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Temperature-dependent band gaps in several
semiconductors: from the role of electron–phonon
renormalization
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Journal of Physics: Condensed Matter
J. Phys.: Condens. Matter 32 (2020) 475503 (7pp)
https://doi.org/10.1088/1361-648X/aba45d
Temperature-dependent band gaps in
several semiconductors: from the role of
electron–phonon renormalization
Yiming Zhang1 , Ziyu Wang1 , Jinyang Xi1
,2
and Jiong Yang1,2
1
Materials Genome Institute, Shanghai University, 99 Shangda Road, Shanghai 200444, People’s
Republic of China
E-mail: jinyangxi@t.shu.edu.cn and jiongy@t.shu.edu.cn
Received 18 May 2020, revised 2 July 2020
Accepted for publication 9 July 2020
Published 31 August 2020
Abstract
Temperature dependence of band gap is one of the most fundamental properties for
semiconductors, and has strong influences on many applications. The renormalization of the
band gap at finite temperatures is due to the lattice expansion and the phonon-induced atomic
vibrations. In this work, we apply the recently-developed electron–phonon renormalization
(EPR) method to study the temperature-dependent band gap in some classical covalent
(diamond, Si, and SiC) and ionic semiconductors (MgO and NaCl). The contributions from
both the lattice expansion and the phonon-induced atomic vibrations at finite temperatures are
considered. The results show that the band gaps Eg all decrease as temperature T increases,
consistent with the experiments and other theoretical studies (e.g., from 0 K to 1500 K, the
reductions are ∼0.451 eV for diamond and ∼1.148 eV for MgO, respectively). The covalent
compounds investigated show weaker temperature dependences of Eg s than the ionic
compounds, due to the much weaker lattice expansions and therefore low contributions from
these. The zero-point motion effect has greater influence on the band gap in semiconductors
with light atoms, such as diamond (reduction ∼0.437 eV), due to larger atomic displacements.
By decomposing the EPR effect into respective phonon modes, it is found that the
high-frequency optical phonon vibrations dominate the temperature-dependent band gap in
both covalent and ionic compounds. Our work provides the fundamental understandings on the
temperature-dependent band gaps caused by lattice dynamics.
Keywords: temperature-dependent band gap, electron–phonon renormalization, phonon
vibration, lattice expansion
S Supplementary material for this article is available online
(Some figures may appear in colour only in the online journal)
and γ-ray detection material; the semiconductor can be used
as transparent conductive materials if the band gap value is
greater than 3.0 eV [2, 3]. In experiments, it is found that the
band gap value is not only related to the structure of materials,
but also the temperature. In 1960s–1970s [4, 5], people found
that the band gaps of some classical semiconductors decrease
with temperature, and empirical formulas were fitted accordingly. For instance, the reductions of the band gap values are
∼0.083 eV for Si from 25 K to 410 K, and ∼0.431 eV for
1. Introduction
The band gap is one of the most fundamental properties for
semiconductors, and it plays a very important role in many
applications. For example, semiconductor with a band gap
value of 1.0–1.5 eV is good candidate for solar absorber [1];
as the band gap value is 1.6–2.5 eV, it can be used as x-ray
2
Author to whom any correspondence should be addressed.
1361-648X/20/475503+8$33.00
1
© 2020 IOP Publishing Ltd
Printed in the UK
J. Phys.: Condens. Matter 32 (2020) 475503
Y Zhang et al
GaAs from 21 K to 973 K, respectively, following the relation
for the temperature T dependence of the band gap Eg = E0
− aT 2 /(T + b) (E0 is its value at 0 K, a and b are constants).
This is a well-known effect in semiconductor physics, and is
often referred as the ‘Varshni effect’ [4]. The redshift of the gap
with temperature is seen in many albeit not all semiconductors.
