Page 1
Temperature
K. P. O’Donnell and X. Chen
llniversity of Strathclyde, Glasgow, G4 ONG Scotland, United Kingdom
dependence of semiconductor band gaps
(Received 5 November 1990; accepted for publication 27 March 1991)
The application of a simple three-parameter fit to the temperature dependence of
semiconductor band gaps is justified on both practical and theoretical grounds. In all trials
the fit is numerically better than that obtained using the widely quoted Varshni
equation. The formula is shown to be compatible with reasonable assumptions about the
influence of phonons on the band-gap energy. Approximate analytical expressions are derived
for the entropy and enthalpy of formation of electron-hole pairs in semiconductors.
In this letter we advocate the use of a new three-pa-
rameter fit to the temperature dependence of semiconduc-
tor band gaps. This fitting improves upon the semi-empir-
ical Varshni equation* both numerically,
better fits to the data, and theoretically, since the parame-
ters of the fit may be related to an intrinsic interaction of
semiconductors, namely the electron-phonon
Similar expressions to ours have appeared in the litera-
ture2T3 but the practical and theoretical justification of this
kind of data fit have not previously been worked out in
detail. We emphasize that our approach is empirical: we
aim simply to describe the data as well as possible with the
minimum number of free parameters.
The Varshni relation for the temperature dependence
of semiconductor band gaps is
since it gives
coupling.
Eg(T)=Eo--cYT2/(T+pA
(1)
where cr and fi are fitting parameters characteristic of a
given material.’ The theoretical basis of this much-used
relation’15 is unfortunately rather weak, since /3, which is
supposed to be related to the Debye temperature, may in
certain important cases be negative.’ Moreover, at low
temperature, Eqm ( 1) predicts a quadratic temperature de-
pendence, whereas experiment finds (an approximate)
temperature independence at very low temperatures. This
point was noted by Manoogian and Leclerc’ who devised
the equation
&=-J-$,(1 +AT”) +I+, coth(; j +t),coth( 2 j].
The first term on the right is included to describe the effect
of lattice expansion on the band gap, by means of the pa-
rameters A and X. The coth terms represent contributions
from electron-phonon coupling, with acoustic (0,) and op-
tical (0,) terms being averaged separately. 8 = hv/k is the
mode energy expressed as a temperature. With six param-
eters to fit, the procedure is somewhat tricky, but fits5*6 to
experimental data on the group-IV semiconductors seem to
yield reasonable values of 8, and 8,. In addition the lattice
contribution was found to vary as Te.’ for all three semi-
conductors C, Si, and Ge. A simplified form of Eq. (2),
which sets x = 2/3 and considers only one average phonon
energy is used by Donofrio et al.’
Viiia et al3 fit data on the critical-point
germanium to the Varshni relation (their Table I), but
point out that they can also lit these data to an expression
energies of
(3)
which represents a decrease in the energy thresholds “pro-
portional to Bose-Einstein statistical factors for phonon
emission plus absorption.” The term within the brackets in
Eq. (3) readily reduces to coth( 8/2T) so that Eq. (3) can
be written in a form
E=a - b coth( 8/2T),
which is comparable with Eq. (2).
Recently, Collins et al* have used an empirical fit with
a distribution of phonon energies. They write
(3a)
E&T) =E’ + J’dwf (~1 [n(w,T) + :I
- a(cll + 2cr2)Al?T)/3Vc.
(4)
The last term on the right accounts for lattice expansion.
The phonon term is numerically integrated using the ap-
proximation f (w) = cwg(w) where g(w) is the empirical
density of phonon states and n(o,T> is the Bose-Einstein
occupation number. This approach owes a great deal to
theoretical work by Allen and co-workers.‘,”
experiment is obtained at the expense of a high computing
overhead in the work of Collins et al.’ Rather poor fits are
obtained using the full theoretical treatments.
In this letter we advocate the equation
A good fit to
Eg(T)=Eg(O) -S(fiw)[coth((fizu)/2kT) - l] (5)
as a direct replacement of the Varshni equation. Eg(0) is
the band gap at zero temperature, S is a dimensionless
coupling constant, and (ti) is an average phonon energy.
