Structural stabilities, mechanical and thermodynamic properties of chalcogenide perovskite ABS3 (A = Li, Na, K, Rb, Cs; B = Si, Ge, Sn) from first-principles study

In this study, first-principles calculations have been used to study the mechanical and thermodynamic properties of chalcogenide perovskite ABS3 (A = Li, Na, K, Rb, Cs; B = Si, Ge, Sn) in the triclinic phase. The structural stabilities of perovskite were investigated through Goldschmidt’s tolerance factor (t) and phonon dispersion. It was indicated that all of the investigated materials construct stable perovskite structures. The mechanical properties of chalcogenide perovskites ABS3 were systematically investigated by density functional theory (DFT). The DFT method was considered within the meta-generalized gradient approximation revTPSS. 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. The independent elastic constants Cij have been used for the prediction of mechanical properties like bulk modulus (B), Shear modulus (G), Young’s modulus (E) Poisson’s ratio (ν), and the universal anisotropic index (AU). The mechanical stability, brittleness, and ductility behaviors of materials were discussed. The covalent, ionic, and metallic nature of the materials were also discussed. The thermodynamic parameters including heat capacity, entropy, enthalpy, and free energy were also computed and discussed with a wide range of temperatures (0–1000 K).


Introduction
The general perovskite crystal structure is a primitive cube, with the A-larger cation in the corner, the B-smaller cation in the middle of the cube, and X anions commonly in the center of the face edge as shown in Fig. 1. The cubic perovskite structure ABX 3 has a greater A-X separation than the B-X, and their configuration is affected by temperature, pressure, and chemical content [1]. As shown in Fig. 1, each B cation is bounded by 6 anions (X), while each A cation is bordered by 12 anions. The valance state of cation and anion are used to categorize perovskite. There are three categories of simple compounds: A +2 B +4 X 3 (2,4), A +3 B +3 X 3 (3,3), and A +1 B +5 X 3 (1,5). The cubic perovskite structure belongs to Pm-3 m crystallography space group. The equivalent positions of the A and B sites cations in the ideal unit cell are (0, 0, 0) and (1/2, 1/2, 1/2) respectively, and for the X anions (1/2, 0, 0). The ionic radii in such a structure are related through the following Eq. (1), where R A , R B , and R x are ionic radii of A, B, and X, respectively. The perovskites on a number of occasions were found to be distorted from the ideal structure. The degree of distortion from the ideal cubic perovskite can be quantified by the tolerance factor [1], Based on the analysis of tolerance factor value, in the range of 0.813 < t < 1.107 indicates ideal cubic structures or a distorted perovskite structure with tilted octahedra. Structures having t values outside the range cannot be considered stable perovskites [3][4][5]. The mechanism of the distortion from ideal cubic perovskite can be classified into five versions [6]: (i) distortion of the BX 6 octahedron, e.g., by the Jahn-Teller effect, (ii) off-center displacement of the B cations in the BX 6 octahedron, this effect is the basis for possible ferroelectricity, (iii) so-called tilting of the octahedron framework, usually occurring as a result of a too-small A cation at the cuboctahedral site, (iv) ordering of more than one kind of cations A or B, or vacancies, and (v) formation of oxygen vacancies. The distortions as a consequence of the substitution of the A-and/or B-site cation can be used to fine-tune physical properties exhibited by perovskites. Substitution of the A-and /or B-site cation can be achieved by different sizes and charges of a e-mail: mtaha@psas.bsu.edu.eg (corresponding author) Eur. Phys. J. Plus (2022) 137:1006 Fig. 1 The cubic crystal structure of perovskite [2] metal ions. Moreover, it is possible to make a partial substitution of the A or B sites giving the formation of double perovskite. Where the cations in the A or B site are two different elements with strongly differing sizes ( High stability, environmentally friendly, strong optical absorption, good carrier mobility, and a direct bandgap of several chalcogenide perovskites make them candidates for renewable energy applications [7,8]. Chalcogenide perovskites are promising solar-cell materials. Several chalcogenide perovskites have been successfully investigated theoretically and synthesized experimentally [9][10][11]. Previous theoretical studies by Sun et al. [12] predicted that the band gaps of CaTiS 3 (1.0 eV), BaZrS 3 (1.75 eV), CaZrSe 3 (1.3 eV), and CaHfSe 3 (1.2 eV), and they are ideal for making single-junction solar cells. Ju et al. [13] in their theoretical studies, predicted that the band gaps of SrSnSe 3 , CaSnS 3 , and SrSnS 3 in the range of 0.9-1.6 eV making these perovskites are suitable for making single-junction solar cells. Recently, perovskite chalcogenide has attracted increased attention in different applications. SbCrSe 3 and SiCrTe 3 are promising for infrared detection application [14]. LaYS 3 is a promising candidate for photoelectrochemical water splitting [15]. Compared to perovskite oxides, chalcogenide perovskites have more favorable thermoelectric properties. The thermoelectric performance of perovskite oxides is limited due to their high lattice thermal conductivity and large band gaps. With the replacement of oxygen with chalcogen elements, chalcogenide possess lower thermal conductivity and lower band gaps. These features are desirable attributes for high-temperature thermoelectric applications [16]. For high-temperature thermoelectric BaZrS 3 , Ba 3 Zr 2 S 7 , Ba 2 ZrS 4 , a-SrZrS 3 and b-SrZrS 3 are candidates [17]. Previous work indicated the important role of mechanical and thermodynamics properties of perovskites in energy applications [17,18]. In general, density functional theory (DFT) calculations are a promising approach for predicting physicochemical properties of materials that closely match the experimental results [19][20][21][22]. Many recent reports employing DFT calculations have predicted mechanical and thermodynamics properties for chalcogenide perovskite materials [10,[23][24][25]. Elastic properties explain a compound's deformation under external pressure. Mechanical properties of a material, such as stiffness, stability, and even phase structure, are influenced by pressure. As a result, before using a compound in device production, it is critical to understand its mechanical behavior. Mechanical properties show how microcracks can form in a compound during the growing process and indicate whether it is suitable for industrial applications [26]. In the present work, the Table 1 Goldschmidt's tolerance factor (t), lattice parameters, unit cell volume (v) and thermotical density (D) for LBS group, NBS group and ABS group using mGAA-revTPSS method

