Heterojunction band offsets and effective masses in III-V quaternary alloys
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Semicond.
Sci.
Technol.
6
(1991)
2731.
Printed
in
the
UK
Heterojunction band offsets and effective
masses in
Ill-V
quaternary alloys
M
P
C
M
Krijn
Philips
Research Laboratories,
PO
Box
80000,
5600
JA
Eindhoven.
The
Netherlands
Received
23
April
1990,
in
final
form
30
August
1990,
accepted for publication
17
SeDtember
1990
Abstract.
Estimates
of valence-band and conduction-band offsets for lattice-
matched and pseudomorphic strained heterostructures of
six
technologically
important
Ill-V
quaternary alloys are presented. Valence-band offsets are obtained
via interpolation
of
the theory-based results of
Van
de
Walle's 'model-solid'
approach for
the
binary constituents. Estimates for
band
gap
differences
are
obtained via interpolation of
the
experimental
band
gap energies of
the
ternary
constituents.
Adding
the
valence-band offset and
band
gap difference
gives
an
estimate
of
the conduction-band offset. Band-edge effective masses at
r
are
determined from
a
linear interpolation
of
the
effective masses of
the
binary
constituents, obtained from self-consistent ab initio
band
structure calculations.
Results
are shown to agree
well
with
the outcome of experiments.
1.
Introduction
The design and modelling
of
optoelectronic devices re-
quires a knowledge of various material parameters. What
is essential is a knowledge of the heterojunction band
offset (also known as band discontinuity, band alignment
or
band line-up), i.e. the relative position of the band
edges of the semiconductors constituting the heterojunc-
tion. This relative position determines the confinement
of
the electrons and holes and depends on the composition
of the constituents and the amount of strain due to lattice
mismatch. Equally important are the band gap energy
and the conduction-band and valence-band effective
masses. The hand gap energy determines the operating
wavelength whereas the effective masses determine the
electron and hole mobility
[IJ.
The commercially attractive
0.5-4
pm wavelength
range may be covered by heterojunction devices
constructed from 111-V quaternary alloys. Notably,
AIGalnP, AIGaInAs, AIGaAsSh, GaInPAs, GaInAsSb
and InPAsSh. These quaternary alloys include most
ternary alloys.
A
lack of knowledge of the dependence
of
the various material parameters
on
alloy composition
has hampered a systematic approach to the design
of
tailor-made optoelectronic devices. The determination
of
band offsets in particular poses
a
problem.
Recently it has been recognized that, for the class
of
111-V semiconductor heterojunctions, the band offset is
largely determined by the hulk properties of the semicon-
ductors constituting the heterojunction; band offsets are
found to be
orientation independent
and
transitiue
[2-41.
However, it is
in
general not possible to measure
or
0268-1242/91/010027+05
$03.50
@
1991
IOP
Publishing
Ltd
calculate hulk band-edge energy levels on an
absolute
scale.
Van de Walle
[SI
has circumvented this problem
by relating the energy levels in hulk semiconductors
which he obtained from self-consistent
ab initio
band
structure calculations, to a common reference level. This
reference level was chosen to he the average electrostatic
potential in
a
semi-infinite 'model-solid'. The model-
solid
was built
up
from
a
superposition of neutral atoms
which were required to mimic the bulk electron density.
This approach also makes it possible to calculate the shift
of energy levels due to strain. The model-solid results for
the binary constituents and estimates for some
of
the
ternary constituents
of
the aforementioned quaternary
alloys were shown to agree well with the outcome of
experiments
[2,
51.
Band structure calculations of material properties
require the presence
of
ordering, which is absent from
or
unknown in most alloys (some alloys exhibit ordering in
the
(1
11)
directions, however
[6-81).
Consequently,
there is as yet
no
reliable method available to calculate
the dependence
of
the various material parameters on
alloy composition. The best one can do is to combine the
available theoretical and experimental data
on
the binary
and ternary constituents
of
the quaternary alloys and
employ an interpolation scheme to obtain an estimate
for
the various material parameters.
