Diameter and Chirality Dependence of Exciton Properties in Carbon Nanotubes

We calculate the diameter and chirality dependences of the binding energies, sizes, and bright-dark splittings of excitons in semiconducting single-wall carbon nanotubes (SWNTs). Using results and insights from {\it ab initio} calculations, we employ a symmetry-based, variational method based on the effective-mass and envelope-function approximations using tight-binding wavefunctions. Binding energies and spatial extents show a leading dependence with diameter as $1\mathrm{/}d$ and $d$, respectively, with chirality corrections providing a spread of roughly 20% with a strong family behavior. Bright-dark exciton splittings show a $1\mathrm{/}{d}^2$ leading dependence. We provide analytical expressions for the binding energies, sizes, and splittings that should be useful to guide future experiments.