Curvature Effects on the E$_{33}$ and E$_{44}$ Exciton Transitions in Semiconducting Single-Walled Carbon Nanotubes

We discuss recent measurements of the E$_{33}$ and E$_{44}$ transitions of small diameter (0.7 to 1.2 nm) single-walled carbon nanotubes using deep blue (415 to 465 nm) resonance Raman spectroscopy and photoluminescence excitation spectroscopy in the UV and blue regions (280 to 480 nm). Individual radial breathing mode features, as well as Raman and photoluminescence excitation maxima, are assigned to specific nanotube chiralities. Transition-dependent trends in RBM intensities are discussed. We present a scaling law analysis of transition energies and show that energies for nanotubes with diameter less than 0.9 nm are not explained by previous scaling law descriptions for larger diameter nanotubes. This new behavior at small diameters is interpreted in terms of both a crossing-over of the E$_{33}$ and E$_{44}$ trend lines for a given 2n+m branch, and a chirality dependence in the many-body exciton effects that becomes significant at high curvatures.