Lorentzian Gas and Hot Electrons

The problem of hot electrons in a nonpolar crystal is reconsidered using the Lorentzian gas model more accurately. Scattering by acoustical phonons alone is considered first. The new results are (1) an asymptotic formula for the moments of the velocity distribution which permits calculation of the deviation from the square root law at high fields, and (2) a recursion system allowing the calculation of any velocity polynomial in terms of the average energy, random velocity, and mobility of the electrons. Scattering by ionized impurities in addition to acoustical phonons is considered next and the distribution function is derived. The proportionality constant relating the change in the low-field mobility to $E^2$ is shown to be highly sensitive to ionized impurity scattering. Thus, appreciable changes from its value for pure lattice scattering occur for $\frac{{\mu }_0}{{\mu }_I}$ as low as ${10}^{-3\mathrm{}}$. ($E$ is the field strength and ${\mu }_0$ and ${\mu }_I$ are the low-field lattice and impurity mobilities, respectively.) It is pointed out that substantial deviations from results obtained using a Maxwellian distribution do occur.