Dark excitons in single-walled carbon nanotubes investigated by electroabsorption spectroscopy

We report electroabsorption (EA) spectra in micelle-wrapped single walled carbon nanotubes (SWNT). By applying the high electric field (up to 85kV/cm), the absorption spectra of semiconducting SWNTs show field-induced change in the region of $E_{11}$ and $E_{22}$ transitions. The EA spectra are essentially reproduced by the second derivative curves of the absorption spectra. Such spectral features indicate that the bright (one-photon allowed) exciton and dark (two-photon allowed) exciton for each chiral index are nearly degenerate. The closer scrutiny of the EA spectra reveals that the dark excitons for several chiral indices are located on the higher energy side of the bright states.