  High-field magneto-optical spectroscopy was conducted on highly-selected
chiral (6,5) specific single-walled carbon nanotubes. Spectra of phonon
sidebands in both 1st and 2nd sub-bands were observed to be unchanged by the
application of an external magnetic field up to 52 T. Our analyses led to the
conclusion that both phonon sidebands in respective sub-band originate from the
dark K-momentum singlet (D-K-S) excitons. Moreover, while the relative ordering
between the bandedge bright exciton and its zero-momentum anti-bonding
counterpart was found to be opposite for the 1st and 2nd sub-bands, the
relative ordering between the D-K-S exciton and the band-edge bright exciton
was clarified to be the same for both sub-bands. Energy of these D-K-S excitons
was estimated to be ~ 21.5 and ~ 37.3 meV above the band-edge bright exciton
for the 1st and 2nd sub-bands, respectively.
