β
and the crystalline axis is about 0.3 at Β % 41.5 kG. Such differences are well
within the accuracy of SFR measurements and could be observed in the precise
measurements of Vdovin and Skok (1986). They employed the SFR gain
method with magnetic field modulation. The resonance peak is shifted by
crystal rotation around the [110] crystallographic axis (fig. 10). The anisotropy
can be expressed by (Golubev et al. 1985)
Ag = 7 0 l - ^ s i n 2 0 ( l + c o s 2 0 )
J 9(E
g
(47)
+ A)2 '
and agrees well with the experimental data with γ0 = 0.090 ± 0.004.
The dependence of the effective g-factor on the energy or fez-state can
qualitatively be demonstrated by studying SFR scattering under nonequilibrium conditions. This is achieved, e.g., if a short intense electric field pulse
is applied to the sample. After a first experiment of Mooradian et al. (1972)
concerning stimulated SFR, scattering gain measurements have been performed
by Richter et al. (1981).
For an explanation of the electric field induced change in the Raman gain in
the left part of fig. 11, the £(fc z)-dispersion for a spin-split Landau level is
schematically drawn. At zero field, all electrons with fez-vectors within the
hatched Δ/c-interval are allowed to take part in a spin-flip process. The full
Mo]k
r
(7701
Ί
[717]
[770]
77777
[700]
α
b
Fig. 10. (a) Anisotropic correction to the g-factor of the InSb conduction band, (b) Magnitude of
anisotropy along the marked directions.
H.G. Hqfele
236
ΛΕ
4 Ε
Fig. 11. Spin-split Landau levels with Fermi edge under the influence of an electric field and
absorption of radiation (from Richter et al. 1981).
curves in the right part of this picture give the Fermi distribution functions at
two different electron temperatures. The disturbed electron distribution enlarges
the interband absorption of pump laser radiation during the electric pulse
leading to an increase in the number of free electrons in the conduction band.
During the time the electric field is operating one gets, therefore, both a changed
electron temperature and a Fermi level, dynamically shifted to higher energies
(dashed curve on the right). For this reason, the center of the Afcz-interval for
possible spin-flip processes is shifted to higher energies and because of the
nonparabolicity of the conduction band, the average effective g-value decreases.
Hence, the gain curve is displaced to higher magnetic fields and the difference
between two gain curves with and without an electric field is recorded.
The effective g-factor, as derived from SFR gain measurements with nonde­
generate n-InSb samples (n = 8 χ 1 0 13 c m - 3) is different from that of samples
with the electrons being in the degenerate regime (Pascher et al. 1976). Two
separate g-values are found and extrapolation to Β = 0 leads to absolute values
considerably smaller than 51.2, which is the expected value for conduction
electrons at this carrier concentration. The results can be understood if the SFR
scattering is assumed to be caused by electrons localized in isolated donor states.
5.1.1.4. Line shape and line width. In the line shape of SFR-spectra information
appears on the dynamics of carriers. The broadening of the line has two
constituent parts, an homogeneous broadening and an inhomogeneous one. In
the absence of inhomogeneous broadening mechanisms, the line shape is just a
simple Lorentzian broadened by spin relaxation and dephasing of electron-spin
precession. The line width results in
A ch o m
=
( ! 2/ T i +
\/T\om)
«
2/T\om.
(48)
Spin-flip Raman scattering
237
(a)
(b)
Fig. 12. InSb models used in the line-shape calculations: (a) inhomogeneous nonparabolicity
broadening and (b) inhomogeneous Doppler broadening. (qz is the component of the scattering
wave vector q = k L — k s along the magnetic-field direction (from Brueck et al. 1973).)
Fig. 13. Spin-flip light scattering line width as a function of magnetic field Η for q- ΗΦ0
geometry. n = l x l 0 1 c6 m - 3 (from Brueck et al. 1973).
and q-H=0
geometry
H.G. Hafele
238
F o r S F R scattering from m o b i l e electrons i n h o m o g e n e o u s b r o a d e n i n g usually
d o m i n a t e s , d u e t o b a n d n o n p a r a b o l i c i t y a n d D o p p l e r effect (diffusion). T h e s e
i n h o m o g e n e o u s c o n t r i b u t i o n s define the time T 2 n ,h related t o the i n h o m o g e n e o u s l y b r o a d e n e d line w i d t h
A o / nh = 2 / T 2 n .h
(49)
T h e t w o i n h o m o g e n e o u s m e c h a n i s m s are s k e t c h e d in fig. 12.
O b v i o u s l y , the D o p p l e r b r o a d e n i n g d e p e n d s o n the a m o u n t
of the
qz
c o m p o n e n t , i m p a r t e d by the p h o t o n s . Since the carriers c a n diffuse relatively
freely a l o n g the m a g n e t i c field, the line s h a p e is sensitive t o the scattering
g e o m e t r y . T h i s w a s d e m o n s t r a t e d b y Brueck et al. (1973) w h o o b s e r v e d the
s p o n t a n e o u s S F R spectra as a function of m a g n e t i c field for different g e o m e t r i e s
(fig. 13).
T h e results in the t w o g e o m e t r i e s are very different from o n e a n o t h e r . W i t h
b o t h the incident a n d scattered light p r o p a g a t i n g perpendicular t o the field
(q*H=Q),
n a r r o w line w i d t h s ( 0 . 2 - 0 . 4 c m " 1 ) are f o u n d w h i c h r e m a i n a l m o s t
c o n s t a n t in the m a g n e t i c field range applied.
T h e line w i d t h s w i t h q a l o n g the m a g n e t i c field ( # · / / # 0) are m u c h larger,
a n d d r o p from 6.5 c m - 1 at 21 k G t o less t h a n 0.3 c m - 1 at 80.5 k G . In later
i n v e s t i g a t i o n s , the R a m a n g a i n t e c h n i q u e , w h i c h a u t o m a t i c a l l y i n v o l v e s the
collinear q*H = 0 configuration, w a s u s e d (Brueck a n d M o o r a d i a n 1973).
L o w e r d o p e d s a m p l e s (n = 8 x 1 0 1 4c m ~ 3 ) exhibit line w i d t h s b e t w e e n 2 a n d
50 G at m a g n e t i c fields b e t w e e n 0 a n d 10 k G . At the m e a s u r i n g t u n i n g rate of
67.5 M H z / G this c o r r e s p o n d s t o frequency w i d t h s of 135 M H z t o 3.375 G H z or
4.5 χ 1 0 _ 3c m ~ 1 t o 0.11 c m " 1 .
Pascher et al. (1978) performed g a i n m e a s u r e m e n t s in the low-field r e g i o n
near the q u a n t u m limit w h e r e the n o n p a r a b o l i c i t y b r o a d e n i n g b e c o m e s evident.
T h e curves in fig. 14 d e m o n s t r a t e the great difference of the line w i d t h in the
q u a n t u m limit for the t w o temperatures. At 4.2 Κ there is a n a l m o s t linear
increase w i t h the m a g n e t i c field, w h e r e a s at 1.8 Κ it remains essentially c o n s t a n t .
E v e n at 1.8 K, the electron temperature T e is o b v i o u s l y t o o high t o fulfill the
c o n d i t i o n kTe <ζ EF. T h e p h e n o m e n o l o g i c a l t h e o r y of Brueck a n d B l u m (1972)
predicts variation of the line w i d t h w i t h the m a g n e t i c field as 1/B2 for Τ = 0, a n d
p r o p o r t i o n a l t o B2 for kTt ^ EF.
T h e increase of the line w i d t h at small fields, the relative m a x i m u m at a b o u t
2 k G , a n d the m a x i m u m at a b o u t 3 k G for the l o w t e m p e r a t u r e c a n be
understood qualitatively by the magnetic field dependence of the occupation of
the L a n d a u levels.
It is n o t e w o r t h y that the line s h a p e s o b s e r v e d w i t h I n S b s a m p l e s h a v i n g a
degenerate electron distribution in all cases were s y m m e t r i c . T h e general
characteristics of line w i d t h a n d line s h a p e h a v e b e e n studied by Brueck et al.
(1973) a n d Y u e n et al. ( 1 9 7 4 a ) . In w h a t follows, the first treatment is considered,
b e c a u s e the latter o n e fails t o e x p l a i n s o m e essential e x p e r i m e n t a l features.
Spin-flip Raman scattering
239
20001—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—Γ
0 1 2 3 4 5 6 7 8 9
10 11 12 13141516
magnetic field (kG)
Fig. 14. Spontaneous SFR line width versus magnetic field at two temperatures, η = 1.35
χ 1 0 15 c m - 3. The quantum limit lies above 6.6 kG (from Pascher et al. 1978).
A simple theory is necessarily expected to yield broad asymmetric line shapes,
due to the combined effect of nonparabolicity and the sharp structure in the
density of states. The symmetric line shape can be explained by the effect of
orbital collisions via motional narrowing. These collisions are assumed to alter
the kz states of the spin excitation with a characteristic relaxation time τ ρ, but
they do not alter the phase of the spin excitation. Then at each τ ρ collision, kz
and the spin-flip frequency co SF are changed randomly. If these collisions occur
sufficiently frequently, the averaging gives rise to a line width proportional to τ ρ.
The line shapes calculated according to this concept are shown in fig. 15.
A comparison of the three curves demonstrates the striking effect of the τ ρ
collisions. In the absence of collisions ( τ ρ = oo) the line shape is determined by
the sharp peak of the density of states and a long tail towards smaller frequency
shifts, due to the nonparabolicity broadening. Decreasing τ ρ down to τ ρ
= 1 0 ~ 1 2s (a value comparable to usual collision times evaluated from dc
mobility measurements) obliterates these effects almost completely. It gives rise
to a narrow symmetric line with the peak shifted towards the centre of the
inhomogeneous frequency distribution.
The good agreement with experimental results, represented in fig. 13, is
obtained with t p = 4 x 1 0 " 1 3s for both geometries (the two homogeneous
contributions within the brackets of eq. (48) are combined in the quantity T s in
(Brueck et al. 1973).
H.G. Hafele
240
60
50
ζ
ο
α 40
UJ
to
to
CO
§ 3 0
u
UJ
>
5
20
UJ
10
0
-3
-2
-1
FREQUENCY
0
( c m - )1
Fig. 15. Effects of orbital collisions on the line shapes in the q · H = 0 geometry, n = l x l 0
Η = 35 kG, and τ 8 = 1 0 " 10 s (from Brueck et al. 1973).
1 6
cm
- 3
,
5.1.1.5. Spin-relaxation
times. In SFR scattering from conduction-band
electrons in InSb the spin-lattice relaxation time Tx refers to decay from the
upper (spin-down) to the lower (spin-up) state in the η = 0 Landau level. This
quantity affects the dynamics of the SFR process and is a function of carrier
concentration, temperature and magnetic field. To determine Tx experimentally,
a variety of quite different methods have been employed. It is interesting to
remark that the range of values quoted in literature extends to about three
orders of magnitude.
The occupation of the two spin states involved in the Raman process is
determined by the balance between excitation and relaxation. With each Stokes
photon produced in the sample, the population of the upper spin state is
increased by one. As the number of excitations available for stimulated Raman
scattering may be small in comparison to the number of incident laser photons,
spin saturation will occur and limit the conversion efficiency. Furthermore, the
attenuation of the pump beam by energy transfer to the Stokes beam (so called
pump depletion) is no longer negligible at high conversion rates. Both spin
saturation and pump depletion have been accounted for in a rate-equation
model put forward by Wherrett and Firth (1972) in continuous, and by Firth
(1972) in pulsed SFR laser operation. From this analysis Wherrett and Firth
Spin-flip Raman scattering
241
obtained the spin-relaxation time 7\ ~ 12 ns for η = 1 0 1 6 cm 3at 36 kG and Tx
~ 1.3 ns for n = 1 0 1 5c m " 3 at 16.9 kG.
To determine 7\ directly, double-pulse techniques have been used. With a
strong SFR pulse induced by a first pump pulse a nonequilibrium population of
the upper spin level is created, which decays with the spin-relaxation time to the
thermal equilibrium population. As the Raman gain decreases with increasing
occupation of the upper spin level, this occupation can be monitored by the SFR
output power of a second pump pulse applied with a variable time delay
between pulses.
In an experiment, where a sequence of C 0 2 laser pulses was applied to the
sample, Brueck and Mooradian (1976) obtained T t = ( 6 0 ± 2 0 ) ns for n= 1.2
χ 1 0 1 6c m ~ 3 .
Using a two-pulse method with CO lasers, Pascher et al. (1976a) determined
the spin-relaxation time in n-InSb for different carrier concentrations and
different temperatures as a function of the magnetic field. The results for an
n = 1.35 χ 1 0 15 c m " 3 InSb sample are shown in fig. 16. At 4.2 K, TY varies
between 40 and 75 ns with a flat maximum at 6 kG. At the lower temperature of
1.8 K, it increases to about 130 ns at 7 kG. The maxima of the spin-relaxation
time have a clear relation to the quantum limit which is above 6.6 kG for this
electron concentration.
These results contrast with the much lower value of 1 ns reported by Nguyen
et al. (1976) for a sample in the 1 0 15 c m " 3 range. In this four-photon mixing
experiment the determination of Τγ is based on intensity measurements, as
discussed in section 3.6.
As shown in section 3.8, SF resonance of conduction-band electrons in InSb is
Fig. 16. Spin-flip relaxation time in InSb for Τ = 1.8 Κ and Τ = 42 Κ versus magnetic field (from
Pascher et al. 1976a).
242
H.G. Hafele
indicated b y a c h a n g e in t h e electrical conductivity. T h e c o n d u c t i v i t y signal in
the presence of S F R scattering p r o v e d t o b e t h e s u m of a n i n t r a b a n d a b s o r p t i o n i n d u c e d c o n t r i b u t i o n a n d a spin-flip i n d u c e d o n e . F r o m t h e tail of t h e total
signal, after the e n d of t h e a b s o r p t i o n c o n t r i b u t i o n , a characteristic relaxation
time w a s derived a n d identified with the s p i n - l a t t i c e relaxation time b e t w e e n t h e
Oj a n d 0 | L a n d a u sublevels (Grisar et al. 1976). T h e Ύγ values o b t a i n e d for a
s a m p l e of n = 1.57 χ 1 0 1 6c m ~ 3 e x t e n d from 2 5 0 t o 1 5 0 n s at m a g n e t i c field
strengths b e t w e e n 5 0 a n d 135 k G .
Studenikin a n d S k o k (1986) p o i n t e d o u t that t h e c o n d u c t i v i t y under S F R
scattering c a n be c h a n g e d b o t h b y redistribution of electrons b e t w e e n spin
states, a n d b y e l e c t r o n - g a s heating. Their m e a s u r e m e n t s suggest that with a
s a m p l e of η = 0.73 χ 1 0 1 5 c m " 3 a n d T = 2 K , S F transitions influence m a g netoresistance primarily b y the h e a t i n g m e c h a n i s m w h e n the excess energy
g*μBB after spin relaxation is distributed b e t w e e n electrons at the l o w e r spin
level. Q u a n t i t a t i v e e v a l u a t i o n yields a n energy relaxation time τ Ε » 0.9 χ 1 0 " 6 s.
T h e o r i e s of the spin relaxation time of t h e c o n d u c t i o n electrons in I n S b h a v e
b e e n presented b y Chazalviel ( 1 9 7 5 ) , B o g u s l a w s k i a n d Z a w a d z k i ( 1 9 8 0 ) a n d
others (see P i k u s a n d T i t k o v 1984). Here, w e follow the paper of B o g u s l a w s k i
a n d Z a w a d z k i , where spin-relaxation times b e t w e e n the t w o l o w e s t spin s u b b a n d s are calculated a n d c o m p a r e d w i t h t h e S F R e x p e r i m e n t s discussed a b o v e .
T h e p o t e n t i a l s o f i o n i s e d impurities a n d p h o n o n s are c o n s i d e r e d a s perturb­
ations. T h e s e p e r t u r b a t i o n s are spin i n d e p e n d e n t , a n d the spin-flip transitions
are a l l o w e d b e c a u s e of the spin m i x i n g of t h e electron w a v e f u n c t i o n s in t h e
three-band m o d e l . Scattering b y i o n i s e d impurities a n d b y a c o u s t i c p h o n o n s
( d e f o r m a t i o n p o t e n t i a l interaction) is s h o w n t o be t h e d o m i n a n t m e c h a n i s m s in
d e t e r m i n i n g the spin lifetime at n o t t o o high m a g n e t i c fields. T h e matrix e l e m e n t
of the impurity scattering p o t e n t i a l turns o u t t o b e p r o p o r t i o n a l t o kz, a n d T^p
ocEz = (h2/2m*)k2.
S o spin-flip transitions from t h e b o t t o m of the upper s u b b a n d are forbidden.
T h e probability for spin relaxation d u e t o t h e d e f o r m a t i o n p o t e n t i a l interac­
tion with a c o u s t i c p h o n o n s d o e s n o t v a n i s h for kz = 0. T h e matrix e l e m e n t for
a c o u s t i c - p h o n o n scattering is p r o p o r t i o n a l t o the m o m e n t u m transfer. Since t h e
m i n i m a l value o f t h e m o m e n t u m transfer, g i v e n b y (2m^g^pBB)l,2/h
is increas­
ing w i t h m a g n e t i c field, the role of the a c o u s t i c - p h o n o n scattering a l s o increases.
Spin-flip relaxation d u e t o optical p h o n o n s is p o s s i b l e o n l y for spin splitting
larger t h a n the p h o n o n energy, w h i c h o c c u r s a b o v e 12.5 Τ in I n S b .
At l o w temperatures, o n l y i o n i s e d impurities are of i m p o r t a n c e . T h e cal­
culated relaxation times for this s i t u a t i o n are o n e order of m a g n i t u d e larger t h a n
the experimental d a t a of P a s c h e r et al. (1976b). B o g u s l a w s k i a n d Z a w a d z k i
(1980) p o i n t e d o u t that calculated v a l u e s of Tx are in the order of the
experimental o n e s if the influence o f i o n i s e d impurities o n electronic states c l o s e
t o the b o t t o m of the L a n d a u levels are t a k e n i n t o a c c o u n t . T h i s c a u s e s the
density of states t o b e c o m e b r o a d e n e d a n d the electronic kinetic energy t o
remain finite e v e n for t h e l o w e s t lying states.
Spin-flip Raman scattering
243
The experiment of Brueck and Mooradian (1976), which was performed at
higher temperatures, higher magnetic fields and higher carrier concentrations
than above, is described quite well by the simple theory. In this situation, the
ionised-impurity mode and the phonon mode contribute with comparable
magnitude.
Systematic investigations of the dephasing time T2 in InSb have not been
performed yet. Taking T2 1 = Γ as the half width of the spontaneous spin-flip
Raman line, Nguyen et al. (1976) derived T2 = 5.3 χ 1 0 " 1 0 s for an InSb sample
with η = 1.2 χ 1 0 15 c m - 3 at about 5.3 kG. Good agreement between experi­
mental line shape and theoretical results was obtained by Brueck et al. (1973) in
their calculations where the parameter T2 ( = T s) = 1 0 " 1 0 was used. The timedelayed CARS measurements by Pascher et al. (1980), as described in section
3.6, point to a dephasing time T2 in the order of 100 ns. The sample used was
non-degenerate (n = 1.35 χ 1 0 15 c m " 3) and the SFR scattering is thought to
occur on account of electrons localized in bound donor states.
5.1.1.6. SFR scattering in p-material.
Following the prediction of Yafet (1966)
of spin-flip transitions in the valence band, Ebert et al. (1977) investigated
magneto-Raman scattering in p-InSb. At magnetic fields above 30 kG and
temperatures not higher than 2.5 Κ they observed stimulated Stokes and antiStokes lines with a Raman shift only slightly depending on the magnetic field
strength. The narrow line width pointed to a spin-flip process. These data have
been re-analyzed by Scott (1980b) in terms of free- to bound-exciton transitions,
rather than SF transitions. Later experiments suggested the Raman scattering to
arise from photo-excited holes without spin reversal. Making use of the valenceband calculations of Trebin et al. (1979), the strange features of the spectro­
scopic results were explained by transitions at non-zero values of the long­
itudinal momentum fez, where anti-crossing of neighbouring magnetic levels
leads to high joint densities of states (Ebert et al. 1981). Further measurements of
stimulated Raman scattering with uniaxial stress applied to the InSb crystals
confirm this interpretation (Wolfstadter et al. 1988). Nevertheless, there are
open questions that still need to be answered.
5.1.2. Indium arsenide (InAs)
The theoretical description of SFR scattering in InAs resembles that of InSb,
because the energy bands have the same structure. Due to the lower spin-orbit
energy Δ (or the lower g-factor), the scattering cross section is expected to be
smaller compared to that of InSb. Spontaneous Raman scattering in InAs
involving an electron spin-flip was first observed by Patel and Slusher (1968) for
magnetic fields of about 50 kG up to 100 kG. Line widths were of the order of
1 c m " 1.
Using a HF-laser pump near the band-gap resonance, the threshold for
stimulated scattering at 85 kG was determined to be less than about 20 W (Eng
et al. 1974). The undoped InAs crystals had a nominal electron density of 1.43
244
H.G. Hafele
χ 1 0 16 c m " 3 at 300 Κ which corresponds to a quantum limit at 0 Κ of about
41 kG. Nevertheless, no stimulated Stokes emission was detected below a
magnetic field of 59 kG.
From the nearly linear tuning rate for the first Stokes component in the range
59-85 kG a zero-field effective g-factor of — 15.28 ± 0 . 1 5 was derived. This is in
agreement with values resulting from the magneto-optical interband-absorption
measurements of Pidgeon et al. (1967).
Pascher (1982) studied SFR scattering in n-InAs by means of resonant fourwave mixing with two Q-switched C 0 2 lasers. This method allowed investiga­
tion of g-values and line shapes in a wide range of the magnetic field between 0.5
and 6.5 T. Figure 17 shows a plot of the effective g-factor as a function of the
magnetic field. Simple extrapolation of the high-field values to zero-field yields a
band-edge g-factor of 15.0. If the energy dependence of g* is taken into account,
one obtains g*(0) = 15.31. The different results at high, intermediate and low
fields are explained by the different population of the Landau states. In the
quantum-limit (high fields) SFR transitions in the η = 0 Landau level at kz = 0
are possible. In the intermediate region only electrons in the η = 0 Landau level
with kz φ 0 can contribute and scattering in the low-field region occurs from
electrons in the η = 1 level at kz = 0.
Line shapes at different magnetic fields are presented in fig. 18. At high
magnetic fields (above 4 T) the lines are symmetric with a full width at half
maximum of about 1.3 kG (0.9 c m " 1) . With decreasing field, the lines become
more and more asymmetric with a steep descent at the high-field side. The
asymmetry increases with increasing power of the lasers. A qualitative expla­
nation of these features follows from the general discussion of four-wave mixing
spectroscopy in section 3.5. The line shape of the CARS intensity in the vicinity
13.5 I
1
1
1
1
1
L_
1.0
2.0
3.0
4.0
5.0
6.0
magnetic
field/Τ
Fig. 17. Effective g-value of n-InAs as a function of magnetic field, η = 1.5 χ 10
(from Pascher 1982).
61 cm
, 3 T= 1.6 Κ
Spin-flip Raman scattering
245
magnetic field
Fig. 18. Typical line shapes of the spin resonant four-wave mixing in InAs at different magnetic
fields (from Pascher 1982).
of the relatively weak Raman resonance is complicated, due to the interference
of resonant and nonresonant contributions to the third-order susceptibility.
Additional terms have to be added to χ ( 3) if not only the initial matter states, but
also higher states are significantly populated. This situation is obviously given
for the SFR-CARS spectra of fig. 18 at low magnetic fields and high pump-laser
powers (Pascher 1984).
5.2. 11-VI-compound
semiconductors
5.2.1. Mercury cadmium telluride
(HgCd)Te
5.2.1.1. Effective g-factor.
The theory of SFR scattering from conduction-band
electrons indicates that crystals of the narrow-gap alloy semiconductor
H g i ^ C d ^ T e should exhibit strong and highly tunable SFR scattering. For
compositions around χ = 0.2 one deals with an InSb-like structure with very
small electron effective masses and very large negative g-factors. The band
structure of this small-gap semiconductor has been treated by Zawadzki (1973)
in terms of the three-band model. A comprehensive review of the properties of
this alloy system is given by Dornhaus and Nimtz (1976). The fundamental gap
E% depends linearly on the composition χ of the alloy system, and this feature
gives the ability to adjust the energy gap so as to achieve resonance enhance­
ment when operating with a C 0 2 pump laser.
For the first time, tunable SFR lasering was obtained by Sattler et al. (1974).
In the consequtive experiments of Weber et al. (1975), where a conventional
grating-tunable cw C 0 2 laser was utilized, the SFR tuning data have been fitted
H.G. Hafele
246
to the theories of Roth et al. (1959) and Lax et al. (1961). They yield a zero
magnetic field g-factor of - 8 2 + 3 for a specimen with χ = 0.234 and n =
(5.4 + 0.5) χ 1 0 1 4c m " 3 . With a Q-switched C 0 2 laser, Kruse (1975) observed
first-Stokes and anti-Stokes radiation when pumped at 10.6 μπι, and firstStokes, second-Stokes, and anti-Stokes radiation when pumped at 10.26 μπι.
As expected, the stimulated SFR radiation was found to be polarized per­
pendicularly to the incident laser. From the tuning characteristic, g-factors of 86
and 87 were derived from the first-Stokes and second-Stokes signals,
respectively.
As shown by Norton and Kruse (1977) the SFR laser output power exhibits
quantum oscillations similar to the oscillations observed on InSb. Maxima in
the signal can be related to the η = 2,1 and 0 Landau levels as they attain the
Fermi level. With increasing pump power, the maxima and minima shift to
higher magnetic fields. It was concluded that local heating of the sample above
20 mW average power reduces the effective g-value.
In some experiments, the spin resonance showed a doubling (Bridges et al.
1979) or even a tripling (Norton and Kruse 1977). Comparison of various
samples suggests the conclusion that different segregated homogeneous regions
with different g-values exist in the material.
For a discussion of the effective g-value, we have to remember that spin-flip
transitions can only occur between occupied and empty levels. In magnetic fields
below the quantum limit, only electrons in the η = 0 Landau level at kz φ 0 in a
limited Afcz interval can take part. This range is marked by the vertical arrows S'
and S" in the inset of fig. 19, which is taken from the paper of Bridges et al.
(1979). For fields equal to or greater than the quantum limit, SF transitions in
the η = 0 Landau level at kz = 0 are possible. The frequencies corresponding to
the transition limits S' and S" have been computed and are plotted in fig. 19
together with the experimental results. Assuming the energies to be
f
Γ
4 /
£ « . ± = i £ . | - l + [l + ^ ( t o ^
h2k2
\ 2 1 1 / 12
J
J,
(50)
one derives for the regime below the quantum limit an average spin-flip energy
of
Εη^-Εη,+
*μΒ^Β^Ι-ψ^.
(51)
This implies a constant effective g-value of g*(l — 2 £ F/ £ g ) , in spite of nonpara­
bolicity of the conduction band. The experimental results agree with this
prediction well.
A constant g-factor, determined by the Fermi energy EF at low magnetic
fields, was also observed in the SFR gain experiments by Hofling et al. (1983).
Below the quantum limit of 0.52 Τ the g-factor was measured to be nearly
constant, about —87. Taking the Fermi energy £ F = 4.2meV of this sample,
a value of — 86.2 has been calculated.
Spin-flip Raman scattering
247
H(kG)
Fig. 19. Tuning curve for spin resonance in H g t _j.Cdj.Te and (inset) conduction-band energies for
En + (spin-up) and E„ _ (spin-down) states of the η = 0 and η = 1 Landau levels for an η-type sample
with χ = 0.234, η = 4.8 χ 1 0 14 c m - 3 at Η = 2 kG. £ F is the Fermi level (from Bridges et al. 1979).
The absolute value of g* decreases to about 60 at a field of 5 Τ This follows
from four-wave mixing measurements of Pascher (1983), where pure as well
combined spin-flip resonances could be observed (Rashba 1979).
5.2.1.2. Line shape and line width. As discussed in connection with sponta­
neous SFR line width in InSb the nonparabolicity would be expected to result in
a large inhomogeneous line broadening. Furthermore, the lines should be highly
asymmetric because of the increase of the density of states towards the band
edge. Both the four-wave mixing investigations and the gain measurements
demonstrated rather small, symmetric lines. The experimental results are in
accordance, also in the fact that the line width increases with the magnetic field
to a relative maximum near the quantum limit and rises steadily with growing
magnetic field (fig. 20).
Absolute values of the line width are between 15 and 40 G (0.06 c m - 1 and
0.17 c m " 1) at low magnetic fields.
The model of motional narrowing was used to explain these features
quantitatively in terms of the phenomenological lifetimes τ 5 and τ Ρ (Brueck and
Mooradian 1973, Brueck and Blum 1972, Brueck et al. 1973, Yuen et al. 1974a,
Davies 1973). A homogeneous background line width is assumed to be due to
H.G. Hafele
248
Fig. 20. Spontaneous SFR line width in H g 0 7 C7 d 0 2T3 e versus magnetic field, η = 1.0 χ 1 0 15 cm
Τ— 1.8 Κ (from Hofling et al. 1983).
,
3
the coherence lifetime T s, where τ 8 has two contributions so that
2
Αω8 = 2/τ δ = 2/[2.5 χ l ( T 1 0( s ) ] + 2£(fcG) 3 / /[2.4
χ 1 0 " 9( s ) ] .
(52)
32
The proportionality to B ' relates the second term to spin relaxation by
acoustic-phonon scattering.
For the motionally narrowed nonparabolicity component, a orbital collision
time τ Ρ = 3.5 χ 1 0 " 1 2 s is used. With these parameters, the theoretical magnetic
field dependence fits the experimental points below the quantum limit, but gives
too slow a fall when the quantum limit is reached. So, explanation of the
discrepancies needs further theoretical investigation.
5.2.2. Cadmium sulphide
(CdS)
5.2.2.1. Origin of SFR spectra.
Unlike the small-gap semiconductors whose gfactors are large, the wider band-gap H-VI-compound semiconductors CdS,
Spin-flip Raman scattering
249
CdSe and CdTe have rather small spin splittings, characterized by \g*\ ^ 2. This
is caused by the fact that their conduction and valence band spin-orbit
interactions are small compared to their band gaps, g-factors near the free
electron value of 2, imply small Stokes shifts, which require experimental
arrangements of high sensitivity and high spectral resolution. SFR scattering in
CdS has been summarized in a variety of reviews, e.g., by Geschwind and
Romestain (1984) and by Scott (1980a).
The first experiments were performed by Thomas and Hopfield (1968). They
observed extremely powerful Raman scattering in the back-scattering geometry
of fig. 3d. The samples in the form of thin platelets were rotated about an axis, so
that variation of the angle θ between the hexagonal c-axis and the magnetic field
was possible. In pure crystals, intense Raman light is produced with θ = 90°. As
θ is reduced from 90° the Raman intensity decreases strongly. The Stokes, and
anti-Stokes lines could be attributed to SFR scattering from the magnetically
split ground states of neutral donors and neutral acceptors, i.e., from bound
electrons and holes.
In other experiments, SFR scattering from free carriers in In-doped CdS with
quite high efficiency has also been detected (Fleury and Scott 1971, Scott et al.
1972). The measurements employed a 90° scattering geometry with the optic
axis along z, as drawn in fig. 3c. The scattering efficiency did not vary when the
sample was rotated and the direction of the magnetic field was changed from
Λ || ζ to B\\x. Moreover, no strong dependence of the scattering efficiency upon
the pump laser frequency was found, when laser wavelengths of 476.5, 488 and
514.5 nm were used. This contrasts strongly to the results with bound-electron
scattering where the cross section was observed to fall by a factor of 130 in going
from 488 to 496.5 nm.
In addition to the normal AS = ± 1 SF processes, CdS also yields multiple
SFR scattering, as first observed by Scott and Damen (1972). These Raman
structures have energy shifts of exact integral multiples of the Zeeman frequency
α
NEUTRAL
DONOR
a)
^
2
S 1 2/ GROUND
STATE
Fig. 21. (a) Band structure of CdS. (b) Bound excitons as excited states of donor in the region of the
laser light (from Geschwind and Romestain 1984).
250
H.G. Hafele
g*pBB ( G e s c h w i n d a n d R o m e s t a i n 1984). T h e strength for t h e AS = n p r o c e s s e s
is strongly m a g n e t i c field d e p e n d e n t a n d line w i d t h s a n d selection rules are
indicative of spin-flip from impurity electrons. T h e o r y e x p l a i n s t h e reversal of
t w o o r m o r e spins b y e x c h a n g e c o u p l i n g a m o n g the intermediate b o u n d - e x c i t o n
state a n d electrons at neighbouring impurity sites ( E c o n o m o u et al. 1972) [see
a l s o multiple S F scattering in Z n T e ( O k a a n d C a r d o n a 1 9 8 1 ) ] .
5.2.2.2. Cross section and selection rules. In t h e f o l l o w i n g d i s c u s s i o n of the
cross section, w e restrict ourselves t o the case of neutral d o n o r s a s treated b y
T h o m a s a n d Hopfield (1968). T h e t h e o r y is b a s e d o n the peculiarities of the
b a n d structure, w h i c h is s h o w n in fig. 21 [ s e e Casella ( 1 9 5 9 ) ] . T h e p-type valence
b a n d is split at fc = 0 i n t o three t w o - f o l d d e g e n e r a t e states n a m e d A, B, a n d C.
S p i n - o r b i t c o u p l i n g separates i n t o J = \ a n d J = § c o m p o n e n t s with t h e (small)
s p i n - o r b i t interaction A = 0.057 eV. T h e J = § c o m p o n e n t is further split b y the
trigonal crystal field i n t o ms = ± | a n d m3 = ±% with s e p a r a t i o n of 0 . 0 1 6 eV.
T h e b o u n d - d o n o r g r o u n d s t a t e is formed from the c o n d u c t i o n b a n d states
near fc = 0. T h e r e are optically a l l o w e d excited states, n a m e l y e x c i t o n s m a d e u p
of h o l e s in the A, B, C valence b a n d s a n d a n electron in the c o n d u c t i o n b a n d ,
b o u n d t o a neutral d o n o r . T h e e x c i t o n from t h e t o p v a l e n c e b a n d o p e r a t e s as a n
intermediate state w i t h a n e x t r e m e l y small energy d e n o m i n a t o r for e x c i t a t i o n
with the 4 8 8 n m a r g o n laser line. T h u s , a one-level a p p r o x i m a t i o n is possible
where o n l y t h e l o w e s t energy intermediate state is k e p t in the s u m of eq. (7).
F o r R a m a n scattering a s s o c i a t e d w i t h spin-flip of t h e b o u n d electrons, a n
external m a g n e t i c field m u s t be applied at s o m e angle 0 w i t h the c-axis. T h e n , the
electron spins in the g r o u n d state will n o t b e resolved a l o n g the c-axis b u t h a v e
eigenstates
a
cos0/2|ST> + sin0/2|S|>,
(53a)
b
-sin0/2|ST> + cos0/2|Sj>.
(53b)
S F R scattering m a y o c c u r from state a t o b o r b t o a. In the one-level
a p p r o x i m a t i o n , the o n l y intermediate states are t h e e x c i t o n s w i t h the basis states
(54a)
(54b)
T h e matrix e l e m e n t for g o i n g from a t o b, via the intermediate state \(X + i 7 ) T >
w i t h t h e p o l a r i z a t i o n of t h e incident light a l o n g the x - a x i s a n d that of the
scattered light a l o n g the y-axis is g i v e n by ( S c o t t 1980a)
-iM
2
cos(0/2) sin(0/2),
(55)
where
Μ = | < ί | £ χ | ( Χ + i Υ ) ΐ > | = \Q\Ey\(X
+ i7)T>|.
(56)
Spin-flip Raman scattering
251
The same result is obtained for the matrix element via intermediate state of
eq. (54b).
Adding and squaring the two contributions yields the total SFR scattering
cross section for this geometry as (Thomas and Hopfield 1968)
with
(58)
w h e r e / i s the oscillator strength and hcoA is the energy of the A exciton. It is the
combination of the very large oscillator strength of the bound exciton ( / ~ 10)
and the condition of almost exact resonance with the A exciton level that gives
rise to the large cross section, in spite of the small spin-orbit interaction.
Without the inequality of the energy resonance denominators due to spin-orbit
interaction, the contributions from the A, B, and C excitons would cancel each
other.
In the calculation of scattering occurring from neutral acceptors, the Zeeman
splitting of the intermediate states must be taken into account. It follows that
SFR scattering by holes is much weaker than by electrons.
The selection rules result from symmetry considerations. Since the site
symmetry of a substitutional point defect is C 3 v, single donors or acceptors will
belong to one of the two irreducible representations of the double group, namely
A6 or AAS. Theoretical predictions are found to agree with the observations.
In the case of delocalized electrons the excited states are no longer bound
excitons with large oscillator strength but electrons and holes in the band
continuum, as illustrated in fig. 2. The observed polarization is that expected for
a free-carrier process and different from that found from bound-electron
scattering. There are only contributions from Raman tensor elements a^, when
i Φ j and either ί|| Β or / | | B. Violation of the free-electron selection rules at low
temperatures was interpreted as arising from bound electrons (Scott et al. 1972).
5.2.23. Effective g-factor.
From the spin splitting in the Raman spectra as a
function of the magnetic field, the g-values were determined. The results for pure
crystals (ΙΟ 1 5—10 16 donors c m - 3) yield a g-value for electrons bound to donors
close to 1.76. Within the experimental accuracy this value was found to be
isotropic. In contrast to this bound electron g-factor, the g-factor of holes bound
to the In acceptor is highly anisotropic. This is shown in the diagram of fig. 22.
The data are obtained both from SFR spectra and from fluorescence spectra and
give a g-value g h = 2.76 cos 0, where θ is the angle between c-axis and the
magnetic field.
The frequency shift of the SFR line from conduction electrons corresponds
H.G. Hafele
252
Fig. 22. Anisotropy of the g-factor of holes in the ground state of the / x transition. The g-value of
electrons ge is indicated (from Thomas and Hopfield 1968).
with a g-factor of g e = 1.86 ± 0.03 (Fleury and Scott 1971). For II-VI semi­
conductors the following relation for the conduction-band g-factor was
proposed (Kurik 1970)
* - ' [ ' - t e - ' k b ]
Using A = 0.07 eV, Eg = 2.57 eV and m*/m = 0.18, one calculates g e = 1.92 for
CdS.
5.2.2.4. Line shape, line width and relaxation times. Whereas the line width of
bound-electron scattering is independent of scattering angle and nearly inde­
pendent of temperature over the range from 2 to 150 K, scattering from free
conduction electrons exhibits strong variation with angle and temperature. For
right-angle geometry the line width increases from 0.05 c m - 1 at 2 Κ to about
4 c m - 1 at 150 Κ (Scott et al. 1972). A series of lines is shown in fig. 23.
Solid curves are observed spectra, dots give the shape fit to a spectral
distribution function 5(ω). This is assumed to be simply related to the free-spin
susceptibility
S((o) =
η(ω) + 1
π
τ
Im χλ(ω) =
ή(ω) + 1
π
χ0
ωΓ
( ω — c o S )F
+1
,
(60)
where ή(ω) = [exp(to//cT) - I ] " 1 . χ0 is the ω = 0 susceptibility, ω δΡ = g*μBB/h
is the Zeeman splitting and Γ is a phenomenological damping constant.
According to the Bloch equation for the magnetization, the parameter Γ is
related to the transverse relaxation time as Γ = 1/T 2. However, the measured
Spin-flip Raman scattering
253
Fig. 23. Line shapes in CdS versus temperature. Right-angle scattering at 56 kG, η = 5 χ 1 0 17 c m - 3.
Solid curves are observed spectra, dots give the shape fit to S(a>) convoluted with the spectral
response function of the spectrometer (from Scott et al. 1972).
widths are not determined by spin lifetimes, but by an inhomogeneous
broadening due to spin diffusion. This interpretation arises from measurements
of the SF line width in dependence with the scattering angle θ or momentum
transfer q (q = 2kL sin Θ/2). There is a dramatic line narrowing for small-angle
scattering (fig. 24). The measured line widths follow a relation of the form
r(e,T)
= A(T) + B(T)q2.
(61)
This angular dependence can be explained by spin diffusion. Basically it is the
effect of motional narrowing in N M R , but in spin diffusion, collisions prevent
the spins from diffusing, whereas in motional narrowing, collisions prevent the
spins from dephasing. In both cases, the line width is proportional to the
collision lifetime T c (see below). Although the qualitative features of the theory
have been experimentally confirmed for CdS, there is a disagreement for the
absolute values of the line widths, which is believed to represent a failure of the
theory.
H.G. Hafele
254
8
0
WAVE-NUMBER
-
8
-16
S H I F T (cm" 1)
Fig. 24. Line-shape data for η = 5 χ 1 0 17 cm ~ 3 CdS at scattering angles θ = 28° and 152°. Τ = 80 Κ,
Η = 80 kG (from Scott et al. 1972).
A R a m a n e c h o experiment as described in section 3.7 has been performed in
C d S in order t o m e a s u r e t h e h o m o g e n e o u s c o h e r e n c e t i m e Γ * 0 01 o f the spin levels
of the b o u n d - d o n o r state ( H u et al. 1 9 7 6 a n d 1978). T h e d e p h a s i n g t i m e T2 in
fig. 25 is f o u n d t o b e a b o u t 2 0 0 n s at m a g n e t i c fields b e t w e e n 7 a n d 17.4 k G .
Another R a m a n scattering experiment in η-type C d S , where d o n o r spins were
c o h e r e n t l y driven b y s i m u l t a n e o u s irradiation w i t h laser light at frequency OJL
and microwaves
at ω 5 s h o w e d
a homogeneous
E P R line w i t h
T2 = 4 n s
( R o m e s t a i n et al. 1974).
5.2.2.5. Special
applications
of
SFR
scattering,
(a) Studies
of
the
insu­
l a t o r - m e t a l transition in C d S . D e t a i l e d m e a s u r e m e n t s o f S F R scattering line
w i d t h s h a v e b e e n utilized t o s t u d y e l e c t r o n d y n a m i c s in c o n n e c t i o n w i t h t h e
i n s u l a t o r - m e t a l transition in C d S ( G e s c h w i n d et al. 1980, G e s c h w i n d a n d
R o m e s t a i n 1984). W i t h d e r e a l i z a t i o n , a D o p p l e r shift is s u p e r i m p o s e d o n a n y
o t h e r l i n e - b r o a d e n i n g m e c h a n i s m . It is g i v e n b y
Δ ω 0 = q-v,
w h e r e ν is t h e e l e c t r o n velocity a n d q = kL — ks
(62)
t h e scattering v e c t o r o r
m o m e n t u m transfer. T h e n , a c o l l i s i o n a l l y n a r r o w e d line h a s t h e w i d t h ( G e s ­
c h w i n d a n d R o m e s t a i n 1984)
Δ ω = 2 ( Δ ω π ) 2 τ ε = 2(q · t>)A2V
TC = 2D C<? 2,
(63)
where a v e r a g i n g is o v e r t h e F e r m i surface, T c d e n o t e s t h e c o l l i s i o n time a n d
Spin-flip Raman scattering
255
10
PULSE SEPARATION (ns)
Fig. 25. Echo intensity as a function of pulse separation τ on a semi-log plot for Β = 7.0 and 17.4 kG.
The inset is echo intensity versus time on linear scale [ H u et al. 1978 (reprinted from Geschwind and
Romestain 1984)].
Dc = l/3vpTc is the diffusion constant. Experimental results on the line width in a
degenerate metallic sample of CdS show a q2 ( ~ sin 2 0/2) dependence of line
width which demonstrates the diffusive motion of the carriers.
(b) Determination of the fc-linear term. Besides spin diffusion there is another
interesting effect modifying the SFR line width in CdS. It is specific for a polar
semiconductor and is due to a /c-linear term in the conduction-band energy
(Hopfield and Thomas 1963). Owing to the presence of an internal permanent
electric field Ε along the crystallographic c-axis in CdS, the spin S of a moving
carrier couples in a manner similar to spin-orbit interaction. The effective
H.G. Hafele
256
magnetic field 'seen' by the electron yields an additional energy l(k χ c) · S
where c is a unit vector parallel to the c-axis, and λ is proportional to the
strength of E. This results in a ^-dependence of the line width as
Aco =
(64)
D(q±q0)\
where D = hk/m*, q0 = Xm*(c χ fi0)/h, and K0 denotes a unit vector along the
external magnetic field. From this, an asymmetry follows between Stokes- and
anti-Stokes line width, which has been confirmed by SFR measurements
(Romestain et al. 1977).
5.2.3. Zinc telluride
(ZnTe)
5.2.3.1. Origin of SFR spectra. ZnTe shows a great variety of SFR features.
One observed scattering from both free holes and free electrons, holes bound to
shallow and deep acceptors (Douglas et al. 1983) electrons bound to shallow
donors as well as multiple SF scattering of electrons (Oka and Cardona 1981)
and spin-flip plus phonon Raman scattering (Hollis et al. 1975). A review as far
as 1980 is given by Scott (1980a).
In p-ZnTe, SF scattering from free holes was observed for the first time (Hollis
et al. 1973). In this Raman process, an incoming laser photon excites an electron
from the valence band to an intermediate state in the conduction band (fig. 26a).
Then, the electron drops back into a state of opposite spin with the emission of a
Stokes photon (fig. 26b). The Stokes shift is equal to the energy splitting of the
spin-reversed states in the valence band. Since the hole scattering is predomi­
nantly produced by photo-excitation, only the shallowest levels are expected to
be involved. Detailed experiments of Hollis and Scott (1977) expanded on the
first experiments in p-ZnTe, excitation wavelength, power, magnetic field,
polarization, and sample characteristics were varied. In these measurements, use
was made of the fact that the energy gap of ZnTe at 2 Κ happens to coincide
almost exactly with the photon energies of krypton laser lines. Typical spectra
are shown in fig. 27.
α
b
Fig. 26. Schematic description of SFR scattering from holes in the valence band of a p-type
semiconductor (from Hollis 1977).
Spin-flip Raman scattering
257
Fig. 27. SFR scattering from heavy holes in pure ZnTe samples. The Krypton laser power is 24 mW
(from Hollis and Scott 1977).
5.2.3.2. Effective g-factor.
With the inclusion of spin-orbit coupling, the
valence band of a zinc-blende structure semiconductor breaks up into a four­
fold degenerate J = j state ( Γ 8) and a two-fold degenerate J = \ state (Γ 7) at the
Brillouin-zone center. The 4 x 4 formalism applied here neglects perturbations
by the split-off band and the conduction band. This approximation is assumed
to be valid in ZnTe, where the split-off band lies at Δ = 0.93 eV and the
conduction band lies at E% = 2.39 eV away from the four-fold degenerate valence
band.
The energy levels for light and heavy holes of ZnTe in a magnetic field have
been computed by Hollis (1977), using the canonical transformation method of
Luttinger and Kohn (1955). The calculated energy difference of the shallowest
heavy-hole levels (n = 2 level) yields a g-value of g£h(2) = -1-0.92 ± 0 . 1 5 . Average
denotes a spherical average over the system in the spherical approximation. In
the theory only holes with zero momentum kH « 0 are regarded.
The measured gyromagnetic ratios in p-ZnTe lie between g = 0.9 and g = 1.1,
depending upon the sample. This agreement between experimental and theoret­
ical g-values allows the assignment as SFR scattering from heavy holes.
Astonishingly, intense SFR spectra were observed in ZnTe:As and ZnTe:Na.
They are due to photo-excited conduction electrons (Toms et al. 1979) and
H.G. Hqfele
258
yielded g = 0.39. Spin-flip of photo-excited electrons is especially emphasized in
the review by Scott (1980a).
A series of studies was made on SFR scattering from electrons and holes
bound to donors and acceptors. Toms et al. (1978) investigated SF-acceptor
scattering and extended their earlier work on nominally pure ZnTe to samples
of ZnTe:As and ZnTe:Ρ (As and Ρ substitutional for Te). This scattering is
associated with SF transitions within the ground-state level, split by the external
magnetic field. Since the shallow acceptor levels take their character from the
four-fold degenerate uppermost valence-band levels, spins will be quantized into
Μ = f, i , — i , and — § states by the magnetic field. The states are unequally split
because of the cubic Jf terms in the impurity spin Hamiltonian
Η = μ^{3χΒχ
+ JyBy + JZBZ) + μ^\ΡχΒχ
+ J3yBy +
J\BZ).
(65)
This suggests, defining two different g-factors
gl/2=g
g3/2
+
= g
+
ig'>
(66a)
(66b)
The Raman features are interpreted as transitions — § — ^ and — \
+ | with
the dominant intermediate state assumed to be s-like levels in the conduction
band. At sample temperatures of about 1.6 Κ only the — § level will be
populated. Transitions with AM = 0, ± 1 , ± 2 are allowed.
From the SFR spectra, the following bound-hole g-values were deduced for
ZnTe:P g 1 /2 = 0.61 ± 0.04 and g 3 2/ = 0.63 ± 0.04. Similar values were found for
holes bound to Li and Na acceptors in ZnTe, which have nearly the same
binding energies (Toms et al. 1979).
By using an argon laser pumped dye laser Oka and Cardona (1981) studied
various resonant aspects for SFR scattering by donors and acceptors. They
observed SFR scattering from electrons bound to donors with high efficiency
when the incident-laser photon is in resonance with the donor-bound exciton
state (D°, X) (see fig. 28). From the frequency shift, the magnitude of the bound
electron g-value |g| = 0.41 ± 0.01 is deduced, in good agreement with the
electron g-factor in ZnTe (g e = 0.39) mentioned above.
The sign of g was determined to be negative ( g = - 0 . 4 1 ± 0 . 0 1 ) . This was
achieved from SF scattering and luminescence measurements, because the
selection rule breaks down in highly resonant scattering. The negative sign is
contrary to calculations within the framework of the three-band k · ρ model. On
theoretical grounds, the g-factor of electrons bound to donors (or in excitons) is
expected to be the same as of conduction-band electrons, and to be given by
(67)
For the gap-energy £ g = 2.38 eV, the spin-orbit splitting Δ = 0.9 eV and the
squared matrix element of linear momentum P2 = 20 eV, one evaluates g e =
259
Spin-flip Raman scattering
τ — Ι — ι — Ι — ι — Ι — ι
J
-0.8
I
-0.6
Ι
I
Ι
-0.4
STOKES
I
-0.2
1 — ι — Ι — ι — Ι — ι
Ι
I
0
SHIFT
L
I
*0.2
Ι
I
Ι
*0.4
ANTI-STOKES
1—Γ
I
+0.6
L
+0.8
SHIFT
(meV)
Fig. 28. SF scattering spectra of the donor electrons. Multiple S F scatterings are indicated as AS =
± 2 , ± 3 . Luminescence due to
X) appears overlapping with them (from Oka and Cardona
1981).
+ 0.47. Extension of eq. (67) within a five-band model including the F 1 5
conduction band, however, yields g e = —0.4 (Cardona 1963).
5.2.4. Other II-VI compound semiconductors (ZnSe, CdSe, CdTe)
Spontaneous SFR scattering from conduction electrons in the wide-band-gap
semiconductor ZnSe was investigated at the same time as in CdS (Fleury and
Scott 1971, Scott et al. 1972). Despite the fact that ZnSe is cubic, the same
selection rules were found to apply to ZnSe as to CdS which is hexagonal.
Contrary to CdS, not all samples exhibited SF scattering. Line widths of indium
doped samples (η ~ 5 χ 1 0 1 7 c m " 3 ) are equal to that of CdS (0.05 c m " 1 at 2 Κ
to 4 c m " 1 at 150 K), whereas quite dramatically different temperature de­
pendences of cross sections were found. A strong decrease of peak intensity in
ZnSe with increasing temperature may be explained by broadening of the
resonant intermediate-exciton states. The Raman frequency shift implies an
effective g-value for the conduction electrons of 1.18 ± 0.03. By pumping with a
high-power cw Nd.YAG laser, electron SFR scattering was also demonstrated
in η-type CdTe and CdSe (Walker et al. 1972). The spectra taken at 4.2 Κ and
magnetic fields near 100 kG yield ^-factors of + 0 . 5 3 + 0.03 and - 0 . 7 5 ± 0.03
for CdSe and CdTe, respectively.
5.3. IV-VI-compound
semiconductors
The cubic lead chalcogenides PbS, PbSe, PbTe and the pseudobinary alloys
P b ^ S n ^ S e and P b ^ S n / T e belong to the interesting group of narrow-gap
H.G. Hafele
260
semiconductors. They have a direct gap at the L-points in the Brillouin zone.
