Abstract
The time required to emit an optical (polar and intervalley) phonon by a nearly-free electron in a semiconductor is evaluated using a nonequilibrium Green's-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 Green's-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 Green's 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.
Highlights
In the traditional treatment of the Boltzmann equation for electron transport in semiconductors, collisions are, in general, assumed instantaneous in time and pointlike in space
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 Green’s function in the momentum variables
We have derived 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 polar-optical or of an intervally optical phonon as a function of time
Summary
In the traditional treatment of the Boltzmann equation for electron transport in semiconductors, collisions are, in general, assumed instantaneous in time and pointlike in space. Emin[1,2] studied the lattice relaxation time in a small-polaron hopping motion, which built upon earlier work in this area.[3,4] Geltman[5] investigated the time evolution of the ionization probability of a simple onedimensional model atom under the influence of a harmonic electric field as well as a linearly polarized light, by the numerical solution of the time-dependent Schrodinger equation He found that the duration of the radiation pulse is of crucial importance, as multiphoton absorption was seen to require a certain minimum time to develop, and that processes that are forbidden in the long-time limit by energy conservation are important at short times. They find that at equilibrium the noninterference of subsequent processes is given by a single characteristic time scale, and that an estimator of this time scale is given approximately by the Landau liquid theory as h/E, where E is measured from the threshold for the process under consideration
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