#### Abstract

The rapid growth of the solar energy industry is driving a strong demand for high performance, efficient photoelectric materials. In particular, ferroelectrics composed of earth-abundant elements may be useful in solar cell applications due to their large internal polarization. Unfortunately, wide band gaps prevent many such materials from absorbing light in the visible to mid-infrared range. Here, we address the band gap issue by investigating the effects of substituting sulfur for oxygen in the perovskite structure ZnSnO_{3}. Using evolutionary methods, we identify the stable and metastable structures of ZnSnS_{3} and compare them to those previously characterized for ZnSnO_{3}. Our results suggest that the most stable structure of ZnSnS_{3} is the monoclinic structure, followed by the metastable ilmenite and lithium niobate structures. The latter structure is highly polarized, possessing a significantly reduced band gap of 1.28 eV. These desirable characteristics make it a prime candidate for solar cell applications.

#### 1. Introduction

Ferroelectrics are materials that possess spontaneous electric polarization. This results from a lack of inversion symmetry; all ferroelectric crystals are noncentrosymmetric. Due to intrinsic polarization, ferroelectrics may serve as light harvesters in photovoltaic devices [1–6]. In a semiconductor p-n junction acting as a photovoltaic device, the built-in potential across the depletion layer is used to separate the photoexcited electron-hole pairs; the maximum open-circuit voltage is thus almost equal to the semiconductor band gap. In a ferroelectric, on the other hand, the separation of the photoexcited pairs is due to the built-in potential induced by the intrinsic polarization; this makes possible the realization of open-circuit voltages that far exceed the band gap [7–17].

Among the most important ferroelectrics are metal oxide perovskites with the general formula ABO_{3}, where A and B are metal cations (B is usually a transition metal). Well-known examples of such ferroelectrics are BaTiO_{3} and LiNbO_{3}. These oxides have relatively large internal electric fields that could be exploited in photovoltaic applications. However, progress in this area has been hampered by the fact that these ferroelectrics have a large band gap (3-4 eV), which makes them unsuitable for efficient light harvesting. The large band gap is due to the strong ionic bonding between the transition metal B and oxygen, which, in turn, is due to the large difference in electronegativity between these atoms. In ABO_{3}, the highest valence band is derived from oxygen 2p orbitals, while the low conduction bands are derived from the transition metal 3d states.

To reduce the band gap in ferroelectrics, different strategies have been implemented. Choi et al. [18] alloyed the ferroelectric Bi_{4}Ti_{3}O_{12}, which has an optical band gap between 3.1 and 3.6 eV [19–21], with LaCoO_{3}, which is a Mott insulator with a small band gap of 0.1 eV [22]. From X-ray diffraction and scanning transmission electron microscopy, they concluded that some La atoms substitute for some Bi atoms at specific sites, and some Co atoms occupy some Ti sites. Since Co is more electronegative than Ti (1.88 versus 1.54), Co-O bond is less ionic than Ti-O bond, and a reduction in the band gap is expected. Indeed, a reduction of up to 1 eV was observed.

In another approach, Grinberg et al. [23] placed two different transition metal cations on the perovskite B-site, with one atom driving ferroelectricity and the other producing a band gap in the visible range. They mixed the ferroelectric oxide potassium niobate (KNbO_{3}) with barium nickel niobate (BaNi_{1/2}Nb_{1/2}O_{3−δ}) so as to introduce Ni^{2+} on the B-site along with an oxygen vacancy which can give rise to gap states in the host KNbO_{3} crystal. The solid solutions thus formed, with to , were ferroelectric, with a direct band gap ranging from 1.1 to 3.38 eV.

In some multiferroics, which exhibit a magnetic order alongside the ferroelectric one, a somewhat smaller band gap exists. In BiFeO_{3} the band gap is 2.7 eV [24, 25]. It is thus expected that multiferroics, such as BiFeO_{3} and Bi_{2}FeCrO_{6} [26, 27], would be more promising candidates for solar cell applications. In Bi_{2}FeCrO_{6} epitaxial thin films, the optical band gap depends on the degree of Fe-Cr ordering. This dependence results from the hybridization of the 3d orbitals in Fe and Cr with the 2p orbitals in oxygen. Nechache et al. [28] investigated the effect of Fe/Cr ordering on the band gap and found that, under the right film growth conditions, the gap can be tuned all the way from 2.7 eV down to 1.5 eV.

