Insight into the Physical Properties of the Chalcogenide XZrS₃ (X = Ca, Ba) Perovskites: A First-Principles Computation

Abstract

This study investigates the structural, mechanical, optical, thermal, and electronic properties of the ionic semiconducting materials XZrS₃ (X = Ca, Ba) within the framework of density functional theory (DFT). Here, the elastic constants, modulus (bulk, shear, Young's), ratios (Pugh, Poisson) and elastic anisotropy for XZrS₃ (X = Ca, Ba) are studied. Furthermore, the electronic, optical, and thermal properties for XZrS₃ (X = Ca, Ba) are regenerated and designed using the values obtained with Cambridge Serial Total Energy Package (CASTEP) software. The calculated lattice parameters show excellent agreement with theoretical and experimental values. The elastic stiffness constants confirm the mechanical stability of both compounds. Although XZrS₃ (X = Ca, Ba) is elastically anisotropic, it has little optical anisotropy. The electronic band structures of the material exhibit direct-bandgap semiconducting behavior, with values of 1.3 eV (CaZrS₃) and 1.1 eV (BaZrS₃) using the generalized gradient approximation (GGA), respectively, which is ideal for solar cell (0.9–1.56 eV) and optoelectronic device applications. Bandgap values of 1.9 eV and 1.6 eV are found for CaZrS₃ and BaZrS₃, respectively, using the Heyd–Scuseria–Ernzerhof HSE06 functional, which is consistent with previous theoretical and experimental bandgap results. The optical properties including dielectric function, refractive index, absorption coefficient, reflectivity, and loss function are characterized using the GGA of Perdew–Burke–Ernzerhof (GGA-PBE) and HSE06 methods and are discussed in detail. Because of the relatively low Debye temperature (D), thermal conductivity of the lattice (kph), and minimum thermal conductivity (K_min), the studied materials can be used as thermal barrier coating (TBC) materials. The capacity of heat, Debye temperature, and thermal coefficient of expansion are all computed.

Introduction

Perovskites of diverse composition and architectures have drawn considerable attention due to their potential application in various industrial and technical domains. Applications include mixed conducting oxides for gas separation and metallicity–insulation, and use as high-temperature electrodes, cathodes, photo-electrodes, and photocatalysts. They also play essential roles in other fields as photodetectors, photovoltaics, solid electrolytes, hydrogen sensors, piezoelectric transducers, thermostat actuators, dielectric resonators, ferromagnetic materials, semiconductors, lasers, superconductors, magnetic memory devices, electro-optical modulators, and ferroelectrics.^1,2,3,4 Chalcogenide perovskite materials (polycrystalline) such as CaZrS₃, SrZrS₃, and BaZrS₃ have garnered less attention than their oxide and halide counterparts; as a result, only a limited amount of information about their physical features has been published. These materials are very efficient in solar energy conversion.^5,6,7,8 Due to their highly ionic nature, perovskites have provided a unique technique for generating semiconducting characteristics for energy harvesting, lighting (solid-state), and sensing. For example, Sun et al.⁹ examined the dielectric constants and the electronic structure of CaZrS₃ and BaZrS₃ compounds in three distinct phases with first-principles calculations. The perovskite materials of organic–inorganic halides demonstrated efficiency of 22.7% as of 2018.¹⁰

Perovskites of organic–inorganic halides have recently been the subject of photovoltaics research. The temperature (T), pressure (P), and dependence of the optical phonons in this chalcogenide perovskite are explored using density functional theory (DFT) and Raman spectroscopy.^{11,12,13,14,15} Previous Raman studies on oxide perovskites¹⁶ established that the impact of temperature on vibrational methods is critical to understanding the polarization states in perovskites, structural stability and bonding, and the processes that cause these traits to vary. In 2015, Sun et al. reported that the optical and electrical features of deformed perovskite structures such as CaHfSe_3, CaTiS₃, CaZrSe₃, and BaZrS₃ are appropriate for single-junction solar cells.^17,18 However, while BaZrS₃ has been studied in the lab, the other compositions with the requisite structure and attributes have yet to be explored.¹⁹ Meng et al. used computational analysis to show that compositions BaZr_1−xTixS₃ and BaZrSe₃ have a bandgap with a theoretic maximal power transfer efficiency of 30%; however, the composite has yet to be practically produced.²⁰ SrSnSe₃, SrSnS_3, as well as other XSnX₃ (X = S, Se) materials with a deformed perovskite structure, such as SrSnSe₃, SrSnS₃, and related admixtures, were evaluated by Ju et al. as possible components for solar cells.⁸

In this work, elastic constants, modulus (bulk, shear, Young's), ratios (Pugh, Poisson), elastic anisotropy, and Mulliken population for XZrS₃ (X = Ca, Ba) are investigated for the first time. Furthermore, the electronic, optical, and thermal properties for XZrS₃ (X = Ca, Ba) are regenerated and designed differently using the data obtained from CASTEP code. These three properties provided more updated values. A study on CaZrO_3−xS_x (x = 0, 1, 2, and 3) was conducted to explore the structural, electronic, and optical properties of CaZrO₃²¹ The authors used ABINIT software to calculate the properties. In this research, CASTEP software was used to determine the structural, electronic, and optical properties of CaZrS₃, and BaZrS₃.²¹

