▎ 摘　　要

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)]. The nonzero chemical potential (mu) 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 selfsimilarity, 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, mu) proportional to mu(2) exp [-(-alpha(D) ln ln r(ss D))(alpha D)] (alpha(D) = 1.82, beta(D) = 0.263, and alpha(D) = 0.955). The Fourier power spectrum function also behaves like ln[S(q)] = -beta(-aS)(S) (ln q)(aS) + 2 ln mu (a(S) = 3.0 +/- 0.1 and beta(S) = 2.08 +/- 0.03) in contrast to the ordinary Gaussian rough surfaces for which a(S) = 1 and beta(S) = 1/2 (1 + alpha)(-1) (a being the roughness exponent). The geometrical properties are, however, similar to the ungated (mu = 0) case, with the exponents that are reported in the text.