An important class of ferroelectric oxides is those that crystallize in the LiNbO_{3} structure (LN-structure) with space group R3c. These crystals are noncentrosymmetric, with a large polarization. LN-ZnSnO_{3}, synthesized under a pressure of 7 GPa [29], has a polarization given by 59 *μ*C/cm^{2}. Other LN-type polar oxides synthesized under high pressure include CdPbO_{3} [30], PbNiO_{3} [30, 31], GaFeO_{3} [32], and LiOsO_{3} [33]. Ilmenite ZnGeO_{3} transforms to an orthorhombic perovskite phase at 30 GPa, and upon releasing the pressure, to the LN-structure [34, 35], with a polarization of about 60 *μ*C/cm^{2} [36, 37]. Several other LN-type oxides, such as MnTiO_{3} [38–41], MnSnO_{3} [42, 43], FeTiO_{3} [39, 43–45], FeGeO_{3} [46], MgGeO_{3} [35, 39, 47], and CuTaO_{3} [48], were similarly obtained during decompression from the perovskite-type phase, which is stable at high pressure. More recently, LN-type ZnTiO_{3}, with a large polarization of 88 *μ*C/cm^{2}, has been synthesized in this fashion [49].

Polar oxide semiconductors of the LN-type have wide band gaps. For example, ZnSnO_{3} has a band gap of 3.3–3.7 eV [50, 51]; such values are typical for this class of materials. However, it has been noted that the band gap in ZnSnO_{3} is very sensitive to variations in the lattice constants, and suggestions have been made to tune the band gap by growing ZnSnO_{3} films on a substrate with some degree of lattice mismatch [52], by substituting sulfur for oxygen [53], or by substitutional doping with calcium or barium [54].

In this work, we carry out first-principles calculations, using density functional theory (DFT), on three LN-type crystals with large remnant polarization, namely, ZnSnO_{3}, ZnGeO_{3}, and ZnTiO_{3}. We show that, upon using the modified Becke-Johnson (mBJ) exchange potential [55], the correct sizes of the band gaps are calculated for these crystals. Having established the validity of this computational method in producing the correct band gaps for this family of compounds, we then focus on ZnSnO_{3} and study the effect of substituting sulfur for oxygen. Previous work has already validated the substitution of sulfur for oxygen and vice versa as a way of tailoring band gaps in other materials [56–58]. Since the structure of the resulting compound, ZnSnS_{3}, is unknown, we carry out extensive calculations using evolutionary algorithms in order to determine its structure. We find that the most stable form of ZnSnS_{3} has a monoclinic structure, followed closely by the ilmenite and lithium niobate (LN) forms. The monoclinic and ilmenite phases are not polar, but the LN-phase is ferroelectric, with significant polarization and a small band gap of 1.28 eV.

#### 2. Methods

To predict the stable and metastable structures of ZnSnS_{3}, two different evolutionary methods, implemented in the codes USPEX and CALYPSO, were employed. The USPEX (Universal Structure Predictor: Evolutionary Xtallography) software [59–61], developed by Oganov, Glass, Lyakhov, and Zhu, features local optimization, real-space representation, and variation operators that mimic natural evolution. The CALYPSO (Crystal Structure Analysis by Particle Swarm Optimization) software [62], developed by Wang et al., uses local structural optimization and the particle swarm optimization (PSO) method to update structures.

The first step, in both methods, is to generate a population of random crystal structures, each with a symmetry described by a randomly chosen space group. Once a space group is selected, appropriate lattice vectors and atomic positions are generated. Each generated structure is optimized using density functional theory, and its free energy (known as the fitness function) is calculated. The structure optimization is carried out using the VASP [63–65] software, which uses a plane wave basis for expanding the electronic wave function. Each structure is optimized in four steps, beginning with a coarse optimization, followed by increasingly accurate iterations. In the last optimization step, the kinetic energy cutoff for plane wave expansion of the wave function is 600 eV. The optimized structures of the initially generated population constitute the first generation, with each member being called an individual. A new generation is then produced, with some of its members being generated randomly while others are obtained from the best structures (those with lowest energy) of the previous generation. In USPEX, new individuals (offspring) are produced from parent structures by applying variation operators such as heredity, mutation, or permutation. In the PSO method, a new structure is generated from a previous one by updating the atomic positions using an evolutionary algorithm. The structures in the new generation are optimized, and the best among them serve as precursors for structures in the next generation. The process continues until convergence to the best structures is achieved.

