Maximum Anisotropy Approximation for Calculating Electron Distributions; Application to High Field Transport in Semiconductors

Abstract

The conventional spherical harmonics expansion of the electron distribution function has been useful in obtaining solutions of the Boltzmann equation under circumstances in which the distribution function is fairly isotropic. This same expansion is here used under conditions in which the distribution is distorted by the presence of a large number of electrons streaming along the direction of the electric field. The innovation is a change in the truncation procedure used to obtain a closed system of equations from the infinite hierarchy of equations which result when the spherical harmonics expansion is inserted into the Boltzmann equation. The new method is applied to the calculation of the electron distribution in a semiconductor in the presence of a strong electric field. The resulting analytic distribution functions and ionization rates are compared with numerical computations in which the Boltzmann equation was solved with no approximations about the angular dependence. Agreement is of the order of 10-15% over a range o fparameters for which the ionization rates vary through three and one half decades.