We adopt this notation from the vibronic model of Huang
and Rhys.” Data taken from the literature’“14
GaAs, Gap, Si, and diamond are to be fitted. Following the
methods of Thurmond” we use Eq. (5) to derive expres-
sions for the thermodynamic functions: the Gibbs energy,
enthalpy, and entropy of formation of electron-hole pairs
in a semiconductor. Finally we justify, without proof, the
applicability of the new equation on the basis of simple
thermodynamics.
Data for GaAs, GaP, Si, and C were obtained from the
literature. Table I shows the best fit values of the parame-
concerning-
2924 Appl. Phys. Lett. 58 (25), 24 June 1991
0003-6951/91/252924-03$02.00 @ 1991 American Institute of Physics
2924
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Page 2
TABLE I. Fit parameters fo? temperature dependence of semiconductor
band gaps.
Temp. range
‘3-Q
f-Q(O)
(eV)
(fiwu)
meV s Ref.
Si
GaAS
GaP
C
O-300
lo-loo0
lo-1300
100-700
1.170
1.521
2.338
5.450
1.49
3.00
3.35
2.31
25.5
26.7
43.6
94.0
13.
12
17
1;
ters Eg(O), S’, and (&)
obtained are shown in Fig. 1. In each case, they are statis-
tically better than Varshni fits to the same data. The silicon
data show some anomalies which will be described in detail
elsewhere.
At high temperatures, kT)(iia)
in each case. The excellent fits
and
Eg( T) -+EG(O) - 2SkT.
(6)
The slope of the Eg vs T curves approach the limiting
value
(7)
It appears from Table I that the limiting values for silicon
and diamond are somewhat smaller than for the other
semiconductors. However, the range of applicability
data fit never exceeds the range of the measurements. A
wider range on the Si data is obviously desirable, while the
diamond data is also seen to have a restricted range if the
large value of the mean phonon energy is taken into ac-
count.
The entropy and enthalpy of formation
hole pairs may be obtained from Eq. (5) following
method of Thurmond who identifies Eg=AE,,
dard Gibbs energy.” Thus,
of a
of electron-
the
as the Stan-
d
A&,(T) = - dT AE,,( T)
S(fim)”
=----
csch( (+iu))/2kT)
2
2k
(
T
(8)
is the entropy of formation of electron-hole pairs and the
enthalpy, AH,, e AE,, + TAS,,, can be evaluated using
Eqs. (5) and (8). At high temperature, Eq. (8) becomes
A&( T) -t - 2Sk.
(9)
The temperature dependence of the entropy for formation
of electron-hole pairs in silicon is plotted by way of exam-
ple in Fig. 2 using parameters from Table I. This plot
differs appreciably from previous estimates16 which used a
polynomial fit to the data of Bludau et aZ.l3 Extrapolation
is always a dangerous procedure but we may be fairly con-
fident that the entropy of formation for e-h pairs in silicon
saturates near room temperature with a value close to 3k.
Finally, we offer a brief justification
namics of the form of Eq. (5). It has been known for a very
long time that temperature-dependent
teractions effectively determine
from thermody-
electron-phonon in-
semiconductor band
0
1.20 [
t
t
-.
I
(c) Si
I
>
t
%
c!J
P
3
1.10 -I
0 300
5.45.
‘*em
. 00
.
(d) c
5.43 -
%
‘.
.
s
9
l
2
-1
.
5.36-
5.34 i
100
I.
J
700
300 500
TEMP/K
FIG. 1. Energy band gaps as a function of temperature of (a) gallium
arsenide, (b) gallium phosphide, (c) silicon, and (d) diamond.
gaps.” The reasoning is as follows. The band gap reflects
the bond energy. An increase in temperature changes the
chemical bonding as electrons are promoted from valence
band to conduction band. In the intrinsic
range, direct effects due to thermal band-to-band excita-
tions are negligible. The lattice phonons on the other hand
have relatively small energies and are excited in large num-
temperature
2925
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Appl. Phys. Lett., Vol. 58, No. 25, 24 June 1991
K. P. O’Donnell and X. Chen
2925
Page 3
-0.6 I-L-.
0
TEYP/K
FIG. 2. Entropy of formation of electron-hole pairs in silicon, calculated
using Eq. (8).
bers at moderate temperatures. They influence the bonding
through various orders of electron-phonon interaction.“,”
Since any such effect depends on phonon numbers, the
dependence on temperature
W3&m,ons-~(~,T) -
coth(fiiw/2kT).
that the dependence of band gap on lattice expansion takes
nearly the same analytic form. The (static) lattice contri-
bution to the temperature dependence
aa?
