JOURNAL OF APPLIED PHYSICS
VOLUME 90, NUMBER 8
15 OCTOBER 2001
Dispersion-related assessments of temperature dependences
for the fundamental band gap of hexagonal GaN
R. Pässlera)
Technische Universität Chemnitz, Institut für Physik, D-09107 Chemnitz, Germany
共Received 30 March 2001; accepted for publication 11 July 2001兲
We have analyzed a series of data sets available from published literature for the temperature
dependence of A and B exciton peak positions associated with the fundamental band gap of
hexagonal GaN layers grown on sapphire. In this article, in contrast to preceding ones, we use the
dispersion-related
three-parameter
formula
E g (T)⫽E g (0)⫺( ␣ ⌰/2) 关 (1⫹( ␲ 2 /6)(2T/⌰) 2
4 1/4
⫹(2T/⌰) ) ⫺1 兴 , which is a very good approximation in particular for the transition region
between the regimes of moderate and large dispersion. This formula is shown here to be well
adapted to the dispersion regime frequently found in hexagonal GaN layers. By means of
least-mean-square fittings we have estimated the limiting magnitudes of the slopes, S(T)
⬅⫺dE g (T)/dT, of the E g (T) curves published by various experimental groups to be of order ␣
⬅S(⬁)⬇(5.8⫾1.0)⫻10⫺4 eV/K. The effective phonon temperature has been found to be of order
⌰⬇(590⫾110) K, which corresponds to an ensemble-averaged magnitude of about 50 meV for the
average phonon energy. The location of the latter within the energy gap between the low- and
high-energy subsections of the phonon energy spectrum of h-GaN suggests that the weights of
contributions made by both subbands to the limiting slope ␣ are nearly the same. This explains the
order of ⌬⬇0.5– 0.6 as being typical for the dispersion coefficient of the h-GaN layers under study.
The inadequacies of both the Bose–Einstein model 共corresponding to the limiting regime of
vanishing dispersion ⌬→0兲 and Varshni’s ad hoc formula 共corresponding to a physically unrealistic
regime of excessively large dispersion ⌬⬇1兲 are discussed. Unwarranted applications of these
conventional models to numerical fittings, especially of unduly restricted data sets (T⭐300 K), are
identified as the main cause of the excessively large scatter of parameters quoted for h-GaN in
various recent articles. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1402147兴
magnitude as the Debye temperature ⌰ D of the material in
question. Various early applications1,2 of this ad hoc model
to semiconductor materials with moderate widths of the fundamental band gap, E g (0)⬍2.5 eV, seemed to show reasonable behavior. On the other hand, other applications of this
model—also by the author himself— 1 to the cases of diamond and 6H SiC even gave negative values both for the
limiting slope ␣ and the parameter ␤. This was in clear conflict with theoretical expectations. The same result 共␣ ⬍0 and
␤ ⬍0兲 emerged later when Varshni’s formula was applied to
a least-mean-square fitting process3 for the E g (T) dependence measured in a hexagonal GaN layer 共between 2 and
295 K兲. In view of such physically absurd parameter
constellations,1,3 for diamond, SiC, and h-GaN, one could
have questioned the applicability of Varshni’s formula to
wide band gap materials, E(0)⬎2.5 eV.
Comprehensive theoretical investigations performed during the past 20 yr have unanimously shown that the contributions of the individual phonon modes to the total shrinkage
of the fundamental band gap E g (0)⫺E g (T) are generally
proportional to the average phonon occupation numbers
n̄(T)⫽(exp(ប␻/kBT)⫺1)⫺1 共cf. Ref. 4 and theoretical articles cited therein兲. Temporarily neglecting the dispersion of
phonon energies, one could thus expect the variation of the
band gap width to be nearly proportional to a corresponding
Bose–Einstein occupation factor,5–10 (E(0)⫺E(T))
⬀(exp(⌰B /T)⫺1)⫺1, where ⌰ B ⬅ប ␻ eff /kB represents some
I. INTRODUCTION
The temperature dependence of the fundamental energy
gap is a basic empirical property of semiconductor materials.
A detailed knowledge of this E g (T) dependence is of great
importance, above all, for applications in optoelectronic devices that are intended for operation within a relatively large
temperature interval. This concerns in particular GaN and its
alloys with other III–V nitrides 共AlN and InN兲, which are to
be used, among other things, for fabrication of devices operating above room temperature. Good design of such devices
thus requires, among other things, a detailed experimental
study of the E g (T) dependence in these materials from relatively low up to high T 共i.e., beyond 300 K兲 in combination
with a physically adequate analytical representation and accurate numerical fitting of the corresponding data sets.
An analytical expression, which has been frequently
used during the past 30 yr for numerical fittings of E g (T)
dependences reported for various semiconductor materials,
was first suggested by Varshni1 in the simple form E(T)
⫽E(0)⫺ ␣ T 2 /( ␤ ⫹T). In this model, the parameter ␣ represents the magnitude of the limiting slope of the corresponding E(T) curve, and ␤ is a physically undefinable temperature parameter believed to be of the same order of
a兲
Electronic mail: passler@physik.tu-chemnitz.de
0021-8979/2001/90(8)/3956/9/$18.00
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© 2001 American Institute of Physics
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J. Appl. Phys., Vol. 90, No. 8, 15 October 2001
R. Pässler
effective phonon temperature. Yet, this model function shows
a plateau in the cryogenic region (T⬍⌰ B /5), which contradicts experimental observations. This applies in particular to
hexagonal GaN layers, where the E g (T) dependence in this
region is repeatedly found to be given in rather good approximation by a quadratic function, (E g (0)⫺E g (T))⬀T 2
共from 0 up to about 200 K; see below兲.
