Electron-phonon scattering

In electron-phonon scattering, an Electron changes its energy by $\pm \hslash {\omega }_{\mathbf{q}}$ while its crystal momentum changes by a phonon wavevector modulo a reciprocal lattice vector. Charge-transport relaxation weights a scattering angle $\theta$ by $1-\cos \theta$. Small-angle events can therefore relax heat efficiently while relaxing electrical current weakly. At temperatures above the phonon excitation scale, scattering becomes approximately elastic in the electron thermal shell; residual elastic disorder often dominates at very low temperature.

For a spherical Fermi surface of radius $k_F$ lying inside a zone with shortest reciprocal lattice vector magnitude $G>2k_F$, a translated neighbouring Fermi sphere is separated from it by $q_U=G-2k_F$. An electron-phonon Umklapp scattering process satisfying $\mathbf{k}+\mathbf{q}={\mathbf{k}}^{\prime }+\mathbf{G}$ therefore needs a phonon wavevector of at least $q_U$. For an approximately linear acoustic dispersion its activation scale isAt $T\ll T_U$ these phonons are sparsely occupied; once they become thermally available, momentum-relaxing scattering can grow rapidly. Normal small-angle scattering and residual disorder need not vanish below this scale, so it is a crossover estimate rather than a sharp universal conductivity threshold.

For acoustic modes ${\omega }_q\simeq c_{s}q$, thermally active wavevectors have $q_T\sim k_{B}T/(\hslash c_s)$. When $q_T\ll k_F$, a nearly elastic event on a spherical Fermi surface has $1-\cos \theta =q^2/(2k_F^2)$, suppressing momentum relaxation. In the clean three-dimensional acoustic model, the thermal relaxation rate scales as $T^3$ while the electrical transport rate gains two extra powers and scales as $T^5$. With electronic heat capacity proportional to $T$, this gives $K_e\propto T^{-2}$ and electrical resistivity proportional to $T^5$ in the phonon-limited regime. Residual elastic scattering cuts off these behaviours as temperature falls further.^[ref]