Evolution of non-equilibrium electron distribution functions at hydrodynamic scales in multi-valley semiconductors

Abstract

The concept of the physical evolution process for the distribution function is used to derive a non-equilibrium hydro-kinetic transport theory. The hydro-kinetic distribution that is interpolated between the kinetic and hydrodynamic levels is introduced to elucidate the physics of evolution for the distribution. The evolution scales of the distribution are decomposed into characteristic scales of hydrodynamic parameters, such as carrier density, energy and momentum characteristic times. The coarseness of the hydro-kinetic distribution function is determined by scales of the chosen hydrodynamic parameters. The hydro-kinetic distribution is used to close the infinite set of moments and to determine the rate coefficients in the closed set of hydrodynamic equations. In this paper, the hydro-kinetic distribution at the energy characteristic scale is applied to study evolution of the electron energy/momentum distribution and transport parameters, including inter-valley transfer, in GaAs subjected to a fast varying electric field. Monte Carlo (MC) simulations are also included to illustrate the difference between evolution scales of the kinetic and tepsilon -scale hydro-kinetic distributions. The study indicates that the electron distribution is strongly dependent on the mean energy but weakly on the average momentum. In GaAs subjected to a rapid increase in field, effects of the momentum dependence is enhanced only near the peak of strong velocity overshoot, such as the overshoot in the Gamma valley. The Gamma -valley energy scale hydro-kinetic (energy dependent) distribution thus appreciably deviates from the kinetic distribution near the peak of strong overshoot. As a result, the hydro-kinetic model leads to a smaller overshoot in the Gamma valley than the Me method. In the case of less pronounced velocity overshoot, the energy scale hydro-kinetic distribution can reasonably follow the evolution scale of the kinetic distribution function.