Conformal Invariance and Shape-Dependent Conductance of Graphene Samples

TL;DR

This paper explores how the conductance of graphene samples depends on shape and quantum Hall states, using analytic models to predict conductance features for different geometries and layers.

Contribution

It introduces a shape-invariant conductance analysis for graphene, combining analytic rectangle models with the semicircle transport model to interpret experimental features.

Findings

Conductance depends on shape similarly for monolayer and bilayer graphene.

Predicted conductance plateaus at neutrality point match recent experiments.

Conductance maxima and minima correspond to quantum Hall states, varying with sample width.

Abstract

For a sample of an arbitrary shape, the dependence of its conductance on the longitudinal and Hall conductivity is identical to that of a rectangle. We use analytic results for a conducting rectangle, combined with the semicircle model for transport coefficients, to study properties of the monolayer and bilayer graphene. A conductance plateau centered at the neutrality point, predicted for square geometry, is in agreement with recent experiments. For rectangular geometry, the conductance exhibits maxima at the densities of compressible quantum Hall states for wide samples, and minima for narrow samples. The positions and relative sizes of these features are different in the monolayer and bilayer cases, indicating that the conductance can be used as a tool for sample diagnostic.