Quantum theory of impact ionization in coherent high-field semiconductor transport

Generation of carriers in semiconductors by impact ionization is studied under the influence of a constant, arbibrarily high electric field. Using the density-matrix approach a system of equations for the coherent dynamics of electrons and holes in the presence of impact ionization and Auger recombination is derived, which extends the semiconductor Bloch equations by the inclusion of impact-ionization density-correlation functions as additional dynamic variables. From these equations we recover the pure (Zener) and the photon-induced (Franz-Keldysh) carrier tunneling rate and derive an expression for the field-assisted impact-ionization scattering rate. Different levels of approximation of the kinetic equations are discussed. It is shown that in contrast to the semiclassical treatment in the presence of an electric field, a fixed impact-ionization threshold does no longer exist, and the impact-ionization scattering rate is drastically enhanced around the semiclassical threshold by the intracollisional field effect. The close connection of field-assisted impact ionization to the Franz-Keldysh effect is emphasized.