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

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 ${(E}_{\mathrm{ii}}^{\exp }),$ and then we compare these $E_{\mathrm{ii}}^{\exp }$ with the $E_{\mathrm{ii}}$ values obtained with theoretical predictions ${(E}_{\mathrm{ii}}^{\mathrm{cal}}).\mathrm{}$ In such an approach, the nanotube $\mathrm{}(n,m)\mathrm{}$ structure identification depends on the theory parameters, but the experimental determination of $E_{\mathrm{ii}}^{\exp }$ 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 ${\omega }_{\mathrm{RBM}}$ and $E_{\mathrm{ii}}^{\exp }$ for specific tubes, and we then performed the $\mathrm{}(n,m)\mathrm{}$ 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 ${(d}_t<1.1\mathrm{}\mathrm{nm}),$ where there are very few $\mathrm{}(n,m)\mathrm{}$ possibilities for SWNTs that can be in resonance with the appropriate laser energy $E_{\mathrm{laser}}.$ This allows an experimental determination of $E_{\mathrm{ii}}^{\exp }$ values to be made for a variety of $\mathrm{}(n,m)\mathrm{}$ SWNTs. Our experimental results indicate that: (i) the large curvature in small diameter tubes induces a $\sigma -\pi$ hybridization, thus lowering the electronic band energies, and (ii) the simple formulation of the tight binding model $({\gamma }_0=2.89\mathrm{}\mathrm{eV})$ to determine $E_{\mathrm{ii}}$ starts to deviate from $E_{\mathrm{ii}}^{\exp }$ for tubes with $d_t<1.1\mathrm{}\mathrm{nm},$ but the deviation $\Delta E_{22}{=E}_{22}^{\exp }-E_{22}^{\mathrm{cal}}$ remains smaller than 20 meV for $d_t>~0.83\mathrm{nm}.$ A comparison between $E_{\mathrm{ii}}^{\exp }$ data obtained from Raman and photoluminescence is made, and a comparison is also made between $E_{\mathrm{ii}}^{\exp }$ data for SWNTs and double-wall carbon nanotubes.