Fundamental relationship between the Herring and Cantrell-Butcher formulas for the phonon-drag thermopower of two-dimensional electron and hole gases

Abstract

We show that Herring's formula for the phonon-drag thermopower ${S}^{g}$ is equivalent to the standard Cantrell-Butcher (CB) formula for a two-dimensional electron (or hole) gas. The CB approach provides a rigorous theoretical framework which is remarkably successful in explaining the measured ${S}^{g}$ in two-dimensional systems. Herring's approach, although phenomenological, shows particular interest because it leads to an elegant and simple formula for ${S}^{g}$ in terms of the phonon-limited mobility. The equivalence of the two formulas justifies the use of Herring's expression for the estimation of the phonon-limited mobility directly from the phonon-drag thermopower data and vice versa. Consequently, we exploit both the accuracy of the CB formula and the simplicity of Herring's formula. Our analysis leads to the derivation of an optimum average for the inelastic carrier relaxation rate ${\ensuremath{\tau}}_{\mathrm{ac}}^{\ensuremath{-}1}$ associated with the acoustic-phonon scattering. This average is free of the conventional approximations made in the literature and explains with remarkable accuracy mobility data in both n- and p-type ${\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{A}\mathrm{s}/\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}$ heterostructures.