Dark Excitons in Carbon Nanotubes

We investigate the lowest many-body excited states in carbon nanotubes by means of ab initio calculations. On the basis of these calculations and an additional theoretical analysis of the excitons, we demonstrate that the splitting between the dark and bright spin-singlet states is due to the direct Coulomb electron-hole interaction, that the dark exciton always has gerade symmetry, and that it is always lower in energy relative to the bright exciton. Applying these studies to 0.4-0.8-nm-diameter tubes, we find a gerade-ungerade splitting of several tens of an meV, and a singlet-triplet splitting of comparable value. Our symmetry analysis of the origin of the singlet splitting indicates possible startegies to increase the emission efficiency of nanotubes.