Dynamical polarization of graphene at finite doping

TL;DR

This paper provides an exact calculation of graphene's polarization at finite doping, revealing how doping affects susceptibility, Friedel oscillations, and the behavior of plasmons and phonons, with implications for graphene's electronic and vibrational properties.

Contribution

It presents an exact polarization calculation for doped graphene within the random phase approximation, analyzing static and dynamic responses for arbitrary parameters.

Findings

Static susceptibility saturates at low momenta with doping.

Friedel oscillations decay with the same power law as Thomas-Fermi screening.

Doping causes significant stiffening of longitudinal acoustic phonons.

Abstract

The polarization of graphene is calculated exactly within the random phase approximation for arbitrary frequency, wave vector, and doping. At finite doping, the static susceptibility saturates to a constant value for low momenta. At $q=2 k_{F}$ it has a discontinuity only in the second derivative. In the presence of a charged impurity this results in Friedel oscillations which decay with the same power law as the Thomas Fermi contribution, the latter being always dominant. The spin density oscillations in the presence of a magnetic impurity are also calculated. The dynamical polarization for low $q$ and arbitrary $\omega $ is employed to calculate the dispersion relation and the decay rate of plasmons and acoustic phonons as a function of doping. The low screening of graphene, combined with the absence of a gap, leads to a significant stiffening of the longitudinal acoustic lattice vibrations.