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 ∼ 1350 cm −1 , D ′ at ∼ 1600 cmand D ′′ at ∼ 1100 cm −1 ), and two-phonons ones (such as in the 2D, 2D ′ , or D + D ′′ 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 kind of idealized defects. For an excitation energy of ǫL = 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 ′ , 2D and 2D ′ lines; the line shapes; the presence of small intensity lines in the 1800, 2000 cm −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 2D ′ and 2D lines can be used to determine experimentally the electronic linewidth. The intensity ratio between the D and D ′ 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 K→ Γ direction (inner phonons) and not to the K→M one (outer phonons), as usually assumed. The small 2D line width at ǫL = 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. ǫL = 3.8 eV, the 2D line becomes broader and evolves in an asymmetric double peak structure.

Confocal Raman spectroscopy has emerged as a major, versatile workhorse for the non-invasive characterization of graphene. Although it is successfully used to determine the number of layers, the quality of edges, and the effects of strain, doping and disorder, the nature of the experimentally observed broadening of the most prominent Raman 2D line has remained unclear. Here we show that the observed 2D line width contains valuable information on strain variations in graphene on length scales far below the laser spot size, that is, on the nanometre-scale. This finding is highly relevant as it has been shown recently that such nanometre-scaled strain variations limit the carrier mobility in high-quality graphene devices. Consequently, the 2D line width is a good and easily accessible quantity for classifying the crystalline quality, nanometre-scale flatness as well as local electronic properties of graphene, all important for future scientific and industrial applications.