Materials Research Express

PAPER • OPEN ACCESSElectronic properties and crystal structures of double-perovskites,
Ba2BiIIIBiVO6,Ba2PrBiO6, and Ba2PrSbO6: First-principles study
To cite this article: Kazume Nishidate et al 2020 Mater. Res. Express 7 065505

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Mater. Res. Express 7 (2020) 065505

https://doi.org/10.1088/2053-1591/ab97e5

PAPER

OPEN ACCESSElectronic properties and crystal structures of double-perovskites,
Ba2BiIIIBiVO6,Ba2PrBiO6, and Ba2PrSbO6: First-principles study

RECEIVED

17 March 2020
REVISED

26 May 2020

Kazume Nishidate1 , Achy Adiko1, Michiaki Matsukawa1, Haruka Taniguchi1 , Arisa Sato1,
Akiyuki Matsushita2, Satoru Tanibayashi3 and Masayuki Hasegawa4
1

ACCEPTED FOR PUBLICATION

29 May 2020
PUBLISHED

10 June 2020

2
3
4Faculty of Science and Engineering, Iwate University, Morioka 020-8551, Japan
National Institute for Materials Science, Ibaraki 305-0047, Japan
Department of Engineering for Future Innovation, National Institute of Technology, Ichinoseki College, Ichinoseki 021-8511, Japan
Professor Emeritus, Iwate University, Morioka 020-8550, Japan

E-mail: nisidate@iwate-u.ac.jp
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the terms of the Creative
Commons Attribution 4.0
licence.
Any further distribution ofthis work must maintainattribution to theauthor(s) and the title of
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and DOI.

Keywords: double-perovskite, band structure, hybrid functional

Abstract
In recent experiments, a signiﬁcant band gap widening was observed when Sb was substituted for Bi in
the double-perovskite Ba2PrBiO6. In this work, we study a series of double-perovskites, Ba2BiIIIBiVO6,
Ba2PrBiO6, and Ba2PrSbO6 using the ﬁrst-principles density functional theory with the HeydScuseria-Ernzerhof hybrid functional to investigate the substitution effect on the structural andelectronic properties. We ﬁnd that the two topmost valence bands are disappeared on the substitutionof PrIII for BiIII, and the two bottommost conduction bands are disappeared on the substitution of SbV
for BiV, causing the signiﬁcant band gap widening. Further, our calculation suggests that theBa2PrPrBiO6 is a possible candidate as a source of the PrIV signal observed in the experiment. We ﬁndthat the B-site disordering atomic conﬁguration, Ba2B″VB′IIIO6, are restored to those of the original
structures. On the other hand, our results also suggest the importance of the partial B-site disorder toexplain the experimentally observed band gaps.

1. Introduction
TiO2 is well known as an efﬁcient photo catalytic material and has a wide range of application area from airpuriﬁer to antimicrobial coating [1–6]. However, TiO2 uses only the ultraviolet ray which is less than 3% of thewhole sunlight pouring on the Earth surface. Therefore extensive studies have been devoted to develop photocatalytic materials sensitive to visible light that makes up 50% of the sunlight. A large number of doubleperovskite oxides, A2B′IIIB″VO6, have been studied due to their intriguing physical and chemical properties
originating in their mixed valence nature [7]. Above all, Ba2PrBiO6 was found to show an efﬁcient photocatalytic activity to dissolve water molecules into oxygen and hydrogen gases under the visible light [8, 9]. Recent
experiment on the Ba2PrSbO6 suggests the presence of the typical B-site ordering and the band gap modiﬁcation[10, 11]. However, detailed theoretical analysis on the electronic structures and the crystal geometries of the
double-perovskites has not been performed. In this work, we study the structural and electronic properties of
Ba2BiIIIBiVO6, Ba2PrBiO6, and Ba2PrSbO6 double-perovskites using the ﬁrst-principles density functional
theory. The Heyd-Scuseria-Ernzerhof hybrid functional was applied to the calculations to handle the strong
electron-correlation. We ﬁnd that the BiIII 6s states at the top of valence band of Ba2BiIIIBiVO6 vanish on the Pr
substitution for Bi at B¢ III-site. When Sb is substituted for Bi at B″V-site, the BiV 6s states at the bottom of the
conduction band vanish causing additional widening of the band gap. Further, our calculation suggests that theBa2PrPrBiO6 is a possible candidate as a source of the PrIV signal observed in the experiment. We ﬁnd that the Bsite disorder atomic conﬁgurations, Ba2B″VB′IIIO6, are easily restored to those of the original structures. This
demonstrates the stability of the B-site ordering in the double-perovskite framework. On the other hand, ourresults also suggest the importance of the partial B-site disorder in the double-perovskites to explain theexperimentally observed band gaps.
© 2020 The Author(s). Published by IOP Publishing Ltd

