Figure 1 shows the optimized unit cell of Ca₃PbO with Wyckoff positions given as Ca (0, 0.5, 0.5), Ca (0.5, 0, 0.5), Ca (0.5, 0.5, 0), O (0.5, 0.5, 0.5) and Pb (0, 0, 0). The structures of Ca₃SnO and Ca₃GeO are all similar to that of Ca₃PbO with the position of Pb, Sn and Ge being the same. The convergence was achieved at a pressure of zero GPa. The equilibrium lattice constants, bulk modulus and derived bulk modulus have been obtained by minimizing the total crystal energy. This minimum energy was calculated for different values of lattice constants using second order Birch-Murnaghan equation of states. The calculated values are represented in Table 1 for cubic phase of space group Pm3 m ¯ (#221) for Ca₃BO (B = Pb, Ge, Sn). The lattice constants are 4.87Å, 4.86Å, and 4.84Å respectively for Ca₃BO (B = Pb, Ge, Sn) compounds which are found to compare well with other studies.

The three crystals show very close similarities in their sizes because of the similarity in the atoms in each crystal. The value of lattice parameters for all the three crystals are in agreement with other theoretical and experimental studies available. The values of bulk modulus and volume of single crystal also agrees

with the data of other studies.

Figure 2 shows the optimization of energy against lattice parameter. The lattice parameter was found to be 9.199 Bohr (4.87 Å) with volume of 115.35Å3 for Ca₃PbO. The optimization follows for all the three crystals.

In this section, the partial density of states and band structures of Ca₃BO (B = Pb, Ge, Sn) are discussed. The Fermi energy for all these crystals was set at zero and they all show semi-metallic behaviour. In Figure 3, Ca₃PbO shows that above the Fermi level O-s states and Pb-s states are dominant even though other states also appear. Below the Femi level Ca-s-p states are more dominant. Ca₃GeO has Ge-s and Ca-s states in the conduction band and for Ca₃SnO there are Sn-s and Ca-s states. It is clearly seen that the electrons of metallic atoms are majorly responsible for the conduction of energy in the compounds. Ca₃PbO has

more electrons on the conduction band making it better conductor compared to the rest. This could be attributed to the position of Pb in the periodic table that makes it contribute more valence electrons.

From Figure 3, it is seen that s-states of Ca and Ge are dominant in the conduction band but p-states are the ones dominant for Oxygen in the conduction band. This is expected to be the case for Ca₃SnO and Ca₃PbO compounds where oxygen is expected to behave in the similar way as in Ca₃GeO.

From Figure 4, the band structures are plotted and all the three compounds show semi-metallic nature with direct band gap for all the compounds. The high symmetry points are all similar with strikingly similar band structures. The only differences is the Fermi energy of each of the compounds for example for Fermi energy for Ca₃SnO is 8.64 eV and that of Ca₃PbO is 9.16 eV.

The mechanical properties of Ca₃PbO were also simulated using thermo_pw code which is a post-processing code in quantum espresso to find the stability nature of this material. The cubic crystal has only three independent parameters which are C₁₁, C₁₂ and C₄₄. For a solid under strain the elastic energy is given as [19] ,

Δ E V = 1 2 ∑ i = 1 6 ∑ j = 1 6 c i j e i e j (1)

where V is the undistorted lattice cell volume, ΔE is the change in energy from the strain vectors e = (e, e₂, e₃, e₄, e₅, e₆) and c is the matrix of the elastic constants [20] . The Born’s stability criteria [21] at zero pressure requires that the elastic constants are related such that: C₁₁ >0, C₁₁ - C₁₂ > 0, C₄₄ > 0, C₁₁ + 2C₁₂ > 0 and C₁₁ > B > C₁₂. These constants are calculated by using the obtained second-order elastic constants of Equation (2); the results are given in Table 2.

C i j = 1 V 0 ∂ 2 E t o t ∂ ξ i ∂ ξ j (2)

By using Born’s stability rule, it can be seen that these compounds are stable. This is important in mechanical application where pressure is to be applied on the crystal like in thermoelectric energy generation [22] .

The Zener anisotropy factor (A), Poisson ratio (õ), and Young’s modulus (E) that are important parameters in understanding elastic properties are also calculated using these formulas [12] ;

A = 2 C 44 C 11 − C 12 (3)

δ = 1 2 { B − 2 3 G B + 1 3 G } (4)

E = 9 G B G + 3 B (5)

where G is the isotropic shear modulus as a function of crystal orientation and is given as G = ( G V + G R ) / 2 , herein G_V is Voigt’s shear modulus (which is related

to the upper bound of G values) and G_R is Reuss’s shear modulus (which is related to the lower bound of G values) and can be written as G V = ( C 11 − C 12 + 3 C 44 ) / 5 and 5 / G R = 4 / ( C 11 − C 12 ) + 3 / C 44 , respectively.

Table 3 shows calculated values of Voigt’s isotropic shear modulus, Reuss’s shear modulus, ratio of G/B, Young’s modulus E, maximum and minimum Poisson’s ration (n_max, n_min) and compressibility factor β for all the three crystals. It is seen that the isotropic shear modulus are such that the value of E_V and E_R increases from Ca₃PbO, Ca₃SnO and Ca₃GeO respectively. The ratio G/B is called Pugh’s ratio which shows whether a material is brittle or ductile. For G/B > 0.5 shows brittle nature of the material but G/B < 0.5 shows that the material is ductile. For the three crystals, G/B values calculated are 0.82, 0.80 and 0.83 for Ca₃PbO, Ca₃SnO and Ca₃GeO respectively. This shows that these crystals are brittle. It can also be concluded that Ca₃GeO has the largest Young’s modulus which is an indication that it is stiffer compared to the other two compounds.

For Poisson’s ratio less than 0.25, the material shows strong covalent bonds and value larger than 0.25 indicates ionic bond [23] [24] [25] [26] [27] . As can be seen from our computations all our Poisson’s ratios are less than 0.25 which is an indication that all these atoms have strong covalent bonds. For bulk materials, elastic anisotropy is determined by Zener anisotropy factor A which if is 1, shows that the material is isotropic otherwise the material is anisotropic. All three materials show factor of A as 1.42, 1.39 and 1.09 for Ca₃PbO, Ca₃SnO and Ca₃GeO respectively. It can therefore be concluded that these compounds are anisotropic. This therefore implies that direction of pressure application is critical for maximum performance of these materials in thermoelectric design.

The compressibility is a measure of elasticity defined from the relation;

β = C 11 − C 12 Ω (6)

Ω = ( C 11 + C 12 ) C 11 − 2 C 12 2 (7)

The compressibility values are found to be 0.0064 GPa⁻¹, 0.0058 GPa⁻¹ and 0.0061 GPa⁻¹ for Ca₃PbO, Ca₃SnO and Ca₃GeO respectively. Compressibility is the inverse of bulk modulus and therefore high value of β means low value of bulk modulus.

Phonon dispersion relations are depicted in Figures 5-7 below. The vibration frequency is directly proportional to the energy.

The primitive cell of the atoms has five atoms which gives 15 phonon branches with 3 acoustic and 12 optical modes for all the crystals. In all the crystals, there is no gap between the acoustic and optical branches of these modes. The energy of dispersion is for Ca₃SnO is the least followed by Ca₃GeO and then Ca₃PbO. This is not proportional to atomic mass unit of the metal in the crystal. Figure 7 can be related to Figure 4 where there is lack of overlap between the conduction and valence band. This shows that tin will require more energy for the conduction electrons to fully participate in conduction compared to the other two crystals.