Structural, mechanical and thermodynamic properties of Cu$_{2}$CoXs$_{4}$ (X = Si, Ge, Sn) studied by a density functional theory method

In the paper, we will investigate the basic physical properties of Cu$_{2}$CoXS$_{4}$ (X = Si, Ge, Sn) by using the density functional theory approach. The calculated lattice parameters are in good correspondence with the theoretical data, and the chosen theoretical method is proved to be reliable. In the first part, the Mulliken population analysis indicates the bonds between $S$ atoms and other three atoms in Cu$_{2}$CoXS$_{4}$ (X = Si, Ge, Sn) exhibit the feature of covalent bond. And then, the calculated elastic constants prove the mechanical stability of Cu$_{2}$CoXS$_{4}$ (X = Si, Ge, Sn) in I $\bar4$2m structure. Results are given for B/G and $A^{U}$ reveal Cu$_{2}$CoXS$_{4}$ (X = Si, Ge, Sn) can behave as a ductile and elastic material. Finally, the pressure and temperature dependence of heat capacity, thermal expansion, entropy and Debye temperature in the rang from 0 to 1000 K and from 0 GPa to 50 GPa are also reported in this study.