E33 and E44 optical transitions in semiconducting single-walled carbon nanotubes: Electron diffraction and Raman experiments

By combining, on the same freestanding single-walled carbon nanotubes, electron diffraction and Raman experiments, we were able to obtain the resonance energy of unambiguously (n,m) -identified single-walled carbon nanotubes. We focus on the analysis of the first optical transition of metallic tubes (E11M) and the third and fourth transitions of semiconducting tubes (E33S and E44S, respectively) in comparison with calculated values using a nonorthogonal tight-binding approach. For semiconducting tubes, we find that the calculated energies E33S and E44S have to be corrected by non-diameter-dependent (rigid) shifts of about 0.43 eV and 0.44 eV, respectively, for tubes in the 1.4-2.4-nm -diameter range. For metallic tubes in the 1.2-1.7-nm -diameter range, we show that a rigid shift (0.32 eV) of the calculated transition energy also leads to a good estimation of E11M. The rather large and non-diameter-dependent shifts for the third and fourth transitions in semiconducting tubes question a recent theoretical study, which relates the shifts to electron-electron correlation and exciton binding energy and suggest that the exciton binding is very small or missing for the higher transitions E33S and E44S, contrary to the lower transitions E11S and E22S.