We calculate the exciton binding energy in single-walled carbon nanotubes
with narrow band gaps, accounting for the quasi-relativistic dispersion of
electrons and holes. Exact analytical solutions of the quantum relativistic
two-body problem are obtain for several limiting cases. We show that the
binding energy scales with the band gap, and conclude on the basis of the data
available for semiconductor nanotubes that there is no transition to an
excitonic insulator in quasi-metallic nanotubes and that their THz applications
are feasible.Comment: 11 pages, 3 figures. Several references and an additional appendix
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