The conduction band is a multivalley structure consisting of four cigar-shaped
ellipsoids along the <111> axis. This implies the existence of longitudinal and
transversal masses and g-values. The widening interest in these alloys comes
from the possibility of producing semiconducting crystals with customertaylored energy gaps. A review on the magneto-optical properties of the IV-VI
compounds was given by Bauer (1980).
5.3.1. Lead telluride (PbTe) and lead tin telluride
(PbSn)Te
Spontaneous SFR scattering from electrons in n-PbTe was observed by Patel
and Slusher (1968) using a C 0 2 laser and magnetic fields up to 105 kG.
Measurements were performed in bulk samples with three different orientations
of the magnetic field with respect to the crystalline axis. The g-value,
obtained at an angle 0 between Β and the < 111 > direction of the electron valley,
is related to the longitudinal and transversal ^-factors g, and qt through the
expression
g(9)* =
(g2cosi 2 0 + g t 2 s i n 2 0 ) 1 / .2
(68)
These authors were the first to observe gt near the band edge in PbTe directly.
The g-values are in good agreement with later investigations by means of
electric-dipole-excited electron-spin-resonance (EDE-ESR) (Schraber and
Doezema 1979). This generally electric-dipole forbidden transition can be
observed in the small-gap semiconductor PbTe because k'p interaction gives
rise to mixed wave functions in the presence of spin-orbit coupling.
SFR spectra of n- and p-PbTe epitaxial films on B a F 2 substrates were
obtained by optical four-wave mixing with two C 0 2 lasers (Pascher 1982a,
1984). The energy of the spin-split Landau levels as a function of magnetic field
and the transition scheme for Raman-resonant four-wave mixing are drawn in
fig. 29. The large third-order susceptibilities in narrow-gap semiconductors
allow for CARS measurements, even in thin epitaxial layers down to 5 μιη
thickness. Due to the narrow line width of the SFR resonances in the CARS
spectra the data on the effective g-values are very precise.
The measured effective g-factors turn out to be gf b = 66.1 ± 0.5, g\b = 65.6
± 0.5, g tcb = 16.7 ± 0.2 and g? = 13.5 ± 0.2. In the k ρ treatment of Mitchell and
Wallis (1966) these g-factors are related to the band parameters by
(69a)
(69b)
where 2P\\m
denotes the interband matrix element in the two-band model,
Spin-flip Raman scattering
261
Fig. 29. Scheme of Landau levels versus magnetic field (left); the arrows indicate transitions allowed
in Voigt configuration. Energy level diagram for spin-flip transitions in the valence band (right); b, c
and d indicate intermediate states, co ca = co SF (after Pascher 1984).
and acy = PJP\\. g{~ and gt~ account for the contribution of the far bands to the
g-factors. They are found to be relatively small in PbTe.
From the CARS spectra, which also exhibit combined spin flip 0 +
1" and
using additional results of interband absorption measurements Pascher (1984)
was able to calculate a revised consistent set of band parameters for PbTe.
The principal effect of alloying SnTe into PbTe is a gradual variation of the
2 e has an energy
L-point energy gap with composition. A crystal of P b 0 8 8S n 0 1 T
gap of about 122 meV. In such a sample with nt = 1 0 17 c m " 3 , stimulated SFR
scattering could be observed in the 10 μιη region. The high pump intensity of a
TEA CO 2 laser was needed to generate Stokes output power with an external
conversion efficiency of only 3 χ 1 0 " 6% .
From the experiments, the g-value of conduction electrons was estimated in
magnetic fields between 10 and 40 kG. The g-factor extrapolated to zero
magnetic field amounts to 61.7.
5.3.2. Lead selenide (PbSe)
SF resonances in n-PbSe have been studied by optical four-wave mixing
spectroscopy in fields of up to 7 Τ (Pascher et al. 1983b, Pascher 1984). From the
CARS spectra as drawn in fig. 30 and additionally observed interband tran­
sitions the band parameters were calculated. The analysis yields the band-edge
262
H.G. Hafele
η - PbSe
I
ι
ι
ι
4.0
50
60
magnetic field
/
I
Τ
Fig. 30. CARS intensity versus magnetic field for a n-PbSe epitaxial layer. 2f||<211>. 1 cb, 0"
- > 0 +, 0 = 1 9 . 4 7 ° ,
2 c b , 0 " - 0 + , 0 = 61.81°, 3 cb, 0 " - 0 + , 0 = 90°, and
4
ν ο , 0 +- 0 " ,
0 = 90° (from Pascher 1984).
values of g-factors gf b = 41.08 ± 0.2, g t cb = 32.5 ± 0.2 and g* b = 30.6 ± 0 . 3 .
Measurements show the known approximate equality of the conduction- and
valence-band g-factors, indicating that the band parameters are determined
mainly by the interaction of the two adjacent bands (two-band model).
Nevertheless, the contributions of the far-bands to the g-factors turn out to
reach 25% in PbSe.
6. Experimental
and theoretical
semiconductors
(DMS)
results of diluted
magnetic
SFR scattering has proven a useful tool in studying electronic properties of
diluted magnetic semiconductors. As known, this class of alloys results from
the substitution of a magnetic ion ( M n 2 + , F e 2 + ) for the cation in a II-VI
compound semiconductor. These semimagnetic semiconductors possess a
sizeable exchange coupling between the spins of the mobile band carriers and
the local magnetic moments of the magnetic ions incorporated in the lattice. The
Spin-flip Raman scattering
263
interaction Hamiltonian is of the Heisenberg type and has the form
(70)
Ν
where r and s are the position and the spin operators of the mobile carriers,
respectively, and RN and SN position and spin of the Nth paramagnetic ion,
respectively (Gaj et al. 1978). Thanks to this strong interaction, the crystals
exhibit a variety of novel magneto-optical effects. Here, we are confronted with a
giant Zeeman splitting of conduction-band electrons. This can be accounted for
by introducing effective Lande factors, which in some (CdMn)Se samples reach
values as large as g e ff ~ 200, compared to g* = 0.5 in CdSe. The results on D M S
have been discussed in reviews by Furdyna (1982) and Furdyna and Kossut
(1988).
SFR scattering has been first observed in the narrow-gap η-type compound
H g 0 . 8 9 M no . i i T e by Geyer and Fan (1980) and in the wide-gap material
Cd x .^Mn^-Se by Nawrocki et al. (1980,1981). We restrict ourselves to the widegap group and start with a description of (CdMn)Se, because SFR scattering has
been most extensively studied in this material.
6.1. Cadmium manganese selenide [(CdMn)Se]
[(ZnMn)Se']
and zinc manganese selenide
6.1.1. Origin of SFR spectra
At low Mn concentrations Cdl .^Mn^Se has the hexagonal wurtzite lattice and
the band structure resembles that of CdSe. Samples not intentionally doped,
exhibit η-type conductivity with 1 0 1 6- 1 0 1 8 electrons c m - 3 (Peterson et al.
1985). As in (CdMn)S and (CdMn)Te the dominant SF scattering in (CdMn)Se
is caused by electrons bound to shallow donors. This is evidenced by the large
scattering cross section, the selection rules and by a finite energy shift at zero
magnetic field. The exchange interaction between the spin s of a loosely bound
electron and the spins SN of the Μ η 2 + ions appreciably alters the energy of the
spin state of the electron. In a magnetic field, the net alignment < 5 N> gives rise to
a spin splitting, which totally dominates the direct Zeeman splitting g*pBB.
Moreover, the inverse process takes place and an electron localized at a donor
polarizes the M n 2 + ions within its orbit and produces a ferromagnetic spin
cloud around the impurity. The resultant bound state which is influenced by the
thermodynamic fluctuations of magnetization is termed bound magnetic
polaron (BMP) (Golnik et al. 1980, Died and Spalek 1982, 1983, Heiman et al.
1983a, Wolff and Warnock 1984, Warnock and Wolff 1985).
Thus, the total spin-flip energy for conduction-band electrons has three
contributions (Shapira et al. 1982, Heiman et al. 1983b, 1984).
AE = g * ^ BB + x a N 0< S z > + £ B M (PB , T).
(71)
264
H.G. Hafele
The first term is the energy of the direct interaction between the applied field Β
and the electron spin in the absence of magnetic ions. It is small, since the
conduction-band g-factor in pure CdSe is g* = 0.50. The second term describes
the exchange contribution to the Zeeman splitting (Gaj et al. 1978, 1979). χ is
the mole fraction of the M n 2 + , N0 the number of cations per unit volume,
α = <S|J\S} the exchange integral for the conduction band, where the function
S denotes the periodic part of Bloch functions of s symmetry. <5 Z> is the
magnitude of the thermal average of Μ η 2 + spin component along B. The use of
this average corresponds to the standard molecular-field approximation and is
justified since the mobile electrons interact with a large number of manganese
ions. In consequence of the electron-Mn 2 + exchange interaction, the Raman
shift is closely related to the magnetisation of the crystal (Shapira et al. 1982).
Thus, it depends not only on the applied magnetic field, but also on temperature
and manganese concentration.
The third term is due to the B M P interactions. It takes into account the
difference between the local magnetization within the donor orbit (arising from
~ 10 2 M n 2 + ) and that of the bulk. £ B MP is positive and vanishes at very high
fields (Dietel and Spalek 1983, Heiman et al. 1983a). Expression (71) describes
well the experimental spin-flip Stokes energy in (CdMn)Se, which was deter­
mined from spontaneous Raman scattering as shown in fig. 31.
Apart from the lowest-field region AE rises linearly with field and saturates at
Fig. 31. Spin-flip Stokes energy AE versus applied magnetic field Β in Cdj-^Mn^Se, χ = 0.01 for
various temperatures. A backscattering configuration was used with incident and scattered
polarization (xz), Jf||x,c||z. The light source was a tunable dye laser (from Heiman et al. 1983a).
Spin-flip Raman scattering
265
high fields. This is a consequence of saturation of the manganese magnetization.
It follows roughly a Brillouin-type law described by the expression
(72)
where S0 is a fitting parameter, B5/2 is the Brillouin function of index f, f # μ Β
= 5μ Β is the magnetic moment of M n 2 + ions and k is the Boltzmann constant.
The second fitting parameter 7^ F is a measure of the average antiferromagnetic
interaction between the M n 2 + spins, depending on χ (Heiman et al. 1984b). SFR
experiments in an extended magnetic-field range show some systematic devi­
ations from a Brillouin function. It turns out that the addition of a phenomenological exchange temperature 7^ F to the argument is an insufficient corrective
measure (Douglas et al. 1984).
By comparing the SFR shift AE with magnetization data of the same sample,
the exchange-energy parameter aiV 0 for conduction-band electrons can be
determined. The analysis yields a value of (258 ± 5) meV in C d 1 _ x M n x S e for
0 . 0 5 ^ x ^ 0 . 2 at 1.9 Κ (Heiman et al. 1984b) and ( 2 4 3 ± 1 0 ) m e V for
Z n i ^ M ^ S e with χ = 0.03 at the same temperature (Heiman et al. 1984a).
SFR measurements at very low magnetic fields reveal an anomalous
behaviour in as the spin splitting persists even in the absence of the field (fig. 32).
3
>
Ε 2
LU
<
•
T=1.9K
ο
3.4
χ
6.9
Δ
12.8
+
18.0
•
28.3
0
ο
6
2
8
10
Β (kG)
Fig. 32. Spin-flip Stokes energy AE as a function of magnetic field B, for different temperatures. The
solid curves describe theoretical fits with two adjustable parameters (from Heiman et al. 1983a).
266
H.G. Hafele
The finite spin-flip energy at zero field is given by E B MP and is caused by thermal
fluctuations in the local magnetization and the bound magnetic polaron
(Peterson et al. 1985). The solid curves in fig. 32 were generated from the theory
and give good agreement with the experimental points.
Other interesting Raman features with g « 2 have been observed in (CdMn)Se
(Petrou et al. 1983) and in (ZnMn)Se (Douglas et al. 1984). They are associated
with SF transitions within the Zeeman multiplet of the Μ η 2 + 3d electrons and
will be treated with CdMnTe (see section 6.2).
6.1.2. Cross section and selection rules
Cross section and selection rules appropriate to donor-bound electrons with the
wurtzite symmetry have been determined in CdS by Thomas and Hopfield
(1968) (see section 5.2.2.2). These authors have shown that in η-samples the
important intermediate states near the absorption edge are the free-exciton
states and donor-bound exciton states made up of holes in the A, Β and C
valence bands (/£, 1* and /ξ)· I n the one-level approximation only the boundexciton state I* is taken into account, because it has the large oscillator strength
of bound excitons and the laser energy matches the excitation energy.
Then, the cross section near resonance is given by (Heiman et al. 1983a)
(73)
where/denotes the oscillator strength of the exciton bound to neutral donor 7 2,
Ej is the I2 energy for the A gap and Γ is a phenomenological broadening
parameter.
The dramatic variation of the measured scattering strength on the photon
energy ftcoL is described by eq. (73) with El = (1942 ± 5) meV and Γ between
5 and 10 meV. The polarization results are in good agreement with the selection
rules of Thomas and Hopfield (1968). However, in a semimagnetic semi­
conductor, the electron-spin-quantization axis is not aligned along the applied
magnetic field at moderate field strengths. Thus, the degree of polarization
depends on the degree of alignment and is a function of magnetic field and
temperature (Alov et al. 1983, Peterson et al. 1985).
6.1.3. Effective g-factor
If magnetic fields are considered beyond the B M P range, where the Raman shift
varies linearly with field, an effective g-value can be defined as
AE = §μΒΒ.
(74)
In the mean-field approximation (Krivglaz 1974) g is given by (Heiman et al.
1983a)
g= g
35x(xN0
1 2 f c ( T + T A F)
(75)
Spin-flip Raman scattering
267
where g* is the bare g-value, g* = 0.5, gMn = 2.0 and χ is an effective x-value; the
exchange energy ocN0 and 7^ F were already introduced in eqs(71) and (72),
respectively. Since T AF is small ( « 1 - 3 K), it follows that the Stokes shift is
approximately proportional to B/T. The curves of fig. 31 yield g = 70 for
χ = 0.01 and 7 = 1 . 9 K; with χ = 0.1 samples, g = 170 is measured at the same low
temperature.
Even though the electron-Mn 2 + interaction is conventionally described by an
enhanced g-factor, Heiman et al. (1983a) pointed out that the effect is better
viewed as a magnetic-field amplification. The magnetization of the M n 2 + ions
provides an 'effective field' which acts on the spins of the band electrons
(Komarov et al. 1977).
6.1.4. Line width
In semimagnetic semiconductors a homogeneous broadening stemming from a
finite spin-relaxation time adds to the usual sources of line broadening. The
perturbation due to the exchange interaction leads to additional SF transitions
of band electrons (Walukiewicz 1980). SFR-line shapes have been obtained by
various authors (Ryabchenko and Semenov 1983, Dietel and Spalek 1983, and
Heiman et al. 1983a). Here, we follow the paper of Heiman et al., who derived an
expression for the SFR spectrum from the spin-spin correlation function. For
the zero-field case it has the form
(76)
where
3 5 χ ( α Ν 0) 2
9 6 π Κ ) 3Ν 0'
and αξ denotes the donor Bohr radius. Spectral line shapes of B M P are shown in
fig. 33. The solid curves are best fits from theory with W0 increasing from
0.53 meV (for χ = 0.05) to 1.04 meV (for χ = 0.30) and T AF from 1.16 to 4.04 K.
Obviously, theory provides a very good description for χ ^ 0.10. The spectra of
the χ = 0.20 and χ = 0.30 samples show additional width, which may be caused
by spatial fluctuations in the Mn concentration (Peterson et al. 1982).
Numerical results of the SFR-line shape without the assumption of constant
exchange coupling have been presented by Thibblin et al. (1986). For each
temperature, the calculated spectrum consists of a high-energy broad maximum
and a low-energy sharp peak. At present, it is not clear whether this peak
explains the additional width of the Raman features in fig. 33.
6.2. Cadmium manganese telluride (CdMn)
Te
The compound (CdMn)Te is viewed as the prototype for wide-band-gap D M S .
Doping by accident or intentional doping with Ga produces enough neutral
H.G. Hafele
268
Fig. 33. Zero-field SFR spectra for the Cdj _ xM n xS e , χ = 0.05, 0.10, 0.20, and 0.30 samples at
T= 1.8 K. The scans are recorded in the right-angle scattering geometry with (σ+, z) polarization
(from Peterson et al. 1985).
donors to allow the observation of scattering associated with the spin-flip of
electrons bound to donors (Peterson et al. 1982, 1985).
A theory of SFR scattering from electrons has been presented and applied to
(CdMn)Te by Walukiewicz (1980). He included the exchange interaction into
the existing approaches for solving the eigenenergy problem for the effectivemass Hamiltonian (Jaczynski et al. 1978, Gaj et al. 1978). Since the energy gap
and the spin-orbit splitting energy are large in (CdMn)Te {E%~ 1.54eV, Δ
± 1.0 eV), the usual simplifications can be introduced and the Γ 6 conduction
band as well as the Γ 7 spin-orbit split-off valence band become parabolic. The
effect of the exchange interaction appears in the effective g-factor for the
conduction band as
gc = gi + 2ocN0(Sz}/hcoc.
(77)
The energetic structure is similar to the one showed in fig. 5, however, the spin
splittings of all bands are comparable and are much larger than the orbital
splittings. Since pure CdTe displays relatively large effective mass and small gvalue (m c = 0.1 m, g* = —0.7), the Stokes shift resulting from the conduction
band spin splitting is approximately given by
fuoL - hws = gcpBB
~ aJV 0<S z>.
(78)
Besides SFR scattering from electrons bound to donors, scattering due to SF
transitions within the Zeeman multiplet of the M n 2 + 3d electrons have been
observed and extensively studied in the paramagnetic phase in (CdMn)Te
(Peterson et al. 1985, Petrou et al. 1983). This Raman mechanism involves
interband transitions in conjunction with the exchange interaction between
band electrons and M n 2 + . As shown by Petrou et al. (1983), the interaction
Hamiltonian, eq. (70), induces synchronous changes in the spin states of the
band electrons on the one hand, and the M n 2 + ions on the other. Thus,
Spin-flip Raman scattering
269
transitions between adjacent sublevels of the M n 2 + and a spin-flip of the
electron occur simultaneously. A possible Stokes process is illustrated by fig. 34.
The energy level scheme shows the Γ 8 valence band and the Γ 6 conduction
band, magnetically split into four sublevels (with m3 = — f, — j , + ^, + f ) and
two sublevels (with m 7 = + i , — ^), respectively. An incident photon with σ+
polarization creates an electron-hole pair with the electron excited to the mj = j
level (Am 7 = 1, single arrow). This electron interacts with a Μ η 2 + ion raising the
spin of this and lowering its own spin, according to
| w s > M n ^ | w i j > e= » | m s + l >
M 2
n +|
m j - l>e
(79)
The scattered photon possesses ζ polarization, since in the last step of this
Raman process the electron returns to its initial state with Amj = 0. The
frequency shift ω ΡΜ is determined by the energy separation between adjacent
sublevels of the Zeeman multiplet of Μ η 2 + in the paramagnetic phase, therefore,
hωPM = AE = gμBB.
(80)
Within experimental errors the frequency shift is measured to be linear in Β with
the g-value g = 2.01 ± 0.02. A spectrum is drawn in fig. 35. The intensity of the
ω ΡΜ line grows by several orders of magnitude as the laser photon energy
approaches that of the C d 1 _ x M n x T e band gap. Close to resonance, also lines
with 2 ω Ρ Μ, 3 ω Ρ Μ, 4 ω ΡΜ and combination lines with a > L Ol ± ω Ρ Μ, ω ί θ2 + ω Ρ Μ,
2 e o L Ol + ω ΡΜ and c o L Ol + c o L 02 + ω ΡΜ are present, where LOx and L 0 2 are the
CdTe-like and MnTe-like LO phonons, respectively (Peterson et al. 1985).
6.3. Cadmium manganese sulfide \_(CdMn)S~\
The Raman spectra of the hexagonal (CdMn)S simultaneously exhibit SF
scattering from shallow donors interacting via s-d exchange with the Mn ions
and M n 2 + local SF scattering as in (CdMn)Se and (CdMn)Te. The first SFR
experiments in (CdMn)S have shown that the exchange interaction splits the
spin state of the donor electron in the absence of the magnetic field (Alov et al.
1981).
+ 1/2-
CB
VB
(£>-
(ΓΒ)
1 / .2
- 3/2
-1/2
+ 1/2
+ 3/2
·
-
Fig. 34. Raman mechanism for internal S F transition within the Zeeman multiplet of the ground
state of Μ η 2+ involving the band electrons. The double arrow refers to the de-excitation of the
electron and the simultaneous excitation of the M n 2+ -ion (from Petrou et al. 1983).
H.G. Hafele
270
40
Λ
A
Cd
σι, ζ
T = 300K
Μη Te
o.6
o.4
1
H = 6 0 kG
-5
χ 10
30
s
AS!
ii
ii
ii
ii
_
-
ii
ii
ii
ii
Ι |·
I ι
i i
ii
ij
i i
-
j i
i i
i i
j i
/
ι
ι
ι
1
ΙΟ
5
0
I
ι *
5
Γ­
10
RAMAN SHIFT (cm')
Fig. 35. Stokes (S) and anti-Stokes (AS) Raman lines at ω ΡΜ resulting from Am s = ± 1 SF
transitions within the Zeeman multiplet of M n 2+ in C d ^ ^ M n ^ T e , χ = 0.40 (from Petrou et al.
1983).
Extending measurements to higher magnetic fields enables noticeable devi­
ation from f-Brillouin functional dependence of the SF energies to be observed.
Douglas et al. (1984) suggest that M n - M n correlations should be introduced in
a more rigorous way to account for the experimental observations.
Heiman et al. (1983b) and Nawrocki et al. (1984) determined both the spin-flip
energy AE of donor-bound electrons and the magnetization in (CdMn)S alloys.
The analysis of the data at magnetic fields near saturation give the exchange
energy αΛΓ0 = (217 ± 1 1 ) meV for an χ = 0.023 sample. Alov et al. (1984) have
investigated SFR scattering in (CdMn)S:In with donor impurity densities higher
than the Mott value (N0 — NA) > 1 0 18 c m - 3. In that region, screening prohibits
bound states and one is concerned with a degenerate electron gas (see fig. 2).
Here, no zero-field splitting as with bound magnetic polarons is observed.
Due to the large spin splitting (according to g % 90), electrons from states
Spin-flip Raman scattering
271
Θ
135°180°
5170
5160 _ . 5 1 5 0
λ,Α
sinM0/2)
Fig. 36. (a) SFR spectra of Cdj _ xM n xS at various scattering angles ( Γ = 1.4 Κ, Η = 36 kOe). (b)
SFR line width Γ versus the momentum transfer squared g 2 ocsin 2(0/2) (from Alov et al. 1984).
deep below the Fermi surface can contribute to SFR scattering. The intensity
turns out to be proportional to the spin polarization Ρ of the electron subsystem
and increases linearly with the spectral shift AE.
lccP
=
3AE
n+ + nl
4~7
(81)
where n e+ and n~ are the electron densities in the two spin states and μ is the
chemical potential. This behavior of the free-electron gas is quite different from
scattering by the electrons bound to donors (Alov et al. 1983, Douglas et al.
1984). As demonstrated with CdS, the Doppler shift of the scattered light
frequency reveals information on the dynamics of free delocalized carriers.
According to eq. 63, dynamic narrowing produces a Lorentzian profile of the
SFR line with a width, proportional to q2, in contrast to the pure Doppler
broadening, where Δ ω 0 oc q.
The dependence of the SFR line width on the angle θ between the directions of
the incident and scattered light is shown in fig. 36. Variation of the angle θ from
zero to 180° changes the SFR line width by more than a decade, from 0.4 to
4.5 meV. The slope of the line in fig. 36b yields a collision time of T c « 3
χ 10 - 1 4 ,
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CHAPTER 6
Magnetoplasma Effects in IV-VI
Compounds
G. BAUER
Institut fur Physik, Montanuniversitat
Leoben, A-8700 Leoben,
Landau Level
© Elsevier Science Publishers B.V., 1991
Austria
Spectroscopy
Edited by
G. Landwehr and E.I. Rashba
Contents
1. Introduction
279
2. General properties of IV-VI compounds related to magnetoplasma effects
280
3. The classical oscillator model
283
3.1. Single-valley systems
285
3.2. Many-valley systems
285
3.2.1. The Faraday configuration, B\\ [001]
287
3.2.2. The Voigt configuration, B\\ [ 0 0 1 ]
287
3.2.3. The Faraday configuration,
fl||[lll]
3.2.4. The Voigt configuration, B\\ [ 1 Ϊ 0 ]
4. Numerical data on the dielectric function, and transmission and reflectance spectra
288
288
291
4.1. The Faraday configuration, # | | [ 1 1 1 ]
291
4.2. The Voigt configuration, B\\ [ Π 0 ] for £ | | Β and E1B
297
5. Magnetoplasma effects in two-dimensional systems
305
5.1. Inversion layers
305
5.2. Quantum wells
316
5.3. Doping super lattices
318
6. Magnetoplasma effects in strained semiconductor layers
320
7. Magnetoplasma reflection using the strip line technique
324
8. Dynamical conductivity in the frequency range of coupled LO-phonon-plasmon
excitations
9. Linear-response theory
328
330
10. Conclusions
338
References
339
1.
Introduction
The subject of the microwave and infrared properties of semiconductors has
been treated in considerable detail. Perhaps the most comprehensive review on
this topic is still the one by Palik and Furdyna (1970).
In a semiconductor, the host lattice and whether it has infrared-active
phonons or not, together with the free carriers, determine the response to
electromagnetic radiation at frequencies below the fundamental band edge. The
infrared-active phonons have characteristic frequencies in the range of
1 0 0 - 5 0 0 c m - 1 for most semiconductors and induce an opaque range in the
Reststrahlen region. On the other hand, the free-carrier concentration JV, the
effective mass m and the high-frequency dielectric constant
determine an
effective plasma frequency (ω*) 2 = N ^ 2/ e 0m e 00 below which, for small carrier
damping, the semiconductor has a reflectivity close to one. By applying
magnetic fields the regions where the samples are opaque are influenced by
additional resonances such as cyclotron resonances, and the wave propagation
is determined by the properties of the semiconductor and the orientation of
magnetic field with respect to the crystal axes and the propagation direction of
the light.
In the past, magnetoplasma effects have been studied in considerable detail
both experimentally and theoretically in single- and many-valley semiconduc­
tors. The classical description based on Lorentz oscillators for the infraredactive phonons and on the Drude-like model dielectric function was mainly used
and the relevant procedures can be found, e.g., in the reviews by Palik and
Furdyna (1970), Pidgeon (1980), Zawadzki (1974) and Grosse (1979). Plasma
effects in 2 D systems were reviewed by Ando et al. (1982) and in these volumes
by Volkov and Mikhailov. Especially in narrow-gap materials, the simple
effective-mass picture has to be replaced by more complicated analyses which
take nonparabolicity into account as shown by Zawadzki (1974).
In addition often in narrow-gap semiconductors the free-carrier concen­
tration is quite high. Then characteristic propagation thresholds for
electromagnetic waves like reflection minima, regions of evanescent waves etc.
can only be determined by a complete analysis of the total dielectric function.
The description of the response of a narrow-gap semiconductor to far-infrared
radiation provides a lot of information about the band nonparabolicity and
more subtle details like inversion asymmetry splitting (II-VI compounds),
wavefunction admixture and so on. These features can induce new resonances
such as spin-flip (SF) and combined spin-flip (CSF) transitions. In an extension
of the simple Drude model, Pastor et al. (1981) and Witowski et al. (1982) have
added additional oscillator terms to the Drude-like expression. Although these
kinds of transitions have no simple classical analogue like cyclotron resonance
in terms of electron orbits, such a procedure leads to some insight, e.g., in which
geometry a plasma-shifted spin resonance occurs (E1B) and in which it does
280
G. Bauer
not (E\\B). However, the oscillator strength is a quantum mechanical property
and the transmission and reflection spectra will be determined by transition
probabilities and the occupancy of initial and final states.
In this review we shall concentrate on the description of narrow-gap IV-VI
compounds with the main emphasis on obtaining information about band
structure parameters. The IV-VI compounds have been investigated for quite a
long time and many excellent papers on microwave magnetoplasma effects, also
dealing with nonlocal eiiccts have already appeared before 1970 (e.g., Nii 1964,
Bermon 1967, Numata and Uemura 1964, Perkowitz 1969). Due to the
introduction of the strip line technique these materials with quite high carrier
concentrations then became accessible for further investigations in the farinfrared region. Von Ortenberg (1980) has summarized these methods. Another
addition to the field was the advent of epitaxial techniques through which rather
high-quality films with comparatively low carrier concentrations became
available.
Since thin films are important for other semiconductor systems as well, we
shall mainly present data on thin-film work where the apparent spectra are
influenced by the substrate properties as well.
This chapter is organized as follows: we start with a brief description of the
general properties, followed by the classical-model dielectric function for both
the Faraday and Voigt geometries, giving explicit examples for special configurations of the magnetic field. Then magnetoplasma effects in 2 D systems are
described followed by a section on magnetoplasmon effects in strained layers. A
section on the investigation of magnetoplasma effects in the strip line configuration is followed by a discussion about the dynamical conductivity in the
frequency range of coupled LO-phonon-plasmon oscillations. Pinning phenomena occurring in this range were treated by Vogl (1977) and Vogl and Kocevar
(1978), who outlined the new effects for degenerate statistics (Fermi energy
within the bands), which are not included here. Finally there is a section on the
quantum mechanical formulation of the dielectric function problem. This
question was pioneered by the work of Quinn and Rodriguez (1962), Quinn
(1964) and Wallace (1970). Recently Wallis and Balkanski (1986) have summarized in their book Many Body Aspects of Solid State Spectroscopy the effort
which has been directed towards the quantum mechanical formulation of the
dielectric function so that just the specific problem of IV-VI compounds will be
dealt with.
2. General properties of IV-VI
magnetoplasma
effects
compounds
related
to
The cubic IV-VI compounds have a NaCl crystal structure and the conduction
as well as valence bands have multiple extrema at the L points of the Brillouin
zone. The energy gaps are small (PbTe ~ 190 meV; PbSe ~ 140 meV;
Magnetoplasma effects
281
P b ^ S a J e , from 190 meV to OmeV for χ = 0.35, at T = 4.2K). Therefore
nonparabolicity is quite important. Due to the NaCl structure there is one
infrared-active phonon and the parameters are given in table 1. The static
dielectric constants are extremely high compared to other semiconducting
compounds. The Fermi surface consists of four elongated ellipsoids oriented
along the [111] direction for electrons and holes. The Brillouin zone is shown
in fig. 1, together with the variation of the Fermi energy with applied field for
n-PbTe (8 χ 1 0 1 6 c m " 3) . For the calculation of the Landau states a ( 4 x 4 )
matrix Hamiltonian has to be solved (Mitchell and Wallis 1966, Adler et al.
1973, Dimmock 1971, Burkhard et al. 1979, Bauer 1980). For the calculation of
the plasma frequency the model outlined by Zawadzki (1974) can be used which
starts with a generalized ellipsoidal energy band with arbitrary nonparabolicity
y(E) =
aapkakp,
where y(0) = 0, and the shape of the ellipsoids does not change with energy.
The function y(E) describes the nonparabolicity of the band and the tensor ααβ
is symmetric.
In a simple expansion, the first-order nonparabolic energy bands of the IV-VI
compounds are described by
1
(h
+
j L
Y \ 2 +2 n h=
2
(
k
(
kx
t?(k2x + k*)
2m±
+
2
k
^ _
)
+
2m7
+ K ) +
|
E
^^.jL (k2
\ (
){
E
2
+
2m*
k
_ 2^ _ )
E
)
2m/
h2k2
2m ||
Table 1
Model fit parameters
PbTe
PbSe
BaF2
21
218
38 c m " 1
119 c m " 1
2 cm"1
2.16
7.2
185.5 c m "
Γ
33-35
1330-1350 (4.2 Κ)
17.5-18.5 c m " 1 (4.2 Κ)
114 c m - 1
2-3 cm"1
Free carriers:
η, ρ
5 χ 1 0 1 ,6 . . . , 5 χ Ι Ο 18 c m " 3
1 χ 1 0 1 7,
1 χ 1 0 1 c8 m - 3
0.02 l m 0
(band edge value)
-10
<1,...,4
0.024m 0
-11
0.037m 0
Lattice parameters:
ε*
ω το
m t(n)
Κ = melmx{n)
ωτ
mt{p)
K(p)
-1.9
2-8
0.036m 0
-1.8
5-10 cm"
(Γ-dep.)
G. Bauer
282
MAGNETIC FIELD (T)
Fig. 1. Dependence of the Fermi energy of n-PbTe (n = 8 χ 1 0 1 c6 m - 3) on magnetic field for
β || [111]. The indices h and 1 refer to Landau states of the obliquely oriented valleys <Tl 1 > and those
oriented parallel to β, [111]. Inset: Brillouin zone of the cubic IV-VI compounds.
where
1 _ 2P\
m1
m § £ g'
1 _
m||
2P\
mlE^
where PL and Ρ () are respectively the transverse and longitudinal interband
momentum matrix elements and the terms m*, mf represent the far band contribu­
tions to the transverse and longitudinal effective masses in the valence ( + ) and
conduction (— ) bands, respectively.
The effective plasma frequency is then given by
e2NTrA7T
—
3
\m
where A = det ααβ and an average inverse mass can be derived from ω* (the
'plasma mass').
In semiconducting lead compounds the minimum carrier concentration is of
the order of 1 0 16 cm ~ 3 and thus the relevant frequencies, the plasma frequency,
the lattice mode frequencies and the cyclotron resonance frequencies are all of
the same order of magnitude. The complete dielectric function which consists
_
ω*2 =
Magnetoplasma effects
283
of all these contributions is necessary for the description of experiments.
The main features observed in transmission and reflection, in the Faraday and
Voigt geometries are described reasonably well by simple classical oscillators if
the free-carrier damping is taken into account. The shortcomings of such a
procedure are listed in the following:
(i) In a narrow-gap semiconductor the effective masses are energy dependent.
(ii) In a quantizing magnetic field the e(k) relationship is split into Landau
states which are occupied to finite values of kB (momentum in the β-field
direction) according to the Fermi distribution. The classical cyclotron frequency
corresponds to transition between subsequent Landau states with equal spins.
In nonparabolic bands, at a given magnetic field, a single transition energy is not
observed but instead an entire distribution of energies, those for kB = 0
corresponding to the maximum energy and those for kB = fcBmax to the minimum
energy.
(iii) Apart from cyclotron resonance transitions, in a narrow-gap semi­
conductor due to various mechanisms spin-flip and combined resonance
transitions are possible with a finite oscillator strength. In a classical Drude
model these transitions can only be included by additional 'artificial' oscillators.
(iv) Transitions between Landau states depend on the dipole oscillator
strength and the occupancy of the initial and final states. The total free-carrier
concentration Ntot which enters into the plasma frequency and which determines
the oscillator strength in the classical Drude model is not an adequate
description for the quantum mechanical transition probability and the dif­
ferences in occupancy.
(v) The occupancy of a given Landau state depends on the carrier con­
centration and on the temperature; in a many-valley semiconductor or for
warped surfaces of constant energy it also depends on the direction of the
applied magnetic field with respect to the crystal axes.
In the following we start with the description of the magnetoplasma
phenomena in IV-VI compounds using the Drude model. The principal features
and the main physical content of the magneto-optical spectra can be derived.
The necessary generalization, a quantum mechanical model originally given by
Wallace (1970), is used to exemplify the essential differences.
3. The classical
oscillator
model
In order to obtain the relevant optical constants of a given material in a certain
direction of the applied magnetic field and for a certain polarization of the
electromagnetic radiation the tensor components of the dielectric function have
to be known.
284
G. Bauer
The dielectric function consists in general of three susceptibility contributions:
£
(1
Xvalence electrons
Zpolar phonons) ^
Xfree electrons
(1)
The first contributions are given by
1
Zvalence electrons
^ao
the so-called high-frequency dielectric constant,
( g s- 0 < a | o
Xpolarphonon
^
_
2ω _
^
W
where ε 5 denotes the static dielectric constant, ω τ ο the transverse optic mode
frequency (q-+0) and Γ the phonon mode damping parameter. The oscillator
strength depends on the difference between the static and high-frequency
dielectric constants, and the longitudinal optic mode frequency coLO is deter­
mined by the Lyddane-Sachs-Teller relation: (ct> L 0A% 0) 2 = e j e ^ .
The free-carrier contribution Xfc is magnetic field dependent. In order to
determine its tensor components we start with a classical equation of motion
±(mv)
at
= e(E+vxB)-—.
(3)
τ
The semiconductor properties are determined by the effective-mass tensor m
and by the relaxation time of the carriers τ (usually energy dependent, but for
simplicity assumed to be determined by the Fermi energy in a degenerate
semiconductor), ν is the velocity of the carriers. The magnetic field Β corre­
sponds to the externally applied field (B0) since the contribution of the radiation
field can be neglected. The electric field is the radiation field:
Ε=ΕτΆά{ω)~Εέ«*.
The drift velocity ν has then the same time dependence as Erad. Using eq. (3), the
frequency-dependent current density /(ω) = nev can be obtained from eq. (3)
(Palik and Furdyna 1970, von Ortenberg 1980) to be given by
j=Ne2[eE((D)
χ (mB)
+ ie2B(E-
- i || m||(m)~ιΕ(ω)(ω
Β)/(ω + ίω τ)
+ ϊωτ)^2ΒηιΒ-
(ω + ϊωτ)2 \ \ m | | ] ~ \
(4)
where ω τ = 1/τ, ( m ) " 1 is the inverse of the effective-mass tensor m and \\m\\
denotes the determinant.
From j = dijEj and χ{0 = ίσ/ε 0ω we obtain finally the tensor components of the
free-carrier susceptibility %.
Magnetoplasma effects
3.1. Single-valley
285
system
In epitaxial layers, due to strain effects the fourfold degeneracy of the L states of
the Brillouin zone can be lifted. For sufficiently small carrier concentrations the
Fermi energy can be smaller than the strain-split energy AEC (see section 6).
Then a single-valley situation occurs and the magneto-optical properties will be
determined by this fact.
For an angle θ between the orientation of the applied magnetic field Β (Β || ζ)
and the main axis of the ellipsoid, the tensor component of %ic are then given
(e.g., von Ortenberg 1980) by
Ne2
Xxx
ω + ϊωτ
mK£0(D (ω + ί ω τ) 2 + ω 2 '
Ne2
Xyy
mxs0a)((u
Ne2
Xzz
ω + ίω τ
mg c o s 2 θ + m t s i n 2 θ
+ ia)T)2 — (u2
m{
ω + ίω τ
m t c o s 2 6 + me s i n 2 θ
ί
m te 0cu \^(ω + ί ω τ) 2 — ω2
e2B2/mtmt(oj
(ω2 + ί ω 2) iNe2
Xxy
Xyx
Xyz
Xzy
Xzx
Xxz
mts0a)
Ne2
+
m(
ίωτγ
(5)
ω2
(eB/mt)[(m,
cos2θ + m t sin 20)/m,]
(ω + ΐ ω τ) 2 — ω 2
(ω + ίω τ)
(m t — m^) sin θ cos θ
mts0(D (ω + \ωτ)2 — ωΐ
iNe2
el?/mt
(m t — m,) sin θ cos θ
mte0co (ω + icoz)2 — ω2
m(
where the cyclotron frequency ω 0 is given by
e2B2 me c o s 2 θ + m t sin 2 0
ω2 = —
ί
^
(6)
m,
m tz
and m t,
denote the transverse and longitudinal effective masses, respectively.
3.2. Many-valley
systems
In order to obtain the conductivity or susceptibility tensor for the four-ellipsoid
model one needs to add the current contributions from all four valleys in a
common coordinate system. We shall present the components explicitly for two
orientations of high symmetry, namely 2f||[100] and Z f | | [ l l l ] in Faraday
configuration and for B\\ [100] and B\\ [ Π 0 ] in the Voigt configuration. If the
orientation of the magnetic field is confined to a high-symmetry plane, the (010)
286
G. Bauer
plane, the tensor components are given (von Ortenberg 1980) by:
Ne2
Xxx =
A ( 2 / m t + l/m,) + sin(2<5)[j(l/m, - 2/m t)]
\
+ 1C°t) 1
2ε0ω ω
- ^ΓΖ(
ω 2. — (ω + ί ω τ) 2
Ate 2
-
2
ω
+ — (ω + ίω τ)
-^ΓΖ
j ( 2 M Hh llmA ~ sin(2(5)[j(l/m, - 2/m t)]
X
Zyy =
τ
Λ
2ε0ω
Ν
/ i(2M+l/m,)
(
* 2/
Ζζζ= ~ ~
2 ε 0ω
ν(
ω
+
ί
τ )ω
" 1τ
\ω
+
7
ΓΤ—72
— (ω + ΐω τ)
_!.·
Υ - β 2 β 2 Μ > , ( ω
ω + ι ω Τ)
t 7
,
^M+l/m,)
+
ω ι — (ω + ίω τ)
+ ΐ ω τ) 2 +
^(2Μ+1Μ)
ω + — ( ω + ί ω τ) 2
\
sin(2<5)[i(l/m,-l/m,)]
|
ω+ — (ω + ί ω τ) 2
+
• e2B2lm2m,(cj
+ ί ω τ) 2 + j(2/m t + 1/m,) - sin(23)[&(l/m, - l/m t)]
ω ΐ - (ω + ί ω τ) 2
iiVe 2 eB
Xxy
Xyx
A ( 2 m t + m,) + s i n ( 2 ^ ) [ i ( m , - m t ) ]
2
ω 2 — (ω + ίω τ)
2 ε 0 ω m me \
j(2m t + m ^ ) - s i n ( 2 ( 3 ) [ i ( m ( , - m t ) ]
|
ω ΐ — (ω + ί ω τ) 2
2 ε 0ω ν
\
11
ω2+ — (ω +
ίωτ)2
c o s ( 2 ^ ) [ i ( l / m , - l / m t) ]
ω 2. — (ω + ί ω τ) 2
iiVe 2
e£
2 ε 0ω mt2m^
cos(2£) [|(m, - m t)]
ω + — (ω + ί ω τ)
cos(2<5) ft(m, - m t)]
2
ω 2. — (ω Η- ί ω τ) 2
(7)
where ω ± is given by
2
_ e2B2
+
/(2m
2mt t +
Λ
3
sin(2<5)(m, -— mra
t) t)\
sin(2d)(m;>
±
6
J
and δ denotes the angle between the magnetic field orientation and the [001]
axis. For arbitrary magnetic field orientation in the (010) plane there are two sets
of ellipsoids with different tilt angles δ resulting in two different cyclotron
resonance frequencies. For Β along one of the <100> directions just one
cyclotron frequency occurs.
Magnetoplasma effects
287
For arbitrary magnetic field orientation the phenomenon of'tilted orbits' also
occurs: whereas in momentum space the cyclotron orbits are always per­
pendicular to Β in coordinate space, the orbit can be tilted with respect to Β
unless the field is parallel to one of the principal axes of the ellipsoid.
Thus a carrier contributes to a component of an oscillatory current parallel to
Β at coc(S) and couples resonantly to an electric field of such a frequency
polarized parallel to B.
3.2.1. The Faraday configuration, 2?||[001]
In the Faraday configuration for B\\ [001] the free-carrier susceptibility is given
by (Burkhard et al. 1976)
±
_
2
±(Ne
/e/0)(l/m,
+• 2/m
ω,)
t)(co + Ίωτ ±
j v —
-υ/ν-/·-ν
- / — ι / ν • - ~ τ ^ - ~ ι /
_ / _ 22 —
_ _ω22 — ω
2 2 _l + 2ίωω τ)\
ω(ω
Afc
/Q\
'
yyf
where
co. —
(2/m,+
\/mt)eB
(l/m, + 2/m t)m t
and the cyclotron frequency is given by
coc = eB\ — 4 — - )
.
(10)
3.2.2. The Voigt configuration, £ | | [ 0 0 1 ]
In the case of the Voigt geometry one has to distinguish between two cases,
either El Β (extraordinary mode) or E\\B (ordinary mode).
The refractive indices are given by
~2
nl
_
c ,
— xx b 1"
e
xyeyx
~
εχχ
_
—
-β β
Ί+
~ Z
+ "+" ε -
>
8
~2
w
|| — £zz>
where
_
S zz
~ ^
(7Ve 2^ 0)[(m t + 27?v)/3mtwv](a>2 — ω 2 + 2ϊωωτ — 3e2B2)/(mt
2
2
ω (ω - ω
2
+ 2m^)mt
2
- ω + 2ίωω τ)
(Π)
(ε, denotes the lattice contribution due to ε^ and the polar phonon mode).
The resonance behaviour in the ordinary mode is due to a tilted-orbit
G. Bauer
288
resonance. The behaviour in the extraordinary mode is much more complex:
η\ = ε(
_ {Ne2/e0){mt
+ 2fty/3m tm < f)[co 2 - ω 2 + 2ίωω τ - 3m /(2m t + m,/(mt + 2 n y ) 2c o 2]
ω 2 ( ω 2 - ω2 + 2ίωω τ - ω 2 )
(12)
There will be structure due to a resonant behaviour in εχχ at coc (due to
incomplete screening of the internal polarization field). The second resonance
occurs when εχχ in the second term vanishes, i.e., for zeroes of εχχ. This leads to
the so-called plasmon-shifted cyclotron resonance at approximately ( ω * 2 +
3.2.3. The Faraday configuration, Β || [111]
In this configuration two cyclotron resonance signatures appear, one due to
carriers of the valley oriented with its main axis parallel to the applied B-field
and the second one due to the resonance in the three valleys inclined by the same
angle to B.
The free-carrier susceptibility contribution is given by
±
*
(Ne2^0)(\/mt)(w
=
fc
+ ω0ι - ίω τ ι)
4ω[(ω + ω 0 1) 2 + ω 2 ]
(Ne2fc0m0)[a((u
+ ίω ΐ 2) ± ^ ω 6 ι] ( ω 2 - ω02 - ω22 2
ω[(ω - ω
where ω0ί =
ω 02 =
2
2
2 2
τ 2
2
2
- ω ) + 4 ω ω τ 2]
2\ωωτι)
(13)
eB/mv
^ / [ 1 / ^ ( 8 Κ + 1 Μ ) ]
a = ^2(4/m, +
^ ϊ ΐ ( 8 Μ +
1
/
,2
5/mt)m0,
l M ) m
0
,
where ω τι and ωΧ2 denote the free-carrier damping in the [111] valley and the
three obliquely oriented valleys, respectively. Equation (13) was derived by
Burkhard et al. (1976), including damping which was omitted by Wallace (1965,
1966) in earlier treatments.
It is evident from the structure of the second term of eq. (13) that there will
also be a resonant behaviour for the wrong sense of circular polarization due to
carriers in the oblique valleys (Burkhard et al. 1976, 1979).
3.2.4. The Voigt configuration, # | | [ 1 Ϊ 0 ]
In order to obtain the dielectric function it is important to consider the
propagation direction of light. For epitaxially grown {111} oriented samples, the
wave vector of the light will be parallel to the [111] direction whereas for a {100}
oriented field, the wavevector will be parallel to <100> directions.
Magnetoplasma effects
289
We treat both cases, £ | | [ 1 1 1 ] and £ | | [ 0 0 1 ] in some detail in order to
demonstrate the necessary steps in the derivation of the free-carrier
susceptibility.
Starting with the equation of motion
m-v
= e(E+ ν χ Β) -
ωτητν
and using the explicit time dependence of the velocity ν one obtains
(ωτ - ίω) mv = e(E + ν χ Β).
The current j = Nev is then given by
j = Νβ\{ωτ
l
eb~] E=
- \(D)m-
(14)
σΕ,
where
/
0
-By\
0
-B2
Bx
-Bx
0
/
For each of the four valleys at the L points the conductivity tensor is calculated
in the coordinate system x', y', z' where the mass tensor is diagonal:
Thus
Ι(ωτ-Ίω)τηχ
eB$>
\
(15)
(ω, — ico)mt
ft(s) = ne 2
( w t- i c o ) m ^
where s = 1, . . . , 4 .
The next step is a transformation in the system of coordinates (jc, y, z) where
[110] || ex, [ 1 Ϊ 0 ] || e}, and [001] || es. This transformation is performed for all four
valleys. This leads to the components of %fc (Krost et al. 1985):
Ne2
Xxx
1
2ηιιε0ω (ω + ίω τ)
Ne2
2τηιε0ω
Ne2
(ω + Ίωτ)
1 +
j ( l + 2mt/mM(Q + ί ω τ) 2 (ω +
ίωτ)2-ω^
i ( + 2m t/m,)
2
(ω + ί ω τ) - ω
2 + mJmf
.
2
+ (ω +
ω2
1
ίω τ) — ω2
(16)
G. Bauer
290
( ω + \ωτ)2
— c o 2a
(ω + ί ω τ )
2
— ω ,Cb
(ω + ί ω τ) 2 — co 2b
2^η ιε 0ω \ ( ω + ί ω τ) 2 - ω 2 β
where
ωο
Β =
7T72
τ
and
a)^
=
eB[
(m tm,) 1 /2
'
.
(17
V 3 mt m / /
In this coordinate system the dielectric function for the two Voigt configu­
rations is given for a magnetic field oriented parallel to [110] and with the
propagation direction of light parallel to [001].
For the ordinary Voigt mode (n^=n0 the refractive index)
η2
=
(1
+
Zoo +
Xph
+
Xxx)
=
β/ +
Χχχ·
(18)
A resonance occurs at the tilted-orbit resonance co C b, the two valleys oriented by
θ = 90° with respect to Β do not contribute.
As far as the extraordinary Voigt mode (n e = nL) is concerned, ne is given by
« = β, + 2„-Μ*,
(19)
Xzz
where the appropriate expressions [see eq. (16)] have to be inserted in order to
obtain n e.
For {lll}-oriented epitaxial films the coordinate system x,y,z has to be
rotated around ex = ex into the (x, y, z) system where ez is parallel to the [111]
direction and thus normal to the film surface.
The result of this rotation is then the following susceptibility tensor (Krost et
al. 1985)
/x*
X=
0
0
\
0
hXyy
-
Xn
0
+ ΊΧζζ
+ %y/2(Xz2
Xyz + ^ X z z - X y y )
~ Xyy)
for (Zs||e x||lf) and the extraordinary (E±B\\ey)
W e = Syy
Syz^zyfezz'
\
iXzz
+ hyy
(20)
)
modes:
(21)
It is important to note that zeros in the real part of εζζ are responsible for
resonant structures in n\ (which depends both on χζζ and xyy).
There are three resonances in w 2, one due to incomplete screening close to a > C ,b
a hybrid resonance which occurs approximately at ω = ^(ω 2 + ω 2 ) 1 /2 if the
Magnetoplasma effects
291
lattice contribution to the dielectric function is assumed to be constant,
otherwise an appreciable shift of this resonance occurs towards co C b. The third
resonance is a plasma-shifted cyclotron resonance.
Since the total n\ is also determined by the contribution coming from eyy and
by the off-diagonal elements eyz and ezy, which are different for the [111]
propagation direction of light, additional resonances appear: those close to co C a,
a>Cb and ω τ ο. The resonance close to coCb is cancelled by a term of the opposite
sign from syzszy/szz.
Therefore n\ exhibits five resonant structures.
For constant-frequency and magnetic-field sweeps the resonances of the TO
phonon mode and the last one, due to zeros in ε ζζ are not seen (for Β IE). Three
resonances remain: one being the hybrid resonance, the second one which is
close to coCa and the third one a resonance due to the zero of εζζ between co Ca and
the phonon frequency. For E\\B only the resonance at <yCb remains.
4. Numerical
reflectance
4.1. The Faraday
data on the dielectric
spectra
function,
and transmission
and
configuration,
In fig. 2 the real and imaginary parts of the total dielectric function (including all
lattice contributions) in the Faraday geometry ε( + ) are shown as functions of
frequency for a fixed magnetic field. For the cyclotron resonance active (CRA)
sense of circular polarization two resonant structures appear which are
associated with the resonant absorption due to carriers in the [111] valley and
the three oblique <Tll> valleys. For the inactive sense of circular polarization
(CRI) the three oblique valleys also exhibit a resonant absorption, since the
eigenmode is an elliptic one which can be excited even for the CRI mode. For the
dispersive measurement Fourier transform spectroscopy is usually used, for
which no circular polarization of the far-infrared radiation is possible. Thus in
an attempt to fit reflectivity or transmission spectra which are taken with
linearly polarized light the absorption for the CRI sense contributes to the total
signal. For the [111] valley at the maximum, 50% of the incoming intensity can
be absorbed at the resonance frequency, since for that valley only the CRA mode
contributes to the absorption for J f | | [ l l l ] .
