Intermediate-frequency Raman modes for the lower optical transitions of semiconducting single-walled carbon nanotubes

A new class of intermediate-frequency modes (IFMs) associated with the $E_{22}^S$ and $E_{11}^S$ optical transitions of bundled HiPco single-walled carbon nanotubes (SWNTs) have been investigated via tunable laser (700–$985\mathrm{nm}$) resonance Raman spectroscopy. “Steplike” dispersive behavior was observed for these IFMs, along with associated clusters of radial breathing mode (RBM) overtones at higher frequencies. While the excitation profiles of both RBM and RBM overtones follow a classical behavior predicted by resonance Raman theory, significant differences are observed for the IFM excitation profiles. The observed IFM maxima were found to obey a resonance behavior based on a combination of the $E_{22}^S$ and$E_{11}^S$ transition energies, scaled by the inverse diameter of the respective nanotube. Only IFMs for the $\mathrm{mod}(n-m,3)=2$ nanotubes are visible, with intensities found to obey the family-based chiral angle dependence similar to previously reported electron-phonon interaction patterns observed for the RBM.

Figure 1

(Color online) (a) Two-dimensional intensity map of laser excitation energy versus Raman shift for the radial breathing modes (RBMs), intermediate-frequency modes (IFMs), and RBM overtones for a bundled SWNT HiPco sample. RBM features are labeled according to $(2n+m)$ families. (b) Representative four spectra obtained with $E_{\mathrm{laser}}=1.646$, 1.570, 1.464, and $1.331\mathrm{eV}$, respectively. Dashed lines are to guide the eye with respect to corresponding features in (a) and peak positions in (b). The intensities above $380{\mathrm{cm}}^{-1}$ have been multiplied by a factor of 10 in both (a) and (b) graphs to facilitate the visualization of the weak IFM and RBM overtone features, in relation to the stronger RBMs.

Figure 2

(Color online) (a) Schematic mapping of the observed Raman transitions assigned to IFMs (red crescentlike features) and RBM overtones (blue ellipsoids) from Fig. 1a. Numbers in brackets indicate the dominant peak position in ${\mathrm{cm}}^{-1}$. (b) Optical transition energies $\left(E_{ii}^S\right)$ of bundled SWNTs as a function of RBM Raman shift (bottom abscissa) and nanotube diameter (top abscissa). Scaling was based on the following relation: ${\omega }_{\mathrm{RBM}}=223.5∕d_t+12.5$ (Ref. 25). Empty and solid symbols stand for semiconducting $\mathrm{mod}(n-m,3)=1$ and $\mathrm{mod}(n-m,3)=2$ nanotubes, respectively. Curves connecting various $E_{ii}^S$ transitions belong to the same $(2n+m=\mathrm{const})$ families, depicted by the number in boxes. Vertical fading thick lines indicate the $E_{22}^S\text{−}{E}_{11}^S$ locale where the highest IFM intensities were observed for the zigzag or near zigzag nanotubes (shown as number pairs in brackets).

Figure 3

(Color online) IFM Raman shifts as a function of inverse carbon nanotube diameter. The solid line is a linear fit. Bottom left inset illustrates a representative spectrum with $E_{\mathrm{laser}}=1.644\mathrm{eV}$ (or $754\mathrm{nm}$) and the deconvolution to its constituent peaks. Experimental data is shown as open circles, with the solid line being the combined fitted spectrum.

Figure 4

Experimentally obtained IFM relative Raman cross sections (shown with “$∎$” and “엯” symbols) for the (9,1) and (8,0) nanotubes, respectively. Solid lines are the corresponding Lorentzian fits.

Figure 5

Excitation profile of the IFM (엯), RBM $(\times )$, and RBM overtone $(◻)$ for the (12,1) SWNT. The solid curves are the Lorentzian fits. The strongest resonant positions for both RBM and RBM overtone appear to coincide with $(E_{22}^S+1∕2E_{\mathrm{ph}})$ and $(E_{22}^S+E_{\mathrm{ph}})$, respectively. The Raman cross sections for IFM and RBM overtones have been multiplied by a factor of 10 to better illustrate these features.

Figure 6

(a) $E_{\mathrm{IFM}}$ $(●)$ as a function of inverse diameter. $E_{11}^S$ $(▵)$ and $E_{22}^S$ $(▿)$ are also plotted for comparison, along with the calculated $(\times )$ IFM excitation profile maxima based on Eq. (4). (b) IFM resonance Raman cross sections as a function of nanotube diameter for different $(n,m)$ SWNTs. Nanotubes belonging to the same $(2n+m)$ families (indicated by the number in squares) are plotted using the same symbol. Solid lines represent the fit using Eq. (5).