A physically adequate description of E g (T) dependences
for a large variety of semiconductors, including wide band
gap materials, can be given only on the basis of a sufficiently
general analytical framework,4,10–12 one that accounts in a
reasonable way for the basic features of the phonon dispersion actually found in a given material. In Sec. II we present
numerical analyses of a variety of data sets13–22 reported in
recent years on the temperature dependence of absorption or
luminescence peak positions due to A 共and B兲 excitons in
h-GaN layers grown on sapphire (Al2O3) substrates. The
corresponding least-mean-square fitting processes were performed using a somewhat simpler version 共three-parameter
formula11兲 of the analytical apparatus reported in Refs. 4 and
12, which is well adapted to the particular dispersion regime
repeatedly found in hexagonal GaN layers. The results are
discussed in Sec. III. Various aspects of the inappropriateness
of Varshni’s formula and expressions of Bose–Einstein type
are considered in detail in Sec. IV.
II. DISPERSION-RELATED MODEL SPECIALIZED FOR
HEXAGONAL GaN
A detailed numerical analysis of the E(T) data sets presented in Refs. 13–22 for h-GaN layers grown on sapphire
by an elaborate dispersion-related analytical model 共such as
the ␳ representation4,12 or the two-oscillator model23 which
is, however, beyond the scope of the present article兲 shows,
among other things, that the material-specific phonon dispersion coefficient4
⌬⬅ 冑具 共 ប ␻ ⫺ប ␻
¯ 兲2典
冒
ប␻
¯,
共1兲
is on the order of ⌬⬇0.5– 0.6. This means, in particular, that
the ⌬ values for h-GaN samples are as a rule of the same
order as 共slightly smaller than兲 the critical magnitude4,24 of
⌬ c ⫽3 ⫺1/2⫽0.577 35 共the latter corresponding to the characteristic boundary between the regimes of moderate and large
dispersion兲.4,24 This situation, for hexagonal GaN, generally
enables reasonable numerical fittings and theoretical interpretations of E(T) data sets on the basis of a relatively
simple analytical three-parameter expression of the form11
E 共 T 兲 ⫽E 共 0 兲 ⫺
␣⌰
2
冉冑 冉 冊 冉 冊 冊
4
1⫹
␲ 2 2T
6 ⌰
2
⫹
2T
⌰
4
⫺1 .
共2兲
Here we have denoted as usual by ⌰⫽ប ␻
¯ /k B the effective
共average兲 phonon temperature,4,11,12 and the parameter ␣ represents the limiting slope4,11,12 共entropy2,10兲, ␣ ⬅
⫺dE(T)/dT) 兩 T→⬁ , of this E(T) curve. 关Note that Eq. 共2兲 is
coincident with the ␳ →1 limit of the more general
dispersion-related formula used in Refs. 4 and 12. This corresponds to fixing the dispersion coefficient ⌬ at a magnitude
3957
of ⌬ c ⫽3 ⫺1/2.兴 For sufficiently low temperatures, T⬍⌰/4
共i.e., here T⬍150 K兲, Eq. 共2兲 can be seen to reduce in very
good approximation to a quadratic dependence of the form
E(T)→E(0)⫺( ␲ 2 ␣ /12⌰)T 2 共cf. Sec. IV兲. At sufficiently
high temperatures, T⬎⌰, Eq. 共2兲 tends to the linear asymptote E(T)→H(⬁)⫺ ␣ T, where H(⬁)⫽E(0)⫹ ␣ ⌰/2 represents the limiting enthalpy.2,10
We have performed least-mean-square fittings of the
E(T) data sets given in Refs. 13–22 for h-GaN layers grown
on sapphire by means of Eq. 共2兲. 共Four examples are shown
in Fig. 1.兲 The resulting empirical parameter values are listed
in Table I. In cases where temperature dependences have
been available for the energy positions of both the A and the
B exciton lines 关cf. Figs. 1共b兲 and 1共d兲兴, we have fitted the
corresponding E A (T) and E B (T) data sets simultaneously, by
one and the same set of parameters ␣ and ⌰ 共in combination
with separately adjusted T→0 positions E A (0) and E B (0)
⬅E A (0)⫹(E B ⫺E A )兲. The corresponding constant distances
E B ⫺E A are listed in the fourth column of Table I.
III. DISCUSSION
Viewing the parameter values listed in Table I we observe that the limiting slopes for the h-GaN samples are
throughout of order ␣ ⬵(5.8⫾1.0)⫻10⫺4 eV/K. The associated effective phonon temperatures are ⌰⬵(590⫾110) K.
Observing that the relevant 共high-temperature limiting兲 magnitude of the Debye temperature25 of h-GaN amounts to
⌰ D ⬇870 K we satisfy ourselves that the typical 共average兲
ratio between both quantities is on the order of ⌰/⌰ D ⬇2/3
共in accordance with the common trend for most other III–V
compounds, as well as group IV and II–VI materials兲.4,11 A
physically plausible interpretation of this typical effective
phonon temperature of ⌰⬇600 K 共in combination with a
dispersion coefficient of order ⌬⬇0.5– 0.6兲 for h-GaN can
be readily given when the phonon density of states is considered. From Fig. 1 of Ref. 25 we see that the phonon spectrum of h-GaN consists of:
共1兲 a relatively broad low-energy subband extending
from 0 up to about 45 meV 共comprising three acoustical and
three low-energy optical branches兲, which shows a relatively
broad maximum centered at ប ␻ 1 ⬇23 meV 共corresponding to
a phonon temperature, ⌰ 1 ⬅ប ␻ 1 /k B , of about 270 K兲 and
共2兲 a high-energy subband extending from about 65 to
about 95 meV 共comprising six high-energy optical branches兲,
which is strongly dominated by a peak located at about
ប ␻ 2 ⬇75 meV 共corresponding to a phonon temperature, ⌰ 2
⬅ប ␻ 2 /k B , of about 870 K兲.
A prominent feature of this phonon spectrum for h-GaN is
the existence of a rather broad gap of about 20 meV between
the upper edge of the low-energy subband and the lower
edge of the high-energy subband 共i.e., between about 45 and
65 meV兲. Our fittings yield values for the effective phonon
¯ ⬅k B ⌰ that range within this phonon energy gap
energy ប ␻
共except for those of Refs. 14 and 17兲. This means, first of all,
that both subbands are obviously making significant contributions to the E(T) dependences. Moreover, we can easily
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3958
J. Appl. Phys., Vol. 90, No. 8, 15 October 2001
R. Pässler
FIG. 1. Fittings to various measured temperature dependences 共see Refs. 13–15 and 21兲 of exciton peak positions of h-GaN layers grown on sapphire
by qualitatively different analytical models: Dispersion-related three-parameter expression 关Eq. 共2兲兴, Bose–Einstein function 关Eq. 共3兲兴, and Varshni’s
formula 关Eq. 共6兲兴.
