Electron-phonon interaction in impure polycrystalline metals

Abstract

We calculate the electron-phonon scattering rate in polycrystalline metals, e.g., ${\mathrm{Ti}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x},$ in the limit of dilute impurity concentration. We consider the additional contribution due to the Umklapp process of impurity scattering, which has been neglected in all previous nearly free-electron calculations but is important for the present problem. We find that, as a result of including the Umklapp process, the scattering rate in the dirty limit ${q}_{T}l\ensuremath{\ll}1$ is enhanced by the disorder due to substitutional impurities and random lattice shift of crystallites, where ${q}_{T}=$thermal phonon wave vector and $l=$electron mean free path. Specifically, we obtain the scattering rate $1/{\ensuremath{\tau}}_{\mathrm{ep}}\ensuremath{\sim}{T}^{2}{l}^{\ensuremath{-}1},$ where $T=\mathit{temperature},$ in agreement with previous experiments both in order of magnitude and in functional dependence. This work satisfactorily explains the long-standing discrepancy between theories and experiments regarding the effect of disorder on electron-phonon scattering, for the case of polycrystalline metals with dilute impurity concentration.