Scaling properties of monolayer graphene away from the Dirac point

The statistical properties of the carrier density profile of graphene in the ground state in the presence of particle-particle interaction and random charged impurity in zero gate voltage has been recently obtained by Najafi et al. [Phys. Rev. E 95, 032112 (2017)2470-004510.1103/PhysRevE.95.032112]. The nonzero chemical potential (μ) in gated graphene has nontrivial 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 also 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 behaves like D_{2}(r,μ)∝μ^{2}exp[-(-a_{D}lnlnr^{β_{D}})^{α_{D}}] (α_{D}=1.82, β_{D}=0.263, and a_{D}=0.955). The Fourier power spectrum function also behaves like ln[S(q)]=-β_{S}^{-a_{S}}(lnq)^{a_{S}}+2lnμ (a_{S}=3.0±0.1 and β_{S}=2.08±0.03) in contrast to the ordinary Gaussian rough surfaces for which a_{S}=1 and β_{S}=1/2(1+α)^{-1} (α being the roughness exponent). The geometrical properties are, however, similar to the ungated (μ=0) case, with the exponents that are reported in the text.