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J. Appl. Phys., Vol. 90, No. 8, 15 October 2001
R. Pässler
3959
TABLE I. Empirical parameter values resulting from numerical fittings 关using Eq. 共2兲兴 of data sets representing the temperature dependence of exciton peak
positions 共associated with the fundamental band gap兲 of hexagonal GaN layers grown on sapphire (Al2O3). To show the qualitative differences of the present
versus conventional three-parameter models we have also listed the values of parameters obtained via a function of Bose–Einstein type 关Eq. 共3兲兴 and Varshni’s
formula 关Eq. 共6兲兴.
Ref.
T fitted
共K兲
E (A) (0)
共eV兲
E B ⫺E A
共meV兲
␣ /10⫺4
共eV/K兲
⌰
共K兲
k B⌰
共meV兲
c⬅ ␲ 2 ␣ /12⌰
共10⫺6 eV/K2兲
␣⌰/2
共eV兲
a B ⬅ ␬ /2
共eV兲
⌰B
共K兲
␣ B /10⫺4
共eV/K兲
␣ V /10⫺4
共eV/K兲
␤
共K兲
c V⬅ ␣ V / ␤
共10⫺6 eV/K2兲
13
14
15
16
17
18
19
20
21
22
10 to 626
15 to 300
2 to 1067
10 to 300
4 to 300
15 to 475
10 to 475
5 to 300
13 to 260
9 to 310
3.476
3.485
3.472
3.490
3.479
3.482
3.495
3.487
3.495
3.488
—
8.9
—
—
5.8
9.3
—
6.8
9.3
7.9
6.77
4.82
6.28
6.24
4.84
6.39
6.03
5.79
5.94
5.31
602
487
622
630
479
569
694
557
567
533
52
42
54
54
41
49
60
48
49
46
0.92
0.81
0.83
0.81
0.83
0.92
0.71
0.86
0.86
0.82
0.204
0.117
0.195
0.197
0.116
0.182
0.209
0.161
0.168
0.146
0.155
0.058
0.162
0.072
0.051
0.112
0.111
0.065
0.055
0.054
500
315
540
349
285
416
467
325
294
297
6.18
3.67
5.98
4.13
3.55
5.39
4.77
4.01
3.74
3.64
9.37
9.96
7.3
24.3
11.0
9.77
11.8
13.5
20.1
9.27
772
1072
594
2866
1200
842
1414
1420
2196
1129
1.21
0.93
1.23
0.85
0.92
1.16
0.83
0.95
0.92
0.93
estimate from Table I that a typical magnitude for the effective phonon energy is ប ␻
¯ ⬇50 meV. 共Note that the ensemble
average of the series of k B ⌰ values listed in Table I is 49.5
meV.兲 This value is nearly coincident with the arithmetic
mean (ប ␻ 1 ⫹ប ␻ 2 )/2⬇49 meV of the low- and high-energy
phonon peak positions. Thus we can assume that the relative
weights,23 W 1 and W 2 , of the low- and high-energy contributions are nearly the same, W 1,2⬇0.5. At the same time we
see that the distance of each peak ប ␻ 1,2 from the average
¯ is about 兩 ប ␻ 1,2⫺ប ␻
¯ 兩 ⬇25 meV. These
phonon energy ប ␻
qualitative considerations lead to an estimate for the disper¯ 兩 /ប ␻
¯ ⬇0.5 关Eq. 共1兲兴. This
sion coefficient ⌬⬇ 兩 ប ␻ 1,2⫺ប ␻
order-of-magnitude estimation is in agreement with samplespecific results ⌬⬇0.4– 0.6 that follow from more detailed
analyses, e.g., on the basis of a two-oscillator model.23 关Note
that such rigorous estimations of ⌬ values, which are beyond
the scope of the present article, provide a further justification
for the application of the relatively simple three-parameter
expression,11 Eq. 共2兲, to preliminary numerical analyses of
most E A/B (T) data sets available up to now for hexagonal
GaN layers.兴
IV. COMPARISON WITH CONVENTIONAL MODELS
In the majority of experimental articles on the temperature dependence of the fundamental energy gap or the associated exciton peak positions in hexagonal GaN layers, numerical fittings of the E(T) data have been performed by
invoking either Varshni’s ad hoc formula1,2 or an analytical
expression of the Bose–Einstein type.5–11 These two conventional models refer, however, to the limiting regimes of either extremely large dispersion 共⌬⬇1, cf. below兲 or completely vanishing dispersion (⌬→0), respectively. Thus they
contradict physical reality for most group IV, III–V, and
II–VI 共including wide band gap兲 materials, whose dispersion
coefficients range from 0.3 to 0.7 between these extreme
values.4,23,24 The consequence of this is a significant uncertainty in the derived parameters as well as systematic misfits
in the cryogenic region.10–12,23,24 This uncertainty is especially pronounced for III–V nitrides. This is because the cutoff temperature for experimental measurements T max is often
chosen near 共or even significantly below兲 room temperature
共cf. Table I for GaN/Al2O3兲, whereas the effective Debye
temperature ⌰ D ⬇870 K for GaN 共Ref. 25兲 共in analogy to
⌰ D ⬇1020 K for AlN 共Ref. 26兲 and ⌰ D ⬇700 K for InN兲27 is
a factor of ⬃3 higher.
A. Comparison with models of Bose–Einstein type
The low temperature sections of Bose–Einstein models,
(E(0)⫺E(T))⬀(exp(⌰B /T)⫺1)⫺1 关dashed curves in Figs.