Chalcogenide perovskites, containing group IVB and group IIA elements whose constituent components are Sr, Ba, Ca and Hf, Zr, Ti, respectively, have emerged as promising alternatives to the hazardous lead-based halide perovskites, with more ecologically benign and moisture-stable properties, along with their enhanced bandgap, absorption coefficient, and stability. The study proposes to compare the characteristics of halide and chalcogenide perovskites using first-principles calculations, inspired by previous work on these materials. Chalcogenide perovskites have recently been regarded as new semiconductor materials with the general pattern ABX₃, where A = Mg, Ca, Sr, Ba, B = Ti, Zr, Hf, and X = S or Se. Perovskites have a higher ionic content than typical semiconductors but lower than oxides or halides. These perovskites have been produced for over half a century but have received little attention,²² so their physical characteristics are unknown. This situation will change after the theoretical research is evaluated.²³

The investigation revealed many materials with high bandgap, significant absorption, and excellent carrier mobility, enabling them to be examined as promising optoelectronic materials. Different chalcogenide perovskites, including BaZrS₃, SrTiS₃,²³ and SrHfS₃,^19,24 have been synthesized as a result of experimental studies. XZrS₃ (X = Ca, Ba) is a chalcogenide perovskite with high light absorption that solidifies in an orthorhombic phase with the Pnma space group.²⁵ This work uses density functional theory (DFT) to investigate the unknown characteristics of the CaZrS₃ and BaZrS₃ perovskites including their optical, mechanical, hardness, and elastic properties, which have not been studied before. It also aims to propose suitable applications for these materials.

Computational Methods

DFT, as developed using the CASTEP (Cambridge Serial Total Energy Package) program, is used in all of the calculations in this article.²⁵ The advantage of DFT is that it does not attempt to compute the many-body wave functions (complex). Instead, the overall energy of the system is expressed in electron density. This study followed the empirical crystal structure described by two researchers, Lelieveld and Ijdo.¹⁷ The orthorhombic (space group Pnma) structure of XZrS₃ (X = Ca, Ba) compounds is comparable to CsSnI₃. Figure 1 shows the crystal structure of the XZrS₃ (X = Ca, Ba) compound. Theoretical approaches in matter physics have become one of the most active study disciplines. It entails employing ab initio approaches to identify and predict the characteristics of materials.¹⁹ The structural and electrical characteristics of XZrS₃ are calculated using CASTEP software. For all scenarios addressed here, for the planar wave expansion, the kinetic cutoff energy is 500 eV, maximal stress is 0.02 GPa, and maximal displacement is 5 × 10⁻⁴. Customized k-point sampling is used to integrate the first Brillouin zone. The stress–strain approach in CASTEP software is used to derive elastic constants. Mulliken population analysis is used to compute the Vickers hardness (H_v). The system achieved the ground state by self-consistent computation using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization approach when the total energy is steady within a factor of 5010⁻⁵ eV atom⁻¹, and force of about 10⁻² eV atom⁻¹. The elastic constants and moduli are calculated using a series of finite homogeneous crystal deformations based on the stress–strain techniques implemented in the CASTEP algorithm. The CASTEP data estimate the imaginary part of the complex dielectric function. The quasi-harmonic Debye model explores the thermodynamic characteristics of XZrS₃ compounds.²⁶ We employed the Heyd–Scuseria–Ernzerhof HSE06 hybrid functional in Quantum Espresso, utilizing a 4 × 4 × 4 k-point grid and a kinetic energy cutoff of 60 Ry for electronic and optical properties.²⁷

Fig. 1

Crystal structures of (a) CaZrS₃and (b) BaZrS₃.

Results and Discussion

Structural Properties

The investigated crystal structure of XZrS₃ (X = Ba, Ca) is displayed in Fig. 1. The chalcogenide perovskite XZrS₃ (X = Ba, Ca) crystalizes into the orthorhombic phases with the Pnma space group. The energy of the ground state uses Murnaghan's equation for calculation²⁸ of the unit cell volume. The equilibrium crystal structure was first sketched using the experimentally reported lattice constants for CaZrS₃, a = 7.030, b = 9.589, and c = 6.536, as well as the Wyckoff locations (0.090, 0.25, 0.033) for Ca, (0, 0, 0.5) for Zr, (0.0345, 0.25, 0.6252) for S1, and (0.1897, 0.0626, 0.8185) for S2. The Wyckoff positions for BaZrS₃ are (0.0376, 0.25, 0.0069) for Ba and (0, 0, 0.5) for Zr.²⁵ The optimized structural parameters of XZrS₃ (X = Ba, Ca) are summarized in Table I. It is seen from Table I that the calculated parameters agree with the corresponding experimental results,^29,30 and the structural properties for both structures are similar in terms of cell volumes besides lattice constants.

