First–Principles Investigation of the Structural, Elastic, Electronic, and Optical Properties of α– and β–SrZrS3: Implications for Photovoltaic Applications

Transition metal perovskite chalcogenides are attractive solar absorber materials for renewable energy applications. Herein, we present the first–principles screened hybrid density functional theory analyses of the structural, elastic, electronic and optical properties of the two structure modifications of strontium zirconium sulfide (needle–like α–SrZrS3 and distorted β–SrZrS3 phases). Through the analysis of the predicted electronic structures, we show that both α– and β–SrZrS3 materials are direct band gaps absorbers, with calculated band gaps of 1.38, and 1.95 eV, respectively, in close agreement with estimates from diffuse–reflectance measurements. A strong light absorption in the visible region is predicted for the α– and β–SrZrS3, as reflected in their high optical absorbance (in the order of 105 cm−1), with the β–SrZrS3 phase showing stronger absorption than the α–SrZrS3 phase. We also report the first theoretical prediction of effective masses of photo-generated charge carriers in α– and β–SrZrS3 materials. Predicted small effective masses of holes and electrons at the valence, and conduction bands, respectively, point to high mobility (high conductivity) and low recombination rate of photo-generated charge carriers in α– and β–SrZrS3 materials, which are necessary for efficient photovoltaic conversion.


Introduction
Perovskite materials have attracted significant attention from researchers due to their potential in various applications. In photovoltaic applications, inorganic-organic halide perovskite materials represent a great breakthrough in the development of solar cell materials with the power conversion efficiency rising from 3.6 to about 24%, since the first application in 2009 by Kojima et al. [1]. The recently reported efficiencies are comparable to the most advanced thin film solar cells, such as CdTe, GaAs, as well as the silicon-based solar cells. In addition to this unprecedented improvement in power conversion efficiency are the ease and low-cost of synthesis of the materials, as well as the fabrication of the solar cell device. Notwithstanding the rapid increase in their efficiencies, the toxicity of the lead content and the intrinsic instability of the bulk lead-halide perovskite materials and their interface heterostructures remain significant drawbacks to their large-scale applications and commercialization [2][3][4][5]. These concerns have motivated the search for new absorber materials, that are lead-free but with similar electronic and optical properties as lead-halide perovskite materials. Efforts to replace the Pb in lead halide perovskites with low toxic cations including Sn (II) [6], Ag (I) [7],

Computational Details
The density functional theory (DFT) were performed within periodic boundary conditions as implemented in the Vienna Ab initio Simulation Package (VASP) [21,22]. The Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) functional [23] was used for geometry optimizations, while for electronic structures and optical calculations, the screened hybrid functional HSE06 with 25% Hartree−Fock exchange and screening parameter of ω = 0.11 bohr −1 was employed [24]. The projected augmented wave (PAW) method was used to describe the interactions between the valence electrons and the cores [25]. Long-range van der Waals (vdW) were accounted for using of the Grimme (DFT-D3) method [26]. A plane-wave basis, set with a kinetic energy cut-off of 600 eV, was tested to be sufficient to converge the total energy of the α-SrZrS 3 and β-SrZrS 3 phases to within 10 −6 eV and the residual Hellmann-Feynman forces on all relaxed atoms reached 10 −3 eV Å −1 . The Brillouin zone of the α-SrZrS 3 and β-SrZrS 3 phases was sampled using 5 × 7 × 3, and 5 × 3 × 5 Monkhorst-Pack [27] K-points mesh, respectively.
The optical properties of α-SrZrS 3 and β-SrZrS 3 were determined from the complex dielectric function, ε(ω) = ε 1 (ω) + i ε 2 (ω) within the independent-particle formalism [28], where the the imaginary part of the dielectric function is calculated in the long wavelength q→0 limit as, where, Ω and w k are the volume of the primitive cell and k-point weights, respectively. The ck ( vk ) are k-dependent conduction (valence) band energies, u vk , u ck are the cell periodic part of the pseudo-wave function and e α,β are the unit vectors along the Cartesian directions. From Kramers-Kronig transformations, the real part of dielectric function can be determined from the relation, where P denotes the principle value. A small value of 0.1, which is acceptable for most calculations was used for the complex shift (η) to smoothen the real part of the dielectric function. The optical parameters, such as absorption coefficient, are based on ε 1 and ε 2 . The absorption coefficient (α abs ) is calculated using the following relation: Local field and excitonic effects have been neglected in the present study as they are not accurately treated in the independent-particle formalism. These effects may be accounted for by using expensive methods, such as Bethe-Salpeter equation (BSE) and time-dependent DFT with proper exchange-correlation kernels [29].
The elastic stiffness constants were calculated using the stress-strain method [30]. The strain (σ) and stress (ε) are related according to Hook's law by σ i = C ij ε j , where C ij are the elastic stiffness constants. For an orthorhombic system, there are nine independent elastic constants viz; C11, C22, C33, C12, C13, C23, C44, C55 and C66. These elastic constants are used to predict the bulk modulus (B V ) and the shear modulus (G V ). Whereas, Young's modulus (E) and the Poisson's ratio (v) are in turn predicted from the calculated bulk and shear moduli. The bulk and shear moduli measure the material's resistance to uniform compression, and shearing strains, respectively. The bulk and shear moduli were calculated using the Voigt approximation [31] which for orthorhombic structures, can be simplified as follows: The Young's modulus (E) and Poisson's ratio (v), which are characteristic stiffness properties of a material were obtained using the relations: E = 9BG (3B+G) and ν = (3B−2G) 2(3B+G) .