1共a兲–1共d兲兴 show an asymptotic plateau behavior,10,11 (E(0)
⫺E(T))⬀exp(⫺⌰B /T), which is obviously not in accordance with experimental observations. As opposed to this,
the E(T) dependences following from Eq. 共2兲 关solid curves
in Figs. 1共a兲–1共d兲兴 tend in this region to quadratic asymptotes (E(0)⫺E(T))⬀T 2 , and actually provide relatively
good fits to the entire of E(T) data sets, from 0 up to T max .
共This was found to be true for all h-GaN data sets.兲13–22 This
is reasonable given that Eq. 共2兲 corresponds to the transition
region between moderate and large dispersion ⌬⬇0.5– 0.6,
while the Bose–Einstein models5–12 apply to the limit of
very small 共vanishing兲 dispersion ⌬→0.
The relatively large scatter in Bose–Einstein parameter
values for hexagonal GaN layers 共seen in Table I兲 seems to
have been largely overlooked in earlier studies. This might
be due 共at least partly兲 to qualitatively different definitions
for model-specific parameter sets. 共Quantitative comparisons
between results of different authors are hampered by a multiple diversification of parameter sets.兲 In order to overcome
this impediment, we rewrite here the characteristic E B (T)
dependence in three equivalent 共alternately used兲
representations:5–11
冋
E B 共 T 兲 ⫽E B ⫺a B 1⫹
2
exp共 ⌰ B /T 兲 ⫺1
⫽E 共 0 兲 ⫺
␬
exp共 ⌰ B /T 兲 ⫺1
⫽E 共 0 兲 ⫺
␣ B⌰ B
.
exp共 ⌰ B /T 兲 ⫺1
册
共3兲
关Note that in Ref. 9 the parameter ␬ had been denoted by ␭.
Further one can readily show by means of the equality10,11
(exp(x)⫺1)⫺1⫽(coth(x/2)⫺1)/2 that the formally different
E B (T) expressions given, e.g., in Refs. 10, 11, and 28 –31,
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3960
J. Appl. Phys., Vol. 90, No. 8, 15 October 2001
R. Pässler
some of which involve even other definitions for empirical
parameters, are physically equivalent to Eq. 共3兲兴. Here we
have denoted as usual by ⌰ B ⬅ប ␻ B /k B 关in analogy to ⌰ in
Eq. 共2兲兴 the phonon temperature associated with the single
共‘‘phantom’’兲 oscillator under consideration. The parameter
␣ B represents 关in analogy to ␣ in Eq. 共2兲兴 the limiting slope,
␣ B ⬅⫺dE B (T)/dT) 兩 T→⬁ . The more frequently quoted coupling parameters5–9 a B ⬅ ␬ /2(⬅␭/2) follow immediately
from a comparison of the proportionality factors in the different representations of Eq. 共3兲 to be connected with the
limiting slope11,12 ␣ B by the equation a B ⬅ ␬ /2⫽ ␣ B ⌰ B /2.
This means, conversely,
␣ B ⫽2a B /⌰ B ⬅ ␬ /⌰ B
共4兲
for the limiting slope ␣ B 关as expressed in terms of the conventional coupling parameters, a B or ␬ 共⬅␭兲, and the associated phonon temperature ⌰ B 兴. Equation 共4兲 is especially
useful in direct comparisons of theoretical estimates of limiting slopes estimated via a Bose–Einstein function 关Eq. 共3兲兴
with Eq. 共2兲 共or Varshni’s formula;1,2 see below兲. Further
note that the high temperature asymptote associated with Eq.
共3兲 is given by E B (T)→H B (⬁)⫺ ␣ B T, where H B (⬁)⬅E B
⫽E(0)⫹a B ⫽E(0)⫹ ␣ B ⌰ B /2 represents the point of intersection of this asymptote with the energy axis.
Various numerical fittings of E(T) data sets using Eq.
共3兲 have already been performed.16,17,21 Here we have done
analogous fittings for the other experimental data. The results
for the coupling parameters a B ⬅ ␬ /2, Bose–Einstein temperatures ⌰ B , and limiting slopes ␣ B 关calculated via Eq. 共4兲兴
are listed in Table I. From the latter we see that in general
␣ B⬍ ␣
B. Comparison with Varshni’s formula
共5兲
and
This can be easily understood by the following. As we
have concluded in Sec. III, the upper subband of optic
phonons is making strong contributions to the limiting slopes
␣ ⬅⫺dE(T)/dT) 兩 T→⬁ of the E(T) dependences in h-GaN.
This contribution is clearly observable at sufficiently high
temperatures, Tⲏ⌰⬇600 K 关as shown for Refs. 13 and 15
in Figs. 1共a兲 and 1共c兲兴. On the other hand, due to the unusually high magnitude of the corresponding phonon temperature (⌰ 2 ⬇870 K), the high-energy optic modes are only
weakly activated from low T up to the vicinity of room temperature. Consequently, for data sets restricted to T
⭐300 K, a model function of Bose–Einstein type as Eq. 共3兲
is—owing to its a priori ignorance of dispersion—incapable
of properly detecting and quantifying this weakly activated
competitor of the primary contribution made by the lowenergy subband. This means that the contribution made by
high-energy optic phonons will be underestimated. Consequently, the parameters ␣ B (⫽2a B /⌰ B ⬅ ␬ /⌰ B ) and ⌰ B of
Eq. 共3兲 will be less than their dispersion-related counterparts
␣ and ⌰ of Eq. 共2兲.
It is obvious that the high-energy subband can fully display its competition with the low-energy subband only at
temperatures comparable with, or even higher than, ⌰ D .
This means that, for a wide band gap material like GaN 共in
analogy to AlN and InN兲, a Bose–Einstein model might give
physically reasonable limiting slopes and effective phonon
temperatures only when the E(T) data extend up to the Debye temperatures. For this reason we have found that ␣ B
⬇ ␣ and ⌰ B ⬇⌰ 共in analogy to Ref. 32, for zinc chalcogenides兲 only for the data of Refs. 13 and 15.
⌰ B ⬍⌰.
