The CaNiH₃ ternary hydride, typically containing a single molecule, crystallizes in the perovskite-type cubic structure with (#221) space-group. The Wyckoff positions of the atoms (Ca, Ni, and H) inside the unit cell are 1a (0,0,0), 1b (1/2,1/2,1/2), and 3c (1/2,0,0), respectively. In order to calculate the ground state properties of the CaNiH₃ hydride, the total energies are calculated for various volumes around the equilibrium position, and then the obtained results are fitted to the Murnaghanʼs equation of state.^[16] The calculated total energy versus the unit cell volume of CaNiH₃ is shown in Fig. 1. The calculated lattice parameter a₀, bulk modulus B₀, and its first derivative for this ternary compound are listed in Table 1, together with the available experimental and other theoretical results. From Table 1, it is clear that our results are in good agreement with the previously reported experimental and theoretical results. Our calculated equilibrium lattice constants within the PBE-GGA (PW-LDA) are found just 0.39% (2.28%) lower than the experimental value,^[6] indicating the reliability of our calculations. Our computed lattice constants are also in good agreement with those calculated by Sato et al.^[8] Similarly, our predicted bulk modulus at the level of PBE-GGA is also somehow smaller than that obtained within PW-LDA. However, the bulk modulus obtained at the level of PW-LDA is consistent with the previously reported theoretical one.^[8] On the whole, by comparing the results obtained at the levels of PBE-GGA and PW-LDA, we find that the results obtained for the structural parameters by PBE-GGA are improved over those by PW-LDA.

Fig. 1.

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	Fig. 1. Calculated total energy versus volume for CaNiH3 using GGA approximation.

Table 1.

Table 1.

Table 1. Calculated lattice constant a0, bulk modulus B0, its pressure derivative , and elastic constants Cij for CaNiH3 compared with other theoretical and experimental results. . Work Method a0/Å B0/GPa C11/GPa C12/GPa C44/GPa Present LDA 3.443 113.080a 4.279 186.13 76.54 59.28 113.079b Present GGA 3.530 94.223a 3.951 144.38 68.43 54.24 93.752b – – – – Exp. – 3.544[6] – – – – – Other Calc. GGA 3.526[8] 93.2[8] a From Murnaghanʼs equation of state. b From .

Table 1. Calculated lattice constant a0, bulk modulus B0, its pressure derivative , and elastic constants Cij for CaNiH3 compared with other theoretical and experimental results. .

The study of the elastic parameters is crucial to getting knowledge about the bonding among the nearest neighbors, structural stability, and anisotropic character of a material, and these parameters are vital to understanding the mechanical behavior. Moreover, studying the aforementioned parameters at the high pressure also provides information about the material response, mechanical stability, and strength under compression. As our system is cubic, it is well recognized that the three independent elastic constants are C₁₁, C₁₂, and C₄₄. To calculate the elastic constants, the Thomas Charpin method as integrated within the WIEN2k package is used^[12] and the obtained elastic constants for the CaNiH₃ compound are given in Table 1. These are the first results concerning the elastic properties of this compound. From our calculations, we find that the calculated elastic constants C_ij are positive. The stability criteria for mechanical properties^[17] are satisfied, that is, ( ( , alongside satisfying the bulk modulus B₀ criterion, i.e., . From Table 1, one can easily see that the bulk modulus B₀ calculated from the elastic constants has nearly the same value as the one obtained from the fitting of the data into the Murnaghanʼs equation of state (EOS). This might be an estimate of the reliability and accuracy of our predicted elastic constants for CaNiH₃. Furthermore from Table 1, it can also be noticed that the unidirectional compression found in terms of the elastic constant (C₁₁) in the principal crystallographic direction is much higher than that of C₄₄, demonstrating a weaker resistance to pure shear deformation in comparison with the resistance offered to unidirectional compression.

