Ionic screening of charged impurities in electrolytically gated graphene

We present a model for dual-gated, single-layer graphene, with a back gate separated by a layer of oxide, and the top gate potential applied through a thick layer of liquid electrolyte that contains mobile ions in a diffuse layer, described in the Debye-Huckel approximation, which is separated from graphene by a charge-free Stern layer. After deriving a nonlinear equation for the average charge carrier density in graphene in terms of the gate potentials, we use the Green's function of the Poisson equation to express the fluctuating part of the electrostatic potential in the plane of graphene in terms of the distribution function for fixed charged impurities in the oxide. By using both the Thomas-Fermi and the random phase approximations of graphene's response at nonzero temperature, we show that the presence of mobile ions in the electrolyte significantly increases graphene's screening ability of the in-plane potential for a single impurity, accentuates Friedel oscillations in that potential, and gives rise to a linear plasmen dispersion in doped graphene at long wavelengths. In the case of multiple charged impurities in the oxide, the increasing ion concentration in the electrolyte causes a reduction in the autocorrelation function of the fluctuating in-plane potential when the impurities are uncorrelated. However, when the impurities are correlated, the relative effect of the increased ion concentration in the electrolyte is drastically reduced, while the autocorrelation function in this case takes negative values in a range of interimpurity distances.