Scale-invariant puddles in Graphene: Geometric properties of electron-hole distribution at the Dirac point

TL;DR

This paper investigates the geometric and statistical properties of electron-hole puddles in graphene at the Dirac point, revealing self-similar and non-Gaussian features influenced by Coulomb disorder and interactions.

Contribution

It introduces a detailed analysis of the spatial inhomogeneity of graphene's electron gas, highlighting the self-similar and non-Gaussian nature of puddles using the Thomas-Fermi-Dirac framework.

Findings

Electron-hole puddles exhibit self-similar statistical properties.

Charge field is non-Gaussian with unique Kondev relations.

Disorder potential remains Gaussian, but resulting charge distribution does not.

Abstract

We characterize the carrier density profile of the ground state of graphene in the presence of particle-particle interaction and random charged impurity for zero gate voltage. We provide detailed analysis on the resulting spatially inhomogeneous electron gas taking into account the particle-particle interaction and the remote coulomb disorder on an equal footing within the Thomas-Fermi-Dirac theory. We present some general features of the carrier density probability measure of the graphene sheet. We also show that, when viewed as a random surface, the resulting electron-hole puddles at zero chemical potential show peculiar self-similar statistical properties. Although the disorder potential is chosen to be Gaussian, we show that the charge field is non-Gaussian with unusual Kondev relations which can be regarded as a new class of two-dimensional (2D) random-field surfaces.