Characterization of nanographite and carbon nanotubes by polarization dependent optical spectroscopy

The optical absorption for ? electrons as a function of the electron wavevector k is investigated by first order perturbation theory in graphite and single wall carbon nanotubes (SWNTs). The matrix element connecting two states in the valence and conduction bands is found to be significantly anisotropic in k -space and polarization dependent. In the case of graphite, the absorption shows a node around the equi-energy contour, and in the case of SWNTs we obtain selection rules that allow only transitions between certain pairs of subbands. The strength of the optical absorption is not only diameter dependent but also chirality dependent. The implications of the optical absorption matrix element on the resonant conditions are discussed.