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MODERN PROBLEMS IN CONDENSED MATTER SCIENCES V o l u m e 27.1 V.M. Moscow, A.A. AGRANOVICH USSR MARADUDIN Irvine, California, USA Advisory board editorial F. Abeles, Paris, France F. Bassani, Pisa, Italy N. Bloembergen, Cambridge, MA, USA E. Burstein, Philadelphia, PA, USA I.L. Fabelinskii, Moscow, USSR P. Fulde, Stuttgart, F R G M.D. Galanin, Moscow, USSR V.L. Ginzburg, Moscow, USSR H. Haken, Stuttgart, F R G R.M. Hochstrasser, Philadelphia, PA, USA LP. Ipatova, Leningrad, USSR A.A. Kaplyanskii, Leningrad, USSR L.V. Keldysh, Moscow, USSR R. Kubo, Tokyo, Japan R. Loudon, Colchester, U K Yu.A. Ossipyan, Moscow, USSR L.P. Pitaevskii, Moscow, USSR A.M. Prokhorov, Moscow, USSR K.K. Rebane, Tallinn, USSR J.M. Rowell, Red Bank, NJ, USA NORTH-HOLLAND AMSTERDAM · OXFORD · NEW YORK · TOKYO LANDAU LEVEL SPECTROSCOPY Volume editors G. L A N D W E H R Wiirzburg, Germany E.I. R A S H B A Moscow, USSR 1991 NORTH-HOLLAND AMSTERDAM · OXFORD · NEW YORK · TOKYO © Elsevier Science Publishers Β .V., 1991 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the written permission of the Publisher, Elsevier Science Publishers B.V., P.O. Box 211, 1000 AE Amsterdam, The Netherlands. Special regulations for readers in the USA: This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the publisher. No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. ISBN 0 444 88874 8 0 444 88535 8 0 444 88873 X North-Holland Elsevier Science Publishers P.O. Box 211 1000 A E Amsterdam The Netherlands (Set) (Vol. 27.1) (Vol. 27.2) B.V. Sole distributors for the U S A and Canada: Elsevier Science Publishing Company, Inc. 655 A v e n u e of the Americas N e w York, N Y 10010 USA Printed on acid free paper M O D E R N P R O B L E M S IN C O N D E N S E D M A T T E R SCIENCES Vol. 1. S U R F A C E P O L A R I T O N S V.M. Agranovich and D . L . Mills, editors Vol. 2. EXCITONS E.I. Rashba and M . D . Sturge, editors Vol. 3. E L E C T R O N I C E X C I T A T I O N E N E R G Y T R A N S F E R IN CONDENSED MATTER V.M. Agranovich and M . D . Galanin Vol. 4. SPECTROSCOPY A N D E X C I T A T I O N D Y N A M I C S O F C O N D E N S E D M O L E C U L A R SYSTEMS V.M. Agranovich and R.M. Hochstrasser, editors Vol. 5. LIGHT SCATTERING N E A R P H A S E TRANSITIONS H.Z. Cummins and A . P . Levanyuk, editors Vol. 6. E L E C T R O N - H O L E D R O P L E T S IN S E M I C O N D U C T O R S C D . Jeffries and L.V. Keldysh, editors Vol. 7. T H E D Y N A M I C A L J A H N - T E L L E R EFFECT IN L O C A L I Z E D SYSTEMS Y u . E . Perlin and M. Wagner, editors Vol. 8. OPTICAL O R I E N T A T I O N F. Meier and B . P . Zakharchenya, editors Vol. 9. S U R F A C E EXCITATIONS V.M. Agranovich and R. Loudon, editors Vol. 10. E L E C T R O N - E L E C T R O N I N T E R A C T I O N S IN D I S O R D E R E D SYSTEMS A . L . Efros and M. Pollak, editors Vol. 11. M E D I U M - E N E R G Y ION REFLECTION FROM SOLIDS E.S. Mashkova and V . A . Molchanov Vol. 12. N O N E Q U I L I B R I U M S U P E R C O N D U C T I V I T Y D . N . Langenberg and A . I . Larkin, editors M O D E R N PROBLEMS IN C O N D E N S E D M A T T E R SCIENCES Vol. 13. PHYSICS OF R A D I A T I O N EFFECTS IN CRYSTALS R . A . Johnson and A . N . Orlov, editors Vol. 14. I N C O M M E N S U R A T E P H A S E S IN DIELECTRICS (Two volumes) R. Blinc and A . P . Levanyuk, editors Vol. 15. U N I T A R Y T R A N S F O R M A T I O N S IN SOLID