Demonstration of one-parameter scaling at the Dirac point in graphene

TL;DR

This paper numerically demonstrates one-parameter scaling of conductivity at the Dirac point in graphene, showing the conductivity increases logarithmically with sample size without reaching a fixed point, challenging previous predictions.

Contribution

It provides the first numerical evidence of one-parameter scaling at the Dirac point in graphene, clarifying the scaling behavior in the absence of intervalley scattering.

Findings

Conductivity exhibits one-parameter scaling with sample size.

Scaling flow has no fixed point for conductivities studied.

Conductivity increases logarithmically with sample size.

Abstract

We numerically calculate the conductivity $\sigma$ of an undoped graphene sheet (size $L$) in the limit of vanishingly small lattice constant. We demonstrate one-parameter scaling for random impurity scattering and determine the scaling function $\beta(\sigma)=d\ln\sigma/d\ln L$. Contrary to a recent prediction, the scaling flow has no fixed point ($\beta>0$) for conductivities up to and beyond the symplectic metal-insulator transition. Instead, the data supports an alternative scaling flow for which the conductivity at the Dirac point increases logarithmically with sample size in the absence of intervalley scattering -- without reaching a scale-invariant limit.