  We studied the asymmetry between valence and conduction bands in
single-walled carbon nanotubes (SWNTs) through the direct observation of
spin-singlet transverse dark excitons using polarized photoluminescence
excitation spectroscopy. The intrinsic electron-hole (e-h) asymmetry lifts the
degeneracy of the transverse exciton wavefunctions at two equivalent K and K'
valleys in momentum space, which gives finite oscillator strength to transverse
dark exciton states. Chirality-dependent spectral weight transfer to transverse
dark states was clearly observed, indicating that the degree of the e-h
asymmetry depends on the specific nanotube structure. Based on comparison
between theoretical and experimental results, we evaluated the band asymmetry
parameters in graphene and various carbon nanotube structures.
