Properties of the type I Ge-based clathrates Ba$_{8}$Al$_{13}$Ge$_{33}$ and Ba$_{8}$Al$_{16}$Ge$_{30}$

The type I clathrate lattice is simple cubic with 46 atoms per unit cell. The cages in this lattice can host ``guests'' and the framework can have substituted atoms. Here, we focus on the ``alloy'' system Ba$_{8}$Al$_{x}$Ge$_{30-x}$ (x is an integer; 0$<$x$<$15). The Ba are guests and Al substitutes for some Ge framework atoms. Using the local density approximation (LDA), we have calculated some properties of the type I clathrates Ba$_{8}$Al$_{13}$Ge$_{33}$ and Ba$_{8}$Al$_{16}$Ge$_{30}$. Our calculations of the equilibrium structures predict that Ba$_{8}$Al$_{16}$Ge$_{30 }$and Ba$_{8}$Al$_{13}$Ge$_{33}$ have approximately the same lattice constant and that Ba$_{8}$Al$_{13}$Ge$_{33}$ is expected to be slightly more stable than Ba$_{8}$Al$_{16}$Ge$_{30}$. Our band structures and electronic density of states results predict that Ba$_{8}$Al$_{13}$Ge$_{33}$ is metallic and that Ba$_{8}$Al$_{16}$Ge$_{30}$ is a semiconductor with an indirect fundamental band gap of 0.3 eV. The vibrational spectrum predicts low frequency rattling modes caused by the Ba guests that are loosely bound in the Al-Ge framework cages. Such modes may scatter the heat-carrying acoustic vibrational framework modes, potentially reducing the thermal conductivity.