Screening of Coulomb Impurities in Graphene

TL;DR

This paper provides an exact calculation of vacuum polarization around Coulomb impurities in graphene, revealing how impurity charge and electron interactions influence criticality, with implications for impurity screening.

Contribution

It offers an exact Green's function approach to analyze Coulomb impurities in graphene, including self-consistent electron-electron interactions in the subcritical regime.

Findings

Impurity with Z=1 remains subcritical for all α.

Impurities with Z=2, 3, can become supercritical at certain α.

Results are exact in the parameter Zα, accounting for electron interactions.

Abstract

We calculate exactly the vacuum polarization charge density in the field of a subcritical Coulomb impurity, $Z|e|/r$, in graphene. Our analysis is based on the exact electron Green's function, obtained by using the operator method, and leads to results that are exact in the parameter $Z\alpha$, where $\alpha$ is the "fine structure constant" of graphene. Taking into account also electron-electron interactions in the Hartree approximation, we solve the problem self-consistently in the subcritical regime, where the impurity has an effective charge $Z_{eff}$, determined by the localized induced charge. We find that an impurity with bare charge Z=1 remains subcritical, $Z_{eff} \alpha < 1/2$, for any $\alpha$, while impurities with $Z=2,3$ and higher can become supercritical at certain values of $\alpha$.