Excitonic optical transitions characterized by Raman excitation profiles in single-walled carbon nanotubes

We examine the excitonic nature of the $E_{33}$ optical transition of the individual free-standing index-identified $(23,7)$ single-walled carbon nanotube by means of the measurements of its radial-breathing-mode and $G$-mode Raman excitation profiles. We confirm that it is impossible to determine unambiguously the nature of its $E_{33}$ optical transition (excitonic vs band to band) based only on the excitation profiles. Nevertheless, by combining Raman scattering, Rayleigh scattering, and optical absorption measurements on strictly the same individual $(23,7)$ single-walled carbon nanotube, we show that the absorption, Rayleigh spectra, and Raman excitation profiles of the longitudinal and transverse $G$ modes are best fitted by considering the nature of the $E_{33}$ transition as excitonic. The fit of the three sets of data gives close values of the transition energy $E_{33}$ and damping parameter ${\mathrm{\Gamma }}_{33}$. This comparison shows that the fit of the Raman excitation profiles provides with good accuracy the energy and damping parameter of the excitonic optical transitions in single-walled carbon nanotubes.

Figure 1

(a) HRTEM (inset) and electron diffraction (ED) pattern of the $(23,7)$ SWNT, experimental ED pattern (left) and simulated ED pattern (right). (b) Absorption spectrum of the $(23,7)$ SWNT in the 1.4–2.4 eV range; the horizontal double arrow indicates the excitation energy range used in the Raman experiments. (c) RBM and $G$-mode ranges of the Raman spectra of the $(23,7)$ SWNT excited at 1.85 eV (top) and 1.66 eV (bottom). Each spectrum is normalized on the intensity of the LO $G$ mode. The vertical offset of the spectra is for clarity. Dots are experimental data; brown and red solid lines show the fit of the spectra recorded at 1.85 and 1.66 eV, respectively; black dotted (dashed) line is the result of the fit of the LO (TO) $G$ mode by a Lorentzian.

Figure 2

REPs of the RBM (top), LO $G$ mode (middle), and TO $G$ mode (bottom). Circles are experimental data, with the red dotted line being the best fit for the band-to-band model [expression (1)] and the blue solid line being the best fit for the excitonic model [expression (2)]. $M_{33},E_{33}$, and ${\mathrm{\Gamma }}_{33}$ are free parameters in the fit (see text). The three REPs were simultaneously fitted in each model. The full calculation of the REP within the band-to-band model is given for comparison (green dashed line).

Figure 3

(a) Rayleigh spectrum in the 1.45–1.8 eV range. Circles are the experimental data. The blue solid line is the fit of the $E_{33}$ peak in the excitonic model; the red dotted line is the best fit of the experimental data in the simplified band-to-band approach. The Rayleigh spectrum obtained by full calculation in the band-to-band model, using a fixed damping parameter of 16 meV and after scaling and shifting the spectrum, is given for comparison (green dashed line). (b) Absorption spectrum of the $(23,7)$ SWNT in the 1.45–1.8 eV range; the symbols are the same as in (a).