The aim
of
this paper is to present estimates
of
the
relative position
of
conduction-hand and valence-band
edges and effective masses as a function of alloy composi-
tion and lattice mismatch. Results are tabulated
so
that a
systematic search for device characteristics and trends is
straightforward,
27
M
P
C
M
Krijn
2.
Theory
We
closely follow Van de Walle
[SI
in describing the
dependence
on
strain of the valence- and conduction-
band edges in
111-V
semiconductors.
Pseudomorphic
(or
commensurate) growth of
strained layers
on
a substrate subjects these layers to a
biaxial strain
ell
parallel to the plane of the interface and
a uniaxial strain
e,
perpendicular
to
it:
Here,
a,
denotes the lattice constant
of
the substrate and
a
the equilibrium lattice constant
of
the layer material.
D
depends
on
the elastic constants
c
of the layer material
and
on
the interface orientation:
Spp
ta.h!e
!
for
a
!is!
gf
!he
eqni!ibinm !actice ccnstzfits
and elastic constants of the relevant
111-V
binary semi-
conductors.
The effect of strain
on
energy levels can be decom-
posed into hydrostatic and shear contributions.
In
the
following we separately discuss the effect
of
both contri-
butions to the conduction band and valence band
for
zincblende-type semiconductors
at
r.
The hydrostatic strain component leads to a shift
of
the average valence-band energy
E,,,,
=
(Ehh
+
E,,
+
EJ3,
i.e.
of
the average of the energies
of
the heavy-hole
(E,,;
J
=
3/2,
m,
=
+3/2),
light-hole
(E,,;
J
=
3/2,
m,
=
k
1/2),
and spin-orbit split-off
(Eso;
J
=
1/2,
m,
=
f
1/2)
bands:
Wav
=
a,(2q
+
~d
(44
AEtY
=
a,(2q
+
cl),
(46)
where
a,
and
a,
are
the hydrostatic deformation poten-
tials for the valence band and conduction band, respec-
tively.
The shear contribution couples
to
the spin-orbit
interaction and leads to an additional splitting
of
the
valence-hand energies.
In
the case
of
growth
on
a
(001)
or
(111)
substrate, the energy shifts relative to
E,,
=
E,,,,
+
A0/3 (Ao
is
the spin-orbit splitting in the absence
of
strain) are given by
[S,
91
and similarly
for
the conduction band energy
=
-
+SESh
(54
AE;;
=
-
+Ao
+
+SEsh
++[Ai
+
A0SE'"
+
~(6EPh)2]1'Z
(56)
AE;:
=
-
$Ao
+
+&ESh
-
+[Ai
+
AOSEsh
+
q(6E
sh)2]1'2.
(Sc)
The strain-dependent shift
SESh
depends
on
the interface
orientation:
-
ell)
(64
(6b)
The quantities
b
and
d
are the tetragonal and rhombo-
hedral shear deformation potentials, respectively.
In
the
absence of strain, equation
(5)
leads to the correct
spin-orbit splitting
Ao.
Conduction bands at
r
are not
affected
by
the shear contribution to the strain.
Once
E,,,,
is
known
on
an
absolute
scale (only
physically meaningful relative to
E",
in related semicon-
ductors), the valence-band and conduction-band edge
~~0Ol.sh
=
,jElll.sh
=
Z
j$dSd(&i
-Ell).
Table
1.
and elastic constants cI,,
Cl2
and c, (in
IO"
dyn cm-';
[15]).
Valence-band average
potentials a, and a,(T) as calculated within Van de Walle's model-solid approach (in eV;
[5]).
Spin-orbit splittings
Ao,
band
gaps
€,(I-).
EJX),
EJL)
(at room temperature) and shear deformation potentials band
d
(in eV;
[151
and
(221,
except where
indicated).
