Resonance Raman spectroscopy(n,m)-dependent effects in small-diameter single-wall carbon nanotubes

This paper presents an accurate analysis of (i) the electronic transition energies $E_{22}^S$ and $E_{11}^M$, (ii) the radial breathing mode (RBM) frequencies ${\omega }_{\mathrm{RBM}}$, and (iii) the corresponding RBM intensities from 40 small-diameter single-wall carbon nanotubes (SWNTs) in the diameter range $0.7<d_t<1.3\mathrm{nm}$. The electronic transition energies $\left(E_{ii}\right)$ are initially considered from nonorthogonal tight-binding total-energy calculations. To account for $d_t$-dependent many-body effects, a logarithmic correction, as proposed by Kane and Mele, is applied to both $E_{22}^S$ and $E_{11}^M$. The remaining discrepancies between the experimental and theoretical $E_{ii}$ values are shown to be proportional to the chirality-dependent effective masses of electrons and holes, as obtained from the electron energy dispersion relations. Chirality dependent screening effects are also identified in metallic SWNTs. For the RBM frequencies, a small deviation from the linear $1∕d_t$ behavior is observed, and this deviation is analyzed based on a chirality-dependent mode softening effect due to nanotube curvature. For those interested in sample characterization, the $(n,m)$ dependence of the resonance intensities is also addressed, the experimental results being compared with theoretical predictions based on matrix elements calculations. This analysis suggests that the (7,5), (7,6), and (6,5) SWNTs are more abundant in sodium dodecyl sulfate wrapped HiPco SWNTs in aqueous solution, in agreement with results previously reported for SWNTs grown by the CoMoCAT or alcohol methods.