Data obtained from computational studies are crucial in building the necessary infrastructure for materials informatics. This computational foundation supplemented with experimental observations can then be employed in the extraction of possible hidden structure–property relationships through machine learning. There are limited attempts to sample the materials configuration space, even for the simplest chemical formulas. Advances in computational methods have now made it possible to accomplish this task. In this study, we analyze four chemical formulas, i.e., BSb, AlSb, MgSi2, and Sn3S, using first-principles computations. We show that numerous thermodynamically more stable crystal structures can be predicted computationally for these relatively simple chemical formulas, while the configuration space can be significantly and effectively mapped out. This approach allows for the prediction of new ground state structures, thereby expanding the available data on these materials. It also provides an understanding of the underlying potential energy topography and adds quality data for materials informatics.

The rapid development of materials informatics has triggered an enormous demand of high quality and reliable materials data. During the last decade, some major materials databases such as Materials Project,1 Open Quantum Materials Database,2 AFLOWLIB,3 and NoMAD were developed. These databases are remarkably diverse in terms of chemistry, providing hundreds of thousands to millions of data records. Numerous materials with exotic (unusual) stoichiometries can be found on these databases; many of them were initially predicted computationally and then realized experimentally.4–6 By collecting, standardizing, and disseminating materials information, these databases have become an invaluable resource and major reference for materials scientists, especially in computational studies.7–9 Within this context, materials databases are expected to also be sufficiently diverse in terms of geometries, or configurations, as well.

Considering boron antimonide (BSb), a solid material for which the zinc-blende F4̄3m structure is the only entry available in Materials Project, as an example, we will further elaborate this argument. In fact, this atomic structure was first hypothesized for BSb in 1998 for a computational study,10 and it has then been used as the initial input for numerous subsequent modeling work during the next 20 years.10–30 Structural properties,13 elastic properties,16 electronic structure properties,12,13,21 optical properties,15 and thermal conductivity20 of BSb in the F4̄3m structure were computed and documented in some handbooks.31–33 Surprisingly, experimental data34–37 on BSb and its structure are rather ambiguous and inconclusive.33,38,39 In a review article,38 Ackland has noted that “the zinc-blende structure is chemically unstable at ambient conditions.” This is a typical example of the significant value the materials databases could offer if more diverse data are available.

We then collected some statistics of Materials Project, and a quick snapshot is shown in Fig. 1. Presently, a total of 126 300 atomic structures of 89 431 chemical formulas are accessible. This means that on average, there are less than two atomic structures for a given formula. In particular, about 83% (74 624) of the total number of formulas have one atomic structure, another 11% (9633) have two atomic structures, and there are about 500 formulas, each of which has more than ten atomic structures. While the remarkable diversity of the Materials Project database in the chemical composition space is obvious, Fig. 1 also implies that for most of the formulas, the configuration space could be better explored and sampled.

FIG. 1.

“Data size” distribution of the accessible chemical formulas available in the Materials Project database. Data obtained on November 25, 2020.

FIG. 1.

“Data size” distribution of the accessible chemical formulas available in the Materials Project database. Data obtained on November 25, 2020.

Close modal

In fact, comprehensively exploring the configuration space by experimental approaches is exceedingly challenging. The configuration space terminology we use herein is a general term that includes all the possible choices of the atomic positions within a given unit cell, which can also vary with 6 degrees of freedom. Therefore, for a unit cell containing N atoms, the dimensionality of the configuration space is 3N + 6. In this respect, advanced computational tools are particularly suitable in order to examine what are possible atomic configurations for a given chemical formula.4,44 During the last decade, modern materials structure prediction methods have become more efficient and sustainable, leading to numerous discoveries of new materials.4 While the finding of these materials may originally be driven by possible novel properties desired,45 the atomic structures discovered can significantly contribute to the diversity in the configuration space of the existing materials databases and assist in further discovery of materials with complex compositions.

In the present contribution, we discuss the possibility of exploring the materials configuration space by computations at the level of density functional theory (DFT),46,47 somehow addressing the aforementioned limit of materials data. The main focus of this case-study work is a set of four inorganic chemical formulas, such as BSb, AlSb, MgSi2, and Sn3S. As summarized in Table I, each of them has a few crystal structures originally reported in the existing databases. Using a modern computational structure prediction method, we demonstrate how such small datasets can be efficiently and significantly diversified, uncovering new thermodynamically stable structures while enlarging the accessible configuration space characterized by a new (unusual) atomic environment. While details related to BSb were somehow focused, the cases of AlSb, MgSi2, and Sn3S were considered in order to illustrate the generalizability of the proposal. We believe that this approach is crucial to improving the quality of already established databases targeted for materials informatics.

