Analysis of models for quantum transport of electrons in graphene layers

TL;DR

This paper develops and analyzes two mathematical models for quantum electron transport in graphene, proving local well-posedness using advanced dispersive estimates and boundary condition adaptations.

Contribution

It introduces a new model for electrons in bounded domains with Dirichlet conditions and extends dispersive estimates to mixed quantum states.

Findings

Proved local existence and uniqueness for the Dirac model in unbounded domains.

Established well-posedness of the boundary-constrained model in fractional Sobolev spaces.

Generalized Strichartz estimates for mixed quantum states.

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 $\RR^2$, the model takes the form of a system of massless Dirac equations coupled together by a selfconsistent 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 $H^s(\RR^2)$, for $s > 3/8$ which includes in particular the energy space $H^{1/2}(\RR^2)$. The main tools that enable to reach $s\in (3/8,1/2)$ 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 $\Omega$. In order to take into account Dirichlet boundary conditions which are not compatible with the Dirac Hamiltonian $H_{0}$, we propose a different model built on a modified Hamiltonian displaying the same energy band diagram as $H_{0}$ near the Dirac points. The well-posedness of the system in this case is proved in $H^s_{A}$, the domain of the fractional order Dirichlet Laplacian operator, for $1/2\leq s<5/2$.