Theoretical Study of the Properties of the Type II Clathrates AxB$_{136}$(A=alkali atom;B=Si,Ge,Sn0 $\le $ x $\le $ 24)

Type II clathrate semiconductors have cage-like lattices in which Group IV atoms are tetrahedrally-coordinated and sp$^{3}$ covalently bonded. The cages can contain ``guest'' atoms; usually alkali or alkaline earth atoms. These materials are of interest because of their thermoelectric properties. Motivated by recent experimental and theoretical interest [1,2] in the x dependence of properties of the Si and Ge-based Type II clathrate materials A$_{\mathrm{x}}$Si$_{136}$ and A$_{\mathrm{x}}$Ge$_{136}$ (A $=$ alkali atom) we are carrying out a systematic theoretical study of the properties of the Type II clathrate systems A$_{\mathrm{x}}$B$_{136}$(A $=$ alkali atom; B $=$ Si, Ge, Sn). Recent powder X-ray diffraction experiments have found the very interesting result that in Na$_{\mathrm{x}}$Si$_{\mathrm{136}}$, for increasing x in the range 0 $\le $ x $\le $ 8 a lattice contraction occurs and that as x is increased further (8 $\le $ x $\le $ 24), a contrasting lattice expansion results. These observations have motivated us to study the behavior of the lattice constant and other properties as a function of guest concentration in several Type II clathrates. We present results of a density functional based study of the properties of A$_{\mathrm{x}}$B$_{136}$ as a function of x. Results are discussed for the x dependence of the lattice constant and for other structural and electronic properties of these materials. [1] S. Stefanoski and G. Nolas, Cryst. Growth Des. 2011, dx.doi.org/10.1021/cg200756r [2] M. Beekman, E. Nenghabi, K. Biswas, C. Myles, M. Baitinger, Y. Grin, G.S. Nolas, Inorg. Chem. 49 2010, DOI: 10.1021/ic1005049