STATE PHYSICS M. Wagner Vol. 16. N O N E Q U I L I B R I U M P H O N O N S IN N O N M E T A L L I C CRYSTALS W. Eisenmenger and A . A . Kaplyanskii, editors Vol. 17. SOLITONS S.E. Trullinger, V.L. Pokrovskii and V . E . Zakharov, editors Vol. 18. T R A N S P O R T IN P H O N O N SYSTEMS V.L. Gurevich Vol. 19. C A R R I E R SCATTERING IN METALS A N D SEMICONDUCTORS V.F. Gantmakher and L B . Levinson Vol. 20. SEMIMETALS - 1. G R A P H I T E A N D ITS C O M P O U N D S N . B . Brandt, S.M. Chudinov and Ya.G. Ponomarev Vol. 21. SPECTROSCOPY OF SOLIDS C O N T A I N I N G R A R E E A R T H IONS A . A . Kaplyanskii and R.M. Macfarlane, editors Vol. 22. SPIN WAVES A N D M A G N E T I C EXCITATIONS (Two volumes) A . S . Borovik-Romanov and S.K. Sinha, editors Vol. 23. OPTICAL PROPERTIES OF M I X E D CRYSTALS R.J. Elliott and LP. Ipatova, editors M O D E R N P R O B L E M S IN C O N D E N S E D M A T T E R SCIENCES Vol. 24. T H E DIELECTRIC F U N C T I O N O F C O N D E N S E D SYSTEMS L.V. Keldysh, D . A . Kirzhnitz and A . A . Maradudin, editors Vol. 25. C H A R G E D E N S I T Y WAVES IN SOLIDS L.P. Gor'kov and G. Gruner, editors Vol. 26. H E L I U M T H R E E W.P. Halperin and L.P. Pitaevskii, editors Vol. 27. L A N D A U L E V E L SPECTROSCOPY G. Landwehr and E.I. Rashba, editors In preparation HOPPING T R A N S P O R T IN SOLIDS B. Shklovskii and M. Pollak, editors NONLINEAR SURFACE ELECTROMAGNETIC PHENOMENA G. Stegeman and H . E . Ponath, editors MESOSCOPIC P H E N O M E N A IN SOLIDS B.L. Altshuler, R. Webb and P . A . L e e , editors ELECTRONIC PHASE TRANSITIONS W. Hanke and Yu. Kopaev, editors ELASTIC STRAIN FIELDS A N D D I S L O C A T I O N MOBILITY V.L. Indenbom and J. Lothe, editors Oh, how many of them are in the fields! But each flowers in its own way - there In this is the highest achievement of a flower! Matsuo Β as ho 1644-1694 P R E F A C E T O T H E S E R I E S Our understanding of condensed matter is developing rapidly at the present time, and the numerous new insights gained in this field define to a significant degree the face of contemporary science. Furthermore, discoveries made in this area are shaping present and future technology. This being so, it is clear that the most important results and directions for future developments can only be covered by an international group of authors working in cooperation. "Modern Problems in Condensed Matter Sciences" is a series of con­ tributed volumes and monographs on condensed matter science that is pub­ lished by North-Holland Physics Publishing, a division of Elsevier Science Publishers. With the support of a distinguished Advisory Editorial Board, areas of current interest that have reached a maturity to be reviewed, are selected for the series. Both Soviet and Western scholars are contributing to the series, and each contributed volume has, accordingly, two editors. Monographs, written by either Western or Soviet authors, are also included. The complete series will provide the most comprehensive coverage available of condensed matter science. Another important outcome of the foundation of this series is the emer­ gence of a rather interesting and fruitful form of collaboration among scholars from different countries. We are deeply convinced that such international collaboration in the spheres of science and art, as well as other socially useful spheres of human activity, will assist in the establishment of a climate of confidence