Quantum Correction to the Semiclassical Electron-Phonon Scattering Operator

A quantum kinetic equation approach is adopted in order to incorporate quantum effects such as collisional broadening due to finite lifetime of single particle states, and collisional retardation due to finite collision time. A quantum correction to the semiclassical electron distribution function is obtained using an asymptotic expansion for the quantum electron-phonon collision operator in its weak formulation. Based on this expansion, the evolution of a highly peaked, nonequilibrium distribution function in Si and Ge is analyzed. It is shown that in Ge and Si, where the electron-phonon interaction is weak, the quantum correction due to the finite collision time leads to an extra broadening of new replicas of the initial distribution function. As the observation time exceeds the collision duration, the quantum correction starts to diminish and the semiclassical solution for a particular replica is recovered.