First principles calculations of the thermal properties of tin clathrate materials

Using a Local Density Approximation (LDA) approach, we have studied the energetics and the thermal properties (free energy, specific heat, lattice vibrational entropy) of the tin-based Type I semiconductor clathrates Sn$_{46}$, K$_{8}$Sn$_{46}$, K$_{8}$Sn$_{44}{\rm M}_{2}$ (${\rm M}$ is a Sn vacancy), Cs$_{8}$Ga$_{8}$Sn$_{38}$ and Cs$_{8}$Zn$_{4}$Sn$_{42}$. The clathrate lattices are open framework, cage-like structures. Our results predict that K$_{8}$Sn$_{46}$ is slightly less stable than K$_{8}$Sn$_{44}{\rm M}_{2}$, in agreement with other theories.\footnote{L. Mollnitz et. al., J. Chem. Phys. 117, 3 (2002)} We have optimized the geometry of each structure and have calculated the phonon density of states. The thermodynamic properties have then been calculated as a function of temperature. The localized vibrational (``rattler'') modes of the guests Cs and K have been calculated and their Einstein temperatures obtained. The Debye temperatures of each host clathrate have also been calculated. We use our results to help to explain the observed difference\footnote{G. S. Nolas et. al., Phys. Rev. B 53,165201 (2002)} in the lattice thermal conductivities of some of these materials.