Past exam of the quantum condensed matter course of the University of Cambridge / 2019 / Section B / 4 / e / Solution

For a spherical Fermi surface inside a primitive cubic lattice zone, the shortest reciprocal vector has magnitude $G=2\pi /a$. The nearest translated Fermi sphere is separated by the electron-phonon Umklapp thresholdA phonon of this wavevector can connect nearly opposite-velocity Fermi states using $\mathbf{k}+\mathbf{q}={\mathbf{k}}^{\prime }+\mathbf{G}$, transferring reciprocal-lattice momentum to the crystal and efficiently relaxing electrical current. With the supplied sound speed, its thermal scale isFor $T\ll T_U$ such phonons have exponentially small occupation. As they become populated, Umklapp scattering can make the electrical conductivity fall rapidly. Lower-energy normal scattering and residual disorder still give resistance below this scale; this is a crossover estimate, not a sharp transition. The numerical estimate extrapolates the linear acoustic dispersion to $q_U$ and assumes the spherical Fermi surface implicit in giving one $k_F$.

Using $2k_F$ alone as the phonon wavevector misses this mechanism: direct backscattering would need $1.4\pi /a$, beyond the first-zone edge along this direction, while the reciprocal-lattice transfer permits it with $0.6\pi /a$.