and peace. The publishing house "Nauka" publishes the volumes in the Russian lan­ guage. This way the broadest possible readership is ensured. The General Editors of the Series, V.M. Agranovich A . A . Maradudin 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 References Adams, E.N., and T.D. Holstein, 1959, J. Phys. Chem. Solids 10, 254. Aoki, H., 1987, Rep. Prog. Phys. 50, 655. Bangert, E., and G. Landwehr, 1986, Surf. Sci. 170, 593. Bloch, E, 1928, Z. Phys. 52, 555. Burstein, E., G.S. Picus and H.A. Gebbie, 1957, Phys. Rev. 105, 1123. Chakraborty, T., and P. Pietilainen, 1988, The Fractional Quantum Hall Effect, Springer Series Solid State Sciences, Vol. 85 (Springer, Berlin), de Haas, W.J., and P.M. van Alphen, 1930, Proc. Amsterdam 33, 1106, Leiden Comm. 212a. Dingle, R.B., 1951, Proc. Int. Conf. on Very Low Temperature (Oxford) p. 165. Dorfman, J., 1951, Dokl. Akad. Nauk USSR 81, 765. Dresselhaus, G., A.F. Kip and C. Kittel, 1953, Phys. Rev. 92, 827. Elliott, R.J., and R. Loudon, 1959, J. Phys. Chem. Solids 8, 382. Gross, E.F., B.P. Zakharchenya and N.M. Reinov, 1954, Dokl. Akad. Nauk USSR 97, 57. Gross, E.F., B.P. Zakharchenya and P.P. Pavinskii, 1957, Zh. Tekh. Fiz. 27, 2177. Grosse, P., 1979, Freie Elektronen in Festkorpern (Springer, Berlin). Kanskaya, L.M., S.I. Kokhanovskii and R.P. Seysyan, 1979, Fiz. Tekh. Poluprovodn. 13, 2424 [Sov. Phys.-Semicond. 12, 1420]. Landau, L., 1930, Z. Phys. 64, 629. Landwehr, G., ed., 1987, High Magnetic Fields in Semiconductor Physics, Springer Series Solid State Sciences, Vol. 71 (Springer, Berlin). Landwehr, G., ed., 1989, High Magnetic Fields in Semiconductor Physics II, Springer Series Solid State Sciences, Vol. 87 (Springer, Berlin). Lax, B., and J.G. Mavroides, 1967, Appl. Opt. 6, 647. Lax, B., H.J. Zeiger, R.N. Dexter and E.S. Rosenblum, 1954, Phys. Rev. 23, 1418. Mott, N.F., and H. Jones, 1936, The Theory of the Properties and Metals and Alloys (Clarendon Press, Oxford). Palik, E.D., and I.K. Furdyna, 1970, Rep. Prog. Phys. 33, 1193. Patel, C.N., and A . D . Shaw, 1970, Phys. Rev. Lett. 24, 451. Peierls, R., 1933, Z. Phys. 80, 763. Pidgeon, C.R., 1980, Handbook of Semiconductors, ed. M. Balkanski (North-Holland, Amsterdam) p. 229. Pidgeon, C.R., and R.N. Brown, 1966, Phys. Rev. 146, 575. Prange, E., and M. Girvin, eds, 1987, The Quantum Hall Effect (Springer, Berlin). Rashba, E.I., and V.B. Timofeev, 1986, Sov. Phys.-Semicond. 20, 617. Shockley, W, 1953, Phys. Rev. 90, 491. Shoenberg, D., 1939, Pro. R. Soc. A 170, 341. Shoenberg, D., 1984, Magnetic Oscillations in Metals (Cambridge University Press, Cambridge). Shubnikov, L., and W.J. de Haas, 1930, Leiden Comm. 207, a, c, d,; 210, a. von Klitzing, K., G. Dorda and M. Pepper, 1980, Phys. Rev. Lett. 45, 494. Wolff, PA., 1966, Phys. Rev. Lett. 16, 225. Yafet, Y., 1966, Phys. Rev. 152, 858. 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. 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Fischer, 1972, Phys. Rev. Β 6, 2100. Waldman, J., D.M. Larsen, P E . Tannenwald, C.C. Bradley, D.R. Cohn and B. Lax, 1969, Phys. Rev. 2 3 , 1033. Weiler, M.H., 1970, Solid State Commun. 8 , 1 0 1 7 . Yafet, Y., 19xx, in: Solid State Physics, Vol. 14, eds F. Seitz and D . Turnbull (Academic Press, N e w York) p. 1. Yoshihara, T , 1971, Master Thesis (Graduate School of Science, Osaka University) unpublished. 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). 