Electron-phonon coupling and Raman spectroscopy in graphene

We show that the electron-phonon coupling in graphene, in contrast with the nonrelativistic two-dimensional electron gas, leads to shifts in the phonon frequencies that are nontrivial functions of the electronic density. These shifts can be measured directly in Raman spectroscopy. We show that depending whether the chemical potential is smaller (larger) than half of the phonon frequency, the frequency shift can negative (positive) relative to the neutral case (when the chemical potential is at the Dirac point), respectively. We show that the use of the static response function to calculate these shifts is incorrect and leads always to phonon softening. In samples with many layers, we find a shift proportional to the carrier concentration, and a splitting of the phonon frequencies if the charge is not homogeneously distributed. We also discuss the effects of edges in the problem.