Computational method
All calculations have been performed using the DMol 3 program [27], which is based on DFT. The CsSnS 3 (triclinic, P-1,2) was taken from the material project website (mp-561710), and the Cs ion were replaced by Li, Na, K, and Rb, while the Sn ion were replaced by Si and Ge ion. The geometry optimization and the mechanical and thermodynamics calculations of these structures were performed within the meta-generalized gradient approximation (m-GGA) revTPSS method [28]. The K point set in the Brillouin zone was 2 × 2 × 2 k. For core treatments, all-electron relativistic was used. All-electron Kohn-Sham wave functions are expanded in a Double Numerical basis with polarized orbital (DNP). The elastic constants [29] are essential parameters for knowing the mechanical properties, such as bulk modulus (B), shear modulus (G), Young's modulus (E) Poisson's ratio (ν), and the universal anisotropic index (A U ). Based on the obtained elastic constants, the bulk modulus and shear modulus can be calculated according to the Voigt-Reuss-Hill method [30]. The Voigt approximation Eur. Phys. J. Plus (2022) 137:1006 Fig. 3 Variation of the a bulk modulus and unit cell volume b shear modulus and unit cell volume c Young's modulus and unit cell volume for the LBS group, as obtained from the mGGA/revTPSS calculations assumes a uniform strain throughout a crystalline material [31] and the Reuss approximation is based on the assumption of uniform stress applied on the crystalline material [32]. The bulk (B V ) and the shear modulus (G V ) can be obtained in the Voigt theory, as while in the Reuss theory, where S ij is the elastic compliance matrix. The B, G, B/G, E, V and A U [33] can be calculated as: The phonon frequencies are carried out under the CASTEP module in the Materials Studio package, using the linear response method, since DMol 3 does not compute the phonons. CASTEP is an ab initio quantum mechanics calculation program based on DFT and plane wave pseudopotential method. The generalized gradient approximation GGA-PBEsol was used to deal with the exchange-correlation energy between electrons. Norm conserving pseudopotential was applied to describe the interaction potential between the real ion and the valence electron.

