Competing spring constant versus double resonance effects on the properties of dispersive modes in isolated single-wall carbon nanotubes

We report a study of the disorder-induced D band in the resonance Raman spectra of isolated single-wall carbon nanotubes (SWNTs). We show that the D-band frequency ${\omega }_D$ depends directly on the nanotube diameter $d_t$ and also on the magnitude of the wave vector for the quantized states $k_{\mathrm{ii}},$ where the van Hove singularities in the density of states occur. These two effects are manifested in the D-band frequency through the ${\omega }_D={\omega }_D^0{+C/d}_t$ functional form, but with C negative (positive) for the spring-constant- (double-resonance-) dependent processes, thereby indicating that the spring constant softens and the double resonance stiffens the D-band frequencies. In the case of the spring constant effect, ${\omega }_D^0$ is the frequency observed in two-dimensional graphite. The outcome of the softening versus stiffening competition depends on the nanotube diameter range. When plotted over a wide $d_t$ range, the diameter dependence of ${\omega }_D$ $\mathrm{}(C<\mathrm{}0\mathrm{})\mathrm{}$ arises from the softening of the spring constants due to the nanotube curvature, but within a single interband transition $E_{\mathrm{ii}},$ whereby the $d_t$ variation is small, the D-band stiffening $\mathrm{}(C>\mathrm{}0\mathrm{})\mathrm{}$ due to the double-resonance condition becomes the dominant effect.