Résumé
We calculate the long-wavelength static screening properties of both neutral and doped graphene in the framework of density-functional theory. We use a plane-wave approach with periodic images in the third dimension and truncate the Coulomb interactions to eliminate spurious interlayer screening. We carefully address the issue of extracting two dimensional dielectric properties from simulated three-dimensional potentials. We compare this method with analytical expressions derived for two dimensional massless Dirac fermions in the random phase approximation. We evaluate the contributions of the deviation from conical bands, exchange-correlation and local-fields. For momenta smaller than twice the Fermi wavevector, the static screening of graphene within the density-functional perturbative approach agrees with the results for conical bands within random phase approximation and neglecting local fields. For larger momenta, we find that the analytical model underestimates the static dielectric function by $\approx 10%$, mainly due to the conical band approximation.