Structural Properties
SrZrS 3 crystallizes in two structure modifications; α and β-phases, both in the orthorhombic crystal system with space group Pnma ( Figure 1). The α-SrZrS 3 has five independent lattice sites; one Sr, one Zr, and three S sites, whereas in the β-phase, there exist only four positions (one Sr, one Zr, and two S sites) [32]. The α-SrZrS 3 structure consists of one-dimensional double chains of edge-sharing ZrS 6 octahedra along the b axis, with Sr atoms in nine-fold coordination with S atoms, forming tri-capped trigonal prisms ( Figure 1a) that are interconnected via common triangular faces [32]. Based on powder neutron diffraction analysis, the β-SrZrS 3 is said to have a distorted perovskite structure [33]. Unlike the α-phase, the structure of β-SrZrS 3 is constituted by three-dimensionally connected corner-sharing Zr octahedra, and Sr atoms which are eightfold coordinated in a bi-capped trigonal prism ( Figure 1b). Summarized in Table 1 are the calculated lattice parameters obtained from a full structure optimization at the using PBE and HSE06 functionals, which show good agreement with known experimental data. 32 In the α-phase, each of the ZrS 6 octahedra shares two opposite edges with one another, forming a linear chain which in turn, interconnect in pairs through additional edge-sharing. The Zr-S bonds form the shortest, having a length of 2.45 Å, whereas the longest bond having a length of 2.65 Å, which involves an S atom bonding to three Zr atoms, occurs opposite to the shortest bond. The Sr-S bonds in the α-SrZrS 3 phase ranges from 3.06-3.09 Å. Conversely, the ZrS 6 octahedra of the β-phase have all their corners interconnected forming a three-dimensional network, with Zr-S bonds ranging from 2.54 to 2.57 Å, in close agreement with experimental values (Table 1). Based on the optimized structures, we have simulated X-ray diffraction (XRD) spectrum of α-SrZrS 3 and β-SrZrS 3 using the VESTA Crystallographic Software as shown in Figure 2a. We can see that clear differences between the assigned peaks of α-SrZrS 3 and β-SrZrS 3 , which is consistent with the difference in their lattice parameters. All the peaks in the simulated DFT spectrum match very closely with the experimental XRD measurement from the work of Niu et al. [34], as shown ( Figure 2b). We consider that the assigned reflection peaks in the DFT XRD spectrum may become useful in clarifying future experiments, for instance to distinguish between the α-SrZrS 3 and β-SrZrS 3 phases.

