Theory of Raman enhancement by two-dimensional materials: Applications for graphene-enhanced Raman spectroscopy

We propose a third-order time-dependent perturbation theory approach to describe the chemical surfaceenhanced Raman spectroscopy of molecules interacting with two-dimensional (2D) surfaces such as an ideal 2D metal and graphene, which are both 2D metallic monolayers. A detailed analysis is performed for all the possible scattering processes involving both electrons and holes and considering the different time orderings for the electron-photon and electron-phonon interactions. We show that for ideal 2D metals a surface enhancement of the Raman scattering is possible if the Fermi energy of the surface is near the energy of either the HOMO or the LUMO states of the molecule and that a maximum enhancement is obtained when the Fermi energy matches the energy of either the HOMO or the LUMO energies plus or minus the phonon energy. The graphene-enhanced Raman spectroscopy effect is then explained as a particular case of a 2D surface, on which the density of electronic states is not constant, but increases linearly with the energy measured from the charge neutrality point. In the case of graphene, the Raman enhancement can occur for any value of the Fermi energy between the HOMO and LUMO states of the molecule. The proposed model allows for a formal approach for calculating the Raman intensity of molecules interacting with different 2D materials.