Diffusion equation for hot electrons

The diffusion equation for hot electrons is derived starting from the Boltzmann equation. It is shown that only under restricted conditions the equation may be simplified to a form analogous to the low-field form and a diffusion constant may be defined. The formula for the constant as derived for these conditions is found to be different from those given earlier. The formula indicates that the constant may be even negative at high fields in some materials and also that Monte Carlo calculations do not give its proper value. The hot-electron diffusion equation is also found to have an extra significant term, which does not occur in the low-field equation. Considering the conditions, required to be satisfied for the validity of the simplified form of the diffusion current, it is also concluded that the hot-electron diffusion cannot be analyzed by using the common phenomenological diffusion equation for the majority of diffusion experiments and hot-electron devices.