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.