Mechanical Properties
The elastic properties of materials give the data necessary in understanding the bonding property between adjacent atomic planes, stiffness, bonding anisotropic and structural stability of the material [35,36]. Shown in Table 2 are the calculated single crystal elastic constants of the αand β-SrZrS 3 materials, all of which satisfy the Born's mechanical stability criteria for orthorhombic structures [37], indicating that both materials are mechanically stable under ambient conditions. The predicted trend of C 11 > C 22 > C 33 for the α-SrZrS 3 phase, indicates greater stiffness in the [100] direction than in the [010] and [001] directions. For the β-SrZrS 3 phase, C 22 > C 33 > C 11 which implies that β-SrZrS 3 is stiffest in the [010] direction and least stiff in the [100] direction. The high elastic stiffness of the αand β-SrZrS 3 could be attributed to strong Sr-S and Zr-S chemical bonding [35]. Summarized in Table 2 are the calculated Bulk (B), shear (G) and Young's (E) moduli. A higher bulk modulus is predicted for the β-SrZrS 3 phase (79.9 GPa), similar to the value of 77.35 predicted by Oumertem et al. 15 , compared to the α-SrZrS 3 (61.7 GPa) phase. Similar bulk modulus values were obtained by fitting a third-order Birch-Murnaghan (3 rd BM) equation of state (EOS) [38,39] to the DFT-PBE obtained total electronic energy (E) vs unit cell volume (V) data ( Figure 3) based on the equation, where E 0 and V 0 are the equilibrium energy and volume, B 0 is hydrostatic bulk modulus and B' is the pressure derivative of the bulk modulus at T = 0 K and P = 0 GPa. The bulk modulus and its pressure derivative are calculated at 66.73 GPa and 2.33 for α-SrZrS 3 and 83.75 GPa and 2.93 for β-SrZrS 3 , respectively. This result suggests that the β-SrZrS 3 phase is more rigid and less prone to compressive deformation than the α-SrZrS 3 phase. Conversely, the α-phase has higher shear and Young's moduli than the β-phase implying more resistance to shear and tensile deformation. The Poisson's ratios (v) are predicted at 0.244 and 0.436 for the α, and β-phases, respectively. The Poisson's ratio allows us to test the ductility/brittleness of material. A material is characterized as ductile if v < 0.5, otherwise it is classified as brittle [40]. Based on this criterion, we conclude that both αand β-SrZrS 3 are ductile materials, which implies that any strain-induced defects at the interface, when these materials are deposited on substrates may relax over a relatively short distance.

Electronic Properties
Shown in Figure 4a,b are the electronic band structures of the α-SrZrS 3 and β-SrZrS 3 phases, respectively, with the corresponding density of states projected on the Sr-d, Zr-d and S-p states in Figure 5a,b. Analysis of the band structures reveal that both αand β-SrZrS 3 are direct band gaps absorbers seeing the bottom of the conduction band and the top of the valence band are located at the same crystal momentum (Γ) points on the Brillouin zone. The band gap energies of αand β-SrZrS 3 are predicted at 1.38, and 1.95 eV, respectively. The predicted values are in close agreement with the estimated optical band gaps from diffuse-reflectance measurements: α-SrZrS 3 (1.52 eV) and β-SrZrS 3 (2.05 eV) [16]. It is evident from the projected density of states (Figure 5a,b) that the conduction band edge, in both phases, is dominated by Zr-d states. Whereas, the contribution from S-p orbital dominates the valence band edge.  The charge carrier effective masses (m*) which is intricately linked to the diffusion coefficient (D) and mobility (µ) of charge carrier in a semiconductor via the relations D = k B T e µ and µ = e τ m * , were also calculated for the αand β-SrZrS 3 materials. Small photo-carrier effective masses result in increased µ and D. The conductivity of charge carrier effective mass dictates the electrical resistivity and optical response of solar devices; hence their calculation is important [41]. Whiles effective masses can be quite difficult to obtain experimentally, accurate DFT calculations [42,43] can complement experiments by computing these properties. The effective masses can be obtained by fitting the energy of the valence band maximum (holes) and conduction band minimum (electrons) to a quadratic polynomial in the reciprocal lattice vector k based on the equation In Table 3 the calculated electron and hole effective masses for the αand β-SrZrS 3 materials are summarized in some selective directions of the Brillouin zone. The predicted smaller, and therefore, the light mass charge carriers correspond to a high mobility of the electrons and holes at the conduction, and valence bands, respectively, and consequently point to high conductivity. The high conductivity also demonstrates the efficient separation of photo-generated charge carriers, which give rise to high-efficiency fabricated solar cell devices. In general, we found that the holes have higher effective masses than electrons in both the αand β-SrZrS 3 materials, suggesting that the electrons do tunnel much readily than the holes. The large effective mass difference between the electrons and holes is an important factor in minimizing their recombination rate [44]. By computing the ratio of the hole to electron effective masses (D = m * h /m * e ), the recombination rate of photo-generated charge carriers can be assessed [45]. Higher D values generally signify higher mobility and a lower recombination rate of the photo-generated charges [46,47]. As shown in Table 3