Table I Unit cell volume V (ų) with optimized parameters for the lattice (a, b, c) of XZrS₃ (X = Ca, Ba) materials

Elastic Constant

Elasticity in a solid connects physical and spatiotemporal action under stress, providing opportunities for commercial use. As described in CASTEP, the stress–strain approach is employed to derive elastic constants. Orthorhombic crystals have nine distinct single-crystal elastic constants considering symmetry, all listed in Table II for XZrS₃ (X = Ca, Ba). To investigate the mechanical durability of a crystal structure, elastic constants can be employed.^32,33 The study of mechanical properties is organized as follows. First, the strain–stress approach is used to compute the independent elastic constants.^34,35,36 Applying Hill's estimation, which represents the median values of the highest and lowest elastic moduli,³³ the multifaceted moduli of elasticity are derived using the obtained elastic constants. Brittleness, anisotropic toughness, and hardness are assessed via elastic moduli in the studied chalcogenide system in addition to elastic constants. To be mechanically stable, Eqs. 1, 2, and 3 of the Born criteria for an orthorhombic structure must be fulfilled.

$$ C_{{{11}}} > \, 0;C_{{{11}}} C_{{{22}}} > {\text{C}}^{{2}}_{{{12}}} $$

(1)

$$ C_{{{11}}} C_{{{22}}} C_{{{33}}} + { 2}C_{{{12}}} C_{{{13}}} C_{{{23}}} - C_{{{11}}} {\text{C}}^{{2}}_{{{23}}} - C_{{{22}}} {\text{C}}^{{2}}_{{{13}}} - C_{{{33}}} {\text{C}}^{{2}}_{{{12}}} > \, 0 $$

(2)

$$ C_{{{44}}} > 0;C_{{{55}}} > 0;C_{{{66}}} > 0 $$

(3)

Table II The calculated elastic constants, C_ij (in GPa), of XZrS₃ (X = Ca, Ba) compounds

The calculated nine elastic constants have satisfied the abovementioned Born criteria, which indicates that the compounds under study are mechanically stable. These stiffness constants are also utilized to compute the polycrystalline elastic moduli using well-known formalisms. Hill's approximation³⁷ is used to determine the bulk and the shear moduli, and it commonly expresses the mean of the highest limit (Voigt³⁸). The reduced limit of B and G, Y, υ, H_micro, H_macro,, and K_IC are calculated from C_ij using Eqs. 4, 5, 6, and 7.

$$Y=\frac{9BG}{3B+G}$$

(4)

$$\upupsilon =\frac{3B-Y}{6B}$$

(5)

$$\mathrm{Hmacro }=2{[(\frac{G}{B})^2G]}^{0.58}-3$$

(6)

$$\mathrm{Hmicro }=\frac{(1-2\upupsilon ){\text{Y}}}{6(1+\upupsilon )}$$

(7)

The elastic constants are used to obtain the bulk modulus B and shear modulus G. G and B are determined using Hill's estimation, which means the greater (range of Voigt³⁷) and smaller values (Reuss bounds) of elastic moduli are used to obtain the actual elastic moduli. The Voigt limits of bulk (B_V) and shear (G_V) moduli for orthorhombic crystals are computed using Eqs. 8 and 9³⁹

$$ {\text{B}}_{{\text{v}}} = \left[ {{\text{C}}_{{{33}}} + {\text{ C}}_{{{12}}} + {\text{ C}}_{{{11}}} + {2}\left( {{\text{C}}_{{{23}}} + {\text{ C}}_{{{13}}} + {\text{ C}}_{{{12}}} } \right)} \right] $$

(8)

$$ {\text{G}}_{{\text{v}}} = \left[ {{\text{C}}_{{{33}}} {-}{\text{C}}_{{{23}}} + {\text{C}}_{{{22}}} {-}{\text{C}}_{{{13}}} {-}{\text{C}}_{{{12}}} + {\text{ C}}_{{{11}}} + {3}\left( {{\text{C}}_{{{66}}} + {\text{C}}_{{{55}}} + {\text{C}}_{{{44}}} } \right)} \right] $$

(9)

The Reuss limits of shear (G_R) and bulk (B_R) moduli for orthorhombic crystals are derived using Eqs. 10, 11, and 12:⁴⁰

$$ {\text{B}}_{{\text{R}}} = {\text{x}}\left[ \begin{gathered} \left( {{\text{C}}_{{{22}}} {-}{\text{ 2C}}_{{{23}}} + {\text{ C}}_{{{33}}} } \right){\text{ C}}_{{{11}}} + \left( {{\text{C}}_{{{33}}} {-}{\text{ 2C}}_{{{13}}} } \right){\text{ C}}_{{{22}}} \hfill \\ + \left( {{\text{2C}}_{{{23}}} - {\text{ C}}_{{{12}}} } \right){\text{ C}}_{{{12}}} - {\text{ 2C}}_{{{33}}} {\text{C}}_{{{12}}} + \, \left( {{\text{2C}}_{{{12}}} - {\text{ C}}_{{{13}}} } \right){\text{ C}}_{{{13}}} \hfill \\ + \left( {{\text{2C}}_{{{13}}} - {\text{ C}}_{{{23}}} } \right){\text{ C}}_{{{23}}} \hfill \\ \end{gathered} \right]^{{ - {1}}} $$

