Modeling of structural, elastic, mechanical, acoustical, electronic and thermodynamic properties of XPdF₃ (X = Rb, Tl) perovskites through density functional theory

The previous studies [36–42] on halide perovskites reveal the accuracy of GGA for calculations of structural parameters in comparison to other exchange correlation potentials. Therefore, we got motivation for calculation of lattice constant (α_o), bulk modulus (B), rate of change of bulk modulus with respect to pressure (B^/) and ground state energy (E_o) for XPdF₃ (X = Rb, Tl) using PBE-GGA exchange potential. First, the ground state stability of these compounds was tested among non magnetic (NM), ferromagnetic (FM) and anti ferromagnetic (AFM) phases. The resulted ground state energy was found to be less in NM phase. Hence our calculations predict that these compounds are most stable in NM phase in ground state as shown in figures 2 and 3. Therefore, we calculated the structure parameters α_o, B, B^/ and E_o only in NM phase by art of fitting variables using Birch-Murnaghan equation (equation (3)) [43]. Where, using data of minimization of energy process, energies versus volume was re-plotted with help of equation (3). The constants V_o (α_o × α_o × α_o), B, B^/ and E_o were set to fit the re-plotted curves with those in figures 2 and 3. These constants are the obtained structural parameters.

$$\begin{eqnarray}&&\begin{array}{l}E\left(V\right)={E}_{o}+\displaystyle \frac{9{V}_{o}{B}_{o}}{16}\left[\left\{\left.\left(\displaystyle \frac{{V}_{o}}{V}\right)-1\right\}\right.{B}_{o}^{/}\right.\\ \,+\left.\left\{{\left.{\left(\displaystyle \frac{{V}_{o}}{V}\right)}^{\displaystyle \frac{2}{3}}-1\right\}}^{2}\left\{\left.6-4{\left(\displaystyle \frac{{V}_{o}}{V}\right)}^{\displaystyle \frac{2}{3}}\right\}\right.\right.\right]\end{array}\end{eqnarray} \tag{ 3 }$$

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The calculated values are repotted in table 1. The available results of experimental [26, 27] and theoretical [28–30] studies on XPdF₃ (X = Rb, Tl) are also given in the table for comparison. These confirm the reliability of our present results. Further, the hardness of TlPdF₃ is larger than RbPdF₃ because of comparatively high bulk modulus. Also the lattice constant of TlPdF₃ is greater than RbPdF₃ because of grater size of Tl atom than Rb atom.

Elastic constants C_ij (C₁₁, C₁₂, C₄₄) play significant role in deducing the mechanical behavior of a cubic material. These provide vital information about mechanical stability, ductility or brittleness, nature of binding interactions. These constants also provide enough stuff for calculations of thermodynamic parameters such as melting temperature, Debye temperature [44] etc. In present study C_ij were calculated from the relation of strain as a function of unit cell volume using Charpin scheme [45] as employed in Wien2k code. In theoretical calculations, the cubic symmetry of a material in homogeneous expansion can be retained by convergence of elastic constants. Therefore, C₁₁, C₁₂ and C₄₄ were calculated with respect to increasing k points. All three elastic constants (listed in table 2) were converged at 1000 k points. Due to lack of experimental data, our reported elastic constants can be used as reference in future studies about XPdF₃ (X = Rb, Tl). However, we confirm the stability of these compounds in cubic phase from following condition [46, 47].

$$\begin{eqnarray*}&&\begin{array}{l}{C}_{44}\gt 0,\,{C}_{11}\gt B\gt {C}_{12},\displaystyle \frac{1}{3}\left(2{C}_{12}+{C}_{11}\right)\gt 0,\\ \displaystyle \frac{1}{2}\left({C}_{11}-{C}_{12}\right)\gt 0\end{array}\end{eqnarray*}$$

Various materials are subjected to loads or forces, when they are in services. In such situations it is essential to know the properties of the material. So, that one may design a structure from those materials which may bear the estimated loads without any deformation and fracture. The mechanical performance of a material replicates the connection between its response and deformation to an applied force or load. Key mechanical design properties are strength, hardness, stiffness, toughness and ductility [48]. These can be tested from calculated results of shear modulus (G), young modulus (E), Poisson's ratio (υ), B/G, and Cauchy pressure (C₁₂–C₄₄) (table 2). These were calculated using by help converged values of elastic constants using below relations [49, 50].