Material parameters for various zincbiende-type semiconductors. Lattice constant a (in
A)
at room temperature
and hydrostatic deformation
AIP
AlAs
AiSb
GaP
GaAs
GaSb
inP
lnAs
lnSb
5.451 1.32 0.63 0.62
-8.09
0.07' 3.58 2.45
5.660
1.25 0.53 0.54 -7.49 0.28 2.95 2.16'
6.136
0.88
0.43 0.41
-6.66
0.65
2.22 1.61'
5.451 1.41 0.62 0.70 -7.40
0.08
2.74 2.26
5.653 1.18 0.54 0.59 -6.92 0.34 1.42 1.91'
6.096
0.88
0.40 0.43 -6.25 0.82 0.72 1.05'
5.869 1.02 0.58 0.46 -7.04 0.11 1.35 2.21"
6.058 0.83 0.45 0.40 -6.67 0.38 0.36 1.37'
6.479
0.66
0.36 0.30 -6.09
0.81
0.17 1.63''
3.11. 3.15
-5.54
-1.6'
2.80' 2.47 -5.64
-1.5'
2.21' 1.38 -6.97 -1.4 -4.3
2.63 1.70 -7.14 -1.5 -4.6
1.73b 1.16 -7.17 -1.7 -4.6
0.76' 0.79 -6.85 -2.0 -4.8
2.05' 1.27 -5.04 -1.6 -4.2
1.07*
1.00
-5.08 -1.8 -3.6
0.93" 0.36 -6.17 -2.1
-5.0
a
Present
work
1161,
'1171.
=
1211.
*IlO].
28
Heterojunction band offsets and eflective
masses
energies
E,
and
E,,
respectively, are obtained
on
an
absolute scale from
A,
E,
=
E,,,,
+
-
+
AE;:*"
+
max(AEL!,
AE;:) (7a)
3
(7b)
A,
E,
=
E,,,,
+
-
+
E,
+
AE4Y
3
where
E",*",
A,,
and
E,
(band gap energy) refer
to
unstrained bulk properties. The strain contribution is
incorporated in
AE:Yav, AE;;, AE;,"
and
AEF.
Values for
E,,,,
as calculated
by
Van de Walk
together with values for
A,,
E,,
a,,
a,,
b
and
d
are listed in
table
1.
These data enable one
to
calculate the band offset
at the interface by directly comparing
the
values of
E,
and
E,
of the semiconductors constituting the hetero-
junction.
Equations
(4),
(5)
and
(6)
were arrived at by treating
the effect of strain as a small perturbation, and are
therefore only valid in the case
of
small lattice mismatch.
The mismatch attainable in pseudomorphic heterojunc-
tions composed of 111-V alloys is expected
to
fall within
this range
of
validity.
3.
Interpolation scheme
Material parameters of quaternary alloys
(Q)
with com-
positions
of
the form
AB,C,D,_,-,
or
A,B,-,C,D,_,
can be estimated by interpolating the parameters of the
binary
(E)
or ternary
(T)
alloys, depending
on
which
information is available. Estimates
of
Q
are readily
obtained from a linear interpolation of the
E's.
This is
known to be a valid procedure for
the
variation of the
lattice constant with alloy composition (Vegard's law
[l]).
Parameters such as
e.g.
E,
and
A,
in
ternary alloys
of the form
AB,C,_,,
can be approximated well by the
quadratic expression
TABC(X)
=
XBAn
+
(1
-
XPAC
t
x(X
-
1)cAnc
(8)
In
the case of
AB,C,D1_,_,,
Q
can be expressed as
Q(x,
Y)
=
XBAB
f
YEA,
+
(1
-
x
-
YWAD
-
xYcABC
-
x(l
-
x
-
Y)cABD
-
Y(I
-
-
Y)cACD
(9)
Equally, in the case of
A,B,-,C,D,-,
[IO]
In
the latter case,
Q
reduces to
Ton
the
quaternary plane
boundaries
(x
=
0,
1;
y
=
0,
1)
and
to
the average of the
Ts
at the midpoint
(x
=
0.5;
y
=
0.5).
The
constant
C
in
equation
(8)
is
known
as the bowing parameter. Experi-
mentally determined bowing parameters for
E,
and
A,
are listed in table
2.
In ternary alloys constituted from
Table
2.
ting
bowing parameters.
Band
gap
(at
r)
and spin-orbit
split-
C(E,)
C(A,)
Reference
AIAs.Sb.
I
0.84 0.15
1181
"
I-^
AI,Ga,
_,P
0.0
0.0
izoj
AI,Ga,-,As
0.37
0.0
[17,
271
AI,Ga,_,Sb
0.47
0.30
[21]
Alh
-
.P
0.0
0.0
122,
181
Al,ln,
_,AS
0.70
0.15
[22,
181
GaP,As,-.