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Figure 1. Ba2BiIIIBiVO6 with the C2/m symmetry projected on the (010) plane. Bi (Ba) atoms are large dark-purple (light-green)
spheres. Small spheres are the oxygen atoms. A C2/m unit cell contains four BaBiO3 chemical units and is the region indicated withthe letters p, q, r, and s, while an I2/m unit cell is the region indicated with the letters q, t, r, and s.

2. Calculation method
We use the density functional theory (DFT) as implemented in the Vienna ab-initio Simulation Package (VASP)[12] and the projector augmented wave (PAW) potentials [13] to study the structural and electronic properties ofthe double perovskite. It has been known that standard exchange-correlation functionals, such as the local
density approximation (LDA) and the generalized gradient approximation (GGA), predict the electronic
properties of double perovskites as metal or semi-metal with a very narrow band gap [14]. Theoretical study
considering the electron correlations through the GGA+U method showed an improvement on the optimized
structural and electronic properties of Ba2BiIIIBiVO6 [15]. However, since the method utilizes specially tuned
parameter U, application of the method to the other double-perovskites is not straightforward. On the other
hand, recently developed Heyd-Scuseria-Ernzerhof hybrid functional (HSE06) [16, 17] is known to have an
ability to account for the strong correlation effects and has been successfully applied to analyze the structural andelectronic properties of the double-perovskites [18]. Therefore we use the HSE06 functional throughout ourstudy. We use an energy cutoff of 500eV for plane wave basis set together with a 6×6×6 k-point grid. The
rather high energy cutoff and dense k-point grid are necessary to predict correctly the crystal structures in the
HSE06 computations. Equilibrium crystal structures were achieved so that the maximum force component was
smaller than 1 meV/Å3, and the maximum stress component smaller than 1 meV/Å3. Optimized crystal
structures were visualized using the VESTA [19].

3. Results and discussion
First of all, we evaluate the structural and electronic properties of Ba2BiIIIBiVO6 as it gives the basis in analyzing
the double perovskite crystals [20, 21]. Unit cell contains four BaBiO3 chemical units where the BiIII octahedra
and the BiV octahedra are arranged alternately forming the B-site ordering. It has the monoclinic I2/m
symmetry at room temperature [22], and which is equivalent to the conventional setting of C2/m No.12 [23, 24].
The I2/m cell is outlined on the C2/m supercell in the ﬁgure 1.
Where the distortions of octahedra caused by the mixed valencies are characterized by breathing distortion δ
and tilting angle f as depicted in ﬁgure 2.
Structural optimization has been done for the room temperature phase of C2/m (table 1). Lattice constants
optimized within PBE are signiﬁcantly overestimated thereby causing the overestimation of v0. On the other
hand, HSE06 correctly predicts not only the lattice constants but also the internal distortion parameters, δ and f.
The difference is even clear when we calculate the electronic property of the system: Ba2BiIIIBiVO6 is semimetallic in the PBE functional while it opens band gap signiﬁcantly when we use the HSE06 functional consisting2

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Figure 2. Distorted octahedra of Ba2BiIIIBiVO6. Where f is deﬁned as the average tilting angle of the neighboring BiIIIO6 and BiV O6
1
from the dotted line which is equal to the c-axis of the I2/m cell. d = 2 (BiIIIO - BiVO) is deﬁned as the breathing distortion, with
V
III
V
III
Bi O and Bi O are average Bi -O and Bi -O distances, respectively [14].

Table 1. Structural parameters of Ba2BiIIIBiVO6. Lattice parameters (a, b, and c) and the angle β in the
monoclinic I2/m cell extrapolated from those of the C2/m cell are presented. Cell volume per chemical formula
unit is v0. δ and f are the breathing distortion and the tilting angle, respectively.

