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 m
agnetic 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.
