Distribution Functions and Ionization Rates for Hot Electrons in Semiconductors

The distribution of electrons in a semiconductor at high electric field is governed by a Boltzmann equation which describes the effects of the field, the phonons, and the ionization processes on the electrons. This equation can be converted to an integral equation for the space and energy dependent collision density by performing the angular integrations. The integral equation is solved numerically to obtain alpha, the ionization rate per unit path length. The resulting alpha shows a dependence on field strength $\mathcal{E}$ which is roughly $\mathrm{exp}(\ensuremath{-}\frac{b}{\mathcal{E}})$ at low fields and $\mathrm{exp}(\ensuremath{-}\frac{c}{{\mathcal{E}}^{2}})$ at high fields, but there are significant differences from the earlier calculations of Wolff and Shockley. These differences result from the approximations used by the earlier workers to treat the angular dependence. We present graphs of $log\ensuremath{\alpha}$ vs ($\frac{1}{\mathcal{E}}$) for a wide range of material parameters. These graphs are useful in interpreting measurements of charge multiplication in terms of the properties of the material supporting the transport process.