(10)

$$ {\text{G}}_{{\text{R}}} = {15}\left\{ \begin{gathered} \left[ {{\text{C}}_{{{33}}} + {\text{ C}}_{{{23}}} + {\text{ C}}_{{{22}}} } \right){ 4} + \left( {{\text{C}}_{{{13}}} {-}{\text{ C}}_{{{12}}} + {\text{ C}}_{{{33}}} + {\text{ C}}_{{{33}}} {\text{C}}_{{{12}}} } \right){\text{ C}}_{{{22}}} + \, \left( {{\text{C}}_{{{12}}} + {\text{ C}}_{{{23}}} } \right) \hfill \\ - \left( {{\text{C}}_{{{13}}} + {\text{ C}}_{{{12}}} } \right){\text{ C}}_{{{13}}} - \, \left( {{\text{C}}_{{{23}}} + {\text{C}}_{{{13}}} } \right){\text{ C}}_{{{23}}} ]/{3}\left( {{\text{C}}^{{ - {1}}}_{{{55}}} + {\text{C}}^{{ - {1}}}_{{{66}}} + {\text{ C}}^{{ - {1}}}_{{{44}}} } \right) \hfill \\ \end{gathered} \right\}^{{ - {1}}} + {\text{X}} $$

(11)

$$ {\text{X}} = \left( { - {\text{ C}}_{{{13}}} {\text{C}}_{{{22}}} + {\text{ C}}_{{{12}}} {\text{C}}_{{{23}}} } \right){\text{ C}}_{{{13}}} + \left( {{-}{\text{ C}}_{{{23}}} {\text{C}}_{{{11}}} + {\text{ C}}_{{{12}}} {\text{C}}_{{{13}}} } \right){\text{ C}}_{{{23}}} + \left( {{-}{\text{ C}}^{{2}}_{{{12}}} + {\text{ C}}_{{{11}}} {\text{C}}_{{{22}}} } \right){\text{ C}}_{{{33}}} $$

(12)

G = (G_V + G_R)/2 and B = (B_V + B_R)/2 are the Hill's averages of G and B.⁴⁰ Poisson's ratio and Young's modulus are then calculated using the G and B values obtained.^41,42 The resistance to a change in crystal volume is marked by B, whereas the resistivity to a crystal change form is indicated by G. The Y values represent the stiffness of the solid. Elastic moduli (B, Y, and G) are significant for defining solid mechanical properties and determining the hardness of materials. The most straightforward and reliable elastic constant for estimating the hardness of solids is C₄₄.⁴³

The values in Table III demonstrate that CaZrS₃ is more rigid and stronger than BaZrS₃. Considering the elastic moduli, the shear modulus (G) is the best predictor of hardness, which agrees with the prior conclusion. These claims can be supported by the computation of the hardness indices (H_micro and H_macro), which are given in Eqs. 6 and 7. Table III shows the stated values, which are compatible with the predictions depending on C₄₄ and G values. The resulting hardness and elastic moduli, H_macro (using Chen's equation), correspond well to the earlier findings. H_macro, on the other hand, appears to be higher than H_micro computed using Chen's method. The parameters in the formulas are connected to the changes in the hardness values. Depending on the computed hardness using the Chen formula, Cherednichenko et al. reported that all the compounds tested would be hard materials.⁴³ The raw materials are thought to be tough. Mazhnik et al. used Chen's method to compute the toughness of several compounds and discovered that the theoretical and experimental values were reasonably close.

Table III Computed bulk modulus, B (GPa), shear modulus, G (GPa), Young's modulus, Y (GPa), Pugh's ratio (B/G), Poisson's ratio (υ), hardness, H_micro and H_macro, and Cauchy pressure (C₁₂-C₄₄) for XZrS₃ (X = Ca, Ba) compounds

Furthermore, the density of states (DOS) of these molecules, as shown in Fig. 4, explains why the estimated hardness (micro and macro) values of CaZrS₃are somewhat greater than those of BaZrS₃. Whenever the electronic states of the Ca and Ba atoms hybridized, creating powerful covalent bonds between them, the maximum DOS was created. The energy of its highest level determines the hardness of the chemical bonds; the lower the energy zone, the stronger the covalent bonds. As a result, the hardness characteristics of CaZrS₃ are greater than those of BaZrS₃. Other ceramic materials⁴² have yielded similar findings. The reported values of 0.14 and 0.15 show that chemical bonding in BaZrS₃ and CaZrS₃ is mostly covalent.⁴⁰ Furthermore, the interatomic forces may remain non-central even when values of υ fall between 0.25% and 0.50%. The fact that the values obtained do not fit within the previously provided range demonstrates that the interatomic forces predominant in BaZrS₃ and CaZrS₃ are not central.