For instance, copper halides [6] and lead halide perovskites [7]
exhibit an ‘inverse Varshni’ effect, which is a blueshift of the
gap with temperature. Therefore, in order to obtain the band
gap values consistent with the experiments, the temperature
effect should be taken into account.
The temperature dependence of band gap usually has two
major sources: (1) lattice expansion; (2) phonon-induced deviations from atomic equilibrium sites. Generally, the change of
electronic structure caused by phonon vibrations is called electron–phonon renormalization (EPR) [8–12], which includes
the effect of zero-point vibration at zero temperature, and
thermal vibration at finite temperatures. Thus, the change of
electronic energy ΔEnk (T) for band n and wave-vector k at
temperature T should follow the relation:
VIB
VIB
LE
(0) + ΔEnk
(T) + ΔEnk
(T) .
ΔEnk (T) = ΔEnk
on band gaps and optical spectra in semiconductors and insulators [10, 20–22].
In this work, we adopt the ‘one-shot’ method to examine the temperature-dependent band gaps for several classical semiconductors, including covalent (diamond, Si and SiC)
and ionic semiconductors (NaCl and MgO). The lattice expansion is considered by using the self-consistent quasi harmonic
approximation (SCQHA) method [23, 24]. Therefore, the
explicit contributions of lattice expansion, zero-point vibration, and thermal vibration will be all included in details. The
ionic semiconductors in our study show much stronger temperature dependence of band gaps than covalent ones, mainly due
to the influence of the lattice expansion. Besides, the modedecomposed EPR will also be discussed, and some optical
phonons are responsible for the sizable band gap variations.
This work provides a comprehensive EPR method that can
study the semiconductor’s band gap at any given temperature
accurately and efficiently.
2. Model and computational details
(1)
The complete calculation process in this study is as follows:
VIB
VIB
(0) and the second term ΔEnk
(T) on
The first term ΔEnk
the right are due to the zero-point vibration at 0 K and thermal
LE
(T) is
lattice vibration at T, respectively; The third term ΔEnk
due to the lattice expansion at T. In the literature, the phononinduced renormalization draws great attention. In some cases,
the effect of zero-point renormalization (ZPR) on band gap
is substantial. For example, the zero-point vibration of diamond causes the direct band gap change of nearly 600 meV
[9]; Antonius et al [13] found the GW quasiparticle corrections can increase the ZPR of the diamond’s direct band gap
by as much as ∼200 meV; Bester et al [14] pointed out that
ZPR is about 0.3–0.9 eV for the band gap in carbon and silicon nanoclusters. Besides, the effect of thermal vibration at
finite temperature T is also very important: Giustino et al [9]
found the direct band gap decreases ∼200 meV from 0 K to
700 K for diamond; Angelis et al [15] observed a gradual
blue-shift of UV–vis absorption spectrum for cubic phase perovskite MAPbI3 from 310 K to 400 K and predicted the band
gap increases ∼50 meV from 320 K to 650 K; Zhang et al [16]
studied the temperature-dependent band gap in PbTe by firstprinciples molecular dynamics (MD) and found the band gap
increases ∼200 meV from 0 K to 200 K.
There are three commonly used theoretical methods to
study the phonon-induced band gap variations: (1) the
temperature-dependent eigenenergies can be computed as a
time average of the band gaps for MD snapshots [10, 15–17].
(2) Though the second-order perturbation theory (PT) within
the adiabatic or non-adiabatic harmonic approximations, such
as the Allen–Heine–Cardona (AHC) theory [8–12, 18, 19],
the thermal shift of the electronic energies can be determined.
(3) By a Bose–Einstein weighted sum of the contribution
from each phonon eigenvector, an effective structure under
thermal vibration can be obtained for further calculations.
This is known as the ‘one-shot’ or frozen-phonon method.
Compared with MD and PT methods, it shows accurate yet
computationally efficient results on temperature dependences
(a) Through SCQHA [23, 24], the volume–temperature
(V –T) relationship can be obtained.
