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 d_t and also on the magnitude of the wave vector for the quantized states k_ii, where the van Hove singularities in the density of states occur. These two effects are manifested in the D-band frequency through the ω_D=ω_D⁰+Cld, functional form, but with C negative (positive) for the spring-constant- (doubleresonance-) 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, ω_D⁰ 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 ω_D (C<0) arises from the softening of the spring constants due to the nanotube curvature, but within a single interband transition E_ii, whereby the d_t variation is small, the D-band stiffening (C>0) due to the double-resonance condition becomes the dominant effect.