In fig. 3 magnetoreflectivity data are shown, calculated using eq. (13) for five
different magnetic fields and for linearly polarized light. The main features are
the following, starting from low frequencies:
(i) a dielectric anomaly associated with the zero in Re ε between the TO mode
frequency and the resonance of the three oblique valleys at ω02;
(ii) a structure related to a>C2 which manifests itself as a dip in the reflectivity
due to the positive part of Re ε for frequencies below a>C2;
G. Bauer
292
1500
3000
ι
·
2700
900
300 -
-300
-600
900
I
1500
1200
900
ί
ί
/
1500
30
— ,4 0. ^ 50
>
- 1200
-
A
1800
70
ι
FREQUENCY
610 "
9 ι0
8ι0
( c m 1)
I
100
Γ
I
600
-
V
- ?
110
1
120
130
300
0
Fig. 2. Real part (ε*: full curve) and imaginary part ( ε | : chain curve) of the dielectric function for
two polarizations ( ε ί > ,2 CRA; ε ί > ,2 CRI) according to eq.(13) for £ = 2.78 Τ, £ II [111] for n-PbTe
parameters. The CRI mode exhibits a resonant structure just for the oblique valleys, in ε! and ε 2.
(iii) dielectric anomalies associated with co Cl which induce structures in R for
frequencies below coCl;
(iv) a plasma edge which is split at higher magnetic fields.
The parameters used for these model calculations are summarized in table 1.
Depending on the carrier concentration the masses are taken at the Fermi
energy. Multiple-reflection and interference effects in the semiconductor film
and the substrate are considered with the equations as given by Burkhard et al.
(1976).
The main features present in the model calculations are also seen in the
experimentally observed magnetoreflectivity as a function of frequency. As an
example data are shown for an n - P ^ _ x M n x T e sample at Τ = 5 Κ in fig. 4. Due
to the larger gap of Ρ ^ _ χ Μ η χ Τ β as compared with PbTe the effective masses
are somewhat smaller.
The magnetic field dependence of the real part of the dielectric function for the
CRA and CRI modes at a fixed laser frequency (λ = 119 μιη) is shown in fig. 5
(after Schaber 1979c). The resonance positions, as well as the dielectric ano­
malies, are indicated. The calculated reflectivity spectra for the parameters as
given, for both the CRA and CRI modes are shown as well. Finally experimental
Magnetoplasma effects
1.0
0.8
>- 0.6
>
>····>-»Γ,
ί^'
ν
\
η-PbTe
Bllkll [111]
Β=0Τ
t : 0.4
293
—
1Τ —·
2Τ
3Τ
cr
0.2
4Τ -
-
5Τ
αΌ
100
200
300
1
WAVENUMBER (cm" )
Fig. 3. Model calculation of R(v) for the Faraday configuration with linearly polarized light
[R = (R + + K ~ ) / 2 ] for various fields ( ω τ = 3 α η - 1, η = 5.1 χ 1 0 17 c m " 3, ^ = 0.025,Μ0, Κ =10,
d = 3.7 μπι), according to eq. (13). Arrows indicate ω0ί and wC2 for Β = 5 Τ.
Ρ^.χΜηχΤε
χ=0.012
—Β=0Τ
B=1T
--B=2T
«™«B= 3 Τ
--B=4T
- B=5T
100
200
FREQUENCY (cm1)
300
Fig. 4. Magnetoreflectivity spectra of PbMnTe showing structures due to dielectric anomalies
associated with cyclotron resonances as well as the plasma edge splitting. Structures beyond
200 c m " 1 are dominated by the sandwich structure
Pbl-xMnxTe/BsLF2.
G. Bauer
294
>Η-
>
d P b =8.7pm
Te
N ™ = 1 . 5 M 0 16
1>
16
Ν < 1 1=1.88χ10
mc =
1 0.0226m 0
m C2 =0.0513 m 0
ur=60
1
>-
0
2
4
MAGNETIC FIELD (T)
6
Fig. 5. Magnetic spectroscopy: Re ε* (full curve and broken curve) as a function of Β for fixed laser
wavelength {λ = 1 1 9 μπι). D A denotes the position of the dielectric anomalies. The calculated
reflectivity spectra are shown in the center part of this figure, the measured reflectivity is shown in the
bottom part. In the derivative dR/dB a spin resonance becomes visible (after Schaber 1979c).
data on the reflectivity of an n-PbTe sample with a total carrier concentration of
3.44 χ 1 0 1 6c m " 3 are shown, again for both circular polarizations. In the
derivative of the reflectivity with respect to the magnetic field an additional
structure appears at magnetic fields in between ω** 1 11 and co<2f1 ° , attributed to a
spin-flip resonance in the [111] valley. The g-factor derived from the resonance
position, g , ^ 5 7 . 1 , is somewhat smaller than the value obtained by Pascher
(1984) from a four-wave resonant mixing experiment which is more accurate.
Using p-type samples Schaber and Doezema (1979a) were able to see spin
resonance for carriers in the [111] and the oblique <Tll> valleys and also thus
to obtain information about the transverse g-factor (fig. 6). In an ellipsoidal
approximation g(6) = (g2 cos2 θ + gf s i n 2 0 ) 1 /2 holds.
Magnetoplasma effects
295
1) 11111
CRl l 1 S
R
'
0
2
4
6
MAGNETIC FIELD (T)
8
Fig. 6. Reflectivity as a function of magnetic field for n- and p-PbTe. Spin resonances for both types
of valleys are visible and the resonant behaviour for CRI polarization in the oblique valleys. From
the polarization dependence the sign of the electron g-factor is positive (after Schaber and Doezema,
1979a and 1979b).
Magnetotransmission experiments at constant laser wavelength are com­
plementary to the magnetoreflectivity data. In fig. 7 a sample with a mobility
μ = 1.5 χ 10 6 c m 2 V " 1 s " 1 ( T = 2 K) is used which exhibits additional structures
close to the cyclotron resonance of the oblique valleys, interpreted as evidence
for quantum effects: both the 0 ~ - l " as well as 0 + - l + transitions are observed.
The wavelength dependence of magnetotransmission for an η - Ρ ^ _ χ Μ η χΤ 6
sample is shown in fig. 8. Whereas at λ = 96.5 μιη the cyclotron resonances are
still accompanied by dielectric anomalies, since the real part of sL is negative for
this and for longer wavelengths, for shorter wavelengths (70.6 μπι and 57 μπι)
for which Re sL > 0, transmission minima are actually observed. This is
generally the case if a > L a rs ^e ω£ο where:
For laser frequencies larger than ω χ ο, but smaller than ω^ 0, the model
calculations always yield enhanced transmission for Β > BTes (fig. 9).
296
G. Bauer
Fig. 7. Magnetotransmission as a function of magnetic field for n-PbTe. Additional structures are
due to quantum effects caused by the nonparabolicity of the band: the 0 ~ - » l ~ and 0 +- > l +
transitions do not coincide.
The change from transmission windows close to the cyclotron resonance
frequencies associated with dielectric anomalies and transmission minima for
shorter wavelengths is also well accounted for by the model oscillator fits based
on eq. (13) as shown in fig. 10 for the wavelengths λ = 96.5 μπι, 70.6 μπι and 57 μπι
for a 3μιη thick sample with n = 1 χ 1 0 1 7c m ~ 3 . The sample thickness is of
crucial importance as shown by the broken curve for λ = 51 μπι which
corresponds to a thickness of 3000 A. For the same parameter set, the influence
of sample thickness is also illustrated for a laser frequency well below ω^ 0, thus
corresponding to R e e L < 0 . Whereas for the 3 μπι thick sample dielectric
anomalies appear, a sufficiently thin (d = 3000 A) sample exhibits minima at the
two resonant magnetic fields (fig. 11). In fig. 12, it is demonstrated experimen­
tally that in quantum well systems this behaviour is actually observed: for a PbTe
layer of d = 70 A sandwiched between PbEuSeTe barriers, the holes in the [111]
valley cause a minimum in the transmission spectrum at λ = 118.8 μπι. Kim et al.
(1987) did not observe the oblique valley resonance, most probably due to the
lack of occupancy of the three <Tl 1 > valleys in this particular sample.
Returning to the three-dimensional bulk case, in fig. 13 the effect of varying
the carrier concentration on the transmission spectra is demonstrated (for
d = 5 μπι). Since the magnitude of the maxima in the positive region of Re ε
Magnetoplasma effects
297
Fig. 8. Magnetotransmission as a function of field for η - Ρ ^ _ χΜ η χΤ β (<Ζ = 3.5μηι, η = 1 . 2 χ
1 0 17 c m - 3) for several laser wavelengths. For sufficiently short wavelengths Re s L > 0 and thus the
resonances cause transmission minima without dielectric anomalies.
depends critically on the carrier concentration, the magnetotransmission spectra
reflect these changes quite dramatically.
4.2. The Voigt configuration, £ | | [ l T 0 ] / o r E\\B and Ε LB
In fig. 14 the results of model dielectric function calculations according to
eqs (20) and (21) are shown for dispersive spectroscopy for magnetic fields of 3 Τ
(η = 1 χ 1 0 17 c m - 3) for the ordinary mode (a), which exhibits a resonance at
298
G. Bauer
2
4
6
8
MAGNETIC FIELD (T)
10
Fig. 9. Calculated magnetotransmission as a function of field for PbTe {d = 5 μπι, ωτ = 1 cm
n = 5x 1 0 1 c6 m - 3) .
,
o)Cb(E\\B) and for the extraordinary mode (b) (E±B), for Λ||[111]. In the
ordinary mode, the oblique valley resonance induces a resonant behaviour,
apart from the structure close to ω χ ο.
The real part of n2 is much more complex [see eqs (16)-(21)], since it is
determined by the resonances of eyy and the zeros of szz as discussed before. For
the magnetic spectroscopy in fig. 15, the real part of n2 and n2 are shown as a
function of Β for λ= 163 μπι. The parameter of the curves is the free-carrier
concentration: 5 χ 1 0 16 c m " 3 , 1 χ 1 0 17 c m " 3 and 5 χ 1 0 17 c m " 3 . For n2 three
!
299
Magnetoplasma effects
η-PbTe
ΒΙΙΠ11]
λ=96.5μιη
Fig. 10. As fig. 9 but for d P
be X=
1
3 u m , ω, = 3 c m
and n = l x l 0
4ι>τ 6 = 3 0 0 θ Α . ( > 1 = 5 7 μ π ι ) .
Fig. 11. As fig. 10 but for λ = 118.8 μπι.
1 7
cm
.
3
Broken curve:
G. Bauer
300
1
1
1
PbTe QW
I
hole-CRA
mode
T = 4.2K
1
%
10
hi/
1
/P b E u S e T e
-PbTeQ.W.
^PbEuSeTe
X= 118.83 μπ\
— (III) Βα F 2
1
I
Chromel Film
1
4
6
8
V
1
0
2
MAGNETIC FIELD (T)
Fig. 12. Magnetotransmission of a single quantum well structure, i f P beX= 7 0 A between two
Pbo.gEuo.! S e 0 . o 9 6 T e 0 . 9 40 barriers (d = 2 μηι), after Kim et al. (1987) [Phys. Rev. Β 35 2501 (1987)].
1.0
η-PbTe
Bllkll[T11]
"
"
\ = 118.8 pm
Λ
Ntot=5x1016cm"3 —
/
1
1x1017cnv3
ι
1
2x10 1 7c m- 3—
z
1
0.5
A
Ο
OO
/
\
/
/
1
\
MAGNETIC
FIELD
\
\
\
\
\
ν
1
\
Ii
/
/
/'··.
/
/
/
\
f
\ /
(T)
Fig. 13. Influence of the carrier concentration on the dielectric anomalies associated with CR in
magnetotransmission (d = 5 μπι, ω τ = 4 c m - 1) .
Magnetoplasma effects
301
2000
0
50
FREQUENCY
100
(cm"1)
Fig. 14. Frequency dependence of the real part of the square of the refractive index of the ordinary
mode of ε ζζ and of the extraordinary mode (s yy - e zye yz/e zz)
for the Voigt geometry with
_ ? | | [ l l 0 ] (B = 3T) and it || [ 1 1 1 ] with n = 8 χ 1 0 16 c m " 3, ω τ = 1 c m - 1 (from Krost et al. 1985).
resonant structures appear: the hybrid resonance at approximately ω = \
2
a resonance close to coc& and for higher fields the resonance due
to a zero of εζζ between coCa and the phonon frequency.
For the E\\B geometry, the model calculation for a PbTe film with η = 1 χ
is shown for 118.8μπι (fig. 16), for several free1 0 1 7c m ~ 3 and ά=5μτη
carrier damping parameters according to eqs (16)—(21). The experimentally
observed magnetotransmission spectra for PbTe as shown in fig. 17 closely
resemble the model calculations.
However, for the dilute magnetic semiconductor (Furdyna, 1988)
Pbi-^Mn^-Te an additional rather strong spin-flip resonance occurs which is
associated with a transition within the two valleys oriented at 35° with respect
to B. The resonance structures change of course their position with wavelength
( c o Ca + a > c b) 1 /,
302
G. Bauer
2000
•c:
01
cr
-2000
8
2000
!i
10
\
\
\
•χ
QJ
QC
λ
-2000
,
, ϋ .
2
1
\
x
,
4
, \ .
6
ι
8
ι
10
MAGNETIC FIELD (T)
Fig. 15. Magnetic field dependence of nl and nl for the parameters of fig. 14, with carrier
concentrations: full curves, 5 x l 0 1 6c m - 3; dotted curves, l x l 0 1 7c m - 3; chain curves,
5 χ 1 0 17 c m - 3 (from Krost et al. 1985).
as shown in fig. 18. The experimentally observed signatures for the E\\B and
Ε LB configurations are compared with model calculations in a Pbi_j.EUj.Te
sample (x = 1%, η = 4 χ 1 0 16 c m " 3 , d = 9.8 μιη, ωτ = 5 c m - 1) in fig. 19.
All the main structures actually appear, e.g., for the Ε LB mode the three
resonances associated with the transmission windows. As shown in fig. 20a the
carrier concentration has a considerable influence in the concentration range of
303
Magnetoplasma effects
π-PbTe
ΕΙΙΒΙΙΠΪΟ]
λ=118.8μιη
ZD
o
oo
oo
ux=1cm
—
2cm
—
4cm —··
8cm
—
__
00
<
2
4
MAGNETIC FIELD
6
(T)
Fig. 16. Magnetotransmission of n-PbTe for the ordinary Voigt mode (n = 1 χ 1 0 17 c m - 3,
d = 5 μπι) for several carrier dampings.
interest on the apparent spectra and it is quite intriguing to deduce the effective
masses from the experiments in the Ε IB geometry alone. In fig. 20b the
dependence of the magnetotransmission [for η = 5 χ 1 0 16 c m " 3 , ωτ = 1 c m " 1 ,
otherwise identical parameters with fig. 20a] on the laser wavelength for n-PbTe
is shown.
The experimental and calculated data in the dispersive geometry are com­
pared with each other in fig. 21. Krost et al. have investigated n-Pbi-^Eu^Te
(x = 0.01, 5 χ 1 0 16 c m " 3 , d = 5.5 μιη). The overall agreement between experi­
ment and theory is not very good, the considerable damping of the small
structure close to 130 cm ~ 1 for Ε J_ Β is due to a strong frequency dependence of
ωτ which is discussed in the section on dynamical conductivity and which was
not taken into account in the model calculations. All structures beyond
180 c m " 1 are due to the combined properties of the PbTe and B a F 2 sandwich
structure.
McKnight and Drew (1980) have investigated n- and p-PbTe in the Voigt
304
G. Bauer
ΒIIΕ 11(110]
X=118.8pm
T=11K
η-PbTe
|
3
d
uo
uo
CR
A
β
n-Pbi-χΜηχΤθ
x = 0.012
<
SF
CR
0
1
2
3
4
5
MAGNETIC FIELD IT]
6
7
Fig. 17. Experimental data for magnetotransmission on a n-PbTe sample ( n = l χ 1 0 1 c7 m ~ 3,
d = 3.5 μm) in comparison to η - Ρ ^ _ χΜ η χΤ β (sample of fig. 4) which exhibits an additional spinflip resonance (from von Ortenberg et al. 1985 and Pascher et al. 1989).
geometry using a derivative reflection technique, and thus enhancing the
resonant structures as well as the dielectric anomalies. For /?||[001] and
Β || [110] both modes Ε IB and Ε \\ Β were excited simultaneously and thus the
spectra are quite complicated.
Figure 22 shows as an example data taken at λ = 119 μιη by McKnight and
Drew on a set of different samples. The signatures denoted by c are thought to
have been caused by cyclotron resonances, those with h by hybrid resonances.
The field is oriented along the [110] direction of p-type (110) plane samples. The
splittings observed at higher concentrations are interpreted as being caused by
cyclotron resonance transitions involving different initial and final Landau
states.
In the derivative spectra special care must be taken with any surface layers
since weak structures are considerably enhanced.
305
Magnetoplasma effects
n-PbTe
BIIEII [1Ϊ0]
λ=163.5μίτι
ZD
ο
oo
oo
Σ:
λ=118.8μπι\
00
<
V
ll
λ=96.5μΓη
0
ι\
ι Υ
2
4
6
MAGNETIC FIELD (Τ)
8
Fig. 18. Magnetotransmission in the ordinary Voigt mode, with the wavelength as the parameter
(sample parameters as fig. 16).
5 . Magnetoplasma
effects in two-dimensional
systems
Systems with reduced dimensionality are also of interest in narrow-gap IV-VI
compounds. Cyclotron resonance experiments were performed by Schaber and
Doezema (1979b) on inversion layers on p-PbTe, by Pichler et al. (1985, 1987a)
and by Murase et al. (1985), Shimominov et al. (1990) on PbTe/PbSnTe
quantum well structures (QW) and by Pichler et al. (1987b) and Pichler (1988)
on PbTe doping superlattices.
5.7. Inversion
layers
Schaber and Doezema induced electrons on the surface of a p-type bulk PbTe
sample using a metal-insulator semiconductor structure, with a metallic gate
(NiCr), transparent in the far-infrared region.
306
G. Bauer
Fig. 19. Experimental data on n-Pbj _ χΕ ι ι χΤ β (n = 6x 1 0 1 6c m , 3 </ = 5.5μπι, cot = 5 c m *) for
£||H||[1T0] and in the £ ± £ | | [ 1 Ϊ 0 ] mode (at T= 1 5 K (—), calculated data: (
).
Magnetoreflectivity measurements at fixed laser frequencies were performed
using circularly polarized radiation. The influence of the inversion layer
electrons on the magnetoreflectivity spectra was detected by measuring the
differential magnetoreflectance, i.e., by chopping the gate voltage and thus
changing the inversion layer density. For the measurement of the inversion layer
spin resonance the derivative of the reflectivity with respect to the applied
magnetic field was recorded.
Since Schaber and Doezema used [111]-oriented epitaxial films all experi­
ments in the Faraday geometry are for Z ? | | [ l l l ] . Therefore two cyclotron
resonances are expected, one for the [111] valley and the second one for the
three oblique <Tll> valleys whose major axes form an angle of 70.53° with the
surface normal.
Magnetoplasma effects
307
n-PbTe
Ε1ΒΙΙΠΪ0]
.
a,
n = 5xX)16cm'3: —
1x1017cm-3:
2x1017cm"3 —
λ=118.θμιη
/
/
ο
uo
uo
\
<
G_
4
6
8
10
MAGNETIC FIELD (T)
4
6
8
10
MAGNETIC FIELD (T)
12
14
16
(a)
(b)
Fig. 20. (a) Influence of carrier concentration on magnetotransmission in n-PbTe (d = 5 μπι,
ω τ = 4 α η _ )1 in £"_Ι_/?|| [ 1 Ϊ 0 ] . (b) Magnetotransmission in the extraordinary Voigt geometry,
with laser wavelength as parameter (n = 5 χ 1 0 16 c m - 3, ω τ = 1 c m "
G. Bauer
308
Fig. 21. Reflectivity as a function of frequency for Ε IB and B\\ Β with Β = 3 Τ for Pbj _ xE u xT e
(x = 0.01), sample parameters as in fig. 19. Full curves, experimental data; dotted curves,
calculated data.
However, in the strictly two-dimensional limit the 2 D oblique valley res­
onance should be considerably higher than the cyclotron mass for the 3 D case
(B|| [111]) (see table 2),
m*D = imt(l
+ Sm,/mt)1,2>
in comparison with
(22)
Magnetoplasma effects
20
30
40
50
MAGNETIC
60
70
309
80
90
KX)
FIELD (T)
Fig. 22. Derivative reflection as a function of Β for l f | | [ 1 1 0 ] of (110) plane p-PbTe samples,
unpolarized radiation. The resonances c and h refer to cyclotron and hybrid resonances. The
splitting of the CR peaks is attributed to Landau level population effects, e.g., 1 + - • 2 + and 2 " - * 3 ~
transitions for ρ = 2.1 χ 1 0 18 c m " 3) . After McKnight and Drew, Phys. Rev. B21 3447 (1980).
Table 2
Two-dimensional mass parameters (after Stern and Howard 1967) gv is the valley degeneracy factor.
Orientation
roz
£v
[001]
[111]
3m,/(l + 2m,/m t)
(a) m(
(b) 9m,/(l + 8 m , M )
4
1
3
^ c y c l o t r o n ~~
DW
m t[ i ( l + 2 m , / m t) ] 1/ 2
im t[(l + 8 m , M ) ] 1 2/
310
G. Bauer
This fact is shown in fig. 23 where the 3 D and 2 D surfaces of constant energy are
compared with each other for the Β || [100] and Β || [111] cases. The correspond­
ing 2 D density of states for B\\ [111] is shown in fig. 24.
In addition the expressions for the free-carrier contribution to the suscepti­
bility are somewhat different for the 2 D case. In the Faraday geometry for
# | | [ 1 1 1 ] we obtain (Pichler et al. 1987b):
+
col m0 .
_
.
ν _
Zf* = - 7 T — ( ω + ω 0 ΐ+ ι ω τ )
4ω mt
ι
•
co2p
VQ
9
*\mt
T,
2
(ω — ω
ω
\
mt + SmJ2
2
./
c
\m
18
i 8
t +
\
SmJ
^
2
— ω + 2ίωω τ)
where
cocl = (e/mt)B
and
ω ο 2 = (e/m2D)B.
(25)
The consequences for the real part of the dielectric function as a function of the
magnetic field are shown in fig. 25 where the shift of the oblique valley resonance
to higher magnetic fields can be seen according to this classical model.
GROWTH DIRECTION
[112]
Fig. 23. Three- and two-dimensional surfaces of constant energy for cubic n- and p-type IV-VI
compounds with relevant orientations (after Bauer et al. and Kriechbaum, 1987).
Magnetoplasma effects
311
[121]
[110]
b
c(b)
>
α»
Ε:
C(Q)
-20^|
L 4 C
i
I
>-
i
ρ(α)
-10^
T C
3
i
r(b)
t c
1
T C
2
....
1
r
p(Q) J
o DENSITY
c
ti
10"
2χ10 1'
1
1
.,.
OF STATES (cm- 2 m e V 1
Fig. 24. 2 D density of states and electric sub-bands for a rectangular PbTe quantum well with
ζ|| [111].
For {111} surfaces two electric sub-band systems exist, one for the [111]
valley (a) and one for the < T l l > valleys (b), where the former ones are closer
spaced in energy than the latter ones (see table 2 and fig. 24).
The experimental data and parameter fits for differential magnetoreflectivity
are shown in fig. 26 together with the MIS sandwich and a schematic view of the
band diagram for an inversion layer. The closely spaced peaks arise due to the
nonparabolicity of systems which have slightly different cyclotron masses for
each occupied electric sub-band.
The cyclotron masses deduced from the fits at various gate voltages are shown
in fig. 27. These data, taken with a laser photon energy of about 10 meV show
that the oblique valley resonance mass is much smaller than expected for a 2 D
behaviour. It is worth mentioning that in the derivative of the reflectivity with
respect to the magnetic field at constant gate voltage an electric-dipole-excited
312
Table 3
Experimental studies on magnetoplasma effects in IV-V I compounds
n-PbTe
n-PbTe
n, p-PbSe
n, p-PbTe
n-PbTe/BaF 2
n-PbTe/BaF 2
n-PbSnTe/BaF 2
)
n, p-PbTe
n, p-PbSnTe J
n-PbTe
n, p-PbTe
n, p - P b T e / B a F 2
j
n, p - P b S n T e / B a F 2j
n-PbTe
)
p-PbSnTe J
Microwave/FIR
Faraday/Voigt
Magnetic (M),
dispersive (D),
spectroscopy
FIR
MW(70GHz)
MW(70GHz)
MW(70GHz)
FIR
FIR
FIR
F
F, V
V
F/V
F
F
F
D
Μ
Μ
Μ
D
D,M
Μ
FIR
F,V
Μ
Kawamura et al. (1978)
MW
FIR
. F,V
V
Μ
Μ
Foley and Langenberg (1977)
McKnight and Drew (1980)
FIR
F
Μ
Lewis et al. (1980, 1983), Lewis (1980)
FIR
F, V/CSF
Μ
Ichiguchi et al. (1980)
Author
Buss and Kinch (1973)
Perkowitz(1969)
Bermon (1967)
Nii (1964)
Burkhard et al. (1976, 1979)
Bauer (1980)
Bauer (1978, 1980)
G. Bauer
Material
η, p-PbMnTe
η, p - P b T e / B a F 2
η, ρ-PbSe
F, V/SF
F/SF
V
Μ
Μ
Μ
von Ortenberg et al. (1985), Gorska (1984)
Schaber(1979)
McKnight(1972)
M W ( 5 0 GHz)
F
FIR
FIR
FIR
FIR
FIR
FIR
FIR
FIR
FIR
FIR
FIR
FIR
FIR
FIR
Μ
Nishi et al. (1980)
F/V
F
F/V
F
F/N
F/SF
F
F/V
F/V
F
F/V
F/SF
F/V
F
D/M
Μ
Μ
D
Μ
Μ
D
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Krost et al. (1985)
Kim et al. (1987)
Murase et al. (1985)
Pascher et al. (1988)
Pichler et al. (1987a, b)
Schaber and Doezema (1979a, b)
V o g l e t al. (1979)
von Ortenberg et al. (1975), von Ortenberg (1980)
von Ortenberg et al. (1985)
Bangert et al. (1985)
Kawamura et al. (1978)
Pichler et al. (1987a)
Bauer et al. (1987)
Lewis et al. (1982)
Magnetoplasma effects
n-PbTe
1
p-PbSnTe J
n-PbEuTe/BaF 2
p-PbTe S Q W / B a F 2
PbTe/PbSnTe M Q W / B a F 2
n-PbSe/BaF 2
PbTe/PbSnTe M Q W / B a F 2
PbTe/inv. layer/BaF 2
n-PbSnTe/BaF 2
n, p-PbSe
n-PbSnTe/BaF 2
n, p - P b G e T e / B a F 2
n, p - P b G e T e / B a F 2
PbTe doping S L / B a F 2
PbTe/PbSnTe/BaF 2
PbTe/central cell/BaF 2
and bulk
FIR
FIR
FIR
313
G. Bauer
314
η
λ = 118.8μπϊ
BIlRll [111)
)
)
3dim
2 dim
)
\
)
\
0
1
2
3
1
.
5
MAGNETIC
6
7
8
9
10
FIELD (T)
Fig. 25. R e e + for a 3 D PbTe sample (broken curve) in comparison to a 2 D sample (full curve) with
otherwise identical parameters for B | | [ l l l ] . The difference occurs in the oblique valley (b) resonance
(from Pichler et al. 1987a).
0
1
2
3
MAGNETIC
4
5
FIELD (T)
6
7
Magnetoplasma effects
315
[1121
*
—
•
—
QO
•
_
m
1
v o (
—
Q
~ » α 2
'"CQ
002'
H)
1
L
1
1
*—'
200 400 600 800 1000
V G - V t h( V )
Fig. 27. Inversion layer cyclotron masses for several sub-bands as deduced from experiments as
shown in fig. 26. The observed masses for the oblique (b) valley are much smaller than the '2D' mass
(see inset) (after Schaber and Doezema (1979), see fig. 26).
spin-flip resonance is observed for carriers in the [111] valley. The g-factor is
somewhat smaller than the bulk value.
In order to investigate the quasi-2D system of the inversion layer in PbTe
further experiments with tilted magnetic fields were performed. Figure 28
shows an example of an observed spectrum (reflectivity for VG = Khreshoid a *n
VG=VG(NS).
After tilting the magnetic field, the oblique valley resonance splits into three
Fig. 26. Magnetospectroscopy on inversion electrons in p-PbTe, for β | | [ 1 1 1 ] for several gate
voltages. Fine structure in the resonance is due to the population of more than one sub-band in the
[111] and oblique valleys. Inset: schematic band diagram at the PbTe surface [after Schaber and
Doezema, Phys. Rev. Β 20, 5257 (1979)].
c
316
G. Bauer
0
[111] ,
•
cr
Β
,— [101]
II
ρ-PbTe
ο
cr
λ =
119 μιη
T=4.2K
N s ~ 2 x 1 0 1 2c m " 2
0
1
2
!
3
MAGNETIC
U
5
FIELD (Τ)
6
7
Fig. 28. Difference reflectivity as a function of magnetic field for inversion layer electrons. The
parameter 0 denotes the tilt angle of the sample, the splitting into three resonances is a remarkable
deviation from a '2D' behaviour [after Schaber and Doezema, Phys. Rev. Β 20, 5257 (1979)].
resonances, i.e., the triple degeneracy of the three valleys is lifted and no
(cos Θ) ~1 dependence is observed as anticipated for a simple '2D' behaviour.
5.2. Quantum wells
Apart from the experiments with carriers confined to a triangular potential well,
experiments on samples with square wells (using multiquantum wells) were also
performed as well as on samples with parabolic wells (doping superlattices).
Results on FIR transmission on PbTe/PbSnTe M Q W structures (dPbTe =
90 nm, d P b S ne T= 27 nm) in the J ? | | [ l l l ] geometry (B perpendicular to the
layers) clearly show a transition from a '2D' to a '3D' behaviour with increasing
laser photon energy and hence higher resonance fields for the oblique valley
resonance (fig. 29). In the classical oscillator fits a '2D' behaviour was assumed.
However, as shown by Kriechbaum et al. (1988), Benet et al. (1987) for high
magnetic fields the Landau fan charts are essentially superimposed on the
electric sublevels. The field at which a transition from '2D'- to '3D'-like behavi­
our takes place can be estimated from a comparison of the binding length ζ
[ζ ~ 50 nm for the triangular potential well in the data of Schaber et al. (1979b)
and 27 nm in the experiments of fig. 29] and a magnetic length / =
(h/eB)l/2.
The effect of the confining potential is much more clear in experiments with
Magnetoplasma effects
317
ΒιιΚιι [mi
T=2K
λ = 118.8
Pbv XSn xTe (d=27nm)
x = 0.135
i-
PbTe(d=90nm)
CYCLOTRON ORBIT
I
nfif
1211]
\
/
/11111
^
λ=305μιη
[112]
0 1 2 3 4 5 6 7 8 9 10
MAGNETIC FIELD ( T )
Fig. 29. Magnetotransmission on a multiquantum well PbTe/PbSnTe sample with 25 periods.
Experimental data: full curves, for Β|| [111], broken curves: calculated data assuming a strictly 2 D
behaviour for the oblique valleys (after Bauer et al. 1987).
the magnetic field parallel to the layers. In fig. 30, for E\\B\\ [ U O ] , mag­
netotransmission data obtained with two laser wavelengths are shown. For
A = 57 μιη at Β ~ 4 Τ a clear resonance is observed, interpreted as the 0~->
1" transition within the [TlT] valley. For larger wavelengths (λ = 118.8 μιη) the
resonant structure close to 2 Τ is barely observable due to the fact that in the B^
geometry there is a transition from the electrically dominated carrier motion at
low magnetic fields to Landau states at higher magnetic fields, as shown in the
inset of fig. 30.
The calculation of the eigenstates in the M Q W structure for the B^ geometry
was carried out within the framework of an envelope function theory by
Kriechbaum et al. (1988).
318
G. Bauer
E i l Bll [110]
T=5K
MQW A
|PbJSnJe(d=36nm,25x)
i-x X
11 1 1 1
^
VbTe(d=117nm,25x)
00
cr
MAGNETIC FIELD (T)
0
1
2
3
MAGNETIC FIELD (T).
4
5
Fig. 30. Magnetotransmission on a PbTe/PbSnTe M Q W sample in the B (( geometry indicating the
effect of the confining potential on CR. Full curves, experimental data; fine curves, model
calculations. Inset: results of envelope function calculations on Landau states in MQW. [After
Kriechbaum et al. (1988)].
5.3. Doping
superlattices
The quasi-parabolic band edge modulation in doping superlattices (Dohler
1986) causes approximately equidistant electric sub-bands. Again for PbTe with
[111] ||z, two sub-band systems occur associated with Landau sets. For FIR
laser energies within the Reststrahlen region the cyclotron resonances are
accompanied by dielectric anomalies. Due to nonparabolicity the cyclotron
transitions originating from Landau levels associated with different electric subbands occur at slightly different fields as shown in fig. 31 for a PbTe doping
superlattice (SL) sample. Another interesting phenomenon is the occurrence of
both electron and hole cyclotron resonances in doping SLs which are illuminated with band gap radiation. Due to the built-in potential, the nonequilibrium
carriers are separated in real space and their carrier lifetime is drastically
enhanced. With circularly polarized radiation, for the [111] valley, electron and
Magnetoplasma effects
319
ΒΙΙΠ11]
0
1
2
3
4
MAGNETIC
5
6
FIELD
7 0 2 4 6 8
10
IT)
Fig. 31. Magnetotransmission on a PbTe doping superlattice showing the effects of population of
more than one electric sub-band in CR transitions. Right-hand side: Landau levels associated with
electric sub-bands obtained from self-consistent calculation of a doping SL potential (after Pichler
et al. 1987b).
hole active resonances can be distinguished as shown in fig. 32. For the oblique
valleys the CRI modes also induces resonant structures (Pichler et al. 1987b,
Pichler 1988).
These data provide some interesting clues to the quasi-2D behaviour of
carriers in the doping SLs as well as the peculiarities of such structures caused by
their indirect band gap in real space.
In order to summarize the '2D' far-infrared cyclotron resonance effects we
conclude that:
(i) evidence for the population of several sub-bands was found from CR
experiments on inversion electrons on the surface of p-PbTe.
(ii) In M Q W PbTe/PbSnTe, as well as in SQW PbTe/PbEuTe (Kim et al.
1987) samples, cyclotron resonance data were useful in obtaining information
about carrier confinement.
(iii) The analysis of magnetotransmission and magnetoreflection data is quite
complicated due to multiple-reflection and interference effects in many-layer
320
G. Bauer
DP=65NM(10x)
Bllkll[111]
X = 1 1 8 (8 / J M
'BUFFER^
^BQR
320nmf
T = 5 K
D N=83NM(10x
CIRCULAR
(HOLE
0
ACTIVE)
1
2
3
MAGNETIC
4
5
6
7
FIELD (T)
Fig. 32. Magnetotransmission as a function of field for a doping SL sample illuminated with band
gap radiation. Nonequilibrium carriers can be detected with the aid of circularly polarized radiation.
The large difference in damping parameter for the electrons (n) and holes (p) is visible from the
hole active sense for the oblique valley resonance (upper trace): the resonance denoted C R , m c, n
(CRI) is much sharper than C R , m c, ρ (CRA). A spin-flip resonance is observed as well for holes in
the a-valley.
samples (see, e.g., Pichler et al. 1987a for the use of a transfer matrix formalism
for the calculation of the complex refractive indices of a multilayer stack).
6. Magnetoplasma
effects in strained
semiconductor
layers
As investigations of epitaxial layers become more and more of interest, the effects
of strain, due to lattice mismatch film/substrate to thermal expansion coefficient
differences, have to be considered in their consequences for magnetoplasma
effects. In a semiconductor, the strain shifts the energy levels according to the
deformation potential tensor components and the amount of strain. In a manyvalley semiconductor even the degeneracy of equivalent band states can be
Magnetoplasma effects
321
lifted. Under such conditions the population of different valleys may be different
and therefore magneto-optical reflection or transmission data are influenced by
more than a single plasma frequency.
For IV-VI systems either the [111] or [100] directions are chosen as the
growth directions for epitaxial films. Since the epitaxial films are usually of the
order of several microns thick and the lateral dimensions and the substrate
thickness are of the order of millimeters the thin-film approximation holds. We
follow the description by Kriechbaum et al. (1984) and by Singleton et al. (1986).
For the (111) case we use a coordinate system χ || [ Π 0 ] , y || [112], ζ || [111]. The
stress tensor components are determined by the fact that σζζ = 0 as well as σ 0 for
i Φ j whereas σχχ = oyy. The strain tensor is related to the stress tensor by the
elastic stiffness tensor cijkl:
(26)
oij = cijkfikl
where
i=j=k=l
C
22
i=j^k=l
C 44
i = l*j
c
ijkl *=
= k
(27)
otherwise
.0
(for cubic crystals).
The strain tensor component are given by
/εχχ
ο
o\
0
exy
0
hi —
\0
(28)
0 sJ
zz
(εχχ = eyy since in the plane of the film the strain is isotropic).
In order to transform the elastic stiffness tensor cijkl (with respect to the cubic
axes of the semiconductor film) we use
C
mnop
=
^mi^nj^ok^p^ijkl-
(29)
This transformation then yields the relation between ε 3 3 and ε ι ι (for σ 3 3 = 0):
~~2(c 11 + 2cl2 — 2 c 4 4)
( c 11 + 2 c 1 2 + c 4 4)
'»·
)
The strain tensor (which is diagonal in the coordinate system x, y , z) has to be
transformed in the cubic axes system ε^:
&ij = ^im &jn8mn»
where ocJm = ocmi.
( ^ 1)
G. Bauer
322
One obtains for this system ([100], [010], [001]):
f i ( 2 f i 1 1+ f i 3 3) = 6 D ;
Ii(E33-eii) =
i= j
fis;
For the strain as given, the four equivalent L states become nonequivalent and
the [111] and the three oblique < T l l > valleys are shifted by an amount <5£(cs)v:
c, ν denote the conduction and valence bands, respectively, and s the valley:
δΕ</» Ν=Σ^/·<%.
(33)
In the cubic-axis system, the deformation potential tensor is given by
D?j v' ( s) = Dcdv6u
+ Dc^u\s)uf
(34)
where u\s) and uf are the direction cosines of the angles between the xt crystal
axes and the sth ellipsoid major axis, and D d ,v and D„'v are the dilatation and
uniaxial deformation potential constants defined by Herring and Vogt (1956).
The components of Di} for the [111] valley are given by
D
n
= D22 = D33 = Dd + i D u ;
Dl2 = Dl3 = D23 = $DU.
For the oblique valleys, e.g., [ I l l ] :
D1 X=D22
= D33 = Dd + i D u ;
D 1 2 = ^ D u,
D23 = Dl3 =
-±DU.
The resulting energy shifts are then as follows:
δΕ[\ι1] = (3Dr
+ DS' v)e d + 2DcSes
(35)
δ£<^> = (3Dr
+ ^ v) e
(36)
and
d
- f D^ss.
Since the D c ,v are similar both for the conduction and valence bands of PbTe,
both bands are shifted in the same direction with respect to those of other valleys
but the band gaps differ only slightly, see fig. 33.
In case of [100]-oriented substrates all four valleys are equivalent, i.e., all the
major axes of the ellipsoids make the same angle with the surface normal. The
axes which make the strain tensor diagonal are the crystal axes: [100], [010] and
[001]. The energy shifts are given by
δ ^ 5 )ν = 2ε1
i
~
c
( D S ' v' (
( C l lC l 2 )
ll
s)
+
j ^ ' v ' ( s ) ).
(37)
Experimental investigations of the effect of strain on far-infrared transmission
data were carried out by Ramage (1978), Lewis (1980), Vuong et al. (1985),
Bauer et al. (1983), Kriechbaum et al. (1984, 1988) and Singleton et al. (1986).
The effect of a transfer of carriers from the three < T l l > valleys to the [111]
valley in n-PbTe with increasing biaxial tensile strain is shown in fig. 34. For a
323
Magnetoplasma effects
STRAIN SPLITTING
Fig. 33. Schematic diagram illustrating the effect of a biaxial tensile strain (e.g., PbTe film on (111)
BaF, substrate at liquid helium temperatures) on the band structure. The shifts AE, and AEv are
given by eqs (35) and (36).
n- PbTe
2
4
6
MAGNETIC FIELD ( T )
8
Fig. 34. Model calculation on the magnetotransmission demonstrating the consequences of biaxial
tensile strain, i.e., a transfer of carriers from the oblique valleys to the [ l l 13 valley in n-type samples
and resulting changes in the dielectric anomalies (parameters: N,,, = 1 x 10'' ~ m - o,
~ =, 1 cm-',
d=5pm).
324
G. Bauer
total carrier concentration N t o =t 1 χ 1 0 1 7c m ~ 3 the relative importance of the
two dielectric anomalies accompanying the two cyclotron resonances for
Β||[111], in the Faraday geometry, is drastically altered if
is changed
from Ν 1 1 1 1 1 = iNtot to JiV t o .t These calculated data demonstrate (i) how sensitive
the anomalies react to changes in the carrier concentration and (ii) that the
transmission maxima and their shapes depend in a rather complicated way on
the total carrier concentration.
7. Magnetoplasma
reflection
using the strip line
technique
The application of the strip line technique for the measurements of mag­
netoplasma effects in narrow-gap semiconductors has been summarized by von
Ortenberg (1980).
Depending on the relative orientation of the magnetic field Β and the
propagation vector k in the strip line, three configurations can be distinguished:
2
BP
3
MAGNETIC FIELD (T)
Fig. 35. Experimental data (full curves), on strip line transmission with £ | | [ 0 0 1 ] for n-PbSe and
calculated transmission data (broken curves), (after von Ortenberg et al. (1975)).
Magnetoplasma effects
325
(i) parallel: B\\k (quasi-Faraday),
(ii) perpendicular: Β Ik, but Β in the plane of the sample (quasi-Voigt),
(iii) surface: B\\k but Β perpendicular to the surface plane of the sample.
The detailed analysis of the phenomena in the different configurations was
given by von Ortenberg (1980). It is important to mention that minima in the
k II [001]
n-PbTe
Bll kll [001]
ΒΙΙΠ10]
513pm
b)
513 μπι
433pm
on
Σ.
00
<
on
2
3
MAGNETIC
0
1
FIELD (T)
3
Fig. 36. Strip line transmission in n-PbTe in the quasi-Voigt configuration left-hand side (lhs) and
in the quasi-Faraday configuration right-hand side (rhs). In the lower half, oscillator fits are shown
with dampings of 1 c m " 1 (phonon damping 3.5 c m - 1) . The main structures for the right-hand side
are due to dielectric anomalies associated with CRI and CRA absorptions as well as a dip (arrows)
interpreted as a combined spin-flip resonance (e.g., 0 + - • 1"). In the quasi-Voigt configuration three
dielectric anomalies are visible, one originating from the hybrid, the second from a cyclotron and a
third from the LO-phonon-coupled magnetoplasma resonance. [After Ichiguchi et al. Solid State
Commun. 3 4 309(1980).]
326
G. Bauer
strip line transmission do not coincide with the resonance positions which are
instead related to the turning point in the slope at magnetic fields smaller than
those corresponding to the minima (fig. 35).
Von Ortenberg and Schwarzbeck et al. have analyzed n- and p-PbSe in
several strip line configurations. For the field in the [001] direction, rotating the
sample in the (010) plane and for a tilt angle of 0° one resonance appears. For
arbitrary tilt angles the resonance in general splits since two different sets of
ellipsoids become effective.
The Osaka group has successfully used the strip line technique for the
investigation of PbTe (Ichiguchi et al. 1981), Pb^^Sn^Te and Pb^^Ge^Te
(Kawamura et al. 1978).
For the configuration B\\ [001] Ichiguchi et al. have detected, apart from the
cyclotron resonance, an additional structure appearing at a fixed frequency at
higher magnetic fields which was interpreted as evidence for a combined spinflip resonance [ n + - > ( n + + l ) ~ , e.g., 0 + -> 1"]; Ichiguchi's data are shown in
fig. 36 for several wavelengths.
PbTe
ω=297cm- 1
g P1 7 5
4kg,:: 60.0
t Voigt
30 -
\ E I I B -- Ε
if
Β
20
/ / f
' ///
- 10
Έ
" α)
^
>- 0
§30
C3
_l.
C
"CRA
IR
,17 -3
N= * 2 M ) " C I T I
Λ
Bll [110]
qll [001]
i_
_ .l
Faradaiy conf. /
\
" \
E_
mt=0.0218
K=11.0
εί = -780
J
1
. ι
.i
/
/
,»
/
"
~
LU
cr
-•pK-
20
ω το "
!
10
/
' /
" b)
\
;
'£
1
/ '
BlklKOOU
V
/
L
0
MAGNETIC
L
1
1
FIELD
J
2
(T)
i
l
l
0'
[001]
30°
60*
90*
[010]
Fig. 37. Lhs: transmission minima (circles) and resonance positions (broken lines) as well as CSF
dips (triangles) in the quasi-Voigt as well as the quasi-Faraday configuration. Rhs: angular
dependence from Faraday to Voigt configuration of the dips and additional CSF structures
(triangles). Full curves and broken lines, calculated data. [After Ichiguchi et al. Solid State Commun.
3 4 309(1980).]
Magnetoplasma effects
327
In fig. 36 the data are also shown for the configuration with Z?||[110] and
perpendicular to the propagation direction of the submillimeter waves. In this
configuration, a quasi-Voigt one, there are three dips which result from three
dielectric anomalies associated with the hybrid resonance, the heavy cyclotron
resonance and the LO-phonon-coupled magnetoplasma resonance, in that order
for fixed frequency and increasing magnetic field. The transmission dips and
resonance positions in the ω-Β plane are shown in fig. 37 for PbTe and fig. 38
for P b 0 7S n 0 Te.
3
The use of the strip line configuration can be quite helpful for the detection of
resonances not only with samples which have rather high carrier concentrations
but also for the detection of rather weak structures. Recently, von Ortenberg et
al. (1985) have investigated n - P b ^ M n ^ T e samples which exhibit in the
configuration B\\ [ Π 0 ] , for k || [lTO] and a (111) sample surface, in addition to
the oblique valley resonance, also a spin-flip resonance (see fig. 39). These two
resonances are visible for direct transmission experiments as well. However, in
the strip line transmission experiments, additional lines are visible which were
interpreted by von Ortenberg et al. as being associated with impurities. In this
context, Lewis et al. (1983) have claimed to see evidence for shallow bound
states from magneto-optical transmission experiments in p-PbTe, explained by
central cell effects.
The expressions for the analysis of strip line experiments were described in
detail by von Ortenberg (1980) and Ichiguchi (1981) so that those equations are
not given here.
Fig. 38. Angular dependence of dielectric anomalies in strip line transmission on p-PbSnTe (1.3
χ 1 0 17 c m - 3) together with model parameters [after Ichiguchi et al. Solid State Commun. 34 309
(1980)].
328
0
G. Bauer
5
10
MAGNETIC FIELD [ T ]
Fig. 39. Strip line transmission spectrum for # | | £ " [ 1 ΐ 0 ] , sample surface (111), of n-Pbj _ xM n xT e
(x = 0.19) for λ= 118.8 μπι. The inset shows the calculation for the oblique valley CR and spinflip transitions in comparison with experimental data and the unidentified impurity transitions I t ,
I 2 and I 3 (after von Ortenberg et al. 1985).
8. Dynamical conductivity in the frequency
phonon-plasmon
excitations
range of coupled
LO-
Already in 1972 Mycielski et al. (1972) had observed that the free-carrier
damping exhibits a strong frequency dependence in PbSe in the region close to
the plasma frequency. Experiments carried out in the Voigt geometry have
indicated that ωτ also depends on the applied magnetic field.
In subsequent theoretical analyses, Mycielski and Mycielski (1978) have
attributed the rapid increase of ωτ close to ω ρ and its decrease at higher
frequencies to a photon-ionized impurity (or defect)-plasmon process. The
presence of ionized impurities or defects in polar semiconductors mediates an
absorption of electromagnetic radiation and the generation of collective plasma
oscillations (plasmons or magnetoplasmons) according to Mycielski. In the
case of the lead salts the interaction occurs via two perturbations: one connected
with the electron potential energy in the presence of vacancies, the other in the
field of the polarization charge density. The net power absorption is due to the
creation of a plasmon excitation. From this power absorption the dynamical
conductivity and thus ωτ(ω) can be derived, which is shown in its functional
Magnetoplasma effects
329
dependence for PbTe in fig. 40. There is a 'bump' structure in the theoretical
expression whereas the experimental data derived from the infrared reflectivity
measurements are somewhat smeared out.
In this context it is worth mentioning that the experimental data on ωτ(ω) as
obtained from a Drude fit to the reflectivity data have to be corrected. A
frequency-dependent damping necessarily requires a frequency-dependent res­
onance frequency in order not to violate causality and the Kramers-Kronig
transformation of Re χ and Im χ. Therefore for χ{0 an expression according to
Xtc = - ω * / ( ω [ ω + {(ω) + ίω,(ω)])
(38)
should be used where
2 Γ 00
x/(x)
ξ(ω) = -\
π Jo -^P^dx,
x -co
(39)
2
/(*)=
Γ - ω τ( ο ο ) ,
0^χ^ω^ο
, _
, _
+
Ιω τ(χ) — ω τ( ο ο ) ,
(40)
ω{0 < x < ο ο .
The corresponding function ξ(ω) is also shown in fig. 40 as calculated by
Burkhard et al. (1978).
Mycielski (1974) has developed his formalism for ωτ(ω) for a plasma
frequency much higher than the LO phonon mode frequency. Katayama and
Λ
•
V
PbTe-BaF2
2385
T=5K
FREQUENCY
(cm' 1)
FREQUENCY
(cnr
Fig. 40. Left-hand side; frequency dependence of the damping parameter ωτ = l/τ according to the
Mycielski model and its Kramers-Kronig transform ξ{ω) (after Burkhard et al. 1978). Right-hand
side, reflectivity against wavenumber ν (full curve, experimental data; · , calculated values) for
n-PbTe in the region of coupled LO-phonon-plasmon excitations.
G. Bauer
330
0
25
50
M A G N E T I C FIELD
(T)
75
Fig. 41. Calculated damping time ωτ against magnetic field and frequency in comparison with the
measured data of Mycielski et al. [after Katayama et al., Phys. Rev. Β 1 9 6513 (1979)].
Mills (1979) and Katayama et al. (1983) have re-examined the problem of the
high-frequency relaxation time in polar materials and the influence of a
magnetic field on this for PbSe, as well as for PbTe, applying their theory to data
obtained by Burkhard et al. (1978) and Mycielski (1974).
Katayama et al. calculate the relaxation time due to electron-ionized impurity
scattering and electron-LO-phonon-plasmon scattering using a Kubo-type
formalism. Their main conclusion is that the scattering of electrons by the
coupled LO-phonon-plasmon modes is the dominant contribution to ωτ and
that it also provides the mechanism for the dependence of ωτ on the magnetic
field strength. In fig. 41 the results of Katayama et al. are shown in comparison
with the experimental data on ω τ(ω, Β) obtained by Mycielski and Mycielski
(1978).
It should be noted that Sommer (1979, 1980) has given a generalized treat­
ment of the Mycielski process including the effects of nonparabolicity and
multivalley conduction band structures.
9. Linear-response
theory
Wallace has already presented in 1970 a systematic study of magnetoplasma
effects in many-valley narrow-gap semiconductors like PbTe. An extension of
Magnetoplasma effects
331
this work taking into account the full complexity of the IV-VI semiconductor
band structure was given later by Vogl et al. (1979) as well as by Bangert et al.
(1985).
Wallace (1980) has derived the electron contributions to the dielectric tensor
in external magnetic fields using the second quantization formalism expressing
the results in terms of the polarization representation. For magnetic fields
oriented along a symmetry axis the dielectric tensor is diagonal in this
representation. The general procedure adopted by all authors and outlined in
detail by Wallace (1980) starts with a calculation of the Landau states and their
matrix elements for the current operator. The frequency-dependent conductivity
is calculated using a general theory of linear response with which a
Peierls-Greenwood-type formula is derived. With this technique the interaction
of electrons with phonons, plasmons etc. can also be taken into account, in
principle.