(b) According to the V –T relationship, we construct an m × m
× m supercell with the expanded volume V(T ), and then
the phonon spectrum for the supercell is calculated.
(c) At each T, the distorted atomic configuration can be
obtained by displacing the atoms from their equilibria
by an amount Δτ κα (κ and α indicate the atom and the
Cartesian direction, respectively) [21],
1 Δτκα = Mp /Mκ 2
(−1)υ−1 eκα,υ συ,T .
(2)
υ
Here, M p is the proton mass, M κ is the κth nucleus’s mass,
and the Gaussian widths σ υ,T for the υth normal mode and
temperature T is given by [21]
(3)
σ2υ,T = /2Mp ωυ 2nυ,T + 1 ,
−1
where
nυ,T = exp ωυ /kB T − 1
is
the
Bose–Einstein distribution. This is the so-called
‘one-shot’ structure.
(d) Based on the temperature-dependent ‘one-shot’ structure,
the band gap can be determined from the effective band
structure (EBS) of the primitive cell that is obtained by
unfolding from the supercell’s band structure.
All the calculations were performed using the Vienna ab
initio simulation package (VASP) [25–27], based on density
functional theory [28, 29]. The generalized gradient approximation of Perdew–Burke–Ernzerhof (PBE) type was used
as the exchange–correlation function [30]. The projectoraugmented wave method [25] was applied with the plane wave
cutoff energies of 520 eV (diamond, MgO, and SiC), 320 eV
(Si) and 350 eV (NaCl), respectively. The energy convergence
accuracy of 10−5 eV and the forces acting on atoms of 10−5 eV
2
J. Phys.: Condens. Matter 32 (2020) 475503
Y Zhang et al
Table 1. The volumetric thermal expansion coefficients αV for diamond, Si, SiC,
MgO and NaCl at different temperatures and corresponding values from experiments.
This work
−6
Diamond
Si
SiC
MgO
NaCl
Experiments
−1
3.645 × 10 K at 300 K
14.811 × 10−6 K−1 at 1000 K
9.490 × 10−6 K−1 at 293 K
7.958 × 10−6 K−1 at 300 K
38.194 × 10−6 K−1 at 300 K
36.928 × 10−6 K−1 at 283 K
173.773 × 10−6 K−1 at 300 K
3.0 × 10−6 K−1 at 300 Ka
13.2 × 10−6 K−1 at 1000 Ka
8.778 × 10−6 K−1 at 293 Kb
8.322 × 10−6 K−1 at 300 Ka
29.52 × 10−6 K−1 at 283 Kc
—
a
Reference [36].
Reference [37].
c
Reference [38].
b
Figure 1. The primitive cell’s band structures of (a) diamond and (e) MgO, as well as the EBSs and phonon spectra at 0 K and 1050 K for
(b)–(d) diamond and (f)–(h) MgO, respectively. The VBM is set as the zero of energy and the high-symmetry points are Γ (0 0 0), X (0.5 0
0.5), W (0.5 0.25 0.75), K (0.375 0.375 0.75), L (0.5 0.5 0.5), U (0.625 0.625 0.625), F (0 0.5 0), Q (0 0.5 0.5). The band gap values are also
shown in orange.
Å−1 were adopted for self-consistence calculation and structural optimization. In order to obtain the V –T curve, the 4 ×
4 × 4 supercell (128 atoms) and 2 × 2 × 2 Monkhorst–Pack
grids [31] of k-point mesh were used. In the EPR calculations,
we used 6 × 6 × 6 supercell (432 atoms) and 4 × 4 × 4 kpoint grid. The phonon spectra were obtained by phonopy code
[32] and the V –T curves based on SCQHA code [23, 24, 33].
