Special Issue Paper
Journal of Electronic Materials, Vol. 24, No. 9, 1995
Temperature Dependence of Band Gaps in HgCdTe
and Other Semiconductors
SRINIVASAN KRISHNAMURTHY,* A.-B. CHEN, t A. SHER,* and
M. VAN SCHILFGAARDE*
*SRI International, Menlo Park, CA 94025
tPhysics Department, Auburn University, Auburn, AL 36349
Band-edge shifts induced by the electron-phonon interaction are calculated for
HgCdTe alloys and various semiconductor compounds starting from accurate
zero-temperature band structures. The calculated t e m p e r a t u r e variation of gaps
agrees with experiments to better than 10% in all materials except InAs and
InSb where the deviation is about 50%. While the simple picture t h a t the intra
(inter)-band transitions reduce (increase) the gap still holds, we show that both
the conduction band edge E c and valence band edge E v move down in energy.
These shifts in E v affect the valence band offsets in heterojunctions at finite
temperature. The t e m p e r a t u r e variations of valence band offset and the electron
effective mass are also reported.
K e y w o r d s : III-V semiconductors, band offset, electron-phonon interactions,
HgCdTe and alloys, t e m p e r a t u r e - d e p e n d e n t band structures
INTRODUCTION
The t e m p e r a t u r e (T) dependence of energy gaps of
9 semiconductors is of great physical and technological
interest. The quantities such as band offset and effective m a s s depend sensitively on the t e m p e r a t u r e
variation of band edges. Numerous theoretical 1-9 and
experimentapO 19 studies have been u n d e r t a k e n to
obtain both qualitative and quantitative variations of
various gaps in semiconductors. The gap decreases
' with increasing temperature in medium-gap and widegap semiconductors, and it increases in small-gap
materials such as HgCdTe, PbS, PbSe, and PbTe. The
thermal expansion of the lattice and electron-phonon
interactions are usually considered causes for the
t e m p e r a t u r e variation of the band structures. Thermal expansion always reduces gaps.
In a perturbation-theory t r e a t m e n t of electronphonon interactions, the intraband transitions reduce the gap whereas interband transitions increase
it, and the net shift in the gap can be positive or
negative. Here we calculate the gap variation in
(Received October 4, 1994; revised January 8, 1995)
Hgl_xCdxTe alloys, GaAs, InAs, InSb, InP, and CdTe
compounds, starting from accurate band structures,
wave functions, proper phonon dispersion relations,
and taking account of matrix elements of the electronphonon interactions. The contributions from each
phonon branch to each electron band have been obtained to assist physical understanding of the underlying causes of the variations. We show that both
conduction and valence band edges move down in
energy. When the valence band edge moves more t h a n
the conduction band edge, the gap increases with T, as
in the case of some Hgl_xCdxTe alloys with x < 0.5. The
reverse occurs for all III-V compounds studied and
Hgl_xCdxTe with x > 0.5. This observation has an
important effect on our understanding of the variation of band offsets in semiconductor heterojunctions.
In addition to the gap, other features of the band
structure change with t e m p e r a t u r e and will affect the
spectral variations of the absorption coefficient and
transport properties.
METHOD
Our calculation of the t e m p e r a t u r e dependence of
the band gap starts with accurate band structures.
1121
1122
Krishnamurthy, Chen, Sher, and van Schilfgaarde
120
'
I
'
100
80
I
'
I
'
.
Theory
-
Ref.16,17
-
Ref. 18
I
'
I
'
I
where V 1 and V 2are the first two terms in the Taylor
expansion of the total electron-phonon potential in
powers of atomic displacements ~. In the TB formalism, Eq. (1) can be written in terms of the matrix
elements
'
"j,,~/J
/7"/
J
"~///
>
and
r
4o
0
< l'j'alV2[lj(~ >i~ 2[~l'j"
1
"(V) 2V a , ( d l l ~,
,).~l,j, ,
(3)
////if"
,
I
40
,
I
80
,
I
120
I
I
160
,
I
200
,
I
240
,
where d~i is the position vector connecting atomic
sites l, species (anion or cation) j and site l', species j',
and V,J(d~i) is a HPTB matrix element between the
orbitals ~ and a' located on those atoms. From the
q u a n t u m theory of harmonic crystals, the atomic
displacements ~ can be expressed in terms of normal
modes; that is, phonons. We have
I
280
T(K)
Fig. 1. Change in the band gap of Hgo.~sCdo.22Tewith temperature.
