Next-to-Leading Order Ab Initio Electron-Phonon Scattering

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.