Electron-electron interaction effects on cross-polarized optical absorption in semiconducting single-walled carbon nanotubes (S-SWCNTs)

Within the tight binding theory of S-SWCNTs optical transitions polarized transverse to the nanotube axis, E$_{12}$ and E$_{21}$, are degenerate, and occur at (E$_{11}$ + E$_{22}$)/2, where E$_{11}$ and E$_{22}$ are the optical transitions polarized along the nanotube axis. Electron-electron interactions split the degeneracy of the transverse transitions, giving new eigenstates that are a redshifted optically forbidden odd superposition and a blueshifted (by several tenths of 1 eV) optically allowed even superposition of the one-electron excitations \footnote{H. Zhao and S. Mazumdar, Phys. Rev. Lett. {\bf 93}, 157402 (2004)}. Recent experiments \footnote{Y. Miyauchi, M. Oba and S. Maruyama, Phys. Rev. B, accepted (2006)} have confirmed our qualitative prediction. Here we report quantitative calculations of the longitudinal versus transverse optical absorptions in the four S-SWCNTs studied by Miyauchi {\it et al.}, within a $\pi$-electron Hamiltonian with long range Coulomb interactions \footnote{Z. Wang, H. Zhao and S. Mazumdar, Phys. Rev. B {\bf 74}, 195406 (2006)}. We make detailed comparisons between experiments and theory. We also comment on the role of electron hoppings beyond nearest neighbor and the binding energy of the transverse exciton.