Density Functional Calculations of the Mechanical, Electronic and Dynamical Properties of Antiperovskite Ca_{3}BO (B = Pb, Ge, Sn) ()
1. Introduction
Inverse Perovskites or antiperovskite compounds have two types of anion and cation positions reversed. Their general formula is given as A_{3}BO where A sites are occupied by an electropositive ion while B sites are occupied by different anions whose size is different from A. Antiperovskite has displayed interesting properties which include magnetism, ionic conductivity, superconductivity and thermoelectricity, [1] among others. The discovery of superconducting properties of cubic antiperovskite MgCN_{i3} compound [2] has motivated research on more of these antiperovskites. These findings have prompted a new round of development of functional antiperovskites. A new class of antiperovskite compounds A_{3}BO where A = Ca, Sr, Ba and B = Pb have been reported to crystallize into the cubic inverse perovskite structure. Because of the varied properties of Antiperovskite materials, interests have shifted to their unusual high charge carrier mobility which leads to many technological applications [3] . There has been in the recent past growing research interest in band topology, leading to a broad classification of non-trivial topological metals classified as Weyl Semimetal, Dirac Semimetal and Nodal line Semimetal (NLS)^{ } ^{[4]} . These classified metals have their conduction band and valence hand crossing due to band inversion.
Three Dimensional (3D) Dirac fermion systems is a class of topological materials in which bulk conduction and valence bands with linear energy momentum dispersion relations meet at finite points or along curves in K-space. In order to verify the presence of 3D Dirac fermions theoretically, it is necessary to study electronic structure and in particular band structure to identify a Dirac point in 3D momentum space. The band structure of the inverse perovskite Ca_{3}PbO, through soft x-ray angle-resolved photoemission spectroscopy and cone like band dispersions are observed which is in close agreement with the predictions of electronic structure calculations [5] .
The band structure calculation of Ca_{3}BO and its family using Wein2k have found that three-dimensional massive Dirac electrons exist at their Fermi energy. The present work is based on first principle calculations and demonstrates inverse perovskite phase of Ca_{3}BO (B = Pb, Sn, Ge) as the new candidates for realizing Dirac fermionic system. An investigation of structural, electronic and dynamical properties of Ca_{3}BO is reported using DFT-based simulation code Quantum Espresso (QE). The electronic band structure and the projected density of states (PDOS) have also been studied. The present computed results are found to be in good agreement with the already available experimental and theoretical results.
The rest of this paper is arranged such that Section II is Computational methodology, Section III is discussing results and Section IV are Conclusions.
2. Computational Methodology
Calculations in this work were done in the framework of density functional theory using for the exchange-correlation functional, the generalized gradient approximation method of Perdew-Burke Ernzerhof (PBE) [6] . The pseudopotentials were extracted from Quantum espresso database [7] . The total energy convergence in the iterative solution of the Kohn-Sham equations [8] was set at 2 × 10^{−8} Ry. Optimized cell dimensions were fitted in Murnaghan fitting methodology of second order. The k-points and the kinetic energy cut-off values were properly optimized to convergence at the ground state energy. The valence configuration used for Ca_{3}PbO was 4s and 3p for Ca, 6s^{2} for Pb and 2s^{2} for O. Similar criteria is used for the other two compounds where Pb is replaced with Ge and Sn. The Brillouin sampling was based on the Monkhost scheme [9] [10] [11] . The K-point mesh in the irreducible high symmetry points in the Brillouin zone used was 13 × 13 × 13. For the elastic constant calculation, the “quasi-static” approximation was used where the elastic constants were computed at zero absolute temperature and saved as elastic constants.
3. Results
3.1. Structure Properties
Figure 1 shows the optimized unit cell of Ca_{3}PbO 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 Ca_{3}SnO and Ca_{3}GeO are all similar to that of Ca_{3}PbO 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
$\text{Pm3 m \xaf}$ (#221) for Ca_{3}BO (B = Pb, Ge, Sn). The lattice constants are 4.87Å, 4.86Å, and 4.84Å respectively for Ca_{3}BO (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
Figure 1. Optimized crystal structure of Ca_{3}PbO as visualized in xcrysden (Colour online) shows grey for O, light blue for Ca and red for Pb respectively. The central red at (0, 0, 0) is the position of Pb. It is this position that varies for Ca_{3}SnO and Ca_{3}GeO to be occupied by Sn and Ge respectively.
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 Ca_{3}PbO. The optimization follows for all the three crystals.
3.2. Electronic Structure Properties
In this section, the partial density of states and band structures of Ca_{3}BO (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, Ca_{3}PbO 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. Ca_{3}GeO has Ge-s and Ca-s states in the conduction band and for Ca_{3}SnO 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. Ca_{3}PbO has
Table 1. Calculated lattice parameter (a), volume (V), bulk modulus (B) and pressure derivative of bulk modulus (dB/dP).