For the Faraday geometry the results are the following:
Zfrree electrons =
Σ
< 7 ± () s
(41)
where a±{s) is the high-frequency conductivity in a quantizing magnetic field due
to carriers in the valley (s) for right and left ( ± ) circular polarizations of the
FIR radiation. In the long-wavelength limit (g->0) it is given by
σ ± (*>(ω, q^O) = i ^ [ a
m0a>
+
t
- i ( a t - a,) s i n 20 s ]
_|_ y y
hcokykB
± M U(KkB,a,s)-f{En',kB,o,s)]
h~
( £ „ ' , j k B, ( ,Ts - ω - ΐω τ'
)
1
where/denotes the Fermi function, η and ri are Landau level quantum numbers,
σ denotes the spin quantum number, kB the carrier wave vector in the direction
of Β and ωτ the damping parameter.
0S is the angle between the magnetic field Β and the 5th main valley axis. The
transition matrix elements in the approximation used by Wallace are given by
(for CR transitions only)
Mi = ( e >
0
) K V + i ) ^
n
^ +
y ^
n
M l = ( e 2/ m 0) [ y sV + 1 ) ^ - 1 + y ; 2 ^ ,
X
where
(En',kB,a,s - EntkB,ats)>
y, = W £ +
>/£),
=
+
]1
]
w + 1
(43)
(44)
( 4 2
332
G. Bauer
with
( s i n 20 s = § for Β || [001] and 0 or | for Β || [111],
(45)
and the summation £ f c ,y f cB is replaced by
(46)
Ns is the carrier concentration in the valley s, it depends on the magnetic field Β
through the magnetic field dependence of EF(B) which is obtained from
(47)
As far as the broadening is concerned a factor ωτ is introduced in the linearresponse formulation. The problems associated with such an assumption were
discussed by Wallis and Balkanski (1986).
Extending the original ideas of Wallace and considering the strong nonparabolicity the mass factors a t and a, have to be changed in order to use the masses
at the Fermi energy:
(48)
according to Bangert et al. (1985).
The Wallace model is already a substantial improvement as compared to the
classical oscillator models: it takes into account the occupancy of the initial and
final states, the correct density of states function and the kB dependence of the
Landau states. However, it does not take into account some more subtle effects
of the IV-VI semiconductor band structure which eventually also lead to spinflip and combined resonance transitions. In order to calculate the relevant
transition matrix elements in the electric dipole approximation the expression
(49)
is used where α = ( α 1 , α 2 , α 3 ) is the polarization unit vector of the incident
radiation and v = (vl9 v2, v3). His the two-band or multiband Hamiltonian used
for the calculation of the Landau states. In the expression for the dielectric
function the transition matrix elements Μ 2 are then calculated by
(50)
where i and / denote the initial and final states respectively.
Magnetoplasma effects
333
The dipole operator has to be presented in the same Bloch function basis as
used for the Landau level calculation.
For the Landau levels a (4 χ 4) matrix Hamiltonian based on the work of
Mitchell and Wallis (1966) is used which takes into account exactly the twoband interaction between the conduction and valence bands, and includes the
interactions with two distant conduction and two valence levels in the k-p
approximation. Only for the B\\ [111] direction and for the [111] valley oriented
with its main axis parallel to Β the calculation is straightforward. For an
arbitrary direction of Β a. unitary transformation is necessary and the eigenstates
of Η have to be expanded in a series of oscillator functions leading to a matrix
representation of dimension 4m χ 4m. A numerical diagonalization gives the
energies and wavefunctions of the Landau levels. The values of the dipole
moments are therefore also calculated numerically and depend of course on / , /
and also on the magnetic field.
i
Fig. 42. Schematic E(kB)
E(k )
relationship indicating the kB dependence of the CR and SF transition
energies for nonparabolic bands.
334
G. Bauer
The Wallace-type expression for the conductivity is then given by
σ<ί+ (ω) = — [<xt - i ( a t - a,) s i n 2 0 s ] JV<*>
+
i f !
K/
-H.>P[/(,»-/(,)]
;l
ω
£
E
Ef-Ei-ηω-ιη/τ
Using the numerically calculated variation of the Fermi energy with magnetic
field for a given direction, it is then possible to determine the free-carrier
susceptibility contribution to the total dielectric function. The initial |*> and
final states < / | can be any Landau states since the nonparabolic band structure
makes spin-flip and combined spin-flip transitions possible. The summation
over the electron momenta along the direction of the applied magneticfieldΒ
800
1400
600
1200
4 0 0
1000
200
-800
L— - J
_I
1— - 6 0 0
- 2 0 0
400
-400
200
-600
FREQUENCY
(cm1
)
Fig. 43. Real (elt full and broken curves) and imaginary parts ( ε 2, chain curves) of the dielectric
function in the linear-response model for n-PbTe, ωτ = 1 c m " l , EF = 5.7 meV for Β = 2.78 Τ, after
Burkhard et al. (1979).
335
Magnetoplasma effects
also includes automatically the nonparabolicity effects in the density of states.
For a fixed value of B, the transition energy is largest for kB = 0 and diminishes
for larger kB and reaches a minimum for kB = k¥ as shown schematically in
fig. 42.
In fig. 43 the real (ε χ) and the imaginary parts (ε 2) of the total dielectric
function for a magnetic field of 2.78 Τ are shown together with the data on the
free-carrier susceptibility χ 1 .
These data deviate considerably from those calculated with a classical
oscillator model, as also shown for the same field in fig. 42. In particular the
asymmetry of the structures arising from non-parabolicity is much more pro­
nounced in the case of the linear-response model.
In fig. 44 a direct comparison of the two models is presented in their
consequences for the fits of magnetoreflectivity in PbTe (after Burkhard et al.
1979) as well as for reflectivity and transmission data on Pbj .^Sn^Te (x = 0.05)
after Vogl et al. (1979) (fig. 45).
The shortcoming of the approach is still a rather phenomenological damping
parameter cox which is taken to be constant, independent of Β and kB as well as of
frequency. In principle the old problem of the cyclotron resonance linewidth
enters into the calculation of ωτ.
FREQUENCY
FREQUENCY
Fig. 44. Comparison of model fits using the classical oscillator model to represent magnetoreflec­
tivity on n-PbTe (8 x 1 0 16 c m - 3) for Β = 2.48 Τ (upper half) together with data obtained from the
linear-response model (lower half); ωτ for both 1.5 c m " 1, fields Bx = 2.48 Τ and B2 = 4.11 Τ (after
Burkhard et al. 1979).
G. Bauer
336
10
>
075
>
1Λ
τ
υ
LU
£
P bi
0 5
-,
Snje
χ = 0. 35
H = 5<1.3 kG
025
ν
J
AO
LO
CO
CO
<
α:
1
80
120
160
FREQUENCY [crrr 1]
Fig. 45. Magnetoreflectivity and transmission spectra on n-Pbj _ χ8 η χΤ ε (χ = 0.05) for Β = 5.93 Τ.
Δ , experimental data; full curves, calculated data with linear-response model (after Vogl et al. 1980).
Apart from the cubic PbTe or PbSnTe case, the quantum mechanical
description of the free-carrier susceptibility will be important for the description
of semimagnetic specimens as well as for quantum well systems. It has already
been used to describe magneto-optical transitions in PbGeTe, which undergoes
a phase transition from the high-temperature cubic to the low-temperature
rhombohedral phase. Indeed, many transitions forbidden in the cubic phase
become allowed in the rhombohedral structure as a consequence of the band
structure terms being linear in k and the resulting level-mixing. Magneto-optical
transitions like cyclotron resonance harmonics (0a -> 2a, a standing for one of
the components of the Kramers pair, α,β) combined spin flip (L0/J->Lla) and
spin flip (Τ0β->Τ0α) have dipole moments which are already comparable to
cyclotron resonance transitions (e.g., L0a->Lla). In fig. 46 the real and
imaginary parts of the dielectric function at λ = 96.5 μπι is shown as a function
of magnetic field together with the experimental and calculated transmission
spectra (after Bangert et al. 1985). The notation Τ and L has the following
Magnetoplasma effects
4
6
MAGNETIC
337
8
10
FIELD (T)
Fig. 46. Upper part: real and imaginary parts of the dielectric function of n-PbGeTe as a function of
the magnetic field for λ = 96.5 μπι. Central part, magnetotransmission: full curve, experimental data;
· , calculated data. Lower part: variation of Fermi energy with magnetic field (after Bangert et al.
1985). The Τ valley is shifted with respect to the L valleys due to strain as well as T< Tc and thus
a rhombohedral phase transition has occurred.
meaning: due to the phase transition the four equivalent L states in the Brillouin
zone are converted into a Τ state (along the rhombohedral c-direction) and
three equivalent L states. The fit is not ideal since band parameters taken from
magneto-optical interband transitions were used and therefore all the important
parameters were fixed, particularly the carrier concentration and thus the
population of the initial Landau states. In such a complex situation a classical
oscillator fit would require a large number of free parameters to adjust the
338
G. Bauer
experimental data. The parameters would then, however, not be at all related to
the physics being investigated.
As a function of temperature, the resonant structures change considerably due
to the nearly second order of the phase transition in PbGeTe. The intraband
magneto-optical transitions, yield, analyzed with the aid of the linear-response
model, information on an additional matrix element related to a band structure
parameter linear in k which is connected to the primary order parameter u, the
sublattice shift in rhombohedral PbGeTe. The Fermi surface is so complicated
(see Bangert et al. 1985) that an effective mass picture is no longer useful.
In such a complicated system there is no other possibility of extracting useful
information from magnetotransmission or reflection data than of adopting the
procedure outlined in this section.
Even for cubic systems, all transitions related to spin flip (SF, CSF) should
be treated within the linear-response model, in table 3 the experimental studies
of IV-VI compounds are summarized.
10.
Conclusions
The techniques of far-infrared (FIR) spectroscopy such as optically pumped
FIR lasers as light sources for the magnetic spectroscopy or Fourier transform
spectroscopy for the dispersive case are by now quite mature. An energy region
from less than 1 meV to about 100 meV can be covered, a range in which many
elementary excitations in semiconductors are found.
Whereas in wide-gap materials, for small carrier concentrations, mag­
netotransmission data can be immediately used to obtain information about the
effective masses, electronic ^-factors, impurity states etc., in narrow-gap
materials a simple-minded interpretation of the spectra might lead to errors in
the assignment.
In many narrow-gap materials the free-carrier concentration is so high that
the Fermi energy, the plasma energy hcop9 and the phonon energies are
comparable. In addition the nonparabolicity is quite important. In such systems,
in principle, all the information about the dielectric function is necessary, in order
to extract the relevant information from the data. Whereas for the infraredactive phonons the Lorentz oscillator model is quite adequate, for the free carriers
a linear-response model is much more appropriate for the description of χ .
This is especially true for spin-flip and combined resonance transitions for
which no classical analogue in terms of cyclotron orbits exists.
Since studies on narrow-gap multi- or single-quantum well samples are of
increasing interest, the quantum mechanical description will also become more
popular in comparison to the Drude oscillator one for χ ί ο. In particular, for
dilute magnetic (semimagnetic) semiconductors where the oscillator strength of
the spin-flip transitions can be quite large, such a treatment will actually provide
useful information about the band structure and enable us to exploit the
powerful FIR techniques.
Magnetoplasma effects
339
Recently Luo et al. (1987) have already investigated H g ^ M ^ T e epilayers.
The narrow-gap II-VI and IV-VI compounds are obvious candidates on which
to use the methods outlined in the previous section.
There is an alternative to the use of model fits to measured reflectivity and
transmission spectra, which has already been discussed in some detail by Palik
and Furdyna (1970): using a Kramers-Kronig transform it is possible to extract
the dielectric function directly from the .experimental data, with some compli­
cations if the dielectric tensor is antisymmetric (Stimets and Lax 1970). This
approach has already been used by Bauer et al. (1985) to extract band structure
information from magnetoreflectivity data on zero-gap H g ^ ^ M ^ T e .
In this review the emphasis has been on lead compounds because, due their
many-valley band structure and their narrow gaps, these materials serve as a
kind of model substance to describe all the intricate problems which can arise in
the data analysis.
Acknowledgments
I thank E. Bangert, H. Burkhard, H. Clemens, M. Kriechbaum, P. Pichler,
J. Oswald, P. Vogl, M. von Ortenberg and W. Zawadzki for helpful discussions
on various topics presented in this work, and L. Zeder, A. Tappauf and
M. Wieser for technical assistance with the preparation of the manuscript.
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CHAPTER 7
Interband Magneto-optics of
Semiconductors as
Diamagnetic Exciton Spectroscopy
R.P. SEISYAN and B.P. ZAKHARCHENYA
A.F. Ioffe Physical-Technical
Institute
USSR Academy of Sciences
194021 Leningrad, USSR
Landau Level
© Elsevier Science Publishers B.V., 1991
Spectroscopy
Edited by
G. Landwehr and E.I. Rashba
Contents
1. Introduction
347
2. Modern theoretical fundamentals of interband magneto-optics
349
2.1. The exciton and hydrogen atom in magnetic
fields
350
2.2. Formation of the bound (exciton) state in a strong magnetic field for an arbitrary
attractive potential
351
2.3. Semiconductor in a strong magnetic field: formation of diamagnetic excitons
involving Landau sub-bands
353
2.4. Diamagnetic excitons in diamond-like semiconductors
2.5. Diamagnetic excitons in 'intermediate'
358
fields
367
3. The exciton nature of oscillatory magnetoabsorption spectra
371
4. Experimental data on oscillatory magnetoabsorption spectra
380
4.1. Diamagnetic excitons in crystals with a strongly pronounced Wannier-Mott
exciton ground state
381
4.2. Diamagnetic excitons in crystals with a well-developed Wannier-Mott exciton
spectrum
4.3. Diamagnetic excitons in crystals with suppressed Wannier-Mott exciton states
404
421
5. Conclusion: band parameter calculation from diamagnetic exciton spectra
434
References
440
1.
Introduction
Gross and Zakharchenya (1954) were the first to reveal the influence of magnetic
fields on the optical spectra of semiconductors. In the first experiments the
Zeeman effect on a narrow forbidden line corresponding to the ground state was
observed in the exciton spectrum of C u 2 0 . Subsequently one observed for the
same crystal Zeeman splitting and a strong diamagnetic shift of the terms of the
Wannier-Mott hydrogen-like series corresponding to excited states of large
radius (Gross and Zakharchenya 1956). It was found that in the presence of a
magnetic field the diamagnetic shift for large-radius excitons becomes so large as
to exceed the separation between the terms of the series with large quantum
numbers η = 6 , . . . , 9, with the line spectrum extending beyond the ionization
limit where hv > e g. Such a direct transition from the exciton spectrum to a
discrete spectrum in the region of the continuum made Gross et al. (1957), who
observed absorption oscillations in C u 2 0 in a magnetic field, suggest that
excitons contribute substantially to this effect in semiconductors.
Practically simultaneously with this work, Burstein and Picus (1957) and
Zwerdling and Lax (1957) reported on absorption coefficient oscillations in the
region of the continuum which were observed for indium antimonide and
germanium, respectively, in the presence of magnetic fields. There were no
indications at that time of the exciton structure in the edge absorption spectrum,
and therefore the observed phenomenon was interpreted in the subsequently
published theoretical papers (Roth et al. 1959, Burstein et al. 1959) as a direct
manifestation of optical transitions between Landau sub-bands in the conduc­
tion and valence bands of the semiconductor. Thus the effect called oscillatory
magnetoabsorption (OMA) was considered similar to the cyclotron resonance
(CR) discovered shortly before and studied intensively, and began to be
considered as a major interband magneto-optical phenomenon in
semiconductors.
It became possible to consider other magneto-optical phenomena, such as the
interband effects of Faraday, Voigt and so on, on a common physical basis as
originating from the optical transitions occurring in absorption.
Magneto-optical effects in semiconductors turned out to be a major experi­
mental means of validating the principles of the band theory of semiconductors
and for accurately determining its parameters. The discovery of O M A opened
up new horizons in the magneto-optics of semiconductors primarily by
broadening the number of materials and the range of experimental conditions
under which this effect could be reliably observed. Apart from this, studying the
OMA promised to yield additional information - compared with cyclotron
resonance - such as accurate values of the energy gaps, g-factors, deviations
from the parabolic dispersion relations, etc.
All this resulted in a rapid accumulation of extensive experimental and
theoretical material. The data obtained in studies of oscillatory magnetoabsorp-
348
R.P. Seisyan and B.P. Zakharchenya
tion and its interpretation as due to transitions directly between free-carrier
states in Landau sub-bands entered into handbooks and thus have made up the
bulk of our knowledge concerning the band structure parameters of various
semiconductors.
B. Lax with co-workers of the F. Bitter National Magnetic Laboratory played
a major role in the extensive investigation of the O M A and associated
phenomena. Their work which is well known from review talks given at
international conferences on the physics of semiconductors in the period from
the late 1950s to the 1970s have become an integral part of monographs and
textbooks.
However, a deeper analysis of the problem has revealed contradictions in the
accepted interpretation of the oscillatory magnetoabsorption and brought to
light its relation with another fundamental phenomenon in semiconductors; the
formation of a bound state of electrons and holes, that is excitons. Improved
methods of crystal growth and progress in semiconductor technology have
resulted in the discovery of exciton states at the absorption edge of germanium a
semiconductor with a relatively narrow gap. Subsequently it became possible to
observe the discrete structure of the Wannier-Mott exciton already in most of
the available crystalline semiconductors. While in diamond-like semiconductors
one could not observe an as well developed exciton series as in cuprous oxide,
nevertheless it was readily seen that the Coulomb interaction between the
electron and the hole created in the absorption of a photon not only is essential
within the energy gap but also contributes substantially to the absorption
coefficient in the continuum.
An important role in the further development of this problem was played by
the theoretical work of Elliott and Loudon (1959) who showed that the whole
OMA spectrum could actually be due to excitons. Indeed, at a certain magnetic
field each pair of the Landau sub-bands involved in optical transitions becomes
the ionization limit of an exciton series, and the oscillator strength for
transitions to the discrete ground state of such excitons may exceed by far that
for transitions to the continuum or to unbound states. These theoretical ideas
were later developed by Hasegawa and Howard (1961). Gor'kov and Dzyaloshinskii (1967), Zhilich and Monozon (1968). Johnson (1967) succeeded in finding
experimental verification of the validity of these ideas by observing a structure
which was interpreted as an excited state of the exciton in a magnetic field. The
authors of the present chapter, together with co-workers, obtained reliable
evidence for the excitonic nature of the O M A spectrum of germanium crystals as
a whole (1968-1969).
It should be proper at this point to recall the work of Gross, Zakharchenya
and Pavinskii (1957) where the first observation of O M A in cuprous oxide was
described as a discovery of 'diamagnetic levels of the exciton'. Nevertheless, the
subsequent developments took such a turn that we continue to witness
publication of otherwise serious theoretical and experimental studies where the
idea of including the exciton nature of the spectrum is either ignored altogether
Diamagnetic exciton spectroscopy of semiconductors
349
or considered in a formal way. On obtaining a qualitative and sometimes even
quantitative agreement with experiment, of a theory which does not take into
account the Coulomb interaction the authors of these publications draw
conclusions about the accurate values of the band structure parameters, and in
doing so, they quite frequently introduce corrections into the figures derived, for
instance, from cyclotron resonance experiments.
The appearance of reliable results supporting the exciton concept and
permitting its extension to various crystalline semiconductors, as well as an
evaluation of the scale of disagreement with the previously accepted interpretation of the phenomenon have called for further progress in the techniques of
spectroscopic studies and new theoretical work.
The present status of our knowledge apparently does not leave any doubt that
practically any spectrum of oscillatory magnetoabsorption is actually a spectrum of exciton states of a particular kind which appears in semiconductor
crystals placed in a magnetic field. We have called such states 'diamagnetic
excitons' (Zakharchenya and Seisyan 1969). Diamagnetic excitons modify
dramatically the spectral shape of the fundamental absorption edge of crystalline semiconductors by transforming monotonic behavior into line spectra.
Strictly speaking, the previously used term Oscillatory magnetoabsorption' is
not absolutely correct, so that even in a phenomenological description it could
properly be used only for the diamagnetic exciton spectrum at large diamagnetic
quantum numbers when the linewidth becomes comparable with the line
separation. (Nevertheless, we shall follow the tradition by using it as a
descriptive equivalent to the 'diamagnetic exciton spectrum'.)
Diamagnetic exciton spectra contain a wealth of information on the electronic
band structure of crystals and in this respect can, in some cases, offer more
potential than the spectroscopy of semiconductors which does not use magnetic
fields. In a certain sense the spectroscopy of diamagnetic excitons combines the
possibilities of such well established methods as cyclotron and paramagnetic
resonance and complements the data that can be obtained by these techniques.
The concept of the exciton nature of interband magneto-optical spectra
together with the vast factual material analyzed from this standpoint have found
adequate reflection in the recently published monograph of Seisyan (1984).
In the present chapter we are going to generalize briefly the ideas underlying
the diamagnetic exciton spectroscopy while presenting some new data on the
subject.
2. Modern
theoretical
fundamentals
of interband
magneto-optics
The present day approach to the interband magneto-optics of semiconductors is
inseparable from the magneto-optics of the exciton state, since the primary
optical absorption event always creates an electron and a hole at the same point
in quasimomentum space, while a strong magnetic field favors their binding
R.P. Seisyan and B.P. Zakharchenya
350
irrespective of the kind of interaction potential involved and the depth of the
potential well. Therefore we will first consider the general aspects of the exciton
behavior in a magnetic field.
2.1. The exciton and hydrogen atom in a magnetic field
The problem of the energy spectrum of the Wannier-Mott exciton in a magnetic
field for simple spherical bands with a quadratic dispersion relation is similar to
that of the hydrogen atom spectrum in the presence of a magnetic field. The
difference lies in that the exciton gas in solids is characterized by an effective
reduced electron and hole mass μ rather than by the free electron mass m. Apart
from this, for large-radius excitons the interaction between the electron and the
hole is weakened by the dielectric constant of the medium κ0 which is
substantially higher than that of the vacuum.
As already pointed out, Elliott and Loudon (1959) were the first to reveal a
number of essential features in the exciton spectrum obtained in a strong
magnetic field. A certain shortcoming-of their work was the use of approxima­
tions with parameters containing an implicit dependence on the parameters of
the problem. Hasegawa and Howard (1961) succeeded in solving this problem
without imposing such limitations, but for the case of a very strong magnetic
field only. Zhilich and Monozon (1968) analyzed the problem of the adiabatic
approximation in the formulation of Hasegawa and Howard (1961). The
equation of relative exciton motion in a strong magnetic field was derived in a
straightforward way by Gor'kov and Dzyaloshinskii (1967) who also included
the effect of the total exciton momentum on the relative motion of the electron
and hole.
Consider briefly a general approach to the problem of the Coulomb
interaction in the presence of a magnetic field for the case of'simple' bands. As is
well known, the energy and wavefunctions of a large-radius exciton in a crystal
can be derived with a sufficiently high accuracy by solving the two-particle
equation in the effective-mass approximation. Introducing the coordinates of
relative motion r, and of the center of mass, R, one can now carry out canonical
transformations equivalent to representing the exciton wavefunction Ψ in the
form
¥>(*, r) = exp {iIK - (c/he) A(rft R} φ{τ)9
(1)
where Κ is the exciton wavevector and A(r) is the vector potential of the
magnetic field. Taking a Coulomb potential, one now obtains for the relative
motion of the electron and hole, which is described by a function </>(r), a
Schrodinger equation:
[ - ( 6 2 / 2 μ ) V2 + e2/K0\r\
2
+ (e ^c)
2
A (r)2 2
+ \(eh/c)(m*-1
-m*-l)A(r)
V
1
(2eh/c)(m? + m*) ~ A(r) Κ ] φ ( ή
= [e - (h K /2)(m*
+ m * ) " x] 4>{r).
(2)
Diamagnetic exciton spectroscopy of semiconductors
351
Besides the first two terms describing exciton states in the absence of a magnetic
field, the equation contains three more terms which depend on the field.
The first of them is the usual Zeeman term originating from the exciton's
having a magnetic moment which is associated with the orbital motion of the
charged particles:
i(eh/c)(m*-l-m*~l)A(r)
V= - ( ^ / 2 c ) ( m c* - 1 - m * " l ) B £ > ,
(3)
where S£\r, \hV~] is the orbital moment operator.
The second additional term is actually diamagnetic energy and for a magnetic
field Β parallel to ζ can be rewritten as (x2 + y2) e2Β2βμο2.
The third term is due
to the Lorentzian force acting on the electron and hole moving in a magnetic
field. For the exciton its effect is equivalent to that of an electric field acting only
in the center-of-mass system:
- (1/c) [hK(m* + m*)" 1 χ Β] r = - (1/c)(V χ Β) r
= -(2h/c)(m*
+ m*)-lA(r)K,
(4)
where hK(m* + m*) = V is the exciton velocity.
These three extra terms may be treated by perturbation theory, considering
the free-exciton spectrum to be unperturbed when they are small compared with
the Coulomb term. In the case of strong magnetic fields this approach is
naturally invalid, and one should consider the Coulomb potential as a
perturbation. N o w the solution of the problem in the absence of perturbation
will be a wavefunction equal to the product of the wavefunctions of Landau
states for the electron and the hole, the unperturbed spectrum corresponding to
transitions between the Landau sub-bands.
Thus the problem of the behavior of a hydrogen-like system in a magnetic
field has two domains which can be analyzed, to wit, the domains of a weak and
a strong magnetic field compared with the Coulomb potential. These are
radically different solutions, dependent on the behavior of three-dimensional
and one-dimensional systems, respectively. The intermediate field domain
cannot be described analytically.
2.2. Formation of the bound (exciton)
arbitrary attractive potential
state in a strong magnetic field for an
In magnetic fields sufficiently strong that the interaction energy between the
electron and hole is less than their cyclotron energies the interaction becomes
increasingly more one dimensional. The motion in the plane perpendicular to
is totally
the magnetic field occurs in cyclotron orbits of radius L = (eh/cB)1,29
governed by the magnetic field and does not depend on the carrier effective
mass. At the same time motion along the magnetic field will be determined by a
one-dimensional potential obtained, for instance, by averaging the threedimensional Coulomb potential with the wavefunctions of transverse motion. It
is not always possible to describe the interaction between the electron and hole
R.P. Seisyan and B.P. Zakharchenya
352
at Β = 0 with the simple Coulomb potential e2/K0\r\. Moreover, in the absence of
magnetic field the potential well may sometimes turn out to be too shallow for
binding to occur. One can readily see, however, that in a strong magnetic field
the motion becomes quasi-one-dimensional thus resulting in the formation of
bound exciton states even in the case of non-Coulomb potentials which are
weak compared with the energy of the particle motion in the magnetic field.
The spectrum of bound states for a weak arbitrary potential U(r) in the
presence of a strong magnetic field is usually found by adiabatic separation of
variables in the Schrodinger equation with the Hamiltonian (written here for the
sake of convenience in cylindrical coordinates)
1
8
8
2
8
1 δ2
. 1 δ
ρ2
,
r-2
5\
Presenting the wavefunctions in the form
Ψ = (2π)" 1/2eiMtpR(p,
ζ) W(z)9
(6)
which assumes the motion along the z-axis to be adiabatically slow compared
with that in the xy-plane, as well as neglecting the 'nonadiabaticity operator'
Q2R
QRQW
we obtain after separation of the variables two differential equations:
Γ h2 (
1 θ
8
Μ2
ρ2
M\
/ τ τ
Ί
χ
= V(z)R(p,z)9
2
h
ά
2m* d z 2
(8)
+ V(z)-e
)W(z)=0.
(9)
The one-dimensional adiabatic potential V(z) is found in the first order of
perturbation theory. Assuming in the first approximation the potential U(r) to
be zero, we obtain a spectrum corresponding to the spectrum of transitions
between Landau levels:
V0(z) = β*(21 + Μ + \M\ Η- 1),
(10)
where β* = hQ/2U(r), Ω = eB/pc, μ is the reduced electron and hole mass, / is the
Landau quantum number. For V(z) we can write
V(z) = V0(z) +
dp p\Rnpm(p)\2U(Jp2
+ z2) = V0(z) + bVnpm{z).
(11)
The energy spectrum of bound states is dominated by the term 6V(z) which
depends on the z-coordinate only. As is well known, with a one-dimensional
Diamagnetic exciton spectroscopy of semiconductors
353
attractive potential, no matter how weak it is, there always exists at least one
bound state. Therefore, even shallow potentials which do not produce a bound
state without a magnetic field (\U\ <^ h2/m*a2, where a is a characteristic range of
the potential) become binding in the presence of a magnetic field.
The one-dimensional eq. (9) with the potential of eq. (11) yields the binding
energy of such states:
(12)
For states with Μ = 0, the expression for the binding energy in a weak arbitrary
attractive potential can be written in the following general form (Bychkov 1960):
(13)
where ν is the volume of localization of the potential U(r). Thus, the weakness of
interaction without a magnetic field does not hinder bound-state formation in
the strong-field domain. This becomes the more valid the weaker is the
interaction potential and, hence, the greater is β*. It is the extent to which β*
exceeds unity that serves as a measure of validity with which one can consider
the motion along the field as one dimensional.
2.3. Semiconductor in a strong magnetic field: formation of diamagnetic
excitons involving Landau sub-bands
In the case of the interaction with a Coulomb potential, the strong-field criterion
can take the form of the following inequality (the Elliott-Loudon criterion):
β = (a*/L)2 = hQ/2R* = (φ2/'μ2e3c)
4
2
B>\.
(14)
Here R* = μβ /2Η κΙ is the effective Rydberg of the Wannier-Mott exciton and
is the effective ground-state radius. This particular problem
α* = (η2/β2)(κ0/μ)
can be solved only in the limiting case of / ? - • oo. For a finite field the solution
turns out to be more or less approximate. Consider in a general way the major
techniques used to solve the problem of the diamagnetic exciton and evaluate,
wherever possible, the scale of the error introduced.
The Hamiltonian of relative motion of this problem for Κ = 0 in the case of
simple bands is similar to eq. (5) with U(r) replaced by 2/(p 2 + z 2 ) 1 / .2 It is
appropriate also to use dimensionless energy, magnetic field and length
measured in units of the bound state ionization energy R*, ratio β and the radius
a* of the state, accordingly.
The first simplification consists in using the adiabatic approximation, which
reduces to representing wavefunctions in the form of eq. (6) assuming slow
R.P. Seisyan and B.P. Zakharchenya
354
'hydrogen-like' motion along the z-axis. As shown by comparison with numer­
ical calculations, the adiabatic approximation is already good for exceeding
unity only slightly, β ^ 1.
An equation of type (8) is solved by perturbation techniques assuming in the
first approximation the Coulomb term to be zero. Such a spectrum will
correspond to the spectrum of transitions between Landau sub-bands, eq. (10),
where β* should be replaced by β from eq. (14). The eigenfunctions of this
equation are
C , Me
R(p,z)-p
\
F
_
{
l
_
M
+
h
ix
)
M
j
*
<
Q
·
(15)
Here χ = \βρ2,
F(/, M, x) is the degenerate hypergeometric function. The
effective potential V(z) can be found by averaging the Coulomb potential with
radial functions Rlm:
V(z)=V0(z)-VlM(z):
°°
Γ
\RlM(p)f--f==dpdcp.
(16)
The solution of a one-dimensional equation of type (9) can be written in a
general form
% v = %
~
(17)
«
where slM = V0R* is the spectrum of transitions between Landau sub-bands, and
^ΐΜν(β)
is the binding energy of the exciton series associated with each pair of
magnetic sub-bands involved in the optical transitions.
For the general form of the potential VlM(z) we can write
VIM(Z)
=
C
2
M
^ 2
Q~XXMF2{U
\M\
+ 1, x)(x + βζ2/2)~ί/2
dx.
(18)
N o w the behavior of each Coulomb series and the exciton binding energy Μ can
be derived by solving a one-dimensional equation with potential (18).
Unfortunately, neither potential (18) nor, even more, the solution of eq. (9)
can be obtained analytically.
Elliott and Loudon (1959) proposed a solution with an approximated
potential
K(z)=
S
- +
A2ael
(19)
1 '
κ 0( α + | ζ | ) + K0(a + \z\r
1 }
derived from a numerical analysis of the problem. Here the parameters a and A
depend on the quantum numbers / and Μ and, implicitly, on B. Nevertheless, by
properly selecting the potential, Elliott and Loudon succeeded in reaching a
good agreement with experiment down to relatively small β > 1. Other authors,
for example, Hasegawa and Howard (1961), calculated the potential (18)
Diamagnetic exciton spectroscopy of semiconductors
355
analytically. For Μ = 0 and / = 0 they assumed
V00(z) = {2πβγΐ21\
- <Kzyft/2n
e x p ( / ^ 2/ 2 ) ,
(20)
which follows from the general form of eq. (18) (φ(ζ9 β) is the probability
integral). One could introduce other simplifications, however, their validity
becomes ever more limited and, moreover, requires very strong fields β> 1. As
demonstrated by comparison with numerical calculations, eq. (20) provides a
fairly low accuracy of solution for small β > 1. The accuracy can be improved by
solving the one-dimensional equation numerically with the potential (18). The
use of any potential approximating V[z) represents the third simplification.
Finally, the solutions of the one-dimensional equation (17) involving the
potential (19) or (20) for large z> L when the potential becomes proportional to
z _ 1 are well known and can be written in terms of Whittaker's functions
WVtl/2{z).
The spectrum of the diamagnetic exciton series can be derived from
equations
*K,i/2(0) = 0,
w
'v,
1/2(0) =
0,
for odd states,
(21)
for even states.
(22)
However, while eqs (21), (22) do in principle solve the problem of the energy
spectrum, their exact solution is again possible only in the limit
00.
For an infinite field, the motion along the z-axis occurs in a one-dimensional
Coulomb potential e2/rc0\z\. A problem of this kind was solved by Loudon
(1959). For Β = oo the energy spectrum forms a hydrogen-like series similar to
the spectrum of the three-dimensional Wannier-Mott exciton, but with a onedimensional Coulomb quantum number v:
R„=-{p^l2h2K2)v-29
v = 0,l,....
(23)
Here the ν = 0 state has an infinite binding energy corresponding to the particle
falling onto the center.
The solution for the quasi-one-dimensional case of a strong but finite
magnetic field can be written in a form similar to eq. (23) by including
corrections to the quantum numbers, the so-called 'quantum defects':
ΜΒ = (p„eV2h2K2)(v
+ 5 v u g) - 2 ~ R$(v
+ 8 v u g) " 2 .
(24)
The quantum defects 8 v ug representing a correction to the finiteness of Β in the
spectrum of the one-dimensional problem tend to zero for Β oo. For infinite B9
all levels except the ground state are doubly degenerate in parity (even, g; and
odd, u). In a finite magnetic field the degeneracy is removed, and all levels except
the ground one become doublets each with an even and an odd component.
The condition of quantization of the even components which can become
involved, for instance, in the spectrum of oscillatory magnetoabsorption for
R.P. Seisyan and B.P. Zakharchenya
356
dipole-allowed transitions is given by the transcendental equation derived from
eq.(22):
2C + φ(\ - δν) + (2<5V)"1 - Ιη(δ2νβ/2)1/2
+ ± J Ψ2(ηρ, Μ; χ)In χ dx = 0.
(25)
Here C = 0.5772 ... is Euler's constant, Ψ(η, Μ, χ) is the transverse component
of the wavefunction, ^(x) = d ln(F(x))/dx is the logarithmic derivative of the
gamma function, np is the radial quantum number, <5V = ν + 8v u g. The integral in
eq. (25) can be expressed in terms of degenerate hypergeometric functions and
reduced to
j Ψ2{ηρ, Μ; χ)In χ dx = φ(ηρ + Μ + 1).
(26)
The roots of the transcendental eq. (25) describe the ground and excited states of
the exciton below the corresponding Landau level with the number
/ = n p + i(A# + |A#|).
(27)
As the magnetic field increases, δν for excited states tends to integer values:
1, 2, .... The ground-state binding energy in the very high field limit (In β> I)
grows logarithmically, since
δ0^Ιβη(β/2).
(28)
Thus in optical interband transitions occurring in a high magnetic field below
each pair of the Landau sub-bands between which transitions are allowed an
exciton series appears which consists of one singlet (v = 0) and a number (v > 0)
of doublet levels. For sufficiently large ν they form a Coulomb series. As the
magnetic field Β increases, the quantum defects 8 v ug decrease which, in its turn,
results in an increase of the binding energy of the vth state.
Such a spectrum can be presented in a generalized form by an expression
differing substantially from that for the Wannier-Mott exciton spectrum:
ε* = s
- R<J>/(v + 6 v u g) 2 ,
l M
ν = 0, 1 , . . . .
(29)
Here εΙΜ are given, for instance, by eq. (10) and represent the energy spectrum
of transitions between Landau levels without the inclusion of the Coulomb
interaction; the quantum defect δν representing a correction for the finiteness of
the magnetic field also depends on the Landau number /, moment projection Μ
and magnetic field B. The quasidiscrete spectrum of such exciton states extends
far into the domain of ε > e g rather than being limited by the absorption edge ε 8
as it is in the three-dimensional case.
As the Landau number / increases, the depth of the one-dimensional Coulomb
well decreases which produces a contraction of the higher lying series. Since
Diamagnetic exciton spectroscopy of semiconductors
357
eq. (29) includes, besides the discrete spectrum, the continuum as well, all
Coulomb series except the lowest (longest wavelength) one lie superposed on the
background of the continuum branching off, as it were, from each of the
preceding Landau levels.
Spectrum (29) is very peculiar in that while it is essentially an exciton
spectrum it possesses the intrinsic features of the spectrum of transitions to freecarrier states in the diamagnetic Landau sub-bands. It is such a pattern that we
call the diamagnetic exciton spectrum. Speaking about the orbital motion of the
'diamagnetic' exciton under conditions where the criterion of Elliott and
Loudon is met it should be stressed that the orbit of such excitons tends to be
elongated in the magnetic field direction. Establishing an unambiguous corre­
spondence between the quantum numbers ν and n0 (the quantum number of the
three-dimensional exciton for Β = 0) presents a difficulty since the behavior of
the Wannier-Mott exciton in the intermediate field domain where the spacing
between the Landau levels is comparable with R*, or where β « 1, does not lend
itself to theoretical description.
Elliott and Loudon (1959) calculated numerically the oscillator strengths for
transitions to the states corresponding to the diamagnetic exciton. They showed
that the intensities of such transitions may dominate those of the transitions
immediately under the Landau sub-bands and increase superlinearly with
increasing B. Diagrams illustrating the formation of the diamagnetic exciton
spectrum are presented in figs 1 and 2.
oo
3
Ζ
V-Sand
777777777777777777777777777
β=0
Fig. 1. Schematic representation of the formation of the diamagnetic exciton spectrum. The energy
diagrams for excitons and electrons are conventionally matched. Dashed arrows specify the
transitions to excited states of the longest wavelength series of the diamagnetic excitons and to their
ionization limit.
358
R.P. Seisyan and B.P. Zakharchenya
Fig. 2. Spectrum of diamagnetic exciton series near two adjacent pairs of optically coupled Landau
sub-bands. The shaded region identifies the beginning of the series continuum. Dashed curves:
density of states without the inclusion of the Coulomb interaction. Absorption spectra for the case of
experimentally resolved states with (a): ν ^ 2 and (b): ν ^ 1 are shown.
2.4. Diamagnetic excitons in diamond-like
semiconductors
Zhilich (1971) was the first to attempt to solve the problem of the exciton in a
high magnetic field in diamond and zincblende semiconductors using the
adiabatic approximation. He initially diagonalized the Hamiltonian corre­
sponding to the electron and hole in a magnetic field without the Coulomb
interaction, and only after this introduced the Coulomb term averaged with the
wavefunctions of the problem for transitions between the Landau sub-bands.
This method provides the possibility of studying the exciton associated with a
pair of sub-bands without taking into account possible admixing of states from
the adjacent sub-bands. As was shown subsequently, this approach can be fully
valid only in the case of transitions from the light hole sub-bands with large /.
An adiabatic solution for crystals with a complex valence band was developed
by Rees (1972). Rees included consideration of the Coulomb term before
diagonalization and thus took correctly into account the admixing of states
from different Landau sub-bands. However, in extending the solution from
acceptor states to the diamagnetic exciton he automatically introduced an
inaccuracy in the initial effective-mass Hamiltonian by omitting the term
describing the relative motion of the electron and the hole, (eh/m*c)(J?B), where
i f is the orbital momentum,
= [r, p]h~x; ρ is the momentum operator.
Subsequently the problem was treated in a more general way by Lipari and
Altarelli (1974) and Martin and Wallis (1974). Regrettably, they did not obtain
an analytical result which would be convenient for analysis and calculations.
Therefore we shall consider here the solution in the version developed by
Gel'mont et al. (1977).
In this calculation, the warping of the energy surfaces and band nonparaboli­
city were neglected. The Hamiltonian describing the behavior of the electron and
Diamagnetic exciton spectroscopy of semiconductors
359
hole (exciton at rest) in a magnetic field in this approximation has the form
J f = ( f t 2/ 2 m ) [ ( 7l + | y + m/m?) J f
- (he/mc)K(/B)
+ {he/m*c)(B&)
2
- (y/2) £
{ « } { A / „ } ]
+ i g * / i B( * B ) ~
(30)
=
Here 7 = y 2 7 3 > 7 i » ^ are the Luttinger parameters of the valence band, X =
\Vr - (e/ch)A(r) is the generalized momentum, A(r) =
r], fx,fy,
fz are the
numerical matrices of projections of the moment f = §, {afr} = [(af?) + (6a)]/2,
σ Χ ) 7ζ> are the Pauli matrices and, μ Β is the Bohr magneton. The spectra of the
Landau sub-bands are obtained by equating the potential V(r) to zero. For dia­
mond-like semiconductors such a spectrum can be written as,
tiMX
=
&ω 0[(/ + i)(m/m*) ± ^g* + ε , _ Μ+ 1
/ 2 ] ;, Λ
(31)
+ ( ! - Α 2)[(α ΛΑ + α π Λ )ι 2 / 4 + 3η(η + λ + ^ ) y 2 ] 1 / ,2
β ΜΛ = (α.Α +
2
2
(32)
where Αχ = 4Λ(2 - f )/3, α„Λ =
+ (γχ - Κ)λ + ( / I - f )(Η Η- >l)y and hco0 is the
free-electron cyclotron energy heB/mc. The index A assumes in the general case
four values: + § , + 1 , — \, — f, with λ = + § corresponding to the heavy hole, and
λ = +1, to the light one.
In practice eq. (31) turns out to be oversimplified. For more accurate
calculations of the Landau sub-bands one should preferably use the determinant
equations derived by Pidgeon and Brown (1966) based on the Kane model.
These equations yield solutions appropriate for calculating the Landau subband energies of the conduction band and three valence bands (with two series
for each band, ε*, ε 1 , ε*, ε 8ο , respectively). In the quasiclassical limit the subbands form practically equidistant sequences which Luttinger termed 'ladders'.
The modified equations of Pidgeon and Brown contain as band structure
the following
quantities:
EP = (2m/h2)P2,
where
P=
parameters
— i(h/m)(S\pz\Zy
is the interband momentum matrix element for the conduc­
tion and valence-bands, F is a similar quantity for the interaction of the con­
duction band with the higher lying bands, as well as Nu y l 9 y 2 , y 3 , K, q. Apart
from this, these equations also contain the energy gaps ε 8, and A0 and θ which is
the angle between the magnetic field direction and the c 4-axis in the (110) plane.
Despite the wide use the equations of Pidgeon and Brown have found in
calculations we shall not write them out here and shall instead refer the reader to
the chapter in this volume written by Pidgeon or to the paper of Weiler (1979).
Note only that the above set of parameters describes adequately by the wellknown relations of the modified Kane model the energy spectrum of the
electrons and holes for all diamond-like semiconductors in the vicinity of k = 0.
(With the possible exception of InSb and solid solutions of the type C d xH g ! _ XT
with χ < 0.3 where eg is small and one should take into account the potential
nonlocality effects associated with the exchange interaction.) The problem of the
determination of these parameters from oscillatory magnetoabsorption data
360
R.P. Seisyan and B.P. Zakharchenya
should thus reduce to a 'reconstruction' of the spectrum of transitions directly
between the Landau sub-bands described by the equations of Pidgeon and
Brown by adding the diamagnetic exciton binding energies ^ιάη^ν(β)
to the
positions of each of the observed spectral lines. Note that the selection rules in
the Landau numbers / and η for optical interband transitions between the
Landau sub-bands and to diamagnetic exciton states totally coincide to a first
approximation*.
An essential feature in the energy level system of diamond-like semi­
conductors in high magnetic fields is the substantial difference of the longi­
tudinal masses characterized by the curvature of the one-dimensional hole
sub-bands
m i
2
( ^ = [ ( d 2 / d C 2 ) e 1 ± (2 n , C ) | ^ 0 ] - 1
(33)
(we have here a dimensionless wavevector ζ = kzL) from the conventional
'three-dimensional' heavy- and light-hole masses, m hh and m l h. In a general case
the energy in the vicinity of ζ = 0 can be written as
fii.2(",
0 =
*li(n, 0) + h2C2/2L2mUn>
0-
(34)
The behavior of the longitudinal mass was studied numerically by Wallis and
Bowlden (1960) and, subsequently, more accurately by Evtukhov (1962) who
also took into account the interaction of the v 1 2 bands with the i; so band
separated by the spin-orbit coupling energy A0.
Analytical expressions for the longitudinal masses were obtained recently by
Gel'mont et al. (1977) in the approximation which does not include band
warping. The longitudinal mass depends on the Luttinger number η and ladder
index λ. The analytical expression for η ^ 2 is presented here for the first time
—
= y 1 +2y—
mnX
χ
(η + 1)(βπΑ ~
- +
2εηλ - αηλ 1/2)(£„Λ ~
12y2
αηλχ
«,,,-3/2) +
(η ~ !)( £nA ~ *η,ΖΙ2)(?>ηλ
~ ^η,Ι/ΐ) ~ 6y2w(tt2 ~ 1)
(35)
Table 1 lists the values of mass and energy for η = — 1,0, 1 where the total
number of ladders is less than four, and table 2, light-hole longitudinal masses
for some A 3 B 5 compounds.
As follows from eq. (35), the longitudinal mass of the hole can be negative
T h e r e is some confusion in the literature concerning the sub-band numbering. There exist
valence band numberings of Luttinger, Roth et al. and of Aggarwal. We shall use here the numbering
of Aggarwal (1973) in which the selection rules for the Landau number / are written as Al =
± 1 = / c - / v. To simplify the notation, we shall denote the Landau-Luttinger hole sub-bands by η
while omitting for the electron the subscript c on / c.
361
Diamagnetic exciton spectroscopy of semiconductors
Table 1
Hole magnetic sub-band energies and reciprocal masses in diamond-like semiconductors for
n= - 1 ; 0 ; 1.
η
μ
-1
ε
f
0
- I
-!
1
\
- I
- |
mjmnk
(y!+y)/2-3fc/2
yx-2y
- 2 y + 1 2 y 2/ ( | a 2 - a 3 yl )i
7 i + 2y - 1 2 72/ ( Ι « 2 - « 3| )
( y -i 7 ) / 2 - / c / 2
3 ( yi + y)/2 - 3/c/2
Zyx+2y-k
+ [(27! + 37 - 2/c) 2/4 + 6 7 2 ] 1
3(7! - γ)/2 - k/2
37!+27-/c
- [(27ι + 37 - 2k)274 - 6 7 2 ] 1
= (2η - 1)(7ι - 7)/2 + Λ/2,
αι
2/
2/
V I + 2 y ( « I - Α 3) / ( 2 ε 1 /- A
2 T - A 3)
+ 2 4 72( ε 1 2/ - )Α/ Ι( Β 1 2/ - α 2) ( 2 ε 1 2/ - ιβ - α 3)
7ι + 2γ + 2 4 72( α 2 - α) /ι[ ( α 2 - α 3) ( α 2 - Λ) ι- 6 y2]
7 ι + 2 7 ( α ι - α 3 ) / ( 2 ε _ 3 / - 2α ! - α 3 )
+ 2 4 72( ε _ 3 2/ - )Α/ Ι( Β _ 3 2/ - α 2) ( 2 ε _ 3 2/ - Α , - Α 3)
α 2 = (2η + l)(y t - y)/2 - k/2, a 3 = (2n + 3 ) ( 7l + y)/2 - 3fc/2
Table 2
Light-hole effective longitudinal masses for some A 3B 5 compounds.
Material
GaSb
GaAs
InSb
InAs
InP
Hole
level
series
a
b
a
b
a+
b+
a+
b
a+
b
+
+
+
+
+
+
Landau-Luttinger quantum number
-1
0
1
2
3
4
0.12
0.29
0.16
0.37
0.13
0.40
0.17
0.47
0.26
0.66
0.074
0.054
0.145
0.139
0.018
0.016
0.036
0.029
0.208
0.115
0.045
0.054
0.090
0.091
0.018
0.015
0.027
0.028
0.094
0.099
0.043
0.045
0.080
0.086
0.015
0.015
0.027
0.027
0.089
0.095
0.042
0.045
0.079
0.084
0.015
0.015
0.027
0.027
0.089
0.093
0.042
0.044
0.078
0.083
0.015
0.015
0.027
0.027
0.089
0.092
even for γχ > 2y (the condition for the hole mass to be positive in the absence of
magnetic field). Moreover, for any relative magnitude of the constants yl9 y, k the
heavy-hole mass of one of the Luttinger bands starting from a certain number η
becomes negative. To show this, one can study the behavior of mnX for η ^> 1. The
light-hole levels for η > 1 are characterized by a mass
m/m±U2,n = yi + 2 y > 0 .
For heavy-hole levels with λ = ± | for η > 1 the mass is
m ± 32/ , n
= 7i — 2y±2nyz—
(2y +
—
-.
y1-k)(k-y1-y)
(36)
R.P. Seisyan and B.P. Zakharchenya
362
Whence one readily sees that for sufficiently large η the curvature of one of the
valence bands at the center of the Brillouin zone indeed becomes negative.
Although it does not follow directly from eq. (35) for the longitudinal masses,
calculations show that for InSb, InAs, InP, GaAs, GaSb and Ge this should
occur starting from η = 2, that is from the moment of the appearance of two
'heavy' levels with the same n. Since the separation between two heavy hole
levels with the samerc,2 Δε = ε 3 / 2, , ι — ε 1 / 2η is small and decreases with increasing
level number as n~2, they should be considered as degenerate. The structure of
the heavy-hole spectrum can be derived from the equation
ε„,3/2
+ ( 2 / 2 Μ π , 3 / -2 ε
bnC
hi
0
=
+ £ 2 / 2 Μ η , _ 3 / -2 ε
εη,-3/2
Here
- ( n - l ) 1/ 2C
M„"3/2
M~i3/2
= - 2 F O/ (2 £N , 3 / 2
M,
_
1 / ( 2|
)Q_
~ «π.-3/2)
+
3 2
/
(-f)],
Φ,.,3/2»
= ~ 2 / > 2/ ( ε „ , _ 3 /2 - ε„, 3 / )2 + m / m W _) 3 / .2
Equation (37) yields the structure of the heavy-hole spectrum e W F ± 3(/ 02 for
any diamond-like semiconductor. In the vicinity of ζ = 0 the spectrum repre­
sents typically a two-humped curve for ε„ _ 3 / (2 / c z) (fig. 3).
The diamagnetic exciton binding energy can be obtained by solving the onedimensional equation obtained by adiabatic separation of variables in
Schrodinger's equation with the Hamiltonian (30). It is similar in form to eq. (9),
however, the mass characterizing longitudinal motion in the exciton will be
Fig. 3. Schematic
representation
of the heavy-hole Landau
semiconductors.
sub-bands
in
diamond-like
Diamagnetic exciton spectroscopy of semiconductors
363
dominated by the induced mass μηλ which depends on the number of the
Landau-Luttinger level, n = l — Μ + J, of the hole involved in exciton
formation:
™/μηλ = m/m* + m/mnX.
(38)
The potential U(z) will depend on the quantum numbers λ9 /, Μ:
Γ
π2
UxlM(z)=-(e2/K0L)
k 0L )
άφ
Ι
= - (e2/K0L)(2C
<
+
\ΦΐΜ(ξ,
z2/L2y
/ 2 β1- « / 2 ( 2 πΐ ! ) - ^ ( 2 ε ηλ
Χ [(ε»Α " a w A )l ^ - MF 2 ( - U
χΓ
+ ζ2/υ)-^
φ,λ)\2(ξ
άξ
- α ηλ - a w A )l " 1
- Μ + 1; ξ ) Γ ~ 2( λ - Μ + 1)
( / - Μ + Α + 1 ) + (ε„λ - α η ΛΚ λ>" MF 2 ( - U
_ 1
t
- Μ + 1; ξ)
χ Γ ~ 2( Α 1 - Μ + 1)Γ~*(/ - Μ - λχ + 1)] άξ.
