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

GW is nowadays the most accurate ab-initio method to determine electronic bands. So far GW has never been used to determine neither the electron-phonon coupling (EPC) nor phonon dispersions. We show that GW approach [1] can be used to compute the EPC and the phonon dispersion. In particular, in graphene and graphite, standard DFT (LDA and GGA) underestimates, by a factor of 2, the slope of the highest optical branch at the zone boundary (K) and the square of its EPC by almost 80\%. On the contrary, GW reproduces the experimental phonon dispersion near K, the value of the EPC, and the electronic band dispersion, in agreement with phonon dispersions from inelastic x-ray scattering and Raman spectroscopy. Comparing these results with other computational methods, the B3LYP hybrid functional gives phonons close to GW but overestimates the EPC at K by about 30\%. Within Hartree-Fock, the graphene structure displays an instability under a distortion following the A'1 phonon at K. [1] M. Lazzeri et al., Phys. Rev. B 78, 081406(R) (2008).