Deviation from Matthiessen's Rule in Dilute Nonmagnetic Alloys

It is shown how the low-temperature deviations from Matthiessen's Rule observed in a wide variety of dilute nonmagnetic alloys are a consequence of phonon drag in the regime where $\lambda$, the electron mean free path, is determined by electron-impurity scattering [i.e., $g^2{\left(\frac{T}{\theta }\right)}^3\ll {\left(k_{\mathrm{F}}\lambda \right)}^{-1\mathrm{}}$ where $g^2\sim \frac{v_S}{v_{\mathrm{F}}}$]. For a spherical Fermi surface ${\Delta }_T\sim g^2{\left(\frac{T}{\theta }\right)}^3\times {\left(\lambda k_{\mathrm{F}}\right)}^{-1\mathrm{}}\ln \left(\lambda k_{\mathrm{F}}\right)+g^4{\left(\frac{T}{\theta }\right)}^5\ln \left(\frac{{\rho }_0}{{{\rho }_T}^{\mathrm{pure}}}\right)$, where ${\Delta }_T={\rho }_T-{\rho }_0-{{\rho }_T}^{\mathrm{pure}}$ with ${{\rho }_T}^{\mathrm{pure}}$ being the resistivity of the alloy (host) at temperature $T$. For a cylindrical Fermi surface ${\Delta }_T\sim g^2{\left(\frac{T}{\theta }\right)}^3\ln \left(\frac{{\rho }_0}{{{\rho }_T}^{\mathrm{pure}}}\right)$. It is also shown that the singular behavior noted by Mills is spurious.