(39)
Here φΙΜ(ξ, φ, λ) are the eigenfunctions of the problem for the transition
between the magnetic sub-bands with Hamiltonian (30) for V(r) = 0 which can
be written in the form
3/2
ΦΐΜ(ξ,
λ) =
Ψ,
^Σ
χ ΓΙΙ2(1
i ( M " μ)φξ(μ - M)/2Q- e ξ/2
3Ι2
^ - Μ
+
1 / 2 ; ^ )
Γ
(
_ μΜ
+
1
)
(
2
2π
- Μ + μ + 1) F( - Ζ, μ - Μ + 1; ξ)χμ9
;
!
)
1
/
(40)
where Γ(ζ) is the gamma function. For the coefficients we have
'
"» -
C
(A)
( 4 1 )
where μ just as λ, runs through four values: + §, + i , — ^, - f ; 6pq are the
Kronecker symbols, ξ = ( χ 2 + y2)/2L2,
F( — /, μ — Μ + 1; ξ) is the degenerate
hypergeometric function, and χμ is the eigenvector of the matrix #z {/ζχμ = μχμ).
Similar to the way this was done in the case of a 'simple' semiconductor, it can
be shown that the energies of an exciton formed from an electron and a light
hole in a high magnetic field are determined by means of a transcendental
equation:
2€ + ψ(\-δν)
+
±δν-±\η(δ2νβ/2)
f 2;
'qo
lnCd£
o
\φΙΜ(ξ,φ,λ)\2άφ
= 0.
(42)
Jo
The quantity δν = ν + dvug which has to be found from eq. (42) determines the
diamagnetic exciton binding energy ^ d e through the expression & dc =
β4μηλ/2Η2κ2)δ2.
The integrals in eq. (42) can be expressed in terms of ^-functions.
R.P. Seisyan and B.P. Zakharchenya
364
As a result, eq. (42) can be rewritten for η ^ 2 in the form
2C + φ(1 - <5V) + ( 2 ^ v) " 1 - iln((5 v 20/2)
+ i f e a " α„Λ,)</Ί^ + η + i + i ( M + λ + |M - λ\)](2εΛλ - α πλ - α ^ ) " 1
+ i f o a ' ηλ)Ψ
α
Ιλχ + η + i + i ( M χ ( 2 ε η λ- α η λ- α η
)-1
Λ ι
+ |M - Ax |)]
(43)
For the lower levels and for - 1 ^ η ^ 2 the integral in eq. (42) reduces to
^(1 + M - f ) for n= - 1 (only the λ = \ level exists), to ^[A + i + £ ( M - A
+ | Μ - / ί | ) ] for n = 0 (levels with A = f and i exist), and to ^ [ 2 + i ( M - i
+
for n = 1 (A = i); for the states with A = f and A = - i eq.(43)
holds.
Equation (43) also describes the energy levels of a diamagnetic exciton formed
from an electron and a heavy hole in very high fields where β > 1. If the magnetic
fields are not sufficiently high that the separation between Landau-Luttinger
levels is larger than the exciton binding energy, then in the analysis of the
problem of the exciton levels one should take into consideration both heavyhole Luttinger levels. The need for taking into account both levels in the exciton
formation arises also for any fields in the case of sufficiently large n. As already
shown, the mass at the top of the heavy-hole Landau band becomes negative
starting from a certain level and decreases with increasing n. Therefore when
determining exciton levels for η > 1 one should take into consideration the
nonmonotonic nature of the dependence of the heavy-hole energy on kz (fig. 3).
Thus the binding energy of a diamagnetic exciton formed from an electron
and a light hole at η Ρ 1 can be readily found from the expression
^ d e = e^J2h2K20S2v,
(44)
where μηλ is the induced exciton mass, δν is the root of eq. (43) depending on the
moment Μ with which the exciton is born. The reduced mass of such an exciton
can be calculated from eqs (35) and (38) without including the nonparabolic
effects.
Figures 4 and 5 display the binding energies of the diamagnetic excitons
formed from light holes as a function of magnetic field Η and Landau number /.
The binding energy grows logarithmically weaker (only through £ v) with
increasing field. The binding energy depends on / primarily through the mass
mnX and at Ζ = 0, when the mass of the light hole compares with that of the heavy
hole in the quasiclassical limit m?h(0) = m/(y1 — 2γ), reaches a maximum. As η
increases, the limiting value is reached which corresponds to the light-hole mass
in the quasiclassical limit, m?h(co) = m/(yl + 2y). After that the binding energy
practically remains constant retaining only a weak implicit dependence on /
through (5V.
An analysis of the diamagnetic exciton involving heavy-hole sub-bands is
complicated by the mixing of states from two ladders. The binding energy of
Diamagnetic exciton spectroscopy of semiconductors
0
0
3
τ
3.0
6
3
iZ
i5
ft
ι
1
1
1
Γ
610
80
4.0
365
β
BJ
Fig. 4. Binding energy variation of the diamagnetic excitons formed in diamond-like semi­
conductors (exemplified by GaAs) by light holes plotted against magnetic field: 1-3; for the ν = 0
ground state, transitions involved, b +( - l ) b c ( 0 ) , a +( - l ) a c ( 0 ) , a b +( 3 ) a b c( 4 ) , accordingly; 4 - 6 ;
same for the ν = 1 first excited state. Dashed lines: variational calculations (by Gel'mont et al. 1977).
I
o
1
e
1
4
1
1
6
8
i
Fig. 5. Binding energy variation of the diamagnetic excitons formed in diamond-like semi­
conductors (exemplified by GaSb) by light holes plotted against the Landau quantum number /:
1, for the ν = 0 ground state; 2, for the ν = 1 first excited state, transition involved, b +( — l ) b c( 0 ) ;
3, same as 1, including nonparabolicity but neglecting quantum effects. Broken lines are drawn to
aid the eye (by Gel'mont et al. 1977).
such a diamagnetic exciton could be found for each sub-band and any magnetic
field Β by the variational technique. However, in the case of high magnetic fields
and for a sufficient separation between the upper and lower heavy-hole subbands with the same n, the exciton could exist in each of them separately for
small n. This case is of particular interest since, by eq. (35), for the ladder with
λ = — f the hole mass at the zone center can be negative and one could thus
expect manifestation of the corresponding effects. The heavy-hole masses at the
366
R.P. Seisyan and B.P. Zakharchenya
zone center in both ladders are small and close in absolute magnitude but
opposite in sign and decrease rapidly with increasing η becoming less than the
electron mass m*. At the same time the mass of the side maxima of the λ = — §
ladder is only slightly less than its quasiclassical value for small η and
approaches it for η > 3. Note that the separation between two ladders with the
same η decreases from a value of the order of the free-electron cyclotron energy
hco0 down to 0.03fta>0 for η = 10.
In cases where the magnetic field is sufficient for excitons to exist separately in
each sub-band of a pair of sub-bands with the same n, one could expect the
'upper' sub-band to become involved in the formation of a 'direct' diagmagnetic
exciton with a very small (at the expense of the longitudinal hole mass for ζ = 0)
reduced mass. The binding energy of such exciton is small. At the same time the
lower sub-band can participate in the formation of two 'indirect' excitons with a
mass determined by the curvature of ε(ζ) at the maximum and close to that of
the heavy hole in the quasiclassical limit. These maxima are separated from the
central maximum ε 3 /2 by a value Ask of the order of the free-electron cyclotron
energy. Such sub-band structure may affect the relative intensities of the
absorption lines produced by transitions from the 'heavy'-hole levels a" and b ~ .
For the η = 2 transitions the reduced exciton mass is still not negative but has
already become very large, much greater than the electron mass. This will affect
the intensity ratio strongly. Quite frequently, however, the lines corresponding
to the transitions from the a" and b " series overlap, the halfwidth of the
resultant line becoming such as to enable the use of the average energy ε Η ν. For
the binding energy in this case we have
^ d e = eV2h2K2062lmr1
+ ( y t - 2y)/m],
(45)
with δ0 found from eq. (43). The binding energies obtained in this way without
the inclusion of nonparabolic effects depend only weakly on h and / (only
through δ0).
The problem of finding the binding energy for the excited states of the
diamagnetic exciton formed by light-hole ladders is solved simply by using an
expression of type (44) where δν is the vth root of an equation of type (43). The
binding energy of an excited state formed by the diamagnetic exciton with the
heavy-hole ladder could also be determined by the variational technique.
However, the condition for the existence of excited states should be written as
0t{ll <ζ hco0 Δε*„, where Δε^„ is the energy gap between the nearest magnetic subbands of the hole with the same η in dimensionless units. This condition is
readily met for light holes and is difficult to satisfy for heavy holes.
We conclude by pointing out another feature of the diamagnetic exciton
spectrum. Inclusion of the Coulomb interaction removes forbiddenness from
transitions and thus results in the appearance of lines which do not have
counterparts in the spectra of transitions between Landau sub-bands. Such
transitions are specified by the selection rules ΔΖ = 0, 2 in addition to the
Diamagnetic exciton spectroscopy of semiconductors
367
selection rules for the allowed transitions, Δ / = ± 1 , and their intensity is
proportional to the ratio ^^/Ηω0 Δε^„. Judging from the energy difference in
the denominator, one could expect these transitions to be more intense for heavy
holes, however, the anomalously large mass of the light holes at the η = 0
Landau-Luttinger level makes this transition also significant.
2.5. Diamagnetic excitons in intermediate
fields
As already pointed out, the Schrodinger equation for the exciton in a magnetic
field can be solved analytically even for simple bands only in the limits Β = 0 and
oo. The experimenter is most interested, however, in the 'intermediate' field
domain where condition (14) is only weakly met {β> 1).
For instance, in C u 2 0 , S n 0 2 , CdS and GaSe crystals where the binding
energy is fairly high the strong-field criterion β > 1 is difficult to satisfy. At the
same time their magneto-optic spectra reveal features characteristic of the
diamagnetic excitons even for β < 1. As this will be seen from what follows, there
are certain grounds for classifying such observations as the case of'intermediate'
rather than 'weak' fields.
This relates also to such crystals as GaAs or InP although modern techniques
provide means for the realization for these materials of the β> 1 condition.
In analyzing possible theoretical approaches to this problem it should be
stressed that most of the attempts to solve it make use in some way or other of
the adiabatic separation of variables. As already mentioned, experimental data
show the adiabatic approximation to be sufficiently valid already for not very
large / ? , ( £ > 1).
Moreover, it turns out that the adiabatic approach is valid over a substan­
tially broader magnetic field range.
Zhilich and Monozon (1968) drew attention to the fact that the quasiclassical
frequencies of exciton motion in an excited state along the magnetic field
direction are defined by the separation between the corresponding levels and are
Whence a conclusion was drawn that the adiabatic approxi­
ωηο&R*/nlh.
mation can be used to analyze the excited states already at βη$ > 1.
Monozon and Zhilich (1968) proposed to use the approach developed by
Hasegawa and Howard (1961) for the description of the part of the onedimensional series lying between the nearest energies for transitions to the
Landau sub-bands, that is for the exciton levels with numbers n0 satisfying the
condition R*/2nl <^ ΗΩ, or βη\ > 1. If this condition is met, the quasiclassical
frequencies of motion along the z-axis turn out to be much lower than the
frequencies of transverse motion in a magnetic field, ω ε = eB/m*c. Unfortunate­
ly, within this approach the question of the behavior of the ground state in each
diamagnetic exciton series, that is of its position and the oscillator strength,
remains unanswered.
Note that these criteria turn out to be much less stringent than eq. (14) which
368
R.P. Seisyan and B.P. Zakharchenya
makes the region of applicability of the diamagnetic exciton model formally
unbounded. Indeed, one can always find a number n0 for which this criterion is
met.
Experiments show, however, that in practice the real existence of a discrete
structure with the corresponding n 0 or its flare-up in the magnetic field are
required. The spectra of absorption oscillations which may be identified with
those of the diamagnetic exciton appear even when criterion (14) is not met but
the separation between the Landau sub-bands involved becomes greater than
the binding energy of the exciton excited states observed at Β = 0. In other
words, it is always sufficient in practice that
ftO/2K*'(n0)=KM,
R*'(n0) = R*/n2.
(46)
Obviously enough, this includes β < 1 suggesting that the behavior of the
ground state with n0 = 1 can at the same time be described by the relations for
the Zeeman effect and diamagnetic shift. Thus the magnetic field at which the
diamagnetic excitons will be observed depends on the actual binding energy for
Β = 0 and the number of terms in the exciton series.
Gantmakher et al. (1982) have rigorously shown that the spectra of a
hydrogen-like atom in intermediate magnetic fields can be described analytically
below the Landau levels with a large number Z. Also, for these levels the
adiabatic approximation is valid when a criterion less stringent that eq. (14) is
satisfied:
βΙ>1.
(47)
This also includes β ^ 1. It has also been shown in the quasi-classical approxi­
mation Ζ > 1 that the exciton wavefunction can be written in the form
R„fM = (npL) -
' θ;Μ {ρ,
ι2
2
L) s i n Q " 2 θ ζΜ(ρ, L) d p ) ,
(48)
where
0,.M(P> L) = (2epL2 ~yMy = (m*-
M2L2/p2
m*)/(m* + m*),
-
p2/4L2y2,
μ = m*m*/(m* + m*)
and the rotation points ργ and p 2 are given by the expression
p
\
2
= 4L2{21 - Μ + 1 Τ [(2/ + 1)(2Z - 2M + 1 ) ] 1 / }2 .
(49)
Substituting eq. (48) in eq. (16) yields the adiabatic potential δ V(z):
δ V(z) =
/
Υ
Μ/2
κ(( 4^4Y Y
/2
(50)
πκ0{ρ2 + ζ2)112
\\PI-z2)
y
where K(x) is a complete elliptical integral of the first kind. The characteristic
scale of motion along ζ varies depending on the relative values of Μ and I.
Diamagnetic exciton spectroscopy of semiconductors
369
Indeed, for Ζ > \M\ the asymptotic form of the potential corresponding to
Pi ^ Pi> is appropriate:
πκ0α*
4p2
For |M| > Ζ with Μ < 0 and l — Μ <ζΙ
for the potential assumes the form
δΚ(ζ) = — f - 1
with Μ > 0, the asymptotic expression
+ ^ Λ
(52)
coinciding with that found by Zhilich and Monozon (1968). The ground-state
energy can be determined using a one-parameter wavefunction
g(z) = ( c 0 t ) 1 / e4 x p ( - a z 2 / 2 ) ,
(53)
by means of the variational technique. For / > Μ for the ground-state energy we
shall have
ε = hQU - (\M\ + yM)/2] + -ξ —(ΐ -^η
+ 1 - A.
(54)
μα*ρ2π \
256p 2
/
The characteristic scale of the longitudinal motion for Μ = 0 will be ζ γ =
(p 2 a) 112 . For the inverse inequalities |M| <^ Ζ or I — Μ <ζ I the energy levels are
given by the expression
e2
ε = * > [ / - (|M|
+
M
T )/2] - _
h2(N±i)
+ -±-^
t
= ο,N 1 , . . . .
(55)
In this limit this coincides with the result of Zhilich and Monozon (1968), with a
characteristic scale of longitudinal motion z 2 = ( p | a * ) 1 / .4 In both cases the
separation between the bottom of the Landau sub-band and the bound-state
energy level is small compared with ηΩ, that is the spacing between the Landau
levels. Thus the adiabatic approximation turns out to be valid for Ζ > 1 with the
inequality βΐ > 1 which is weaker than β ρ I. In the case where criterion (47) is
satisfied the exciton wavefunction becomes lens shaped with the axis of rotation
directed along ζ (for β
1 it is cigar shaped and oriented along B). The
characteristic scales z 1 >2 here are small compared with the cyclotron radius
Ly/i.
Exact values of the ground-state binding energy for small Ζ in the case of
'intermediate' fields can be obtained only by the variational technique. With the
known solutions for Β = 0 and Β = oo a 'good' variational function is construc­
ted in such a way that its symmetry type be as close as possible to that of the
eigenfunction of the given state. However, the solution of this problem first
requires a solution of the problem of line correspondence in the spectra of the
Wannier-Mott exciton and the diamagnetic exciton.
370
R.P. Seisyan
and B.P.
Zakharchenya
Two rules of correspondence were discussed, one of them based on the
principles of conservation of the number of nodal surfaces, and another, on the
principle of term non-crossing.
Exciton states in a zero magnetic field are described by the same quantum
numbers as the states of the hydrogen atom, namely: n 0, the principal quantum
number; Z0, the orbital moment; M, the projection of angular moment on the
z-axis. States in a high magnetic field are specified by four quantum numbers: M,
the projection of angular moment on the z-axis; Z, Landau quantum number; v,
one-dimensional Coulomb quantum number; and parity under reflection
through the plane perpendicular to z, that is g, u (or ' + ',' —'). Parity and the
projection of the angular moment are 'good' quantum numbers for any
magnetic field.
Shinada et al. (1970), in developing Kleiner's rule mentioned for the first time
by Elliott and Loudon (1959) based their classification on the idea of
conservation of the number of nodal surfaces. This idea gained many followers.
The non-crossing rule following from general principles of quantum mechanics
and group theory yields in a number of cases a different result. Boyle and
Howard (1961) are among the proponents of this principle. The proof of its
validity was given by Lee et al. (1973).
In solving the problem of correspondence, it is appropriate to divide states
into two groups of truly bound and metastable states. Indeed, since M, the
projection of angular moment, is a 'good' quantum number for a hydrogen-like
system in any magnetic field, the minimum energy of a free electron-hole pair of
moment Μ in a magnetic field will be
em
ni
= fti2[(|M| + M
7 )/2 + i ]
(56)
[here γ = (m* — m*)/(m* + m*)]. Therefore, all states of the exciton of moment
Μ involving Landau levels with Zc = (|M| + M)/2 and Zv = (|M| — M)/2 are
always bound. It is these levels that have counterparts in the Wannier-Mott
exciton spectrum at Β = 0, and their energies can be calculated by variational
techniques.
The table of correspondence for these levels can be constructed using the noncrossing rule for any Μ. On the other hand, the states lying above e m ni are
metastable against decay to the continuum and do not have counterparts for
Β = 0. These states arise from the continuum.
According to the non-crossing rule, all states of a given moment and parity
(for Β = 0) will transfer, as the energy increases, to states of the same Μ and
parity for Zc = 0 (B - • oo).
Cabib et al. (1972) performed a comprehensive numerical calculation of the
behavior of Is and 2s states in intermediate fields. The results of their
calculations of binding energies and oscillator strengths can be applied to
advantage in evaluating the intermediate field case for the diamond-like
semiconductors as well, provided the reduced longitudinal masses, eq. (35), are
used.
371
Diamagnetic exciton spectroscopy of semiconductors
A variational calculation for diamond-like semiconductors was carried out by
Dolgopolskii et al. (1977) for the lc = 0 transitions in GaAs. The results obtained
agree well with experiment over a broad field range of Β = 0 to Β — 20 Τ. There
is, however, no description of the behavior in intermediate fields of higher
diamagnetic exciton states characterized by Landau quantum numbers I = 1, 2.
The genesis of the exciton levels in the high-field limit and their corre­
spondence to the three-dimensional exciton levels which we have touched on
only briefly remains a radical and still unsolved problem. Despite the simplicity
and attractiveness of the non-crossing rule and its validity for the description of
the behavior of the lower levels which do not have a continuum background, its
application to levels with / c ^ 1 requires invoking additional assumptions. This
becomes particularly evident when analyzing the spectra of diamagnetic
excitons, for instance, in C u 2 0 , CdSe and GaSe.
3. The exciton
nature of oscillatory
magnetoabsorption
spectra
Diamagnetic exciton spectra exhibit many features of the spectrum of oscillatory
magnetoabsorption for optical transitions directly between the Landau subbands. This apparently accounts for the fact that the results of some earlier
observations of the diamagnetic exciton spectra in various crystals were
successfully interpreted as spectra of transitions between the Landau sub-bands.
We should like to stress here a general property of the spectrum of
diamagnetic excitons which is used as a basis for the most consistent, that is
many-particle, approach to the problem of optical transitions. The ionization
edges of the exciton series ε^(β) and, hence, the energies of transitions to the
free-carrier states lying directly below the Landau sub-bands are not identified
by any spectral features no matter how many terms there are in the diamagnetic
exciton series. (The same is true for the boundary of the Wannier-Mott exciton
series for Β = 0 corresponding to the energy gap.) Accordingly, the spectrum
does not contain features associated with transitions between the Landau subbands, and the expression for the absorption coefficient in these regions
represents a linear dependence on photon energy:
Μ) = α£[1-Λ(ε-ε^)],
(57)
here oc# are the absorption coefficients at the ionization limits ε™Μ.
In this section we shall consider the features which discriminate the spectrum
of diamagnetic excitons created in a magnetic field from that of'pure' transitions
between the Landau sub-bands.
If we disregard the Coulomb interaction, then, as is well known, extrapolation
of the positions of the O M A maxima to Β = 0 should yield the magnitude of the
energy gap ε 8. It was assumed that such an extrapolation gives an accurate width
for the energy gap. This opinion became widespread and entered textbooks. In
actual fact, however, this idea is wrong, and extrapolation to B = 0 yields
R.P. Seisyan and B.P. Zakharchenya
372
energies close to the positions of the ground (n0 = 1) state of the Wannier-Mott
exciton in cases where criterion (14) is met. In relatively wide-gap semi­
conductors, and when the criterion βη^ > 1 is satisfied, extrapolation to zero
intersects the region of excited states of the Wannier-Mott exciton (fig. 6).
Edwards and Lazazzera (1960) were the first to find that the Β = 0 point does not
coincide with sg.
It should be pointed out that, strictly speaking, the linear extrapolation
procedure is inapplicable here since the magnetic field dependence of the
positions of the maxima is nonlinear (smax(B) oc Bm, m φ 1). It is superlinear
(m > 1) for low Β because of quadratic terms and sublinear (m < 1) for large Β
and / as a result of the nonparabolicity which is usually present. The index m
may be different for different temperatures and because of the screening by free
carriers.
It is essential for understanding the nature of the spectrum that the longest
wavelength lines vary quadratically in relatively weak fields. This is due to the
diamagnetic shift of the exciton absorption which is substantial for the largeradius exciton:
2 *2
2
/
\3
^^ '~i^Ai)*B
mK
,58)
Obviously enough, the larger the shift, the bigger is the exciton radius a*xc(Aedia
n
oca*x2c) a d
the smaller the reduced exciton mass ( A e d i oa c ^ ~ 3) . The quadratic
shift is well described within the framework of perturbation theory up to β ^ 0.4.
For β ^ 1 the slope of the lines in the magnetic field dependence is already close
to linear which corresponds to the Landau sub-bands. Calculations carried out
using eq. (58) or other expressions that reflect more accurately features of the
0
α
δ
0
~6
7
Fig. 6. Schematic of the fan diagram, i.e., the dependence of the positions of magnetoabsorption
maxima on magnetic field: (a) for semiconductors with only the ground state of the Wannier-Mott
exciton series clearly pronounced; (b) for relatively wide-band semiconductors with a well-developed
exciton structure. Dashed lines show the same dependence for the transitions between Landau levels,
i.e., to free carrier states.
Diamagnetic exciton spectroscopy of semiconductors
373
real band structure yield satisfactory estimates of the reduced masses, while
extrapolation on a quadratic scale gives 'good' values of ε 8. The accuracy can be
improved by also taking into account the linear Zeeman term which dominates
for Β -> 0, although in semiconductors with medium values of ε 8 and narrow-gap
materials the Zeeman components are unresolvable.
In most pure semiconductors the ground state (n0 = 1) of the Wannier-Mott
exciton is clearly observed in the absorption edge spectra already at Β = 0. In
such cases the first O M A line is detected at any field. At the same time, analysis
of magneto-optical spectra suggests that this line should be assigned one of the
indices nlMv characterizing all spectral lines, otherwise a 'shortage' of lines will
be revealed in the classification. Apart from this, for crystals exhibiting exciton
maxima in the absence of magnetic fields it seemed appropriate to differentiate
the spectrum by separating in it the structure associated with 'pure' transitions
between the Landau sub-bands from the exciton structure. From the viewpoint
of the concept in question such differentiation has no sense. It should be pointed
out that some attempts to find an additional structure in the spectrum which
could be attributed to excitons irrespective of transitions between the magnetic
sub-bands were based on a misunderstanding, since the actual reason for this
structure was deformations in the thin sample cemented onto a substrate which
would contract or expand under cooling.
The unique pattern of the spectrum as a spectrum of diamagnetic excitons is
revealed in fig. 7 which shows schematically the formation of OMA. It becomes
clear that the long-wavelength line in the spectrum belongs to the diamagnetic
exciton state described by the lowest Landau numbers. Otherwise the line with
d
0
β j
0
Fig. 7. Schematic of the formation of an oscillatory magnetoabsorption spectrum for crystals with a
pronounced Wannier-Mott exciton ground-state maximum: (a) spectrum for Β = 0; (b) fan diagram;
(c) spectrum for β f> 1.
R.P. Seisyan and B.P. Zakharchenya
374
such quantum numbers should be behind the 'exciton' line which would be at
odds with the correct sequence of the spectral lines.
As follows from eq. (29), an essential feature of the diamagnetic exciton
spectrum is the possibility of observation of excited states with ν ^ 1. Johnson
(1966) was the first to observe an excited state near the first O M A line with an
electronic Landau quantum number Zc = 0. However, as already pointed out, the
assignment of the first line to the exciton is quite frequently fully obvious.
Therefore, detection of excited states near shorter wavelength lines with lc ^ 1
which are already superposed on the continuum from the preceding lines
appears to be of particular importance. Such excited states with ν = 1 were later
observed in crystals of germanium, indium arsenide, gallium antimonide, and,
finally, indium antimonide. In crystals with a wider energy gap one detected
series of diamagnetic excitons with ν > 1, for example, in GaAs or CdSe. Note,
however, that the assignment of a weak line to an excited state with ν ^ 1
requires careful experimental checking.
Varfolomeev et al. (1968), and Seisyan and Zakharchenya (1969) proposed a
technique to detect excited states making use of the possibilities inherent in
differential (modulation) magnetospectroscopy. They showed that large-radius
excitons possess a higher sensitivity to electric fields. When parallel electric and
magnetic fields are applied to the crystal, the signal in the differential spectrum
due to the excited state increases becoming comparable with that coming from
the ground state. This relative variation of intensities in the spectrum is typical
of bound states and can be used to advantage for the unambiguous assignment
of the line origin. The use of parallel fields comes from the need of avoiding
interference associated with the flareup of additional lines in the crossed-field
geometry. This technique was employed to detect excited states for spectral lines
with Zc = 1, 2. Note that the dependence on the dc component of the electric field
E0 passes through a maximum.
An increase in the signal due to the excited states was also detected in
magnetic-field-modulated spectra. This effect is connected with the steeper
dependence of the position of excited states on B, which for low fields is
proportional to the square of the radius of the state in question.
Electric fields, in both parallel and crossed geometries, result in a damping out
of the oscillatory structure, that is a smoothening of the spectrum. Such washing
out of the lines in transitions to free states could be accounted for by the
Franz-Keldysh effect. It was, however, shown by Zakharchenya et al. (1968)
that this damping occurs sometimes at electric fields more than an order of
magnitude lower than that required for the Franz-Keldysh washing out to
occur, which can be derived from the relation
Λ 0 Ρ Κ> Δ ε ί Γ,
(59)
2
1/3
is the characteristic Franz-Keldysh frequency, and
where 0 FK = [(eE) /2ph]
Αεη> is the separation between the neighboring maxima in the spectra. Experi-
Diamagnetic exciton spectroscopy of semiconductors
375
ment shows that the fields should be either such that
( 6° )
ft0FK>«de,
or sufficiently high to produce direct ionization of the exciton states:
\E\>ajeal.
(61)
Impact ionization of excitons occurring at fields such that
e\E\l*0>®dQ
(62)
can also be used. Here
is the mean-free path of the free carriers ionizing the
exciton. If the observation is carried out under conditions favoring a noticeable
concentration of free carriers that are not bound in excitons, the magnitude of ZJ
is given not by the mean-free path Z0 in the direction of the drift but rather by the
highest energy part of the mean-free path distribution function.
Apart from this, the sequence of damping corresponds to the magnitude of the
reduced longitudinal masses entering the binding energy ^ d e, and for the
damping to occur at small / ^ 3 it may require a higher field for transitions from
the light- than the heavy-hole states which is completely at odds with the ideas
concerning transitions to the free-carrier states.
Note a remarkable feature of the diamagnetic exciton spectra in that they
permit observation of the 'inversion' effect associated with the 'quasielectric'
field:
ξ=
1c ν χ Β = c—^—
Κ χ Β.
M
(63)
e xc
The line position in the spectrum depends on the orientation of magnetic field
with respect to the direction of observation through the Stark effect in the field ξ.
The shift Ainy is
Aiw = -£-p-(KxB)
(64)
where ρ is the exciton dipole moment vector. When an electric field is applied,
there will always exist a dipole moment ρ = χΕ, where χ is the exciton
polarizability. Then
^ n V = ^LE^(KxB).
cMe
(65)
xc
The effect depends on the scalar-vector product of E9 Κ ana Β in such a way that
the sign of z l i nv reverses as one of these vectors changes orientation by π
(inversion). Monozon et al. (970) put forward an interesting idea on the
possibility of observation of the effect in the differential spectrum. This
possibility is connected with the fact that here, in contrast to the general case of
absorption depending only on the modulus of £, the linear term with the first
R.P. Seisyan and B.P. Zakharchenya
376
derivative with respect to the field, (δα/5Ε)0Εί,
is also nonzero (El is the
amplitude of the ac component of the modulating electric field).
Such effects can be observed only in the spectrum of free electron-hole pairs.
Note also that the effect is particularly strong for excited states and, as shown by
Gross and Agekyan (1968), depends on the quantum number n0 as O C W Q .
The Coulomb interaction in the exciton can be effectively screened by free
charge carriers. It is necessary that
0?xc > r S C
r,
(66)
where r s cr is the screening radius for which one can take the Debye-Huckel
radius. Screening can be provided by fixed charges too, with the only require­
ment that
a:i*
l
>r
(67)
exc ^ ' ι
where rA D is the average separation between ionized acceptors or donors defined
a s f A, D* ( i V A
r 1 / .3
> D
Turning on the magnetic field suppresses these effects and under certain
conditions, as shown by Seisyan et al. (1968), exciton absorption increases
greatly in magnetic fields. This effect can be observed, for instance, in
germanium with the impurity concentration sufficiently high to damp exciton
absorption at Β = 0 due to screening. Application of a magnetic field not only
restores the discrete (line) structure of the O M A spectrum but also favors a
general increase of the average absorption level to the values typical of undoped
crystals. The same effect can be observed also in impurity-induced breakdown,
while the O M A spectrum of germanium at temperatures low enough to freeze
out the carriers is seen to consist of sufficiently narrow lines which disappear
after breakdown when carriers appear at a noticeable concentration. Interest­
ingly enough, flareup in CdTe was also observed in the case where the discrete
exciton structure is suppressed because of the presence of a high concentration
of ionized impurities.
In their interpretation of the flareup, D'yakonov et al. (1969) suggest that a
discrete state must exist in a one-dimensional Coulomb well appearing in the
presence of a strong [e.g., in the sense of eq. (14)] magnetic field. D'yakonov et al.
(1969) obtained a one-dimensional potential VlM(z) by averaging with radial
functions a screened Coulomb potential of the type
(68)
and calculated the energy and length of the screened diamagnetic exciton
(69)
The results should be considered separately for four regions on the screening
Diamagnetic exciton spectroscopy of semiconductors
377
radius scale r s cr (fig. 8). In region I where r s cr > a{B\ the screening for states with a
quantum number equal to or greater than ν can be neglected. In region II where
a(Bv^ υ > rSCT > a{B0) there are no excited states for any field but the ground state is
fixed. Finally, in region III where L < r s cr < a(B] the binding energy of the
diamagnetic exciton ground state decreases logarithmically, while in region IV
where r s cr < L it drops rapidly to zero as ( r s c /r L ) 4.
Thus in a high magnetic field the screening condition (66) already becomes
insufficient. Moreover, the more stringent condition a(B0) > r s cr still maintains a
bound character for the motion. At the same time region IV can seldom be
realized even in narrow-gap or doped semiconductors.
Lineshape analysis may yield valuable information on the exciton nature of
the spectrum. For large values of Ωτ ρ 1 the lineshape of transitions between
free-carrier states should approximate the shape of the density of states function
and exhibit characteristic sawtooth features with steep long-wavelength edges
and smooth short-wavelength fall-offs.
In actual fact all maxima in the O M A spectra are fairly symmetrical. In cases,
however, where asymmetry is observed its pattern is opposite to what should be
expected (fig. 9). This pattern is explained by invoking ideas concerning electricfield-assisted tunnelling from discrete states to the continuum, which is similar
to the Franz-Keldysh effect for free carriers. However, in this case it would be
more appropriate to use the theory of optical transitions in an electric field in the
presence of the Coulomb interaction between the electron and the hole
developed by Merkulov and Perel' (1974). This theory was modified to be
applicable to the case of diamagnetic excitons by Monozon et al. (1975) and
showed complete quantitative agreement with experiment.
In cases where the conditions of observation are close to optimal, that is in the
absence of fields perturbing the spectrum, the lines are perfectly symmetrical and
narrow. In certain experiments the halfwidth of some lines does not exceed
Fig. 8. Binding energy of different states of the diamagnetic exciton series as a function of screening
radius: curve 1 ground state; curves 2 and 3, first and second excited states. Regions: I, r s cr > a£t2);
II, aB < r s cr < 4 1 2 ;) III, L < r s cr < α Β; IV, r s cr < L (by D'yakonov et al. 1968).
378
R.P. Seisyan and B.P. Zakharchenya
a)
Fig. 9. Schematic diagram of the lineshape in oscillatory magnetoabsorption spectra for a simple
semiconductor: (a) reduced density of states in conduction and valence bands; (b) the corresponding
absorption spectrum for direct transitions between the Landau sub-bands in the case of Ωτ° v ^ 1; (c)
schematic experimental spectrum for Ωτ > 1, dashed curve: spectrum in the presence of electric field
initiating tunnelling transitions to exciton states.
0.1 meV (e.g., in experiments on GaAs). Thus, there can be no doubt that one
has here large Ωτ*ρ 1, where τ* is the characteristic relaxation time of the
process. In cases, however, where τ* is small and the maxima become broad, a
quantitative calculation of the spectrum for noninteracting carriers should yield
a shift of the maxima toward higher energies by ~0.58Λ/τ*. For small Ωτ* ^ 1
this could produce a noticeable short-wavelength shift of the lines. At the same
time the spectral lines are always shifted longward of the energies for transitions
between the Landau levels by the binding energy of the diamagnetic exciton.
There are many observations made under conditions where the requirement
Ωτ0 > 1 was not met (here τ 0 = ( τ " 1 + τ ν - )1 - 1 is the mean relaxation time of
electrons and holes). The optimum sample temperature for spectral studies not
always coincides with the temperature of the maximum mobility and, hence, of
the maximum mean relaxation time τ 0 . On the other hand, there are converse
Diamagnetic exciton spectroscopy of semiconductors
379
cases where the condition Ωτ0 > 1 is met whereas no O M A spectrum is observed
in sufficiently high fields when β > 1. These observations can be accounted for
by a different temperature dependence of the exciton lifetime r e x .c The condi­
tions of scattering and annihilation are connected with the electrical neutrality
of this quasiparticle. At the same time it is the condition QTCXC > 1 that is
essential for observation of the diamagnetic exciton spectrum. On the other
hand, it should be pointed out that a quantitative analysis of oscillatory absorp­
tion or the Faraday effect led Korovin and Kharitonov (1965) to a seemingly
paradoxical conclusion that, in a high magnetic field, scattering at the carrier
energies corresponding to the positions of the electron and hole Landau levels
is facilitated. Apart from this, with increasing Β the relaxation time tends to
values which are smaller by more than an order of magnitude than of the
conductivity without the magnetic field present.
Finally, straightforward arguments in favor of the exciton nature of the O M A
spectrum can be derived from the conventional present day procedure of
obtaining a consistent set of band parameters for a semiconductor by computer
fitting the theoretical spectrum to the experimental data. The sum of the squared
deviations of the experimental points from the theoretical values, just as the
standard deviation per point, turns out to be substantially smaller for the
theoretical spectrum corrected for the exciton binding energy. This becomes
particularly obvious when one fixes the origin of the spectrum at the position of
the exciton maximum for Β = 0, obtained from direct absorption measurements.
A typical fitting graph is shown in fig. 10. The spectrum calculated without
taking into account the exciton binding energies has its own set of band
\
1
2
P'
P.
7
Ρ
Fig. 10. Summed squares of the deviations, Σδ 2, of the calculated from the experimental spectrum as
a function of one of the fitting parameters P: curve 1 without taking into account the diamagnetic
exciton binding energy; curve 2 using calculated values of ^ d :e P' is a wrong and P0 is the correct
value of the parameter.
380
R.P. Seisyan and B.P. Zakharchenya
parameters. These parameters, however, are shifted with respect to the true
values and the fitting curve is smoother whereas the fitting including the binding
energy yields sharper minima, and the parameters obtained in this way agree
well with the data derived from the intraband magneto-optics, for example, from
cyclotron resonance experiments. At the same time the literature data on
cyclotron resonance and magnetoabsorption oscillations calculated without the
inclusion of Coulomb interaction disagree to a large extent. It is essential that
the sum of the squared deviations can be minimized also by varying the static
dielectric constant κ0 which enters only the binding energy ^ d e, and ^ d e = 0
corresponds to κ0 = oo. The value of κ0 obtained in this way for InSb agrees
sufficiently well with the values specified in the literature. The need in taking into
consideration the binding energy in band parameter calculations becomes
particularly obvious in calculations by the first two maxima involving the light
hole. Here the value of 3$de and its variations with η, Ζ and Β are particularly
large which makes an analysis disregarding the exciton totally fruitless.
4. Experimental
data on oscillatory
magnetoabsorption
spectra
Experimental and theoretical analyses of interband magneto-optical phenom­
ena in semiconductors give us grounds to believe that practically all experi­
mental data accumulated up to now, from the discovery of magnetoabsorption
oscillations to present day studies, should be considered within the framework
of the spectroscopy of diamagnetic excitons. This is clearly seen by considering
the results of magnetospectroscopic measurements carried out over a broad
range of semiconductors, from narrow- to broad-gap materials.
In analyzing the interband magneto-optical spectra of semiconductors as
exciton spectra, it is only natural to attempt to classify crystals by the original
exciton characteristics of the fundamental optical absorption edge in the
absence of a magnetic field. Three different cases are essential here. The first case
is typical of most moderate-gap semiconductors: one observes in their spectra at
Β = 0 only one discrete Wannier-Mott exciton maximum. In the second case,
characteristic of relatively broad-gap semiconductors, one can detect at Β = 0,
besides the ground, also excited states. The third case relates to crystals where
the Β = 0 exciton maximum is not observed for some reason. We will classify
these cases, respectively, as related with (1) a clearly pronounced ground state,
(2) with a well developed spectrum, and (3) with suppressed states of the
Wannier-Mott exciton.
Note that while being useful in the practical sense, this classification is to a
large measure conventional. Indeed, there are semiconductors which in earlier
times exhibited at Β = 0 only one, the ground state of the Wannier-Mott exciton
while later, when crystals of higher purity became available, it has become
possible to observe higher excited terms of the exciton series as well. Gallium
Diamagnetic exciton spectroscopy of semiconductors
381
arsenide, for instance, can be placed into this class. Sometimes at Β = 0 the
exciton is not observed at all, however, exciton absorption occurs already in
very weak magnetic fields. Until recently indium antimonide was considered to
belong to this category. Among materials with suppressed exciton states could
be categorized also sufficiently heavily doped germanium.
Thus the classification adopted here is seen to characterize a concrete sample
under concrete conditions of a spectroscopic study rather than the semi­
conductor material generally. Nevertheless the common character of the
magneto-optical spectra, of the conditions conducive to observation of the
effect, and of the problems arising in attempts at an adequate description of
the results obtained make this classification convenient and useful.
4.1. Diamagnetic excitons in crystals with a strongly pronounced
Mott exciton ground state, case studies
Wannier-
Crystals with allowed direct transitions and a clearly pronounced
Wannier-Mott exciton ground state turn out to be the most convenient subjects
for experimental checks of the major propositions concerning the diamagnetic
excitons. At Β = 0 one observes in them at the absorption edge one n0 = 1
maximum which crosses over smoothly to a continuum.
As the necessary and sufficient criterion of observation of the diamagnetic
excitons serves here the criterion of Elliott-Loudon namely, β$> 1. It is met
relatively easily in the cases of Ge, InSb, InAs and GaSb.
Germanium
Germanium is one of materials on which the phenomenon of oscillatory
absorption in magnetic fields was first observed (in 1957). Therefore, the O M A
effect in this material is among the most comprehensively studied. Macferlane et
al. (1958) were the first to reveal the exciton structure of the fundamental edge in
Ge in direct interband transitions Γ£
F f which is represented by one n0 = 1
maximum.
Figure 11 displays the shape of the spectrum at the absorption edge in direct
transitions in the coordinates rectifying the dependence for direct allowed
optical transitions from the valence band into the quasicontinuum of the
conduction band. Interestingly enough, exciton absorption manifests itself
sufficiently clearly already at room temperature while there is no pronounced
maximum.
The most comprehensive study of the O M A spectra in Ge for direct
transitions was carried out by Zwerdling et al. (1959a), and later by Seisyan et al.
(1968a). In the latter work attention was focused on removal of mechanical stress
in samples and on the proof of the exciton nature of the spectrum. Details of the
germanium band structure were established by Aggarwal (1970) in experiments
on piezoreflectance in magnetic field. Finally, Varfolomeev et al. (1977) per-
382
R.P. Seisyan and B.P. Zakharchenya
0.88 \
Fig. 11. Absorption edge of germanium for direct transitions. Curves 1,2, 3, experiments at different
, Τ = 77 Κ;
temperatures; curve 4, experiment on screening involving carrier injection with η > nSCT
curves 5, 6, theory, absorption and oscillatory magnetoabsorption {B = 3.5 T) for simple bands
neglecting the Coulomb interaction, only for 4.2 Κ (by Seisyan et al. 1968b).
formed an analysis of the magnetoabsorption spectrum in this crystal as a
spectrum of diamagnetic excitons including the determination of the contri­
bution coming from the light-hole nonparabolicity. In fig. 12 the spectra
obtained in the above work are compared with those calculated by the theory of
Gel'mont et al. (1977). Comparison of the experimental and theoretical 'noexciton' spectra reveals the interesting fact that a satisfactory agreement between
the two can be reached by shifting the theoretical spectrum toward lower
energies by an amount constituting a fairly large fraction of the spacing between
the maxima which exceeds by far the experimental error and is of the order of
the diamagnetic exciton binding energy 0tlM(B) for / > 1. Agreement is achieved
both in the positions of the first six or seven maxima and in their structure, the
first maximum included.
Note that to bring into agreement the experiment performed at 7.9 Τ and the
theory in the 4no-exciton' approximation, Zwerdling et al. (1959a) contracted the
experimental spectrum by about 1% and in this way produced a substantial
shift (by 1 0 - 1 eV) towards shorter wavelengths.
In the absence of appreciable mechanical stress, experimental O M A spectra
agree quite well in structure with the theoretical spectrum for transitions
between the Landau sub-bands drawn by taking into account the specific
features of the germanium valence band structure and shifted by the binding
energy of the diamagnetic exciton. All absorption maxima (i.e., minima in the
relative transmission curves, fig. 12) correspond to transitions to the v = 0
diamagnetic exciton states involving the Landau sub-bands. Note that there are
Diamagnetic exciton spectroscopy of semiconductors
383
loe£te¥
0.90
0.»
0.%
0.96
0.98
toe 6,eV
400
AO
10
80
10
090
£5"
Tt.
t
0.90
09e
0.9*
Ιτ Τ
0.9Ί
"ax
Jr.
196
We
ϊ .
Too
ίτ
0.98
t
too
ToTeJv
ί
, It
<a?£eK
Fig. 12. Oscillatory magnetoabsorption of germanium as a spectrum of diamagnetic excitons at
Τ = 4.2 Κ, Faraday configuration; (a) σ~ spectrum, (b) σ+ spectrum. The theoretical spectra include
the diamagnetic exciton binding energy corrected for the electron and light-hole nonparabolicity.
Transitions involved: I, a + - > a c; II, b + - > b c; III, a - - > a c; IV, b~ - • b 0. The maxima corresponding to
diamagnetic exciton excited states are specified by arrows (by Varfolomeev et al. 1977).
no maxima corresponding to transitions directly between the Landau subbands. Thus the positions of the sub-bands become revealed only through the
corresponding exciton states.
Figure 13 displays the most typical dependences of the position of the
absorption maxima e m ax on magnetic field. At all temperatures most of the
maxima reveal a trend to converge to the exciton peak at the absorption edge
observed at low temperatures in the absence of a magnetic field. These
dependences are best fitted by the expression e m ax = ε 0 + ocBm where α and m turn
out to be functions of the serial peak number and ε 0 ~ ε(6Χ!..
The same relationships are presented in fig. 14 on a logarithmic scale in the
form of Δε = ε ^ — ε 6Χ|. plotted against Β. For the Ν = \ peak at 4.2 Κ this
relationship has a slope corresponding to m = 2 which is characteristic of the
diamagnetic shift of the Wannier-Mott exciton. As the serial number Ν of the
maximum increases, m decreases rapidly becoming of the order of unity. Its drop
384
R.P. Seisyan and B.P. Zakharchenya
Η
0
ι
ι
ι
,
ι
ι
ι
ι
0.5 (0 4.5 2.0 8.5 3.0 3.5
Fig. 13. Positions of the oscillatory magnetoabsorption π spectrum maxima in Ge plotted against
magnetic field. Solid lines: e m xa = ε 0 + OLB"1 where α and m are fitting parameters. Points: experiment
(by Seisyan et al. 1968a).
*€,mtV
•
-Z93K
77/f
Δ - k2K
χ
_l
I I
-
I I I
Q5Q6Q8
{0
{5
2.0 i.53.035
*/T
Fig. 14. Magnetic field dependence of oscillatory magnetoabsorption maxima on a log scale, with
the maxima numbered as in fig. 13 (by Seisyan et al. 1968a).
Diamagnetic exciton spectroscopy of semiconductors
385
down to m < 1 for large /, due to the nonparabolic effects, is apparently the
reason behind the overestimated values ε 0 > e e xc which quite frequently are
erroneously interpreted as ε 0 = ε 8 , just as m > 1 leads to ε 0 < sexc if ε 0 was
obtained by linear extrapolation.
A distinctive feature of Ge lies in the existence of indirect transitions to
exciton states which occur at photon energies lower than ε ° χ .ο As is well known,
these transitions do not produce maxima and form only absorption bands, their
positions being determined as eexc = s'g — R*' ± sph where e'g is the energy gap for
indirect transitions, /?*' is the corresponding binding energy, and ε ρ 11 is the
energy of the phonons participating in the transitions. For indirect allowed
transitions the exciton band in the absence of a magnetic field can be written as
α oc (hv — e e x ) c1 / . 2 The maxima permitting determination of the positions of the
corresponding bands can be observed in differential spectra.
Button et al. (1959) and Lax et al. (1960) were the first to perform interband
absorption studies for indirect transitions in germanium. Such transitions are
observed to occur in the interval between ε 8 , that is the energy gap for indirect
transitions from the top of the valence band to the absolute minimum of the
conduction band, L 6 , and the energy gap for direct transitions, ε 8 (corrected for
the magnetic shift of the bands). Indirect transitions in a magnetic field involve
the absorption or emission of phonons which results in a number of specific
spectral features. The first of them consists in that one observes absorption steps
rather than maxima, and the second, that the selection rules in the Landau
quantum number I are listed. Finally, because of the many-valley structure of the
conduction band at the L point the electron cyclotronfrequency for the < 111 >
and <110> directions has two values at each and is determined in terms of the
effective mass tensor components
and the angular coefficients.
Just as in the B = 0 case, clearly pronounced features in the spectrum of
indirect interband magneto-optical transitions can be observed in the differen­
tial spectra. Indeed, sharp maxima were seen by Aggarwal et al. (1969) in
piezomagnetoabsorption spectra (fig. 15), and by Aronov et al. (1971) - in
electric-field-modulated magnetoabsorption spectra and in Λ,-modulated spec­
tra. Extrapolation of the energy dependence of the effective reduced mass to
s'c = 0 yields a value of m? L(0) which differs somewhat from the most reliable
value obtained in cyclotron resonance experiments. An essential feature of these
data is their deviation from linearity at low energies. Aggarwal et al. (1969)
believe these inconsistencies to result from the unaccounted for exciton nature of
the phenomenon.
Taking into account the intensity ratio for the ground and first excited states,
Varfolomeev et al. (1968,1977) chose for their experiments, which were aimed at
detecting a fine structure in the spectrum associated with the excited states of the
diamagnetic exciton for Zc = 0, relatively thick crystals of d= 10-15 μιη. Such
crystals could provide a m a dx = 1 to 2 for transitions to the ν = 1 states. The
structure of the magnetoabsorption in the region of the first maxima in the
R.P. Seisyan and B.P. Zakharchenya
386
•4
~~
Ge,
Τ = 30 Κ
3
5'8.89T^
n
/|
/1
e
/ ι
/ ι
ί\/\
- 4
0
\\ Ι
Ί
y
~~~7Ί
_ Ι
160
L I
*
. . If. J ,
800
Ιν
ι,
Ι I, . '
840
ν
.
I
B.meV
ιι
IU
880
Fig. 15. Indirect interband magneto-optical transitions in the spectra of differential piezoabsorption
of germanium for Β || [111] (by Aggarwal et al. 1969).
Fig. 16. Excited states in the π-spectra of the diamagnetic exciton in germanium: (a) transmission;
(b) differential absorption; (c) theoretical spectrum of transitions between Landau sub-bands
displaced for matching (by Varfolomeev et al. 1968).
Diamagnetic exciton spectroscopy of semiconductors
387
π-spectrum obtained on stress-free germanium crystals at 4.2 Κ is presented in
fig. 16.
Observation of the fine structure in several short-wavelength maxima with
15* 1 in germanium was made possible by the use of the electroabsorption
technique. The choice of the parallel orientation of the electric and magnetic
fields was determined by the specific features of the spectrum for the ground
states (v = 0) for Ε parallel to B. In such a geometry, one has a simple
superposition of very weak effects of the electric field and of the effects of the
magnetic field. At the same time the effect of the £-field on the excited states is
substantial, namely, the odd states corresponding to the quantum defect 5v u
flare up, and the line pairs v ug belonging to the same ν undergo a linear Stark
effect: Αε(Ε) =
3ϋ2Εκ0ν2/μβ.
Figure 17 presents an electroabsorption curve for germanium obtained in the
region where the first five main transitions in magnetic field are observed. The
minima in the electroabsorption curve labelled 1 through 5 originate obviously
from transitions to the corresponding exciton ground states and coincide with
the main minima in fig. 16. In addition to these minima, one clearly sees minima
1", 2", 3" whose positions in the spectrum are very close to the energies
corresponding to the transitions directly between the Landau sublevels. The
relative intensity of the additional minima grows with increasing magnetic field.
As the dc electric field E0 increases, one initially also observes a fast growth in
the intensity of these minima. However, starting from certain electric fields
the intensity stops growing and starts to decrease (see, e.g., fig. 18). At some
900
910
900
910
6 . m*V—~
Fig. 17. Differential electroabsorption spectra in germanium near the first absorption maxima for
/ c = 0 and 1 in the presence of a magnetic field parallel to Ε for different values of the electric field
d.c. component: Ex = 1 5 0 V c m - 1, 8 0 0 H z , 3.4T, e\\B (by Seisyan and Zakharchenya 1969).
388
R.P. Seisyan and B.P. Zakharchenya
160
HO
60
40
0
0.2
OA E^KV/cm
Fig. 18. Electroabsorption amplitude in germanium near the extremum coinciding with the exciton
absorption line plotted against the d.c. component of electric field (by Seisyan 1984).
values of the electric field the additional minima disappear altogether, appar­
ently as a result of ionization of the corresponding exciton states. These fields
assume different values for different minima at £ = 3.4T,
= (3.5-5.0)
χ ΙΟ 2 V c m " x , and decrease with increasing serial number. The observation of
excited states 2" (3") argues for the exciton nature of the oscillatory magneto­
absorption maxima connected with higher Landau quantum numbers.
