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 Eᵢᵢ experimentally (Eᵢᵢᵉˣᵖ), and then we compare these Eᵢᵢᵉˣᵖ with the Eᵢᵢ values obtained with theoretical predictions (Eᵢᵢᶜᵃˡ). In such an approach, the nanotube $(n,m)$ structure identification depends on the theory parameters, but the experimental determination of Eᵢᵢᵉˣᵖ does not, and depends only on the experimental AS/S intensity ratio and the laser energy Eₗₐₛₑᵣ used in the experiment. We measured the radial breathing mode frequency ωRBM and Eᵢᵢᵉˣᵖ for specific tubes, and we then performed the $(n,m)$ identification by using the dₜ diameter dependence of the electronic transitions. We present such an analysis for a wide nanotube diameter range, focusing primarily on small diameter SWNTs (dₜ<1.1nm), where there are very few $(n,m)$ possibilities for SWNTs that can be in resonance with the appropriate laser energy Eₗₐₛₑᵣ. This allows an experimental determination of Eᵢᵢᵉˣᵖ 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=2.89eV) to determine Eᵢᵢ starts to deviate from Eᵢᵢᵉˣᵖ for tubes with dₜ<1.1nm, but the deviation ΔE₂₂=E₂₂ᵉˣᵖ-E₂₂ᶜᵃˡ remains smaller than 20 meV for dₜ>~0.83nm. A comparison between Eᵢᵢᵉˣᵖ data obtained from Raman and photoluminescence is made, and a comparison is also made between Eᵢᵢᵉˣᵖ data for SWNTs and double-wall carbon nanotubes.