Electron-electron interactions and doping dependence of the two-phonon Raman intensity in graphene

Raman spectroscopy is a fast and nondestructive means to characterize graphene samples. In particular, the Raman spectra are strongly affected by doping. While the resulting change in position and width of the $G$ peak can be explained by the nonadiabatic Kohn anomaly at $ensuremath{Gamma}$, the significant doping dependence of the $2D$ peak intensity has not been understood yet. Here we show that this is due to a combination of electron-phonon and electron-electron scattering. Under full resonance, the photogenerated electron-hole pairs can scatter not just with phonons but also with doping-induced electrons or holes, and this changes the intensity. We explain the doping dependence and show how it can be used to determine the corresponding electron-phonon coupling. This is higher than predicted by density-functional theory, as a consequence of renormalization by Coulomb interactions.