E33andE44optical transitions in semiconducting single-walled carbon nanotubes: Electron diffraction and Raman experiments

Abstract

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 $0.43\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ and $0.44\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, respectively, for tubes in the $1.4--2.4\text{\ensuremath{-}}\mathrm{nm}$-diameter range. For metallic tubes in the $1.2--1.7\text{\ensuremath{-}}\mathrm{nm}$-diameter range, we show that a rigid shift $(0.32\phantom{\rule{0.3em}{0ex}}\mathrm{eV})$ 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 ${E}_{22}^{S}$.