Stokes and anti-Stokes Raman spectra of small-diameter isolated carbon nanotubes

Abstract

By measuring the anti-Stokes (AS) and Stokes (S) Raman spectra on the same isolated single-wall carbon nanotube (SWNT), we here determine the electronic transition energies ${E}_{\mathrm{ii}}$ experimentally ${(E}_{\mathrm{ii}}^{\mathrm{exp}}),$ and then we compare these ${E}_{\mathrm{ii}}^{\mathrm{exp}}$ with the ${E}_{\mathrm{ii}}$ values obtained with theoretical predictions ${(E}_{\mathrm{ii}}^{\mathrm{cal}}).$ In such an approach, the nanotube $(n,m)$ structure identification depends on the theory parameters, but the experimental determination of ${E}_{\mathrm{ii}}^{\mathrm{exp}}$ does not, and depends only on the experimental AS/S intensity ratio and the laser energy ${E}_{\mathrm{laser}}$ used in the experiment. We measured the radial breathing mode frequency ${\ensuremath{\omega}}_{\mathrm{RBM}}$ and ${E}_{\mathrm{ii}}^{\mathrm{exp}}$ for specific tubes, and we then performed the $(n,m)$ identification by using the ${d}_{t}$ diameter dependence of the electronic transitions. We present such an analysis for a wide nanotube diameter range, focusing primarily on small diameter SWNTs ${(d}_{t}<1.1\mathrm{nm}),$ where there are very few $(n,m)$ possibilities for SWNTs that can be in resonance with the appropriate laser energy ${E}_{\mathrm{laser}}.$ This allows an experimental determination of ${E}_{\mathrm{ii}}^{\mathrm{exp}}$ values to be made for a variety of $(n,m)$ SWNTs. Our experimental results indicate that: (i) the large curvature in small diameter tubes induces a $\ensuremath{\sigma}\ensuremath{-}\ensuremath{\pi}$ hybridization, thus lowering the electronic band energies, and (ii) the simple formulation of the tight binding model $({\ensuremath{\gamma}}_{0}=2.89\mathrm{eV})$ to determine ${E}_{\mathrm{ii}}$ starts to deviate from ${E}_{\mathrm{ii}}^{\mathrm{exp}}$ for tubes with ${d}_{t}<1.1\mathrm{nm},$ but the deviation $\ensuremath{\Delta}{E}_{22}{=E}_{22}^{\mathrm{exp}}\ensuremath{-}{E}_{22}^{\mathrm{cal}}$ remains smaller than 20 meV for ${d}_{t}>~0.83\mathrm{nm}.$ A comparison between ${E}_{\mathrm{ii}}^{\mathrm{exp}}$ data obtained from Raman and photoluminescence is made, and a comparison is also made between ${E}_{\mathrm{ii}}^{\mathrm{exp}}$ data for SWNTs and double-wall carbon nanotubes.