In particular, the difference between the ensemble averages
of limiting slopes ␣ B ⬇4.5⫻10⫺4 eV/K and ␣ ⬇5.8
⫻10⫺4 eV/K is more than 20%. This correlates with a difference of more than 30% between the ensemble average of
phonon temperatures estimated for Bose–Einstein oscillators, ⌰ B ⬇380 K, and its dispersion-related counterpart of
⌰⬇575 K. The differences between the separations between
the zero-temperature positions E(0) and the points of intersection H B (⬁) or H(⬁) 共⫽limiting enthapies兲,2,4,10 of the
high temperature asymptotes H B (⬁)⫺ ␣ B T and H(⬁)⫺ ␣ T
with the energy axis are especially large. The ensemble averages of these separations are H B (⬁)⫺E(0)⬅E B ⫺E(0)
⫽a B ⬅ ␬ /2⬇0.09 eV and H(⬁)⫺E(0)⫽ ␣ ⌰/2⬇0.17 eV.
This corresponds to an average underestimate of more than
40% if Eq. 共3兲 is used.
Most importantly, when we compare the parameter sets
listed for the individual h-GaN samples in Table I we become aware that the differences between the values obtained
for comparable parameters are relatively small for Refs. 13
and 15, where the E(T) data extend beyond 600 K 关cf. Figs.
1共a兲 and 1共c兲兴 and are rather large for Refs. 14, 16, 17, and
20–22, where the E(T) data sets are limited to T⭐300 K 关cf.
Figs. 1共b兲 and 1共d兲兴.
Let us finally consider Varshni’s formula1,2
E V 共 T 兲 ⫽E 共 0 兲 ⫺
冉
冊
␣ VT 2
␣V 2
T
⫽E 共 0 兲 ⫺
T 1⫺
, 共6兲
␤ ⫹T
␤
␤ ⫹T
which has been frequently used in the last 3 decades for
fittings of E(T) data for a large variety of materials, including h-GaN. We denote here by ␣ V the limiting slope ␣ V ⬅
⫺dE V (T)/dT) 兩 T→⬁ , and ␤ is a physically undefinable temperature parameter 共see below兲.
With respect to Refs. 13, 15, and 19 we have listed in
Table I the original results quoted in these articles. 关Note that
for these three cases our numerical fittings using Eq. 共6兲 give
nearly the same ␣ V and ␤ values.兴 For Ref. 16 we could not
confirm the result given there of ␣ V ⫽5.84⫻10⫺4 eV/K with
␤ ⫽465 K 共Fig. 5 of Ref. 16兲. Our least-mean-square fitting
process resulted in a significant reduction of the residual
variance 共by a factor of 2.5兲 at a fully relaxed parameter
constellation of ␣ V ⫽24.3⫻10⫺4 eV/K with ␤ ⫽2866 K.
关Note that this cannot be due to an inaccurate redigitization
of those authors’ data, as they provided us with their original
digitally recorded E(T) data.兴 Such physically unreasonable
parameter values like the latter 共see Table I兲 are a typical
feature24 of Varshni’s ad hoc formula, Eq. 共6兲, when applied
to restricted E(T) data sets 0⬍Tⱗ⌰/2 共i.e., here 0⬍T
ⱗ300 K; for more details see Ref. 24 and below兲. With respect to Refs. 14, 17, 18, and 20–22 we emphasize that the
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J. Appl. Phys., Vol. 90, No. 8, 15 October 2001
R. Pässler
parameters listed in Table I result from simultaneous fittings
of E A (T) and E B (T) data by one and the same set of ␣ V and
␤ values 关cf. Figs. 1共b兲 and 1共c兲兴. This is in contrast to the
original fittings reported in Refs. 14, 17, 18, and 20, where
the individual E A (T) and E B (T) data sets were fitted independently, with different parameter sets. In any case, we find
that the present ␣ V and ␤ values are of the same order of
magnitude as those given in Refs. 14, 17, 18, and 20–22.
As can be seen from Table I, the values of ␣ V and ␤ vary
widely, with an uncertainty comparable to that shown in recent articles18,20,33–36 for a larger variety of h-GaN samples.
The uncertainty in the parameters estimated via Varshni’s
formula is larger than that found from applications of the
Bose–Einstein model 关Eq. 共3兲兴. This has several causes.
At high temperatures, TⰇ ␤ , we see that Eq. 共6兲 tends to
a linear asymptote of the form E V (T)→E(0)⫺ ␣ V (T⫺ ␤ ).
When one compares this with the dispersion-related counterpart E(T)→E(0)⫺ ␣ (T⫺⌰/2) 关from Eq. 共2兲兴, one might
naively expect that the limiting slopes ␣ V and ␣ are nearly
the same, and that the parameter ␤ is comparable with ⌰/2.
Table I shows, however, that this never happens.
Comparing in more detail the individual ␣ V and ␣ values
we see, on the one hand, that the difference is relatively
small 共⬍20%兲 for the special case of Ref. 15, where the
E(T) data extend beyond 1000 K 关Fig. 1共c兲兴. On the other
hand, this difference is very large ( ␣ V / ␣ ⬃3) for Refs. 16
and 21, where the E(T) data sets are limited to T⭐T max
⫽300 K and T⭐T max⫽260 K, respectively 关cf. Fig. 1共d兲兴.
The dramatic difference in ␣ V of a factor of 3 between Refs.
15 and Refs. 16 and 21 is accompanied by a similarly large
increase 共a factor on the order of 3– 4兲 for the parameter ␤.
Our analytical and numerical studies, reported in Ref.
24, showed that these large discrepancies in Varshni parameters are closely connected with the variation of the experimental cutoff temperatures T max . The ratio of T max to ⌰ D is
about 1 in Ref. 15 and about 1/3 in Refs. 16 and 21. This
relationship between T max and the derived ␣ V and ␤ can be
better understood after considering the low temperature behavior of the E(T) data.
