Quantum Transmission Conditions for Diffusive Transport in Graphene with Steep Potentials

Abstract

We present a formal derivation of a drift-diffusion model for stationary electron transport in graphene, in presence of sharp potential profiles, such as barriers and steps. Assuming the electric potential to have steep variations within a strip of vanishing width on a macroscopic scale, such strip is viewed as a quantum interface that couples the classical regions at its left and right sides. In the two classical regions, where the potential is assumed to be smooth, electron and hole transport is described in terms of semiclassical kinetic equations. The diffusive limit of the kinetic model is derived by means of a Hilbert expansion and a boundary layer analysis, and consists of drift-diffusion equations in the classical regions, coupled by quantum diffusive transmission conditions through the interface. The boundary layer analysis leads to the discussion of a four-fold Milne (half-space, half-range) transport problem.

Mathematical modelling of charge transport in graphene heterojunctions

“…This point, which was only briefly mentioned in Ref. [4], is fully developed in the Section 3 of the present paper. We obtain in this way an explicit expression of the interpolation coefficient for the specific case we are interested in, that is the case of a potential barrier and of purely electron transport.…”

“…In Section 2 we review the main results of Ref. [4]. In particular, we show how the densities at both sides of a quantum interface are connected by diffusive transmission conditions (Theorem 2.2).…”

“…The aim of this paper is to apply the theory of diffusive quantum transmission conditions, developed in Ref. [4], to the mathematical modelling of a device of this kind.…”

“…The theory has been recently revisited in the case of graphene in Ref. [4]. The main novelty in such a case comes from the fact that electrons in graphene feature a conical intersection between the conduction band and the valence band and, therefore, the behaviour of charge carriers is well described by a Dirac-like equation [8].…”

“…In this section we briefly review the transport model across quantum interfaces in graphene, model that has been developed in Ref. [4].…”