TABLE I.

A summary of the existing and predicted structure data of BSb, AlSb, MgSi2, and Sn3S. Among the lowest-energy predicted structures, the R3̄m structure of MgSi2, the Pmn21 structure of Sn2S, and the C2/c structure of BSb are significantly lower than the previously known structures.

Existing dataPredicted data
Mater.No. of structsStable structReferenceNo. of structsStable structReference
BSb F4̄3m 35  61 C2/c This work 
AlSb P63mc 40  124 P42/mnm This work 
MgSi2 Imma 41  65 R3̄m 42  
Sn3Fm3̄m 1  39 Pmn21 43  
Existing dataPredicted data
Mater.No. of structsStable structReferenceNo. of structsStable structReference
BSb F4̄3m 35  61 C2/c This work 
AlSb P63mc 40  124 P42/mnm This work 
MgSi2 Imma 41  65 R3̄m 42  
Sn3Fm3̄m 1  39 Pmn21 43  

Among four solid materials examined, MgSi2 and Sn3S are essentially hypothetical materials, catching attention mostly from computations41–43,48 but little experiments.49 On the other hand, AlSb is a computationally and experimentally well-studied40,50,51 semiconductor with an indirect bandgap of about 1.6 eV. The last material, i.e., BSb, as discussed above, is somewhere in between. Despite numerous computational10–13,15–26 and experimental studies,34–36 our understanding on this material, and specifically its ground state structure, remains essentially limited, unclear, and ambiguous.38,39 For any of these materials, a few atomic structures can be found in the literature, clearly indicating that the accessible domain of their configuration space is limited. A summary of their available data is given in Table I.

Our first-principles calculations were performed using the DFT formalism as implemented in the Vienna Ab initio Simulation Package (vasp).52,53 We used a basis set of plane waves with kinetic energy up to 600 eV to represent the Kohn–Sham orbitals, the Perdew–Burke–Ernzerhof (PBE) functional54 to approximate the exchange-correlation energy, and the Monkhorst–Pack k-point meshed (XC) no less than 7 × 7 × 7 to sample the Brillouin zone. Convergence in optimizing the structures was assumed when the atomistic forces become less than 0.01 eV/Å. The reported electronic bandgap was computed on top of the structures optimized at the PBE level using the Heyd–Scuseria–Ernzerhof (HSE) XC functional55 with the exact exchange to the generalized-gradient exchange in a ratio of 1:3 and a screening parameter of 0.2 Å−1. The symmetry of the examined structures was determined using findsym56 at a tolerance of 0.05 Å, while some of them were visualized using vesta.57 

The low-energy structures of AlSb and BSb were predicted using the minima-hopping method,59–60 the same approach previously used for MgSi2 and Sn3S.42,43 Given a chemical formula, its multi-dimensional energy landscape is constructed from the DFT-computed energy and it is then explored. Because this method allows for unconstrained searches with strong bias toward the low-energy domains of the configuration space, it is powerful in identifying low-energy structures of solids, specifically those with exotic/unusual structural motifs at the atomic level,43,45,61,62 i.e., those that can generally not be obtained from prototype structures. Given its nature, this method, just like any reliable materials structure prediction methods, is computationally expensive, especially for soft materials and/or those with large unit cells. More details on this aspect will be discussed in Sec. IV.

In principle, the dynamical stability of materials structures predicted by computations should be examined via phonon calculations. If a structure is found to be dynamically unstable, one can follow the unstable vibration modes in order to reach a slightly distorted but stable structure. Although this procedure is computationally very expensive, it does not drive the unstable structure very far in the configuration space and, thus, does not alter our conclusion. Therefore, we skip the phonon calculations in this work.

