Theory of phonon-limited resistivity of metals including the effects of anharmonicity, Debye-Waller factor, and the multiphonon term

The theory of phonon-limited resistivity $\rho$ of metals is extended to include the effects of anharmonicity, Debye-Waller factor, and the first term of the multiphonon series. The double-time temperature-dependent Green's-function approach is used. All the relevant Green's functions involving two-, three-, and fourphonon operators are obtained exactly. The contribution to $\rho$ from the third-order correlation functions are identified with the interference term. The contribution to $\rho$ from the fourth-order correlation functions are identified with the Debye-Waller factor and the first term of the multiphonon series. The anharmonic contributions to $\rho$ arise from the cubic and quartic shifts of the phonons and the phonon width, which are obtained from the full anharmonic one-phonon Green's function. The interference term represents the explicit cubic anharmonic contribution to $\rho$. Our expressions are valid for all temperatures. In the high-temperature limit all these contributions to $\rho$ are found to vary as $T^2$. Thus the formula for $\rho$ in the high-temperature limit is found to be $\rho =AT+B{T}^2$, where the linear term arises from the harmonic theory.