Electron transmission through step- and barrier-like potentials in graphene ribbons

TL;DR

This paper derives exact formulas for electron transmission in graphene ribbons with step and barrier potentials, revealing how these formulas adapt from classical tunneling to account for graphene's unique electronic properties.

Contribution

It provides the first exact expressions for transmission coefficients in armchair graphene ribbons with step-like and barrier-like potentials, extending classical tunneling formulas to graphene-specific scenarios.

Findings

Transmission described by modified textbook formulas with energy-dependent barriers

In metallic ribbons, the effective barrier approaches zero for the lowest electron modes

In semiconducting ribbons, the effective barrier remains nonzero for the highest hole modes

Abstract

The list of textbook tunneling formulas is extended by deriving exact expressions for the transmission coefficient in graphene ribbons with armchair edges and the step-like and barrier-like profiles of site energies along the ribbon. These expressions are obtained by matching wave functions at the interfaces between the regions, where quasiparticles have constant but different potential energies. It is shown that for an $U_0$ high barrier and low-energy electrons and holes, the mode transmission of charge carriers in this type of ribbons is described by the textbook formula, where the constant barrier is replaced by an effective, energy-dependent barrier, $U_0\to U(E)$. For the lowest/highest electron/hole mode, $U(E)$ goes, respectively, to zero and nonzero value in metallic and semiconducting ribbons. This and other peculiarities of through-barrier/step transmission in graphene are discussed and compared with related earlier results.