Optical Properties
The calculated real (dispersive, ε 1 ) and imaginary (absorptive, ε 2 ) parts of the dielectric function for α-SrZrS 3 and β-SrZrS 3 are shown in Figures 6 and 7, respectively. The dielectric constant is predicted at 9.36 for α-SrZrS 3 (Figure 6a) and 15.75 β-SrZrS 3 (Figure 7a). The predicted high dielectric constants are desired properties for photovoltaics applications as a dielectric constant value of 10 or more is good enough to obtain exciton binding energy (E b ) value lower than 25 meV at room temperature [48]. The absorbance of α-SrZrS 3 (Figure 6b) starts at around 1.5 eV, close to its band gap, but with negligible absorption until after 2 eV. The absorption of β-SrZrS 3 (Figure 7b) starts at around 2.0 eV, which corresponds to it fundamental band gap. Owing the orthorhombic crystal symmetry of αand β-SrZrS 3 , the absorption coefficient, reflectivity and refractive index plotted along the three the crystallographic directions: x, y, and z directions ([100], [010], and [001]) are found to be anisotropic. The absorption coefficient of a material represents its light harvesting ability, which is very necessary as it has great effects on the power conversion efficiency of the resulting solar cells. The high absorption coefficient is desired in solar cell applications. The calculated absorption coefficient, reflectivity and refractive index are shown in Figures 6 and 7. A high absorption coefficient in the order of 10 5 cm -1 is predicted for both αand β-SrZrS 3 in the visible light region, which make them suitable for the photovoltaic application. The β-SrZrS 3 phase shows stronger absorption around 2 eV than α-SrZrS 3 phase. The reflectivity and refractive index are two important parameters necessary for solar applications. The reflectivity gives a measure of reflecting light or radiation. When it comes to solar cells, the less the material surfaces reflect a sun's rays, the more energy can be generated. The refractive index of a material, on the other hand, shows its transparency. The optical reflectivity of α-SrZrS 3 (Figure 6c) shows a high reflectivity in the y-direction, starts at about 30.2% and reaches a maximum value of about 68.2% at an energy of about 3.8 eV. The static reflectivity in the x, y, and z directions are predicted at about 23%, 30%, and 26%, respectively. This indicate that the least reflection occurs in the z direction. The calculated static reflectivity of β-SrZrS 3 (Figure 7c) is predicted to be about 30%, 34% and 38% in the y, z, and x directions, respectively. The reflectivity reaches a maximum (82.2%) at an energy of about 3.5 eV. The high reflectivity in the visible region could cause a significant loss in solar cell efficiency of devices fabricated. Optical losses from reflected light from the front surface can be reduces through surface texturing and anti-reflection coatings. The refractive index is predicted in the range of 2.8-3.5 for α-SrZrS 3 (Figure 6d) and 3.5-4.0 for β-SrZrS 3 (Figure 7d). The calculated refractive indexes for αand β-SrZrS 3 are similar to the refractive index of Si (η = 3.4 at 550 nm) [49].

Summary and Conclusions
We performed comprehensive first-principles GGA and hybrid DFT investigations the structures and properties α-SrZrS 3 and β-SrZrS 3 transition metal chalcogenide perovskites. Both αand β-SrZrS 3 materials were demonstrated to be mechanically stable at ambient conditions, based on their calculated single-crystal elastic constants. The predicted electronic structures show that both αand β-SrZrS 3 are direct band gaps absorbers with band gap of calculated at 1.38, and 1.95 eV, respectively, in close agreement with estimates from diffuse-reflectance measurements. Based on the dielectric functions obtained, we show that αand β-SrZrS 3 have strong light absorption in the visible region, as reflected in their high optical absorbance (in the order 10 5 cm −1 ). The β-SrZrS 3 shows a stronger absorption around 2 eV than α-SrZrS 3 . The first theoretical prediction of effective masses of photo-generated charge carriers, in αand β-SrZrS 3 materials, suggest s high-mobility (high-conductivity) and low recombination rate of photo-carriers in these materials, making them attractive for solar cell and other optoelectronic applications.
Author Contributions: H.I.E. performed the DFT simulations and data analysis, and wrote the paper. E.N. and N.Y.D. contributed to the study design and scientific discussion of the results. All co-authors contributed to the manuscript. All authors have read and agreed to the published version of the manuscript. Acknowledgments: H.I.E. acknowledges African University of Science and Technology (AUST) for studentship. This work also made use of the facilities of ARCHER (http://www.archer.ac.uk), the UK's national supercomputing service via our membership of the UK's HEC Materials Chemistry Consortium, which is funded by EPSRC (EP/L000202). We also acknowledge the used of computational facilities of the Advanced Research Computing at Cardiff (ARCCA) Division, Cardiff University, and HPC Wales. Information on the data that underpins the results presented here, including how to access them, can be found in the Cardiff University data catalogue at http://doi.org/10.17035/d.2020.0099250231.

Conflicts of Interest:
The authors declare no conflict of interest.