Impact of the electron-electron correlation on phonon dispersion: Failure of LDA and GGA DFT functionals in graphene and graphite

We compute the electron-phonon coupling (EPC) of selected phonon modes in graphene and graphite using various ab initio methods. The inclusion of nonlocal exchange-correlation effects within the GW approach strongly renormalizes the square EPC of the A(1)' K mode by almost 80% with respect to density-functional theory in the LDA and GGA approximations. Within GW, the phonon slope of the A(1)' K mode is almost two times larger than in GGA and LDA, in agreement with phonon dispersions from inelastic x-ray scattering and Raman spectroscopy. The hybrid B3LYP functional overestimates the EPC at K by about 30%. Within the Hartree-Fock approximation, the graphene structure displays an instability under a distortion following the A(1)' `phonon at K.