  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.