Figure 2. Optimized energy versus lattice parameter for Ca_{3}PbO. The minimum energy corresponding to the lattice parameter is shown. The behaviour of convergence is similar for all the other two crystals.
Figure 3. Projected density of states in electrons per electron-volt for the compound Ca_{3}BO. S-states, d-states and p-states are labelled.
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 Ca_{3}SnO and Ca_{3}PbO compounds where oxygen is expected to behave in the similar way as in Ca_{3}GeO.
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 Ca_{3}SnO is 8.64 eV and that of Ca_{3}PbO is 9.16 eV.
3.3. Mechanical Properties
The mechanical properties of Ca_{3}PbO 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 C_{11}, C_{12} and C_{44}. For a solid under strain the elastic energy is given as [19] ,
$\frac{\Delta E}{V}=\frac{1}{2}{\displaystyle \underset{i=1}{\overset{6}{\sum}}{\displaystyle \underset{j=1}{\overset{6}{\sum}}{c}_{ij}{e}_{i}{e}_{j}}}$ (1)
where V is the undistorted lattice cell volume, ΔE is the change in energy from the strain vectors e = (e, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}) 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: C_{11} >0, C_{11} - C_{12} > 0, C_{44} > 0, C_{11} + 2C_{12} > 0 and C_{11} > B > C_{12}. These constants are calculated by using the obtained second-order elastic constants of Equation (2); the results are given in Table 2.
${C}_{ij}=\frac{1}{{V}_{0}}\frac{{\partial}^{2}{E}_{tot}}{\partial {\xi}_{i}\partial {\xi}_{j}}$ (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] ;
$A=\frac{2{C}_{44}}{{C}_{11}-{C}_{12}}$ (3)
$\delta =\frac{1}{2}\left\{\frac{B-\frac{2}{3}G}{B+\frac{1}{3}G}\right\}$ (4)
$E=\frac{9GB}{G+3B}$ (5)
where G is the isotropic shear modulus as a function of crystal orientation and is given as
$G=\left({G}_{V}+{G}_{R}\right)/2$ , herein G_{V} is Voigt’s shear modulus (which is related
Figure 4. Bands of antiperovskite Ca_{3}PbO against the high symmetry points. The Fermi energy has been set at the origin. The diagram shows semi-metallic band behaviour. This behaviour is also depicted for the Ca_{3}SnO and Ca_{3}GeO crystals.
Table 2. Elastic constants, bulk modulus and stability.
Table 3. Voigt’s shear modulus (G_{V}), Reuss’s Shear modulus (G_{R}), Voigt’s Young’s modulus (E_{V}), Reuss Young’s modulus (E_{R}), Poisson ratio (n), Zener anisotropy factor (A) and compressibility (β).
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
${G}_{V}=\left({C}_{\text{11}}-{C}_{\text{12}}+\text{3}{C}_{\text{44}}\right)/5$ and
$\text{5}/{G}_{R}=4/\left({C}_{\text{11}}-{C}_{\text{12}}\right)+\text{3}/{C}_{\text{44}}$ , 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 Ca_{3}PbO, Ca_{3}SnO and Ca_{3}GeO 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 Ca_{3}PbO, Ca_{3}SnO and Ca_{3}GeO respectively. This shows that these crystals are brittle. It can also be concluded that Ca_{3}GeO 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 Ca_{3}PbO, Ca_{3}SnO and Ca_{3}GeO 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;
$\beta =\frac{{C}_{11}-{C}_{12}}{\Omega}$ (6)
$\Omega =\left({C}_{11}+{C}_{12}\right){C}_{11}-2{C}_{12}^{2}$ (7)
The compressibility values are found to be 0.0064 GPa^{−1}, 0.0058 GPa^{−1} and 0.0061 GPa^{−1} for Ca_{3}PbO, Ca_{3}SnO and Ca_{3}GeO respectively. Compressibility is the inverse of bulk modulus and therefore high value of β means low value of bulk modulus.
3.4. Dynamical Properties
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 Ca_{3}SnO is the least followed by Ca_{3}GeO and then Ca_{3}PbO. 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.
4. Conclusion
Study of antiperovskite Ca_{3}BO (B = Pb, Ge, Sn) using density functional theory
Figure 5. Dispersion relation of Ca_{3}PbO with the highest frequency at about 350 cm^{−1}.
Figure 6. Dispersion relation of Ca_{3}GeO with highest frequency at about 400 cm^{−1}.
Figure 7. Dispersion relation of Ca_{3}SnO with the highest frequency at 620 cm^{−1}.
is here reported. The electronic structure studies show that these materials are conductors with similarities in properties as depicted by their band structures and projected density of states. There are also indicating stability from elastic constants study and derived properties. Phonon dispersion also points at the metallic nature of compounds CaBO. These crystals are therefore stable in their cubic states and are suitable for pressure-induced applications including thermoelectric applications
Acknowledgements
I acknowledge the support of computational resources provided by Centre for High Performance Computers, of Cape Town, South Africa.