Let us first recall that, at sufficiently low temperatures,
the internally consistent model curve 关Eq. 共2兲兴 tends to a
quadratic asymptote
E 共 T 兲 →E 共 0 兲 ⫺cT 2 ,
共7兲
where the quantity c 共representing the curvature of this asymptote兲 follows from Eq. 共2兲 to be given in terms of ␣ and
⌰ by
c⫽
␲ 2␣
⬵ 共 0.82⫾0.11兲 ⫻10⫺6 eV/K2
12⌰
共8兲
共see ␣, ⌰, and c values in Table I兲. In Figs. 1共a兲–1共d兲 we
have represented the low-temperature asymptotes Eq. 共7兲 by
dotted curves. We see that from 0 up to about 200 K, these
quadratic asymptotes, Eq. 共7兲, are almost indistinguishable
from the original E(T) curves calculated according to Eq. 共2兲
共bold curves兲. However, the same is true for the curves derived from Varshni’s formula Eq. 共6兲. 共dash-dotted curves兲.
This close similarity is plausible particularly in view of the
second representation of Eq. 共6兲. From the latter we see that,
3961
for TⰆ ␤ , the E V (T) curve also tends to a quadratic dependence 关Eq. 共7兲兴, but where the parameter c is given by the
ratio
c V⫽
␣V
⬵ 共 1.0⫾0.2兲 ⫻10⫺6 eV/K2
␤
共9兲
of the parameters ␣ V and ␤. The moderate difference of only
about 20% between c and c V explains the close approaches
between the low-temperature sections (T⬍200 K) of the
dispersion-related E(T) curves and of Varshni’s E V (T)
curves.
Yet, such apparently ideal quadratic low-temperature dependences (E(0)⫺E(T))⬀T 2 , which were found to be a
good approximation for all E(T) data sets under study
within an unusually large temperature interval 共0–200 K兲,
has consequences for least-mean-square fittings of restricted
data sets E(T⬍300 K) by Varshni’s formula. To illustrate
this we write Varshni’s E V (T) function Eq. 共6兲 as a Taylor
expansion
再
E V 共 T 兲 ⫽E 共 0 兲 ⫺c V T 2 1⫺
冉冊 冉冊 冎
T
T
⫹
␤
␤
2
⫺
T
␤
3
⫹...
共10兲
共which is convergent within an interval of 0⭐T⬍ 兩 ␤ 兩 ). The
limiting case of complete coincidence of an E V (T) curve
with a parabola 关Eq. 共7兲兴 corresponds thus, trivially, to a
limiting transition 兩 ␤ 兩 →⬁ for the parameter ␤. The latter
involves, according to definition 共9兲, a simultaneous limiting
transition 兩 ␣ V 兩 →⬁ also for the parameter ␣ V ⬅c V ␤ 关occurring in Eq. 共6兲兴. Actually, when using Varshni’s formula Eq.
共6兲 for least-mean-square fittings, one is frequently concerned with large parameter jumps,24 or even a simultaneous
order-of-magnitude floating24 of ␣ V and ␤ towards infinity,
when the experimental cutoff temperature T max is very low.
We have tested within the present study this parameter floating effect24 in detail. 关Note that both representations for
E V (T) curves in Eq. 共6兲 give, of course, precisely the same
results provided that a sufficiently large number of fitting
cycles is performed in the least-mean-square fitting processes. We also point out that the second representation of
Eq. 共6兲, where the ratio ␣ V / ␤ has been substituted by the
unique proportionality factor c V Eq. 共9兲, is more suitable for
such a delicate numerical study, since c V remains almost
constant even in the course of an order-of-magnitude floating
兩 ␤ 兩 →⬁ of the parameter ␤.兴 In this way we have found,
among other things, that such a floating of the parameters ␤
and ␣ V (⬅c V ␤ ) extends over more than 3 orders of magnitude when, e.g., the data sets of Refs. 14, 16, 17, 20, and 21
are truncated at 200 K. This finding implies that the necessary condition for the fitting process to tend at all to some
finite magnitudes, i.e., values on the order of 10⫺3 eV/K for
␣ V and 103 K for ␤, consist of choosing a cutoff temperature
above 200 K. This basic requirement is fulfilled for all data
sets investigated here. At the same time we see from the
corresponding figures in Refs. 14, 16, 17, and 20–22 that the
experimental E(T) data are relatively sparse and as a rule not
very accurate 关see Figs. 1共b兲 and 1共d兲兴 exactly in the interval
between 200 and 300 K, which is crucial for the actual adjustment of the parameters ␣ V and ␤ to some finite magnitudes.
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3962
J. Appl. Phys., Vol. 90, No. 8, 15 October 2001
R. Pässler
Additional test fittings of various subsets of E(T) data
points, which can be readily obtained by sporadic truncations
of the original data sets somewhere within the crucial interval T⬇200– 300 K, are repeatedly found to result in largely
different values of ␣ V and ␤. This implies that even slight
differences between slopes and/or curvatures, which are inherent to data sets presented hitherto by different investigators, as well as experimental uncertainties of the usual order
共of about 1–2 meV兲 in this crucial region T⬇200– 300 K,
lead as a rule to enormous differences between the adjusted
sets of ␣ V and ␤ values.
Of course, the inherent sample-specific differences
and/or experimental inaccuracies also affect the results of
fittings that use Eq. 共2兲. However, we see clearly from Table
I that the uncertainty in ␣ and ⌰ is an order of magnitude
smaller than in ␣ V and ␤. This is because the unconventional
model expression Eq. 共2兲 is an offspring of a physically reasonable semiempirical theory,4,11 which is well adapted to the
typical dispersion regime frequently seen in hexagonal GaN
layers (⌬⬇0.5– 0.6). Varshni’s ad hoc formula Eq. 共6兲,
while being deceptively good at reproducing experimental
data for various h-GaN layers within a limited temperature
interval (T⬍300 K), cannot be brought into physically reasonable connection with actual features of phonon dispersion.