To analyze the electronic structure with lattice expansion and
vibration, the band unfolding technique within BandUP code
[34, 35] was employed that transformed the large supercell’s
eigenstates into an EBS in the primitive cell.
volume thermal expansion coefficients αV can thus be
extracted (figure S2 of SM). These results are reasonably
comparable with experiments [36–38] (table 1). Comparing
the covalent and ionic compounds, it is found that the αV s in
latter ones are much larger than those in former ones (e.g., at
300 K, 38.194 × 10−6 K−1 for MgO and 3.645 × 10−6 K−1
for diamond), indicating the stronger lattice expansion effect
for ionic compounds.
In the main text, the results of diamond and MgO have
been demonstrated and compared; others are shown in the
SM. The band structures of diamond and MgO primitive cells
without EPR are shown in figures 1(a) and (e), respectively.
Under the PBE functional, diamond is indirect-gap (band gap:
∼4.091 eV) with the valence band maximum (VBM) locating at Γ-point and the conduction band minimum (CBM)
locating along the Γ–X direction. For MgO, it is a Γpoint direct-gap semiconductor with the value of ∼4.414 eV.
Figures 1(b), (c) and (f ), (g) show the EBSs of diamond and
MgO at 0 K and 1050 K, respectively, and those for all temperatures are shown in figures S3 and S4 of SM. Compared
3. Results and discussion
The diamond, Si, and SiC all crystallize in the diamond lattice, while MgO and NaCl are both in the
NaCl (rocksalt) structure. Based on the optimized structures, the V –T relations for all systems investigated are
shown in figure S1 of the supplementary material (SM)
(https://stacks.iop.org/JPCM/32/475503/mmedia), and the
3
J. Phys.: Condens. Matter 32 (2020) 475503
Y Zhang et al
Figure 2. The temperature-dependence of (a) diamond’s indirect-gap and (b) MgO’s direct-gap from EBSs (black blocks), respectively. The
corresponding values of diamond from references [11, 21, 39] (blue stars, orange down triangles and green upper triangles) are also shown
for comparison. The band gap at lowest temperature is set as the reference. The different lines are the corresponding non-linear fittings by
the Varshni equation of ΔE g = aT 2 /(T + b). The absolute values of coefficients |a| and |b| are also shown.
fitting from 600 K to 1500 K in our work is about 4.189 ×
10−4 eV K−1 , which is similar with that of Poncé et al (4.350
× 10−4 eV K−1 ) [11] and other experiment (5.400 × 10−4 eV
K−1 ) [12]. Therefore, our model is reasonable and EPR effect
at finite temperatures is nonnegligible. Similarly, according
to the EBSs in figure S4 of SM, the case of direct-gap MgO
are also examined and shown in figure 2(b). Different from
that of diamond, the band gap decreases much more strongly
at middle-high temperature range. The |a| is 9.372 × 10−4
eV K−1 and the reduced gap value is about 1.148 eV from 0
K to 1500 K (the corresponding reduced value for diamond
are ∼0.451 eV). Besides, there exists a better linear relation
between Eg (T ) and T even at lower room temperature for MgO
with smaller |b| (|b| = 301.0 K for MgO and 643.7 K for
diamond, respectively). The reasons are due to (1) the lower
phonon frequencies of MgO (figure 1(h)), resulting in more
excited phonons and (2) the larger volumetric thermal expansion coefficient of MgO (table 1), indicating more contribution
from lattice expansion (see below).
The band structure of primitive cell, EBSs at different temperatures and the Eg (T )–T curve for Si, SiC, and NaCl are
shown in figures S5–S11 of SM, and the extracted band gaps
at different temperatures for all five systems are summarized
in table S1 of SM. It’s remarkable that the Eg (T )–T curve of
Si by our EPR model also fits well with other experiments
[40, 41] and theoretical calculations [11, 21, 42] as shown in
figure S9 of SM. The ZPR is the band gap difference between
primitive cell and EBS at 0 K, which are summarized for all
five systems and shown in table 2. The corresponding values
from others [11, 12, 19, 43–45] are also shown for comparison. Taking the case of diamond as an example. The ZPR
(∼−0.437 eV) by our model is close to the values from experiments (−0.370 eV [12], −0.410 eV [43] and −0.364 eV [44])
as well as from calculations by using AHC theory (−0.379 eV
[11]) and frozen-phonon method (−0.325 eV [45]). In particular, because the atomic displacement is inversely proportional
to its mass (equation (2)), the ZPR for diamond is largest with
the band gap reduction ∼0.437 eV while it has little effect for
Si (the ZPR reduction ∼0.075 eV).