The following equations may be used to compute the Cauchy pressure, C_P = C₂₃–C₄₄, C₁₂–C₆₆, and C₁₃–C₅₅. Although the ductile solid has a Cauchy pressure, it is evident from Table III that both compounds are expected to be ductile in nature as the Cauchy pressure offers positive values. Furthermore, the substantial negative amount of C_P is used to predict covalent bond directionality.⁴³ As a result, the significant negative coefficients (substantial) of C_P for these materials imply that they feature much controlled covalent bonding within their framework.

Elastic Anisotropy

Elastic anisotropy, in particular, displays variations in the atomic organization in various directions and constitutes one of the most significant characteristics of materials. Additionally, it helps to enhance measurements for describing the tensile strength of solids.⁴⁴ Therefore, investigation into elastic anisotropy in solids is crucial for crystal physics and materials engineering. A wide range of anisotropy parameters may describe solids' mechanical anisotropy in response to varied loads. Equations 13, 14, and 15⁴⁴ can be used to calculate the factors of shear anisotropic coefficients A₁, A₂, and A₃ for different surfaces in orthorhombic substances such as CaZrS₃ and BaZrS₃:

$${A}_{1}=\frac{4{C}_{44}}{{C}_{33}-2{C}_{13}+{C}_{11}}$$

(13)

$${A}_{2}= \frac{4{C}_{55}}{{C}_{33}-2{C}_{23}+{C}_{22}}$$

(14)

$${A}_{3}=\frac{4{C}_{66}}{{C}_{22}-2{C}_{12}+{C}_{11}}$$

(15)

When A₁ = A₂ = A₃ = 1, the solids are referred to as isotropic, and values beyond unity show characteristics of shear anisotropy. Also, the reported CaZrS₃ and BaZrS₃ values demonstrate the anisotropy of these chemicals. The anisotropy of shear A_G and A_B, which expresses compressibility, are defined in Eqs. 16 and 17.⁴⁴

$${A}_{G}=\frac{{G}_{V}-{G}_{R}}{{G}_{V}+{G}_{R}}$$

(16)

$${A}_{B}= \frac{{B}_{V}-{B}_{R}}{{B}_{V}+{B}_{R}}$$

(17)

A_B and A_G represent the maximal anisotropy and isotropy of crystals, corresponding to zero and 100%, respectively. The low magnitude of A_B and A_G expresses the anisotropic character of CaZrS₃ and BaZrS₃. The connection between the higher and lower limits of the bulk and shear modulus can potentially be used to compute the universal anisotropy index A^U in Eq. 18.⁴⁵

$${A}^{U}=5\frac{{G}_{V}}{{G}_{R}}+\frac{{B}_{V}}{{B}_{R}}-6\ge 0$$

(18)

A non-zero A^U value indicates mechanical anisotropy. Table IV summarizes the various anisotropic indices of XZrS₃ (X = Ca, Ba) compounds, including shear anisotropic factors (A₁, A₂, and A₃), percentage anisotropy (A_B and A_G), and universal anisotropy index A^U. Poisson's ratio, shear modulus, and Young's modulus fluctuations are along dissimilar crystallographic orientations that must be addressed to illustrate the elastic anisotropy in greater depth. In addition to the volume of CaZrS₃ and BaZrS₃, this research estimated the 2D plane-sensitive fluctuations and 3-dimensional variations of K, G, and Y using the elastic stiffness constants C_ij (GPa) in equations to explore this anisotropic nature. The mechanical anisotropy properties of the compounds are almost the same, with minor changes in magnitude, as seen in Fig. 2. The anisotropic nature of XZrS₃ (X = Ca,Ba) is clearly seen in Fig. 2, which shows the 2D and 3D views of Young's modulus (Y), shear modulus (G), and Poisson's ratio (υ) to observe the anisotropic nature. To distinguish the behavior in distinct planes, two-dimensional graphing has been used. Y has greater anisotropy across the xy planes. The xz axis shows that the peak values have the lowest anisotropy when the horizontal and vertical axes intersect at an angle of 45°.

Table IV Different anisotropic factors: A₁, A₂, and A₃ (shear anisotropy), A_B, A_G (rate of anisotropy), and A^U (universal anisotropy index) of CaZrS₃ and BaZrS₃ compounds

Fig. 2

2D and 3D views of Young's modulus (Y), shear modulus (G), and Poisson's ratio (υ) for CaZrS₃ and BaZrS₃ to observe the anisotropic nature.