Figure 19 demonstrates the magnetic field dependence of the position of the
maxima 1-3 and 1", 2". The straight line e m a( x£ ) for 1" intersects the vertical axis
Fig. 19. Positions of differential electroabsorption π-spectrum extrema of germanium plotted
against magnetic field. Lines labelled L: theory for transitions to free carrier states involving Landau
sub-bands, with the numbers corresponding to notation of fig. 18 (by Seisyan 1984).
Diamagnetic exciton spectroscopy of semiconductors
389
at a point lying very close to the position of e g(0). This agrees with theoretical
estimates of the behavior of the excited states and offers a possibility to evaluate
the exciton binding energy and its dependence on the magnetic field.
Gallium antimonide
Oscillatory magnetoabsorption in GaSb was first observed by Zwerdling et al.
(1959b), who interpreted it as a result of optical transitions between Landau subbands in the valence and conduction bands. The data obtained on the band
parameters were later improved by Halpern (1965) who employed higher
magnetic fields. Interband magnetoabsorption in GaSb was subsequently
studied by the differential technique of electroreflectance by Pidgeon et al.
(1967) and, finally, detailed differential piezoreflectance spectra in magnetic
fields were obtained for this crystal by Reine et al. (1972) by the piezoreflectance
technique. In all these studies the exciton nature of the oscillatory magneto­
absorption spectrum was practically not taken into account at all. Just as in the
case of germanium, neglect of the Coulomb interaction may partially be justified
by the fact that the ground state of the diamagnetic exciton forming the
spectrum follows to within the binding energy the positions of the energies for
transitions between the corresponding Landau bands, the magnitude of the
binding energy being usually very small compared with the energy gap width.
The scale of the inconsistency originating from such an approach is not clear.
This provided a motivation for Varfolomeev et al. (1976) performing a study on
GaSb crystals to carry out a detailed comparison of the approach to data
treatment based on assuming the effect to be due to optical transitions between
the Landau sub-bands and the approach taking into account the exciton nature
of the spectrum.
A study of the fundamental absorption edge without a magnetic field reveals
the presence of a maximum lying at ε = 0.8102 eV. This maximum was first
observed by Johnson and Fan (1965) and corresponds to absorption by free
exitons. The calculated binding energy of the Wannier-Mott exciton is 1.6 meV.
Studies of the magnetic field dependence of the O M A maxima show that the
experimental values can be made to agree with the theoretical ones for the 'noexciton' model only for high Β and large /. It should be stressed, however, that
such an 'agreement' can be reached only by using £ e x (0)
in place of ε 8 and,
c
accordingly, by shifting the theoretical spectrum towards lower energies.
Correct construction with the line converging to ε 8 reveals, however, an obvious
inconsistency of the spectra for low Β and Z. Apart from this, for the lowest levels
one observes a quadratic behavior of e m a(2?)
characteristic of the Wannier-Mott
x
exciton diamagnetic shift. The inclusion of the Coulomb interaction raises the
energies of the possible transitions between the Landau sub-bands by the
amount equal to the diamagnetic exciton binding energy. The theory developed
by Gel'mont et al. (1977) offers a possibility of calculating the corresponding
energies which should subsequently be subtracted from the calculated positions
390
R.P. Seisyan and B.P. Zakharchenya
of the transitions between the Landau sub-bands. The theoretical and experi­
mental spectra can now be matched without invoking an arbitrary shift which is
frequently introduced by an artificial choice of e g . The band parameter set thus
obtained differs appreciably from the one calculated neglecting the exciton
effects while yielding nearly coinciding values of the effective masses in the
quasiclassical limit /, n$> 1. These band parameters also coincide exactly with
the data obtained from cyclotron resonance studies.
A careful study of the minimum between the first and second maxima reveals
the existence of another maximum which cannot be accounted for by transitions
between the Landau sub-bands. This maximum appears only in the σ~ spec­
trum, its position depending linearly on magnetic field. Its intensity falls off
with decreasing Β and at Β = 2 Τ the maximum becomes barely distinguishable
(fig. 20). Extrapolation of the positions of this maximum to Β = 0 yields for the
main transitions values of ε'(0) greater than sexc(0) but less than ε°. The above
observations are similar to those made earlier for Ge and give grounds to
assume that the maximum belongs to the first excited state of the diamagnetic
exciton formed by the electron and light hole, b + ( —1). The conditions under
Fig. 20. Fine structure of oscillatory magnetoabsorption σ~ -spectrum of GaSb at 4.2 K. Transitions
involved: 1, a +( - l ) a c ( 0 ) ; 2, b +( - l ) b c ( 0 ) ; 3, a +( 0 ) a c( l ) ; 2', diamagnetic exciton excited state (by
Gel'mont et al. 1977).
Diamagnetic exciton spectroscopy of semiconductors
391
which such states can be observed are easily met for light holes and are not
reached for heavy ones. This accounts for the existence of an excited state in the
σ~ spectrum where transitions from light-hole states dominate, and their
absence in the σ+ spectrum dominated by heavy holes. The magnitude of the
binding energy calculated for transitions to the first excited state correlates with
experimental values.
Indium arsenide
Optical phenomena near the absorption edge of InAs have enjoyed considered
attention. Despite an extensive study of magneto-optical phenomena including
O M A (Zwerdling et al. 1957b, Pidgeon et al. 1967), no sufficiently accurate data
on the width of the energy gap in this material existed for a long time.
Varfolomeev et al. (1975) studied optical absorption of thin single-crystal
plates of InAs at 4.2 K. The samples of n-InAs were 6 - 8 μιη thick and were
prepared by a technique excluding the formation of appreciable mechanical
stress. The observed shape of the fundamental absorption edge in InAs is
presented in fig. 21. One clearly sees a maximum at ε 0 = 0.4163 eV similar to
those detected earlier in semiconductor crystals with a broader gap. The
maximum can be assigned to absorption involving ground states of the
Wannier-Mott exciton. Evaluation of the free exciton binding energy in
InAs yields K* = 1.7meV, whence for the energy gap at 4.2 Κ we obtain
e g = 0.4180 eV.
By placing InAs crystals exhibiting discrete exciton structures into a super­
conducting solenoid, Varfolomeev et al. (1977) succeeded in observing in
04/0
OkZQ 0.430
a)
0M8
0M0
8)
Fig. 21. Absorption edge of InAs: (a) wide gap, the maximum corresponds to a Wannier-Mott
exciton n0 = 1 ground state; (b) narrow spectral slit, the step corresponds to exciton-impurity
complex absorption (by Varfolomeev et al. 1975).
R.P. Seisyan and B.P. Zakharchenya
392
relatively weak magnetic fields absorption edge oscillations in magnetic fields
and, after increasing the field, in also detecting a fine structure in the spectrum
which, just as in the case of Ge and GaSb, can apparently be attributed to the
first excited state of the diamagnetic exciton.
By subtracting the calculated binding energies of the diamagnetic exciton (of
the ground and excited states) from the spectrum of the transitions connecting
the Landau sub-bands one can derive the theoretical positions of the maxima for
the diamagnetic exciton spectrum in indium arsenide. Figure 22 presents a
spectrum of oscillatory magnetoabsorption in InAs near the first absorption
maxima and its assignment.
An exact calculation of the corresponding longitudinal effective hole masses
m& and mfh in the quantum limit shows that the longitudinal light-hole masses
for Zv+ = — 1 are substantially heavier than the heavy masses for k = 0 and Z~ = 1,
becoming comparable only at the side maximum of the b~ series for ( m a .x It thus
follows that the true value of β under the conditions of this experiment is small
both for the heavy holes for l~ = 1, and for the light ones for Zv+ = — 1, a more
accurate calculation of the binding energy by the variational technique being
required in both cases.
The results of a variational calculation carried out for the transitions a +
( — l)b c(0), a +( 0 ) b c ( l ) and also, approximately, for b ~ ( l ) a c( 0 ) , are shown in
fig. 22 as dashed lines superposed on the theoretical spectrum. The new binding
energies obtained for transitions 1 and 2 improve substantially the agreement of
the theory with the experiment. As for the correction to the second light-hole
transition it is insignificant, which is in line with the dramatic decrease of the
I(S)ll(0)
e
T~42K
Fig. 22. Fine structure of diamagnetic excitons in the long-wavelength part of the π-spectrum of
InAs at Β = 2.4 Τ, Τ = 4.2 Κ. The corresponding diamagnetic exciton series are shown schematically
under the figure. Pointing downwards are the ground (v = 0) and first excited ( v = 1) states, and
upwards (dashed), the positions of ionization limits (v = oo) or transitions between Landau subbands. Dashed lines in theoretical spectra: positions of the ground states calculated by variational
technique (by Varfolomeev et al. 1977).
Diamagnetic exciton spectroscopy of semiconductors
393
longitudinal light-hole mass already found for / = 0 and with its approach to the
quasiclassical mass limit.
A study of oscillatory magnetoabsorption spectra at Τ % 2 Κ and Β % 8.0 Τ
when β > 1 was carried out by Kanskaya et al. (1983). The σ+ and σ~ spectra
obtained for two sample orientations, Β parallel to [111] and Β parallel to [100],
are presented in fig. 23. Shown below the experimental spectrum is a theoretical
spectrum calculated under the assumption that the absorption maxima corre­
spond to transitions to the ground state of the diamagnetic exciton.
Computer fitting of the spectra taking into account the exciton nature of the
effect yielded a set of InAs band parameters which exceed the earlier values in
accuracy. These energy parameters agree unexpectedly well with the data of
Pidgeon et al. (1970) who did not take into consideration the exciton nature of
if
i
\
M . \ \ n
i Λ if .1 ttjjt
C
)
β
Ii
HO0]
Fig. 23. Diamagnetic exciton spectra in (a, b) σ+ and (c,d) σ~ polarization in InAs. Transitions
involved: I, a + ( / ) a c( / + 1); II, b + ( / ) b c( / ± 1); III, a " ( / ) a c( / ± 1); IV, b ~ ( / ) b c( / ± 1) (by Kanskaya et
al. 1983).
394
R.P. Seisyan and B.P. Zakharchenya
OMA. This agreement is apparently due to the choice of a wrong value for the
energy gap in InAs (0.410 eV) which cancels fairly accurately the difference
between the theoretical and experimental spectra corresponding to the 'mean'
values of the binding energies of diamagnetic excitons under the conditions of
this experiment. As a result of this, however, one apparently had to assume a
smaller value of the energy parameters Ep in order to obtain a correct electron
mass and g-factor.
It is of interest to consider the fine structure of the spectrum in fig. 22 formed
by weak maxima and steps A, Β and D. The maxima A, Β and D lie in the
immediate vicinity of the calculated excited-state energies of the diamagnetic
exciton for the maxima 1, 2 and 4 connected with the ground states of the
diamagnetic exciton, that is for b " ( l ) a c( 0 ) , a + ( - l ) b c ( 0 ) and a + ( 0 ) b c ( - l ) ,
respectively. The theory of Gel'mont et al. (1977) gives more accurate binding
energies for the excited than for the ground states thus providing a possibility for
a reliable determination of the true values of the energy corresponding to the
ionization limits of the diamagnetic exciton, or to transitions between the
Landau sub-bands. The fine structure of the spectrum in InAs corresponding to
the excited states of the diamagnetic exciton includes Zc = 1 transitions and is
certainly one of the 'most representative' among the ones observed up to date;
also, it appears in a relatively weak field.
Indium phosphide*
The InP absorption edge exciton structure was first detected by Turner et al.
(1964). Only one maximum belonging to the ground state of the Wannier-Mott
exciton was observed.
Abdullaev et al. (1973) were apparently the first to observe O M A in this
material. A magnetic field of 3 Τ resulted in the appearance in the π-spectrum of
not less than seven practically equidistant maxima for hv > ε°. A number of
high-field experiments on InP were carried out by Emlin et al. (1974). These
authors applied pulsed magnetic fields of up to 20 Τ with oscillographic signal
recording. For each magnetic field pulse the dependence of absorption on Β was
photographed on the oscillographic tube for a fixed wavelength, and an emax(B)
plot was constructed (fig. 24). Unfortunately, the analysis of the results obtained
on InP made use of simplified concepts concerning the diamagnetic exciton
which do not take into account the valence band degeneracy. An accurate
determination of the parameters of the energy spectrum in InP in a magnetic
field requires the calculation of the longitudinal hole masses and of the
diamagnetic exciton binding energies. Nevertheless even such a simplified
approach permitted one to reach a quite satisfactory agreement between the
experimental data and calculations. An experiment on crystals with the well
developed structure of the Wannier-Mott exciton, which is made possible by the
present day quality of epitaxial layers of InP, appears promising.
* See note added in proof
Diamagnetic exciton spectroscopy of semiconductors
395
Fig. 24. Positions of O M A maxima in InP plotted against magnetic field. Faraday configuration,
77 Κ (by Emlin et al. 1974).
Indium antimonide
The discrete structure belonging to the n 0 = 1 ground state of the
Wannier-Mott exciton in InSb was recently observed for the first time at 2 Κ by
Kanskaya et al. (1979).
An analysis of the conditions under which excitons should be observed at
Β = 0 predetermined the choice for this experimental study of p-InSb with a hole
concentration p = 6x 1 0 1 2c m " 3 and of n-InSb with n = 6 χ 1 0 1 3c m " 3 . The
study was carried out on samples 6-13 μιη thick prepared by polishing and
subsequent chemical etching. The samples were placed directly in liquid helium,
free of the substrate.
The absorption maximum which was observed in diamond-like semicon­
ductors in previous studies and can be assigned to the n0 = 1 state of the
Wannier-Mott exciton lies in InSb at e e xc = 0.2363 eV (fig. 25). Using the
calculated binding energy R* = 0.00050 eV we obtain a fairly accurate value of
sg for InSb: ε 8(2 K) = 0.2368 eV.
By placing InSb samples which at Β = 0 exhibit the Wannier-Mott exciton
into a strong magnetic field one obtains the possibility of observing a unique
pattern of oscillations. The spectrum contains a number of features, either
completely new or observed earlier but particularly clearly pronounced in
samples of this type (fig. 26).
One can write a general expression for the diamagnetic exciton binding
energy in diamond-like semiconductors
@£Μν
= (β4μλ1„/2ίι2κ2)δ;2.
(70)
Here, the electron and hole masses appear twice, namely, in an explicit form in
the reduced electron and hole mass, μλ1η (for series λ and sub-band numbers η
R.P. Seisyan and B.P. Zakharchenya
396
Fig. 25. Discrete structure of the Wannier-Mott exciton at the absorption edge of indium
antimonide: T= 2 Κ, η = 6 χ 1 0 13 c m " 3, d = 1 μπι (by Kanskaya et al. 1979).
and /), and implicitly through the quantum number <5V = ν + δ ν 1 . Indeed, the
number δν with the quantum defect δ v± is one of the roots of the equation
where
= afJL2 depending on μλ1η is a parameter: αλ1η =
h2K0/e2pxln.
Already in studies of the diamagnetic exciton spectra in germanium the need
in taking into account the nonparabolicity of the electron and light hole in the
calculation of μλ1η became recognized. The corrected ^nclv(B)
data yielded better
agreement with experimental spectra. In such narrow-gap semiconductors as
InSb the variation of the electron and light-hole masses with energy is so large
that the inclusion of nonparabolicity effects is necessary also in the calculation of
δυ.
Using the values of the electron and light-hole masses for the conduction
band bottom and the top of the valence band obtained within the framework of
Kane's model one can readily derive the variation of the inverse longitudinal
masses caused by nonparabolicity: Δ [ Η Ι / Η Ι * ( £ , / ) ] , A[m/m^(B, η)-]. (Here the
indices ± correspond to the two series of the electron and light-hole sub-bands.)
N o w both in the expression for &*&1ν(Β) and in the equation for the modified
quantum numbers δν one can take into account the corrected reduced
longitudinal mass
m/mMn(B)
= m/m°Un + Δ[*ι/ιιι±(Β, /)] + A[m/m&(B, w)].
(71)
Studying the behavior of InSb samples which reveal the Wannier-Mott
exciton at Β = 0 demonstrates a very interesting structure of oscillations. Figure
27 displays the first two groups of absorption maxima in the σ~ spectrum
obtained in a magnetic field of 4 T. The sharp peaks assigned to the ground state
of the diamagnetic exciton (v = 0) are clearly separated from several lines
Diamagnetic exciton spectroscopy of semiconductors
•rV
InSe,
1
B = 7.7<iT, G-,5\\i({Q
J_t-L
Li
« It •
6-,
*Tt
T.T
t
o-M
\
. ft
τΤ,τ
5 " , BIlUOO]
ίτ.
is, η
ISJO
.fr
• frf.
.trf
•
ο
L_L
• ί •
.irf.
*-r
enLuol
I)~f4
Λ
397
• Ttf.
• tJ.
.tU.
Τΐ *
.ΤΪ
Λ
Fig. 26. Diamagnetic exciton spectra in InSb for Faraday configuration for principal orientations of
crystallographic axes relative to magnetic field. Transitions involved: I, a + (/)a c(/ + 1); II, b + ( / ) b c
( / ± 1); III, a ~ ( / ) a c( / ± 1); IV, b " ( / ) b c( / ± 1). The theoretical spectra include the binding energy of
the diamagnetic exciton ground state (v = 0). The maxima corresponding to diamagnetic exciton
excited states are specified by arrows. The additional transitions appearing in the β | | [ 1 1 1 ]
orientation are indicated by dashed arrows. T= 1.8 Κ (by Efros et al. 1982).
R.P. Seisyan and B.P. Zakharchenya
398
v= 0
ΙηΪΒ
T'-ZK
β=3.87Τ
β=0
n
0
ν"
V =σο
\r>FsL
/
1
\
j
1
J
k A -
J> vo-tg
11
236
—J_K
1
.1
238
\
sc tt—i
/
| χ -7/
—1
246
^—1
250
—1
25*1
Λ
1
^1 ί
—
258 €,meY
1
Fig. 27. Lineshape of the first oscillatory magnetoabsorption maxima in the σ ~ spectrum of InSb at
Τ = 2 Κ. Shown below are the theoretical positions of the maxima for the diamagnetic exciton series
originating from the transitions: I, a + ( - l ) a c( 0 ) ; II, b + ( - l ) b c( 0 ) (by Kanskaya et al. 1981).
making up a fine structure of the transitions involving the two upper light-hole
levels a + ( — l)a c(0) and b + ( — l)b c(0) (projections of the moment Μ = \, f ) and
the lower electron levels with Zc = 0. The theoretical positions of the lines of the
diamagnetic exciton series specified on the energy axis permit one to assign to
the ν = 1 excited state of the diamagnetic exciton only the weak peak superposed
on the steep long-wavelength side of the Q-maximum whose peak agrees well
with the ionization edge (v = oo). Thus, the assignment by Johnson (1968) of the
short-wavelength absorption band to the ν = 1 excited state is only partially
correct. Considerable absorption in the band beyond the diamagnetic exciton
ionization edge was revealed first by Kanskaya et al. (1982) and requires further
investigation. As already pointed out, the probability of bound-state formation
generally increases with increasing magnetic field whereas the available theo­
retical descriptions of diamagnetic excitons suggest a monotonic behavior of
spectral intensity as one crosses the ionization edge. Here, on the other hand, the
Q-band appearing near the ionization edge and extending shortward by an
amount exceeding the binding energy @tat is seen clearly already at fields Β > 3 Τ
and exhibits a tendency to grow rapidly in intensity as the magnetic field is
increased still further. The quantity β characterizing here the extent to which the
strong-field condition is met takes on values in InSb for Β > 3 Τ which
practically cannot be reached in other, broader gap semiconductors.
Detection of a maximum coinciding with transitions between the Landau subbands could offer the possibility of a precise experimental determination of the
diamagnetic exciton binding energy. Figure 28 presents energy intervals between
the ν = 0 ground-state line, the ν = 1 first excited state and the maximum of the
β-band in addition to the theoretical value of ^ } e For diamond-like semiconductors the Wannier-Mott exciton states represent
Diamagnetic exciton spectroscopy of semiconductors
399
Δ-Ι
4
3
<?
/
J
eo
I
40
I
so
L
β,γ
Fig. 28. Diamagnetic exciton binding energy in InSb plotted against magnetic field. Solid lines:
theory. Experimental points: intervals between Q-band maximum and the (v = 0) ground state of the
diamagnetic exciton involving: I, heavy holes of a~ and b~ series; II, b + light holes; III, a + light
holes. Dashed lines show the experimentally determined intervals from the ground to the ν = 1 first
excited state of the diamagnetic exciton and to the Ρ maximum [for a +( — l ) a c( 0 ) transitions (by
Kanskaya et al. 1981)].
a mixture of states formed by light and heavy holes. Therefore the excited-state
binding energy of the diamagnetic exciton involving in the case of fig. 27 the
longitudinal light-hole mass does not fully correspond for β > 1 to the threedimensional exciton binding energy
= 0.5 meV ( n 0 = 1). The use of the
longitudinal masses calculated by eqs (35) and (71) yields for the light holes a
limiting value which should show practically no variation because δ ν - > 0
already at fields Β > 2 Τ. Then from the positions of the excited-state maxima
one could determine fairly accurately the energy 0tat and compare it with
calculations based on the position of the Q-maximum. Figure 28 shows that the
position of the β-band maximum fits the theoretical ionization edge of the
exciton series well, and that the gap between the ground and first excited states is
comparable with the binding energy of the Wannier-Mott exciton, if its reduced
mass is made up of the electron mass and the longitudinal light-hole mass. Thus
the observed Q-band is apparently of a complex nature connected with the
quasi-one-dimensionality of the diamagnetic exciton while having a maximum
which practically coincides with the position of the exciton series ionization
edge.
A comprehensive study of the behavior of this band in a strong magnetic field
reveals nonmonotonic absorption at the smooth short-wavelength side of the
band. A spectrum of the band obtained in a strong magnetic field is presented in
fig. 29. One clearly sees nearly equidistant maxima, their spacing growing
proportionately to magnetic field. One may construct the dependence of the
positions of the maxima on magnetic field measuring the energy from the
diamagnetic exciton ground state. Such a dependence is displayed in fig. 29b.
Extrapolation of the maximum positions to Β = 0 yields for ε(0) a value of the
400
R.P. Seisyan and B.P. Zakharchenya
InSd
r=2K
ι
0.255
0.860
ι
0.265
ι
0.270
£,eV
Fig. 29. 'Microcyclotron' resonance in InSb. T = 2 K , B = 7.74Τ, σ~ spectrum, the ground state
(v = 0) is connected with a + ( - l ) a c( 0 ) transition. Inset: magnetic field dependence of the energy
intervals between the ground and ν = 1 excited state (1) and the corresponding 'microcyclotron'
resonance lines (2-5) (by Seisyan 1984).
order of 1 meV. A comparison of the energy spacings between the fine-structure
maxima and the theoretical values of the Landau-Luttinger hole level energies
in InSb shows the observed transitions to be close to the energy spacings in the
heavy-hole sub-bands; note also that by exciting transitions from level a~(l) to
the diamagnetic exciton states a ~ ( l ) a c( 0 ) we observe simultaneously, as it were,
a series of transitions within or between the sub-bands a~ and b ~ . The observed
effect, called 'microcyclotron' resonance, certainly deserves further study.
Another remarkable feature of the oscillatory magnetoabsorption spectra in
InSb consists in a clearly pronounced triplet structure of the first absorption
maximum in σ-polarization for Β parallel to [111] (fig. 30). Pidgeon and
Groves (1969) were the first to observe a similar structure when studying the
magnetoreflectance of InSb. The short-wavelength satellite was assigned to
additional transitions induced by inversion asymmetry ( Δ / = — 2, b~(2)a c(0)).
Such assignment permitted turning over to the determination in InSb of the
Dresselhaus constant c* which describes the changes in a crystal's band
structure originating from the absence of inversion symmetry. Since the position
of a line is governed by the magnetic sub-band spectrum and the selection rules,
the line intensity ratio in the triplet was chosen as a major experimentally
Diamagnetic exciton spectroscopy of semiconductors
401
Fig. 30. Fine structure of oscillatory magnetoabsorption spectrum of InSb near the first absorption
maxima of the σ+ spectrum (solid line) and σ~ spectrum (dashed line). Shown below is the
corresponding reflectance spectrum taken from Pidgeon and Groves (1969): Τ = 2 Κ, Β || [ 1 1 1 ] (by
Kanskaya et al. 1982b).
measured parameter in the determination of c*. The triplet structure of the
a " ( l ) a c( 0 ) transition was observed reliably in absorption by Kanskaya et al.
(1982) starting from fields as weak as 1-2 T. The data obtained by them suggest
that this assignment of the short-wavelength line, even when corrected for its
exciton nature, as well as the subsequent calculations of c*, apparently requires
revision. Note also that the reflectance spectra in the exciton region require
particularly careful treatment because of the superposition of effects in the 'dead'
exciton-free layer. In particular, the line intensity ratios here may vary
depending on surface condition and exciton binding energies for the actual
transitions: in this variation, they pass alternately through maximal and
minimal values and thus cannot be used in calculations without a complicated
preliminary treatment. Apart from this, as follows from the above analysis, in all
diamond-like semiconductors, including crystals with inversion symmetry, the
heavy-hole sub-band has a peculiar structure with maxima at kz Φ 0 in a highmagnetic field because of the interaction between the hole sub-bands. The
R.P. Seisyan and B.P. Zakharchenya
402
observed fine structure of the lines will possibly find an adequate explanation
after the interaction of the magnetic hole sub-bands and their behavior for kz -> 0
has been studied taking due account of the exciton nature of the states.
Another remarkable feature in the fine structure of the oscillatory magneto­
absorption spectrum in InSb is seen primarily in the σ+ spectrum for all crystal
orientations and corresponds to the earlier observed 'polaron anomaly' of
Larsen-Johnson. Here likewise there are observations which suggest that the
traditional interpretation of the effect should be reconsidered. First of all, a
study of the spectrum leads to arguments for the exciton nature of the lines in the
'polaron anomaly', so that one should assume the longitudinal optical (LO)
phonon to be coupled not only with the electron but with the exciton as a whole.
The corresponding lines extend far from the resonance absorption points, retain
appreciable intensity up to magnetic fields two times the critical limit, have a
complicated fine structure defying straightforward explanation, and are also
observed with higher Landau number transitions than is the case with the usual
ones corresponding to lc = 1 for transitions from the heavy-hole sub-bands
(fig. 31). Anomalies with typical pinning are seen to occur, although with a lower
intensity, also in transitions from the light-hole states.
2
4
6
BJ
2.
4
6
Bj
Fig. 31. Anomalies in the magnetoabsorption spectra of InSb originating from the interaction of
excitons and optical phonons. Β || [100], Τ = 2 Κ. (a) σ+ spectrum; (b) σ" spectrum (by Kanskaya et
al. 1982b). Transition notations: (a) 1, a"(l)a c(0); 2, b~(l)b c(0); 3, a~(2)a c(l); 4, a + (l)a c(0);
5, b~(2)b c(l); 6, a~(3)a c(2); 7, a"(l)a c(0) 4- LO; 8, b ~ ( l ) b c( 0 ) + LO; 9, a"(2)a c(l) + LO: (b) 1,
a + ( - l)a c(0); 2, b +( - 2 ) b c( 0 ) ; 3, a + (0)a c(l); 4, b + ( - l)b c(0) + LO.
Diamagnetic exciton spectroscopy of semiconductors
403
Efros et al. (1982) showed that on the whole oscillatory magnetoabsorption
spectra in InSb are formed almost exclusively by transitions to the diamagnetic
exciton states satisfying the main selection rules Δ/ = ± 1 . The assignments are
specified in fig. 26. In the experimental determination of InSb band parameters,
Kanskaya et al. (1982) used for the energy gap width ε 8(0) = 0.2368 eV, derived
from a direct observation of the discrete exciton structure at the absorption edge
of InSb in zero magnetic field. Neglecting the exciton effect which is equivalent
to using the dielectric constant κ0 = oo worsens substantially the sum of squared
deviations and affects the band parameter set. It also turns out that much
smaller changes of κ0 may also be essential. Kanskaya et al. (1982) studied the
dependence of the magnitude of κ0 obtained by fitting the theoretical spectrum
to experimental data on the dielectric constant. This dependence shown in fig. 32
reveals a fairly deep minimum at κ0 which agrees well with the accepted value of
κ0 = 17.9.
Taking into account the binding energy of the electron and hole not only
permits reaching a substantially better agreement between theory and experi­
ment (the standard deviation per maximum reducing by almost a factor two)
but also improves markedly the accuracy of the band parameters for this crystal.
Note that the value of electron mass obtained from the analysis, m* =
0.0139m, correlates well with the cyclotron resonance mass. It is also essential
that the 'no-exciton' analysis yields m* = 0.0145m which coincides with the value
derived by Pidgeon and Brown (1966) who did not include in the calculation the
Coulomb interaction in interband transitions.
to*
1.5 23.0 83.5
2k0
16
Π
35
65
\* \ /
60
\ λ ;/
55
50
/
55
54
36
37
33
V lh
53
13
15
Π
19
κ,χ0
0
0.8
OA 16 \fyrt4\
Fig. 32. Sums of squared deviations between experimental oscillatory magnetoabsorption spectrum
and theory for the diamagnetic excitons in InSb under variation of the energy band parameters £ p,
F> 7i,2,3>
1> Nt and dielectric constant κ0 obtained by computer fitting (by Kanskaya et al. 1982a).
404
R.P. Seisyan and B.P. Zakharchenya
4.2. Diamagnetic excitons in crystals with a well-developed
exciton spectrum
Wannier-Mott
The exciton series is seen to have a well-developed structure usually in
comparatively wide-gap semiconductors. The largest number of lines is ob­
served in the case of dipole-forbidden direct transitions occurring in such
crystals as C u 2 0 and S n 0 2 . A smaller number of the members of the series can
be detected in wide-gap hexagonal crystals of the type of CdS and CdSe with
simple bands and allowed direct transitions. Spectra with a developed structure
of the exciton series are observed also in crystals with a layered structure of
the type of GaSe. As for the direct-gap diamond-like A 3 B 5 crystals with a
degenerate valence band, here with relatively low binding energies one finds a
steep rise in absorption with increasing energy beyond the gap edge. This makes
observation of high excited states in the exciton series difficult. Among the A 3 B 5
compounds, gallium arsenide remains the only crystal where, as a result of using
fairly pure and perfect epitaxial layers, excited states in absorption have been
observed.
A characteristic feature of all these materials is that the Elliott-Loudon
criterion, β
1, is difficult or practically impossible to reach in them. Neverthe­
less, one observed here clearly pronounced oscillatory absorption edge spectra
suggesting that the criterion βη% > 1 is satisfied.
Gallium arsenide
The first most comprehensive study of the absorption edge in GaAs is due to
Sturge (1962) who observed at low temperatures only one ground-state
maximum (n0 = 1) of the free exciton. Figure 33 reproduces the dependence of
the absorption coefficient on photon energy obtained by Seisyan et al. (1972)
for thin epitaxial GaAs films. In addition to the first sharp n0 = 1 free-exciton
maximum, one clearly sees a weak peak associated with the n0 = 2 excited state.
At room temperature one observes only one clearly pronounced n0 = 1
maximum.
The investigation showed the positions of the n0 = 1 and n0 = 2 exciton
maxima to be unstable and to depend on sample thickness. Relieving the
samples from mechanical stress permitted one to stabilize the positions of the
exciton lines and to observe at liquid helium temperature a developed fine
structure of the absorption edge with a large number of maxima. A typical
absorption edge spectrum of such GaAs crystals studied by Seisyan and
Abdullaev (1973) is presented in the inset to fig. 33. The experimentally obtained
energy of the n0 = 2 state turns out to lie 0.2 meV closer to the ionization edge
than the theoretical value. A good agreement with theory was obtained in the
photoabsorption experiments of Varfolomeev et al. (1976) where the n0 = 2 line
shifts appreciably towards shorter wavelengths under laser illumination and the
n0 = 3 maximum becomes clearer. Since laser illumination straightens out the
Diamagnetic exciton spectroscopy of semiconductors
405
Fig. 33. Absorption edge of pure epitaxial GaAs layers (by Seisyan et al. 1972). Inset shows the fine
structure of the absorption edge of GaAs at 4.2 Κ (Seisyan and Abdullaev 1973).
bands near the surface, it is these positions of the n0 = 2, 3 lines that should be
accepted as the most credible ones.
Hobden (1965) was the first to succeed in an experimental observation of the
effect of magnetic fields on the absorption edge structure in GaAs. Later the
O M A spectrum of GaAs was investigated in considerable detail by Vrehen
(1968). The samples used in the experiments were of high-resistivity oxygendoped n-GaAs. The oscillations were observed in this case starting from Β « 4 Τ.
A comparison of this spectrum with the data of Seisyan et al. (1973) obtained on
stress-free, pure epitaxial GaAs (fig. 34) shows it to be much poorer in structure
despite the stronger magnetic fields used. Also, the absorption maxima observed
were considerably broader. The figure demonstrates how different can be the
O M A spectra of samples of the same semiconductor having a clearly pronoun­
ced ground state and a well-developed structure of the Wannier-Mott exciton.
The techniques of preparing this gallium arsenide crystal and the experi­
mental investigation employed in the latter case favored observation, at helium
temperatures and in relatively weak magnetic fields, of an extremely rich
spectrum of oscillatory magnetoabsorption in GaAs including 150-200 absorp­
tion maxima. It should also be stressed that many of the absorption peaks are
remarkably narrow, their halfwidth not exceeding (1-2) χ 1 0 " 4e V . The spec­
trum reveals a large number of lines which could be assigned to excited states of
the diamagnetic exciton. However, already a preliminary analysis shows that
this conjecture is insufficient to account for the complicated structure of this
spectrum and suggests that besides the Coulomb interaction the contribution of
other effects should be taken into consideration.
406
R.P. Seisyan and B.P. Zakharchenya
Fig. 34. Positions of the O M A π-spectrum maxima of GaAs plotted against magnetic field: (a)
stress-free epitaxial films; (b) high-resistivity material (by Vrehen 1968); dashed lines in (b) identify
the region of the dependence corresponding to (a). Transition notation: I, a b ~ ( / ) b a c( / + 1);
II, a +( / ) b c( / + 1) (by Seisyan 1984).
A substantial fraction of the additional π-spectrum lines which cannot be
directly associated with diamagnetic exciton states using the principal selection
rules can be assigned to the transitions flaring up in the near surface electric
fields Es 1 Β (Seisyan et al. 1973). The oscillatory magnetoabsorption spectra of
GaAs obtained in intermediate fields (β ^ 1) in the Faraday geometry (fig. 35)
have a remarkable sawtooth-shaped maxima dropping steeply towards higher
energies. In such an experimental geometry the electric fields created by the
charge of the surface states and the external magnetic field are parallel (Es || B).
This suggests that one observes here tunneling transitions to states of the
diamagnetic excitons created by the electric field Es (see Monozon et al. 1973).
Figure 36 presents an O M A spectrum obtained by Nikitin et al. (1982) in the
Faraday geometry and circular polarization for Β = 7.45 Τ. With the values
β % 3 reached in this experiment analytical calculations become possible while
Diamagnetic exciton spectroscopy of semiconductors
*05
^
GaflS
T^iiK
xl
407
G'
ο - //
3
ο-ΙΠ
°-IV
(
(SI (52
(53
154
(55
4.56
(67 (S3
(59
(60
(52 (53 (5k (55 (56 (51 (56 (59 (60 €,βΥ
15Ί (52
153 1.5Μ
(55
(56 (57 (56
(53 (60
(52 (53 (3<t (55 (56 {57 (58 (59 <60e,eV
Fig. 35. Oscillatory magnetoabsorption spectra of GaAs in intermediate fields, Β = 2.5 Τ, obtained
in Faraday configuration at 4.2 K. Landau sub-band transition notation on the theoretical spectra:
I, a~ - > a c; II, b~ - > b c; III, a + - > a c; IV, b + - > b c. The theoretical and experimental spectra are shifted
for matching (by Seisyan et al. 1973).
approximate. Using the binding energies calculated by the variational technique
leads to a better agreement for the first diamagnetic exciton maximum
associated with the heavy-hole level while analytical calculations yield over­
estimated values. At the same time the positions of the heavy-hole excited states
agree well with analytical calculations. Transitions from the upper light-hole
sub-band behave in a similar way.
Variational calculations of the binding energy for lc ^ 1 states meet with
formidable difficulties making necessary a search for approximate methods.
Such a method was proposed already by Vrehen (1968) and is based on
assuming the following correspondence
^ d e( B , O ) ^ ^ d e( 0 , / ) ,
(72)
where θ = B/(2l + 1). Whence, knowing the magnetic field dependence of the
diamagnetic exciton binding energy for Ζ = 0 and using (72), we can easily obtain
the same dependence for / ^ 1 by changing the field scale by a factor 21 + 1. In
R.P. Seisyan and B.P. Zakharchenya
408
Δ- /
X- //
ο-Iff
·-IV
(52
(.54
(36
(.56
'160
(62
(64
166
<0
1.52
(5M
(56
(66
1.60
(62
Fig. 36. Diamagnetic exciton spectra in GaAs. Theoretical spectra were drawn taking into account
the diamagnetic exciton binding energy, short vertical lines specify the additional transitions coming
from valence band warping; in some places positions of transitions between Landau sub-bands or
diamagnetic exciton ionization limits are plotted pointing downward from the energy scale. The
spectra match without any shift. Transition notation: I, a +- > a c ; II, b +- > b c ; III, a ~ - » a c; IV,
b " - > b c (by Nikitin et al. 1982).
the case of a semiconductor with a degenerate valence band, however, this
procedure cannot apparently be considered valid and requires improvement.
A simple analysis (see Nikitin et al. 1982) shows that an approximate estimate
of the energy for Ζ > 0 can be made with data for I = 0 by varying Β such that
θ = B/(np + |M| + 1) rather than θ = B/(2l + 1).
The theoretical spectra drawn using the binding energies obtained by Nikitin
et al. (1982) are placed below the experimental spectra in fig. 37. A comparison
shows a fair agreement considering the degree of approximation used which
shows, in the first place, the validity of the adiabatic approach over a wide range
of fields starting from β ^ 1.
A study of the dependence of the Wannier-Mott exciton line shift on magnetic
field reveals the quadratic behavior for β < 1 to become replaced by a linear
Diamagnetic exciton spectroscopy of semiconductors
409
Fig. 37. ^max
{B) plots for σ~ spectrum obtained in Faraday configuration for GaAs at 4.2 K. Square
brackets: assumed diamagnetic exciton series. Dashed lines represent theoretical transitions between
L-levels, a± and b± identify initial hole leaders. Dashed region identifies the ionizing limit or
transition to free-carrier states, the notation symbols specifying the series of holes involved in the
transition (by Seisyan 1984).
relationship for β > 1. At the same time a change in slope and an onset of line
intensity growth occur. The diamagnetic exciton series originating from the
transformation of the n0 = 2 and n0 = 3 excited states of the three-dimensional
exciton are shown in fig. 37 with brackets. Practically all major additional lines
in the O M A σ-spectra forming the fine structure in this experiment can be
explained within the concept of excited states of the diamagnetic exciton with
the Coulomb numbers ν = 1 and 2. Note that the ionization limit ν = oo cannot
be identified with any feature in the spectrum.
Cuprous oxide and tin dioxide
As is well known, exciton effects at the fundamental absorption edge in
semiconductors were first observed in cuprous oxide crystals (Gross and
Karryev 1951). The exciton spectrum of C u 2 0 still remains the richest and most
remarkable to this day which makes this material a popular subject for studies.
To the dipole-forbidden transitions of the longest wavelength 'yellow' series of
C u 2 0 correspond exciton states described by p-type wavefunctions. Accord­
ingly, the series starts from n0 = 2, the number of members in the series
depending on crystal quality. In the most perfect crystals grown by the
hydrothermal technique one clearly detects eight or nine members of the series.
The positions of such exciton maxima can be fitted well by a simple relation,
( 0 ) = 2.17244 - 0.0972no
£exc
2
eV,
(73a)
where n0 = 2 to 9.
The exciton spectrum at the absorption edge of S n 0 2 was first observed by
R.P. Seisyan and B.P. Zakharchenya
410
Nagasawa and Shionoya (1966). For light polarized along a four-fold axis one
detects in S n 0 2 an exciton line series obeying a hydrogen-like relationship
ee
xc
= 3.59568 - 0.0323rco 2 eV,
(73b)
where n0 = 2 to 6. In cases, however, where the electric vector Ε is parallel to the
c-axis, the absorption edge lies close to 3.8 eV with no line structure seen at all.
The exciton spectrum behaves similarly in C u 2 0 compressed along a fourthorder axis, where as the pressure is increased the 'yellow' series begins to
disappear in the direction perpendicular to the axis of compression. The
behavior of the exciton series in S n 0 2 which is typical of dipole-forbidden
transitions suggests an analogy to the optical behavior of the crystals, namely,
that natural anisotropic tetragonal S n 0 2 is similar to isotropic cubic C u 2 0
subjected to maximal compression.
Reaching the strong-field condition (β^> 1) in these crystals is a formidable
problem. Indeed, the critical field Ββ=ί for S n 0 2 is 70 T, and for C u 2 0 , 600 T.
Such fields have not yet been achieved under conditions permitting highprecision spectroscopic measurements. Nevertheless O M A is observed in these
crystals due apparently to the well-developed exciton structure of the spectrum
making it possible to meet the high-field condition βη% > 1 for members of the
exciton series with n0$> 1. Under these conditions the boundary between the
weak and strong magnetic field phenomena may shift so that the possibility of
obtaining oscillatory spectra depends not only on the magnetic field reached
in the experiment but also on the quality of the crystal as well, which is
characterized by the highest experimentally detected number n0max of the
Coulomb state in the Wannier-Mott exciton spectrum.
Gross et al. (1957) studied cuprous oxide crystals at 4.2 Κ in constant
magnetic fields up to 3.4 T. Later oscillatory magnetoabsorption in C u 2 0 was
investigated by Zakharchenya and Halpern (1967) in a constant magnetic field
£ m ax = 9.4T. Zhilich et al. (1969) present a theoretical interpretation of these
results. Later detailed oscillatory magnetoabsorption spectra were obtained by
Sasaki and Kuwabara (1973) who studied C u 2 0 in constant magnetic fields up
to 16 T, as well as by Agekyan et al. (1973) using pulsed fields of up to 18.5 Τ
(fig. 38). Finally, extremely interesting spectra of oscillatory magnetoabsorption
in C u 2 0 were recently obtained by Sokolov and Yakovlev (1986) using a new
technique consisting in obtaining microwave-modulated spectra of C u 2 0
luminescence excitation in a magnetic field (fig. 39).
Agekyan et al. (1971) were the first to observe oscillatory magnetoabsorption
in S n 0 2 . In this experiment, the wavevector of light and the optical axis of the
crystal were in a plane perpendicular to the vector Β directed along the secondorder axis a (the spectrum of oscillatory magnetoabsorption in this case is
polarized along B). The O M A spectrum was observed starting from 1 T, more
than 10 maxima being found at Β = 3.6 Τ in the continuum (fig. 40). In the case
Blc and q\\c (c is the fourth-order axis, q is the wavevector of light) one sees
Diamagnetic exciton spectroscopy of semiconductors
B,T
W.O
Ax
411
ι
Ι
Ι
ι
ι
.
,
Ι
ι .I,... .ι.
8.0
6.0
"lilmill nl WMffisisast
kO
_JkJ
20
I it!'!-
H6
i n
us
i(9
€teV
Fig. 38. Schematic diagram of the experimentally observed formation of the oscillatory magneto­
absorption σ~ spectrum in C u 20 . The oscillatory part of the spectrum consists of doublets
(connected with an envelope). Inclined straight lines connect maxima with presumably the same
numbers for different magnetic fields read off the vertical scale (by Seisyan 1984).
oscillations only in one polarization, namely, for B\\a the oscillations are
polarized with Ε IB, while with the crystal rotated about the c-axis by
the
oscillations reverse polarization ( £ Ί | / ? ) .
An analysis of the general selection rules for the case of magneto-optical
transitions in simple semiconductors with dipole-forbidden direct transitions
leads to the conclusion that only polarization transitions to states with Μ = ± 1
can have nonzero intensity:
eB/l+
1\1
/2
and the selection rules will be
A/ = / c - / v = ± l .
(75)
Here, the signs ' + ' and ' —' relate to the σ~ and σ+ polarizations, respectively.
As for spectra in π-polarization, only transitions to states with AM = 0 are
possible, and for the selection rules we have
A/ = / c - / v = 0.
(76)
The probability of these transitions, however, is proportional to k \ and is
therefore small; the π-spectra (E\\B) form steps a o c ^ ( b - % ) 1 /2 rather than
absorption maxima and have a relatively low intensity.
The situation becomes somewhat different when one attempts to take into
R.P. Seisyan and B.P. Zakharchenya
412
3.6
IS
A.O
A.2
4Λ
A.S
5J
Fig. 39. Microwave-modulated C u 2 0 luminescence excitation spectrum plotted against magnetic
field, Faraday configuration, Τ = 4.2 Κ. Arrows specify the assumed positions of diamagnetic
exciton lines, numbers adjoining the arrows identify the Landau quantum numbers, subscripts are
2v. (a) and (c), σ~ polarization; (b) and (d), σ+ polarization. Orientation: (a) and (b), c 31| B; (c) and
(d): cJB (Sokolov and Yakovlev 1986).
consideration the exciton nature of the effect in a consistent way. For an electron
and hole bound into a diamagnetic exciton the smooth stepwise behavior is
replaced by a delta function absorption maximum also in the π-spectrum:
α ocflv δ[Λω - εη + ^ d e( / , Μ, ν " ) ] ,
(77)
where flv and ^ d e( / , M, v~) are the oscillator strength and binding energy,
respectively, but only for transitions to the v" state of the diamagnetic exciton
(related to the Landau levels with quantum numbers /, M) which is odd under a
z-+—z inversion. In the logarithmic approximation (β$> 1, In β ρ 1) an ex­
pression for the oscillator strength was obtained by Monozon and Turchinovich
Diamagnetic exciton spectroscopy of semiconductors
413
Fig. 40. Oscillatory magnetoabsorption spectrum of S n 0 2 obtained at 3.6 Τ and 4.2 K. Dashed
arrows identify flaring up p 0n 0 transitions (by Agekyan et al. 1971).
(1978),
(78)
face β/α5 δ*9
where <5V= ν + δν", δν" % [2(2/ + 1) In β]/β, ν = 1, 2 , i s the Coulomb quan­
tum number of excited states of the 'one-dimensional' exciton. The deviation of
the values of <5V from the integer values of ν is seen to be small compared with the
corresponding deviations for even levels for which δ ν + ocln _ 1(/?/2). Thus the
odd-level series of the diamagnetic exciton is fairly close to the Coulomb series
with f5 v« ν = 1, 2 , . . . which permits one to calculate from a diamagnetic exciton
spectrum the band parameters of a semiconductor without a preliminary
determination of all the binding energies. Clearly enough, this reasoning does
not apply to the ground, that is single and even, ν = 0 state of the diamagnetic
exciton.
The magnetoabsorption coefficient for the forbidden interband transition
occurring in S n 0 2 also requires taking account of the crystal anisotropy.
Disregarding the Coulomb interaction for the two bands, Γ 3 -»Γί", it can be
written as
oc = A0B2
{cD-aocym\McyJ\2,
Σ
id.
M c v = M 0[ c o s 26(exEx
I = (L/hKQp
— eyEy) — sin 20(exEy
+ (e/c)A\lc>.
— eyEx)~\
(79)
Here the x- and y-axis lie in the plane of the square face of the crystal's
elementary parallelepiped, with the y-axis directed along the magnetic field and
R.P. Seisyan and B.P. Zakharchenya
414
making an angle with the edge of the elementary parallelepiped a. Then for the
oscillation spectrum we have
ε, = h(eB/c)l(le
+ Mmcmcy2
+ (/ + D/K^)
]
' + ε.
v
2
1
β
The quantities Ix and Iy in eq. (79) are nonzero for different transitions: Ix Φ 0
for / c = / v ± ( 2 n + 1), Μ = 0 , 1 , 2 , . . . . The n = 0 transitions dominate for low
crystal anisotropy whereas for transitions with η > 0 due to anisotropy Ix oc
[ ( m C |m
| v - m c m l ; |)| / ( m C m
| t ;i + m C m
i y i )]
i w. The quantity Iy = 0 when / c = / v + 2/t,
under which conditions / yoc(co — ω ζ , ) 1 / .2 As follows from eq. (79), the terms
containing Ix correspond to absorption peaks, and those with Iy9 to absorption
steps which are difficult to distinguish in the spectrum.
An analysis of O M A in C u 2 0 as a spectrum of diamagnetic excitons
permitted Seisyan (1984) to derive correct values of the electron and hole
effective masses which coincide with the present day data obtained from
cyclotron resonance measurements. Sokolov and Yakovlev (1986) carried out
numerical calculations of the diamagnetic exciton binding energies which
opened a way to evaluating a consistent set of the band parameters for C u 2 0
including hole nonparabolicity and anisotropy.
The major result of the experimental O M A study carried out by Agekyan et
al. (1972) on S n 0 2 was the construction of a detailed dependence of oscillatory
magnetoabsorption maximum positions on magnetic field (fig. 41). Increasing
the magnetic field complicates the exciton structure at the absorption edge by
0
AO
&0
1*0
46.0
20.0
Fig. 41. Positions of oscillatory magnetoabsorption maxima of S n 0 2 plotted against Β for 4.2 K.
Numbers adjoining the curves are serial numbers of the maxima, dashed lines are hypothetical plots
for transitions between free-carrier states in Landau sub-bands; here the numbers correspond to
electronic sub-band quantum number / c (by Agekyan et al. 1972).
(80)
Diamagnetic exciton spectroscopy of semiconductors
415
causing the appearance of new maxima. These changes can be interpreted as due
to Zeeman splitting of the exciton levels n0p+ and a flareup of the n0p0 levels. As
n0 increases, new levels which can also be interpreted as higher excited states
with n0 > 6 appear near the ionization limit of the exciton series.
Figure 42 displays the magnetic field dependences of the shift of the major
energy levels drawn on a log scale to rectify the power law relationship. One
clearly sees a crossover from the quadratic dependence characteristic of the
diamagnetic shift if exciton levels to a linear law typical of transitions between
the Landau sub-bands or of the diamagnetic excitons if the magnetic field
dependences of the binding energy tend to saturation. The crossover to the
linear behavior occurs at 4 - 8 T.
One can calculate the critical fields /?* at which the condition η%β* = 1 is met
and compare them with the experimental values obtained from fig. 42 (arrows)
for n0p±, as well as for the n 0s, n0p0 states flaring up in the field. The agreement
reached with experiment is quite satisfactory suggesting that the sufficient
criterion η%β* « 1 is indeed met in practice and that the excited states of the
Fig. 42. Relative positions of oscillatory magnetoabsorption maxima in S n 0 2 plotted against β on a
log scale. Numbers correspond to the serial numbers in fig. 41, arrows identify the regions of
Ββ=ί.
crossover from a quadratic to linear dependence on Β corresponding to
416
R.P. Seisyan and B.P. Zakharchenya
Wannier-Mott exciton do indeed participate in the formation of the oscillatory
magnetoabsorption spectrum as soon as the magnetic energy hQ has reached a
level in excess of the binding energy of the corresponding state.
Most of the spectral lines reach the domain of the linear dependence on Β and,
hence, their positions run practically parallel to the corresponding ionization
energies. Given the condition m* > m*, the slopes of the emax(B) plots will follow
with a constant step the quantum number / c, namely Αε/ΑΒ = hQ(lc + %)/B. The
magnitude of this step can easily be found (fig. 43) from the separation between
the linear sections of the plots. The points falling on this straight line (fig. 43) at
integer values on the horizontal axis will relate to diamagnetic exciton states
having an ionization limit with a given / c, the slope yielding inverse reduced
mass ( μ * ) " 1. One can thus conclude that most of the diamagnetic exciton series
in S n 0 2 for large / c are represented by one member only, the maximal value of / c
observed in the experiment being 13.
Cadmium sulphide and selenide; other A2B6
compounds*
A 2 B 6 semiconductor compounds fall among the crystals with clearly pronoun­
ced exciton effects at the fundamental absorption edge. They were discovered
and widely studied as early as in the 1950s.
The comparatively large binding energies R* and heavy masses of the
electrons and holes shift the critical magnetic fields required to satisfy the
condition β > 1 towards very high levels which are already difficult to reach for
the n0 = 2 excited state.
