Hot electron distribution function for the Boltzmann equation with analytic bands

The shape of the electron energy distribution (EED) is of paramount importance in order to predict disruptive phenomena in semiconductor devices. Monte Carlo simulations with parabolic bands prove that the EED tail is maxwellian [1], [2], and several analytic euristic models have been introduced for modeling the EED in the bulk case [3], [4]. Starting form the Bolzmann transport equation in the bulk case, with quasi parabolic band, scattering with impurities, acoustic and optical phonons, we prove analytically that its EED, for large values of the energy e, is of the following type