Abstract: Van Hove singularities (VHSs) are a hallmark of reduced dimensionality, leading to a divergent density of states in one and two dimensions and predictions of new electronic properties when the Fermi energy is close to these divergences. In carbon nanotubes, VHSs mark the onset of new subbands. They are elusive in standard electronic transport characterization measurements because they do not typically appear as notable features and therefore their effect on the nanotube conductance is largely unexplored. Here we report conductance measurements of carbon nanotubes where VHSs are clearly revealed by interference patterns of the electronic wave functions, showing both a sharp increase of quantum capacitance, and a sharp reduction of energy level spacing, consistent with an upsurge of density of states. At VHSs, we also measure an anomalous increase of conductance below a temperature of about 30 K. We argue that this transport feature is consistent with the formation of Cooper pairs in the nanotube.

Abstract: It is well known that thermal fluctuations and material impurities affect the motion of vortices in superconductors. These effects are modeled by variants of a time-dependent Ginzburg-Landau model containing either additive or multiplicative noise. Numerical computations are presented that illustrate the effects that noise has on the dynamics of vortex nucleation and vortex motion. For an additive noise model with relatively low variances, it is found that the vortices form a quasi-steady-state lattice in which the vortex core sizes remain roughly fixed but their positions vibrate. Two multiplicative noise models are considered. For one model having relatively long-range order, the sizes of the vortex cores vary in time and from one vortex to another. Finally, for the additive noise case, we show that as the variance of the noise tends to zero, solutions of the stochastic time-dependent Ginzburg-Landau equations converge to solutions of the corresponding equations with no noise.