Lucky-drift mechanism for impact ionisation in semiconductors

Abstract

A simple analytic expression for the ionisation coefficient for impact ionisation is derived on the basis of a new approach which exploits the difference between momentum- and energy-relaxation rates for hot electrons. The basic mechanism whereby an electron gains sufficient energy to ionise is lucky to drift in which the electrons relax momentum but not energy. The electrons gain energy by drift and not by ballistic motion, and a few lucky ones reach the threshold. Those which thermalise may also contribute through the lucky-drift mechanism, starting from the average hot-electron energy. Good agreement with Baraff's theory is obtained. It is shown that neither the Schockley lucky electron nor the Wolff thermalised electron contribute significantly, in agreement with Baraff. However, the concept of Schockley's lucky electron is an essential part of the lucky-drift mechanism. The theory is simply extended to accommodate electrons injected at energies above zero, and some calculations are presented on this topic. A discussion is given of the effect of real band structure and it is concluded that the theory based on parabolic bands remains good provided the mean free path is taken as an average quantity over the relevant energy range. It is argued that the theory has wide application to semiconductors with moderate-to-large energy gaps because of the predominance of nonpolar scattering at high energies. A specific model of a nonparabolic band structure is discussed in which the electron distribution function has a Gaussian form, rather than the Maxwellian form associated with parabolic bands, and a weak negative differential resistance is exhibited.