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 ${\tau }^{-1}$ becomes comparable to the Fermi energy $E_F/\hslash$ (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 At such temperatures, the usual description of a metal in terms of ballistically propagating quasiparticles is no longer valid.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 $N^2$ optical phonon modes, which displays a crossover behavior in the resistivity at temperatures where ${\tau }^{-1}\sim E_F/\hslash$. 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.

Figure 1

Resistivity as a function of $cT/\mathrm{\Lambda }$, where $\mathrm{\Lambda }$ is the bandwidth, for a two-dimensional tight-binding model on a square lattice. The resistivities are calculated for filling $n=1/80,1/40,1/3$ per electronic flavor. The red lines are fits to the low-temperature linear behavior. The $n=1/3$ resistivity continues to rise linearly throughout the shown temperature regime, and crosses over to nonlinear behavior only at temperatures comparable to $\mathrm{\Lambda }/c$, as shown in the inset.

Figure 2

To lowest order in $1/N$, the full set of rainbow diagrams contributes to the electron self-energy, denoted by the filled square. The arrows represent bare electron propagators, squiggly lines phonon propagators, and the thick arrow the fully dressed electron propagator.

Figure 3

The imaginary part of the self-energy at the Fermi level. At low temperatures, $cT\ll E_F$, the self-energy increases linearly with temperature, in accordance with the semiclassical result; the linear red line represents a fit to the low-$T$ behavior. At higher temperatures, ${\mathrm{\Sigma }}^{\prime \prime }\left(0\right)$ becomes proportional to $\sqrt{T}$; the second red line is proportional to $\sqrt{T}$, and almost exactly coincides with the numerics. These results are calculated for a two-dimensional tight-binding system at $n=0.51$.

Figure 4

Diagrams contributing to the conductivity to lowest order in $1/N$. The dashed lines are phonon propagators, while the full lines are fully dressed electron Green's functions. The squiggly lines represent the current operator, adding a factor of $ed\epsilon /dk$.