Electronic and thermodynamic properties of layered \textbf{\textit{Hf}}$_{\mathbf{2}}$\textbf{\textit{S}}\textbf{ from first-principles calculations}

Theoretically we explored two stable phases of inorganic fullerene-like structure of the layered dihafnium sulfide (\textit{Hf}$_{2}S)$. We investigated structural and electronic properties of the two phases of \textit{Hf}$_{2}S$ by using first-principles calculations. Our calculation identifies experimentally observed anti-\textit{ NbS}$_{2}$ structure of \textit{Hf}$_{2}S$. Our electronic calculation results indicate that the density of states of anti- \textit{NbS}$_{2}$ structure of \textit{Hf}$_{2}S$ at fermi level is less than that of the other phase of \textit{Hf}$_{2}S$. To study the relative stability of different phases at finite temperature Helmholtz free energies of two phases are obtained using density functional theory and density functional perturbation theory. The free energy of the anti-\textit{ NbS}$_{2}$ structure of \textit{Hf}$_{2}S$ always lies below the free energy of the other phase by confirming the most stable structure of \textit{Hf}$_{2}S$. The phonon dispersion, phonon density of states including partial density of states and total density of states are obtained within density functional perturbation theory. Our calculated zero-pressure phonon dispersion curves confirm that the thermodynamic stability of \textit{Hf}$_{2}S$ structures. For further investigation of thermodynamic properties, the temperature dependency of thermal expansion, heat capacities at constant pressure and volume are evaluated within the quasiharmonic approximations (QHA).