Knowledge of the elastic properties of materials promotes the understanding of the fundamental aspects of the mechanical deformation and structural properties of a crystal. In order to gain a deeper insight into the mechanical properties of the material, we calculate some polycrystalline elastic moduli such as the anisotropy factor A, Youngʼs modulus E, Poissonʼs ratio υ, and shear modulus G by employing the Voigt Reus Hill approach and expressions given in the followings:^[18–21] 1 2 3 4 where G_R is the representation of the Reussʼs shear modulus related to the lower bound values of G, G_V is the representation of the Voightʼs shear modulus related to the upper bound values of G, which are respectively expressed as 5 6 By using these relations, our calculated results are presented in Table 2. As Youngʼs modulus E describes the hardness of a material, meaning that when the value is higher, the material is considered to be stiffer and exhibits covalent bonding.^[22] The elastic anisotropy is another important physical quantity which tells about the structural stability and is decidedly coupled with the induction of the micro-cracks in the relevant material.^[23] If the system is completely isotropic, then gives a value of unity, any deviation from this unity highlights its degree of elastic anisotropy. The computed A for CaNiH₃ is found to be 1.428 (1.082) using the PBE-GGA (PW-LDA), which means that our compound is not elastically completely isotropic. Similarly, another important factor is the ratio B/G which classifies the materialʼs nature as ductile or brittle. Pughʼs criterion^[24] is used, i.e., if , the material will be ductile, otherwise it will be of the brittle nature. By employing this criterion, one can observe from Table 2 that CaNiH₃ is a ductile material (B/G=1.994). Another index of ductility is Cauchyʼs pressure which is obtained by taking the difference, . This difference serves as an indicator of ductility as well.^[25] If for Cauchyʼs pressure is found to be positive (negative), the material is predicted to be of the ductile (brittle) nature. As the calculated results for the Cauchyʼs pressure of CaNiH₃ are positive, hence showing clearly that CaNiH₃ is of ductile nature. Poissonʼs ratio provides more information about the characteristic nature of the bonding forces. It has already been standardized that the values of of 0.5 and 0.25 are upper and lower limits of the Poissonʼs ratio, respectively, for the central force of a solid.^[26] Following this, we see from Table 2 that the interactions in our investigated compound are central because the Poisson ratio is within the mentioned range.^[27] Figure 2 displays the pressure dependence of the elastic constants and bulk moduli of CaNiH₃, which shows a linear dependence for all the plotted curves in the considered range of pressure. Also, we see that C₁₁, C₁₂, C₄₄, and B₀ increase with increasing pressure, and the elastic constant C₁₁ is more sensitive to the change of pressure compared to the other elastic constants. To our knowledge, no experimental measurements or theoretical data for the elastic constants have been reported in the literature for this compound. This may give our results as a prediction for this hydride.

Fig. 2.

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	Fig. 2. (color online) Calculated pressure dependence of elastic constants Cij and bulk modulus B0 for CaNiH3.

Table 2.

Table 2.

Table 2. Calculated density ρ, anisotropy factor A, Youngʼs modulus E, Poissonʼs ratio υ, shear modulus G, (B/G) ratio, and Cauchyʼs pressure for CaNiH3. . Method ρ/g·cm-1 A E υ G B/G GGA 3.842 1.428 120.853 0.285 47.019 1.994 14.196 LDA 4.141 1.082 116.546 0.328 43.873 2.577 17.271

Table 2. Calculated density ρ, anisotropy factor A, Youngʼs modulus E, Poissonʼs ratio υ, shear modulus G, (B/G) ratio, and Cauchyʼs pressure for CaNiH3. .

The role of the Debye temperature in the lattice vibration theory is the same as that of the Fermi temperature in the theory of electrons of metals since both represent a measure of the temperature to separate the low-temperature region (where quantum statistics is used) from the high temperature region (where classical statistics is found to be applicable). This is a very noteworthy parameter which is coupled to many materials’ thermal characteristics, in turn correlating to a lot of their physical properties, for instance, elastic constants, specific heat, and melting temperature. Therefore, the knowledge of this parameter is essential for suggesting a material suitability for the electronic devices.