Band gaps, band structures, and densities of states were calculated using the all-electron, full-potential, linearized, and augmented plane wave method as implemented in the WIEN2K [66] software suite. Here, space is divided into two regions. One region comprises the interior of nonoverlapping muffin-tin spheres centered on the atomic sites, while the rest of the space (the interstitial) forms the other region. The electronic wave function is expanded in terms of a set of basis functions which take different forms in each distinct region. Inside the spheres, the basis functions are atomic-like functions written as an expansion in spherical harmonics up to . In the interstitial, they are plane waves with a maximum wave vector of magnitude . Each plane wave is augmented by one atomic-like function in each muffin-tin sphere. was chosen so that , where is the radius of the smallest muffin-tin sphere in the unit cell. Charge density was Fourier-expanded up to a maximum wave vector of 14 , where is the Bohr radius. Modified Becke-Johnson exchange potential [55], which is known to yield reasonably accurate band gap values in semiconductors, was adopted in our band structure calculations. In the self-consistent field calculations, the total energy and charge were converged to within 0.1 mRy and 0.001 e, respectively. For total energy calculations, a Monkhorst-Pack [67] 8 × 8 × 8 grid of* k*-points in the Brillouin zone is used.

The phase transition mechanism between stable and metastable structures of ZnSnS_{3} is characterized using a combination of the variable-cell nudged elastic band method (VCNEB) [68] and the Gibbs construction [69]. VCNEB is an extension of the nudged elastic band (NEB) technique, a widely used method for determining possible reaction paths and saddle points along the minimum energy path (MEP) between endpoints on a potential surface. NEB determines this pathway by using a user defined initial and final structures to generate a set of intermediate structures or “images.” These images are then iteratively adjusted in order to minimize the increase in energy along the transition pathway. The traditional NEB algorithm is constrained to maintain a constant unit cell throughout its optimization. VCNEB expands this technique by incorporating lattice parameters into the configuration space, thereby making it suitable for the analysis of phase transition pathways. In particular, it is appropriate for estimating the energy barrier between phases at constant pressure.

Likewise, the Gibbs construction is useful for estimating the pressure at which a specific phase transition will occur. The construction is determined by first plotting the unit cell energy versus volume of a material’s stable structures. A tangent line is drawn between the resulting curves. The slope of this line corresponds to the transition pressure between phases of the given material.

We use the VCNEB implementation included with USPEX in combination with VASP to estimate the energy barrier between the stable and metastable phases of ZnSnS_{3}. For the Gibbs construction, we use a combination of VASP and various Python software packages to plot the energies of the scaled ZnSnS_{3} structures.

#### 3. Results and Discussion

As a rough guide to predicting the stable structure of a compound of the form* ABX*_{3}, one usually calculates the Goldschmidt tolerance factor given bywhere , , and are the radii of the* A*,* B*, and* X* ions, respectively. For Zn, Sn, O, and S, the ionic radii are given, respectively, by 0.74 Å, 0.69 Å, 1.4 Å, and 1.84 Å. Using these values, we find that for ZnSnO_{3}, , while for ZnSnS_{3}, . One is thus tempted to conclude that the replacement of oxygen with sulfur should not cause a change in structure. However, the large polarizability of sulfur indicates that the Goldschmidt tolerance factor may not be appropriate for predicting the structure of sulfur-containing compounds [70].

