Exact solution of a transport equation for hot-electron effects in semiconductors and metals

We present and solve a Boltzmann equation for hot-electron transport where the elastic and inelastic collision terms are characterized by separate relaxation times ${\tau }_{\mathrm{el}}$ and ${\tau }_{\mathrm{in}}$, respectively. The model is solved exactly for metals and semiconductors as a function of R=${\tau }_{\mathrm{in}/{\tau }_{\mathrm{el}}}$. The exact solution is compared to two standard approximation schemes: the sp (or diffusive) approximation in which the Legendre expansion is truncated after the second term, and the effective-temperature model. For large R, the sp approximation becomes exact. In metals, both the effective-temperature model and the sp approximation are qualitatively correct for all R. In semiconductors, a strong dimensionality dependence is seen; in one dimension the approximations are valid, but in three dimensions they in general are not. The longitudinal (parallel to the field) and transverse projections of the exact distribution functions are calculated for semiconductors, and in a regime where the sp approximation is poor, the longitudinal function is seen to be approximated well by the one-dimensional solution with renormalized R.