Structural stability
A number of researchers had used Goldschmidt's tolerance factor (t) to study the perovskite stability [34][35][36]. We used Shannon's radii [37] for the ions in ABS 3 (A Li, Na, K, Rb, Cs; B Si, Ge, Sn) for calculated t. The studied perovskites were classified to LiBS 3 (B Si, Ge, Sn) (LBS group), NaBS 3 (B Si, Ge, Sn) (NBS group) and ABS 3 (A K, Rb, Cs; B Si, Ge, Sn) (ABS group). In Table 1, the calculated t values suggest that all studied perovskites form stable perovskite structures except CsSiS 3 is only partially stable ast < 1.107. In ABS, the increase in thermotical density (D) correlates with the increase in unit cell volume (v) which is accompanied by an increase in lattice parameters that result from the increase in the ionic radius of the B-site cation as shown in Table 1. This behavior is followed in the case of the NBS and ABS groups.

Phonon dispersion
Calculation of phonon dispersion curves is a powerful way to reveal the structural stability of materials. Figure 2 shows the obtained phonon dispersions for LBS, NBS and ABS groups, respectively. There was no imaginary frequency mode, which indicated that the dynamical stability of the studied perovskites. One notable feature from Fig. 2 is the band gap in the phonon spectrum. We can clearly see that this band gap was observed for KSiS 3 , RbSiS 3 , RbSnS 3 and CsSiS 3 while very small for other compounds. It has been shown that the larger gap between acoustic and optic phonons often means higher thermal conductivity [38]. Therefore, we can conclude that KSiS 3 , RbSiS 3 , RbSnS 3 and CsSiS 3 have high thermal conductivity compared to the other compounds.

Elastic properties
The optimized structures of the investigated perovskites are shown in Figs. S1-S3 (Supplementary Information). The triclinic crystal symmetry has 21 independent elastic constants C ij (C 11 , C 12 , C 13 , C 14 , C 15 , C 16 , C 22 , C 23 , C 24 , C 25 , C 26 , C 33 , C 34 , C 35 , C 36 , C 44 , C 45 , C 46 , C 55 , C 56 , C 66 ). The values of the elastic constants are given in Tables 1, 2, 3. All the studied perovskites have a higher value of C 11 value than C 44 . This implies that the investigated perovskites will show higher resistance toward unidirectional compression than resistance against shear deformation compression. The Born criterion for the stability of a triclinic symmetry crystal structure is that all eigenvalues of the C should be positive [39]. Tables 2, 3, 4 show that some elastic constants do not satisfy this criterion for mechanical stability in all directions. This indicates that the triclinic phase is mechanically unstable against the shear deformation along the direction of the C 15 , C 16 Table 5. The brittleness and ductility behaviors of materials can be known by Pugh's criterion (B/G ratio), the critical value of B/G equals 1.75. According to Pugh [40], the material possesses brittleness, if B/G is a smaller critical value otherwise. While the material is ductile for B/G is higher than the critical value. From the obtained B/G ratio in Table 5, it was found that LiSiS 3 Figure 3a-c shows the variation of the bulk modulus, shear modulus, and Young's modulus for the LBS group as a function of ionic radius of B-site cation, respectively. Figure 3a shows that LiGeS 3 has the lowest bulk modulus, indicating that LiGeS 3 has low resistance to shape change under pressure. From Fig. 3b, c, it was found that the shear modulus and Young's modulus values for the LBS follow the order LiSiS 3 > LiGeS 3 > LiSnS 3 . The high shear modulus of LiSiS 3 implies the high capacity to resist deformation and volume change under pressure. On other hand, the high Young's modulus material has a high brittleness. Figure 4a-c shows the bulk modulus, shear modulus, and Young's modulus for the NBS group as a function of the ionic radius of the B-site cation. Bulk modulus, shear modulus, and Young's  [41]. From the calculated values of Poisson's ratio in Table  5, it can conclude that for the LBS group, metallic contribution to interatomic bonding is dominant for LiGeS 3 and LiSnS 3 on other hand LiSiS 3 appear ionic behavior as inter-atomic bonding. For the NBS group, metallic, ionic, covalent behavior as interatomic bonding is dominant for NaSiS 3 , NaGeS 3, and NaSnS 3 , respectively. Ionic contribution to interatomic bonding is dominant for all studied perovskites in the ABS group. The degree of elastic anisotropy in the material is essential to understanding the mechanical properties of the crystal. The universal anisotropic index (A U ) is a measure to quantify the degree of elastic anisotropy. For isotropic crystals are obtained when A U 0 otherwise, when A U > 0 anisotropic crystal is formed [42]. From obtained universal anisotropic index (A U ) in Table 5, for the LBS group, LiSiS 3 and LiGeS 3 are anisotropic, while LiSnS 3 is isotropic. For the NBS group, NaSiS 3 and NaSnS 3 are isotropic, and NaGeS 3 is anisotropic. Finally, all studied perovskite in the ABS group is anisotropic.

