Theory of resonant multiphonon Raman scattering in graphene

We present a detailed calculation of intensities of two-phonon and four-phonon Raman peaks in graphene. Writing the low-energy Hamiltonian of the interaction of electrons with the crystal vibrations and the electromagnetic field from pure symmetry considerations, we describe the system in terms of just a few independent coupling constants, considered to be parameters of the theory. The electron-scattering rate is introduced phenomenologically as another parameter. The results of the calculation are used to extract information about these parameters from the experimentally measured Raman peak intensities. In particular, the Raman intensities are sensitive to the electron-scattering rate, which is not easy to measure by other techniques. Also, the Raman intensities depend on electron-phonon coupling constants; to reproduce the experimental results, one has to take into account renormalization of these coupling constants by electron-electron interaction.