HSE06 (This work)
PBE (This work)
HSE06a
PBEa
Exp1b
Exp2c
Exp3d

a (Å)

b (Å)

c (Å)

β (deg)v0 (Å3)δ (Å)

f (deg)

6.201
6.276
L
L
6.18146.19086.1851

6.150
6.228
L
L
6.13606.14506.1322

8.681
8.793
L
L
8.66978.67858.6585

90.35
90.27
90.24
90.16
90.17390.16490.229

82.77
85.93
82.10
85.76
82.21
82.54
82.10

0.10
0.14
0.09
0.07
0.085
0.166
0.180

11.1
12.4
11.9
12.1
11.2
9.43
10.30

Values were calculated in the optimized structure derived from the cubic high temperature phase (Fm3̄m )[14].
b
Experimental data at room temperature [21].
cExperimental data at 295 K [26].
dExperimental data at 200 K [22].
awith the experiment. It has been argued that this shortcoming of the PBE comes from lesser ability in describing
the exchange correlation effect of strongly-correlated electron system [18, 25].
Next, we calculate the electronic band structure. In a monoclinic system, the shape of the Brillouin zone
depends non-trivially on the lattice vectors [27]. In fact, there are ﬁve possible shapes of Brillouin zones for the
monoclinic crystal structure depending on the choice of lattice vectors [28, 29]. We choose the C2/m whichcorresponds to the MCLC1 lattice as deﬁned in [29]. To generate the set of k-points along the edges of theBrillouin zone, we use the code pymatgen [30].
High resolution band structures and electronic density of states curves were obtained thorough the
WANNIER90 package by constructing maximally localized Wannier functions (MLWFs) [31–34] using the
VASP2WANNIER90 interface [35] (ﬁgure 3). The band gap is indirect and Ei (L  Z) = 0.84 eV while thedirect band gap is Ed = 2.02 eV . Electron distributions of the valence band maximum (VBM) and of the
conduction band minimum (CBM) evaluated at the Γ point are shown in the ﬁgure 4. We can see the 6selectrons of BiIII and the O 2p electrons at the VBM state (Figure 4(a)) reﬂecting their valence states while the 6selectrons of BiV are visible at the CBM state (ﬁgure 4(b)).
We substituted Pr for the BiIII to form Ba2PrBiO6. The optimized structural parameters show an excellent
agreement with the experiments (table 2). Electronic band structure is given in the ﬁgure 5. The band gap isindirect Ei (Z  G) = 3.58 eV and the direct band gap is Ed = 3.70 eV . It is visible that the two isolated
valence bands near the fermi level are now vanished. The breathing distortion δ as well as the tilting angle fretain the values before the substitution reﬂecting the preservation of their valence states (Ba2PrIIIBiVO6).
It has been reported that experimentally prepared Ba2PrBiO6 sample exhibits the valence mixing between
PrIII and Pr IV [10, 11]. In the experiment, the valence state PrIV was gradually suppressed as the amount of
substitution of Sb for Bi was increased, and was completely resolved in the Ba2PrIIISbVO6 sample. However, thevalence state PrIV must be compensated locally to keep the charge neutrality. One of the potential candidates forthe compensation is the locally generated B-site disorder. We construct the structure by exchanging the atomic
positions of Pr and Bi (Ba2PrIIIBiVO6→Ba2BiIIIPrVO6) while ﬁxing the framework made by Ba and O atoms.
After the optimization, the structure recovered to that of the Ba2PrIIIBiVO6 by adjusting the distances fromPr and Bi to the nearest oxygen atoms and the valence state PrIV was not be realized. Next, we investigated the3Mater. Res. Express 7 (2020) 065505

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Figure 3. Band structure and density of states curve of Ba2Bi IIIBiVO6. Fermi level is at 0 eV.

Figure 4. The BiIII 6s orbitals are visible at the middle points of the upper and lower sides, and at the four corners of the cell in the VBM
charge density distribution (a) while those of BiV appear at the middle points of left and right sides, and at the center of the cell in the
CBM charge density distribution (b). The wire-frame representation for the C2/m atomic arrangements (ﬁgure 1) is used to improvethe visibility. Where some of the orbitals are showing their sections at the boundaries of the unit cell.

Table 2. Structural parameters of Ba2PrBiO6. The same notations for columns of the table 1
are used.

This work.
Exp1a
Exp2b
a
b

a (Å)

b (Å)

c (Å)

β (deg)v0 (Å3)δ (Å)

f (deg)

6.211
6.204
6.201

6.169
6.169
6.158

8.707
8.701
8.697

90.41
90.27
90.08

83.41
83.20
83.03

0.12

11.1

Reference [11].
Reference [23].

