  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.
