Large Many-Electron Effects in the Temperature-Dependent Electron-Phonon Renormalization of Semiconductor Band Gaps

We investigate from first principles the temperature-dependent electron-phonon (e-ph) renormalization of the fundamental band gaps of diamond, silicon, and gallium phosphide, with many-electron self-energy effects included, using the GW and GW perturbation theory methods for the band energy and e-ph coupling, respectively. Our results show that the e-ph renormalization of band gaps is enhanced by ∼50% at the GW level compared to results obtained with density functional theory (DFT) and density functional perturbation theory for all three materials. Moreover, the temperature dependence of the band gaps predicted by GW and GW perturbation theory shows excellent agreement with experimental measurements, with significant improvements from results at the DFT level. Our analysis reveals nonuniform GW self-energy corrections, both in the e-ph coupling from different phonon modes and in the phonon-induced contributions (the Fan-Migdal and Debye-Waller terms) to the electron self-energy. Our findings emphasize the crucial role of many-electron effects in the temperature-dependent e-ph renormalization of band gaps and highlight the necessity of including GW self-energy effects going beyond standard DFT approaches for accurate descriptions of e-ph phenomena.