Electron drift and diffusion in a high electric field

The Fokker-Planck framework is used to investigate electron transport in a uniform electric field in a solid, under the assumption that the electron be drifting. The drift velocity and diffusion tensor at an arbitrary field strength are expressed as simple integrals involving the exact band structure and electron-lattice interaction. The position-dependent energy distribution is obtained explicitly, and it is found that the leading edge of a carrier packet is hotter than the trailing edge. The predictions are in quantitative agreement with a Monte Carlo simulation of transport in the absence of any adjustable parameter. In addition, Ohmic transport is retrieved within the Fokker-Planck formalism in the limit of a vanishing field strength. It can be envisioned as ``unlucky drift'' in the sense that the field cannot bring about a significant departure from equilibrium statistics. It is further argued that the connection between equilibrium and high-field statistics ingrained in thermodynamic approaches is the reason for their inadequacy in solid-state electronics.