Analyzing the configurational diversity of the materials data is a primary objective of this work. In principle, each atomic structure of a given chemical formula should be represented as a point in the configuration space. However, the canonical Cartesian coordinates of the atoms in a materials structure are not invariant with respect to many mathematical transformations that do not change the materials in any physical and chemical ways, e.g., rigid translations and rotations; thus, they cannot be used in the analysis. For this reason, we used a recently developed fingerprint,63 which captures pretty well the atomic local environment information while preserving the material presentation under such “identity” transformations in the materials space. This is one of the numerous materials fingerprints65–68 developed during the last decade for capturing the atomic details of materials, representing each of them as a numerical vector, and then being used for machine-learning approaches. Within this approach, materials were actually examined in the fingerprint space, demonstrated63,69 to be a good representation of the materials configuration space. For simplicity, we refer to the fingerprint space as the configuration space in the rest of this work.

More than 300 new low-energy structures were computationally predicted for BSb, AlSb, MgSi2, and Sn3S and summarized in Fig. 2. While all the known structures of these materials were successfully predicted, the lowest-energy unknown P42/mnm structure predicted for AlSb is higher than its known wurtzite P63mc ground state by ≃0.06 eV/atom. In addition, many new structures were predicted to be significantly lower than the well-studied structures of MgSi2,42 Sn3S,43 and BSb. In the case of BSb, the predicted ground state adopts the monoclinic C2/c symmetry and is ≃0.10 eV/atom below the well-studied zinc-blende structure. This is a significant amount of energy, which is comparable to the thermal energy of an atom at ≃1, 100 K. Although details on the predicted structures of MgSi2 and Sn3S can be found in Refs. 42 and 43, respectively, some relevant aspects of these structures will be discussed in Sec. IV of this work. The crystallographic information of the P42/mnm structure of AlSb and the C2/c structure of BSb is given in Table II, while all of the predicted structures can be found in the supplementary material.

FIG. 2.

Low energy spectra of BSb, AlSb, MgSi2, and Sn3S. The green, black, and red lines represent the previously known structures, all the computationally predicted structures, and the lowest-energy predicted structures, respectively. The lowest-energy known structure of each material is set to zero in the energy scale.

FIG. 2.

Low energy spectra of BSb, AlSb, MgSi2, and Sn3S. The green, black, and red lines represent the previously known structures, all the computationally predicted structures, and the lowest-energy predicted structures, respectively. The lowest-energy known structure of each material is set to zero in the energy scale.

Close modal
TABLE II.

Crystallographic information of the lowest-energy predicted P42/mnm structure of AlSb and C2/c of BSb.

MaterialPropertyValue
BSb Space group C2/c 
a, b, c (Å) 6.42, 16.20, 6.35 
β (degrees) 151.12 
(x, y, z) B (0.428 79, 0.235 65, −0.448 19) 
(x, y, z) Sb (−0.406 11, 0.096 17, 0.109 25) 
AlSb Space group P42/mnm 
a, b, c (Å) 7.60, 7.60, 4.41 
(x, y, z) Al (0.172 93, 0.172 93, 0.000 00) 
(x, y, z) Sb (−0.184 36, 0.184 36, 0.000 00) 
MaterialPropertyValue
BSb Space group C2/c 
a, b, c (Å) 6.42, 16.20, 6.35 
β (degrees) 151.12 
(x, y, z) B (0.428 79, 0.235 65, −0.448 19) 
(x, y, z) Sb (−0.406 11, 0.096 17, 0.109 25) 
AlSb Space group P42/mnm 
a, b, c (Å) 7.60, 7.60, 4.41 
(x, y, z) Al (0.172 93, 0.172 93, 0.000 00) 
(x, y, z) Sb (−0.184 36, 0.184 36, 0.000 00) 

Figure 3 signifies the fundamental difference between the computationally predicted structures and the previously known structures of all the materials. In the previously known Imma structure of MgSi2, silicon atoms are arranged in a three-dimensional network, forming parallel one-dimensional tunnels, each of them hosts a zigzag line of magnesium atoms. The discovered R3̄m structure, which is ≃0.10 eV/atom below the Imma structure, consists of alternating 2D planes of magnesium and silicon atoms. For BSb, the predicted C2/c structure is formed by 1D zigzag lines of boron atoms buried between 2D layers of antimonide atoms, being completely different from the previously studied zinc-blende F4̄3m structure.10–13,15–26 Profound distinctive features can also be observed clearly for the computationally predicted Pmn21 structure of Sn3S and the P42/mnm structure of AlSb shown in Fig. 3. A comprehensive visual inspection on all of the predicted structures revealed that their atomic environment is remarkably diverse and a vast majority of them do not resemble any well-known prototype structures.

FIG. 3.

