Collision-duration time for optical-phonon emission in semiconductors

The time required to emit an optical (polar and intervalley) phonon by a nearly-free electron in a semiconductor is evaluated using a nonequilibrium Greenșs-function formalism. The leading idea of the work is that the so-called "collision duration" is related to the time required to build up correlation between the initial and the final state, and then to destroy this correlation as the collision is completed. The use of the nonequilibrium Greenșs-function formalism gives us the possibility to evaluate explicitly the effects of the correlations in time. Our approach is based on two crucial assumptions: we build the self-energy from only the polarization field of the polar-optical phonon; that is, the self-energy is a function of a single time and position, and we introduce the electron correlation function between the initial and the final states, written in terms of a generalized less-than Greenșs function in the momentum variables. We derive an analytical expression for the probability for a carrier to end up in a final state k as a consequence of the emission of a phonon as a function of time. We find that the probability rises to the "Fermi golden rule" result within a few femtoseconds. If the total lifetime broadening of the initial state is comparable to the scattering time, the probability oscillates as it approaches the asymptotic value. For larger initial-state broadening (due to more scattering processes), these oscillations disappear.