Effect of Guest Atom Composition on the Structural and Vibrational Properties of the Type II Clathrate-Based Materials AxSi136, AxGe136 and AxSn136 (A = Na, K, Rb, Cs; 0 ≤ x ≤ 24)

Type II clathrates are interesting due to their potential thermoelectric applications. Powdered X-ray diffraction (XRD) data and density functional calculations for Na_xSi₁₃₆ found a lattice contraction as x increases for 0 < x < 8 and an expansion as x increases for x > 8. This is explained by XRD data that shows that as x increases, the Si₂₈ cages are filled first for x < 8 and the Si₂₀ cages are then filled for x > 8. Motivated by this work, here we report the results of first-principles calculations of the structural and vibrational properties of the Type II clathrate compounds A_xSi₁₃₆, A_xGe₁₃₆, and A_xSn₁₃₆. We present results for the variation of the lattice constants, bulk moduli, and other structural parameters with x. These are contrasted for the Si, Ge, and Sn compounds and for guests A = Na, K, Rb, and Cs. We also present calculated results of phonon dispersion relations for Na₄Si₁₃₆, Na₄Ge₁₃₆, and Na₄Sn₁₃₆ and we compare these for the three materials. Finally, we present calculated results for the elastic constants in Na_xSi₁₃₆, Na_xGe₁₃₆, and Na_xSn₁₃₆ for x = 4 and 8. These are compared for the three hosts, as well as for the two compositions.