Known and lowest-energy discovered structures of BSb, AlSb, MgSi2, and Sn3S. The unit cells are given in gray color.

FIG. 3.

Known and lowest-energy discovered structures of BSb, AlSb, MgSi2, and Sn3S. The unit cells are given in gray color.

Close modal

Such a (fundamental) atomic environment diversity is translated into a wide range of physical properties of the materials. While the well-studied zinc-blende F4̄3m phase of BSb10–13,15–26 is a semiconductor, its thermodynamically more stable C2/c phase is metallic. Similarly, our electronic structure calculations using the HSE XC functional55 revealed that the predicted (metastable) P42/mnm phase of AlSb is a semiconductor with a bandgap of 1.40 eV, i.e., about 0.20–0.30 eV lower than that of the ground-state P63mc phase. Among about 200 low-energy structures predicted for AlSb and BSb, metallic and semiconducting polymorphs with the DFT computed bandgap ranging up to ≃1.6 eV can be found.

In order to demonstrate the configurational diversity and its implication on the materials properties in a more quantitative manner, the examined atomic structures were first represented by the selected neighborhood-informed fingerprint.63 Then, the fingerprinted data of each chemical formula were projected onto the 2D manifold spanned by PC1 and PC2, the first two principal axes obtained by a principal component analysis. The 2D projections of the fingerprinted data of MgSi2, Sn3S, AlSb, and BSb are shown in Fig. 4, integrating the DFT-computed energy and bandgap as two examples of materials properties. While analyses of this kind are typically67,71–72 used to qualitatively visualize the high-dimensional materials space, each panel of Fig. 4 is devoted to a single chemical formula, so the configurational diversity of the materials data can be isolated and uncovered.

FIG. 4.

Accessible configuration space of BSb, AlSb, MgSi2, and Sn3S, projected onto the 2D manifold spanned by PC1 and PC2, two first principal axes of the principal component analysis. The known and lowest-energy discovered structures are indicated by arrows, while color codes are used to represent the DFT computed energy (top row) and HSE bandgap (bottom row) of the examined structures (for metallic structures, their bandgap was set to 0 eV).

FIG. 4.

Accessible configuration space of BSb, AlSb, MgSi2, and Sn3S, projected onto the 2D manifold spanned by PC1 and PC2, two first principal axes of the principal component analysis. The known and lowest-energy discovered structures are indicated by arrows, while color codes are used to represent the DFT computed energy (top row) and HSE bandgap (bottom row) of the examined structures (for metallic structures, their bandgap was set to 0 eV).

Close modal

Figure 4 clearly shows that for all four chemical formulas considered, the computational structure searches have significantly diversified the materials data in the configuration space. For each of them, the previously known structures occupied a fairly small domain, and this accessible space has been enlarged by the predicted data. In fact, the configuration space of any material is practically infinite, and unexplored domains may potentially host exotic/unexpected properties. The most illustrative example is the case of AlSb. Four known phases of this material, including the wurtzite ground state, locate at one side of the 2D space. Among them, two semiconducting phases are distant from the other two metallic phases. More than a hundred new structures were predicted, filling various unknown domains that correspond to both metallic and semiconducting phases of AlSb. Specifically, the lowest-energy predicted P42/mnm structure, which is slightly above the wurtzite ground state, is pretty close to this phase in the projected space and is also a semiconductor. Similar assessments apply for BSb, while MgSi2 and Sn3S are metallic everywhere in the configuration space.

A major reason for the current data sparsity is the known challenges of exploring the configuration space either by experimental or computational methods. From the mathematical standpoint, computational structure prediction is essentially a global optimization problem on a massively high dimensional space. Because the objective function of this problem should be computed at the first-principles level, this work is computationally very expensive. In 1988, predicting a simple crystal structure by computations was referred to as “one of the continuing scandals in the physical sciences” by a Nature’s editor, John Maddox.73 Remarkably, structure prediction methods have evolved dramatically since then, thanks to the rapid developments of hardware74,75 and advance algorithms.44 As a result, numerous new materials were predicted by computations and then realized experimentally.4 

In this work, our searches for the low-energy structures of AlSb and BSb were performed on Comet, a cluster featuring Intel Xeon E5-2680v3 processors with 12 physical cores working at a clock speed of 2.5 GHz. Similar to many other methods, structure predictions using the minima-hopping method are performed in serial, i.e., a series of local minima are visited, aiming toward lowering the energy. By considering no more than 16 atoms per unit cell and launching 4 parallel searches starting from different initial configurations, ≃103 structures were obtained after 7–10 days. Extra refinements and duplicate removal follow, finalizing the predicted data at ≃102 new structures. We note that for materials requiring more number of atoms in a unit cell, the complexity grows exponentially. However, we believe that this level of computational cost is reasonable for exploring the materials spaces and gradually generating/accumulating data for materials informatics.

