Theory of the Thermoelectric Power of Semiconductors

The usual theory of thermoelectric power fails to account for the marked rise in this quantity which has been found recently for some semiconductors as the temperature is lowered below room temperature. This paper develops the recently suggested explanation that the thermoelectric power $Q$ is the sum of the usual electronic term $Q_e$, resulting from the spontaneous tendency of the charge carriers to diffuse from hot to cold, and a phonon term $Q_p$, resulting from the drag on the charge carriers exerted by the phonons streaming from hot to cold in thermal conduction. An equivalent description of the term $Q_p$ can be given in terms of the contribution to the Peltier heat flux in an isothermal specimen, due to phonons dragged along by the electric current. As a prelude to the discussion of $Q_p$, the existing theory of $Q_e$ is first subjected to some refinements required by recent developments in semiconductor theory. The theory of $Q_p$ is then formulated in a simple but general way by making use of an approximate proportionality between heat current and crystal momentum in the phonon system. Using recently-derived results on the probability that a very low-frequency phonon will be scattered by other phonons, an explicit expression for $Q_p\mathrm{}\left(T\right)\mathrm{}$ is derived, which should be valid in the range of moderately low temperatures and low carrier concentrations. At lower temperatures, but still far above the range where the thermal conductivity is appreciably size-dependent, $Q_p$ is dominated by the scattering of phonons from the boundaries of the specimen; the theory of this effect is worked out in detail. Although $Q_p$ is independent of carrier concentration when the latter is low, a considerable decrease is predicted at high carrier concentrations, or at very low temperatures, because of a saturation effect. The effect of Fermi degeneracy on all these phenomena is discussed. Available data on $p$ germanium show all these effects and can be fitted by the theory. The comparison indicates that a large proportion of the low-temperature lattice scattering of holes in $p$ germanium is by shear modes. Although $n$ germanium seems less suited for quantitative comparison, it, also, shows all the predicted effects.