First-principles theory of nonlinear long-range electron-phonon interaction

Describing electron-phonon interactions in a solid requires knowledge of the electron-phonon matrix elements in the Hamiltonian. State-of-the-art first-principles calculations for the electron-phonon interaction are limited to the one-electron-one-phonon matrix element, which is suitable for harmonic materials. However, there is no first-principles theory for one-electron-two-phonon interactions, which occur in anharmonic materials with significant electron-phonon interaction such as halide perovskites and quantum paraelectrics. Here we derive an analytical expression for the long-range part of the one-electron-two-phonon matrix element, written in terms of microscopic quantities that can be calculated from first principles. We show that the long-range one-electron-two-phonon interaction is described by the derivative of the phonon dynamical matrix with respect to an external electric field. We calculate the quasiparticle energy of a large polaron including one-electron-two-phonon interaction and show that it can be written in terms of a one-electron-two-phonon spectral function Tαβ(ω). We demonstrate how to calculate this spectral function and its temperature dependence for the benchmark materials LiF and KTaO3, where it turns out that the effect is very small. The first-principles framework developed in this article is general, paving the way for future calculations of one-electron-two-phonon interactions in materials where the effect may be larger.