Thermoelectric properties of anisotropic semiconductors

General effective transport coefficients and the thermoelectric figure of merit $\mathrm{ZT}$ for anisotropic systems are derived. Sizable induced transverse fields on surfaces perpendicular to the current flow are shown to reduce the effective transport coefficients. A microscopic electronic model relevant for multivalleyed materials with parabolic bands is considered in detail. Within the effective-mass and relaxation-time approximations but neglecting the lattice thermal conductivity ${\kappa }_l,$ the thermopower and Lorenz number are shown to be independent of the tensorial structure of the transport coefficients and are therefore isotropic. $\mathrm{ZT}$ is also isotropic for vanishing lattice thermal conductivity ${\kappa }_l.$ A similar result holds in lower dimensions. For nonvanishing but sufficiently isotropic ${\kappa }_l,$ $\mathrm{ZT}$ is ordinarily maximal along the direction of highest electrical conductivity $\sigma .$ More general numerical calculations suggest that maximal $\mathrm{ZT}$ occurs along the principal direction with the largest $\sigma /{\kappa }_l.$ An explicit bound on $\mathrm{ZT}$ is derived. Consideration of the Esaki-Tsu model shows that nonparabolic dispersion in superlattices has little effect on the thermopower at the carrier concentrations which maximize $\mathrm{ZT}.\mathrm{}$ However, strong anisotropies develop when the chemical potential exceeds the miniband width.