Going forward, materials structure predictions can further be improved and accelerated in various ways. In addition to the foreseen hardware development,74,75 the emerging machine-learning approaches can be used for this purpose. As an example, Fig. 4 shows that if the predicted data of a material, e.g., AlSb, can be learned properly, the structure search can then be driven toward the unexplored domains or those with desired properties, i.e., low energy and desired bandgap. Furthermore, developing highly scalable structure prediction methods such as ab initio random structure search76 or polymer structure predictor77 is also a promising approach for better exploiting the developments of computational infrastructure from the algorithm standpoint.74,75 Finally, when computationally exploring the configuration space can be automated, the development of materials data can be significantly accelerated with minimum human intervention.

It is worth mentioning that the boundary between “hypothetical” and “synthesizable” materials may not always be well-defined and, in fact, technically hard to determine. The main reason is that the formation of a material depends not only on the thermodynamics but also on the kinetics of the synthesis route. Under some distinctive conditions and/or starting from some distinctive precursors, a metastable phase that is either separated with the ground state structure by a very high potential barrier and/or very far from the ground state may still be synthesized. The existence of both graphite and diamond, two very different polymorphs of elemental carbon, is a classic example. Therefore, while a diverse dataset is certainly useful for data mining and learning, it may also be the starting point of a whole new line of studies, including synthesizability evaluating, retrosynthesis planning, and, finally, experimentally realizing.79–80 

The main focus of this work is the diversity of materials data and a possibility of using computational approaches for compensation. We found that while the materials chemical space was very well sampled in established databases, the configurational diversity of the data remains limited and should be expanded. We then considered four chemical formulas for solid materials, i.e., BSb, AlSb, MgSi2, and Sn3S, and showed that modern computational materials structure predictions can be used to survey unexplored domains of the space, identifying new atomic configurations and significantly diversifying the existing materials data. Thanks to the recent advances, this task can be done at a reasonable computational overhead. For the particular materials examined, we discovered a new metallic structure of BSb, which is significantly lower than the well-studied zinc-blende ground state in energy. A metastable phase of AlSb was also predicted, having a semiconducting bandgap of about 1.40 eV. With ≃200 new low-energy structures, the configuration space of AlSb and BSb is now better sampled and understood, and the diversified data of these materials will be useful for future studies. Overall, we believe that in the future, the diversity of materials, when being promoted, will make valuable contributions to materials informatics. While advanced computational approaches at the level of first-principles can now be used for this goal, we believe that their role will be significantly elevated in the near future.

The supplementary material is available as a tarball, providing all the structures of BSb, AlSb, MgSi2, and Sn3S, which were predicted and discussed in this work.

Work by T.N.V. was supported by the Vietnamese National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 103.01-2017.24. The authors acknowledge computational support from XSEDE under Grant No. TG-DMR170031.

All the structures predicted for BSb, AlSb, MgSi2, and Sn3S and discussed in this work can be found in the supplementary material.