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Figure 5. Band structure and density of states curve of Ba2Pr IIIBiVO6. Two isolated valence bands near the Fermi level of Ba2BiIIIBiVO6
were vanished due to the B ¢ III -site substitution.

Table 3. Structural parameters of Ba2PrSbO6. The same notations forcolumns of the table 1 are used.

This work.
Expt.aaa (Å)α (deg)v0 (Å3)δ (Å)

f (deg)

6.075
6.047

60.00
60.16

78.65
78.64

0.16

6.5

Experimental value [11].

Vsubstitution defect of PrBi to see if the B-site ordering is preserved by forming the Ba2PrIII PrBi
O6. After the
structural optimization, all the distances between Pr and neighboring oxygen atoms became to be 2.33 Å (δ = 0)
as the result of the single valency formation of PrIV(Ba2PrIV Pr IV
Bi O6)Pr. It has a similar structural feature of the
cubic BaPrIVO3 (high temperature phase, space group Pm3̄m ) [36] except the rather elongated Pr-O bondlength and the residual tilting angles. We found that the total energy of Ba2PrPrBiO6 per BaPrO3 chemical unit islower than that of the cubic BaPrO3. Our calculation suggests that the Ba2PrPrBiO6 is a possible candidate as asource of the PrIV signal observed in the experiment, where the residual Bi may contribute to form
Ba2BiIIIBiVO6.
We now focus on the substitution of the smaller ion SbV for BiV. The stable crystal structure is R3̄. Again, theoptimized crystal structure shows excellent agreement with the experimental one (table 3).
The distances from Pr and Sb to the nearest neighbor oxygen atoms are almost identical to those of theBa2PrBiO6, reﬂecting the formation of the valence state of Ba2PrIIISbVO6. We also tried to create the B-site
disordered structure by interchanging the atomic positions of Pr and Sb in Ba2PrSbO6. Structural optimizations
were performed only for the atomic positions while ﬁxing the cell shape and the cell volume. The system again
restores its crystal symmetry demonstrating the stability of the B-site ordering of the Ba2PrIIISbVO6 framework.
The electronic band structure is shown in the ﬁgure 6. The BiV bands located at the bottom of the conduction
bands of Ba2PrIIIBiVO6 are vanished and the band gap is widened. The calculated indirect band gap is
Ei (Z  L) = 5.90 eV and the direct band gap is Ed = 5.97 eV . Experimentally estimated band gap Eex also
show the similar widening tendency on the substitution of Sb for BiV (Eex (Ba2PrBiO6) = 0.977 eV → Eex(Ba2PrSbO6) = 2.395 eV [11]). However, the experimental values are about half of the theoretical values
obtained in this work.
The hybrid functional HSE06 incorporates the short-range Hartree–Fock type exchange at the fraction of α
(0 < a < 1). It has been reported that the parameter α is somewhat material-speciﬁc and the best value often
deviates from the standard value of α=0.25 [18, 27, 35]. Therefore we calculated the band gaps for α=0.25,
0.1, and 0.05 to see if it is the case for the double-perovskites (table 4).
The band gaps of Ba2BiIIIBiVO6 calculated with the small α signiﬁcantly deviate from the experimental ones
while those of the Ba2PrIIIBiVO6 and Ba2PrIIISbVO6 show decreasing tendency with decreasing the α but are stilllager than the experimental values. We can conclude that the adoption of the smaller α does not improve the5Mater. Res. Express 7 (2020) 065505

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Figure 6. Band structure and density of states curve of Ba2Pr IIISbVO6. Two isolated conduction bands near the Fermi level of
Ba2PrIIIBiVO6 were vanished due to the B″V-site substitution.

Table 4. HSE06 band gaps (eV) calculated for the α=0.25, 0.1, and 0.005. Experimental band gaps are also shown(Expt.). Band gaps of B-site disordered double-perovskites, Ba2 (Pr III « BiV) O6 and Ba2 (Pr III « SbV ) O6 without
structural relaxations, are also shown.
HSE06
α=0.1α=0.25a

Ba2BiIIIBiVO6
Ba2PrIIIBiVO6
Ba2 (Pr III «BiV)O6
Ba2PrIIISbVO6
Ba2 (Pr III «SbV)O6

Ed
2.02
2.07d
3.70
1.14
5.97
1.89

Expt.

