Resonance Raman scattering from phonon overtones in double-wall carbon nanotubes

The resonance behavior and the dispersion of the ${\mathrm{G}}^{\prime }$ line in double-wall carbon nanotubes were investigated by multifrequency Raman scattering. Using a large number of laser lines for excitation, the dispersion for the response from the inner tubes and from the outer tubes was found to be 85 and $99{\mathrm{cm}}^{-1}∕\mathrm{eV}$, respectively. The reduction of the dispersion for the inner tubes is a consequence of their high curvature and suggests a flattening of the phonon dispersion at the $K$ point in the Brillouin zone. The frequency position for the ${\mathrm{G}}^{\prime }$ line of the inner tubes for a given laser energy was likewise strongly reduced as compared to values expected from tubes with standard diameter. This was again assigned to the strong curvature of the small-diameter inner tubes. Finally, the ${\mathrm{G}}^{\prime }$ line scattering cross sections for the outer tubes and for the inner tubes revealed resonances according to the transitions between the corresponding Van Hove singularities. The response from the inner tubes was particularly strong, in good agreement with the expected resonance enhancement resulting from their high curvature.