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Electron-phonon (e-ph) interactions are usually treated in the lowest order of perturbation theory. Here we derive next-to-leading order e-ph interactions, and compute from first principles the associated two-phonon e-ph scattering rates. The derivation involves Matsubara sums of the relevant two-loop Feynman diagrams, and the numerical calculations are challenging since they involve Brillouin zone integrals over two crystal momenta and depend critically on the intermediate state lifetimes. Using random grids and Monte Carlo integration, together with a self-consistent update of the intermediate state lifetimes, we compute and converge the two-phonon scattering rates, using GaAs as a case study. For the longitudinal optical phonon in GaAs, we find that the two-phonon scattering rates are as large as nearly half the value of the leading-order rates. The energy and temperature dependence of the two-phonon processes are analyzed. We show that including the two-phonon processes is important to accurately predicting the electron mobility in GaAs., Comment: 5 pages, 5 figures

While strong topological insulators (STI) have been experimentally realized soon after their theoretical predictions, a weak topological insulator (WTI) has yet to be unambiguously confirmed. A major obstacle is the lack of distinct natural cleavage surfaces to test the surface selective hallmark of WTI. With a new scheme, we discover that Bi$_{4}$X$_{4}$ (X=Br, I), stable or synthesized before, can be WTI with two natural cleavage surfaces, where two anisotropic Dirac cones stabilize and annihilate, respectively. We further find four surface state Lifshitz transitions under charge doping and two bulk topological phase transitions under uniaxial strain. Near the WTI-STI transition, there emerges a novel Weyl semimetal phase, in which the Fermi arcs generically appear at both cleavage surfaces whereas the Fermi circle only appears at one selected surface., 6 pages, 4 figures