A novel pressure variation study on electronic structure, mechanical stability and thermodynamic properties of potassium based fluoroperovskite

The SrKF₃ compound has an ideal cubic structure with space group Pm-3 m (no. 221). In our recent work [14], we have calculated ground state structural parameters by structural relaxation using Birch Murghan's equation of state. The details of optimized lattice parameters such as lattice constants, tolerance factor, bond lengths, bulk modulus, and its pressure derivative at fixed pressure and zero temperature can be found in our earlier published work [15]. However, for the convenience of new reader we have mentioned modified and precise form of that work in table 1. For the lattice constant (a_o) about 4% deviation is observed between experimental and present calculation that is obvious, because the experimental work was done at ambient conditions while the present work is done at zero kelvin and Castro [29], uses large volume of unit cell as compared to the present first principle investigation. Our current task is to analyze effect of pressure on structural and electronic parameters of SrKF₃ compound with 0–25 GPa range with a step size of 5GPa. The hydrostatic pressure of 0.00 GPa is considered as a base for ground state stable structure and increase in pressure further decreases the volume of the crystal structure as shown in table 1. The calculated lattice constants by LDA and GGA approximation schemes and bond lengths are plotted in figures 1 and 2 respectively. It can be analyzed from the figures that both lattice constants and bond lengths are going to decrease as pressure increases in accordance with our previous work/Other work [7, 9, 14, 15] (at ambient pressure). The key issue to discuss the electronic structure of SrKF₃ is the formation and widening of bandgap with the increase in pressure. As it can be observed from table 2 that compression on the system (0 GPa–25 GPa) increases the bandgap from 3.307 eV to 3.845 eV from GGA scheme and 6.799 eV to 7.302 eV from mBj potential as shown in figure 3. This change in the value of bandgap is in continuation with our previous study [15], that GGA and LDA schemes undervalues energy bandgap of wide bandgap semiconductors and insulators. The cause of this underestimation is discussed in detail in [19]. The calculated lattice constants, bond lengths and volume of unit cell compress to a reasonable extent that the constituent polyhedral of SrKF₃ do not become distorted by the change in pressure from 0–25 GPa. In general, the opening of wide bandgap in fluoroperovskites are due to electro-negativity of fluorine ion. Similar to oxide perovskites, fluorine based perovskites also form the weak covalent bond [35–38]. In this regard, the pressure induced behavior of SrKF₃ is in good agreement with previously reported work by Lee et al [39], that decrease in bond length increases the bonding energy as strength of covalent bond decreases, consequently wider bandgap. In this work, we have examined that as compression increases there is decrease in bond lengths both for Sr–F and K–F as shown in figure 2 respectively. While according to table 2, the calculated bond lengths are in reasonable agreement with the previously published work at 0 GPa. The DOS and energy band structures at different pressure ranges from 0–25 GPa are shown in figures 3–5. We have observed that throughout the calculations the position of maxima of valence band F-2p states remains almost same. The total and partial density of states (TDOS & PDOS) occupies energy interval from (E_F  −  30 eV) up to (E_F  +  15 eV) at ambient pressure as shown in figure 5. It can be observed that above fermi level slightly wide peak is dominated by Sr-3d state at about (E_F  +  8 eV). Next energy interval (E_F  −  3 eV) is governed by F-2p state. The upper valence band situated in a range from (E_F  −  12 eV) to (E_F  −  20 eV) is due to hybridized state of Sr-4p, K-2p and F-2s states respectively. So, the cubic SrKF₃ is by virtue, a very wide bandgap modern semiconductor and with the increase in the pressure the strength of hybridization increases with reduced bond lengths which give rise to antibonding phenomenon among bonds. This antibonding creates high energy, which pushed up energy level away from E_f, consequently widening of bandgap occurs which is previously reported in Pseudo potential theory [40, 41] as well.