To resolve this issue, we carried out extensive calculations using the evolutionary algorithm implemented in USPEX and the PSO method used by CALYPSO. For the USPEX calculation, 20 generations were produced, with each generation containing 40 individuals. With the PSO method, we produced 20 generations, each consisting of 50 individuals. In all cases, the unit cell was assumed to contain two formula units. The most stable structures were identified. To further refine our results, sets of 80 random structures were subsequently generated for each of the most stable space groups. Out of each set, the single structure with the lowest energy was selected, thereby determining the optimal structures and symmetries of ZnSnS_{3}.

In Table 1 we list these ZnSnS_{3} structures by order of stability. To more easily compare the relative stability of various structures, the free energy per atom in the most stable structure is set equal to zero. That structure has a monoclinic unit cell with space group P2_{1} (number 4). The metastable structure with lowest energy is the ilmenite, followed by the LN-type structure. By contrast, in ZnSnO_{3}, the LN-phase is the most stable structure.

To evaluate the usefulness of ZnSnS_{3} for solar cell applications, we have calculated the band structures of the three most stable phases of this compound. Since low band gap values for light absorbers are essential in photovoltaic applications, it is important that the computational method yield accurate values for these gaps. Density functional theory, when using the local density approximation (LDA) or generalized gradient approximation (GGA) to approximate the exchange-correlation term, severely underestimates band gaps in semiconductors. The use of the modified Becke-Johnson (mBJ) exchange potential results in a significant improvement in calculated band gap values, comparable to that produced by the much more expensive GW approximation. For example, consider the case of ZnSnO_{3}. It has a measured band gap in the range 3.3–3.7 eV [50, 51]. Our calculations, using GGA, give a band gap of 1.5 eV. On the other hand, using the mBJ exchange potential, we calculate a band gap of 3.4 eV, which is in excellent agreement with experiment. We have also calculated the band gaps in ZnGeO_{3} and ZnTiO_{3} using the mBJ potential and found them to be 3.57 eV and 3.51 eV, respectively. Though experimental values are not available for the band gaps of these two compounds, the calculated values are typical of those encountered in ferroelectric oxides.

In Figure 1 we present the calculated energy bands and density of states of the monoclinic phase of ZnSnS_{3} with space group P2_{1}, using the mBJ exchange potential. The band gap is 1.90 eV. Similar calculations on the ilmenite phase of ZnSnS_{3}, with the space group R-3, feature a band gap of 2.30 eV. These two phases of ZnSnS_{3} are not very useful as light harvesters in solar cells for two reasons: () they do not have small band gaps and (2) they are not polar.

The situation is markedly different for the LN-phase of ZnSnS_{3} with space group R3c (number 161). This phase has a low band gap and a large polarization. In Figure 2 we show the calculated band structure, using the mBJ potential, along with the density of states. Here the band gap is 1.28 eV, in agreement with a previous calculation [53] that used the GW approximation. The upper valence bands are derived mainly from S p-orbitals, while the lower conduction bands result mainly from Sn s-orbitals and S p-orbitals. The polarization, calculated using the Berry phase formalism of the modern theory of crystal polarization [71, 72], is found to be 57 *μ*C/cm^{2}, which is essentially the same as in the ferroelectric ZnSnO_{3}. About one-third of the polarization is electronic; the rest is ionic. The large ionic polarization derives from the particular crystal structure of this phase of ZnSnS_{3}. Though the Sn ion is octahedrally bonded to six S ions, it does not sit at the center of the octahedron; rather, three of the six Sn-S bonds have a length of 2.42 Å while the other three have a length of 2.57 Å. Hybridization between the Sn s-orbitals and S p-orbitals causes a displacement of the Sn ion from the center of the octahedron, thereby lowering the energy of the system.

As indicated earlier, the monoclinic structure is the most stable phase of ZnSnS_{3}, while the LN-phase is metastable. We now consider the possible transition between these two phases. A variable-cell nudged elastic band (VCNEB) calculation is carried out in order to map out the transition path connecting the two phases. The minimum energy path between the monoclinic and LN-type structures is shown in Figure 3. From the figure we see that an energy barrier of 4.52 eV (0.452 eV/atom) separates the phases. The lattice parameters of the path’s terminal structures and two intermediate metastable structures are presented in Table 2. The high energy barrier present at zero temperature and pressure suggests that, while not the lowest energy configuration of ZnSnS_{3}, it is highly unlikely that the LN-phase, once established, would revert to the monoclinic phase at room temperature.