0.21
0.0
PI
Ga,ln,
-,P
0.79
0.0
[22,
281
GaAs,Sb,
~
1.2
0.60
[22,
181
Gah,
-
.As
0.38
0.15
123,
291
i221
Ga,ln,-,Sb
0.42
0.0
InP,As,
~,
0.28
0.10 124,
181
InP.Sb.
~"
1.3 0.75
125,
181
.
InAs,Sb,_,
0.58
1.2
izz,zg]
non-lattice matched binaries, one material should be
regarded as being expanded, whereas the other is com-
pressed. For
E",+,
this leads to a bowing parameter that
depends
on
the hydrostatic deformation potentials
of
the
binary constituents
[ll]
where
Aa,
=
a,(AB)
-
a,(AC)
and
Aa
=
a(AB)
-
a(AC).
For convenience, results for
E,,,,,
A,,
and
E,
will be
presented in the form of tables of coefficients
of
an
expansion
of
Q
in a product polynomial in
x
and
y
In
the case of
A,B,_,C,D,_y,
the coefficients
C,
are
obtained from fitting equation
(12)
to equation
(10)
in
a
least-squares sense. The error introduced by this fitting
procedure is estimated to be
20.025
eV for the 111-V
alloys considered. Similarly, results for
AE?*", AEkY,
and
SEsh
are written as an expansion that is linear in the
lattice mismatch
where
Aa(x, y)
=
a,
-
a(x, y)
is the lattice mismatch
(a(x,
y)
is the equilibrium lattice constant of the alloy
obtained from Vegard's law, whereas
a,
is the lattice
constant of the substrate),
4.
Results and discussion
The coefficients
of
an expansion
of
E,,,,,
A.
and
E,
(at
room temperature) in a product polynomial as repre-
sented by equation
(12),
are listed in table
3.
The effects of
strain are incorporated in
AEFav, AEty
and
6ESh.
The
expansion according to equation
(13)
leads to the coeffi-
cients listed
in
table
4.
29
M
P
C
M
Krijn
Table
3.
according
lo
equation
112).
Coefficients
of
an expansion of the valence-band average spin-orbit splitting
A.
and band gap
EJT)
L“
AI,Ga,ln, -r-vP
-7.040 -0.240 -0.120
AI,Ga,l n,
-
-
,As
-6.670 -0,201 -0.049
AI,Ga,
-
.As,Sb,
-
-6.250 -0.570 -0.100
Ga,ln, _,P,As,
--y
-6.670 -0.341 -0.029
Ga,ln, -,As,Sb,
--y
-6.090
-0.442
-0.138
InP.As,Sb,
-6.090 -0,442 -0.138
C?,
-0.597
-0.496
-0.421
-0,201
-0.075
-0.656
C,,
C,, C2.3
C2,
c22
-0.573 -0.453
-0.378 -0.324
0.035 -0.188 0.011 0.007 -0.013
-0.006
-0.033
-0.049 -0.069 -0,002
-0.162 0.036 -0.085 0.034 0.002
-0.404 -0.294
~
-
A0
AI,Ga,ln, -x-yP
0.110
-0.030
0.000
-0.040
0.000
0.000
AI,Ga,ln, _,_,As
0.380 -0.190 0.150 -0.250
0.300
0.150
&Gal _,As,Sb,
--I
0.823
-1.022
0.540
-0.441 1.803 -1.435 0.269 -1.325 1.067
Ga,ln,-.P,As,-,
0.381 -0.357 0.087 -0.178 0.480 -0.343 0.138 -0.389 0.261
InP,As,Sb,
_”_”
0.810 -1.630 1.200 -1,450 1.850 0.750
Ga,ln, -,As,Sb,
--y
0.814 -1.537 1.109 0.023 2.242 -2.411 -0.015 -1.802 1.920
E&)
AI,Ga,ln,
--i_-y
P
1.350
0.600
0.790 2.230 0.790
0.000
AI,Ga,ln, _,_,As
0.360 0.680 0.380 1.890 0.710
0.700
AI,Ga, _.As,Sb,
--y
0.725 -0.414 1.114 1.071
Ga,ln,
-
.P,As,
-
0.361 0.751 0.242 0.676
Ga,ln, _,As,Sb,
_y
0.173 -0.399 0.588 0.167
InP,As,Sb,
-~-”
0.170 -0.390 0.580 -0.120 1.600 1.300
2.863
1.610
2.560
3.285 -3.158 0.428 -2.961
1.658 -1.667 0.384 -1.271
2.610 -2.053 0.385 -2.606
Consider a pseudomorphic strained heterostructure
A/B
where
B
is the unstrained substrate. A knowledge of
the alloy composition of
A
and
B,
together with the
information contained in table
3
and table
4
suffices to
obtain an estimate of the conduction-band and valence-
Table
4.
