Ab initio study of fundamental properties of XInO 3 ( X = K, Rb, Cs ) perovskites

: The structural, elastic, anisotropic, and lattice dynamical properties of cubic perovskite compounds XInO 3 ( X = K, Rb, and Cs ) are investigated using ﬁ rst - principles calculations. Electronic band structures and state densities revealed that the electronic nature of the studied materials exhibited half - metallicity properties. The existence of O p – d states close to the Fermi level contributes to the half - metallic properties. Moreover, poly - crystalline properties, such as bulk, Young, and shear moduli and Pugh and Poisson ratios, have been deter - mined. As a result of these characteristics, the compounds under consideration exhibited ductility behavior. As far as is known, since this is the ﬁ rst study of XInO 3 ( X = K, Rb, and Cs ) compounds, this work sheds light on future works.


Introduction
Researchers' favorites and prominent in all areas of material sciences are perovskites, which are found in the world's natural formations and whose effective and rare physical features have lately been recognized.Although it is referred to as perovskite, the calcium-titanium-oxygen complex CaTiO 3 is also used to characterize it.The first man who finds this mineral was the Russian mineralogist Gustav Rose.In 1839, researchers recognized other minerals with a comparable crystalline structure called perovskites.These new compounds, which have a general structure of type ABC 3 , were discovered by Russian mineralogists and named Perovskite in honor of Lev Perovski (1792-1856).Perovskite materials' interesting and surprising properties are newly recognized, and they are being pursued by researchers with rising momentum [1][2][3].
The challenge of being able to store energy generated and utilize it later is a matter of importance when resolving energy problems persists.New materials, particularly perovskite oxides as electrochemical energy materials, offer a great advantage to be utilized as a possible host or carrier for applications.This theoretical study covers the latest progress of one group example of ABO 3 perovskite oxides, with no more information in the literature.In this study, the structural, elastic, anisotropic elastic, electronic, and lattice dynamical properties of XInO 3 (X = K, Rb, and Cs) were examined using density functional theory (DFT) implemented by the Vienna ab initio simulation package (VASP).The lattice constant, bulk, Young's, and shear moduli, and Poisson's ratio have been investigated.Current compounds are stable, both mechanically and dynamically.The calculations revealed that these compounds retain some brittleness while also exhibiting ductility.The material may show bending behavior.With this feature, it can be said that it has the potential to be a new material that can be used in eco-friendly wearable technologies [3][4][5][6].
It can be concluded that the lattice constants, volume, and density have increased for the investigated compounds, as categorized in Table 1.Furthermore, the results of the formation energy, calculated by using equation (1), have demonstrated that those compounds can be both synthesizable and thermodynamically stable:

Mechanical properties
Either conceptual or portable applications require mechanical characteristics, which are both beneficial and essential.
As a result, it is necessary to compute the mechanical properties of the compounds considered.Therefore, current compounds must adhere to the Born stability conditions [13], which are as follows: It is possible to see that the compounds meet Born stability criteria, and those compounds are mechanically stable.Thus, they are suitable for conceptual or portable applications.The elastic constants can be used to express the polycrystalline properties such as bulk (B), shear (G), Young's moduli (E), B/G, G/B, Poisson's ratio (ν), and Vicker's hardness (H f ), and they are calculated by using equations (2)-( 6), respectively.In equations (2) and (3), the Voigt [14] and Reuß [15] approximations are symbolized as subscripts V and R, respectively.The protection of the compound to volume modification under applied hydrostatic pressure could be referred to as the bulk modulus, as it is commonly known in [16].As already concluded from Table 2, KInO 3 has the highest bulk modulus, while CsInO 3 has the lowest of the investigated compounds.The relationship between shear stress and shear strain can be used to discover the shear modulus, also known as elastic shear stiffness [17].As can be seen In all aspects, as concluded from Vicker's hardness results, these compounds are not so hard materials.This behavior makes the compounds ductile.

Anisotropic properties
Anisotropic behaviors are beneficial for determining microcracks in the compounds.In crystals with cubic symmetry, the Zener anisotropy (A), given in equation (7), and maximum-minimum polycrystalline properties are enough to explain this nature.Table 3  = − The considered compounds have exhibited anisotropic properties due to the A being far from 1.The A measures that the compounds are far from isotropic, as is known.As can be seen in Table 3, KInO 3 has largest anisotropy, whereas CsInO 3 has the smallest.Furthermore, with the exception of linear compressibility, analyzed polycrystalline properties have inferred an anisotropic nature.The observed isotropic characteristics of linear compressibility are caused by cubic symmetry.The calculated Poisson ratios of these compounds show that they exceed the theoretical upper bound [23,24].As a result, these compounds are likely to exhibit significant elastic deformation under minimally applied strain.In addition, the current results are visible in Figure 2. The parameters for different materials for different purposes have also been investigated, and interesting results have been found [25][26][27][28][29][30][31][32][33][34][35][36].
The thermal characteristics of a compound, such as melting point and thermal conductivity, can be determined using the Debye temperature (θ D ), which is denoted by equation (11), where the Planck constant is h, the Boltzmann constant is k B , the Avogadro number is N A , the density is ρ, the number of atoms in the unit cell is n, and the molecular mass of the compound is M. The compound has a high melting point and thermal conductivity when the θ D is high.Among handled compounds, RbInO 3 has the highest melting point and thermal conductivity due to large θ D , as considered in Table 4.