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Shereghii, E.M., and V.I. Ivanov-Omskii, 1980, Fiz. Tverd. Tela. 2 2 , 863 (Sov. Phys.-Solid State 503). Uritskii, Z.I., and G.V Shuster, 1965, Zh. Eksp. Teor. Fiz. 4 9 , 182. Wachernig, H., R. Grisar, G. Bauer and S. Hayashi, 1977, Physica Β + C 8 9 , 290. Weiler, M.H., R.L. Aggarwal and B. Lax, 1974, Solid State Commun. 14, 299. Zawadzki, W., R. Grisar, H. Wachernig and G. Bauer, 1978, Solid State Commun. 2 5 , 775. 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). 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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. 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Yafet, Y., 1963, in: Solid State Phys., eds F. Seitz and D . Turnbull (Academic Press, New York) 14, 1. Yafet, Y., 1966, Phys. Rev. 152, 858. Zavoisky, E.K., 1945, J. Phys. U S S R 9, 245. Zawadzki, W , and J. Wlasak, 1976, J. Phys. C: Solid State Phys. 9, L663. Zawadzki, W., P. Pfeffer and H. Sigg, 1985, Solid State Commun. 5 3 , 777. 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. 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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. References Adler, M.S., C R . Hewes and S.D. Senturia, 1973, Phys. Rev. Β 7, 5195. Ando, Τ., A.B. Fowler and F. Stern, 1982, Rev. Mod. Phys. 5 4 , 437. Bangert, E., G. Bauer, E.J. Fantner and H. Pascher, 1985, Phys. Rev. Β 3 1 , 7958. Bauer, G., 1978, in: Proc. Appl. High Magnetic Fields in Semiconductor Physics, ed. J.F. 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Rodriguez, 1962, Phys. Rev. 128, 2487. Ramage, et al., 1975. Schaber, H.Ch., 1979c, Ph.D. Thesis (Munich). Schaber, H.Ch., and R.E. Doezema, 1979a, Solid State Commun. 3 1 , 1 9 7 . Schaber, H.Ch., and R.E. Doezema, 1979b, Phys. Rev. Β 2 0 , 5257. Magnetoplasma effects 341 Singleton, J., E. Kress-Rogers, A.V. Lewis, R.J. Nicholas, E.J. Fantner, G. Bauer and A. LopezOtero, 1986, J. Phys. C: Solid State Phys. 19, 77. Sommer, E., 1979, J. Phys. C: Solid State Phys. 12, 1081. Sommer, E., 1980, J. Phys. C: Solid State Phys. 13, 5393. Stern, F, and W.E. Howard, 1967, Phys. Rev. 163, 816. Stimets, R.W., and B. Lax, 1970, Phys. Rev. Β 1, 4720. Vogl, P., and P. Kocevar, 1978, in: Proc. Int. Conf. on Physics of Semiconductors (Institute of Physics, London) p. 1317. Vogl, P., E.J. Fantner, G. Bauer and A. Lopez-Otero, 1979, J. Magn. & Magn. Mater. 11, 113. von Ortenberg, M., 1980, in: Infrared and Millimeter Waves, Vol. 3, ed. K. Button (Academic Press, New York) p. 275. von Ortenberg, Μ., K. Scharzbeck and G. Landwehr, 1975, Coll. Int. CNRS No. 242, Phys. sous Champs Magn. Intenses (Grenoble) p. 305. von Ortenberg, M., G. Bauer and G. Elsinger, 1985, Proc. Int. Conf. Submillimeter Waves (Mar­ seille) p. 127. Vuong, T.H.H., 1985, D. Philos Thesis (University of Oxford, Oxford). Wallace, PR., 1965, Can. J. Phys. 4 3 , 2162. Wallace, PR., 1966, Can. J. Phys. 4 4 , 2495. Wallace, PR., 1970, Phys. Status Solidi 3 8 , 715. Wallace, PR., 1980, Lecture Notes in Physics, Vol. 133 (Springer, Berlin) p. 447. Wallis, R.F., and M. Balkanski, 1986, Many Body Aspects of Solid State Spectroscopy (NorthHolland, Amsterdam). Witowski, Α., Κ. Pastor and J.K. Furdyna, 1982, Phys. Rev. Β 26, 931. Wojtowicz, T, and W Knap, 1984, Solid State Commun. 5 1 , 115. Zawadzki, W , 1974, Adv. Phys. 2 3 , 435. 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. 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