Path Variable Formulation of the Hot Carrier Problem

Abstract

A theoretical treatment of transport phenomena in strong electric fields is presented. Instead of the Legendre polynomial expansion of the distribution function usually employed in solving the transport equation, we transform the Boltzmann equation to a coordinate system determined by the collision-free trajectories of the particles and formulate an integral equation for the distribution function. This method is applied to hot carriers in nonpolar semiconductors, where the relevant transport equation is then reduced to a one-dimensional integral equation. This equation is solved numerically and energy distributions are calculated for $n$- and $p$-type germanium. The calculations for heavy holes in germanium demonstrate the non-Maxwellian nature of the distribution function as well as its strong displacement in momentum space, and are in excellent agreement with experiment. The energy distributions for electrons show weaker deviations from Maxwellian and smaller ratios of drift to rms velocity, this being due to the weaker coupling to optical phonons for electrons as compared to holes.