Ei
0.84
0.84d
3.58
0.78
5.90
1.09

α=0.05

Ed
1.55

Ei
0.50

Ed
1.40

Ei
0.39

2.82

2.77

2.54

2.50

4.94

4.90

4.61

4.55

Ed
2.0b,cEi
0.2c
0.977e2.395eaStandard value used in the HSE06 calculation.
Reference [37].
cReference [38].
dReference [14].
eReference [11].
bdisparity of the band gaps. On the other hand, we focus on the partial disorder of the B-sites as it has been
experimentally found in the Ba2PrIIIBiVO6 sample [23]. Although our calculation indicates that the B-site
disordered double-perovskites are energetically unfavorable, it dose not exclude the possibility of the presence ofpartial disordered B-sites in the ordered double-perovskite matrices. It is well known that the electronic
properties of semiconductors are easily modiﬁed signiﬁcantly by incorporation of small amounts of impurities
or other kind of defects [39]. We evaluated the band gaps of the B-site disordered double-peroveskites using thestandard value for the α (table 4). Structural relaxations were not performed. We found ﬂuctuating behavior
both in the lowest conduction band and the highest valence band thereby signiﬁcantly narrowing their bandgaps. The indirect band gap of the B-site disordered Ba2PrIIIBiVO6 is Ei = 0.78 eV which is comparable to theexperimental value of 0.977eV. Similarly, the direct band gap of the B-site disordered Ba2PrIIISbVO6 isEd=1.89 eV which is also comparable to the experimental value of 2.395 eV. Although the B-site disordering of
the Ba2PrIIISbVO6 has not been observed in the experiments yet, our results indicate that the B-site disordering
atomic conﬁgurations in the double-perovskites may control their band gaps.
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4. Conclusion
Density functional calculations were performed to investigate the structural and electronic properties of the
double-perovskites, Ba2BiIIIBiVO6, Ba2PrBiO6, and Ba2PrSbO6. The B-site ordering in the Ba2BiIIIBiVO6 creates
typical electronic band structure, where the top of the valence bands are composed of BiIII 6s orbitals while thebottom of the conduction bands are composed of BiV 6s orbitals. By substituting the Pr for BiIII, two valenceband originated in the BiIII 6s orbitals are vanished thereby widening the band gaps. Substitution of Sb for BiV
induces additional band gap widening. Further, our calculation suggests that the Ba2PrPrBiO6 is a possible
candidate as a source of the PrIV signal observed in the experiment. The B-site disordering atomic conﬁgurations
are restored to those of the original crystal structures. It demonstrates the stability of the B-site ordering in the
double-perovskite framework. However, it does not exclude the presence of the partial B-site disordering as it
has been experimentally observed in the Ba2PrIIIBiVO6 sample. Calculated band gaps for the B-site disordered
Ba2PrIIIBiVO6 and Ba2PrIIISbVO6 show comparable results with the experiments. Although the B-site
disordering of the Ba2PrIIISbVO6 has not been observed in the experiments yet, our results indicate that the Bsite disordered atomic conﬁgurations in the double-perovskites may play an important role to adjust theirband gaps.

Acknowledgments
This work was supported in part by JSPS KAKENHI Grant No. JP19K04995, Iketani Science and Technology
Foundation, and The Mazda Foundation. The authors thank the Iwate University Super-Computing andInformation Sciences Center, the Cyberscience Center, Tohoku University, and the Supercomputer Center of
the Institute for Solid State Physics, University of Tokyo, for providing the computational resources.