The xy with xz planes have the maximum anisotropy magnitude along the vertical axis, whereas the yz plane has a medium value. Unlike compressibility, Young's modulus and shear modulus have two plates for every plane: the outermost layers display the peak value for each angle, while the inner layer displays the lowest values. In the xy plane, the outermost layer line consists of the highest value on the horizontal side and the lowest value towards the vertical axis. The cyan edge (interior surface) has the greatest amplitude at 30° from the flat direction and the smallest magnitude on the axis across all four dimensions. The highest value on the axes and the minimum value at an angle of 45° inclination towards the axis are shown by the black dashed lines (the outside surface) within the xz plane. The smallest magnitudes are on both sides at a 30° angle with the vertical axis, while the largest magnitude on the vertical plane is on the outside surface of the yz plane. Like the shear modulus, Poisson's ratio contains planes for each axis, each of which has a complex anisotropic structure. Table V displays the highest and lowest values of Y, G, and ν alongside their greatest to least significant ratios. These ratios can be used to compute elastic anisotropy.

Table V The lowest and highest values for Young's modulus, Poisson's ratio, and shear modulus for CaZrS₃ and BaZrS₃

Electronic Properties

The parameterized scalar translational electronic band structure (EBS) and density of states (DOS) of XZrS₃ in orthorhombic aspects are shown in Fig. 3 (using the GGA functional) and Fig. 4 (using the hybrid HS06 functional). In many semiconducting materials, bandgap estimation using local functionals results in inaccurate values and inappropriate positions of the valence band (VB) and the conduction band (CB), causing theoretical values to deviate from the experimental results. The error might arise from strong Coulomb correlations of the material. In most cases, the use of nonlocal functionals could be a fair solution to the aforementioned problem.^46,47 In order to provide an accurate prediction of the EBS as well as the bandgap, we have used a local functional (GGA) and a nonlocal functional (HSE06) to model the exchange–correlation potential. The computed BS and DOS of CaZrS₃ and BaZrS₃ chalcogenide perovskites are given in Figs. 3, 4, 5, and 6, while the partial DOS (PDOS) is shown in Figs. 5 and 6. As shown in Fig. 3, the electronic BS was formed with the high symmetry path Z – A – M – Γ – Z – R– X– Γ. The figure indicates a bandgap (direct) of 1.3 eV for CaZrS₃ and 1.1 eV for BaZrS₃ using the generalized gradient approximation (GGA) functional. The energy gaps using the GGA are found to be significantly lower. It is decayed compared to CaHfS₃ (direct bandgap 1.51 eV) although exaggerated compared to BaHfS₃ and SrHfS₃.⁴⁸ According to this study, the photovoltaic cell's optimal bandgap ranges from 0.9 to 1.56 eV. The EBS shows electron quantum mechanical activity in a compound. CaZrS₃ and BaZrS₃ chalcogenide perovskites have previously been explored as better solar absorbers.⁴⁹ The energy states graph for the collective atom adjustment comprises the level bands representing solids' energy levels, rather than discrete levels. This is made possible by the holes and electrons that move between the valance and conduction bands. The energy gap is a location in the physical bandgap in which there is no electron, indicating anti-bonding as well as bonding modes,⁴⁷ and it is significant in materials classification. The bandgap value of CaZrS₃ is found to be 1.9 eV using the HSE06 functional as shown in Fig. 4, which is consistent with the previously reported theoretical value of 1.96 eV⁵⁰ and experimental⁵⁰ bandgap result. Furthermore, the bandgap was calculated with the Hubbard correction for on-site interaction, yielding a bandgap of 1.88 eV, which is close to the experimental result for CaZrS₃. Similarly, the bandgap of BaZrS₃ is found to be 1.6 eV using the HSE06 functional as shown in Fig. 4, which is nearly the same as the experimental value⁵¹ The electronic band structure along the high symmetry direction of the Brillouin zone was estimated using the GGA of the Perdew–Burke–Ernzerhof (GGA-PBE) functional and the Hubbard correction (GGA+U) for exchange–correlation potential in a previous work.⁵¹ The GGA-PBE functional fails to approximate the exact exchange–correlation potential, since the bandgap value obtained by approximating the exchange–correlation potential with the GGA-PBE functional is 1.23 eV, which appears to be underestimated as compared to the experimental bandgap value of 1.90 eV. The previously reported bandgap of BaZrS₃ was calculated using the hybrid HSE06 functional, which is in excellent agreement with experimentally measured values of 1.73–1.85 eV.⁵³ The direct bandgap of BaZrS₃ at the Γ point on the Brillouin zone is predicted as 1.74 eV, which is consistent with previously reported experimental values (1.74–1.75 eV). Our results are close to the theoretical and experimental values. The band morphology was significantly affected by the chalcogenide element (X element).

Fig. 3

Band structures (GGA) of (a) CaZrS₃ and (b) BaZrS₃.

Fig. 4

Band structures (HSE06) of (a) CaZrS₃ and (b) BaZrS₃.

Fig. 5

Total density of sates (TDOS) and partial density of states (PDOS) of CaZrS₃.

Fig. 6

Total density of sates (TDOS) and partial density of states (PDOS) of BaZrS₃.

