Searching for confined modes in graphene channels: the variable phase method

TL;DR

This paper introduces a variable phase method to analyze confined modes in graphene, demonstrating the existence of truly confined states at the Dirac point and exploring the effects of potential decay profiles.

Contribution

It reformulates the Dirac equation into a nonlinear differential equation and applies it to study confined states and scattering in graphene with arbitrary potentials.

Findings

Confined states exist at the Dirac point in pristine graphene.

Threshold potential strength for confinement depends on the decay profile of the potential.

Power-law decaying potentials support confined states at arbitrarily small strengths.

Abstract

Using the variable phase method, we reformulate the Dirac equation governing the charge carriers in graphene into a nonlinear first-order differential equation from which we can treat both confined-state problems in electron waveguides and above-barrier scattering problems for arbitrary-shaped potential barriers and wells, decaying at large distances. We show that this method agrees with a known analytic result for a hyperbolic secant potential and go on to investigate the nature of more experimentally realizable electron waveguides, showing that, when the Fermi energy is set at the Dirac point, truly confined states are supported in pristine graphene. In contrast to exponentially-decaying potentials, we discover that the threshold potential strength at which the first confined state appears is vanishingly small for potentials decaying at large distances as a power law, but nonetheless further confined states are formed when the strength and spread of the potential reach a certain threshold.