ORCID iDs
Kazume Nishidate
Haruka Taniguchi

https://orcid.org/0000-0002-3003-4020
https://orcid.org/0000-0002-9739-4047

References[1] Fujishima A and Honda K 1972 Nature 238 37[2] Hoffmann M R, Martin S T, Choi W and Bahnemann D W 1995 Chem. Rev. 95 69[3] Ohko Y, Tryk D A, Hashimoto K and Fujishima A 1998 J. Phys. Chem. B 102 2699
[4] Zou Z, Ye J, Sayama K and Arakawa H 2001 Nature 414 625[5] Karkmaz M, Puzenat E, Guillard C and Herrmann J 2004 J. M. Appl. Catal. B 51 183[6] Zhu H, Lan Y, Gao X, Ringer S P, Zheng Z, Song D and Zhao J 2005 J. Am. Chem. Soc. 127 6730
[7] Vasala S and Karppinen M 2015 Prog. in Solid State Chem. 43 1
[8] Hatakeyama T, Takeda S, Ishikawa F, Ohmura A, Nakayama A, Yamada Y, Matsushita A and Yea J 2010 J. Cer. Soc. Jpn. 118 91[9] Matsushita A, Nakane A, Naka T, Isago T and Yamada H 2012 Jpn. J. Appl. Phys. 51 121802[10] Onodera K, Kogawa T, Matsukawa M, Taniguchi H, Nishidate K, Matsushita A and Shimoda M 2018 J. Phys. Conf. Ser. 969 012122[11] Taniguchi H, Matsukawa M, Onodera K, Nishidate K and Matsushita A 2019 IEEE Trans. on Mag. 55 2400404[12] Kresse G and Furthmüller J 1996 Phys. Rev. B 54 11169[13] Blöchl P 1994 Phys. Rev. B 50 17953[14] Franchini C, Sanna A, Marsman M and Kresse G 2010 Phys. Rev. B 81 085213[15] Korotin D, Kukolev V, Kozhevnikov A V, Novoselov D and Anisimov V I 2012 J. Phys., Condens. Matt. 24 415603[16] Heyd J, Scuseria G E and Ernzerhof M 2003 J. Chem. Phys. 118 8207
[17] Heyd J, Scuseria G E and Ernzerhof M 2006 J. Chem. Phys. 124 219906[18] Franchini C 2014 J. Phys. Condens. Matter 26 253202[19] Momma K and Izumi F 2011 J. Appl. Crystallogr. 44 1272
[20] Cox D E and Sleight A W 1976 Solid State Commun. 19 969[21] Cox D E and Sleight A W 1979 Acta Crystallogr. Sect. B: Struct. Crystallogr. Cryst. Chem. 35 1
[22] Kennedy B J, Howard C J, Knight K S, Zhan Z and Zhau Q 2006 Acta Cryst. B 62 537[23] Harrison W T A, Reis K P, Jacobson A J, Schneemeyer L F and Waszczak J V 1995 Chem, Mater 7 2161
[24] Jaffe H W 1996 Crystal Chemistry and Refractivity (Mineola, New York: Dover Publications, Inc.) 220–1[25] Smolyanyuk A, Boeri L and Franchini C 2017 Phys. Rev. B 96 035103[26] Pei S, Jorgensen J D, Dabrowski B, Hinks D G, Richards D R, Mitchell A W, Newsam J M, Sinha S K, Vaknin D and Jacobson A J 1990
Phys. Rev. B 41 4126
[27] Peelaes H and van de Walle C G 2015 Phys. Status Solidi B 252 828[28] Ruzaikin M P and Kudryavtseva N V 1974 Russ. Phys. J 17 934[29] Setyawan W and Curtarolo S 2010 Comput. Mater. Sci. 49 299[30] Ong S P, Richards W D, Jain A, Hautier G, Kocher M, Cholia S, Gunter D, Chevrier V, Persson K A and Ceder G 2013 Comp. Mat. Sci.
68 314[31] Mostoﬁ A A, Yates J R, Lee Y-S, Souza I, Vanderbilt D and Marzari N 2008 Comput. Phys. Commun. 178 685

7Mater. Res. Express 7 (2020) 065505

K Nishidate et al[32] Marzari N and Vanderbilt D 1997 Phys. Rev. B 56 12847[33] Souza I, Marzari N and Vanderbilt D 2001 Phys. Rev. B 65 035109[34] Marzari N, Mostoﬁ A A, Yates J R, Souza I and Vanderbilt D 2012 Rev. Mod. Phys. 84 1419
[35] Franchini C, Kováčik R, Marsman M, Murthy S S, He J, Ederer C and Kresse G 2012 J. Phys.: Condens. Matt. 24 235602[36] Saines P J, Kennedy B J and Smith R I 2009 Mat. Res. Bull. 44 874[37] Uchida S, Kitazawa K and Tanaka S 1987 Phase Transitions 8 95
[38] Kim K H, Jung C U, Noh T W and Kim S C 1997 Phys. Rev. B 55 15393[39] Yu P Y and Cardona M 2005 Fundamentals of Semiconductors 3rd edn. (Berlin: Springer) pp. 159–202

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