Exciton binding energies in metallic single-walled carbon nanotubes are comparable to those in semiconducting ones

Excitons in metallic single-walled carbon nanotubes (M-SWCNTs) have attracted both theoretical \footnote{J. Deslippe {\it et al.}, Nano. Lett. \textbf{7},1626-1630 (2007).} and experimental \footnote{F. Wang {\it et al.}, unpublished} attention recently. It has been claimed that exciton binding energies in M-SWCNTs are an order of magnitude smaller than those in semiconducting single-walled carbon nanotubes (S-SWCNTs). We have investigated M-SWCNTs within a $\pi$-electron Hamiltonian that has previously reproduced quantitatively the absolute energies as well as the binding energies of both longitudinal \footnote{Z. Wang, H. Zhao and S. Mazumdar, Phys. Rev. B {\bf 74}, 195406 (2006).} and transverse \footnote{Z. Wang, H. Zhao and S. Mazumdar, Phys. Rev. B {\bf 76}, 115431 (2007).} excitons in S-SWCNTs. We are able once again to reproduce quantitatively the available absolute exciton energies and the optical absorption spectra of M-SWCNTs with diameters 0.9 - 1.4 nm. While we need a dielectric constant larger than in the S-SWCNTs, our calculated exciton binding energies in this diameter range are 0.2 - 0.3 eV, only slightly smaller than those in S-SWCNTs with similar diameters. \footnote{Supported by NSF-DMR-0705163.}