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 ωD depends directly on the nanotube diameter dₜ and also on the magnitude of the wave vector for the quantized states kᵢᵢ, where the van Hove singularities in the density of states occur. These two effects are manifested in the D-band frequency through the ωD=ωD⁰+C/dₜ 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, ωD⁰ 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 dₜ range, the diameter dependence of ωD $(C<0)$ arises from the softening of the spring constants due to the nanotube curvature, but within a single interband transition Eᵢᵢ, whereby the dₜ variation is small, the D-band stiffening $(C>0)$ due to the double-resonance condition becomes the dominant effect.