Energy-diffusion equation for an electron gas interacting with polar optical phonons

We present a novel method to solve explicitly the Boltzmann equation for highly energetic electrons interacting with polar optical phonons and scattering mainly in the forward direction. In this approach, the collision integral of the Boltzmann equation is reduced to a differential operator which is much easier to manipulate than the integral form and does not require a relaxation-time approximation. The relaxation of the distribution function with time as well as the spatial evolution of highly energetic electrons are calculated and closed-form expressions for the distribution function are given. In both cases the behavior of the electron distribution is characterized by two fundamental parameters: a drift factor which represents the net rate of phonon emission, and a broadening factor which is proportional to the latter and also to time and distance.