V. VARSHNI’S MODEL RELATED TO EXCESSIVELY
LARGE DISPERSION „⌬É1…
One can perform analytical or numerical assessments of
the shape of Varshni’s E V (T) function in order to find a
dispersion regime that could potentially be associated with
Eq. 共6兲. From such assessments 共in analogy to Ref. 23兲 one
can conclude that Eq. 共6兲 is factually related to a regime of
excessively large dispersion ⌬→⌬ V ⬇1. Detailed analytical
investigations of this type are beyond the scope of the
present article. Nevertheless, in view of the obvious importance of this point for a physically plausible explanation of
the notorious numerical inconsistencies involved frequently
by applications of Varshni’s formula Eq. 共6兲 共especially with
respect to wide band gap materials兲, we would like to give
here at least a numerical illustration of this peculiar state of
affairs. To this end let us consider the possibility that the
E g (T) temperature dependence for a certain system might
actually have the form suggested by Varshni’s formula
E V (T), Eq. 共6兲. The question is, what could be, consequently, the magnitude of the dispersion coefficient ⌬ V associated with such a 共hypothetical兲 Varshni type system.
In order to get a characteristic result for ⌬ V we generate
a series of hypothetical E V (T) data points calculated on the
basis of Varshni’s formula 关Eq. 共6兲兴. Choosing for Varshni’s
empirical parameters, e.g., the values ␣ V ⫽5⫻10⫺4 eV/K
and ␤ ⫽100 K we obtain the E V (T)⫺E V (0) data points represented in Fig. 2. 共by open circles兲 for a temperature interval
from 0 to 10␤ ⫽1000 K. Then we can perform careful leastmean-square fittings of these Varshni data points by a general
version of the dispersion-related theory that is applicable to a
larger variety of dispersion regimes 共from ⌬⫽0 up to an
order of ⌬⬇1, say兲. In this way we can determine, among
FIG. 2. Fittings of a set of simulated Varshni data points 关Eq. 共6兲, for ␣ V
⫽5⫻10⫺4 eV/K and ␤ ⫽100兴 using two qualitatively different versions of
dispersion-related models.
other things, the characteristic magnitude of the dispersion
coefficient that is effectively associated with Varshni’s model
function, Eq. 共6兲.
A convenient version of a duly comprehensive,
dispersion-related model is provided, e.g., by the twooscillator model23 共which applies from the ⌬⫽0 limit up to
⌬ values even higher than 1兲. The result of a complete 共unrestricted兲 five-parameter least-mean-square fit using Eq. 共3兲
of Ref. 23 is represented by the solid curve in Fig. 2. The
corresponding parameter values are ␣ ⫽4.976⫻10⫺4 eV/K
for the limiting slope, ⌰ 1 ⫽81.76 K and ⌰ 2 ⫽500.29 K for
the phonon temperatures of the 共hypothetical兲 low- and highenergy oscillators, and W 1 ⫽0.7454 or W 2 ⫽0.2546 for the
relative weights of their contributions to the limiting slope.
Comparing the fitted curve with the original series of Varshni
data points we see that both dependences are practically indistinguishable for temperatures above 100 K. At the same
time we find that, due to the inherent plateau behavior of the
two-oscillator model23 at very low temperatures T⬍20 K,
the T⫽0 level of the fitted curve is located by 0.85 meV
below the original T→0 Varshni curve position. From the
fitted parameters we obtain for the effective 共average兲 phonon energy,23 ⌰⬅W 1 ⌰ 1 ⫹W 2 ⌰ 2 , a value of ⌰⫽188.32 K
and for the associated dispersion coefficient,23 ⌬
⫽ 冑(⌰ 2 ⫺⌰)(⌰⫺⌰ 1 )/⌰ a magnitude of ⌬⫽0.968. The latter confirms the above suggestion of ⌬→⌬ V ⬇1 for the
Varshni model.
It is instructive to consider still another fit of Varshni’s
data set by the theoretical E(T) dependence following from
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J. Appl. Phys., Vol. 90, No. 8, 15 October 2001
R. Pässler
the power-law model,11,12 which is based on an ansatz of the
form w ␯ (␧)⫽ ␯ ␧ ␯ ⫺1 /␧ ␯o , v ⬎0, for the normalized weighting
function 关in Eq. 共2兲 of Ref. 4兴. The corresponding analytical
expression for the dispersion coefficient ⌬⫽( ␯ ( ␯ ⫹2)) ⫺1/2
关as follows readily from Eq. 共3兲 of Ref. 4兴, indicates that this
model is applicable, in principle, to an arbitrarily large magnitude of dispersion 0⬍⌬⬍⬁. A duly accurate fourparameter least-mean-square fit on the basis of Eq. 共5兲 of
Ref. 12 involves rather time-consuming numerical calculations of the corresponding integrals 共which we have performed via summations over 100 000 discrete equidistant
points located between the integration limits兲. For T
⬎100 K the corresponding theoretical curve 共dotted curve兲
turns out again to be practically indiscernible from Varshni’s
curve 共empty circles兲 or from the two-oscillator model curve
共solid curve兲. At the same time we see from the lowtemperature section of the dotted curve 共plotted in the inset
to Fig. 2兲 that, due to the very strong curvature of the limiting power-law dependence at T⬍20 K 关namely (E(0)
⫺E(T))⬀T 1.4, in this case兴 the T⫽0 level of the fitted curve
turned out to be located at about 1.25 meV above the original
T→0 Varshni curve position. Concerning the associated empirical parameters we have obtained the following values:
␣ ⫽4.991⫻10⫺4 eV/K for the limiting slope, ⌰⫽176.45 K
for the effective phonon temperature, and ␯ ⫽0.394 for the
parameter controlling the shape of the underlying weighting
function. 关Note that this value corresponds to a monotonically decreasing weight function, w ␯ (␧)⬀␧ ␯ ⫺1 ⫽␧ ⫺0.606.兴
This ␯ value corresponds to a magnitude of ⌬⫽( ␯ (gn
⫹2)) ⫺1/2⫽1.029 for the dispersion coefficient.
On the basis of this numerical example we satisfy ourselves that both qualitatively largely different versions of
semiempirical dispersion-related models12,23 give almost the
same results for the basic empirical parameters ␣, ⌰, and ⌬
关in accordance with general expectations with respect to different internally consistent analytical models based on Eq.
