Theory of double-resonant Raman spectra in graphene: Intensity and line shape of defect-induced and two-phonon bands

We calculate the double-resonant (DR) Raman spectrum of graphene, and determine the lines associated to both phonon-defect processes (such as in the $D$ line at $\ensuremath{\sim}$1350 cm${}^{\ensuremath{-}1}$, ${D}^{\ensuremath{'}}$ at $\ensuremath{\sim}$1600 cm${}^{\ensuremath{-}1}$, and ${D}^{\ensuremath{'}\ensuremath{'}}$ at $\ensuremath{\sim}$1100 cm${}^{\ensuremath{-}1}$), and two-phonon ones (such as in the $2D$, $2{D}^{\ensuremath{'}}$, or $D+{D}^{\ensuremath{'}\ensuremath{'}}$ lines). Phonon and electronic dispersions reproduce calculations based on density-functional theory corrected with GW. Electron-light, -phonon, and -defect scattering matrix elements and the electronic linewidth are explicitly…