Mott-Ioffe-Regel limit and resistivity crossover in a tractable electron-phonon model

Many metals display resistivity saturation---a substantial decrease in the slope of the resistivity as a function of temperature that occurs when the electron scattering rate τ^-1 becomes comparable to the Fermi energy EF/ (the Mott-Ioffe-Regel limit). At such temperatures, the usual description of a metal in terms of ballistically propagating quasiparticles is no longer valid. We present a tractable model of a large number N of electronic bands coupled to N² optical phonon modes, which displays a crossover behavior in the resistivity at temperatures where τ^-1~EF/. At low temperatures, the resistivity obeys the familiar linear form, while at high temperatures, the resistivity still increases linearly, but with a modified slope (that can be either lower or higher than the low-temperature slope, depending on the band structure). The high-temperature non-Boltzmann regime is interpreted by considering the diffusion constant and the compressibility, both of which scale as the inverse square root of the temperature.