I 1 AV(T)
(AEg)lattice= 7 7 *BF,
”
where [aEg/ap],
is the pressure dependence, B is the bulk
modulus equal to l/3 (cl1 + 2c12) and Vo, AV( T) are the
lattice volume and its change with temperature, respec-
tively. Using a standard thermodynamic result of Grun-
eisen”
should take the form
We now show
(10)
BAV
-=yv
vo
i!?
we find
aEg E
(AEg)latticeE - 7 I 1
[n(w,T> + $]dw (13)
“---
(11)
(12)
where y is the Gruneisen parameter and E is the mean
thermal energy of the sample given by
E=n
s w&Y(w)
which for a single effective mode g( w ) = S( w - wo) is sim-
PlY
it?=fiiw,,[n(woT) + f].
Hence ( AEg) lattice depends on z, which is seen to depend
on the phonon occupation number, and on V, which varies
only weakly with temperature. The simple theory therefore
predicts that the lattice contribution to AEg( T) should
vary in a similar way to the whole shift. This is in fact
found to be the case in practice, in contradiction of the
power law of the fitting Eq. (2) proposed by Manoogian
and Leclerc.’ Since lattice and phonon contributions to the
band-gap shift have very similar (averaged) temperature
dependences, the use of a simple fitting equation is justified.
In summary, we have shown that a reasonable three-
parameter thermodynamic function may be used with good
effect to fit the temperature dependences of the fundamen-
tal band gaps for technologically important semiconduc-
tors with room-temperature gaps in the range from 1.0 to
5.5 ev.
We would like to thank Gordon Davies for helpful
discussions. X. Chen is supported by the University of
Strathclyde ORS fund.
(14)
‘Y. P. Varshni, Physica 34, 149 (1967).
‘A. Manoogian and A. Leclerc, Phys. Status Solidi B 92, K23 (1979).
3L. Viiia, S. Logothetidis, and M. Cardona, Phys. Rev. B 30, 1979
(1984).
“The Varshni relation is quoted by S. M. Sze, in Semiconductor Devices
(Wiley New York, 1985), p. 13 and by Jacques I. Pankove, in Optical
Processes in Semiconductors (Dover, New York, 1975), p. 27.
‘A recent example is Z. M. Fang, K. Y. Ma, D. H. Jaw, R. M. Cohen,
and G. B. Stringfellow, J. Appl. Phys. 67, 7034 (1990).
bA. Manoogian and A. Leclerc, Can. J. Phys. 57, 1766 (1979).
‘T. Donofrio, G. Lamarche, and J. C Woolley, J. Appl. Phys. 57, 1932
(1985).
‘A. T. CoIlins, S. C. Lawson, Gordon Davies, and H. Kanda, Phys. Rev.
Lett. 65, 891 (1990).
‘Philip B. Allen and Volker Heine, J. Phys. C Solid State Phys. 9, 2305
( 1976), and references therein.
‘OP. B. Allen and M. Cardona, Phys. Rev. B 27, 4760 (1983); P. Lan-
tenschlager, P. B. Allen, and M. Cardona, ibid. 31, 2163 (1985).
“K. Huang and A. Rhys, Proc. R. Sot. A 204, 406 ( 1950).
12M. B. Panish and H. C. Casey, J. Appl. Phys. 40, 163 (1969).
“W Bludau, A. Onton, and W. Heinke, J. Appl. Phys. 45, 1846 (1974).
14C.‘D. Clark, P. J. Dean, and P. V. Harris, Proc. R. Sot. A 277, 312
(1964).
Is C. D . Thurmond, J. Electrochem. Sot. 122, 1133 ( 1975).
16V. Heine and J. A. Van Vechten, Phys. Rev. B 13, 1622 (1975).
“H Y. Fan, Phys. Rev. 82, 900 (1951); Harvey Brooks, Adv. Electron.
7;85 (1955).
‘*J M Ziman Electrons and Phonons (Oxford University, Oxford,
196Oj, p. 1541
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Appl. Phys. Let, Vol. 58, No. 25, 24 June 1991
K. P. O’Donnell and X. Chen 2926
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