To calculate the Debye temperature θ_D, a typical technique is derived from the data of elastic constants in which θ_D is estimated by averaging sound velocity v_m by employing the equation^[28] 7 where k_B, h, and ρ denote Boltzmannʼs constant, Plankʼs constant, and the density, respectively. N_A is Avogadroʼs number, n is the number of atoms in a molecule, and M is the molecular weight.

To calculate the average sound velocity, the following relation is used: 8 where v_t and v_l are the transverse and longitudinal sound velocities, respectively. These values are obtained using Navierʼs equation^[29] 9 10 Our calculated results of θ_D, (v_t, v_l, v_m), and ρ of CaNiH₃ employing relations (7)–(10) are tabulated in Table 3. It is found that the longitudinal velocity has a larger value as compared to the above-described velocities because the particle displacement is parallel to the direction of the wave propagation in longitudinal wave. Like the elastic constants, to our knowledge, no prediction or experimental investigation is found yet in the reported literature for comparison. It is expected that future experiments will confirm our reported calculations.

Table 3.

Table 3.

Table 3. Calculated longitudinal vl, transverse vt, average elastic wave velocities vm, and Debye temperature θD for CaNiH3. . Method θD/K GGA 6381.176 3498.307 3900.308 562.55 LDA 6436.889 3254.962 3649.013 539.62

Table 3. Calculated longitudinal vl, transverse vt, average elastic wave velocities vm, and Debye temperature θD for CaNiH3. .

The calculated band structure of the perovskite-type hydride CaNiH₃ along the higher symmetry directions in the Brillouin zone using (PBE-GGA) approach is shown in Fig. 3. The Fermi level is set at zero energy. From this figure, it can be seen that there are some bands that cross E_F, indicating the metallic chiastic of CaNiH₃. For obtaining deeper insight into the electronic structure of CaNiH₃, the total and partial densities of states (TDOS and PDOS) are displayed in Fig. 4. The lowest valence band in the energy range from −10.0 eV to −6.0 eV below E_F is found mainly due to the Ni-s and H-s states with a few contribution from the Ca-s states. The narrow band between −6.5 eV and −5.5 eV is the doubly degenerate level, which is mainly originated from the Ni-d_eg states and is in fact responsible for the bonding of Ni–H. The nonbinding, triplet degenerate Ni-d t2g states are also found to appear at −2.5 eV. At high energies, from −1.5 eV to 1.5 eV, the doubly degenerate state at the Γ point is an antibonding combination of Ni-d_eg and H-s states with Ca-d hybridization, resulting in the only half filling of the band. This half-filled band makes the CaNiH₃ a metallic conductor. The calculated DOS at E_F is around 0.865 states/eV/ atom. These states are mainly the d states of the Ni atoms. However, it is clear from Fig. 4 that there is very small contribution of Ca atomic states to the bands below the Fermi level. Overall the density of states profile and band structure plots are seen almost similar. Our obtained band structures are also found to be comparable with the already reported one by Sato et al.^[8] The authors also predicted that CaNiH₃ is characterized as a metallic conductor.

Fig. 3.

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	Fig. 3. (color online) Band structure alone high-symmetry directions in the Brillouin zone for the CaNiH3 compound.

Fig. 4.

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	Fig. 4. (color online) Calculated total and partial densities of states for the CaNiH3 compound.