with figures 1(a)–(c) and (e)–(g), it shows that the reductions
of band gap due to ZPR for diamond and MgO are ∼0.437
eV and ∼0.281 eV, respectively. The temperature dependent
reductions for the two materials are ∼0.297 eV and ∼0.801
eV, respectively, from 0 K to 1050 K. These reductions under
finite temperatures originate from both the lattice expansion
and atomic thermal vibrations. Comparing the phonon spectra at 0 K and 1050 K (figures 1(d) and (h)), the deviation
of phonon spectrum is small for diamond due to the strong
C–C bonding, while it is more distinct for MgO, especially
the optical branches. The above results indicate the EPR effect
plays an important role on band structure of semiconductor,
which the contributions of ZPR and temperature effect are on
the same magnitude. Especially, the ZPR is more influential
in diamond than that in MgO, while the temperature effect is
more important in MgO.
The EBSs of diamond at each temperature are shown in
figure S3 of SM and the corresponding indirect-gap values
have been then extracted from these curves and plotted as a
function of temperature (black blocks in figure 2(a), the band
gap at 0 K is set as the reference). The corresponding values
from some experiment [39] and theoretical calculations [11,
21] are also shown for comparison. The band gaps decrease
slowly at low temperatures, and strongly drop when the temperature is higher than room temperature. That is because some
critical phonon modes (see below) responsible for the EPR are
not populated at low temperatures. The curves are fitted by
using Varshni equation (ΔEg (T ) = aT 2 /(T + b), ΔEg (T ) is
the band gap difference between at temperature T and at 0 K)
[4] and the coefficients a and b can be obtained. Especially, the
larger absolute value of |a| represents Eg (T ) decreasing with T
more strongly and the smaller one of |b| shows a better linear
relationship between Eg (T ) and T. Our EPR model is qualitatively in good agreement with others [11, 21, 39]. For instance,
the absolute value of second-order coefficient |a| from this
work is 4.338 × 10−4 eV K−1 and the reduced gap value is
∼0.390 eV from 0 K to 1350 K, the corresponding values by
using the AHC theory in non-adiabatic harmonic approximation are ∼0.320 eV with |a| = 3.976 × 10−4 eV K−1 [11];
besides, the absolute value of first-order coefficient by linear
4
J. Phys.: Condens. Matter 32 (2020) 475503
Y Zhang et al
Table 2. The values of ZPR (in eV) in this work and from other experiments and
theoretical results for diamond, Si, SiC, MgO and NaCl.
ZPR (eV)
System
Diamond
Si
SiC
MgO
NaCl
Indirect gap
Indirect gap
Indirect gap
Direct gap
Direct gap
This work
Others
−0.437
−0.075
−0.145
−0.281
−0.098
−0.370a , −0.410b , −0.364c , −0.379d , −0.325e
−0.064a , d , −0.060e , −0.053c
−0.109e
−0.196f , −0.272g
—
a
Reference [12] (exp.).
Reference [43] (exp.).
c
Reference [44] (exp.).
d
Reference [11], the AHC theory in the adiabatic harmonic approximation (cal.).
e
Reference [45], the frozen-phonon method (cal.).
f
Reference [19], the frozen-phonon method (cal.).
g
Reference [19], the AHC theory with dynamical effects (cal.).
b
In order to compare the difference between covalent and
ionic semiconductors, as well as the contributions of lattice
expansion and phonon vibration, the band gap as a function
of temperature caused by (1) only lattice expansion (signed
as LE) and (2) both of lattice expansion and vibration (signed
as LE + VIB) for all five systems are examined and shown
in figure 3 (the band gaps at 0 K are set as the references).
