Defect phonon renormalization during nonradiative multiphonon transitions in semiconductors

As a typical nonradiative multiphonon transition in semiconductors, carrier capture at defects is critical to the performance of semiconductor devices. Its transition rate is usually calculated using the equal-mode approximation, which assumes that phonon modes and frequencies remain unchanged before and after the transition. Using the carbon substitutional defect $({\text{C}}_{\text{N}})$ in GaN as a benchmark, here we demonstrate that the phonon renormalization can be significant during defect relaxation, which causes errors as large as orders of magnitude in the approximation. To address this issue, we consider (i) the Duschinsky matrix connecting the initial-state and final-state phonons, which accounts for the changes in phonon modes and frequencies, and (ii) the off-diagonal contributions in the total transition matrix element, which incorporates the cross terms of electron-phonon interactions between different modes. With this improvement, the calculated transition rates show agreement with experimental results within an order of magnitude. We believe the present method makes one step forward for the accurate calculation of the multiphonon transition rate, especially in cases with large defect relaxations.