Chirality dependence of theK-momentum dark excitons in carbon nanotubes

Using a collection of 12 semiconducting carbon-nanotube samples, each highly enriched in a single chirality, we study the chirality dependence of the K-momentum dark singlet exciton using phonon sideband optical spectroscopy. Measurements of bright absorptive and emissive sidebands of this finite momentum exciton identify its energy as 20--38 meV above the bright singlet exciton, a separation that exhibits systematic dependencies on tube diameter, $2n+m$ family, and chiral index. We present calculations that explain how chiral angle dependence in this energy separation relates to the Coulomb exchange interaction and elaborate the dominance of the K_A₁^' phonon sidebands over the zone-center phonon sidebands over a wide range of chiralities. The Kataura plot arising from these data is qualitatively well described by theory but the energy separation between the sidebands shows a larger chiral dependence than predicted. This latter observation may indicate a larger dispersion for the associated phonon near the K point than expected from finite distance force modeling.