In order to determine the thermal effects at high pressure and high temperature, Gibbs computational code based on the quasi-harmonic Debye model is used.^[30] In this method, Gibbs function in the non-equilibrium form is described in the following way: 11 where is the non-equilibrium Gibbs function, E(V) is the total energy/unit cell, the product PV stands for the constant hydrostatic pressure condition, and vibrational Helmholtz free energy A_vib and is written as^[31,32] 12 Here, is the Debye integral for an isotropic solid, and n is the number of atoms/formula unit. The expression for θ_D is given as^[31] 13 where M is the molecular mass per unit cell and B_s is the adiabatic bulk modulus in the Debye model, which is approximated by the static compressibility as follows:^[30] 14 The comprehensive description of the function f(σ) can be seen in Refs. [33] and [34]; σ is the Poisson ratio which takes the value 0.25.^[35] The Gibbs function , which is a function of (V,P,T), for non equilibrium conditions is minimized with regard to volume V as 15 From the solution of Eq. (15), one can derive the corresponding thermal equation of state V(P,T). From this, the heat capacity (C_V), isothermal bulk modulus (B_T), and the coefficients of the thermal expansion (α) can be obtained as^[36] 16 17 18 where γ is the Grüneisen parameter, which is defined as 19 In the present study, the calculations of the thermal properties are done over a temperature range from 0 to 600 K. It is found that the quasi-harmonic model is most likely applicable in view of the fact that the temperature we considered is far away from the melting temperature. The pressure effects are investigated over a range of 0–6 GPa. Figure 5 presents the lattice parameters of the CaNiH₃-perovskite under different pressures at different temperature. One can easily note that the lattice parameters increase with increasing temperature at a given pressure, however, the increasing rate is very modest. The lattice constants dramatically decrease with increasing pressure at a given temperature. The temperature effect on the bulk modulus is shown in Fig. 6. It can be seen from the figure that the obtained bulk moduli are almost constant from 0 to 100 K, but decrease linearly above 100 K. It is also found that the compressibility decreases with increasing pressure at a given temperature and increases with increasing temperature at a given pressure owing to the fact that the effect of decreasing temperature is same as that of increasing pressure on the material. The specific heat is another important thermodynamic property of the material which measures the loss ability or heat retention. Moreover, it provides the insight into the vibrational and many other obligatory properties of a substance. We plot the heat capacities C_V and C_P versus temperature at various pressures in Figs. 7(a) and 7(b). As it can be seen in these figures that at low-temperature ( ), C_V and C_P increase more rapidly with increasing temperature and then C_V becomes proportional to .^[37] Comparing for different pressures, one can deduce that the influence of the pressure on is significant at low temperature ( K), whereas at high temperature, the effect of the pressure on the variation of with temperature becomes smaller and the heat capacity at constant volume calculated at each pressure tends to approach the Petit and Dulong limit ( similar to most of solids.^[38] At high temperature, continues to increase linearly. But, for the intermediate temperature range, it is found that the specific heat capacity is mainly governed by the atomic vibrations. The calculated C_V and C_P at room temperature (300 K) and zero pressure are and , respectively.

Fig. 5.

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	Fig. 5. (color online) Variation of the lattice constant as a function of temperature at different pressures for CaNiH3.

Fig. 6.

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	Fig. 6. (color online) Variation of the bulk modulus as a function of temperature at different pressures for CaNiH3.

Fig. 7.

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	Fig. 7. (color online) Variation of the heat capacities (a) CV and (b) versus temperature at various pressures.

The variation of the Debye temperature with respect to the pressure is presented in Fig. 8. From the figure, one can easily note that θ_D is almost constant within the temperature range from 0 to 100 K and starts decreasing linearly with the increase of the temperature above 100 K. We also note that with the constant temperature, θ_D approximately increases linearly with respect to the applied pressure. Thus the temperature/pressure dependence of θ_D clearly shows that the atomic thermal vibrational frequency of the CaNiH₃ hydride varies with the temperature and pressure. As it is recognized already that the harder materials demonstrate higher θ_D. It is obvious from Fig. 8 that the θ_D with regard to the pressure confirms the positive trend. From our calculations, we find that the θ_D at room temperature and zero pressure is 574.65 K. We also find that our obtained results are in nice agreement with the obtained results by following the relations of the sound velocities, i.e., 562.55 K. Our work regarding C_V and θ_D is the first attempt about this material as there is no study found in the literature. Hence our work might be the first attempt in this direction.

Fig. 8.

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	Fig. 8. (color online) Variation of the Debye temperature θD with the temperature at different pressures for CaNiH3.