$$\begin{eqnarray}&&E=\displaystyle \frac{9GB}{3B+G}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\upsilon =\displaystyle \frac{3B-G}{2\left(3B+G\right)}\end{eqnarray} \tag{ 6 }$$

Here, G and B were calculated from average of Voigt and Reuss bounds of share (G_v and G_R) and bulk (B_V and B_R) moduli [51].

$$\begin{eqnarray}&&{G}_{V}=\displaystyle \frac{1}{5}\left(3{C}_{44}+{C}_{11}-{C}_{12}\right)\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{G}_{R}=\displaystyle \frac{5\left({C}_{11}-{C}_{12}\right){C}_{44}}{3\left({C}_{11}-{C}_{12}\right)+4{C}_{44}}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{B}_{V}={B}_{R}=\displaystyle \frac{{C}_{11}+2{C}_{12}}{3}\end{eqnarray} \tag{ 9 }$$

To characterize the stiffness of a material, the measurement of bulk's modulus (B) is required. The material will be stiffer if value of B is higher [52]. In present calculations, B results reveal (table 2) the higher stiffness of TlPdF₃ than RbPdF₃. Furthermore, the bonding between constituent atoms of a hard material has covalent nature which can be indicated by higher value of shear modulus [53]. The young modulus (E) determines the degree of resistance of a material to plastic deformation [48]. The present calculations explore RbPdF₃ as higher resistive to plastic deformation than TlPdF₃. The Poisson's ratio (ratio between transverse and longitudinal strains) is another important mechanical parameter which signifies the stability of a compound to shear stress and is indicator of nature of bonding forces. Its possible values are in range of 0 < ʋ < 0.5 [54]. Larger value of ʋ specifies high plasticity of material. Therefore, TlPdF₃ has high plasticity in comparison of RbPdF₃. Moreover, the bonding forces between both the studied compounds are central type, as calculated ʋ is in allowed range (ʋ ≥ 0.25) of such forces.

The ductile or brittle nature can be recognized by Pugh's ratio (B/G) and Cauchy pressure (C₁₂–C₄₄). According to ductility conditions [55], Pugh's ratio is greater than 1.75 and Cauchy pressure is positive both TlPdF₃ and RbPdF3. Therefore, these perovskites are ductile in nature. However, the ductility of TlPdF₃ is large in comparison of RbPdF3 because of larger value Pugh's ratio and Cauchy pressure. The complete understanding of the ductile nature of a material is essential for at least two reasons. First, it gives information to a structure designer about the limit of plastic deformation prior to fracture. Second, it indicates the limit of permissible deformation at stage of fabrication process. We occasionally give preference to the materials of high ductility as they may experience local deformation exclusive of fracture.

The elastic anisotropy (A) is an imperative parameter used for measurement of the degree of anisotropy in materials; in addition, it has an important usage in engineering sciences to examine the possible micro cracks in materials [56].

$$\begin{eqnarray}&&A=\displaystyle \frac{2{C}_{44}}{{C}_{11}-{C}_{12}}\end{eqnarray} \tag{ 10 }$$

For a perfect isotropic material, A takes the value of unity and the difference from unity quantifies the extent of elastic anisotropy [56]. It is apparent from table 2 that A for both the materials deviates from unity; therefore, they have mechanical anisotropic behavior. Furthermore, TlPdF₃ displays more anisotropy than RbPdF₃ because it's 'A' finds relatively high deviation from unity.

Melting temperature (T_M) is a material's specific defined temperature above which the material changes from solid to liquid state. It can be computed in Kelvin by below empirical formula [57].

$$\begin{eqnarray}&&{T}_{M}=\left[554+5.911{C}_{11}\right]\pm 300\end{eqnarray} \tag{ 11 }$$

The obtained T_M through equation (9) for RbPdF₃ [1330.10 $\pm 300$] is greater than TlPdF₃ [1013.28 $\pm 300$]. Our calculated values are closer to other reported values of T_M for other transitional atom containing fluoride perovskites [19, 42, 48, 58].

