Optical absorption matrix elements in single-wall carbon nanotubes

The optical absorption matrix element as a function of one-dimensional (1D) wave vector k, and subband index μ of a single wall carbon nanotube is given analytically for linearly polarized light with polarization parallel to the nanotube axis. For armchair nano-tubes, it is found that the optical transitions for non-degenerate A symmetry bands are forbidden over the whole 1D k region and the transitions for all other bands are also forbidden at the k = 0 point. Near the Fermi level, the absorption for all metallic nanotubes is found to be approximately zero. For both metallic and semiconducting nanotubes, it is found that the absorption matrix element has a maximum absolute value at the van Hove singularity (vHS) k point around the Fermi energy for each band. The absorption dependence on diameter and chiral angle is also presented for semiconducting nanotubes. For light polarization perpendicular to the nanotube axis, on the other hand, the absorption for nanotubes is generally weak near a vHS.