Stochastic Galerkin methods for the Boltzmann-Poisson system

Abstract

We study uncertainty quantification for a Boltzmann-Poisson system that models electron transport in semiconductors and the physical collision mechanisms over the charges, using the stochastic Galerkin method in order to handle the randomness associated with the problem. In this study we choose first as a source of uncertainty the phonon energy, taking it as a random variable, as its value influences the energy jump appearing in the collision integral for electron-phonon scattering. Then we choose the lattice temperature as a random variable, since it defines the value of the collision operator terms in the case of electron-phonon scattering by being a parameter of the phonon distribution. Finally, we present our numerical simulations for the latter case. We calculate then with our stochastic Discontinuous Galerkin methods the uncertainty in kinetic moments such as density, mean energy, current, etc. associated to a possible physical temperature variation (assumed to follow a uniform distribution) in the lattice environment, as this uncertainty in the temperature is propagated into the electron PDF. Our mathematical and computational results let us predict then in a real world problem setting the impact that possible variations in the lab conditions (such as temperature) or limitations in the mathematical model (such as assumption of a constant phonon energy) will have over the uncertainty in the behavior of electronic devices.