Anomalous Electron-Phonon Transport Properties of Impure Metals. I. The Electrical Resistivity

Abstract

The electron-phonon contribution ${\rho }_{\mathrm{ep}}\mathrm{}(T,c)\mathrm{}$ to the resistivity of an impure metal, or dilute metal alloy, can be drastically different from that of the ideally pure metal, ${\rho }_{\mathrm{ep}}^0\mathrm{}\left(T\right)\mathrm{}$, if, in the region of the Fermi energy, the conduction-electron relaxation time ${\tau }_0\left(\epsilon \right)$ for impurity scattering varies with energy $\epsilon$ on a scale comparable to or less than the Debye energy $\hslash {\omega }_D$ of the metal. This effect is a consequence of the sensitivity of the (inelastic) electron-phonon resistivity to any energy-dependent component in the nonequilibrium electron-distribution function. We present a working formula for the effect and indicate several important consequences for nontransitional metals containing magnetic or nonmagnetic transitional impurities. In the limit of small impurity concentrations $c$, the alloy and host electron-phonon resistivities are connected to the electron-diffusion thermopower $S(T,c)$ of the alloy via the simple relation ${\rho }_{\mathrm{ep}}\mathrm{}(T,c)\mathrm{}\simeq {\rho }_{\mathrm{ep}}^0\mathrm{}\left(T\right)\mathrm{} \{1+{\left(\frac{\hslash {\omega }_D}{{\epsilon }_F}\right)}^2{\left[\frac{S(T,c)\mathrm{}}{S_0\mathrm{}\left(T\right)\mathrm{}}\right]}^2\}$, where $S_0$ denotes the "free-electron" thermopower. More generally, ${\rho }_{\mathrm{ep}}(T,c)$, and also ${\rho }_{\mathrm{imp}}(T,c)$, the resistivity resulting from impurity scattering, are expressed in terms of the first and second derivatives of ${\tau }_0$ at the Fermi energy ${\epsilon }_F$. The anomalous electron-phonon resistivity will cause sharp peaks to appear in the atomic-resistivity temperature curves of very dilute magnetic-impurity systems (e.g., $\mathrm{Cu}\mathrm{Fe}$, $\mathrm{Au}\mathrm{Fe}$, $\mathrm{Au}\mathrm{Mn}$). Experimentally, measurements of deviations from Matthiessen's rule should furnish useful information on the energy dependence of the electron-impurity scattering.