Dynamical polarization of graphene at finite doping

Abstract

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