Curvature effects on the $E_{33}$ and $E_{44}$ exciton transitions in semiconducting single-walled carbon nanotubes

The $E_{33}$ and $E_{44}$ optical transitions of small diameter $(0.7–1.2\mathrm{nm})$ semiconducting single-walled carbon nanotubes are probed using deep blue $(415–465\mathrm{nm})$ resonance Raman spectroscopy and photoluminescence excitation spectroscopy in the UV and blue regions $(280–488\mathrm{nm})$. Individual radial breathing mode features, as well as Raman and photoluminescence excitation maxima, are assigned to specific nanotube chiralities. The chiral angle dependence of Raman intensity within a given $2n+m$ branch is found to continue, as does the trend toward increased excitation linewidths and weaker Raman intensities as higher lying transitions are accessed. The semiconducting $E_{33}$ and $E_{44}$ transition energies obtained for the largest observed diameters confirm recent results [P. T. Araujo et al., Phys. Rev. Lett. 98, 067401 (2007)] that show that these transitions reside on a blueshifted scaling-law line, separate from that describing $E_{11}$ and $E_{22}$ behaviors. Energies for nanotubes with diameter less than $0.9\mathrm{nm}$, however, are not explained by any previous scaling-law fits. This behavior at small diameters is interpreted in terms of both a crossing over of the $E_{33}$ and $E_{44}$ energy trend lines for a given $2n+m$ branch and a chirality dependence in the many-body exciton effects that becomes significant at high curvatures.

Figure 1

Representative carbon nanotube RBM spectra for different excitation wavelengths. Individual spectral intensities are normalized with respect to the most intense peak in each spectrum for clarity.

Figure 2

(Color) (a) 2D intensity map showing RBM intensity behavior as the excitation wavelength is changed. The map shows grouping of the main RBM features into $2n+m=\mathrm{const}$ branches. (b) ETB predicted (Ref. 16) features within the same energy and RBM frequency region for mod 2 semiconductors ($E_{33}$ and $E_{44}$, green triangles), mod 1 semiconductors ($E_{33}$, blue circles), and metallics ($E_{11}$, red squares). The highlighted branches correspond to the observed features in part (a) and in Fig. 1.

Figure 3

(Color online) Resonance Raman excitation profiles for four selected nanotube chiralities. Lines are fits according to Eq. (1) of the experimental data (dots), from which $\Gamma$ values are extracted: (a) (13,0), $\Gamma =115\mathrm{meV}$; (b) (11,4), $\Gamma =130\mathrm{meV}$; (c) (11,1), $\Gamma =113\mathrm{meV}$; (d) (14,3), $\Gamma =111\mathrm{meV}$.

Figure 4

Plot of RBM frequency as a function of inverse nanotube diameter $(1∕d)$. The line is a linear fit to the experimental data points (squares), yielding the parameters $A=219.3\mathrm{nm}{\mathrm{cm}}^{-1}$ and $B=14.7{\mathrm{cm}}^{-1}$ from Eq. (2).

Figure 5

(Color) 2D intensity map showing the nanotube photoluminescence intensity as a function of emission energy, as the excitation energy is varied. Chirality assignments for selected $E_{33}$ and $E_{44}$ features are highlighted.

Figure 6

(Color) (a) Corrected $E_{33}$ and $E_{44}$ transition energies $(E_{\exp }-{\beta }_p\cos 3\theta ∕d_t^2)$ as a function of $p∕d$, where $p=4$ and 5, respectively. $E_{33}$ data are obtained from Raman (black squares) and PLE (red circles) measurements, while $E_{44}$ data (green triangles) are obtained from the PLE measurement. Energy assignments to $E_{33}$ and $E_{44}$ features are chosen directly from their respective ordering from the experimental results. Results are compared to scaling-law lines from Ref. 10 for $E_{33}$ and $E_{44}$ (upper line) and $E_{11}$ and $E_{22}$ (lower line behavior), see Eq. (3). Chiralities showing significant scatter from the scaling-law behavior are designated and highlighted by circles. (b) The data of part (a) replotted after correcting the energy ordering of the (14,3), (12,1), (11,3), (10,2), (9,4), (8,3), and (6,4) chiralities for the effect of branch crossover.

Figure 7

(Color online) ETB calculated (Ref. 16) $E_{33}$ and $E_{44}$ transition energies as a function of RBM frequency for mod 1 (circles) and mod 2 (triangles) species. Branch crossover effect of $E_{33}$ and $E_{44}$ energy behaviors is illustrated, with specific chiralities and branches relevant to Fig. 6a highlighted.