Quantum transmission conditions for diffusive transport in graphene with steep potentials

TL;DR

This paper derives a drift-diffusion model for electron transport in graphene with sharp potential barriers, coupling classical regions via quantum transmission conditions, using kinetic equations and boundary layer analysis.

Contribution

It introduces a novel quantum interface model for sharp potential profiles in graphene, connecting classical drift-diffusion equations through quantum transmission conditions.

Findings

Derived drift-diffusion equations for classical regions

Established quantum diffusive transmission conditions

Analyzed a four-fold Milne transport problem

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.