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 calculated. Defect-induced processes are simulated by considering different kinds of idealized defects. For an excitation energy of ${\ensuremath{\epsilon}}_{L}=2.4$ eV, the agreement with measurements is very good and calculations reproduce the relative intensities among phonon-defect or among two-phonon lines; the measured small widths of the $D$, ${D}^{\ensuremath{'}}$, $2D$, and $2{D}^{\ensuremath{'}}$ lines; the line shapes; the presence of small intensity lines in the 1800--2000-cm${}^{\ensuremath{-}1}$ range. We determine how the spectra depend on the excitation energy, on the light polarization, on the electronic linewidth, on the kind of defects, and on their concentration. According to the present findings, the intensity ratio between the $2{D}^{\ensuremath{'}}$ and $2D$ lines can be used to determine experimentally the electronic linewidth. The intensity ratio between the $D$ and ${D}^{\ensuremath{'}}$ lines depends on the kind of model defect, suggesting that this ratio could possibly be used to identify the kind of defects present in actual samples. Charged impurities outside the graphene plane provide an almost undetectable contribution to the Raman signal. The present analysis reveals that, for both $D$ and $2D$ lines, the dominant DR processes are those in which electrons and holes are both involved in the scattering, because of a destructive quantum interference that kills processes involving only electrons or only holes. The most important phonons belong to the $\mathbf{K}\ensuremath{\rightarrow}\mathbit{\ensuremath{\Gamma}}$ direction ($inner$ phonons) and not to the K $\ensuremath{\rightarrow}$ M one ($outer$ phonons), as usually assumed. The small $2D$ linewidth at ${\ensuremath{\epsilon}}_{L}=2.4$ eV is a consequence of the interplay between the opposite trigonal warpings of the electron and phonon dispersions. At higher excitation, e.g., ${\ensuremath{\epsilon}}_{L}=3.8$ eV, the $2D$ line becomes broader and evolves in an asymmetric double peak structure.