Calculation of hot electron phenomena

Abstract

Analytical formulas are generally able to give a better account of a physical phenomenon than can be provided by a numerical description. The latter may be necessary for hot electrons, however, because of the inability of simple physical principles to truly encompass the dynamics of the distribution function, and because of complexity in the band and scattering scheme of the solid, of the particular phenomenon studied, or geometry of the situation. The traditional phenomenon of interest is the steady state with spatial homogeneity, and independent nondegenerate electrons. In suitable conditions the Boltzmann equation then reduces to one in a single variable (energy) which can be solved by relatively simple means. More generally, for this case, we need a numerical computation of the distribution function. Either the latter is represented by its values on a grid of points in the space of the electron variables, and the operators of the Boltzmann equation by operations on this array, or the "history" of a single electron is simulated by a Monte Carlo scheme. The former method has advantages within its range of applicability. Monte Carlo has been extensively applied, because of its greater ability to deal with band and scattering details. It is also applicable, by elaboration of the computational procedures, to a range of phenomena including time dependence; diffusion; impact ionization; effects of carrier-carrier interaction; field-effect surface scattering; thermalization of drifting carriers; semiconductor junctions; effect of degeneracy.