1.
A.
Jain
,
S. P.
Ong
,
G.
Hautier
,
W.
Chen
,
W. D.
Richards
,
S.
Dacek
,
S.
Cholia
,
D.
Gunter
,
D.
Skinner
,
G.
Ceder
, and
K. A.
Persson
,
APL Mater.
1
,
011002
(
2013
).
2.
J. E.
Saal
,
S.
Kirklin
,
M.
Aykol
,
B.
Meredig
, and
C.
Wolverton
,
JOM
65
,
1501
(
2013
).
3.
R. H.
Taylor
,
F.
Rose
,
C.
Toher
,
O.
Levy
,
K.
Yang
,
M. B.
Nardelli
, and
S.
Curtarolo
,
Comput. Mater. Sci.
93
,
178
(
2014
).
4.
A. R.
Oganov
,
C. J.
Pickard
,
Q.
Zhu
, and
R. J.
Needs
,
Nat. Rev. Mater.
4
,
331
(
2019
).
5.
C. J.
Pickard
,
A.
Salamat
,
M. J.
Bojdys
,
R. J.
Needs
, and
P. F.
McMillan
,
Phys. Rev. B
94
,
094104
(
2016
).
6.
I. A.
Kruglov
,
D. V.
Semenok
,
H.
Song
,
R.
Szczesniak
,
I. A.
Wrona
,
R.
Akashi
,
M. M. D.
Esfahani
,
D.
Duan
,
T.
Cui
, and
A. G.
Kvashnin
,
Phys. Rev. B
101
,
024508
(
2020
).
7.
S.
Curtarolo
,
G. L. W.
Hart
,
M. B.
Nardelli
,
N.
Mingo
,
S.
Sanvito
, and
O.
Levy
,
Nat. Matter.
12
,
191
(
2013
).
8.
T.
Sahoo
,
U. A.
Anene
,
S. K.
Nayak
, and
S. P.
Alpay
,
Mater. Res. Express
7
,
035701
(
2020
).
9.
I.
Opahle
,
G. K. H.
Madsen
, and
R.
Drautz
,
Phys. Chem. Chem. Phys.
14
,
16197
(
2012
).
10.
M.
Ferhat
,
B.
Bouhafs
,
A.
Zaoui
, and
H.
Aourag
,
J. Phys.: Condens. Matter
10
,
7995
(
1998
).
11.
B.
Bouhafs
,
H.
Aourag
,
M.
Ferhat
, and
M.
Certier
,
J. Phys.: Condens. Matter
11
,
5781
(
1999
).
12.
B.
Bouhafs
,
H.
Aourag
, and
M.
Certier
,
J. Phys.: Condens. Matter
12
,
5655
(
2000
).
13.
A.
Zaoui
and
F. E. H.
Hassan
,
J. Phys.: Condens. Matter
13
,
253
(
2001
).
14.
H.
Meradji
,
S.
Drablia
,
S.
Ghemid
,
H.
Belkhir
,
B.
Bouhafs
, and
A.
Tadjer
,
Phys. Status Solidi B
241
,
2881
(
2004
).
15.
A.
Zaoui
,
S.
Kacimi
,
A.
Yakoubi
,
B.
Abbar
, and
B.
Bouhafs
,
Physica B
367
,
195
(
2005
).
16.
D.
Touat
,
M.
Ferhat
, and
A.
Zaoui
,
J. Phys.: Condens. Matter
18
,
3647
(
2006
).
17.
S.
Cui
,
W.
Feng
,
H.
Hu
, and
Z.
Feng
,
Phys. Status Solidi B
246
,
119
(
2009
).
18.
O.
Efimov
,
G.
Lee
, and
Y.-G.
Yoon
,
J. Korean Phys. Soc.
61
,
85
(
2012
).
19.
Naeemullah
,
G.
Murtaza
,
R.
Khenata
,
Mazharullah
, and
S. B.
Omran
,
Phase Transitions
87
,
893
(
2014
).
20.
L.
Lindsay
,
D. A.
Broido
, and
T. L.
Reinecke
,
Phys. Rev. Lett.
111
,
025901
(
2013
).
21.
F.
Ersan
,
G.
Gökoglu
, and
E.
Aktürk
,
J. Phys.: Condens. Matter
26
,
325303
(
2014
).
23.
S.
Bounab
,
A.
Bentabet
,
Y.
Bouhadda
,
G.
Belgoumri
, and
N.
Fenineche
,
J. Electron. Mater.
46
,
4805
(
2017
).
24.
H. L.
Zhuang
and
R. G.
Hennig
,
Appl. Phys. Lett.
101
,
153109
(
2012
).
25.
M.
Benchehima
,
H.
Abid
, and
K.
Benchikh
,
Mater. Chem. Phys.
198
,
214
(
2017
).
26.
N.
Bioud
,
X.-W.
Sun
,
S.
Daoud
,
T.