The elastic wave velocities in a material can be calculated from elastic constants. Generally, elastic waves are divided in two modes, longitudinal wave v_L having parallel polarization to the direction of transmission and two shear waves v_T1 and v_T2 with perpendicular polarization to direction of propagation. The computed velocities of elastic waves for XPdF₃ (X = Rb, Tl) in [100], [110] and [111] directions (using equations (12), (13) [59, 60]) are given in table 3.

$$\begin{eqnarray}&&\begin{array}{l}{V}_{l\left[100\right]}=\sqrt{\displaystyle \frac{{C}_{11}}{\rho }},\,{V}_{l\left[110\right]}=\sqrt{\displaystyle \frac{{C}_{11}+{C}_{12}+{C}_{44}}{2\rho }},\\ {V}_{l\left[111\right]}=\sqrt{\displaystyle \frac{{C}_{11}+2{C}_{12}+4{C}_{44}}{3\rho }}\end{array}\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&\begin{array}{l}{V}_{T1\left[100\right]}={V}_{T2\left[100\right]}={V}_{T1\left[110\right]}=\sqrt{\displaystyle \frac{{C}_{44}}{\rho }},\\ {V}_{T2\left[110\right]}=\sqrt{\displaystyle \frac{{C}_{11}-{C}_{12}}{2\rho }},\,{V}_{T1\left[111\right]}={V}_{T2\left[111\right]}\\ \,=\sqrt{\displaystyle \frac{{C}_{11}-{C}_{12}+{C}_{44}}{3\rho }}\end{array}\end{eqnarray} \tag{ 13 }$$

Table 3.  Calculated density ρ (g cm⁻³) and velocities (in m s⁻¹) of elastic waves in [1 0 0], [1 1 0] and [1 1 1] directions for RbPdF₃ and TlPdF₃.

Compound	ρ	${V}_{l\left[100\right]}$	${V}_{T1\left[100\right]}$	${V}_{T2\left[100\right]}$	${V}_{l\left[110\right]}$	${V}_{T1\left[110\right]}$	${V}_{T2\left[110\right]}$	${V}_{l\left[111\right]}$	${V}_{T1\left[111\right]}$	${V}_{T2\left[111\right]}$
RbPdF3	3.85	5877	1338	1338	4739	1338	3602	4430	3041	3041
TlPdF3	7.36	3249	1605	1605	3323	1605	893	3596	1179	1179

From the table it is clear that longitudinal elastic waves are greatest velocity in [100] direction and the shear elastic waves have lowest speed in [110] direction for all the two compounds. These two perovskites have fairly similar acoustic behavior. However, due to relatively heavier mass of Tl in comparison of Rb, TlPdF₃ has lower sound velocities than RbPdF₃.

Equilibrium lattice constants were utilized for calculation of electronic properties. These properties are associated with band structure and density of states (DOS) and are used for predicting of the electronic nature of a material. The calculated band structures with corresponding DOS are given in figures 4 and 5. Since, band structures of the spins up and down channel are replica of each other. Therefore, here we are presenting electronic properties only in spin up channel. It can be seen from figures that for both compounds, the valence band states cross the Fermi level and reaches at 1.3 eV in the conduction band. Therefore, these compounds are metallic. From DOS plots in panel b of figures, it can be deduced that the valence band extends from −7 eV to 1.3 eV, and is mainly populated by d and p states of Pd and F atoms respectively with minor contribution of d states X atom in lower part. The conduction band is majorly occupied by p states of F with minor collaboration of other all states of constituent atoms, and extends from 3 eV to 8 eV.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For further verification of thermodynamic stability of XPdF₃ (X = Rb, Tl) in cubic phase, we have calculated enthalpy of formation (ΔH) by taking difference between total ground state energy of these compounds and their constituent atoms by following equation [16]

$$\begin{eqnarray}&&{\rm{\Delta }}H={E}_{0}\left({{\rm{XPdF}}}_{3}\right)-\alpha {{\rm{E}}}_{{\rm{X}}}-\beta {{\rm{E}}}_{{\rm{Pd}}}-\gamma {{\rm{E}}}_{F}\end{eqnarray} \tag{ 14 }$$