The numerous magneto-optical studies of excitons in these crystals involving
also the use of magnetic fields were usually restricted to spectral regions below
the ionization limit (see, e.g., Gross et al. 1961). One observed here clearly and
investigated both qualitatively and quantitatively the Zeeman effect and
diamagnetic shift of the n0 = 1, 2 levels. As for the diamagnetic exciton spectra,
this very informative spectral domain has not been adequately explored despite
indirect evidence for their observation [with the exclusion, e.g., of the work of
Chah and Damen (1971) where such transitions were interpreted as involving
the Landau sub-bands].
At the same time the extensive information available on exciton spectra in the
domain of magnetic fields where they may be treated within the framework of
the three-dimensional model makes these subjects very attractive for investiga­
tion. Quite recently, detailed spectra of diamagnetic excitons in hexagonal CdS
and CdSe have been obtained by Gel'mont et al. (1987). A 3.1 Τ spectrum and
a fan diagram for CdSe from this work are displayed in figs 44 and 45,
respectively. It is seen that at Β ^ 8 Τ one can detect tens of O M A lines
belonging to the exciton series A and B. Shown in the same figures are the
calculated positions of transitions between the Landau sub-bands calculated for
the case of hexagonal crystals. The presence of substantial linear-in-/c terms in
*See note added in proof
Diamagnetic exciton spectroscopy of semiconductors
0
3
W
417
I
Fig. 43. Slope of the linear part of e m a( xB ) plotted against quantum number / c. The numbers are the
same as in fig. 42 (by Agekyan et al. 1972).
the dispersion relation suggests that besides the main transitions one should also
take into account in these crystals the additional transitions associated with the
modified selection rules. An essential result of this experimental work is the
observation of the many-member series of the diamagnetic exciton which gives
grounds to hope for progress in the solution of the 'intermediate' field problem.
Gallium selenide
GaSe and similar semiconductors represent an interesting subject for study
because of specific features of their crystal structure, namely, these materials
<86
IB7
m
m
hcjfeVj
Fig. 44. Shape of exciton spectrum for CdSe for: 1, Β = 0; and 2,B = 3.1 Τ (by Gel'mont et al. 1987).
418
R.P. Seisyan and B.P. Zakharchenya
1 2
3
1
5
6
7
8
Diamagnetic exciton spectroscopy of semiconductors
419
have a layered structure with a hexagonal unit cell, the layers being bound only
by van der Waals forces. This structure gives grounds to expect a primarily twodimensional behavior of carriers or bound states.
Excitons in GaSe were detected already by Fielding et al. (1958). In this
crystal, one observes an exciton series: besides the strong n0 = 1 ground-state
peak, the n0 = 2 maximum, and, in particularly good crystals, also the n0 = 3
maximum are all clearly pronounced. Thus GaSe falls into the class of crystals
with a developed Wannier-Mott exciton structure. Absorption in GaSe is
strongly polarized: if the electric vector Ε of the light wave is parallel to the
crystal plane and perpendicular to the c-axis, the absorption is weak, and it
becomes strong for Ε perpendicular to the sample plane and parallel to the
c-axis. The normally observed optical transitions from the valence band of GaSe
to the conduction band for ELc are considered to be direct and allowed,
although their intensity is one to two orders of magnitude smaller than that of
direct dipole-allowed transitions under E\\c polarization.
Magneto-optical experiments on GaSe were performed by Halpern (1966),
Aoyagi et al. (1966) and Brebner et al. (1967). These experiments reveal both a
diamagnetic shift and Zeeman splitting of the members of the Wannier-Mott
exciton series and an oscillatory behavior of the absorption edge. Mooser and
Schluter (1973) carried out a quantitative investigation of the magneto-optical
behavior of the Wannier-Mott exciton in GaSe. They disregard, however, the
oscillatory part of the magneto-optical spectrum while considering the
Wannier-Mott exciton series as three dimensional despite the crystal's layered
structure. It is pointed out that the behavior of 3s states does not fit within this
frame to the quadratic dependence for the diamagnetic shift already at fields
Considering the experience gained in the treatment of oscillatory magneto­
absorption data in crystals with a developed structure of the Wannier-Mott
exciton, we could suggest that deviations from the quadratic course of the 2s and
3s states toward linearity should start exactly after the inequalities R*(2s) ^ hQ^
and K*(3s) ^ hQ2 have been reached, which corresponds to fields Bx = 5 - 6 Τ for
the 2s state, and B2 = 2.5-3 Τ for the 3s. This apparently is true both for the
above work and for the recent study of Rasulov et al. (1987).
In fields Β > 3 Τ one observes the appearance of a fine structure in the
oscillations which can belong to other members of the diamagnetic exciton
series. It can also be due to the new selection rules in I originating from a strong
Fig. 45. Positions of magnetoabsorption maxima in CdSe plotted against magnetic field. Points:
experiment for Τ = 2 Κ, Β || c, Β1 q. Solid lines are drawn close to experimental points simply to aid
the eye. Dashed lines - (a) series (band) A: 1, a ' ( l ) a c( 0 ) ; 2, b + ( - 2 ) b c ( - 1 ) ; 3, a " ( 2 ) a c( l ) ; 4, b +
( - l ) b c( 0 ) ; 5, a"(3)a c(2); 6, b +( 0 ) b c( l ) ; 7, a~(4)a c(3); (b) series (band) Β: 1, b ~ ( 0 ) b c( - l ) ; 2, a +
( - l ) a c( 0 ) ; 3, b " ( 0 ) a c( l ) ; 4, b - ( l ) b c( 0 ) ; 5, a + ( 0 ) b c( - 1 ) ; 6, b ' ( l ) a c( 2 ) ; 7, a + ( 0 ) a c( l ) ; 8, b " ( 2 ) b c( l ) ;
9, a + ( l ) b c( 0 ) ; 10, b"(2)a c(3); 11, a + ( l ) a c( 2 ) ; 12, b ' ( 3 ) b c( 2 ) (by Gel'mont et al. 1987).
420
R.P. Seisyan and B.P. Zakharchenya
anisotropy. Interestingly, the Oscillatory' part of the spectrum is relatively
weakly pronounced and has a totally different structure in the spectrum
obtained in the Voigt geometry (fig. 46). The features observed in the 'diamag­
netic' part of the spectrum do not fit well into the framework of a model with
weak anisotropy and parameters calculated in the approximation of Mooser
and Schluter (1973).
Akimoto and Hasegawa (1967) considered the case of very large anisotropy
and high magnetic field (β > 1), that is of the diamagnetic exciton in a twodimensional crystal, as applied to the problem of diamagnetic excitons in GaSe.
This situation is believed not to be realized in GaSe, either in the sense of high
>1 1. A solution for the case
field, β > 1, or in the sense of strong anisotropy,
of limited anisotropy based on the adiabatic approximation in the quasiclassical
limit, which follows the ideas of Zhilich and Monozon (1968) under the
condition η\β > 1, was obtained by Baldereschi and Bassani (1968), however, it
likewise does not yield a satisfactory description.
The problem of diamagnetic excitons in GaSe-type crystals requires further
experimental and theoretical investigation.
*M
Μ
2J5
S,eV
Fig. 46. Oscillatory magnetoabsorption spectra of GaSe for £ l c , T = 2 K , B = 7.5T: (a) npolarization, (b) σ-polarization (by Rasulov et al. 1987).
Diamagnetic exciton spectroscopy of semiconductors
4.3. Diamagnetic excitons in crystals with suppressed Wannier-Mott
states
421
exciton
All the three kinds of subjects with poorly pronounced, well developed and
suppressed states of the Wannier-Mott exciton can in principle be represented
by the same material but placed, for instance, in different temperature con­
ditions or having different concentrations of free carriers and defects interacting
with excitons. Note that the Coulomb interaction of the electron-hole pair
cannot, as a rule, be switched off completely and is present in all cases. Taking
into account the nonstationary behavior of exciton states in the interaction of an
exciton with scatterers, screening charges and electric fields yields us lifetimes of
a state and the corresponding linewidths which predetermine the presence or
absence in the spectrum of lines associated with excited states of the exciton
series, or with bound states generally. Turning on the magnetic field affects the
exciton interaction with free carriers and defects, resulting primarily in stabiliza­
tion of the exciton states. Moreover, in a high magnetic field which reduces the
problem to a one-dimensional model there may even now exist very shallow
bound states. By increasing the magnetic field one will promote the formation of
exciton states until potential wells appear capable of trapping the holes and
electrons separately, which will result in a breakup of the exciton. This may be
brought about, for example, by potential fluctuations in solid solutions or doped
semiconductors, or by the defect potential.
Flareup of exciton states in magnetic fields
In crystals doped to a certain critical concentration at which the exciton
absorption is suppressed to such an extent that the total edge absorption as a
function of the energy of incident radiation becomes substantially lower than is
the case for an undoped crystal, turning on the magnetic field results in a general
flareup of absorption up to the level corresponding to a pure crystal. This is
illustrated by fig. 47 taken from Seisyan et al. (1968) which shows a general
growth of the absorption level with increasing magnetic field in germanium
doped to i V D% 8 x 1 0 15 c m " 3 . Such concentration at Β = 0 is only slightly in
excess of the critical level with respect to free-carrier screening, and exciton
absorption is suppressed. Integrating the absorption coefficient as a function of /
for Β -> 0 brings it to the same form and values as that for Β = 0. The flareup
observed in the presence of a magnetic field results in an increase of integrated
absorption as the field is turned on. The major result of this experiment is the
establishment of the fact that the screening of the Coulomb interaction by freecharge carriers becomes attenuated as the magnetic field is increased.
In the case of fig. 47 where the magnetic field (B % 3.5 T) is not very strong for
Ge we have β = (a*/L)2 « 3.3, and aB « O.Sa*xc which is equivalent to a situation
where at Β = 0 we would reduce the carrier concentration by a factor of 1.5. This
turns out to be sufficient for observing the effect, since exciton absorption decays
422
R.P. Seisyan and B.P. Zakharchenya
G;T=kZK
0.90
β'325Τ
0.92
ξδΥ
Fig. 47. Magnetic-field-induced flareup of exciton absorption in germanium at 4.2 Κ (by Seisyan
et al. (1968b).
dramatically with increasing concentration, and the experiment is sensitive to a
change of the absorption coefficient.
Consider now the O M A observation in doped germanium at low temper­
atures under impurity-induced breakdown conditions. In this situation it
becomes possible, other conditions maintained relatively constant, to vary
within a broad range the concentration of the free carriers which lead to
dissociation of a bound pair because of screening. The configuration of the
experiment was such that the electric and magnetic fields were parallel. This
permits one to use, within certain limits, the electric field as an auxiliary factor
which does not disturb the pattern of the transitions involved in the oscillatory
magnetoabsorption in crystals.
The / - V characteristic of the sample obtained under the conditions of the
OMA experiment is S-shaped, resembling that typical of impurity-induced
breakdown in germanium. The 'breakdown' sets in at an electric field E'hr (its
value increasing somewhat with increasing magnetic field), after which the
voltage drops, the 'breakdown' continuing to develop at EhT <^ E'hr. Depending
on the actual field used, the current grows almost vertically reaching densities
; > 200 A c m " 2 .
Figure 48 presents π-spectra of oscillatory magnetoabsorption obtained
under these conditions. Increasing the electric fields up to the prebreakdown
level (Ε ^ £ b r) does not lead to any changes in the spectrum, either in the
position of the absorption bands or in their intensity. Reaching the quasivertical
branch of the / - V characteristic is accompanied by an appreciable decay of the
oscillatory spectrum. It is also obvious that the suppression of the oscillations is
governed by the current rather than by the voltage applied, since the spectra in
fig. 48 obtained at practically the same electric field but at different currents
differ markedly in oscillation intensity.
Diamagnetic exciton spectroscopy of semiconductors
423
Fig. 48. Breakdown-induced decay of oscillatory magnetoabsorption in germanium at 4.2 Κ: 1,
j = 0, F = 6 0 V c m " 1; 2, (a) j = 120 and (b) j = 160 A c m - 2, F = 2 0 V c m " 1; 3, j = 200 A c m " 2,
F = 2 0 V c m " 1 (by Seisyan 1984).
The decay becomes weaker as the magnetic field increases which can be
seen by comparing figs 48(a) and (b). This corresponds to the experimentally
observed flareup of oscillatory magnetoabsorption in doped germanium as the
magnetic field is applied.
It should also be pointed out that the stabilization of exciton states in
magnetic fields is sufficiently efficient also for other mechanisms of suppression
of the discrete exciton structure, for instance, in the Stark 'quenching' in the
random field of an ionized impurity. Examples of such behavior can be found in
experiments with compensated material doped in such a way that the freecarrier concentration is very low over a wide temperature range while the total
impurity concentration, JVD + JVA, is high. This case which is typical of a
semiconductor in the 'semi-insulator' state is frequently met, for instance, when
working with GaAs or CdTe.
One can easily find CdTe crystals with a very low free-carrier concentration
which do not exhibit a discrete exciton structure. If the total concentration ΝΌ
+ NA is not too high, one can nevertheless achieve the appearance of a discrete
structure in the spectrum by applying a magnetic field (fig. 49). The suppressed
exciton structure of the absorption edge in CdTe with a free-electron con­
centration of about 1 0 1 4c m ~ 3 flares up in magnetic fields Β ^ 5 - 6 Τ where
β->1. Here ΝΌ + ΝΑ can be estimated as 1 0 1 7c m " 3 . In more closely com­
pensated samples with free-carrier concentrations of the order of 1 0 11 c m - 3
such fields become insufficiently high for the exciton spectrum to flare up.
424
R.P. Seisyan and B.P. Zakharchenya
CdTe
Γ=4.2 κ
B'SPT
159
160
Μ
162 6,eY
Fig. 49. Magnetic field-induced flareup of exciton absorption in compensated CdTe in σ+ and σ~
polarizations. n = 1 0 1 c4 m - 3, ND + N A * 1 0 1 c7 m - 3, T= 4.2Κ (by Seisyan 1984).
Another kind of the exciton absorption flareup effect in magnetic fields was
employed by Kanskaya et al. (1979) to reveal the exciton structure of the
absorption edge in InSb. Application of a field of 1 0 " 2 Τ was sufficient for a
clearly pronounced peak of the n0 = 1 ground state to appear at the absorption
edge where no exciton absorption maximum was present before (fig. 50). One
does not see here any more changes in the absorption background. This suggests
that in this case, likewise, screening of the exciton states was not a crucial factor.
By choosing a technique of further, relatively low-temperature, sample treat­
ment, one succeeded in observing the exciton structure at Β = 0 as well. This
treatment could not affect the bulk properties of the material, and therefore
neither the screening level, nor the 'Stark' broadening in the random-impurity
Coulomb field could change. The effect in this case could originate from an
increase in the oscillator strength of the transition to the ground state sufficient
to overcome the strain-induced smear, or from an increase of effective 'exciton'
thickness of the sample due to a change in the density of charged surface states.
A flareup of the exciton absorption maximum in the magnetic field was also
observed by Ivanov-Omskii et al. (1983) in Cd^Hg^^Te quasibinary solid
solutions with χ = 0.3 (fig. 51). We see here how the application of a relatively
weak magnetic field ( ^ 0 . 5 T) converts the structureless monotonic absorption
edge into the typical spectrum of a crystal with a clearly pronounced exciton
ground state. In this case we likewise do not observe any substantial change of
the background. However here, in contrast to InSb, one did not find a proper
thermal treatment of samples which would permit observation of the exciton
Diamagnetic exciton spectroscopy of semiconductors
425
I
235
240
2tf
Fig. 50. Magnetic field-induced flareup of exciton absorption maximum in indium antimonide: I,
before annealing for different magnetic fields Β χ ΙΟ" 3 Τ (1, β = 0; 2, 5.0; 3, 9.0; 4, 12.5); II, after
annealing, no magnetic field, Τ = 2 Κ, ρ = 6 χ Ι Ο 12 c m - 3, d = 13 μπι (by Kanskaya et al. 1979).
Fig. 51. Flareup of a discrete exciton structure in C d 0 3H g 0 T
7 e in a magnetic field at T=
(a) Β = 0; (b) Β = 1.0 Τ (by Ivanov-Omskii et al. 1983).
1.8 K:
maximum with no magnetic field present. Thus one cannot exclude from the
number of possible reasons for the suppression of the discrete ground exciton
state the fluctuation line broadening inherent in solid solutions.
Diamagnetic exciton in cadmium-mercury tellurides (CMT)
The mercury chalcogenides HgTe and HgSe belonging to the class of A 2 B 6
compounds and crystallizing in a sphalerite-type cubic modification, as well as a
R.P. Seisyan and B.P. Zakharchenya
426
certain region of Cd^Hg^^Te-type solid solutions, are considered as semi­
conductors having a zero or negative-energy gap. Studying magneto-optical
effects in these materials proves to be very fruitful, while the application of
strong magnetic fields can lead to a conversion of a material with zero or
negative-energy gap into a semiconductor, thus justifying attempts at applying
to them the concept of the diamagnetic exciton. However, the Coulomb
interaction, even in solid solution compositions possessing a 'positive' energy
gap, is complicated by potential fluctuations inherent in the solid solutions.
Ivanov-Omskii et al. (1983) and Kokhanovskii et al. (1983) observed in
C d xH g 1_ J CT e (x = 0.3), immediately after the flareup of exciton absorption,
oscillatory magnetoabsorption as the field was increased (fig. 52). Note that the
χ = 0.3 composition is close in energy gap width (ε 8 = 0.258 eV) and band
structure to InSb. One could expect also the magneto-optical spectra of these
materials to behave similarly. The same figure presents an O M A spectrum of
'pure' InSb crystals obtained under identical conditions. A comparison of the
spectra of C M T and InSb shows that the magnetoabsorption lines of
Cdj.Hgi_j.Te are broader by more than an order of magnitude than the
KB)
Vol
2.0
15
10
0.5
11
.1
hi . i t t i t
I1
It! !tl . tit It.
15
iO
0.5
%$&
e.o
8=?7T
10
0.0
1. 1 . ΤτΤ
P300
IV
0.35D
lit, l!
OMJO
ι
tl, t.tl J
OHSO
7^T
Fig. 52. Oscillatory magnetoabsorption spectrum in C d 0 3H g 0 T
7 e for T= 1.8 Κ, Β = 7.0 Τ. (a) σ~
polarization; (b) σ+ polarization. Shown for comparison below is σ+ polarization spectrum of InSb
obtained at Β = 7.7 Τ (by Kokhanovskii et al. 1983).
Diamagnetic exciton spectroscopy of semiconductors
All
corresponding lines in InSb, and that the spectrum proper is much shorter,
implying that the oscillations become indistinguishable against the absorption
continuum at substantially smaller Landau quantum numbers /.
The washing out of the band edges caused by compositional fluctuations
leads to a broadening of the exciton line even in zero magnetic field by two
qualitatively different mechanisms. One of them is due to the fact that the lightcreated exciton may become localized as a whole in the wells originating from
compositional fluctuations. In this case it is the spread in the localization energy
that accounts for the linewidth. The second case is typical of an exciton with a
hole of a large mass and occurs in such semiconducting solid solutions where the
variation of gap width with composition is associated primarily with the motion
of the conduction band edge Aec. In this case the exciton resembles a donor
atom in which the electron moving about a fixed hole averages out the potential
fluctuations in the region of its motion. The exciton may be created in different
regions of the crystal, the associated broadening being determined by the rms
fluctuations of the random potential in the region of electron motion:
J = a c [ x ( l - x ) / M z 3 ] 1 / .2
(81)
here a c = AsJAx, and Ν is the number of sites for a metal atom in a unit volume.
Raikh and Al. Efros (1983) showed that if the condition
q = AMa2/h2>(M/p)112.
(82)
is satisfied, then in any magnetic field such that β > 1, the exciton line becomes
'broadened' as the absorption line of a donor atom in a magnetic field, its
linewidth being
AB = ((xJ2)lx(l-x)/NaL2yf2,
(83)
which increases with magnetic field proportionately to B. Here Μ = m* + m*.
At the same time the binding energy of the diamagnetic exciton grows only
logarithmically. Therefore, in a sufficiently strong field, Β > J3*, the exciton will
inevitably break up.
Estimates made for C d j - H g ^ ^ e show that practically throughout the
semiconducting composition region 0.15 < χ < 0.8 the quantity q is greater than
(Μ/μ)112.
Thus the exciton can exist in zero magnetic field, the mechanism of
exciton line broadening in any magnetic field (up to the exciton breakup) being
similar to that of the absorption line broadening for a donor atom in a magnetic
field. The magnitude of the critical field obtained by comparing the width (83)
with the diamagnetic exciton binding energy yields for a sample with χ « 0.3,
β* = 6 Τ.
Raikh and Al. Efros (1983) also showed that the diamagnetic exciton line
undergoes a specifically excitonic broadening associated with compositional
fluctuations. This component of broadening is proportional to the cubed mass
of the density of states, M\M, of the exciton as a whole. Since the diamagnetic
428
R.P. Seisyan and B.P. Zakharchenya
exciton mass in the plane perpendicular to the magnetic field is M B =
p(a/L)2 In β, this broadening grows with increasing magnetic field.
For Landau levels with a large /, diamagnetic excitons may acquire the shape
not of a cigar with longitudinal dimension ( ~ a) exceeding by far the transverse
size (L) but rather of a lens whose transverse dimension (~Lyfl)
is much
greater than the longitudinal one, ~ [ L a ( / ) 1 / ]2 1 / .2 It can be shown that in this
case M B « / i ( a / L ) / 3 /2 implying that the linewidth grows with the Landau
number / as Z3. It follows from this that the broadening of the diamagnetic
exciton related with high Landau levels occurs faster by this mechanism.
Interestingly, a computer processing of the total spectrum obtained at
Β ^ 7.0 Τ showed that the standard deviation between a theoretical and experi­
mental spectrum for one experimental maximum in the case of the 'exciton
model' is, on the whole, considerably smaller than that without including
Coulomb interaction. This suggests that when analyzing experimental data on
Cdj.Hgi_j.Te in the given magnetic field range one should consider the O M A
spectrum as originating from diamagnetic excitons*. It also follows that the
fluctuation mechanism does not result in a total breakup of the exciton states in
the magnetic field range investigated which would thus require higher fields.
Lead telluride
Mitchell et al. (1964) were the first to investigate the oscillatory magnetoabsorp­
tion in this material. Aggarwal et al. (1968) studied the interband magnetooptical spectrum of PbTe by the piezoreflectance technique, and Smith et al.
(1973), magnetoabsorption of the quasibinary solid solutions P b ^ S n i ^ T e .
These investigations, in addition to a series of other experiments including twophoton absorption and photoconduction in the magnetic field, have provided a
foundation for our present day ideas concerning the band structure of PbTe and
of other lead chalcogenides.
In PbTe, just as in other lead chalcogenides, no exciton effects have ever been
observed which may be related with two circumstances. First, the static
dielectric constant in these materials is extremely high. In PbTe it is maximal
reaching κ0 « 1000. This weakens the Coulomb interaction drastically and
makes the binding energies of hydrogen-like states very small. Another essential
circumstance consists in the relatively broad region of homogeneity in the A 4 B 6
system making preparation of fully stoichiometric material a formidable
problem. As a result, one usually has to work with a material which has a high
concentration of intrinsic defects and, accordingly, of free carriers.
For the many-ellipsoid model, where the extrema lie at symmetry axes for
k φ 0 and the constant-energy surfaces represent ellipsoids of revolution, the
binding energies can be obtained by the variational technique. The inclusion of
the anisotropy, dynamic screening and, finally, lattice relaxation which results,
by Enderlein et al. (1982), in a reduction of the effective dielectric constant κ*1
*See note added in proof
Diamagnetic exciton spectroscopy of semiconductors
429
down to 100, maintains nevertheless a fairly low value of the binding energy,
R* « 1 0 " 5 eV, with such a large radius, a* xc % 7000 A, that bound states should
already thermally dissociate at helium temperatures.
The situation changes radically if a magnetic field is applied,
(i) This inequality is met the earlier, the smaller is R*, and for it to be satisfied
in PbTe a field B0 % 1 0 " 2 Τ is already formally sufficient. At the same time, as
already pointed out, in the presence of a magnetic field the condition r s cr < a*xc is
insufficient to screen the Coulomb interaction of the electron and hole.
(ii) Evaluation of the free-carrier concentration required to ensure appreci­
able screening in PbTe from the condition r s cr = a* xc suggests that screening
effects should be inessential for the ground state at
fields
10 Τ for freecarrier concentrations lower than ^ 5 x 1 0 1 5c m ~ 3 . However, even the inequal­
r <
Ba does in no way imply the absence of diamagnetic exciton effects;
ity scr
indeed, the discrete ground state will exist here with any screening.
The large value of β attainable in PbTe would seem to permit reaching the
very strong inequality In β |> 1 thus offering a possibility of using the logarithmic
approximation of the diamagnetic exciton model described by Zhilich and
Monozon (1984) for anisotropic semiconductors. Disregarding the anisotropy
of the dielectric constant, one can obtain the following expression for the
binding energy of the diamagnetic exciton:
^ d e = R*So
2
= R* l n 2[ 4 „ * 2/ ( y a" ^ 2 + y B" ^ 2 ) ] ,
(84)
where δ0 is the 'quantum defect', y a = y B a 2 a 2 / L 4 , y B = ( a 2 + a 2 ) ~ \ a, = L^/k~j,
^ = ( l m_ ; / mc y c i . ; ) 1 / -2 Taking into account the symmetry of the conduction and
valence bands in PbTe, one can now easily reduce Λ to a form convenient for
calculations:
@dc = R* ln 2[2jS*fc/(l + fc)2].
(85)
Note, however, that in anisotropic semiconductors the reduced cyclotron and
longitudinal (exciton) masses are essentially different, and one should take into
account the actual geometry of the experiment in calculating the quantities
R* = e*pB/2h2(Kf)\
a$ = h2(Kf)2/e2pB,
j?* = (a*/L) 2; besides, Q = eB/cp(
Here μ Β is the reduced mass in the direction of the magnetic field, μ Β 1 =
2
2
- 1 m
( m_ c )
+ ( B v ) -> 1 m^j = mfj sin Θ+ mf\j cos
θ; θ is the angle between Β
and the [111] axes, and p~y\x is the reduced electron and hole cyclotron
mass; pjy\x = (w* y cl c) ~ 1 + ( m * y c l)v" m * y c Jl = [ ( m i , ) " 2 c o s 2 0 + (m?;)" V * ; ) ~ 1
sin 2 0] ~ 1 / .2 In crystals of PbTe type the cyclotron mass m*yclj has in the general
case for the most symmetrical directions one or two values corresponding to the
different sections of an ellipsoid by a plane perpendicular to the magnetic field.
An interesting situation arises when, for instance, for the _?||[111] orientation
one of the ellipsoids with the long axis along the field yields a 'light' cyclotron
mass equal to
and also the smallest possible reduced cyclotron mass μ ±.
R.P. Seisyan and B.P. Zakharchenya
430
(Note that the cyclotron mass is determined essentially by the transverse mass
for other directions as well.) At the same time the reduced mass of the
longitudinal motion entering the exciton binding energy, μ Β, corresponds in this
case to 'heavy' masses mf]p thus resulting in a substantial relative increase of the
binding energy.
One readily sees, however, that if we choose the direction to correspond to the
'heavy' longitudinal mass, β* drops dramatically, and we can no longer take
advantage of the simplifications inherent in the logarithmic approximation, and
will have to solve in a straightforward way the one-dimensional equation (9)
with the adiabatic potential V(z).
The theoretical relationships for $dc{B) obtained in this way by Geiman et al.
(1986) are presented in fig. 53, and for fields lying in the range Β = 2 - 7 Τ are in
the region 0.6-0.9 meV with K C" chosen within the range 120 to 140. Note also
that in PbTe the nonparabolic effects are strong and the masses grow rapidly
with increasing photon energy, which likewise heavily affects 0tat.
In their experimental study into the existence of bound states in PbTe,
Kokhanovskii et al. (1986) made use of heteroepitaxial layers prepared on ascleaved (111) faces of B a F 2 with deposition carried out in a high vacuum. The
experiment performed by sweeping the magnetic field at a fixed wavelength in
the range hv = 0.19-0.49 eV revealed up to 20 sharp O M A maxima showing the
high values of ωτ reached (based on the mobility relaxation time τ 0 , the values
reached were coccyc{ τ0 = 150-200). Besides the absorption maxima detected
usually in interband magnetoabsorption experiments and belonging to two
Hi
°
I
1
1 0 A O
1
1
Jo
1
SO
-/
1—
WSJ
Fig. 53. Diamagnetic exciton binding energies in PbTe plotted against magnetic field: 1, 2, 3,
calculated with KE0" = 100,120 and 130, respectively; 5,6, calculated by eq. (9). Experimental points I
for ^ B were obtained by comparing the positions of the O M A maxima with those of the extrema in
the differential spectra. Also plotted are points II showing the dependence of £ J on Β, Ε IB;
III -Εξ for Ε||Β (by Geyman et al. 1986).
Diamagnetic exciton spectroscopy of semiconductors
431
series with different reduced cyclotron masses, μ ^ , one also observed in the
spectrum a number of features which can be assigned to transitions occurring by
the selection rules modified by the crystal anisotropy (fig. 54). (The 'light'
reduced mass superscripted Ί' is formed by two ellipsoids lying along the field Β
in the conduction and valence bands, while the 'heavy' mass superscripted 'h' is
formed by three equivalent ellipsoids in each band oriented at the same angle to
the magnetic field.)
As should be expected, the O M A spectrum did not exhibit any features which
might have shown the presence of exciton states. To detect bound states, an
electric field was applied along the sample by means of gold or indium electrodes
and was perpendicular to the magnetic field in the Faraday geometry. The high
electrical conductivity of the sample did not permit the application of strong
electric fields to avoid sample heating (the authors were careful not to let the
specific power dissipated exceed the level of 0.2 W c m - 2) . It turned out,
however, that a very weak electric field £ 0 = 2 - 8 V c m _ 1 already produces
appreciable damping of the oscillations which saturates at higher fields. This
provided a possibility, by using an alternating component Εγ cos ωί, to record at
the frequency ω = 1 kHz the very remarkable and strong differential spectrum
which is presented in fig. 55. A very essential feature is the nonmonotonic
dependence of the observed signal on the d.c. component E0 (fig. 56) which
passes through a maximum at Εξ = 1.5-3.1 V c m - 1 and subsequently drops to
zero.
The value of Εξ grows slowly with increasing magnetic field (inset in fig. 56)
I/S)
1(0)
3.0
0
PeTe/&aF2
Τ-
ITL
2K
1
Fig. 54. Oscillatory magnetoabsorption in heteroepitaxial films of P b T e / B a F 2, B\\ [111], T = 2 K.
Shown below is a theoretical spectrum calculated for the heavy-mass (O) and light-mass (Δ) series.
Inset: reciprocal transmission versus EQ (by Kokhanovskii et al. 1986b).
R.P. Seisyan and B.P. Zakharchenya
432
2.0
3.0
40
5.0
6.0
B,T
Fig. 55. Differential electroreflectance in P b T e / B a F 2 for E0 < £ $ and E0 > £ J plotted against a
magnetic field 6 | | [ 1 Π ] . λ = 3.39 μm, T = 2K, Εί = 0 . 2 V c m " 1. Values of E0: (a), 3 V c m - 1, (b),
l O V c m - 1. Extremum notation: Ί' and 'h' denote the light- and heavy-mass series, the number
specifying the Landau quantum number for Δ/ = 0 selection rules. Inset on the right compares, on an
enlarged scale, fragments of the electroreflectance spectrum with the oscillatory magnetoabsorption
spectrum for Ε = 0 near the extremum to explain the experimental evaluation of ^? B. Inset on the left
presents magnetic field dependence of: (c) Rydberg exciton and (d) £J (by Kokhanovskii et al.
1986a).
and depends little on the Landau number for a fixed value of B. The
phenomenon, itself of a strong sensitivity of the absorption coefficient to such a
weak field, defies explanation in the context of well-known effects, for instance,
the Franz-Keldysh effect. The turning off of the sensitivity to electric field for
E0 > Εξ suggests that the field breaks down some easily ionizable states which
at the same time contribute substantially to the absorption coefficient over a
broad energy range. Considering the fairly low impurity concentration, one may
conclude that these states belong to the diamagnetic exciton.
From the possible mechanisms of electric field-induced breakdown of exciton
states we first exclude direct ionization since in this case we would obtain
binding energies much smaller than kT. Taking into account in place of the
applied field E0 the Hall field Εξ which in the case of PbTe may be higher than
the former one turned out to be not very essential in the actual conditions of the
experiment because of the Hall field being shorted out. Carrying out experi­
ments in different geometries and on different samples suggests that the reason
for the breakdown of diamagnetic excitons in PbTe when an electric field
E0 > Εξ is applied is impact ionization by free charge carriers.
Diamagnetic exciton spectroscopy of semiconductors
0
1
4
6
8
'
Φ
1Z
*
433
£0V/cm
Fig. 56. Shape of the dependence of differential signal on electric field E0 at the extremum of the
O M A spectrum. The inset presents a fragment of this dependence in the form of 1η(Δ//£5) plotted
against Eq2 plot for m = — § for the light-mass extrema, with B m xa = 5.2 Τ (1, 2, 3 stand for different
samples) and B m xa = 4.26 Τ (4, 5 identify different samples). The calculated values of Εξ obtained
from the slopes of the straight lines are: 1, 2, 3 - 0.5 V c m " *; 4 - 6.2 V c m " *; 5 - 5.8 V c m " 1 (by
Kokhanovskii et al. 1986b).
The possibility of an independent evaluation of ^ d e provides a comparison
between the electric-field-modulated and conventional OMA spectra. Since
modulation in such a weak field involves only the exciton component of
absorption while the total absorption is governed primarily by transitions to the
continuum above the Landau levels, the extrema in the spectra should be shifted
slightly by an amount determined by the exciton binding energy. The values of
^ d e obtained in this way lie within 0.5-0.8 meV and are presented in fig. 53
plotted against magnetic field.
Another possibility of estimating $dc{B) follows from an analysis of the slope
in the dependence of the differential signal on E0. Assuming that in impact
ionization the signal falls off because of the lifetime τ(£), decreasing as
1/τ(£) = (1/τ 0) exp{ - « d e( B ) / k B T [ l + (cfi/ωΒ)] Vh/v,},
P
(86)
one can evaluate $ d e from the slope of the dependence of the log signal on 1/EQ.
(Here ω is the sound velocity, η is a coefficient accounting for the anisotropy and
v ph and Vj are the frequencies of scattering from phonons and impurity ions,
respectively.) This evaluation likewise leads to binding energies in the range
0.7-0.9 meV (inset in fig. 56).
The relative flareup in differential spectra of heavier reduced-mass transitions
for which the excitons have lower binding energies is another interesting
observation.
434
R.P. Seisyan and B.P. Zakharchenya
Figure 53 compares the experiment with the theoretical curves for PbTe
calculated by eq. (9) without the inclusion of screening. One readily sees that
using KQ{ as a fitting parameter it is possible to reach a reasonable agreement
with experiment for Β^ 6T. The best fit was reached with KQ{ = 130.
Thus, the totality of available experimental data and the theoretical estimates
suggest the existence and substantial role of exciton states in optical and
electrical processes under conditions typical of magnetoabsorption experiments
performed on lead telluride crystals.
5. Conclusion: band parameter
exciton
spectra
calculation
from
diamagnetic
The traditional use of magneto-optical experiments in semiconductors to study
their band structure is complicated by the exciton nature of the interband
spectrum. It becomes necessary to calculate the binding energies
^η^Μ,ν±(β)
separately for each line and to 'reconstruct' the spectrum of transitions between
the Landau sub-bands which would directly reflect the structural features of the
energy bands adjoining the energy gap.
It has been seen that binding energy calculations can be easily made with an
accuracy sufficient for these purposes for narrow-gap and medium ε 8 semi­
conductors. As for the wide-gap materials, here the experimenter has to content
himself with the intermediate magnetic field domain which creates radical
difficulties for the calculation of ^ d e. However, even in this case quantitative
evaluation turns out to be possible provided the characteristics of the diamag­
netic exciton spectrum have been correctly included. A successful solution of the
problems involved in the calculation of ^ d e not only reinforces the position of
interband magneto-optics as a valuable tool for the precise investigation of the
band structure of semiconductors but also suggests some new possibilities. This
has been shown convincingly in the consideration of the results of O M A studies
on InSb.
Among the advantages inherent in magnetospectroscopic experiments is the
possibility of a simultaneous study of the conduction and valence bands and of
obtaining in this way a consistent set of band parameters, whereas in order to
derive the same data from other experiments one often has to use specially
doped n- and p-type crystals.
The simplified consideration of the nature of the effect using a 'simple'
semiconductor model which entered a number of textbooks and monographs
leads to a conclusion that O M A has one large drawback as applied to studying
the band structure. This drawback consists in that at first sight it would seem
that the experimenter can obtain conformation only on reduced mass entering
the quantity Ω = ojccycl + ω* χ ε ,1 so that in order to learn the electron and hole
effective masses separately, additional experiments should be carried out. A
Diamagnetic exciton spectroscopy of semiconductors
435
consideration of concrete experimental results shows persuasively that this
opinion is utterly wrong. By analyzing a high-resolution spectrum, one can
isolate energy intervals associated separately with the electrons, light or heavy
holes and thus to calculate in a straightforward way the corresponding masses.
This possibility appears because in practice one never finds an energy level
diagram corresponding to a 'simple' semiconductor where transitions would be
described by Landau number selection rules, Δ/ = 0. In actual fact these
selection rules turn out to be not very rigorous even for the major transitions,
not to mention the additional transitions caused by the band anisotropy,
warping, effect of applied or inherent electric fields, and so on. Indeed, in
diamond-like semiconductors for direct dipole-allowed transitions, in 'simple'
semiconductors for dipole-forbidden transitions, in anisotropic semiconductors
and in crystals of hexagonal symmetry it is possible by properly choosing the
numbering of the hole levels to reduce the selection rules to Δ/ = ± 1 . This means
that by comparing the spectra obtained in different polarizations (or by isolating
lines in the total spectrum corresponding to different polarizations) one can
always find transitions from the same hole state with / v to electronic states with
' c — Ύ = + 1 a dn ζ — Κ = — 1> their energy spacing being equal to two electron
cyclotron energies. This applies also to hole levels. In this sense, O M A
observation and analysis reproduce the results obtained from a cyclotron
resonance observed simultaneously for electrons and holes with substantial
extension to the corresponding bands. The results of such an analysis for O M A
in InSb are shown in fig. 57.
One readily sees that cyclotron masses can be measured up to energies ε =
ec + sg& 2eg for the electron, to a depth of ~ 80 meV for the light-hole band, and
to a considerable depth for the heavy-hole band. This provided a possibility of a
Fig. 57. Dependence of m/m* in InSb on ε 8(2ε ε + ε 8) obtained from a direct analysis of oscillatory
magnetoabsorption spectra (see fig. 26) with isolation of double cyclotron energies for the electron.
The straight line is a least-squares fit (by Kanskaya et al. 1983).
436
R.P. Seisyan and B.P. Zakharchenya
reliable determination not only of effective masses at the band extrema but also
of the pattern of deviations from the parabolic dispersion relation as well. One
can at the same time derive precise data on spin splitting, while extrapolation to
Β = 0, taking into account the binding energies ^ d e( _ ? ) , yields correct values of
the energy gaps.
Such possibilities for analysis may arise due to the additional transitions
made possible, for example, by the inherent or externally applied electric field
oriented perpendicular to the magnetic field [with selection rules Δ/ = ( Δ / ) 0
± 1], by warping, anisotropy etc.
An interesting possibility for band structure studies is offered by analyzing the
behavior of the first two light-hole Landau levels in the σ~ spectrum. Here the
expressions for the energies derived from the Pidgeon-Brown determinant
equations become greatly simplified. As a result, by calculating the mean and
difference energies of two observed doublets and by combining them in a certain
way [see, e.g., Kanskaya et al. (1983) or the monograph of Seisyan (1984)] one
succeeds in determining, besides others, such parameters as k and q characteriz­
ing the 'spin' splitting of hole levels and its anisotropy which are difficult to
measure otherwise. However, since the steepest variation of $ d e as a function of
Landau number and magnetic field occurs exactly at these lines there is no way
to obtain accurate data without a correct evaluation of the binding energies and,
in some cases, of the nonparabolicity as well (see fig. 58).
The most accurate and internally consistent band structure data should come
from computer fitting of the theoretical to the experimental spectrum. In this
procedure the computer performs a large amount of computational work
involving the calculation of the whole spectrum of transitions between the
Landau sub-bands corrected for the binding energy with variation of all the
band structure parameters. By properly choosing the fitting strategy, based on
the theory of experiment design optimization, one can reach in a few hundred
steps an absolute minimum of the sum of squared deviations, with the initial
position chosen, for instance, by the above mentioned technique. In experiments
on InSb where 125 experimental points were obtained such a minimum
Fig. 58. Dependence of dimensionless differences Δε and half-sums έ for light-hole transitions from
ly = — 1, 0 states on e g(2e c + ε 8) for InSb (see maxima 1, 2 of the σ~ spectra in fig. 26). The straight
lines are least-squares fits (by Kanskaya et al. 1983).
Diamagnetic exciton spectroscopy of semiconductors
437
corresponded to a deviation, on the average, of not more than 0.6 meV for the
experimental points. Note that inclusion of the exciton nature of the effect
results in a substantially smaller standard deviation than would follow from a
'no-exciton' analysis, indeed, the minima turn out to be deeper, and the values of
the parameters are shifted substantially. It is also essential that in this approach
the data obtained from an analysis of the O M A and cyclotron resonance
practically do not differ from one another.
Progress in the spectroscopy of semiconductor crystals and the applied optics
of semiconductors provides a basis for the development of integrated optical
circuits; the advent and development of a new type of semiconductor device
based on exciton effects requires an ever increasing accuracy of our knowledge
concerning the band structure parameters of semiconductors. Interband
magneto-optics or diamagnetic exciton spectroscopy remains a powerful tool to
use for these purposes.
Note added in proof
In the time that has passed after the preparation of the manuscript, certain
progress in diamagnetic exciton spectroscopy has been reached. Three groups
of publications which are most significant for the development of magnetooptics should be mentioned here. The first of them relates to the observation
of quasi-Landau absorption oscillations in InP and CdTe crystals in a weak
field where the β > 1 criterion fails. The second is connected with magnetooptics of quasibinary solid solutions in the A 3 B 5 system, namely, Ini_j.Gaj.As
with x ~ 0 . 5 . The third group deals with magneto-optics of 2 D structures quantum wells and superlattices.
1. Quasi-Landau oscillations with a period ~ΗΩ were observed in 'pure'
epitaxial InP layers and very nearly perfect CdTe single crystals (Abdullaev et
al. 1988). Unique spectra with a multiplicity of very narrow lines were obtained
under conditions far from the strong field criterion in samples where already
at Β = 0 one observes not only the ground (n = 1) but excited (n > 2) states of
the Wannier-Mott exciton as well. This situation is very similar to that
described in section 4.2 for GaAs, as well as for other relatively wide-gap semi­
conductors with a well-developed exciton spectrum structure. A detailed investi­
gation of the region of comparatively weak fields shows that the transition from
the weak magnetic field phenomena, such as the quadratic-in-field diamagnetic
shift, to the linear-in-field oscillatory magneto-absorption occurs at fields corre­
sponding to β(η$ + l ) 2 ^ 1, where n% is the highest Wannier-Mott exciton state
observed at Β = 0. Observation of Ν = 3 in InP (Abdullaev et al. 1989a) and
ng = 2 in CdTe (Abdullaev et al. 1989b) can account for the appearance of the
oscillatory spectrum at f ? ^ 0 . 4 T and 1.6 Τ for InP and CdTe accordingly,
which corresponds to (ηξ + l ) " 2 ^ β ^ 1.
438
R.P. Seysyan and B.P. Zakharchenya
2. O n e c a n reveal here a n interesting a n a l o g y w i t h the s p e c t r o s c o p y of
'Rydberg' a t o m s a n d m o l e c u l e s , w h e r e a n u m b e r of n e w p h e n o m e n a sets in if
the c o n d i t i o n s p r o d u c e d in the l a b o r a t o r y or existing in interstellar m a t t e r
b e c a m e favorable for the d e t e c t i o n of excited states with a giant principal
q u a n t u m n u m b e r n 0 ^ 5 0 0 (see e.g., D a l g a r n o (1985)). T h e a t o m in s u c h a highly
excited state has m a c r o s c o p i c d i m e n s i o n s , s o that the o u t e r electron, in m o v i n g
in a giant orbit, m a k e s p o s s i b l e quasiclassical d e s c r i p t i o n of the a s s o c i a t e d
effects, of particular interest in this c o n n e c t i o n b e i n g the electric a n d m a g n e t i c
field p h e n o m e n a . N o t e that the sensitivity t o external field turns o u t t o be the
higher, the greater is n% o b s e r v e d under the e x p e r i m e n t a l c o n d i t i o n s , s o that a
relatively w e a k m a g n e t i c field b e c o m e s s t r o n g e n o u g h t o p r o d u c e a quasiL a n d a u s p e c t r u m near t o c o n t i n u u m , a n d , still further, for hv >
(ε^ is the
d i s s o c i a t i o n limit), of L a n d a u levels.
Since C o u l o m b interaction in a s e m i c o n d u c t o r o c c u r s in a m e d i u m with a
dielectric c o n s t a n t κ0, a n d the reduced m a s s μ is m u c h less t h a n the electron
m a s s in a v a c u u m , o n e s h o u l d i n t r o d u c e , apart from the c h a n g e s in critical
field scales a s s o c i a t e d w i t h ng, a l s o coefficients i n c l u d i n g different p o w e r s of κ0
a n d μ. T h e n the highly excited states of h y d r o g e n w i t h n0 ^ 30 c a n be rightfully
c o m p a r e d w i t h the ' R y d b e r g ' states w i t h n0 = 2, 3, 4 in I n P , G a A s , or C d T e .
3. W h i l e o n e has s u c c e e d e d in recent p u b l i c a t i o n s in s h e d d i n g light o n the
genesis of the q u a s i - L a n d a u s p e c t r u m in s e m i c o n d u c t o r s , the c a l c u l a t i o n of
spectra, in particular, of the d i a m a g n e t i c e x c i t o n b i n d i n g energies for β < 1 still
remains a n o p e n p r o b l e m , since the c o n d i t i o n β > 1 w a s a n inevitable prerequi­
site t o theoretical c a l c u l a t i o n s at several stages of analysis of the S c h r o d i n g e r
equation.
T h e a s s u m p t i o n of the possibility of a d i a b a t i c s e p a r a t i o n of variables, i.e., of
a description w i t h t w o c o u p l e d e q u a t i o n s , (8) a n d (9), underlies the k n o w n
a p p r o a c h e s . M a t h e m a t i c a l l y this s y s t e m c a n be used in place of the corre­
s p o n d i n g e q u a t i o n w i t h inseparable variables in the case where the frequency
of m o t i o n a l o n g the m a g n e t i c field is less t h a n that in the p l a n e perpendicular
t o the field. It s h o u l d be p o i n t e d o u t that the e x p e r i m e n t a l fact proper of
existence of a q u a s i - L a n d a u s e q u e n c e of m a x i m a m a y be c o n s i d e r e d as a n
a r g u m e n t for the real s e p a r a t i o n of the m o t i o n s , w h i c h m a y justify a n a t t e m p t
t o o b t a i n correct values of the b i n d i n g energy by a n a l y z i n g the o n e - d i m e n s i o n a l
Schrodinger e q u a t i o n ( K o k h a n o v s k i i et al. 1990a). T h e n for s o m e hv > ε 8 the
values of Rde c a n be f o u n d by s o l v i n g eq. (9) numerically w i t h o u t i m p o s i n g a n y
constraints o n β. S u c h b i n d i n g energies h a v e b e e n calculated b y variational
t e c h n i q u e using the l o n g i t u d i n a l reduced m a s s (38) a n d p o t e n t i a l (39) typical
of d i a m o n d - l i k e s e m i c o n d u c t o r s , a n d were f o u n d t o be substantially l o w e r t h a n
t h o s e derived analytically. F r o m the theoretical s t a n d p o i n t , the use of these
data c a n be justified o n l y for βΐ > 1 (see eq. (47)), a l t h o u g h this c o n d i t i o n d o e s
in n o w a y represent a physical constraint o n the existence of the q u a s i - L a n d a u
spectrum.
Diamagnetic exciton spectroscopy of semiconductors
439
The solution of the problem dealing with the determination of the real
boundaries for the application of these results was favored by the observation
in the spectra of lines that could be interpreted as excited states of the diamag­
netic exciton with v = 1, 2, for which the strong field condition is met, thus
making analytical calculations possible. Reconstruction in this way of the
hypothetical positions of the dissociation edges or of transitions between the
Landau sub-bands has subsequently revealed that the accuracy with which the
obtained spectra can be coincided with the ground states of diamagnetic
excitons with ν = 0 is not worse than 10% for all spectral lines with / ^ 1. Note
that the 'lowest', / = 0 state remains three-dimensional and is described in terms
of the theory of the Zeeman effect and of the diamagnetic shift. The good
agreement between the calculated and experimental spectra permitted one to
use the advantages of interband magnetospectroscopy lying in the possibility
of observing Landau levels deep in the bands and to obtain a system of selfconsistent band structure parameters of InP which takes into account the effect
of the upper bands.
CdTe reveals remarkable effects of exciton-phonon interaction which
interfere with exact calculations (Abdulaev et al. 1990).
4. Spectroscopic manifestations of the exciton states localized at potential
fluctuations have been widely studied on quasibinary A 2 B 6 solid solutions. As
for the A 3 B 5 solid solutions, the exciton states here are comparatively shallow,
no convincing evidence for the effect of potential fluctuations on them being
available. Progress has been reached here recently with the use of magnetic
fields in interband magneto-absorption experiments on lnl .^Ga^As which was
investigated both in a free form and on InP substrates. One observed here for
the first time narrowing of the magneto-absorption lines which was subse­
quently followed by their broadening, in full agreement with the theory of
Raikh and Efros (1984, 1988), (Kokhanovskii et al. (1990b)). The band param­
eters of this compound with the exact composition and stressed state of the
epitaxial layer taken into account were also obtained (Kakhanovskii et al.
1990c).
5. We will not dwell here on the investigation of the interband magnetooptics of quantum wells and superlattices since they deserve a special consider­
ation. Note only that analysis of new experimental data requires using not
solely the exact theory of magneto-optic absorption including the formation of
minibands (Berezhkovskii et al. 1982; Berezhkovskii and Suris 1984a, b) but
the exciton nature of the absorption maxima as well. Only under these condi­
tions can the possibility of detecting changes in the band structure parameters
caused by lowering of the system's dimensionality be implemented to the full
extent.
440
R.P. Seysyan and B.P. Zakharchenya
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CHAPTER 8
Interband Magneto-optics in
Narrow-gap Semiconductors
CR. PIDGEON
Department of Physics
Heriot-Watt
University
Edinburgh, United Kingdom
Landau Level
© Elsevier Science Publishers B.V., 1991
Spectroscopy
Edited by
G. Landwehr and E.L Rashba
Contents
1. Introduction
447
2. Theory
448
2.1. Zinc-blende semiconductors and semimetals: the quasi-germanium model
3.
448
2.2. The lead salts
454
2.3. Tellurium
458
2.4. The cadmium-arsenide system
460
Experiments
461
3.1. Zinc-blende semiconductors and semimetals
461
3.2. The lead salts
468
3.3. Tellurium
472
3.4. The cadmium-arsenide system
474
References
479
L
Introduction
Since the discovery of cyclotron resonance in semiconductors (Dresselhaus et al.
1953, Lax et al. 1954) and with the advent of well controlled single-crystal
material of sufficiently high mobility, magneto-optical methods have provided
some of the most powerful and precise techniques for determining energy-band
parameters at symmetry points in the Brillouin zone. In general, both intraband
and interband magneto-optical transitions should be studied for a complete
picture. The intraband effect is discussed in an accompanying chapter by
Zawadzki (chapter 9); interband magneto-optical (IMO) studies form the
subject of the present chapter. The field is still too large in scope for a single
chapter, so we confine our attention here to narrow-gap semiconductors thereby largely removing the problem of Coulomb (excitonic) interactions,
which are discussed elsewhere. We follow Zawadzki (1980) in our definition of
'narrow-gap', meaning those materials where IMO experiments can easily be
carried out with electron energies, £ , becoming comparable to the energy gap,
£ g . This, in turn, means a small effective mass and density of states, so that
even a relatively small conduction-electron concentration populates the band
up to quite high energies above the band edge (the so-called 'Burstein-Moss'
effect). The proximity of the valence band is strongly felt, and the E(k) curve
becomes highly non-parabolic.
