Umklapp electron-phonon scattering anisotropy and deviations from Matthiessen's rule

Abstract

The electrical resistivity is obtained exactly for a model of a dilute alloy that incorporates two isotropic (corresponding to impurity and normal phonon) and one anisotropic (corresponding to Umklapp phonon) additive electron scattering probabilities. The deviations from Matthiessen's rule are found to have a weak dependence on the residual resistivity rho ₀ similar to the ln rho ₀ behaviour recently observed at low temperatures in several metals for a wide range of concentrations and variety of nonmagnetic impurities. An interesting aspect of the exact solution is that the Umklapp scattering contribution to the resistivity is much smaller than the normal scattering contribution in contrast to the results given by the variational principle with the simplest trial function where the opposite is true to polyvalent metals; this result suggests that phonon drag may play an important role in the low temperature resistivity of polyvalent metals.