Electronic properties
The electronic behavior of handled compounds could be revealed by calculating their electronic properties.Figure 3 presents the band structures and, relatedly, the density of states (DOSs) for XInO 3 (X = K, Rb, and Cs) compounds along with respective symmetry lines.The investigated compounds have exhibited halfmetallic properties.Current computations show that the spin-up status' semiconductor behavior, whereas the spin-down state has a metallic nature owing to some bands originating from O p-d occurrences, as seen in Figure 4, reaching the Fermi level and giving the compound a half-metallic character by reducing the gap.
It is possible to observe that the X-atom contributions are about −10 eV for K, −8 eV for Rb, and −5 eV for Cs, respectively, according to Figure 4.In addition to the contribution of roughly −15 eV, the O makes a contribution that lends half-metallicity nature, while the In makes no significant contribution.Moreover, as a result of spinpolarized calculations, each compound investigated has a magnetic moment of about 2μ B .Owing to the five atoms that make up their unit cell, there are likely 15 phonon branches, as is consistent with observations.Twelve of these branches are optical modes, whereas the remaining branches are acoustic modes.The considered compounds are also dynamically stable since no soft mode has been observed.

Lattice dynamical and thermodynamical properties
Figure 6 demonstrates the thermal properties of considered compounds.These representations deal with the relationship between temperature and free energy, Table 4: The longitudinal (V l ), transverse (V t ), and average (V m ) wave velocities and Debye temperature (θ D ) for XInO 3 (X = K, Rb, and Cs) compounds  entropy, and heat capacity.In addition, it is evident that as temperature rises, free energy falls dramatically.Entropy, on the other hand, increases when the temperature rises.
The heat capacity grows dramatically at low temperatures and reaches the Dulong-Petit limit, which is constant at high temperatures.

Conclusion
The objective of this examination was to reveal the structural, elastic, anisotropic, electronic, lattice dynamical, and thermal properties of XInO 3 (X = K, Rb, and Cs) compounds using DFT implemented in VASP 544.Moreover, the investigated compounds are both thermodynamically stable and experimentally synthesizable due to having negative formation energy.In addition, these compounds have demonstrated stable lattice dynamical properties.Since the calculated ν max of these compounds is larger than the upper limit of Poisson's ratio, it is estimated that current compounds will show persistent plastic deformation under applied strain.These compounds have exhibited half-metallic behavior.Besides, they have dominant ionic characteristics.Based on our knowledge, this theoretical study, which is the first examination of investigated compounds, may provide insight into future research.
Funding information There is no funding for this article.

Conflict of interest:
The author states no conflict of interest.
Ethical approval: The conducted research is not related to either human or animal use.

Figure 5
Figure 5 depicts the phonon curves and, relatedly, the PDOS for the respective compound under investigation.Owing to the five atoms that make up their unit cell, there are likely 15 phonon branches, as is consistent with observations.Twelve of these branches are optical modes, whereas the remaining branches are acoustic modes.The considered compounds are also dynamically stable since no soft mode has been observed.

Table 2
depicts the calculated elastic constants (C ij in GPa), bulk, shear, and Young's moduli (B, G, and E in GPa), B/G, G/B, Poisson's ratio, and Vicker's hardness (H f in GPa).

Table 2 ,
[20]O 3 has the highest shear modulus, whereas KInO 3 has the lowest one.As an indicator of elasticity, Young's modulus is determined by stress-tostrain ratios caused by uniaxial deformation[18].Among the investigated compounds, KInO 3 has the lowest Young's modulus, while CsInO 3 has the highest.The B/G ratio, an essential parameter for determining the consistency of the compound, indicates brittleness if it is less than 1.75; otherwise, it indicates ductility.As can be noted in Table2, the computed results of the B/G are larger than the vital value (1.75); therefore, it is possible to say that current compounds have shown ductility characteristics within observations[19].The ratio G/B, another serious criterion, helps predict the bonding peculiarities of compounds and is directly related to the bonding nature of compounds.The ratio G/B, which is another decisive criterion owing to helping predict the bonding peculiarity of compounds, identifies that if it is around 0.6, the compound has a predominantly ionic bonding nature.In contrast, if this ratio is about 1.1, the compound has demonstrated covalent bonding predominantly.Since the G/B ratios of these compounds are less than 0.6, the investigated compounds have exhibited dominantly ionic bonding characteristics, as is seen in Table 2[20].
[22]ogous to G/B, the Poisson's ratio is useful for describing bonding properties.Ionic and covalent bonding compounds have Poisson's ratios of about 0.25 and 0.1, respectively[21].As can be seen in Table2, Poisson's ratios for the studied compounds are larger than 0.25; hence, the present compounds have indicated ionic bonding in harmony with G/B results.Vicker's hardness, the last specification in Table2, was computed utilizing the semiempirical equation proposed by Tian et al.[22]standing on G/B known as Pugh's modulus, as follows: