Curvature Effects on the Optical Transitions of Single-Wall Carbon Nanotubes

Optical transition energies are widely used for providing experimental insight into the electronic band structure of single-wall carbon nanotubes (SWCNTs). While the first and second optical transitions in semiconducting carbon nanotubes have already been heavily studied, due to experimental difficulties in accessing the relevant excitation energy region, little is known about higher lying transitions. Here, I present measurements of the third and fourth optical transitions of small-diameter (0.7-1.2 nm), semiconducting single-wall carbon nanotubes via resonant Raman spectroscopy in the visible deep blue region (415-465 nm) and photoluminescence excitation spectroscopy in the ultraviolet and visible blue optical regions (280-488 nm). Diameter-dependent Raman radial breathing mode features, as well as resonant energy excitation maxima determined by Raman and photoluminescence measurements, are assigned to specific (n,m) nanotube species. The Raman intensity within a given 2n+m branch is found to increase with decreasing chiral angle, consistent with similar measurements for lower order optical states. Additionally, increased excitation line widths and weaker Raman intensities are observed as higher lying transitions are accessed for a given nanotube, in agreement with previous Raman measurements. Chiefly, a scaling law analysis that removes the chiral-angle-dependent contribution to the optical transition energy indicates that the third and fourth transition energies exhibit a significant deviation from the energy trend line observed for the first and second optical transitions, when the transition energies are plotted as a function of nanotube diameter. This deviation can be understood in the context of a change in the competition between exchange and excitonic correction terms. Furthermore, for semiconducting SWCNTs with diameters less than 0.9 nm, an additional deviation is observed that is interpreted as the first observation of crossing-over of the third and fourth transition energy trend lines for a given 2n+m branch and a chirality dependence in the many-body excitonic effects that becomes significant at high nanotube curvatures