Theoretical Study of the Structural and Electronic Properties of K$_{x}$Si$_{136}$, (x = 1, 8, 18)

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. Measurements of the lattice constant [1] as a function of Na concentration x in Na$_{x}$Si$_{136}$ (0 $\le $ x $\le $ 24) have shown the interesting property that, as x is increased in the range (0 $\le $ x $\le $ 8), the lattice constant decreases and that as x is increased further in the range (8 $\le $ x $\le $ 24), the lattice constant increases. We note that some measurements of the properties of K$_{17.8}$Si$_{136 }$have also recently been reported [2]. These observations have motivated us to study the behavior of the lattice constant and other properties as a function of guest concentration in other Type II clathrates. In the present paper, we report the results of a theoretical study of the properties of K$_{x}$Si$_{136}$ as a function of x. We have used density functional theory to investigate the properties of this material with guest concentrations of x = 1, 8, and 18. Our results show that, similar to previous results for Na$_{x}$Si$_{136}$, the lattice constant as a function of x has a minimum at x = 8. We also report results for other structural and electronic properties of K$_{x}$Si$_{136}$.\\[4pt] [1] M. Beekman, E. Nenghabi, K. Biswas, C. Myles, M. Baitinger, Y. Grin, G.S. Nolas, Inorg. Chem. 49 2010, DOI: 10.1021/ic1005049\\[0pt] [2] S. Stefanoski and G. Nolas, Cryst. Growth Des. 2011, dx.doi.org/10.1021/cg200756r