Here, ${E}_{0}\left({{\rm{XPdF}}}_{3}\right)$ is the total ground state energy of ${{\rm{XPdF}}}_{3}$ and ${{\rm{E}}}_{{\rm{X}}},$ ${{\rm{E}}}_{{\rm{Pd}}},$ ${{\rm{E}}}_{{\rm{F}}}$ are the energies of distinct atoms in their respective lattice, per unit cell. The numbers of atoms X (Rb or Tl), Pd and F per unit cell are indicated by letters α, β and γ respectively. The calculated values of ΔH for RbPdF₃ and TlPdF₃ are −1.291 274 Ry and −1.1444 Ry, respectively. The negative sign of ${\rm{\Delta }}H$ signify the formation reactions of investigated compounds as exothermic. This assures the stability of the compounds in cubic phase.

By applying Quasi-harmonic Debye approximation [61], RbPdF₃ and TlPdF₃ were investigated for thermodynamic properties. The variation of cell volume, bulks modulus, Debye temperature (θ_D), thermal expansion (α), heat capacity (C_v) and Grüneisen parameter (γ) with respect to pressure and temperature have been plotted and discussed. The temperature and pressure were taken in range of 0–1000 K (with gap of 100 K) and 0–10 GPa (with gap 5 GPa) respectively. Figures 6(a), (b) shows the pressure and temperature dependence of cell volume respectively for RbPdF₃ and TlPdF₃. From these figures, it is clearly observed that unit cell volume increases with rise in temperature while it decreases with increasing pressure. This is due to the compression and relaxation due to change of pressure and temperature respectively. The variation in volume with changing temperature and pressure is a general trend in solids. Figures 7(a), (b) presents the variation in bulk modulus B while changing temperature and pressure for RbPdF₃ and TlPdF₃. From these figures, we observe a decrease in the value of B with respect to temperature whereas B is found to increase with pressure. The computed B values are 66.04 GPa and 67.43 GPa respectively for RbPdF₃ and TlPdF₃ at 0 GPa pressure and 0 K temperature. These values are in accordance to the values obtained from Birch-Murnaghan equation of state and elastic constants. However, it is also clear from thermodynamic calculations that the nature of stiffer resistance of TlPdF₃ is little greater than RbPdF₃. Figures 8(a), (b) presents the specific heat (C_V) as function of pressure and temperature. Under various pressures and temperatures, the nature of C_V diagram for both the compounds is nearly similar. C_V starts to rise quickly within 0 K to 200 K (usually low temperature range), after 200 K a lethargic rise in the value of C_V is observed up to 600 K, and beyond that it becomes constant. Rapid increase in C_V within 0 K to 200 K temperature pursues renowned Debye model $\left[{C}_{V}(T)/{T}^{3}\right].$ The reason behind this may be explained as follow. The modes of vibration of long wavelengths fully occupy the lattice under consideration as continuum [62]. However, under passable high temperature value, it follows $\left[{C}_{V}(T)/3R\right],$ and is in agreement to Dulong and Petit law [63]. The values for C_V at 0 GPa and 300 K were computed to be 116.78 and 112.21 J mol⁻¹ K for RbPdF₃ and TlPdF₃ respectively.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Debye temperature (θ_D) helps to know many properties of solids like melting temperature, specific heat etc directly or indirectly. Figures 9(a), (b) shows the variation of Debye temperature (θ_D) with respect to pressure and temperature for RbPdF₃ and TlPdF₃. The (θ_D) is observed to decrease with rising temperature and increases with rising pressure. Calculated values of (θ_D) at 0 GPa and 300 K are 346 K and 287 K for RbPdF₃ and TlPdF₃ respectively. Further, in figures 10(a), (b), we have presented the variation of thermal expansion 'α' with varying temperature and pressure respectively for RbPdF₃ and TlPdF₃. 'α' for both the compounds was found to increase rapidly up to 200 K and at higher temperature 'α' is found to increase very slowly. The slow rise of 'α' above 200 K may be as a result of saturation in α up to 200 K. Grüneisen parameter (γ) designates the pressure/temperature dependence of phonon frequencies [64]. The computed γ is given in figures 11(a), (b). The value of γ at 300 K and 0 GPa is almost same and equal to 2.428 and 2.421 for RbPdF₃ and TlPdF₃ respectively.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Hence from thermodynamic parameters, we concluded that pressure and temperature have got opposite effects on the calculated parameters.