By using the Gibbs construction, discussed earlier in Section 2, we can make a rough estimate of the pressure which, at zero temperature, will induce a structural phase transition in ZnSnS_{3} from the monoclinic phase to the LN-phase. This is illustrated in Figure 4, which indicates that the phase transition takes place at a pressure of 16 GPa.

Finally, we examine the formation and stability of ZnSnS_{3} by comparing its free energy to that of some potential precursors at zero temperature. We note that ZnSnO_{3} is formed by a solid-state reaction of ZnO and SnO_{2} at high temperature and pressure. At zero temperature and zero pressure, our calculations indicate that the free energies of ZnO, SnO_{2}, and ZnSnO_{3} are, respectively, −5.29 eV/atom, −7.09 eV/atom, and −6.35 eV/atom. The average free energy per atom of ZnSnO_{3} is thus higher than that of a mixture of ZnO and SnO_{2} by 0.02 eV. However, at zero temperature and a pressure of 7 GPa, the per-atom energy of ZnSnO_{3} drops below that of its precursor mixture by 0.005 eV. This helps explain how it is possible to form stable ZnSnO_{3} from compounds with apparently higher stability. Furthermore, when ZnSnO_{3} is returned to ambient conditions, it remains stable, rather than decomposing into smaller, lower-energy compounds.

We apply the same reasoning to ZnSnS_{3}, assuming that it may be formed at high temperature by applying pressure to a mixture of ZnS and SnS_{2}. At zero temperature and pressure, the free energies of ZnS and SnS_{2} are −4.10 eV/atom and −5.00 eV/atom, respectively, yielding a combined per-atom energy of −4.64 eV. For ZnSnS_{3}, on the other hand, the calculated energy is −4.60 eV/atom, which is higher than that of its precursors by 0.04 eV/atom. However, at zero temperature and a pressure of 18 GPa, our calculations indicate that the free energy per atom of ZnSnS_{3} is lower than the combined per-atom free energy of ZnS and SnS_{2} by 0.004 eV. This indicates that it is possible for ZnSnS_{3} to form, albeit at relatively high pressure. Importantly, these energies were calculated at zero temperature and, in practice, the addition of heat energy may further lower the pressure at which ZnSnS_{3} forms. The analogous example of ZnSnO_{3}, along with the high energy barriers between phases of ZnSnS_{3}, leads us to believe that, once formed, ZnSnS_{3} would be stable at ambient conditions.

#### 4. Conclusions

As anticipated, our calculations indicate that the substitution of sulfur for oxygen in ZnSnO_{3} reduces its band gap, particularly for the LN-phase. This reduction is sufficient to bring the absorption band of ZnSnS_{3} into the visible to mid-infrared spectrum, thereby making it suitable for solar cell applications. An unexpected result of the substitution is that a monoclinic phase becomes the most stable structure of ZnSnS_{3}. It is likely that a suitable application of temperature and pressure may transition ZnSnS_{3} to its LN-phase. VCNEB calculations for ZnSnS_{3} suggest that it will not revert from its LN-phase (once attained) to its monoclinic phase at room temperature. Furthermore, the Gibbs construction specifies the transition pressure between phases to be roughly 16 GPa, a pressure that is not difficult to attain in the lab but is sufficiently high to allow for a persistent metastable structure under typical atmospheric conditions.

Further work is needed to characterize the precise conditions under which transition to the polar LN-phase occurs. The application of transition path sampling (TPS) to further evaluate the transition path produced by VCNEB may help to better identify the most likely pathway between stable and metastable states in ZnSnS_{3}. Finally, the effect of oxygen to sulfur substitution in other oxides such as ZnGeO_{3} and ZnTiO_{3} should be evaluated.

#### Competing Interests

The authors report no conflict of interests in this research.

#### Acknowledgments

The authors gratefully acknowledge support by the National Science Foundation under Grant no. HRD-1547723 and the NSF PREM Program: Cal State LA & Penn State Partnership for Materials Research and Education, Award DMR-1523588. The authors also acknowledge partial support by the Materials Simulation Center and the Material Research Institute at Penn-State University.