Atomic configurations for materials research: A case study of some simple binary compounds

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 data^34–37 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.

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.

Among four solid materials examined, MgSi₂ and Sn₃S are essentially hypothetical materials, catching attention mostly from computations^41–43,48 but little experiments.⁴⁹ On the other hand, AlSb is a computationally and experimentally well-studied^40,50,51 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 computational^{10–13,15–26} 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 fingerprints^65–68 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, demonstrated^63,69 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.

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.

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 BSb^{10–13,15–26} 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 typically^67,71–72 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.

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 hardware^74,75 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.