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.
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