Ab Initio Next-to-Leading Order Electron-Phonon Interactions: Two-Phonon Electron Scattering Processes and their Temperature and Energy Dependence

Electron-phonon (e-ph) interactions are a current focus of first-principles calculations. The lowest-order e-ph self-energy is computed in most works, and only recently there were attempts to include higher-order e-ph interactions using the cumulant method. However, diagrammatic approaches beyond the lowest order have not been attempted for computing e-ph interactions ab initio.
Here, we compute the scattering rates of e-ph processes involving two external phonons. Their expression is derived using many-body perturbation theory and the Matsubara technique on the two skeletonically nontrivial second-order diagrams. The numerical calculations are challenging since they involve Brillouin zone integrals over two crystal momenta and the intermediate state lifetime plays a critical role. Using random grids and Monte Carlo integration, we are able to compute and systematically converge such next-to-leading order e-ph scattering rates. Results are discussed for GaAs and SrTiO3, in which we analyze the two-phonon contributions to e-ph scattering as a function of temperature and carrier energy, and compare them with the lowest-order results. We discuss how our formalism can be extended to detect strong e-ph coupling and polaron formation.