Chiral dependence of K-momentum exciton energies in carbon nanotubes

Fifteen of the sixteen excitons in the $E_{11}$ manifold of carbon nanotubes are nominally dark. Of these, the zero-momentum dark singlet has received the most experimental attention because it exhibits magnetic brightening. By contrast, here we focus on the $K$ and $K'$-momentum dark singlets. Absorptive ($X_{2})$ and emissive ($X_{1})$ sidebands appearing at $\sim $0.2 eV above and $\sim $0.13 eV below the bright exciton, respectively, have been interpreted in numerous ways in the literature. Recently, members of our group studied these sidebands in a (6,5) nanotube and concluded they could be used to energetically locate the $K$-momentum excitons (Torrens, et al, PRL 101, 157401 (2008)). Here we use a combination of experiment and theory to study $X_{1}$ and $X_{2}$ in at least ten samples that are highly purified in a single chirality and use these findings to study how the $K$-momentum exciton energy depends on chirality.