Analysis of models for quantum transport of electrons in graphene layers

Abstract

We present and analyze two mathematical models for the self-consistent quantum transport of electrons in a graphene layer. We treat two situations. First, when the particles can move in all the plane ℝ2, the model takes the form of a system of massless Dirac equations coupled together by a self-consistent potential, which is the trace in the plane of the graphene of the 3D Poisson potential associated to surface densities. In this case, we prove local in time existence and uniqueness of a solution in Hs(ℝ2), for [Formula: see text] which includes in particular the energy space H1/2(ℝ2). The main tools that enable to reach [Formula: see text] are the dispersive Strichartz estimates that we generalized here for mixed quantum states. Second, we consider a situation where the particles are constrained in a regular bounded domain Ω. In order to take into account Dirichlet boundary conditions which are not compatible with the Dirac Hamiltonian H0, we propose a different model built on a modified Hamiltonian displaying the same energy band diagram as H0 near the Dirac points. The well-posedness of the system in this case is proved in [Formula: see text], the domain of the fractional order Dirichlet Laplacian operator, for [Formula: see text].