Thermodynamics properties
Specific heat is an essential parameter in solid physics since it gives us information about heat loss lattice, phase transitions, and energy bands. Figure 6 depicts the variations of specific heat capacities with temperature. It is clear that below 400 K, the heat capacity increases rapidly, but at temperatures higher than 900 K, the behavior appears to be slow according to Dulong-Petit limit [43]. In the temperature range of 0-400 K, the fast increase of the C v value results from phonon thermal softening. For the LBS group, LiGeS 3 has the highest value of C v at T 300 K, and LiSiS 3 has the lowest value. The NaSnS 3 has a high value of C v at T 300 K and NaSiS 3 has a low value in the NBS group. The Sn-S perovskites take the highest values of C v at T 300 K in the ABS group. On other hand, Si-S perovskites have the lowest values. The entropy of the studied perovskite increases monotonically with temperature (Fig. 7). The LiGeS 3 has higher entropy, and LiSiS 3 has lower entropy. NaGeS 3 has lower entropy and NaSiS 3 has higher entropy. This difference in entropy value can result from the different bonding characteristics and/or strengths [44]. In the ABS group, Sn-S perovskites have higher entropy, while Si-S perovskites have lower entropy. The enthalpy of studied perovskite increases with temperature (Fig. 8). The LiSiS 3 has higher enthalpy and LiSnS 3 has lower enthalpy. NaSiS 3 has higher free energy and NaSnS 3 has lower enthalpy. The enthalpy of the ABS group follows two different trends. In the temperature range 25-300 K, Si-S perovskites have higher enthalpy and Si-S Perovskites have lower enthalpy. In the temperature range 300-1000 K, all studied ABS perovskites have nearly the same values. The free energy of the studied perovskites decreases with increasing the temperature (Fig. 9). The LiSiS 3 has higher free energy and LiSnS 3 has lower free energy. NaGeS 3 has higher free energy and NaSnS 3 has lower free energy. Si-S perovskites have higher free energy (ABS group), while Si-S perovskites have lower free energy.

Conclusions
The tolerance factor and phonon dispersion can be used to calculate the structural stabilities of perovskite compounds. Changing the ions that produce ABX 3 perovskite crystals can control the t-factor. The investigation of structural stabilities properties of chalcogenide perovskite ABS 3 (A Li, Na, K, Rb, Cs; B Si, Ge, Sn), show that all studied perovskites are stable. The phonon dispersion results have revealed the dynamical stability of these perovskites. Anticipating structural stabilities of the perovskite has long been a challenge in the search for the discovery of novel functional materials for a wide range of applications, including electrocatalysts and photovoltaics. The obtained results show that these materials are promising for solar cells. The elastic and thermodynamics properties of chalcogenide perovskites ABS 3 (A Li, Na, K, Rb, Cs; B Si, Ge, Sn) were investigated using firstprinciples calculations. The calculated bulk modulus (B), Shear modulus (G), Young's modulus (E) show that LiSnS 3 has lower bulk modulus, lower shear modulus, and lower Young's modulus for the LBS group. For the NBS group, Bulk modulus, shear modulus, and Young's modulus of NaGeS 3 are lower than that of NaSiS 3 and NaSnS 3 . For the ABS group, Bulk modulus, shear modulus, and Young's modulus of Ge-S perovskites (AGeS 3 (A K, Rb, Cs)) are high than that of Si-S perovskites (ASiS 3 (A K, Rb, Cs)) and Sn-S perovskites (ASnS 3 (A K, Rb, Cs)). The calculated Poisson's ratio values show that ionic contribution to interatomic bonding is dominant for all studied perovskite in our work except LiGeS 3 , LiSnS 3 and NaSiS 3 appear metallic contribution and NaSnS 3 appears covalent behavior as inter-atomic bonding. The predicted universal anisotropic index (A U ) reveals that all studied perovskites are anisotropic, except LiSnS 3 , NaSiS 3 and NaSnS 3 are isotropic.