A Matrix Quantum Kinetic Treatment of Impact Ionization in Avalanche Photodiodes

Matrix based quantum kinetic simulations have been widely used for the predictive modeling of electronic devices. Inelastic scattering from phonons and electrons are typically treated as higher order processes in these treatments, captured using mean-field approximations. Carrier multiplication in Avalanche Photodiodes (APDs), however, relies entirely on strongly inelastic impact ionization, making electron-electron scattering the dominant term requiring a rigorous, microscopic treatment. We go well beyond the conventional Born approximation for scattering to develop a matrix-based quantum kinetic theory for impact ionization, involving products of multiple Green's functions. Using a model semiconductor in a reverse-biased p-i-n configuration, we show how its calculated non-equilibrium charge distributions show multiplication at dead-space values consistent with energy-momentum conservation. Our matrix approach can be readily generalized to more sophisticated atomistic Hamiltonians, setting the stage for a fully predictive, `first principles' theory of APDs.