Non-linear screening of external charge by doped graphene

TL;DR

This paper develops a nonlinear Thomas-Fermi model to analyze how doped graphene screens external charges, comparing it with linear models and RPA, revealing persistent nonlinear effects even at high doping levels.

Contribution

It introduces a nonlinear integral equation approach for graphene screening that accounts for finite doping, temperature, substrate effects, and compares with existing models.

Findings

Good agreement between nonlinear TF and RPA models in doped graphene.

Nonlinear effects persist at high doping densities and large distances.

Friedel oscillations are observed in RPA but not in TF model.

Abstract

We solve a nonlinear integral equation for the electrostatic potential in doped graphene due to an external charge, arising from a Thomas-Fermi (TF) model for screening by graphene's $\pi$ electron bands. In particular, we study the effects of a finite equilibrium charge carrier density in graphene, non-zero temperature, non-zero gap between graphene and a dielectric substrate, as well as the nonlinearity in the band density of states. Effects of the exchange and correlation interactions are also briefly discussed for undoped graphene at zero temperature. Nonlinear results are compared with both the linearized TF model and the dielectric screening model within random phase approximation (RPA). In addition, image potential of the external charge is evaluated from the solution of the nonlinear integral equation and compared to the results of linear models. We have found generally good agreement between the results of the nonlinear TF model and the RPA model in doped graphene, apart from Friedel oscillations in the latter model. However, relatively strong nonlinear effects are found in the TF model to persist even at high doping densities and large distances of the external charge.