Semiconductor magneto-optics is by now a mature and established subject.
Representative reviews of different aspects have been given, by Mavroides
(1972), Aggarwal (1972), McCombe and Wagner (1975), Pidgeon (1980), Bauer
(1980), Weiler (1981), Dornhaus and Nimtz (1983), Nimtz and Schlicht (1983),
Singh and Wallace (1987) and references therein. The proceedings of the
'International Conference on Narrow-Gap Semiconductors' series give an excellent review of the general properties of these materials, as also do the proceedings of the biennial 'International Semiconductors Conferences'. A glance at the
last three of the latter quickly makes it obvious that the field has moved dramatically from a study of bulk crystals to that of low-dimensional systems, made
possible by molecular beam epitaxy (MBE) and associated growth techniques
(this subject is discussed elsewhere in companion chapters). Thus, we do not
wish to be encyclopaedic here but an attempt has been made to put down the
basic physical ideas, while referring the reader elsewhere for further details. For
ease of reference, the units used are those of the original sources.
Magneto-optical investigations have reached a particularly advanced state of
sophistication in their application to semiconductors which crystallize in the
zinc-blende structure. These are III-V compounds such as InSb, GaAs, and
II-VI compounds, such as HgTe, CdTe, and CdS. The most successful
application has been with the narrow-gap zinc-blende semiconductors (of which
the prototype is InSb) because of the high quality of the samples available. The
quasi-germanium model originally proposed by Pidgeon and Brown (1966)
448
CR. Pidgeon
provides a n excellent description of the b e h a v i o u r of electrons in b o t h the
c o n d u c t i o n a n d the valence b a n d s of I n S b (Weiler 1978). A very successful
a p p l i c a t i o n of m a g n e t o - o p t i c a l t e c h n i q u e s h a s b e e n the d e t e r m i n a t i o n of the
electronic b a n d structure of the alloy s y s t e m H g ^ j - C d ^ T e . It w a s a m a g n e t o optical study of the s e m i m e t a l H g T e by G r o v e s et al. (1967) w h i c h established
that the q u a s i - g e r m a n i u m band-structure m o d e l c o u l d be used t o describe the
energy b a n d s in H g T e as well as in gray tin, but w i t h a negative, or inverted,
energy g a p ( G r o v e s a n d Paul 1963, G r o v e s et al. 1970). T h i s m o d e l has b e e n
used successfully (Weiler et al. 1977, G u l d n e r et al. 1977) t o describe the
electronic energy b a n d s of the alloy s y s t e m H g ^ ^ C d ^ T e for u p t o 3 0 % C d T e
(x = 0.3). O v e r this w i d e range of alloy c o m p o s i t i o n , x, the m o d e l parameters
are essentially c o n s t a n t except for the energy g a p w h i c h c h a n g e s from —0.3 e V
(inverted or semimetallic structure) t o + 0 . 2 e V ( n o r m a l or s e m i c o n d u c t i n g
structure).
T h e earliest i n v e s t i g a t i o n s o n H g x _ x C d x T e were a i m e d at the d e v e l o p m e n t of
infrared detectors a n d L E D or laser sources. T h e s a m e c a n p r o b a b l y a l s o be
said of the n a r r o w - g a p lead salts a n d their alloys. In practice, crystals
of Ug1 _ x C d x T e available n o w a d a y s h a v e carrier densities a b o u t t w o orders of
m a g n i t u d e smaller t h a n the lead salts, s o that the c o r r e s p o n d i n g l y larger
detectivities of the first m o r e t h a n o u t w e i g h the greater difficulty of preparation.
O n the other h a n d , the lead-salt s y s t e m s are n o w preferred as t u n a b l e d i o d e light
sources in the 3 t o 35 μιη region. C o r r e s p o n d i n g l y a great deal of a t t e n t i o n has
b e e n d e v o t e d t o s t u d y i n g the e n e r g y - b a n d structure of the lead salts a n d alloys,
a n d the m a g n e t o - o p t i c a l studies form the s e c o n d part of the present chapter.
O t h e r n a r r o w - g a p s e m i c o n d u c t o r s y s t e m s that h a v e received s o m e a t t e n t i o n
over the years are tellurium a n d the c a d m i u m - a r s e n i d e family. W e deal with
t h e m s o m e w h a t m o r e briefly here.
2.
Theory
2.1. Zinc-blende semiconductors and semimetals: the quasi-germanium
model
T h e energy b a n d s near the f u n d a m e n t a l g a p of a z i n c - b l e n d e s e m i c o n d u c t o r are
illustrated o n the left-hand side of fig. 1. T h e g a p o c c u r s at the z o n e center, or Γ
point, where the states transform a c c o r d i n g t o representations of the crystals y m m e t r y g r o u p T d. W i t h o u t spin, the S-like c o n d u c t i o n b a n d transforms as Γΐ9
a n d the triply d e g e n e r a t e P-like valence b a n d as Γ 4 . I n c l u d i n g spin, the valence
b a n d splits i n t o b a n d s transforming as Γ8 (J = f ) a n d Γ 7 (J = ^) with s p i n - o r b i t
splitting energy A. T h e c o n d u c t i o n b a n d transforms as Γ 6 (J = {ή. In a n applied
m a g n e t i c field, H, the b a n d s are q u a n t i z e d i n t o L a n d a u levels as illustrated o n
the right-hand side of fig. 1 (Weiler 1981), w h i c h s h o w s the density-of-states
peak (kH = 0) for e a c h level. W i t h certain a p p r o x i m a t i o n s , these levels are
Magneto-optics in semiconductors
449
3
(a)
(b)
Fig. 1. (a) Zinc-blende semiconductor energy bands ( H = 0) and (b) the same bands in applied
magnetic field H. Note that vectors are indicated by over arrows in the figure and by bold italic
letters in the text (after Weiler 1981).
c o u p l e d in g r o u p s of four o n l y , s o that calculating the energies i n v o l v e s
d i a g o n a l i z i n g 4 x 4 matrices. T h e α-set levels are the s p i n - u p state in the
c o n d u c t i o n b a n d , w i t h L a n d a u q u a n t u m n u m b e r n, the states m3 — \ (with
η — 1) a n d — \ ( w i t h η + 1) in the v a l e n c e b a n d , a n d the m3 = \ state in the splitoff b a n d (with η + 1). A similar c o u p l i n g o c c u r s a m o n g the states in the b set.
T h e set of c o u p l e d levels are all labelled w i t h the c o n d u c t i o n - b a n d q u a n t u m
n u m b e r in fig. 1. T h u s , the v a l e n c e - b a n d states labelled — 1 are n o t c o u p l e d t o
a n y c o n d u c t i o n - b a n d level. In order t o o b t a i n the c o m p l e t e H a m i l t o n i a n for the
set of c o u p l e d b a n d s illustrated in fig. 1, there h a v e b e e n t w o different
a p p r o a c h e s b a s e d o n g r o u p - t h e o r e t i c a l t e c h n i q u e s . In the first a p p r o a c h ( K a n e
perturbation
1957) o n e finds i n t e r b a n d matrix e l e m e n t s ραβ of the k*p
H a m i l t o n i a n hk 'p^/m, a n d a l s o of the s p i n - o r b i t H a m i l t o n i a n . F o r the case of
InSb, K a n e e n u m e r a t e d all p o s s i b l e c o m b i n a t i o n s of these matrix e l e m e n t s t o
s e c o n d order in k a m o n g the s i n g l e - g r o u p basis states Γ χ ( c o n d u c t i o n b a n d S)
a n d Γ 4 (valence b a n d s X, Y, a n d Z). T h e first-order matrix e l e m e n t s ραβ a n d
s e c o n d - o r d e r c o m b i n a t i o n s i n v o l v i n g i n t e r m e d i a t e states b e l o n g i n g t o different
CR. Pidgeon
450
representations of the T d single group were adjustable parameters of the perturba­
tion Hamiltonian. This Hamiltonian was expressed in terms of linear combinations
of the functions X, Y, and Ζ and of the spin functions f and j , which diagonalize
the spin-orbit interaction. The Hamiltonian involves matrix elements coupling
the various basis states including adjustable parameters multiplying functions to
second order in k. A second method for obtaining this Hamiltonian was that of
Luttinger (1956), who used a group-theoretical analysis to find all allowed
matrix elements of k and k χ k among the valence-band states transforming as
the Γ 8 representation of the double group. His result involved adjustable
parameters which were linear combinations of those of Kane, but included an
additional parameter q which is nonzero only in the presence of spin-orbit
splittings of the intermediate states. Luttinger's results were extended by Roth et
al. (1959) to include the F 7 split-off band. Pidgeon and Brown (1966) included
the Γ 6 conduction band in their analysis, combining the results of Kane and of
Roth et al. They called their model the quasi-germanium model since they
neglected the effects of inversion symmetry so that the model strictly applies
only to semiconductors such as Ge and Si. The use of this model in the analysis
of magneto-optical experiments has been developed by Weiler et al. (1978,1981),
using the following set of basis functions (chosen to make the 4 x 4 matrices
real):
a set:
10 = ^ / 2 . 1 / 2 =
ST,
|3> = ^ 1,2.3/2 = - ( ί / Λ / 2 ) ( Χ + ίΥ)Τ,
|5> = ^ ! , 2 . - i / 2 = " (i/76)[( X + i Y)T +
U>
2 Z
|7> = * _ , 2 . - i / 2 = - ( i / v / 3 ) [ ( X + i Y ) T - Z U ,
(1)
b set:
|2> = ^ , 2 . - 1 / 2 =
Si,
|6> = Φί/2,1,2 = ~ (i/x/6)C(X + iY)i " 2ZT],
|4> = ^ I / 2, - 3 / 2= - ( i / v / 2 ) ( X + iY)|,
|8> = ΦΙ/2ΛΙ2 = ~ (i/x/3)[(X + iY)T + Z j ] .
(2)
These basis states are not identical to those of any of the important earlier
papers on the zinc-blende Hamiltonian. However, this previous work involves
several inconsistent basis sets.
The Γ 8 χ Γ 8 parameters γΐ9 γ2, y 3 , κ and q were shown by Roth et al. (1959)
also to involve the split-off band Γ when one starts with single-group
representations. In the full double-group picture the ΓΊ χ Γ 7 and ΓΊ χ Γ 8
parameters are independent of the Γ 8 χ Γ 8 ones and are denoted by γ\,Υ29 y' 3, κ\
Magneto-optics in semiconductors
451
a n d κ". T h e c o n d u c t i o n - b a n d ( Γ 6 ) effective m a s s p a r a m e t e r F , the 'linear-k'
p a r a m e t e r C for Γ 8 , a n d the Γ 6 χ Γ 8 p a r a m e t e r s Ρ a n d G were defined b y
D r e s s e l h a u s ( 1 9 5 5 ) , K a n e ( 1 9 5 7 ) , a n d D r e s s e l h a u s et al. (1955) in terms of singleg r o u p basis states; the Γ 6 χ Γ 7 a n d Γ 7 χ Γ 8 p a r a m e t e r s are d e n o t e d b y C,
a n d G'. Weiler (1981) a l s o o b t a i n s three n e w parameters: Ni9
N2,
a n d N3.
P',
N1
c o n t r i b u t e s t o the c o n d u c t i o n - b a n d g factor, similar t o κ in the v a l e n c e b a n d s ;
Ν2 a n d N3 represent a d d i t i o n a l c o u p l i n g s b e t w e e n the Γ6 c o n d u c t i o n b a n d a n d
the Γ8 v a l e n c e b a n d s . T h e s e n e w p a r a m e t e r s , like q, arise from the s p i n - o r b i t
splitting of higher b a n d s .
T h e p a r a m e t e r s q a n d μ = j(y3
— y2) are the s o - c a l l e d G e w a r p i n g p a r a m e t e r s
b e c a u s e , if these were z e r o , in a material w i t h i n v e r s i o n s y m m e t r y like G e , the
e n e r g y b a n d s w o u l d b e i s o t r o p i c o r i n d e p e n d e n t of the direction of k. T h e
a n d N3
p a r a m e t e r s C, G, N29
are called the i n v e r s i o n a s y m m e t r y parameters
b e c a u s e they are z e r o in materials w i t h i n v e r s i o n s y m m e t r y like G e b u t n o n z e r o
in the z i n c - b l e n d e materials.
F o r a m a g n e t i c field in a n arbitrary direction, o n e performs a n a p p r o p r i a t e
coordinate
transformation.
The
resulting
3^(0)
separates
into
two
4 x 4
matrices for the a a n d b sets,
Jf(0)
«Γ* Ί .
Lο
(3)
j r j
if o n e n e g l e c t s t e r m s p r o p o r t i o n a l t o kH,
q, C, G, N2,
a n d N3 a n d m o s t terms
p r o p o r t i o n a l t o the w a r p i n g p a r a m e t e r μ = ^(γ3 — y2)
(Pidgeon and
Brown
1966) - see table 1. T h e s o l u t i o n s t o t h e S c h r o d i n g e r e q u a t i o n s
JTJa> = £Ja>,
jr \by = E \b>,
(4)
|6(n)> =
(5)
b
b
are of the for m
0"ΐΦη
|a(n)> =
<Γ3Φ.-ι
ΚΦη+Χ
ΉΦη+Ι
For
w i t h η > — 1 a n d w i t h a,
= b
a 3 = a3
ri^zl there are eight i n d e p e n d e n t s o l u t i o n s [|a(n)>, |fc(n)>] for e a c h n, w h i c h are
d e n o t e d , in order o f d e c r e a s i n g energy, c o n d u c t i o n b a n d [|tf c(n)>, | & c( « ) > ] , h e a v y
hole
[|α~(π)<,
|fe"(n)>], light
hole
[|a + (n)>, |6 + (w)>],
and
split-off
band
[|a s(n)>, |fe s(n)>]. T h e s e states are illustrated in fig. 1. T h e energies of these levels
are c alc u late d numerically, for a g i v e n set of p a r a m e t e r s a n d for a g i v e n value of
the m a g n e t i c field, by s o l v i n g eq. (4), i.e., by d i a g o n a l i z i n g the 4 χ 4 matrices
given in table 1.
452
Table 1
Pidgeon and Brown model Hamiltonians from eqs(4) (after Weiler 1981).
£ g + s[(2n + l)F + N ! + n + l ]
s/insEp
y/i(n
+ \)sEp
-yi(n+l)s£p
ν4(π+1)515,
-s[(«-i)(yi+r') + !^]
— w"y/6n{n + 1)
|3>
_7" v/3n(n -1-1)
-s[(" + i ) ( y i - / ) - K l
( l / v/ 2 ) s [ ( 2 n + 3 ) / - i c - l ]
|5>
—sy"y/6n{n + 1)
(l/V^M^n + W - K - l ]
-J-s[(M + | )
7
-l ^ - i ]
__g + 5 [ ( 2 « + l ) F - i V 1 + , i ]
yJ^sEp
-Ji(n+l)sEp
J^nsEp
ID
sy"y/3n(n + 1)
_
|7>
|2>
- * [ ( » - i ) ( y i - / ) + **]
sy"^3n(n + 1)
- ( 1 / 7 2 ) ^ ( 2 , 1 + l ) y ' + /c+l]
sy"y/3n(n + 1)
+!)(7ι+/)-!*]
sy"^/6n(n + 1)
-(l/V^)s[(2n+l)/ + /c+l]
|6>
sy"y/dn(n + 1)
|4>
- Λ - 5 [ ( ι ι - ± ) ) Ί + κ + £|
|8>
C.R. Pidgeon
-^(n+\)sEp
JinsEp
Magneto-optics in semiconductors
453
Here, the terms involving yl9y',
and γ" in the fourth rows and columns are
etc. Also in table 1,
included in the single-group approximation P* ~P9y'1~yl9
Ρ is included using the interband energy Ep = 2mP2/h2, and s ~ heH/mc is the
free-electron cyclotron energy. Table 1 is now in ordinary energy units, since F g,
Δ, Ep, and s have the dimensions of energy and F, κ, Nu and the y's are
dimensionless.
The Hamiltonians of table 1 are equivalent to those of Pidgeon and Brown
(1966) and of Aggarwal (1972), except for corrected sign errors in some small,
off-diagonal, terms, and when account is taken of the different basis sets.
Similarly, table 1 is also equivalent to eqs. (B-9) and (B-10) of Roth et al. (1959)
for the Γ 8 and Γ 7 band energies.
The strongest allowed optical transitions among the states of eq. (5) are those
proportional to the interband matrix element P; i.e., we find the optical
perturbation,
', by replacing k by k + (eA/hc) in the terms proportional to Ρ in
the Hamiltonian. Here A is the light vector potential in the radiation gauge. The
resulting transition matrix elements are
< α ( π ' ) | Λ Η Φ ) > = (eE/co)jEp/6m
{ [ ^ 3 a%a\, -a\{a\
- J2
αηΊ)
χ έ - < 5 Λ> + 1 + [ Μ ^ η Ί έ + ^ ν , - ι } ,
(b(n')\J>r\b(n)> = (eE/a>)jEp/6m
{[_(bn6 + ^2 bn8')bn2 - ^3
χ ε_δ
<b(n')\J>T\a(n)> = (-eE/w)^Ep/6m
\bn2\^2
Λ > 1+
bn2b\~\
+ [ η ^ η , ] ε + ^ Λ >_ 1 } ,
a\ + αηΊ)
x ( v / 2 f c S ' - f t S V i ] M « > + i.
(6)
where Ε is the optical electric field, ε + and ε_ are the unit polarization vectors
+ ίε 2) for, respectively, right and left circular polarization c r R< 7 L, trans­
verse to the magnetic field H, and ε 3 is a unit polarization vector parallel to Η
referred to as the π or E\\H polarization. Thus the selection rules for both
interband and intraband transitions (before renumbering of the valence bands)
are,
aL: an->an-u
<j R: an^an-u
n:an-+bn+l,
bn-+bn+u
bn-^bn^l9
bn^an.x.
(7)
The selection rules for interband transitions given by Roth et al. (1959) and by
Pidgeon and Brown (1966) were, for oL and σ κ, An = 0 and —2 rather than + 1
and — 1, respectively. This is because they renumbered all valence-band states
n->n + 1, so that the state numbers η correspond to the quantum number of one
of the larger terms [the coefficient of state |5> or |4> in eqs (5)]. Here, we do not
454
CR. Pidgeon
renumber the valence-band states, so that each set of states η is coupled in the
quasi-germanium model (see, e.g., Groves et al. 1970).
A rich spectrum of IMO transitions is allowed from eq. (7) in all polarizations
( a L, σκ and π) and can be analysed to obtain the band parameters of the quasiGe model Hamiltonian for a given material (see section 3.1). Additional
transitions, produced by the warping and inversion asymmetry perturbations,
are also allowed for particular crystalline orientations with respect to H. These
were observed by Pidgeon and Groves (1969) and analysed by perturbation
theory to obtain estimates for the corresponding band parameters in InSb.
Further details were obtained using modulation spectroscopy techniques, as
reviewed extensively by Aggarwal (1972). See also the chapter by Aronov and
Pikus in the present volume, on electric field effects (chapter 10). A more exten­
sive analysis has been given of both one- and two-photon selection rules for
inter- and intraband transitions associated with warping and inversion asym­
metry by Wlasak (1978), Zawadzki and Wlasak (1978), Zawadzki (1978, 1980),
Weiler (1982) and Chen et al. (1985). A study of high-resolution two-photon
magnetospectroscopy was made by Seiler et al. (1982) for n-InSb in fields up to
15 Τ using C 0 2 lasers. Anisotropic effects were observed for the first time in
two-photon transitions. The technique was extended to the photo-Hall effect
to determine the temperature dependence of the energy gap (Littler and Seiler
1985).
2.2. The lead salts
Lead telluride has been of considerable interest for around a quarter of a century
as a material for the construction of infrared solid-state lasers. It has been the
prototype for all work on the lead-salt system, despite discrepancies in basic
band parameters (Dalven 1973, Bauer 1978) and we consider it here. The lead
chalcogenides possess the cubic NaCl structure, which has inversion symmetry
in addition to the symmetry elements of the zinc-blende structure of the III-V
compounds. As a result, due to spin or time reversal, two-fold Kramers
degeneracy is preserved at all points in the Brillouin zone. Whereas III-V
compounds have four-fold degenerate valence bands at Γ with maxima close by,
the lead salts have two-fold Kramers-degenerate valence-band maxima at the L
points. The conduction-band minima are also at the L points, and it is this
symmetry which makes PbTe successful as a laser source.
A schematic representation of the bands considered when discussing the
properties of PbTe is shown in fig. 2, the exact ordering was determined
primarily by pseudopotential calculations (Bernik and Kleinman 1970, Mar­
tinez et al. 1975) and confirmed experimentally by Jedrzejczak et al. (1978). The
ordering cannot be calculated without a proper inclusion of the spin-orbit
mixing of the levels, which also gives the lead chalcogenides their varied shapes
of the constant-energy surface. In its absence, only the longitudinal or transverse
Magneto-optics in semiconductors
L 63:
/
/
-2-
455
L 3i -
/
/
/
\
\
-0.2 eV
y
6 +
L 1 +
-2.6 eV
45 +
' 3 +
6 +
L
3
+
Fig. 2. Band structure of PbTe at the L point, without spin-orbit coupling (left) and with spin-orbit
coupling (right) (after Singleton et al. 1986).
component of the momentum is coupled between a pair of levels at the L point,
and highly anisotropic energy surfaces are expected. This is observed in PbTe,
which has an anisotropy of its ellipsoidal energy surfaces of approximately 11,
whilst in PbS, with its strong spin-orbit coupling, the anisotropy is only 1.1.
The two-band energy-dependent effective-mass approximation proves to be
totally inadequate in fitting experimental data and the six-band model of
Mitchell and Wallis (1966) must be used. In this k 'p type model the interaction
between the lowest conduction ( L f l ) and the highest valence (Lf + ) levels
is treated exactly, with the next four levels being included to order fc2 at the
L point.
The Mitchell-Wallis Hamiltonian matrix for one valley of PbTe in the
presence of a magnetic field is shown in table 2, where the following symbols are
used: M * is the far-band contribution to the electron ( —) or hole ( + )
longitudinal effective mass, Μ * is the far-band contribution to the electron or
hole transverse effective mass, g*, &?are the far-band contributions to the
transverse and longitudinal g factors, M c v gives the contribution to the effective
456
CR. Pidgeon
Table 2
The complete Hamiltonian matrix for the Mitchell-Wallis model of PbTe in a magnetic field (after
Singleton et al. 1986).
2
gfo
—
+
Eg
μ*(Βχ-\Β,)
fa
Mt
eh
Γ
2
T /:
2M C,
h2k2
Β
^ 2 ^
_
ϊ*Γμ*(Βχ + ΊΒν)
2
Γ
2)
Mx
eh
T /2
/
Eg
V1
hk
h2k2
2M n(
Eg
V12
eh
Γ
Ύ12
— Eg
eh
2
M tH
h2k2
2M R+
Γ
eh
11
/
2/
Es
V12
2 - * ? *
\gfr*Bz
+»
h2k2
+ -——
2MZ
masses from the direct interaction between conduction- and valence-bands and
Μ % = Mi(vg/vf). The situation has been reviewed by Singleton et al. (1986).
The matrix is exactly soluble when Β is parallel to [111] (Grisar et al. 1977)
with solutions of the form
Γ
eh
ECZZ ί = \(an - bn) ± i | (an - bn)2 + 4Eg—B(n
Ύ12
+ 1) I
,
Γ
eh
Ύ12
ECVZ= ΐ = i ( c B + dn) ± ±Ucn - dn)2 + 4 £ b — B ( n + 1 ) 1 ,
(8)
where the label c indicates the conduction band, ν the valence band, σ = +
corresponds to spin up or down and n, the Landau level quantum number, runs
from — 1 upwards for the σ = — levels, and from 0 upwards for the σ = + levels.
The a„, bn and dn are defined below:
eft
fl» = i ^ g + ^
bn = -\E%
+
#(n + i ) + k i
eft
μ Β^
B(n + f) - ±£, +μ ΒΒ,
Magneto-optics in semiconductors
457
eh
eh
dn = - \ E % +
B{n + i ) +
(9)
When 2i is not parallel to the major axis of the constant energy surface, the
matrix may either be solved numerically or approximated analytically. If the
contributions due to the interaction with the further bands can be considered
small compared with the direct conduction-valence-band interaction, an
approximate solution may be obtained by including them in second-order per­
turbation theory. This calculation was carried out by Baraff (1965) for the
conduction-band Landau levels in bismuth; the results are directly applicable
and agree with the numerical methods for PbTe to within 5% (Burkhard et al.
1979). To carry out the approximation, the following substitutions must be
made to the above Landau-level energies:
_
g(
g
P|
(Pjj s i n 2 0 + P 2 c o s 2 0 ) 1 / '2
1
M cv
1
M±
s m 2 0 + P ±c o g 2g
| f t
2 P±(Pl
Μ
1
M±
s i n 2 0 + P 2 c o s 2 0 ) 1 /2
Mi
=
}
(11)
Eg
a(Jj
+ f i w t ± / m 1 ± ) sin 2 fl + P 2 c o s 2 Θ]
M±PL(P\
sin 2 θ + Ρ 2 c o s 2 θ)1·2
9
'
where P ± and P| ( are the conduction-band and valence-band momentum matrix
elements, respectively.
The notation here is such that the g factor-split Landau levels are (n + 1) | + >
and η | — >; however, this notation is not in common use and to agree with
previous references, the quantum numbers of the η \ — > ladder should be
increased by one, to give the level ordering 0 " , 0 + , 1", 1 + , . . . in increasing
energy in the conduction band (Burkhard et al. 1979). Selection rules for oneand two-photon transitions were previously given, e.g., by Grisar et al. (1977).
Singleton et al. (1986) make the point that the lowest Landau levels in the
conduction band and the highest valence-band Landau levels have a field
dependence determined solely by the far-band parameters, and that Landau
levels of consecutive quantum number and opposite spin would be generated
were it not for far-band contributions.
A useful guide when comparing sets of band parameters in the six-band model
is to evaluate dE/dB\B=0 and hence calculate the band-edge effective masses and
g factors. In the six-band model these are given, for B\\ [111], by
{
CR. Pidgeon
458
(14)
(15)
(16)
These expressions illustrate the effects of the four far-band parameters M t~,
M t +, gj and g^ which act as almost independent contributions to the
conduction- and valence-band masses and g factors. It is these parameters which
are responsible for the deviations from symmetry of the conduction and valence
bands.
2.3.
Tellurium
Tellurium was the subject of considerable interest during the late 1960s and
early 1970s in laboratories in France, Germany, the USA and Japan (see, e.g.,
Couder and Hulin 1972, Rebmann et al. 1972, von Ortenberg et al. 1972, and
Doi et al. 1972). Figure 3 shows the basic scheme of conduction and valence
bands with interband selection rules at the critical point H, implied by these
authors and also obtained from earlier theoretical work (Hulin 1966) and IMO
experiments (Rigaux and Drilhon 1966, Alpert and Rigaux 1967, Rebmann et al.
1971).
Due to the first-order Ά-linear' term, the valence band has a distinctive
Ε
1 ι
ι
±
//
Fig. 3. Idealized scheme of energy states near the Η point in Te (Rebmann et al. 1972).
Magneto-optics in semiconductors
459
('camel back') structure. Since Te is naturally grown p-type, it has not been easy
to obtain direct information about the conduction band. Indirect information
has been obtained, e.g., from I M O experiments (Doi et al. 1972, Rebmann et al.
1971). The results of the band-structure calculations of the Japanese group
confirmed that the bottom of the conduction band is most likely to exist near the
point H, and that this state is nondegenerate. On the basis of this fact, the
effective Hamiltonian for the conduction band near the point Η is derived with
use of the k ·ρ perturbation method and the result is expressed in 2 χ 2 matrix
form,
VAkl + Bk2z+Nkz
L
Mk+
Mk_
(17)
Bk2z-Nkz
Akl +
The /r-linear terms with Μ and Ν appear from the second-order perturbation
through the k · ρ and the spin-orbit interactions, in contrast with the first-order
^-linear terms in the valence band.
On the basis of the above Hamiltonian, the Landau levels are calculated. In
the case of a magnetic field parallel to the c-axis (H\\c), the Landau levels are
obtained analytically in the following form:
e„cT
(kz) = nhcoc + Bk2 ± j£hcDc
- Nkz + & „ / i Bf f ) 2 + nhcocM2/A,
(18)
(B-l)l
where the plus corresponds to n \ and the minus to (n — 1)J,. The corresponding
eigenfunctions are expressed by
«
T
(n+l)i
(*,) = « £ ί « . - ι | Η i t , kz} +
bfWHd,
K\
(19)
where the plus corresponds to n | and the minus to (n + 1) j , and where w„, is the
nth harmonic oscillator wavefunction. The Landau levels and the corresponding
eigenfunctions of the valence band in this case have been derived exactly as
follows (Nakao et al. 1971):
s:(kz) = - h c o c ( n + i ) + Bk2
+ V^Mil -
i)] + S k ,
2
(n/A)hcoc(n
+
2 2
(20)
and
ψ: = ηη\Η4,Κ}.
(21)
The selection rule between these Landau levels of the valence and conduction
bands is easily obtained from eqs (19) and (21). Namely, for the allowed
transition (Elc):
An = n — n = 0,
= + 1,
strong intensity,
weak intensity,
CR. Pidgeon
460
and for the forbidden transition (E\\c):
An= ± 1,
= 0,
strong intensity,
weak one.
The transitions occur at the critical points at which the transition energies, ε£ Τ4
— ε*, as a function of kz take extrema. Reflecting the feature of the Landau levels
of the conduction band, that one of the ladders eOi(k2) has minima around kz =
— km(= — N/2B) while the other has minima around kz = 0, there appear two
series of transitions, the energies of which converge to different energies as the
magnetic field tends to zero. This difference represents the decrease of energy by
the Ar-linear term with N.
In the case of a magnetic field perpendicular to the c-axis, the Landau levels of
the conduction band are obtained by neglecting the Α-linear term with M. In this
case, the transitions are allowed between any pair of Landau levels of the
conduction and valence bands, reflecting the characteristics of the valence band.
More recently the subject of the valence band has been rationalized by von
Ortenberg and co-workers in an extensive set of cyclotron resonance measure­
ments and analysis (von Ortenberg et al. 1972, von Ortenberg and Button 1977).
They point out that all of the k ·/? parameter models proposed for the valence
band agree on the fundamental 'dumbbell structure', but they differ in their
higher-order terms and in the resultant parameter values. There is a strong
anisotropy of the E(k) relations of the valence bands for wavevectors parallel
and perpendicular to the trigonal c-axis as expressed in the 2 χ 2 Hamiltonian
(Weiler 1970)
'la + bk2z + (c +f)k2±
+ (d + g)k\ +
ek\k\
l 2
-h > kz
h1/2kz
bk2 + (c -f)k2±
2
x
+ (d-g)k*
+ eklk2 J
(22)
2
where k\ = k + k , and kz and kL are the wavevectors parallel and per­
pendicular to the c-axis. It is concluded that all the models (Weiler 1970, Doi et
al. 1972, Couder and Hulin 1972) give reasonable agreement in resonant
positions - i.e., energy eigenvalues - for cyclotron resonance, but further work is
required on transition strengths and lineshapes for different polarization and
field angles to distinguish between them.
2.4. The cadmium-arsenide
system
A general theory of magneto-optical transitions between any of the valence
bands and the conduction band for semiconductors described by the Kane
model, or its anisotropic generalization, the Bodnar (1977) model has been given
Magneto-optics in semiconductors
461
by Singh and Wallace (1983, 1987). Their treatment applies to the cadmiumarsenide system which is an inverted-gap material. A change in sign in the gap
parameter (Blom 1980) allows them to deal with material with a 'normal' gap,
and the model reduces to the Kane model by putting the anisotropy parameters
equal to zero. A partial list of materials on which this theory is applicable is as
follows: positive gap ( C d 3 P 2 , C d 3 A s 2 _ x P x , C d 3 _ x Z n x A s 2 , . . . ) ; inverted gap
( C d 3A s 2, C d 3A s 2_ J CP J C, C d 3 _ x Z n x A s 2 , . . . ) . As pointed out by these authors, the
calculated IMO transition probabilities are very sensitive to anisotropy para­
meters, magnetic field, final Landau-level number and energy. Since at the
present time it appears that experimental work has not been developed to the
degree of sophistication required for detailed comparison with the predictions,
we refer the reader to the original sources for the theoretical analysis. As
discussed in section 3.4, the most recent I M O studies of Aubin et al. (1980,1981,
1987) give good agreement with the Bodnar model for C d 3 A s 2 , apart from some
aspects of the heavy-hole valence band.
3.
Experiments
3.1. Zinc-blende semiconductors
and semimetals
The IMO technique has been used for electronic band structure determination
in a wide variety of narrow gap III-V and II-VI semiconductors - including
InSb, InAs, GaAs and their alloys, and H g x _ x C d x T e , H g 1 _ x C d x S e ; the
semimetals HgTe, HgSe and their alloys with Cd; and the group-IV semimetal
grey tin. Some of the more successful examples will be described here. The
Pidgeon and Brown (1966) model (table 1) was designed to describe this
situation, since it includes both the non-parabolic energy bands, and the
quantum effects resulting from the partial degeneracy of the p-like band, and
this model has been used successfully in all cases. An extremely thorough
analysis was performed in this way for InSb by Weiler (1979) obtaining
consistency for both interband and intraband measurements as shown in figs 4
and 5. It was found necessary to include small exciton corrections to the IMO
data fit to do this.
Other investigations of this type in InSb were performed, e.g., by Grisar and
Wachernig (1977) and, with stress applied uniaxially, by Trebin et al. (1979).
The technique was used successfully for the inverted semimetals HgTe
(Groves et al. 1967) and grey tin (Groves et al. 1970) using the magnetoreflection
method. Due to the strong interaction between the submerged s-like band and
the light-mass conduction p-band, 'forbidden' transitions between the heavyand light-mass p-bands become strongly allowed ( Γ 8 Γ 8) and are observed, as
well as the normal s- to p-band transitions (Γη - > Γ 8 for grey tin and Γ 6 - + Γ 8
for HgTe, respectively) - see the chapter by Zawadzki in this volume (chapter
CR. Pidgeon
462
200
*
0
"
2
1
4
1
6
1
8
10
1
11
1
12
14
MAGNETIC FIELD (T)
Fig. 4. Interband transitions in InSb, / / | | [ 0 0 1 ] , T = 2 4 K , for optical polarizations aL and <rR
(Weiler 1979). The solid lines (theory) show the best fit.
9). Typical magnetoreflection traces are shown in fig. 6, for grey tin (Groves et
al. 1970).
The first interband magneto-optical experiments on H g i ^ C d ^ T e were
carried out by Harman et al. (1961) and extended by Strauss et al. (1962). They
measured magnetoreflection as well as the Hall coefficient and resistivity, on a
number of samples with χ ranging from 0.136 to 0.20, and carrier concentrations
of 2 to 3 χ 1 0 15 cm""3. An appropriate fit was made to interband and cyclotron
resonance reflection peaks observed for a sample with χ = 0.17 at 4.2 K. The
model of Bowers and Yafet (1959) was used without exciton corrections, with
Magneto-optics in semiconductors
463
Fig. 5. Intra-conduction-band transitions in n-InSb for H\\ [ 1 1 1 ] (after Weiler 1979). The data are
from Lee (1979), Favrot et al. (1976) and McCombe and Wagner (1971). The solid lines (theory)
show the best fit.
parameters Eg = 6 meV and Ρ = 8 χ 1 0 " 8 eV cm (Ep = 2mP2/h2 = 17 eV), giv­
ing w c ~ 4 χ 1 0 " 4 m. The fit to the interband data was not very good because
the complexity of the valence-band structure had been neglected.
The first study of interband transitions accompanied by an analysis using the
quasi-germanium model, again without exciton corrections, was carried out by
Groves et al. (1971) on a sample with χ = 0.161 + 0.003 with carrier con­
centration η ~ 9 . 4 χ 1 0 1 4c m " 3 at liquid-helium temperatures. Interband magnetoreflection data were taken with the sample at approximately 25 and 95 K.
The energy gap Eg was approximately —10 meV (semimetallic configuration) at
the lower, and + 1 0 meV at the higher temperature. The other parameters
determined were Ep ~ 18.5 eV and heavy-hole mass m h ~ (0.28 + 0.1) m.
Kim and Narita (1976) studied far-infrared magnetoabsorption on samples of
Hg! _ xC d xT e with χ = 0 (HgTe), 0.07,0.126, and 0.148, all of which had Eg < 0.
They also used the quasi-germanium model without exciton corrections to
analyze their data, which included cyclotron resonance and a number of
interband transitions.
The most extensive magneto-optical studies of H g x _ xC d xT e were the trans­
mission experiments carried out by Guldner et al. (1977) following their earlier
464
CR. Pidgeon
I
•
I
20
t
20
40
i
l
l
40
ι
I
100
100
I ,. I
60
80
I
ι
60
I
80
<
ι
'
I
'
100
100
80
•
•.
'
i
I
60
ι
40
'
I
80
60
ι
I
20
I
40
ι
L
20
MAGNETIC FIELD (kOe)
Fig. 6. Experimental traces of magnetoreflection in grey tin, made by sweeping the magnetic field at
fixed photon energies (Groves et al. 1970). The top trace shows the peak due to Fg -+ Fg transitions
and the bottom trace shows small peaks from Fg - • F g transitions superimposed on a large
Γη - • Fg" peak.
work (Guldner et al. 1973, Tuchendler et al. 1973) on HgTe. They studied
samples of Hg^^Cd^Te with χ ranging from 0.01 to 0.3, spanning the
semimetallic ( F g ~ — 300 meV) to semiconducting (Eg~ + 2 0 0 meV) region.
They studied interband transitions for all their samples, and intraband tran­
sitions for the semimetallic samples, in both the Faraday and Voigt configur­
ations. The transition energies were fit to the model of Pidgeon and Brown
(1966) with F = Nx = 0 and no exciton corrections. Their results are illustrated
in fig. 7, and values for Eg and Ep given in table 3; see Weiler (1981) and Zawadzki
(1980) for a further discussion of inter- and intraband magneto-optics in the HgTe
and HgSe families, and McCombe (1969), McCombe et al. (1970), McCombe and
Wagner(1971,1974,1975), McCombe (1972).
In the formalism of the quasi-germanium model and in the exact solution in
terms of single harmonic oscillator functions, two approximations were made: (i)
inversion symmetry was assumed and (ii) the second-order warping term Dl was
neglected. Usually, these approximations cause negligible error, but if two levels,
which interact through the omitted terms in the Hamiltonian, are nearly
degenerate in energy, considerable admixing of the wave functions can take
place. This, in turn, can cause observable extra transitions associated with
Magneto-optics in semiconductors
0
10
20
30
465
40
50
H(kG)
Fig. 7. Field dependence of magneto-absorption peaks in H g , _ xC d xT e (after Guldner et al. 1977).
The solid lines (theory) show the best fit.
Table 3
Dependence of energy gap and matrix element Ep on
composition for Hgj .^Cd^Te at 4.2 Κ (after Guldner et
al. 1977).
X
£ g( m e V )
£ P( e V )
0.01
0.025
0.05
0.105
0.115
0.15
0.185
0.215
0.25
0.28
-285
-261
-207
-110
-90
-30
35
86
161
208
18.3
18.5
18.8
18.8
18.8
18.8
18.9
19.0
19.0
19.0
466
CR. Pidgeon
warping and inversion asymmetry (see, e.g., Pidgeon and Groves 1969). In order
to study the warping and inversion asymmetry, magnetoreflection measure­
ments were made on pure η-type InSb (Ν = 1 0 1 4 c m - 3) at 1.5 K, in the Faraday
configuration. Results are shown in fig. 8 for two directions of / / , using left and
right circularly polarized light. Due to the large conduction-band spin splitting,
the transitions to the lowest spin up and spin down conduction-band levels are
widely separated, permitting unambiguous assignments of the transitions. The
set of lines between 254 and 260 meV and that between 273 and 277 meV
corresponds to transitions from levels at the top of the valence band to the first
spin-up and spin-down conduction level, respectively. Thus, for H\\ <100> we see
the pairs of allowed transitions (oL and <rR) to these two levels. We see also that
as predicted for / / | | <111>, the lowest σκ line separates into three components warping induced (w), allowed and linear-A:' induced (k) transitions, and the
250
255
260
270
5
PHOTON ENERGY (meV)
Fig. 8. Magnetoreflection spectra for InSb for # | | < 1 1 1 > and # | | < 1 0 0 > in the Faraday configur­
ation, with Η = 84 kOe and 7 = 1 . 5 K. Transitions to the Mj = j , η = 0 conduction-band level occur
on the left-hand side of the break in the energy scale and those to the M} = — \, η = 0 conductionband level occur on the right-hand side. W labels indicate warping-induced transitions and k labels
the 'linear-Ar' induced transition (after Groves et al. 1969).
02
7
Magneto-optics in semiconductors
467
second oL line into two components - allowed and warping-induced (w)
transitions. It is seen that the warping-induced transitions have strengths
comparable with the allowed transitions, but that the 'linear-fc' transition is
about 20 times weaker. From these measurements the warping and inversion
asymmetry parameters were determined.
From the foregoing comparisons of experiment and theory the following
values were obtained for the warping and linear-A: valence band parameters:
( y 3 - y2) = l.o ± 15%, C = 3.0 χ 1 0 " 3 atomic units ± 30%. The valence-band
ε — k dispersion relations for k along <100> and < 111 > directions, using the
parameters determined above, are shown in fig. 9. The maximum upbending of
the heavy-hole band, resulting from the 'linear-fc' term, is about 1 meV.
By using more sensitive techniques associated with modulation spectroscopy,
such as piezoreflectance and electroreflectance it has been possible (as shown in
figs 10 and 11) to obtain IMO spectra with greater resolution and also to see
spectra associated with transitions from the spin-orbit split-off band (see
Aggarwal 1972). In addition, Landau level structure has been observed at the Ex
2.0
0
2.0
4.0
6.0
8.0
10* k(cm'1)
Fig. 9. Valence band dispersion relations for k along <100> and < 111 > directions which result from
the parameters determined here. The numbers indicate the degeneracy of the band. In all other
directions both light- and heavy-mass bands are split. The maximum splitting occurs for the < 111 >
heavy-mass band (after Groves et al. 1969).
CR. Pidgeon
468
1
ι
||0
1
0.9
ι1
I
1.0
ι
I
1.1
ι
Γ
I
1.2
ι
I
I
1.3
PHOTON ENERGY (eV)
Fig. 10. Stress-modulated magnetoreflection spectrum for light-hole, heavy-hole and spin-orbit
split-off valence to conduction band transitions in Ge (H = 84.9 kOe, Τ = 30 Κ, σ κ polarization).
Theoretically predicted transitions are indicated underneath (after Aggarwal et al. 1970).
edge of InSb (Sari 1976, Alibert et al. 1978). Specific additional effects associated
with crossed and parallel electric and magnetic fields are reviewed by Aronov
and Pikus in the present volume (chapter 10).
3.2. The lead salts
The magneto-optical properties of the lead salts and the mixed crystals
Pbl _ xS n xT e and P b x _ xS n xS e have been reviewed by Bauer (1980), Nimtz and
Schlicht (1983) and Pascher (1982). Extensive work has been published, e.g., on
PbTe (Grisar et al. 1977) P b 1 _ x S n x T e (Appold et al. 1978) and P b 1 _ x G e x T e
(Bangert 1981). Interband magnetoluminescence has been measured (Gureev et
al. 1977). Most recently, very complete IMO studies have been reported by
Singleton et al. (1986) on strained and unstrained films of PbTe grown on B a F 2,
NaCl and PbTe substrates. By a careful orientational study of the magneto­
absorption of these films, it has been possible to deduce an accurate set of band
parameters for PbTe, using the six-band Mitchell-Wallis k · ρ model. It was
demonstrated that the inbuilt strain in the layers is an important parameter,
which can cause changes in both the bandgap, and the momentum matrix
element coupling the L f l and Lf + bands. Both uniaxial and hydrostatic
Magneto-optics in semiconductors
0.400
0.500
0.600
469
0.700
PHOTON ENERGY (eV)
Fig. 11. Electric-field modulated magnetoreflectance spectrum for valence- to conduction-band
transitions in InAs ( H = 8 4 k O e , Γ = 1 . 5 Κ , E1H).
The I M O spectrum for split-off band to
conduction-band transitions is shown on the inset, as is the experimental arrangement for
electroreflectance (after Pidgeon et al. 1967).
components of strain were shown to be present, and these were used to deduce
the deformation potentials for PbTe. The anisotropy of the conduction and
valence bands was shown to be independent of the strain. The material does
not lend itself to optical measurements, with high carrier concentrations
( ~ 1 0 1 7c m ~ 3 ) , even in pure PbTe and Landau-level separations lying in the
reststrahl band ( 1 4 - 1 1 4 c m " 1) for magnetic fields up to 10T. For optical
measurements the sample is, therefore, usually grown as a thin film ( ~ 1 μιη) on
a transparent substrate such as NaCl (100) (Schooler and Zemel 1964), S r F 2 or
B a F 2 (111) (Asch et al. 1974). Problems arise when cooling such samples to the
low temperatures where the mobility of free carriers is sufficiently large for the
observation of quantum phenomena. The samples are grown at around 700 Κ
(Clemens et al. 1983) and differential contraction between the substrate and film
below this temperature produces considerable strain in the PbTe. The conse­
quences of induced strain were reported by Mitchell et al. (1964), who observed
CR. Pidgeon
470
that the bandgap of PbTe grown on NaCl was about 10% lower at 77 Κ than
that of PbTe in the form of thin crystals of bulk material. Burke and Carver
(1978) noted depopulation of valleys due to strain effects and many other
authors have observed similar phenomena (e.g., Weber and Yeung 1973) in
PbTe lasers grown on B a F 2.
In the work of Singleton et al. (1986), the sample was rotated in 2-5° steps
between successive sweeps of the magnetic field at constant laser frequency.
Typical results for one of the PbTe on B a F 2 samples are shown in fig. 12: it is
plain that the shape of the transmission spectrum is a very strong function of the
CO
f—
1
ζ
I
>-
RA
cr
A
h-
co
ON
<
τ η
1
CO­
CO
I
AN
CO
ο
<P =
10
QC
—
f
MAGNETIC FIELD (T)
Fig. 12. Interband magnetotransmission spectra for PbTe on B a F 2 as a function of angle, </>,
between Β and [110]. At all times Β is perpendicular to [ 1 1 1 ] , and the photon energy = 0.2077 eV,
Τ = 1.9 Κ (after Singleton et al. 1986).
Magneto-optics in semiconductors
All
orientation. They suggest this is perhaps the reason why previous workers
measuring magnetotransmission at only two or three different orientations
failed to observe strain-dependent effects (Bauer 1978). An error of only 2° in
orientating the sample would result in significant shifts (Jensen et al. 1978) of the
transmission minima: also some of the absorption peaks due to different valleys
overlap at the two or three orientations used, leading to difficulties in
interpreting the results.
Experimental points for the < 111 > valley, parallel to the sample surface
normal, and the <Tll> valleys give good agreement with the Mitchell-Wallis
model fits in fig. 13 which shows data from two samples. The band parameters
0
2
4
6
8
TRANSITION ENERGY (meV)
Fig. 13. Observed interband magneto-optical transitions for the PbTe on B a F 2 samples compared
with Mitchell-Wallis model fits, (a) shows transitions due to the [ 1 1 1 ] valley, perpendicular to the
magnetic field, while (b) shows transitions due to the [ 1 1 1 ] valleys at an angle of 19.5° to the
magnetic field. The difference in band gap due to the strain is plainly visible. Τ = 1.9 Κ (after
Singleton et al. 1986).
472
CR. Pidgeon
used in both cases were exactly the same with the exception of the bandgap,
which is 0.9 meV larger for the [111] valley, an effect of the uniaxial component
of the strain.
3.3.
Tellurium
Valence to conduction band IMO experiments have been carried out by a
number of workers (Rigaux and Drilhon 1966, Alpert and Rigaux 1967,
Rebmann and Rigaux 1970, Doi et al. 1972). The structure of both the H 4 and
the Η5 valence bands was established by inter-valence band magneto­
absorption (Hardy et al. 1971). Results were obtained by Doi et al. (1972) for
valence to conduction band transitions for the two cases of the electric field of
the light, £ , parallel and perpendicular to the c-axis. The magnetic-field
dependence of the energies of the transmission minima for H\\c is shown in
fig. 14. It should be noticed that two series of transitions really appear and this
feature is due to the effects of the /r-linear terms of both the conduction and
the valence bands. In order to clarify the relation between the assignments for
the two configurations H\\c and Hlc, the dependence of the transition energies
on the direction of the magnetic field was investigated. From these measure-
0
10
20
30
40
50
MAGNETIC FIELD (kOe)
Fig. 14. Photon energy of transmission minima against the magnetic field in Te for H\\ c. Solid lines
are theoretical curves for An = 0 and dashed lines for An = + 1 (after Doi et al. 1972).
Magneto-optics in semiconductors
473
merits the conduction band parameters discussed in section 2.3 were determined.
As pointed out in section 2.3, the most extensive series of cyclotron-resonance
measurements on p-type Te (von Ortenberg et al. 1972,1977) has confirmed the
consistency of the basic 'dumbbell' valence-band model, but leaves unresolved
the details of higher-order terms in different models used (Weiler 1970, Doi et al.
1972, Couder and Hulin 1972).
Rebmann et al. (1972) have observed IMO transitions between the spin-orbit
split-off valence band and conduction band using the electroreflectance tech­
nique, for polarized light with El c. Figure 15 shows spectra observed in crossed
ϋ
1
\
450
500
ENERGY (meV)
550
Fig. 15. Electric-field modulated interband magnetoreflectance spectra for Te (E±c, Hlc,
T =
4.4 K). On the inset plots are shown of energy positions of the minima versus magnetic field for
best fitted theoretical curves (after Rebmann et al. 1972).
474
CR. Pidgeon
fields, in the configuration Elc, HLc. Energies of AR/R minima are plotted as
a function of the magnetic field, in the inset of the figure. The observed structures
were identified with H 5 - > H 6 magneto-optical transitions. Neglecting linear-A
terms in the conduction band dispersion relation, interband matrix elements
were calculated in the crossed-field configuration (Aronov 1963) and shown to
agree with experiment. Energies of predominant transitions, corresponding to
even Δη in the weak electric-field approximation, were quantitatively compared
with experimental data. Band-edge effective masses ^/nymt for H 5 and H 6 levels
were obtained from the fit. For the split-off valence band they deduce ^/m^ 5m t5
= 0.1 m 0, which agrees with intervalence band magneto-absorption data previ­
ously reported (Hardy et al. 1971). For the conduction band, the effective mass
yjm(mu
= (0.072 ± 0.004) m 0 is determined. Analysis of interband magneto­
absorption for / / | | c has provided the transverse mass mt = 0.046 m 0, leading to
the longitudinal mass me = (0.12 ± 0.015) m 0.
3.4. The cadmium-arsenide
system
As pointed out in section 2.4, the theoretical situation is somewhat in advance of
experimental possibilities for the cadmium-arsenide family, largely because the
carrier concentration in the material is typically of the order 1 0 18 c m - 3. This
means that for the small gaps and effective masses involved there is a large
Burstein-Moss shift and one tends to measure properties at the Fermi level
rather than at the band edge. Consequently, it is not as yet possible to explore
the details of the predictions of, e.g., Singh and Wallace (1987) in these systems.
Aubin et al. (1977) determined effective-mass values and energy-band structure
information on C d 3 A s 2 from reported measurements of Shubnikov-de Haas,
magneto-Seebeck and Hall effects using the Kane model for the inverted HgTetype band structure. The validity of this model for C d 3 A s 2 was shown by
Wagner et al. (1971) from interband magneto-absorption measurements at 20 Κ
on C d 3 _ x Z n x A s 2 and C d 3 A s x P 2 _ x alloys. In the absence of the four-level model
- which is now accepted from piezo-galvanomagnetic and thermomagnetic
measurements and magneto-optics (Aubin et al. 1980) - they assumed a simple
HgTe-like model with a non-parabolic conduction band. Typical results, for two
compositions (small and large gap) of C d 3 A s x P 2 _ x , are shown in fig. 16. A plot
of the energy gaps deduced as a function of the alloy compositions is shown in
fig. 17, establishing the inverted gap of C d 3 A s 2 .
Aubin et al. (1980, 1981) reported on interband magneto-optical data taken
from the transmission of identically oriented single crystals of C d 3 A s 2 . Similar
measurements had been reported by Haidemenakis et al. (1966) but the data
covered only a small energy interval. Wagner et al. (1971) analyzed their data in
the light of