Scaling properties of mono-layer graphene away from the Dirac point

TL;DR

This paper analyzes how non-zero chemical potential affects the spatial inhomogeneity and scaling behavior of charge density in gated graphene, revealing non-Gaussian correlations and double-logarithmic spatial dependence.

Contribution

It provides a detailed theoretical analysis of the effects of chemical potential on the scaling properties of charge density in graphene using the Thomas-Fermi-Dirac framework.

Findings

Charge correlations become non-Gaussian with non-zero chemical potential.

Two-point density correlation exhibits double-logarithmic spatial dependence.

Fourier power spectrum deviates from Gaussian rough surface behavior.

Abstract

The statistical properties of the carrier density profile of graphene in the ground state in the presence particle-particle interaction and random charged impurity in zero gate voltage has been recently obtained by Najafi \textit{et al.} (Phys. Rev E95, 032112 (2017)). The non-zero chemical potential ($\mu$) in gated graphene has non-trivial effects on electron-hole puddles, since it generates mass in the Dirac action and destroys the scaling behaviors of the effective Thomas-Fermi-Dirac theory. We provide detailed analysis on the resulting spatially inhomogeneous system in the framework of the Thomas-Fermi-Dirac theory for the Gaussian (white noise) disorder potential. We show that, the chemical potential in this system as a random surface, destroys the self-similarity, and the charge field is non-Gaussian. We find that the two-body correlation functions are factorized to two terms: a pure function of the chemical potential and a pure function of the distance. The spatial dependence of these correlation functions is double-logarithmic, e.g. the two-point density correlation $D_2(r,\mu)\propto \mu^2\exp\left[-\left(-a_D\ln\ln r^{\beta_D}\right)^{\alpha_D} \right]$ ($\alpha_D=1.82$, $\beta_D=0.263$ and $a_D=0.955$). The Fourier power spectrum function behaves like $\ln(S(q))=-\beta_S^{-a_S}\left(\ln q \right)^{a_S}+2\ln \mu$ ($a_S=3.0\pm 0.1$ and $\beta_S=2.08\pm 0.03$) in contrast to the ordinary Gaussian rough surfaces for which $a_S=1$ and $\beta_S=\frac{1}{2}(1+\alpha)^{-1}$, ($\alpha$ being the roughness exponent). The geometrical properties are however similar to the un-gated ($\mu=0$) case, with the exponents that are reported in the text.