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 (formula presented) depends directly on the nanotube diameter (formula presented) and also on the magnitude of the wave vector for the quantized states (formula presented) where the van Hove singularities in the density of states occur. These two effects are manifested in the D-band frequency through the (formula presented) 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, (formula presented) 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 (formula presented) range, the diameter dependence of (formula presented) (formula presented) arises from the softening of the spring constants due to the nanotube curvature, but within a single interband transition (formula presented) whereby the (formula presented) variation is small, the D-band stiffening (formula presented) due to the double-resonance condition becomes the dominant effect.