Two-phonon Raman bands of single-walled carbon nanotubes: a case study

It has been long accepted that the second-order Raman bands in carbon nanotubes are enhanced through the double-resonance mechanism. Although separate aspects of this mechanism have been studied for a few second-order Raman bands, including the most intense defect-induced D band and the two-phonon 2D band, a complete computational approach to the second-order bands is still lacking. Here, we propose such an approach, entirely based on a symmetry-adapted non-orthogonal tight-binding model with ab-initio-derived parameters. As a case study, we consider nanotube (6, 5), for which we calculate the two-phonon spectrum. We investigate in detail the 2D band and identify three contributions to it: a non-dispersive one and two dispersive ones, which are found to depend on the electron and phonon dispersion of the nanotube, and on the laser excitation. We also predict two-phonon bands, which are not allowed in the parent structure graphene. The obtained two-phonon bands are in very good agreement with the available experimental data. The symmetry-adapted formalism makes feasible the calculation of the two-phonon Raman bands of any observable nanotube.