Umklapp Electron-Phonon Scattering in the Low-Temperature Resistivity of Polyvalent Metals

Abstract

A study of the effect of umklapp electron-phonon scattering on the low-temperature resistivity in polyvalent metals yields the following results: (i)The exponential [$\exp (-\frac{\Theta }{T})$] behavior in the alkalis is replaced by power laws in temperature; (ii) for single-OPW scattering states (i.e., spherical Fermisurface) a $T^2$ law results if a constant matrix element is used; if the wave-vector dependence of the matrix element in the vicinity of Bragg planes is included, a comparable $T^4$ law occurs; (iii) two-OPW corrections are important qualitatively only at the lowest temperatures, where they reduce the umklapp contribution to resistivity, distorting the power laws and producing the ultimate low-temperature behavior $T^5$. Nevertheless, the umklapp process strongly dominates the normal at all temperatures. The use of a simple trial function in the variational formula for resistivity makes the calculation and these results applicable to the temperature-dependent resistivity of impure metals. Comparison with experiment supports this interpretation.