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The mechanical stability of cubic crystals under pressure can be determined by elastic constants. In order to verify the change in behavior of mechanical parameters under pressure of (0-25GPa), we used to perform calculations at different values of reduced volumes, each of them corresponds to specific value of hydrostatic pressure. The complete optimization against each value of pressure is done with the step size of 5 GPa pressure. Furthermore, for computing stress tensor, we employ Charpin method [42]. For cubic system, there are three independent elastic constants namely C₁₁, C₁₂, and C₄₄ respectively [43]. In this section, our main aim is to analyze effect of pressure variation on tensile strength, shear strength, rigidity, brittle/ductile behavior, and trends of directional/ non-directional bonding, average sound velocities, and Debye temperature in cubic SrKF₃ fluoroperovskite compound. The details of all calculated elastic and mechanical parameters can be found in the following references [44–46]. Figure 6 presents the variation of elastic moduli such as C₁₁, C₁₂, and C₄₄ and B with regard to different values of pressure. From figure 6 and table 3 it can be observed that all elastic constants increase with the rise in compression. However, a prominent increase is observed in C₁₁ elastic constant, which is concerned with elasticity in length and bulk modulus, which is measure of hardness of the solid. The change in the value of C₁₁ is from 98.101 GPa to 238.746 GPa and the change in bulk modulus value is from 38.71 GPa to 119.74 GPa respectively. The increase in C₁₁ can be best explained in relation with bond strength enhancement (Sr–F, and K–F) because at elevated pressures there is decrease in the bond length and increase of charge density as mentioned in the previous sections. The elastic moduli of shape deformation explore less effect against variation in pressure,which indicates that enhancement of bond strength is less effected by shear deformation. To verify the stability of cubic SrKF₃, we calculate elastic constants for strontium based fluoroperovskite as shown in table 4 within pressure range from 0–25 GPa. It is found that SrKF₃ compound obey the modified stability criteria [47] for cubic crystals under finite strain till 20 GPa i.e.

$$\begin{eqnarray}&&{{M}_{1}}=\frac{\left({{C}_{11}}+2{{C}_{12}}\right)}{3}+\frac{P}{3}>0\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{{M}_{2}}={{C}_{44}}-P>0\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{{M}_{3}}=\frac{\left({{C}_{11}}-~{{C}_{12}}\right)}{2}-P>0.\end{eqnarray} \tag{ 3 }$$

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It can be noticed from figure 7 and table 4 that calculated elastic constants do not satisfy stability condition from equation (2) because at 25 GPa pressure stability criteria M₂ (equation (2)) value is lower than zero, which implies that calculated elastic constants cannot satisfy all mechanical stability conditions, hence the compound is not mechanically stable above 20 GPa which is in agreement with our electronic structure calculations that the compound is mechanically stable against pressure less than 25 GPa. The Shear modulus (G_H), and Young's modulus (Y) also increase with the elevated pressure in similar trend with the value of Bulk modulus (B). For example, the value of G_V, G_R, G_H, and Y at 25 GPa is 51.579 GPa, 34.702 GPa, 43.140 GPa, and 115.545 GPa are several times greater than at 0 GPa respectively, indicating that SrKF₃ compound becomes less compressible at higher pressures ranges. In addition, we observed that with the change in pressure from 0 to 25 GPa the transition is observed from brittle to ductile behavior because B/G value shift from 1.39 to 2.80 and it is previously reported in our own work [2] that, the B/G  >  1.75 classify material as ductile, while B/G  <  1.75 classify material as brittle. Similar effect of pressure variation is observed for poison's ratio (ѵ). The angular characteristics of atomic bonding via Cauchy's law or Cauchy pressure [48] demonstrates comparable transition from negative value of  −11.68 to the positive value of 29.87 respectively as with the increase in pressure. But according to widening trend of bandgap this change in sign is just an indicator in reduction of its angular characteristics [49]. The similar kind of pressure induced behavior is observed for other fluoroperovskite of the same series such as BaLiF₃ and SrLiF₃ [30]. Next, we interpret value of elastic anisotropy factor (A), it can be noticed that the degree of deviation of anisotropic behavior is increased with the increase in pressure. The effect of varying pressure on bond bending trend, as given by Kleinman parameter [50, 51], agrees well with the above-mentioned responses. The trend of melting temperature depicts that an increase in pressure induces less tendency of melting extent of SrKF₃ compound, which eventually increases its melting temperature.

