Quantum transport at the Dirac point: Mapping out the minimum conductivity from pristine to disordered graphene

TL;DR

This paper maps the minimum conductivity of graphene from pristine to disordered states using advanced simulations, revealing how device aspect ratio and impurity density influence transport regimes and highlighting the role of Klein tunneling.

Contribution

It introduces a comprehensive phase space mapping of graphene's minimum conductivity considering device geometry and disorder, with novel insights into tunneling effects and conductivity saturation.

Findings

Conductivity trends depend on aspect ratio and impurity density.

Conductivity converges to approximately 4q^2/h in the dirty limit.

Absence of Fabry-Pérot oscillations indicates Klein tunneling effects.

Abstract

The phase space for graphene's minimum conductivity $\sigma_\mathrm{min}$ is mapped out using Landauer theory modified for scattering using Fermi's Golden Rule, as well as the Non-Equilibrium Green's Function (NEGF) simulation with a Monte Carlo sampling over impurity distributions. The resulting `fan diagram' spans the range from ballistic to diffusive over varying aspect ratios ($W/L$), and bears several surprises. {The device aspect ratio determines how much tunneling (between contacts) is allowed and becomes the dominant factor for the evolution of $\sigma_{min}$ from ballistic to diffusive regime. We find an increasing (for $W/L>1$) or decreasing ($W/L<1$) trend in $\sigma_{min}$ vs. impurity density, all converging around $128q^2/\pi^3h\sim 4q^2/h$ at the dirty limit}. In the diffusive limit, the {conductivity} quasi-saturates due to the precise cancellation between the increase in conducting modes from charge puddles vs the reduction in average transmission from scattering at the Dirac Point. In the clean ballistic limit, the calculated conductivity of the lowest mode shows a surprising absence of Fabry-P\'{e}rot oscillations, unlike other materials including bilayer graphene. We argue that the lack of oscillations even at low temperature is a signature of Klein tunneling.