The corresponding coefficients a and b which are from the
non-linear fittings are summarized in table 3. Due to the EPR
effect, the band gaps all decrease as temperature increases (the
data for solid symbols) and the absolute values of coefficients
|a| and |b| (the data for LE + VIB) are all comparable with
these from other experiments [4, 39–41] and theoretical calculations [11, 21, 42]. Comparing the covalent and ionic systems, the band gap values drop more strongly in MgO and
NaCl (green and cyan solid symbol data, respectively) than
these in diamond, Si and SiC (black, red and blue solid symbol data, respectively). The corresponding absolute values of
coefficients |a| for ionic ones (9.372 × 10−4 eV K−1 for MgO
and 17.70 × 10−4 eV K−1 for NaCl) are larger than these in
covalent ones (4.338 × 10−4 eV K−1 for diamond, 2.051 ×
10−4 eV K−1 for Si and 6.038 × 10−4 eV K−1 for SiC).
The differences between covalent and ionic semiconductors
on their temperature dependence of band gaps can be mainly
rationalized by the different magnitude of lattice expansion.
The LE only results in figure 3 (hollow symbols) indicate
that the band gaps are almost unchanged in covalent semiconductors. The absolute values of lattice expansion induced
|a| are 0.259, 0.252, and 0.292 × 10−4 eV K−1 for diamond,
Si, and SiC respectively (table 3), while the same values are
much larger in ionic semiconductors (absolute value: 4.832
and 6.438 × 10−4 eV K−1 , in table 3). Figure S12 of SM
shows the band structure of MgO, considering the same lattice
expansion ratio as that of diamond at 1050 K. Comparing
the band gaps in figure S12 and figure 1(e), it is found that
the band gap reduction of MgO is only ∼0.062 eV when
adopting the same expansion ratio of diamond, and is much
smaller than that from figure 3 (green hollow symbol data at
1050 K, ∼0.497 eV). Furthermore, due to the larger volumetric
thermal expansion coefficients, the absolute values |b| from LE
Figure 3. Band gaps from EBSs of diamond, Si, SiC, NaCl and
MgO as functions of temperature. The hollow points (signed as LE)
represent the band gaps only including the effect of lattice
expansion, while the solid points (signed as LE + VIB) represent
the band gaps both including the effect of lattice expansion and
vibration. The band gap values at 0 K are set as the references.
are smaller in MgO and NaCl (table 3), resulting in the better
linear relationship between band gap and temperature. Therefore, the lattice expansion makes a significant contribution to
the temperature-dependent band gap for ionic semiconductors,
while it is relatively minor in covalent semiconductors. The
larger thermal expansion coefficients may come from the large
lattice anharmonicity of these compounds, which is beyond the
scope of this study.
The one-shot method is convenient for the study of modedecomposed EPR. According to equation (2), we can obtain
the displacements of atoms due to one special phonon mode
and further study this mode-induced band gap change at
temperature T. Taking the cases of diamond and MgO at
1050 K as examples. The band gap changes due to all the
phonon vibrations (1293 modes in supercell, excluding the
three zone-center acoustic phonons with zero THz frequency)
are obtained and shown in figure 4. Here the atomic displacements have been multiplied by five in order to clearly
distinguish the contributions from different phonon modes.
