Deterministic and stochastic solutions of the Boltzmann equation for charge transport in graphene on substrates

A physically accurate model for charge transport in graphene is given by a semiclassical Boltzmann equation whose scattering terms have been deeply analyzed in the last decade. Due to the computational difficulties, the most part of the available solutions have been obtained with direct Monte Carlo simulations. The aim of this work is to simulate a monolayer graphene on a substrate, as, for instance, considered in [3]. We use a numerical scheme based on the discontinuous Galerkin method for finding deterministic (non stochastic) solutions of the electron Boltzmann equation in graphene [4]. The results are compared with those obtained with a recent DSMC approach that properly takes into account the Pauli exclusion principle [4]. In a semiclassical kinetic setting, the charge transport in graphene is described by four Boltzmann equations, one for electrons in the valence (π) band and one for electrons in the conductions (π∗) band, that in turn can belong to the K or K ′ valley,