共2兲 of Ref. 4兴. We see that the limiting slopes ␣, as resulting
from both alternative fits, differ by less than 0.5% from the
originally chosen ␣ V value. Further we see that the ratio ␤/⌰
between the actually undefined temperature parameter ␤ occurring in Varshni’s formula and the effective phonon temperature ⌰ follows from both alternative fits to have a magnitude of ␤ /⌰⬵0.55⫾0.02. This is in reasonable agreement
with a limiting value 共lower boundary兲 of ␤ /⌰→1/2 as follows immediately, e.g., from a comparison of the high temperature asymptotes of Varshni’s model Eq. 共6兲 and Eq. 共2兲;
cf. Sec. IV B. 共Thus, in view of the characteristic range of
0.6⬎ ␤ /⌰⭓0.5 for an E(T) dependence assumed to be of
Varshni type 关Eq. 共6兲兴, the fact that usually the ␤ values
found are significantly higher than ⌰/2 is a strong indication
of the inadequacy of Varshni’s model.兲
Most importantly, the ⌬ values obtained from our fittings
of the hypothetical Varshni curve using qualitatively different
dispersion-related models12,23 clearly show that the corresponding 共model-specific兲 dispersion coefficient is on the order of unity
⌬ V ⬇1.0共 ⫾0.05兲 .
共11兲
Such a dispersion regime, however, appears to be of purely
3963
academic interest. For all III–V compounds 共as well as
group IV and II–VI materials, including ternary compounds,
quantum-well structures etc.兲, which have been analyzed numerically in detail,4,11,12,23,24 the corresponding dispersion
coefficients have been found to be within an interval of 0
⬍⌬ⱗ0.7. The relatively large gap between the latter 共physically realistic兲 ⌬ region and the excessively large magnitude
of the effective dispersion coefficient ⌬ V ⬇1 共11兲, associated
with Varshni’s model Eq. 共6兲, is the reason for multifarious
numerical inconsistencies.
VI. CONCLUSIONS
The primary aim of this article was to significantly reduce the large scatter in the empirical parameter values of the
temperature dependence of the fundamental energy gap
E g (T) in hexagonal GaN. There are objective and subjective
reasons for uncertainties in these parameter values.
共1兲 The main objective reason is shown here to be the
restriction of experimental measurements to regions below
room temperature. Of course such experiments cannot determine the limiting 共high-temperature兲 slope of the E g (T) dependence, due to the relatively high Debye temperature 共of
about 870 K兲 in h-GaN. An additional aggravation is the
relative paucity or insufficient precision of most data sets
within the crucial region between 200 and 300 K, where the
E g (T) dependence deviates markedly from its quadratic lowtemperature asymptote. Thus, unambiguous values for empirical parameters can only be obtained through measurements far beyond room temperature, and in conjunction with
improved data quality near room temperature.
共2兲 The main subjective reason for parameter uncertainties is shown in this report to be due to the conventional use
of oversimplified analytical models like the Bose–Einstein
model and/or Varshni’s formula. Both models are essentially
incompatible with the dispersion regime actually found in
hexagonal GaN. In contrast to numerous earlier applications
of these models to materials with lower Debye temperatures
共where the model-specific parameter uncertainties turned out
to be relatively moderate兲, such uncertainties are found for
h-GaN to be unusually large 共in analogy to a series of recent
experimental papers兲. The Bose–Einstein model significantly
underestimates both the limiting slope and the effective phonon energy. This is because it is not capable of detecting the
actual weight of the contribution of high-energy optic
phonons, which are only weakly activated at room temperature. Varshni’s formula is found to generally overestimate the
limiting slope. This is due to the obviously inadequate structure 共ad hoc nature兲 of this formula.
共3兲 We can infer a general trend for the dependence of
the relative deviations from the true magnitudes of limiting
slopes and effective phonon temperatures. In particular, for
the limiting slope, its underestimation by the Bose–Einstein
model and its overestimation by Varshni’s formula tend to be
larger with lower experimental cutoff temperatures. An
analogous statement can be made about the temperature parameters. This suggests that the values of these parameters
obtained using the conventional models are primarily a function of the arbitrarily chosen experimental cutoff tempera-
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3964
J. Appl. Phys., Vol. 90, No. 8, 15 October 2001
tures, instead of being characteristic physical parameters of
the material under study.
共4兲 We have shown here that physically reasonable interpretations of the data sets can be easily provided on the basis
of a relatively simple analytical expression 共Sec. II兲, which is
obviously well adapted to the dispersion regime actually
present in many hexagonal GaN layers. Numerical fittings by
this expression not only lead to satisfactory numerical fits of
the experimental data but also provided, above all, physically
reasonable parameter values. In particular, we have achieved
here a reduction of parameter uncertainties by an order of
magnitude in comparison with those associated with Varshni’s formula. This suggests, among other things, that possible
variations of the shapes of E g (T) curves within a larger variety of h-GaN layers might be markedly smaller than usually suspected.
共5兲 With respect to forthcoming numerical analyses of
E(T) data sets for hexagonal 共as well as cubic兲 GaN, including their alloys with AlN and InN, it is desirable to use a
more general dispersion-related expression 共in analogy to
Refs. 4, 23, and 24兲 which enables, in principle, even an
assessment of the more or less significant variations of the
dispersion coefficient ⌬ from sample to sample. A more
comprehensive use of these analytical expressions to numerical analyses of E(T) data sets for III–V nitrides is desirable,
especially if forthcoming experimental data will be improved
and extended.
ACKNOWLEDGMENTS
The author would like to thank Dr. G. Teisseyre, High
Pressure Research Center of the Polish Academy of Sciences, Warsaw, Poland; Professor Dr. S. F. Chichibu, Institute of Applied Physics, University of Tsukuba, Ibaraki, Japan; Professor Dr. F. Calle, Dpto. Ingenerı́a Electrónica,
ETSI de Telecommunicación, UPM Ciudad Universitaria,
Madrid, Spain; and PD Dr. K. Thonke, Abteilung Halbleiterphysik, Universität Ulm, Ulm, Germany, for having transmitted to us the data sets published by their groups 共Refs. 13,
16, 17, and 20, respectively兲.
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