Discontinuity in the family pattern of single-wall carbon nanotubes

The higher lying bright exciton energies (E11M, E33S, E44S, E22M, E55S, E66S, E33M) of single-wall carbon nanotubes are calculated by solving the Bethe-Salpeter equation within an extended tight binding method. For smaller diameter nanotubes, some higher Eii excitonic states are missing. In particular, some Eii 's on the one-dimensional Brillouin zone (cutting line) are no longer relevant to the formation of excitons and are skipped in listing the order of the Eii values. Thus the family patterns show some discontinuities in k space and this effect should be observable in Raman G′ band spectroscopy. The higher exciton energies E33S and E44S have a large chirality dependence due to many body effects, since the self-energy becomes larger than the binding energy. Thus the chirality dependence of the higher Eii comes not only from a single particle energy but also from many-body effects.