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The Debye temperature (θ_D) or Debye cut-off frequency is a significant form of temperature, which used to quantify several thermodynamic properties in the solid. It is basically a measure of the vibrational response of the crystal. In actual, it is the temperature above which the crystal behaves classically. There are two main methods to calculate Debye temperature (θ_D) including elastic constant method and specific heat measurement method. In this section, we consider elastic constant method to calculate Debye temperature (θ_D) and associated parameters following the formulas of the reference [52]. It can be observed from table 3 that wave velocities show a quadratic variation which increases monotonically with rising pressure. However, they can be divided into two groups because longitudinal wave velocity increases rapidly with pressure and is greater than the shear wave velocity. As Debye temperature is directly derived from the elastic wave velocity, so the similar trend of rise is also observed for Debye temperature (θ_D). This monotonic increase can be attributed to the increase in the bulk modulus under elevated pressure ranges because it is well known that the Debye temperature is proportional to the bulk modulus and increase in bulk modulus induces a phenomenon of phonon softening under pressure. As per best of our knowledge, there is lack of available experimental data to compare elastic wave velocity and Debye temperature under pressure, and we hope our study can provide useful guidance for future investigations.

The thermodynamic properties of SrKF₃ compound are used to obtained from Quasi-harmonic Debye model as implemented in the Gibbs program. The calculated details can be seen in the previously published work [25]. The pressure effects are determined in the range of 0–25 GPa, in which cubic stability of SrKF₃ fluoroperovskite remains valid and thermal properties are studied in the temperature range from 0 to 600 K where Quasi-harmonic Debye model remains valid [51]. Figures 8(a) and (b) presents the relation between lattice parameters (lattice constant and unit cell volume) and figure 9 presents isothermal bulk modulus with temperature T up to 600 K at P  =  0 GPa, 5 GPa, 10 GPa, 15 GPa, 20 GPa, and 25 GPa respectively. It can be noticed that both temperature and pressure have inverse relation with lattice parameters and bulk modulus because at a given pressure lattice parameters such as lattice constants and volume expansion increases with the increasing temperature and bulk modulus decreases with increasing temperature at a given pressure and vice-versa. In fact, at higher pressure thermal effects becomes weaker and stronger for lattice parameters and bulk modulus respectively, which reveals that effect of pressure and temperature are inversely proportional to each other. So, increase in temperature causes a significant reduction in the hardness of SrKF₃. Next, we explore as important fundamental parameter of Debye temperature θ_D that depicts many physical properties such as specific heat, elastic constants and melting extent of a material [53]. From the plot of Debye temperature θ_D as shown in figure 10. It should be noted that the static values of the Debye temperature (at T  =  0 and P  =  0) calculated from the quasi-harmonic Debye model is 396.989 K which is very close to the value computed from elastic properties (396.512) as listed in table 3. However, to the best of our knowledge there is no experimental data for comparison with our calculated values. One can notice that (a) for a fixed pressure, θ_D decreases with increase in temperature and for a fixed temperature, θ_D increases with the increasing pressure. (b) The process of increase in pressure and decrease in extent of melting temperature will lead to increase Debye temperature (θ_D) and bulk modulus because at higher pressure ranges bulk modulus increases, as a result a phenomenon of phonon softening takes place and θ_D will increase in accordance with previous temperature ranges [54]. At the end, we analyze material's extent to store heat in terms of specific heat capacities at constant volume and constant pressure C_V and C_P respectively. It can be observed from figures 11(a) and (b) that variation of C_V and C_P are similar to each other as temperature increases. However, the trend of C_V increases slowly following a common phenomenon in all solids at higher temperature ranges by approaching towards Dulong–Petit limit (123.7 J mol⁻¹ K⁻¹) [55] where anharmonic effects prominently suppressed. However, C_P behavior deviates from T  >  300 K and does not converge to a constant value. In particular, when pressure vanishes, the values of C_P increases rapidly at about higher temperature. Unfortunately, there is lack of experimental data on the thermodynamic properties to compare with but we are in hope that our work will serve as a reference material for other co-researchers of this filed.

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