~lj =
Empirical pseudopotential form factors are used to
construct a hybrid pseudopotential tight-binding
(HPTB) Hamiltonian. The pseudopotential part of the
Hamiltonian is universal--it applies to all group IV,
III-V, and II-VI compounds. The smaller tight-binding part is expressed in a minimum set of sp 3 Slater
orbitals per atom. This Hamiltonian is then transformed into an orthonormal basis. A site-diagonal
spin-orbit Hamiltonian is then added. Parameters in
the tight-binding perturbation are chosen to finetune the band structures to agree well with experiments. 2~ Various results obtained using these band
structures are found to be quite reliable. 2~-23 The
present study subjects the accuracy of the wave functions as well as the energies to a sensitive test.
The dilation contribution to the band gap reduction
is given 5,9 by 3aTB3E~_~P, where the thermal expansion coefficient of the lattice aT, the bulk modulus B,
and the change in the gap with pressure are obtained
from the literature2 9 The electron-phonon interactions with all phonon branches that cause the band
structure changes are treated in perturbation theory.
The total Hamiltonian is assumed to be a sum of
potentials from single atoms. The atomic potential in
the solid is traditionally expanded in a Taylor series,
with only the leading term retained, and the energy
shifts it causes are evaluated in second-order perturbation theory. However, it has been demonstrated by
a n u m b e r of researchers 3,4,6 that retention of firstorder perturbation terms with a second term in the
Taylor series expansion is necessary to preserve symmetry. We retain both terms. The change in the
energy at a given wave vector k is
,, l
I
n ' k , ,'Vl'nk>
Z~Enk =<~ nklV2[nk > + ~ < ~ :
2
(1)
1
~q~W~q [~q~Zq.
~q ~q
j
L a J
where q and coare phonon wave vector and frequency,
~. denotes phonon branch, a(a +) is a phonon annihilation (creation) operator, M is the atomic mass, and e
is an eigenvector in a diamond or zine-blende structure of the six-dimensional dynamical matrix eigenvalue problem
Mo~e = D(q)e
(5)
Evaluation of the matrix elements given by Eqs. (2)
and (3) requires knowledge of spatial variations of the
interatomic TB matrix elements V , . In Harrison's
universal TB approach, 24these matrix elements scale
as d -2. In our generalization, we assume that V~,
varies as d -m and the repulsive first-neighbor pair
energy, following Harrison's overlap argument, as rl/
d 2m. The two unknowns m and q are determined by
requiring that the calculated equilibrium bond length
and bulk modulus agree well with experiments. This
approach, with electrons and phonons treated from
the same underlying Hamiltonian, has previousl~
been used successfully to explain hot electron transistor characteristics 23 and is also in fairly good agreement with first-principles calculations. 2~The dynamical matrix D is calculated from the valence force field
model. 26
The calculational procedure is as follows. For a
chosen material, m and q are evaluated. Then the first
and second derivatives of all interatomic matrix elements are obtained. The dynamical matrix is diagonalized to obtain o) and e as a function o f q and ~. The
phonon structures and electronic band structures are
used [Eqs. (1) through (4)] to obtain the change in the
band energy at a given k. The polar coupling terms are
Temperature Dependence of Band Gaps in
HgCdTe and Other Semiconductors
1123
i n c l u d e d in t h e l o n g i t u d i n a l optical p h o n o n cont r i b u t i o n s . W h e n we are i n t e r e s t e d in s t u d y i n g t h e
,change in t h e direct gap, k is t a k e n to be zero.
H o w e v e r , w h e n t h e t e m p e r a t u r e v a r i a t i o n of t h e
effective m a s s e s or i n d i r e c t gap are studied, n o n z e r o
k v a l u e s m u s t be u s e d a n d t h e B r i l l o u i n zone s u m m a tion in Eq. (1) s h o u l d be c a r r i e d over t h e e n t i r e zone
w i t h r e d u c e d , or no, s y m m e t r y .
RESULTS
T h e c a l c u l a t e d b a n d - g a p c h a n g e as a f u n c t i o n of T
in Hgo 7sCdo.22Te is s h o w n in Fig. 1. W i t h i n c r e a s i n g T,
the d i r e c t gap i n c r e a s e s in Hgo.7sCdo 22Te. Notice t h a t
t h e c a l c u l a t e d v a l u e s are t y p i c a l l y w i t h i n 10 to 15
m e V of e x p e r i m e n t a l values. 1e-is T h e cross (x) a t T = 0
r e p r e s e n t s t h e c a l c u l a t e d z e r o - p o i n t c o r r e c t i o n to t h e
gap (13.6 m e V for Hgo78Cdo22Te). T h e z e r o - t e m p e r a t u r e b a n d gap c a l c u l a t e d w i t h o u t e l e c t r o n - p h o n o n
i n t e r a c t i o n s s h o u l d h a v e this c o r r e c t i o n s u b t r a c t e d
for c o m p a r i s o n to e x p e r i m e n t a l values.
T h e c h a n g e in t h e gap is t r a d i t i o n a l l y e x p l a i n e d in
t e r m s of i n t e r - a n d i n t r a b a n d i n t e r a c t i o n s . T h e
i n t r a v a l e n c e (conduction) b a n d i n t e r a c t i o n s p u s h t h e
v a l e n c e (conduction) b a n d edge u p (down), t h u s red u c i n g t h e gap. Similarly, t h e v a l e n c e - c o n d u c t i o n
b a n d i n t e r a c t i o n s i n c r e a s e t h e gap. H e n c e , one m i g h t
e x p e c t t h e gap to d e c r e a s e in wide-gap semiconductors a n d possibly i n c r e a s e in small-gap semiconductors. I n addition, a r g u m e n t s b a s e d only on t o t a l density of s t a t e s a n d i g n o r i n g v a r i a t i o n s in m a t r i x elem e n t s will predict t h e v a l e n c e b a n d edge E v move u p
in e n e r g y , b e c a u s e t h e hole effective m a s s is one to two
o r d e r s of m a g n i t u d e l a r g e r t h a n t h e e l e c t r o n mass. As
s e e n f r o m Fig. 2, our detailed calculations of b a n d
edge m o v e m e n t s in Hgo 78Cdo 22Te do n o t s u p p o r t this
t r a d i t i o n a l view. We find t h a t b o t h E c a n d E v (solid
lines) m o v e down in energy. T h i s s a m e t r e n d is obs e r v e d in o t h e r s e m i c o n d u c t o r c o m p o u n d s s t u d i e d
(GaAs, InP, InAs, InSb, a n d GaSb). T h e m o v e m e n t of
t h e v a l e n c e (solid line) a n d c o n d u c t i o n (dashed) b a n d
edges due to i n t e r a c t i o n w i t h o t h e r b a n d s is also
s h o w n in Fig. 2. T h e i n t e r a c t i o n of t h e b a n d edges
w i t h t h e c o n d u c t i o n b a n d s (CBs) is m u c h s t r o n g e r
t h a n w i t h v a l e n c e b a n d s (VBs), a n d c o n s e q u e n t l y
b o t h E v a n d E m o v e down in energy. E v m o v e s m u c h
m o r e t h a n E c a n d t h e gap i n c r e a s e s .
We a n a l y z e t h e s t r e n g t h of i n t e r - a n d i n t r a b a n d
e l e c t r o n - p h o n o n i n t e r a c t i o n s b y p r e s e n t i n g t h e contributions from each band and from each phonon
mode. Table I lists the calculated values for
Hgo.vsCdo22Te at 300K. A l t h o u g h spin is i n c l u d e d in
o u r b a n d s t r u c t u r e calculations, only spin a v e r a g e d
v a l u e s a r e listed. B a n d indices 1 to 4 c o r r e s p o n d to
VBs, a n d t h e o t h e r s to CBs. T h e first two rows show
c h a n g e s in E v a n d E c d u e to i n t e r a c t i o n s w i t h v a r i o u s
b a n d s . C o n t r i b u t i o n s f r o m e a c h p h o n o n m o d e are
listed in t h e r e m a i n i n g rows. T h e lowest VB is a b o u t
12 eV below E a n d E , a n d h e n c e t h e i n t e r a c t i o n does
n o t affect t h e b a n d edges. We see t h a t t h e i n t e r a c t i o n
w i t h o t h e r VBs t e n d s to p u s h t h e b a n d edges u p in
100
I
'
I
I
I
'
I
i
I
'
I
'
I
i
I
o
-ioo
,~ -200
k.tJ
-300
-400
-5000
i
100
200
300
i
400
I
500
I
600
T(K)
Fig. 2, Variation of conduction (dashed line) and valence (solid line)
band edges of Hgo.TBCdo,22Tewithtemperature.
T a b l e I. C a l c u l a t e d C h a n g e i n t h e V a l e n c e (v) a n d C o n d u c t i o n (c) B a n d E d g e E n e r g i e s
(in meV) o f Hg0.7sCd0.22TeA l l o y
Band
1
2
3
4
5
6
7
8
Total
v
c
5.90
1.04
13.99
2.84
56.85
17.27
88.51
34.07
-80.26
-23.81
-92.49
-25.43
-97.45
-43.96
-102.07
-42.48
TA
v
c
4.03
0.26
6.21
0.65
30.06
12.41
41.52
27.47
-49.86
-3.58
-62.17
-14.81
-61.04
-24.50
-49.62
-15.40
LA
v
c
0.91
0.56
3.18
1.29
3.14
1.27
10.53
1.72
-10.63
-14.94
-5.92
-3.00
-11.41
-2.66
-20.59
-7.17
LO
v
c
0.33
0.13
2.11
0.63
6.12
0.75
11.22
0.44
-9.52
-3.78
--4.98
-2.45
-11.80
-1.18
-17.40
-7.31
v
0.63
2.48
17.53
25.24
-10.26
c
0.10
0.27
2.84
4.44
-1.52
Note: Contributions from interaction with various phonon modes are shown in rows 3 to 10.
-19.42
-5.16
-13.20
-15.62
-14.45
-12.59
TO
1124
Krishnamurthy, Chen, Sher, and van Schilfgaarde
•
0.4
d o
~JS '
o
Q
J
I
I
0.2-
j
x Theory ~ dEg/dT
O Expt )
m Theory dE~)dT
(
()
~--~ -0.2
'
oc)
(Mo
](
LM l ' -
"~J-o
-0.4 -
-0.6 -
-0.8 - ['~ i
0
I
0.8
0.4
~
I
1.2
fi
i I
1.6
i
Eg (eV)
Fig. 3. Derivative of direct gap with temperature for various semiconductor compounds and alloys as a function of zero-temperature
gap. The vertical lines represent Hgo.TsCd0.~Te,InSb, InAs, Hg0.~Cdo.sTe,
GaAs, and CdTe, respectively.
Table II. Calculated E~ [meV], 7 [eVil, c [eV] ,and
the Effective Mass Ratio of Hgo.TsCd0~Te Alloy as
Functions of Temperature
m*(T)/
T
E~
7
c
m*(0)
1.00
113.60
47.7656
0.0588
1.0000
10.00
112.67
47.7553
0.0588
1.0005
20.00
112.56
47.7169
0.0592
1.0072
30.00
114.44
47.6421
0.0598
1.0199
40.00
117.15
47.5582
0.0607
1.0361
50.00
120.42
47.4461
0.0615
1.0532
60.00
123.96
47.3310
0.0624
1.0714
70.00
127.65
47.2091
0.0634
1.0904
80.00
131.44
47.0821
0.0643
1.1095
90.00
135.28
46.9418
0.0653
1.1288
100.00
139.17
46.7964
0.0662
1.1483
150.00
158.85
46.1544
0.0712
1.2529
200.00
178.73
45.5930
0.0767
1.3669
250.00
198.68
45.2441
0.0832
1.4938
300.00
218.66
45.1167
0.0908
1.6342
350.00
238.65
45.3460
0.1000
1.7913
400.00
258.66
46.0193
0.1115
1.9672
450.00
278.67
47.3751
0.1263
2.1657
500.00
298.69
49.8338
0.1468
2.3919
550.00
318.71
54.0581
0.1763
2.6491
600.00
338.74
61.7125
0.2238
2.9445
Note: m* (0) is 0.008. The zero point correction is 13.6 meV.
energy, as expected. Note that the top two valence
bands contribute the most to the fundamental bandedge changes. However, interaction of the band edges
with CBs is even stronger and negative. Particularly,
the interaction of E with all conduction bands is
strong. As E~ moves down more than E , the gap
increases in Hgo.78Cdo.22Te. To understand the role of
v
various phonons, we display the contribution from
each mode separately. The phonon-induced changes
in the band edges at 300K in Hgo.78Cdo.22Teare listed
in the third through tenth rows of Table I. We see that
acoustic phonons account for about 75% of the total
change in the valence and conduction band-edge energies. The selection rules wipe out interband matrix
elements between electrons and polar longitudinal
phonons, so in spite of the stronger coupling constant,
they do not dominate this phenomenon as they d o
with mobilities, which depend on intraband matrix
elements.
Our calculations for other compounds show a qualitatively similar role for phonons. In addition to
Hgo.78Cdo.22Te,we studied the band-gap variation with
temperature in GaAs, InAs, InP, InSb, GaSb, and
CdTe compounds and HgCdTe alloys. The gap changes
linearly at high temperatures (>150K). The calculated d E J d T values (circle) are compared with experiments (cross) values in Fig. 3. We see that the calculations produced correct trends in all these materials,
b u t compare less favorably with experiments in InAs
and InSb. However, it is important to note that the
sign of the change is not exclusively determined by
the magnitude of the zero-temperature gap. For example, although Hgo.7oCdo.3oTe and InSb have the
same zero-temperature gap of 0.235 eV, the InSb gap
decreases with T, whereas the Hgo.voCdo.3oTegap increases with T. The combination of gap size, conduction band width, and intervalley separations gives
rise to these interesting variations in the gap with T.
