Relative Ordering between Bright and Dark Excitons in Single-walled Carbon Nanotubes

The ordering and relative energy splitting between bright and dark excitons are critical to the optical properties of single-walled carbon nanotubes (SWNTs), as they eventually determine the radiative and non-radiative recombination processes of generated carriers. In this work, we report systematic high-field magneto-optical study on the relative ordering between bright and dark excitons in SWNTs. We identified the relative energy position of the dark exciton unambiguously by brightening it in ultra-high magnetic field. The bright-dark excitonic ordering was found to depend not only on the tube structure, but also on the type of transitions. For the 1st sub-band transition, the bright exciton appears to be higher in energy than its dark counterpart for any chiral species and is robust against environmental effect. While for the 2nd sub-band, their relative ordering was found to be chirality-sensitive: the bright exciton can be either higher or lower than the dark one, depending on the specific nanotube structures. These findings provide new clues for engineering the optical and electronic properties of SWNTs.