In both cases, the optical phonon modes with high-frequency
dominate the band gap variation. The high-frequency optical
5
J. Phys.: Condens. Matter 32 (2020) 475503
Y Zhang et al
2
Table 3. The coefficients a and b from the non-linear fittings by Varshni equation: ΔE g = aT /(T + b),
(ΔE g = E g (T ) − E g (0 K), E g s are from EBSs) for diamond, Si, SiC, MgO and NaCl. The values consist of
(1) only caused by lattice expansion (signed as LE), (2) caused by both lattice expansion and vibration
(signed as LE + VIB) and (3) other experiments and theoretical results.
a (×10−4 eV K−1 )
System
LE
LE + VIB
Diamond
−0.259
−4.338
Si
SiC
MgO
NaCl
0.252
0.292
−4.832
−6.438
−2.051
−6.038
−9.372
−17.70
b (K)
Others
−1.979a
0.436b
−3.976c
−6.034d
−5.561e
−3.024f
−3.696b
−2.630g
−0.306h
—
—
LE
LE + VIB
88.36
643.7
−87.45
−385.2
−7.559
1.567
228.8
1806
301.0
14.71
Others
−1437a
−962.1b
−223.6c
934.0d
820.9e
552.9f
439.2b
273.7g
−311.0h
—
—
a
Reference [39] (exp.).
Reference [21]. ‘One-shot’ method (cal.).
c
Fitting above 300 K, reference [11], the AHC theory in the non-adiabatic harmonic approximation (cal.).
d
Reference [40] (exp.).
e
Reference [41] (exp.).
f
Reference [42], the AHC theory in semi-empirical calculations (cal.).
g
Reference [11], the AHC theory in the adiabatic harmonic approximation (cal.).
h
Reference [4] (exp.).
b
Figure 4. The band gap changes by different phonon modes (total number is 1293) at 1050 K for (a) diamond (blue hollow circles) and (b)
MgO (red hollow circles), respectively. The band gap value that only including lattice expansion at 1050 K is set as the reference.
modes represent the atoms motions against each other, and thus
the corresponding deformations affect the band edge atomic
bonding and electronic states greatly. The high frequencies of
the critical phonons are probably the reason for the plateau at
low temperatures (figures 2(a) and (b)), since the phonons have
been populated then.
is found that the EPR effect is important for these semiconductors and should not be neglected. Our results are in good
agreement with other experiments and theoretical models,
indicating that the method is reasonable. Compared with covalent and ionic compounds, it shows the band gaps all decrease
as temperature increases, and the band gaps in ionic compounds drop quicker. The covalent compounds investigated
show weaker temperature dependences of band gaps is due
to the much weaker lattice expansions and therefore low
contributions from these. Besides, the ZPR effect can play
an important role in the semiconductor with light atomic
masses, such as diamond. By decomposing the EPR effect
into respective phonon modes, it is found that the main contribution to the band gap change is the optical phonon modes
with high frequency for both covalent and ionic ones. Our
work provides a comprehensive EPR method that can study
the semiconductor’s band gap at any given temperature, and
4. Conclusions
In this paper, we have applied a recently-developed method
to calculate the EPR effect on semiconductor’s band gap.
The method includes the cases of both lattice expansion and
phonon induced atomic vibrations at any given temperature.
The temperature-dependent band gaps for covalent (diamond,
Si, and SiC) and ionic (MgO and NaCl) compounds have been
studied based on this method as examples. According to the
band gaps exacted from EBSs at different temperatures, it
6
J. Phys.: Condens. Matter 32 (2020) 475503
Y Zhang et al
boosts the fundamental understandings on the temperaturedependent electronic structures.
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Acknowledgments
This work was supported by the National Key Research and
Development Program of China (No. 2017YFB0701600 and
2018YFB0703600), the National Natural Science Foundation
of China (Grant Nos. 21703136 and 11674211), and the 111
Project D16002. JYX acknowledges the support from the
Shanghai Sailing Program, China (17YF1427900).
ORCID iDs
Jinyang Xi
Jiong Yang
https://orcid.org/0000-0003-4198-2840
https://orcid.org/0000-0002-5862-5981
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