The observation that both E v and E c move down in
energy has an important effect on band offsets in
heterojunction-based devices. For example, the zerotemperature
valence band offset between
Hgo.78Cdo.22Te and CdTe is believed to be around 350
meV. H o w e v e r , we find t h a t at 300K, E v in
Hgo.TsCdo.22Teand in CdTe moves down by 215 and 30
meV, respectively. If the dipole contribution remains
the same, the valence band offset decreases to 165
meV at 300K. The contention that the dipole contribution is nearly temperature independent stems from
the observation that any shift in the average effective
crystal potential should effectively be screened out,
since these semiconductors are good dielectrics (e >
10). The temperature variation of the bands should be
taken with respect to a fLxed average potential. For
our Hamiltonian, the valence band edge movements
in each side of the junction are calculated with reference to a fLxed average state. Thus, the calculated
temperature dependence of the difference in the VB
edge of the constituent heterojunction materials effectively governs the temperature dependence of the
band offset. In addition to the electron-phonon interactions discussed above, lattice dilation changes the
band edges differently. 9 This effect is not included
here. In any case, this band offset change has important implications for the design of abrupt heterojunction infrared (IR) absorption and confined
well laser devices.
In principle, the band structure at any wave vector
Temperature Dependence of Band Gaps in
HgCdTe and Other Semiconductors
1125
k will change with temperature. With the change in
the fundamental gap, the band curvature (or effective
mass) also changes thus affecting the optical and
t r a n s p o r t properties of the material. The self-energy
calculated in this method will include the effect of
scattering due to phonons and the change in the
t e m p e r a t u r e - d e p e n d e n t band structure self-consistently. In the case a of narrow-gap material such as
HgoTsCd0.22Te, the effective mass alone does not explain the low-energy portion of the conduction band
structure. The lowest CB energy at any k is best
described by a hyperbola, 27 (Tk2 + c2)1/2- c. The calculated band gap, effective mass, 7, and c as functions of
T are given in Table II. The effective mass and c are
directly proportional to the gap and hence monotonically increase with T. This is expected from a k 9 p
theory argument, b u t the magnitudes predicted by
the two theories differ. 7 decreases slightly at lower
t e m p e r a t u r e s and then starts to increase with T. In a
previous publication, 27 we had simulated these temperature variations of 7 and c by adjusting the Hg
concentration in HgCdTe alloys to produce proper gap
at each temperature 9 Those values are in remarkable
agreement with the values reported in Table II. We
conclude that 7 and c (given in Table II) can be
interpolated to considerable accuracy for any positive
gap in the HgCdTe alloys9
important consequence of this observation will be in
the band offsets in semiconductor heterojunction devices. Finally, there is a small and usually negligible
zero-point motion contribution to low-temperature
band gaps arising from electron-phonon interactions.
ACKNOWLEDGMENT
We t h a n k Dr. M. Cardona of the Max Planck Institute, Stuttgart, for pointing us to several references.
Funding from ONR (contract N00014-93-C-0091) and
ARPA (contract MDA972-92-C-0053) is gratefully
acknowledged.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
CONCLUSIONS
Although the calculations produced correct trends
in all materials, the calculated changes in the band
gap of InAs and InSb were about a factor of two
smaller than in the experiments. We find that our
calculated TA phonon frequencies a w a y from zone
center in these compounds were considerably larger
than those found in experiments. As noted from Table
I, a substantial contribution comes from acoustic
phonons Consequently, our theoretical values of E
(T) are smaller than in experiments. Better predictability should result from improvement in the dynamical
matrix calculated from the underlying Hamiltonian.
In addition, at higher t e m p e r a t u r e s higher-order perturbation terms m u s t be included along with finitet e m p e r a t u r e "renormalized" bands rather than the
zero-temperature bands. Such renormalization affects the monotonic change in the gap and introduces
nonlinear terms.
In summary, we have calculated the t e m p e r a t u r e
variations of band gaps in various semiconductors. A
fairly accurate H P T B Hamiltonian is used in the
calculation of electron and phonon structures. The
calculations explain the increase in the band gap of
Hgo.TsCdo.22Te, and the decrease in the band gap of all
III-V compounds studied. We show that acoustic
phonons make the major contribution. Contrary to
traditional thinking based on total density of states
arguments, we find that both the valence and the
conduction band edges move down in energy. One
9
9
11.
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13.
14.
15.
16.
g
17.
18.
19.
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21.
22.
23.
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25.
26.
27.
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