$K$-momentum dark exciton energy in carbon nanotubes*

Phonon sideband optical spectroscopy determines the energy of the dark $K$-momentum exciton for (6,5) carbon nanotubes (CNTs). One-phonon sidebands appear in absorption and emission, split by two zone-boundary ($K$-point) phonons. Their average energy locates the $E_{11} \quad K$-momentum exciton 36 meV above the $E_{11}$ bright level, higher than available theoretical estimates. A model for exciton-phonon coupling shows the absorbance sideband depends sensitively on the $K$-momentum exciton effective mass and has minimal contributions from zone-center phonons, which dominate the Raman spectra of CNTs.