Statistical mechanics of high-field transport in semiconductors.

Abstract

The paper is devoted to the statistical-mechanical description of high-field transport in semiconductors within the semiclassical picture, assuming the band structure and electron-phonon interaction to be known. Our goal is to obtain a physical understanding in simple, universal terms without resorting to specific, simplified models for the band structure and/or the electron-phonon interaction. We first examine the lucky-drift model against an exact (analytical and numerical) solution of the Boltzmann transport equation in the simple parabolic case, and two discrepancies are found. The first stems from an incorrect expression for the drift velocity of a hot electron, while the second is associated with the approximate nature of the statistical device yielding the energy distribution. With the aim of retaining the features of simplicity of the lucky-drift description, another approach to the statistics of transport in the high-field regime is developed, based upon a nonlinear Fokker-Planck equation in energy space. While it gives the exact solution in the parabolic case, it has the ability to take up transport in an arbitrary band structure as well, and accounts for the success of the generalized lucky-drift model. The carrier energy distribution is expressed in terms of integrals over constant-energy surfaces in the Brillouin zone; the integrals involve the electron-phonon interaction and have direct physical meaning. The momentum-space distribution is derived approximately from the energy distribution. A simple comprehensive picture of transport statistics emerges that is not linked to a particular material or to specific assumptions regarding the electron-phonon interaction.