Song
, and
Z.-J.
Liu
,
Mater. Res. Express
5
,
085904
(
2018
).
27.
Y.
Yao
,
D.
König
, and
M.
Green
,
Sol. Energy Mater. Sol. Cells
111
,
123
(
2013
).
28.
J.
Shi
,
C.
Han
,
X.
Wang
, and
S.
Yun
,
Physica B
574
,
311634
(
2019
).
29.
K.
Boubendira
,
S.
Bendaif
,
O.
Nemiri
,
A.
Boumaza
,
H.
Meradji
,
S.
Ghemid
, and
F. E. H.
Hassan
,
Chin. J. Phys.
55
,
1092
(
2017
).
30.
P.
Mishra
,
D.
Singh
,
Y.
Sonvane
, and
S. K.
Gupta
,
AIP Conf. Proc.
2142
,
110019
(
2019
).
31.
O.
Madelung
,
Semiconductors: Data Handbook
(
Springer-Verlag
,
Berlin, Heidelberg
,
2004
).
32.
Springer Handbook of Condensed Matter and Materials Data
, edited by
W.
Martienssen
and
H.
Warlimont
(
Springer-Verlag
,
Berlin, Heidelberg
,
2006
).
33.
C.
Hilsum
and
A. C.
Rose-Innes
,
Semiconducting III–V Compounds: International Series of Monographs on Semiconductors
(
Elsevier
,
2014
).
34.
S. N.
Das
and
A. K.
Pal
,
Bull. Mater. Sci.
29
,
549
(
2006
).
35.
S.
Dalui
,
S. N.
Das
,
S.
Hussain
,
D.
Paramanik
,
S.
Verma
, and
A. K.
Pal
,
J. Cryst. Growth
305
,
149
(
2007
).
36.
S.
Das
,
R.
Bhunia
,
S.
Hussain
,
R.
Bhar
,
B. R.
Chakraborty
, and
A. K.
Pal
,
Appl. Surf. Sci.
353
,
439
(
2015
).
37.
S.
Das
,
R.
Bhunia
,
S.
Hussain
,
R.
Bhar
, and
A. K.
Pal
,
Eur. Phys. J. Plus
132
,
176
(
2017
).
38.
G. J.
Ackland
,
Rep. Prog. Phys.
64
,
483
(
2001
).
39.
Y.
Kumashiro
,
K.
Nakamura
,
K.
Sato
,
M.
Ohtsuka
,
Y.
Ohishi
,
M.
Nakano
, and
Y.
Doi
,
J. Solid State Chem.
177
,
533
(
2004
).
40.
D. F.
Edwards
and
R. H.
White
,
Handbook of Optical Constants of Solids
(
Elsevier
,
1997
), pp.
501
511
.
41.
M. A.
van Huis
,
J. H.
Chen
,
M. H. F.
Sluiter
, and
H. W.
Zandbergen
,
Acta Mater.
55
,
2183
(
2007
).
42.
43.
V. N.
Tuoc
and
T. D.
Huan
,
J. Phys. Chem. C
122
,
17067
(
2018
).
44.
Modern Methods of Crystal Structure Prediction
, edited by
A. R.
Oganov
(
Wiley-VCH
,
Weinheim, Germany
,
2011
).
45.
T. D.
Huan
,
V.
Sharma
,
G. A.
Rossetti
, and
R.
Ramprasad
,
Phys. Rev. B
90
,
064111
(
2014
).
46.
P.
Hohenberg
and
W.
Kohn
,
Phys. Rev.
136
,
B864
(
1964
).
47.
W.
Kohn
and
L. J.
Sham
,
Phys. Rev.
140
,
A1133
(
1965
).
48.
M. A.
van Huis
,
J. H.
Chen
,
H. W.
Zandbergen
, and
M. H. F.
Sluiter
,
Acta Mater.
54
,
2945
(
2006
).
49.
M. I.
Baleva
,
E.
Goranova
,
M.
Marinova
, and
A.
Atanasov
,
ECS Trans.
8
,
151
(
2007
).
50.
C. T.
Lin
,
E.
Schönherr
, and
H.
Bender
,
J. Cryst. Growth
104
,
653
(
1990
).
51.
K.
Seeger
and
E.
Schonherr
,
Semicond. Sci. Technol.
6
,
301
(
1991
).
52.
G.
Kresse
and
J.
Furthmüller
,
Comput. Mater. Sci.
6
,
15
(
1996
).
53.
G.
Kresse
and
J.
Furthmüller
,
Phys. Rev. B
54
,
11169
(
1996
