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. Using results and insights from ab initio calculations, we employ a symmetry-based variational method within the effective-mass and envelope-function approximations using tight-binding wave functions. Binding energies and spatial extents show a leading dependence on diameter as $1∕d$ and $d$, respectively, with chirality corrections providing a spread of roughly 20% with a strong family behavior. Bright-dark exciton splittings show a $1∕d^2$ leading dependence. We provide analytical expressions for the binding energies, sizes, and splittings that should be useful to guide future experiments.