In Fig. 5, the DOS of the GGA and HSE06 functionals are given for CaZrS₃. In the HSE06 functional, Ca-1s, Zr-2d, and S-3s are the dominant elements. The valence band was dominated by Zr-3d and Ca-3p, with lowest dominance of Zr-3p, suggesting immense binding energy, also making nucleon dissociation difficult, according to the PDOS in Fig. 5. As seen in the PDOS image, the anti-bonding Zr-s, Ca-p, and S-p states are obviously related to the conversion between the bands. The conduction band has a significant hybridization involving Zr-3d and S-4s states, with about the same electron yield as Zr-3d, Ca-3p, Zr-3p, Zr-4s, and S-3p states. They have extra molecules inside the conduction band with a lower hybridization tendency that play a crucial role in the band's electron–hole synergy. CaZrS₃ formed a powerful connection with Ca-3p and Zr-3d in hybridized Ca-3p and Zr-3d. Figure 6 shows the TDOS and PDOS of BaZrS₃. For HSE06, Ba-3p, Zr-2d, and S-3s are the predominant states. For the GGA case, Ba-1p, Zr-3p, and S-4s are the predominant states. The BZ center, also known as Γ-point along the Brillouin zone, contains the material's valence band maximum (VB_max) and conduction band minimum (CB_min). As a result, in CaZrS₃ and BaZrS₃, the band corner refers to Γ –Γ or direct type.

Optical Properties

The optical properties are investigated to observe how the chemical behaves under interaction with light. It is possible to determine optical characteristics using photon-induced electronic processes. In the experiment,⁸ a plasma frequency of 10 eV, Drude term damping of 0.05 eV, and Gaussian smearing of 0.5 eV are used. The weighted transitions between filled and unfilled electronic energy levels provide the function of complex dielectric (imaginary part, "ε₂(ω)"). This method is obtained using the CASTEP-supported formula,⁵⁴ from Eq. 19.

$$\upvarepsilon 2\left(\upomega \right)=\frac{2{e}^{2}\pi }{\Omega {\upvarepsilon }_{0}}{\sum }_{k,v,c}|\langle {\uppsi }_{{\text{k}}}^{{\text{c}}}|\widehat{u} \overrightarrow{r}|{\uppsi }_{{\text{k}}}^{{\text{v}}}\rangle |2\updelta ({E}_{k}^{c} - {E}_{k}^{v} - E)$$

(19)

where the incident photon's frequency is the volume of the unit cell, e is the electronic charge, (h/2π)k is the momentum, and k_v and k_c indicate the quantum states for electrons within the bands of conduction and valence. Energy and momentum conservation are guaranteed throughout the transition by the delta function. Each dielectric constant's real and imaginary elements, which signify a causal process, are connected by the Kramers-Kronig conversion. Therefore, CASTEP applies this transformation to separate the analogous imaginary part, ε₂(ω), with the real element, ε₁(ω), which is the dielectric function. The rest of the optical properties can be computed using these two parts that constitute the complex dielectric constant.

Figure 7 presents optical absorption and conductivity for HSE06 and GGA functional. The absorption coefficient spectrum α(ω) reveals the percentage of light intensity attenuation per unit distance. All studied materials exhibit absorption in visible regions, with optical absorption intensity notably increasing within the 2 to 5 eV range. Optical absorbance spectrum are 23 × 10⁵ cm⁻¹ (CaZrS₃) and 18 × 10⁵ cm⁻¹ (BaZrS₃) for HSE06 functionals. Additionally, CaZrS₃ compounds demonstrate higher UV absorption compared to BaZrS₃. Conductivity starts from 1 eV, increasing with the rising energy after decreasing, showing semiconducting behavior. The optical conductivity spectra start at 0.02 eV and 0.01 eV, reaching maximal peaks at 6000 and 5600 σ(ω) for CaZrS₃ and BaZrS₃, respectively. The photoconductivity of the material rises as the photon energy increases.⁵¹ Figure 8 presents the reflectivity and the refractive index. Subsequently, a decrease with certain oscillations occurs, ultimately reaching a minimal value of 0. The behavior of reflectivity R(ω) relates to incident photon energy. In the low-energy range (0–4 eV), radiation reflectivity is 20–30% for CaZrS₃ and BaZrS₃, before increasing to high peaks in the UV region for both HSE06 and GGA functionals. The refractive index curves reach maximum values at around 2.5 and 2.4 eV for CaZrS₃ and BaZrS₃. Following this, the peak rapidly decreases to zero in the ultraviolet (UV) region, suggesting apparent faster-than-light motion for UV photons. The dielectric function from Fig. 9 is used to explain the crystal's optical response at various photon energies. Material with high dielectric constants (or low dielectric strength) are poor insulators and great electricity conductors. CaZrS₃ and BaZrS₃ materials have high dielectric functions, so they will be excellent conductors.⁵⁵ Subsequently, the energy loss function exhibits pronounced peaks in the UV region, correlating with the plasma frequency (ωp) and serving as an interface between metallic and dielectric behavior.

Fig. 7

The real parts of the absorption coefficient α and conductivity σ of CaZrS₃ and BaZrS₃.