).
54.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
,
Phys. Rev. Lett.
77
,
3865
(
1996
).
55.
J.
Heyd
,
G. E.
Scuseria
, and
M.
Ernzerhof
,
J. Chem. Phys.
118
,
8207
(
2003
).
56.
H. T.
Stokes
and
D. M.
Hatch
,
J. Appl. Crystallogr.
38
,
237
(
2005
).
57.
K.
Momma
and
F.
Izumi
,
J. Appl. Crystallogr.
41
,
653
(
2008
).
58.
S.
Goedecker
,
J. Chem. Phys.
120
,
9911
(
2004
).
59.
M.
Amsler
and
S.
Goedecker
,
J. Chem. Phys.
133
,
224104
(
2010
).
60.
S.
Goedecker
,
Modern Methods of Crystal Structure Prediction
(
Wiley-VCH
,
Weinheim, Germany
,
2011
), Chap. 7, pp.
147
180
.
61.
T. D.
Huan
,
V. N.
Tuoc
, and
N. V.
Minh
,
Phys. Rev. B
93
,
094105
(
2016
).
62.
T. D.
Huan
,
M.
Amsler
,
M. A. L.
Marques
,
S.
Botti
,
A.
Willand
, and
S.
Goedecker
,
Phys. Rev. Lett.
110
,
135502
(
2013
).
63.
R.
Batra
,
H. D.
Tran
,
C.
Kim
,
J.
Chapman
,
L.
Chen
,
A.
Chandrasekaran
, and
R.
Ramprasad
,
J. Phys. Chem. C
123
,
15859
(
2019
).
64.
M.
Rupp
,
A.
Tkatchenko
,
K.-R.
Müller
, and
O. A.
von Lilienfeld
,
Phys. Rev. Lett.
108
,
058301
(
2012
).
65.
A. P.
Bartók
,
R.
Kondor
, and
G.
Csányi
,
Phys. Rev. B
87
,
184115
(
2013
).
66.
J.
Behler
,
J. Chem. Phys.
134
,
074106
(
2011
).
67.
T. D.
Huan
,
R.
Batra
,
J.
Chapman
,
S.
Krishnan
,
L.
Chen
, and
R.
Ramprasad
,
npj Comput. Mater.
3
,
37
(
2017
).
68.
A.
Chandrasekaran
,
D.
Kamal
,
R.
Batra
,
C.
Kim
,
L.
Chen
, and
R.
Ramprasad
,
npj Comput. Mater.
5
,
22
(
2019
).
69.
T. D.
Huan
,
R.
Batra
,
J.
Chapman
,
C.
Kim
,
A.
Chandrasekaran
, and
R.
Ramprasad
,
J. Phys. Chem. C
123
,
20715
(
2019
).
70.
K.
Rajan
,
C.
Suh
, and
P. F.
Mendez
,
Stat. Anal. Data Min.
1
,
361
(
2009
).
71.
S.
De
,
A. P.
Bartók
,
G.
Csányi
, and
M.
Ceriotti
,
Phys. Chem. Chem. Phys.
18
,
13754
(
2016
).
72.
H.
Doan Tran
,
C.
Kim
,
L.
Chen
,
A.
Chandrasekaran
,
R.
Batra
,
S.
Venkatram
,
D.
Kamal
,
J. P.
Lightstone
,
R.
Gurnani
,
P.
Shetty
 et al.,
J. Appl. Phys.
128
,
171104
(
2020
).
73.
74.
M. B.
Giles
and
I.
Reguly
,
Philos. Trans. R. Soc. A
372
,
20130319
(
2014
).
75.
G.
Guidi
,
M.
Ellis
,
A.
Buluc
,
K.
Yelick
, and
D.
Culler
, arXiv:2011.00656 (
2020
).
76.
C. J.
Pickard
and
R. J.
Needs
,
J. Phys.: Condens. Matter.
23
,
053201
(
2011
).
77.
T. D.
Huan
and
R.
Ramprasad
,
J. Phys. Chem. Lett.
11
,
5823
(
2020
).
78.
M.
Aykol
,
V. I.
Hegde
,
L.
Hung
,
S.
Suram
,
P.
Herring
,
C.
Wolverton
, and
J. S.
Hummelshøj
,
Nat. Commun.
10
,
2018
(
2019
).
79.
F. T.
Szczypiński
,
S.
Bennett
, and
K. E.
Jelfs
,
Chem. Sci.
12
,
830
(
2021
).
80.
J.
Jang
,
G. H.
Gu
,
J.
Noh
,
J.
Kim
, and
Y.
Jung
,
J. Am. Chem. Soc.
142
,
18836
(
2020
).

Supplementary Material