Band structures and thermoelectric properties of the clathrates Ba8Ga16Ge30, Sr8Ga16Ge30, Ba8Ga16Si30, and Ba8In16Sn30

Density functional calculations in the generalized gradient approximation are used to study the transport properties of the clathrates ${\mathrm{Ba}}_{\mathrm{8}}{\mathrm{Ga}}_{\mathrm{16}}{\mathrm{Ge}}_{\mathrm{30}},$ ${\mathrm{Sr}}_{\mathrm{8}}{\mathrm{Ga}}_{\mathrm{16}}{\mathrm{Ge}}_{\mathrm{30}},$ ${\mathrm{Ba}}_{\mathrm{8}}{\mathrm{Ga}}_{\mathrm{16}}{\mathrm{Si}}_{\mathrm{30}},$ and ${\mathrm{Ba}}_{\mathrm{8}}{\mathrm{In}}_{\mathrm{16}}{\mathrm{Sn}}_{\mathrm{30}}.$ The band structures of these clathrates indicate that they are all semiconductors. Seebeck coefficients, conductivities and Hall coefficients are calculated, to assess the effects of carrier concentration on the quantity $S^2\mathrm{\sigma /\tau }$ (where S is the Seebeck coefficient, σ is the conductivity, and τ the electron relaxation time) which is proportional to the thermoelectric power factor. In each compound we find that both p- and n-doping will significantly enhance the thermoelectric capabilities of these compounds. For p-doping, the power factors of all four clathrates are of comparable magnitude and have similar temperature dependence, while for n-doping we see significant variations from compound to compound. We estimate that room-temperature ZT values of 0.5 may be possible for optimally n-doped ${\mathrm{Sr}}_{\mathrm{8}}{\mathrm{Ga}}_{\mathrm{16}}{\mathrm{Ge}}_{\mathrm{30}}$ or ${\mathrm{Ba}}_{\mathrm{8}}{\mathrm{In}}_{\mathrm{16}}{\mathrm{Sn}}_{30};$ at 800 K ZT values as large as 1.7 may be possible. For single crystals of high quality, with substantially increased scattering times, the power factor of these materials will be significantly higher. Recent experiments are reviewed in the light of these calculations.