Anisotropy in the Electron-Phonon Umklapp Interaction and Deviations from Matthiessen's Rule

Abstract

Detailed calculations and experimental evidence show that significant deviations from Matthiessen's rule arise naturally as a result of anistropy in the low-temperature electron-scattering probability. Such anisotropy can be considerable in the pure metal as a result of the strong orientational dependence of the electron-phonon umklapp interaction at low temperatures. It is shown that elastic electron-impurity scattering can produce a significant impurity-dependent change in the intrinsic resistivity simply through the reduction of these angular distortions. As a consequence calculations made in the usual relaxation-time approximation are more appropriately compared with resistivity data dominated by such isotropic-impurity scattering than with ideal resistivity data obtained in the pure limit (where umklapp-generated anisotropy is usually substantial). Using the variational method, various model pseudopotentials, and a realistic phonon spectrum, the mechanism's effect has been accurately evaluated for the case of potassium in the temperature range below $\frac{{\ensuremath{\Theta}}_{D}}{5}$. A thorough analysis of the role played by the various scattering mechanisms indicates that the effect is strongly dependent on the relative amounts of umklapp versus normal scattering present. The variational results are found to account for a substantial portion of the observed impurity dependence of the intrinsic resistivity of potassium. The predicted deviations in polyvalent metals are expected to be significantly larger and agree, in general form, with observations of such effects in aluminum.