Coefficients
of
an expansion
of
the strain-
induced shift of the valence-band average
AE;:.,,
conduction-band
AE,(r)”y
and
C~E~‘.‘~
(defined in equation
(5))
according to equation
(13).
AEFaW
AI,Ga,ln, -,-,As
0.90
AI,Ga, .As,Sb,
_
0.86 0.40 0.53 1.01
2.13
0.33
0.31 1.86 0.13
Ga,ln, -.P,As,
-”
0.91
0.18 0.35 0.40
GaJn, _.As,Sb,
--y
0.33
0.58
0.52 -0.16
InP,As,Sb,
--x--y
0.33 0.59 0.79 -0.02
AI,Ga,ln P
1.10 0.67
-4.39
-4.62
-7.47
-4.61
-5.67
-5.68
-
6.81
7.51
7.65
-3.17
-3.02
-0.39
0.29
1.10
1.05
-1.11
-
1.07
-1.17
-1.37
-0.43
-1.81 -0.53
0.41 0.95
-3.17 -0.27
-1.78 -1.45
1.29
-0.03
-0.55
0.03
-1.98 0.09
-2.05 1.14
”
.^,
,,
Ga,l n
,
-
.P,As,
-
7.52 -0.72 -1.05 -0.04
Ga.ln,
-
.As.Sb.
-
8.73 -1.22 -1.10
0.06
-.”,
.,
InP,As,Sb,
_I_y
8.74 -1.21 -1.92 -0.01
band ofset: equation
(7)
and the information in table
3
allow one
to
calculate
E:
and
E:,
From Vegard’s law
one
readily obtains the lattice mismatch at the interface.
Together with the information listed in table
4,
one
is
able to calculate
E:
and
E:.
Consequently, the conduc-
tion-band and valence-band
offsets
are simply
AEv
=
E:
-
Et
and
AEc
=
E:
-
E:,
respectively.
As an example we consider InGaAsP lattice matched
to
InP.
For this heterostructure we arrive at
AEJAE,
-
0.41
throughout the entire compositional range. This
agrees favourably with the experimental results
AEJAE,
=
0.39
(capacitance-voltage technique
[lZ]).
As an example of
a
strained heterostructure we consider
In.Ga,
_.~
As/AI,,Ga,,,As (reliable experimental results
on strained quaternary alloy heterostructures are not
available).
We
obtain values for
AEJAE,
varying linearly
from
50.66
for
x
=
0
to
-0.63
for
x
=
0.2.
Debbar
et
al[13]
obtained values varying from
20.58
for
x
=
0
to
20.71
for
x
=
0.2
(extrapolated from their deep-level
transient spectroscopy results). Again there is
a
quantita-
tive agreement.
A
linear interpolation of
qr),
E,(X)
and
E,(L),
as
included in table
I,
is helpful in predicting whether the
alloy under consideration exhibits a direct band gap
or
not.
From equation
(5)
it
follows that the uppermost
valence-band
in
a layer material in biaxial compression
(E,~
<
0)
is heavy-hole-like, whereas in biaxial extension
>
0)
it
is light-hole-like. In semiconductor quantum-
well
lasers,
the heavy-hole subband has a parallel mass
generally smaller than that of the light-hole subband.
Placing the active layer under biaxial compression results
in the lowering of the uppermost light-hole subband with
30
6
1
/
6
100%