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 (E-11(M)) and the third and fourth transitions of semiconducting tubes (E-33(S) and E-44(S), respectively) in comparison with calculated values using a nonorthogonal tight-binding approach. For semiconducting tubes, we find that the calculated energies E-33(S) and E-44(S) 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 E-11(M). 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 E-33(S) and E-44(S), contrary to the lower transitions E-11(S) and E-22(S).