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_ii experimentally (E^exp_ii), and then we compare these E^exp_ii with the E_ii values obtained with theoretical predictions (E^cal_ii). In such an approach, the nanotube (n,m) structure identification depends on the theory parameters, but the experimental determination of E^exp_ii does not, and depends only on the experimental AS/S intensity ratio and the laser energy E_laser used in the experiment. We measured the radial breathing mode frequency ω_RBM and E^exp_ii 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 nm), where there are very few (n,m) possibilities for SWNTs that can be in resonance with the appropriate laser energy E_laser. This allows an experimental determination of E^exp_ii 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 σ-π hybridization, thus lowering the electronic band energies, and (ii) the simple formulation of the tight binding model (γ₀=2.89 eV) to determine E_ii starts to deviate from E^exp_ii for tubes with d_t<1.1 nm, but the deviation ΔE₂₂=E^exp₂₂-E^cal₂₂ remains smaller than 20 meV for d_t⩾0.83 nm. A comparison between E^exp_ii data obtained from Raman and photoluminescence is made, and a comparison is also made between E^exp_ii data for SWNTs and double-wall carbon nanotubes.