Fig. 8

The real parts of the reflectivity R(ω) and refractive index μ(ω) of CaZrS₃ and BaZrS₃.

Fig. 9

The real parts of the dielectric function ε(ω) and loss function L(ω) of CaZrS₃ and BaZrS₃.

Thermal Properties

The Debye temperature is a critical characteristic in materials that affects melting temperature, specific heat, thermal expansion, thermal conductivity, and lattice vibration. The Debye temperature differentiates between a solid's low- and high-temperature zones. Debye temperature can be determined using Eq. 20,⁵⁶ based on average sound velocity and the theory of elastic constants.

$$ \theta _{D} = {\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h {k_{B} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${k_{B} }$}}\left[ {\left( {{\raise0.7ex\hbox{${3n}$} \!\mathord{\left/ {\vphantom {{3n} {4\pi }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${4\pi }$}}} \right){\text{N}}_{A} \rho /{\text{M}}} \right]^{{\frac{1}{3}}} v_{m} $$

(20)

h is the Planck constant, N_A is Avogadro's number, v is the mean sound velocity, and k_B is the Boltzmann constant. The sound velocity v can be calculated using Eq. 21, as transverse (v_t), longitudinal (v_l), and sound rates in an isotropic material:

$${v}_{m}= {[1/3\left(1/{v}_{1}^{3}+2/{v}_{t}^{3}\right)]}^{-1/3}$$

(21)

From the connection developed by Reuss and Voigt, v_t and v_l can be introduced as polycrystalline with bulk modulus (B) and shear modulus (G).

$${v}_{1} {\left[(3B+4G)/3\rho \right]}^{1/2}$$

(22)

$$ v_{m} = \left[ {{\raise0.7ex\hbox{${\text{G}}$} \!\mathord{\left/ {\vphantom {{\text{G}} \rho }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\rho $}}} \right]^{{1/2}} $$

(23)

A progressive increase in D is seen when Ba can be substituted with Ca in the periodic table, as these elements are located at the same group (Table VI). A better D value is generally associated with higher phonon thermal conductivity. CaZrS₃ should thus have higher thermal conductivity. K_min and Grüneisen parameters can be used to obtain the minimum thermal conductivity.⁵⁶ As K_min (CaZrS₃) > K_min (BaZrS₃), it is worth noting that BaZrS₃ has smaller values of D (322.79 K) and K_min (0.52 W m⁻¹ K⁻¹). As a result, BaZrS₃ might be feasibly employed as a thermal barrier coating (TBC) material.^57,58,59 Figure 10 compares the thermodynamic properties of ABO₃

Table VI The evaluated density ρ, transverse sound velocity v_t, longitudinal sound velocity v_l, average sound velocity v_m, Debye temperature θ_D, minimum thermal conductivity K_min, and melting temperature T_m of XZrS₃ (X = Ca,Ba) under pressure

Fig. 10

(a) Debye temperature, (b) melting temperature, and (c) minimum thermal conductivity of XZrS₃ and different compounds.

Conclusion

The structural, elastic, mechanical, electronic, optical, and thermal properties of XZrS₃ (X = Ca, Ba) were explored in this study using first-principles computation. Compared to other chalcogenide perovskites, the results for XZrS₃ (X = Ca, Ba) are competitive, with direct bandgap values of 1.3 eV (CaZrS₃) and 1.1 eV (BaZrS₃) using GGA, respectively, which is ideal for solar cells (0.9–1.56 eV) and optoelectronic device applications. Bandgap values of 1.9 eV and 1.6 eV were found for CaZrS₃ and BaZrS₃, respectively, using the HSE06 functional, which is consistent with previously reported theoretical and experimental bandgap results. These bandgap values are within the experimental range. The structural properties show that the compound is an excellent alternative for a solar absorber, thus offering a solution for the instability of halide perovskites. When compared to other perovskites, these materials have a substantial optical absorbance spectrum of about 23 × 10⁵ cm⁻¹ (CaZrS₃) and 18 × 10⁵ cm⁻¹ (BaZrS₃), which increases within the 0–3 eV visible range, suggesting that the material has significant capacity to absorb light. The reflectivity of XZrS₃ (X = Ca, Ba) ranges from 25% to 30% for CaZrS₃ and BaZrS₃) in the spectrum range of 0–4 eV, indicating that a substantial portion of incident light is absorbed by the material, making it potentially suitable for solar cell applications. The refractive index reaches maximum values at around 2.5 (CaZrS₃) and 2.4 eV (BaZrS₃), which shows that the material exhibits strong interactions with light in this energy range. Materials with a high refractive index can be used to design antireflective coatings for solar cells. The optical conductivity has maximal peaks at 6000 and 5600 σ(ω) for CaZrS₃ and BaZrS₃, respectively. The compounds will be suitable for optoelectronic device applications. Together, the results show that XZrS₃ is a nontoxic, inexpensive compound with readily available components, with additional value in terms of cost and stability, making it ideal for PV applications.