Fluxoid solitons in superconducting tapered tubes and bottlenecks

TL;DR

The paper investigates fluxoid solitons—topologically protected order-parameter defects—that emerge at bottlenecks in thin-walled superconducting tubes under an axial magnetic field. Using self-consistent two-dimensional Usadel theory for diffusive superconductors, it shows that a fluxoid mismatch $\delta n$ between tube sections with different radii creates stationary fluxoid solitons localized at the boundary, whose number equals $|\delta n|$ and which arrange themselves via mutual repulsion (necklace-like in short bottlenecks and geometry-driven chains in tapered tubes). These solitons have cores where the pairing amplitude vanishes and exhibit phase windings of $\pm 2\pi$, with circulating supercurrents and non-quantized flux restricted by geometry (total flux equals $\Phi_+-\Phi_-$). The findings connect to Pearl vortices and Corbino junction solitons while highlighting the need for the full Usadel framework beyond sine-Gordon descriptions, and they suggest experimental routes for imaging and manipulating these boundary modes in full-shell nanowires.

Abstract

A thin-walled tubular superconductor develops a quantized fluxoid in the presence of an axial magnetic field. The fluxoid corresponds to the number of phase windings of the superconducting order parameter and is topological in nature. When the tube has a radius variation along the axial direction, forming a bottleneck structure between sections with different radius, a fluxoid mismatch can appear depending on the applied magnetic field. The bottleneck then becomes a topological boundary and is host to topologically protected solutions for the order parameter, dubbed fluxoid solitons, that are free to move around bottlenecks with cylindrical symmetry. Fluxoid solitons are a new type of vortex with non-quantized flux, loosely related to Pearl vortices in thin superconducting films and fluxons in Corbino Josephson junctions. We characterize their properties as a function of system parameters using the self-consistent quasiclassical theory of diffusive superconductors. We consider both short bottleneck structures and long tapered tubes, where multiple trapped fluxoid solitons adopt elaborate arrangements dictated by their mutual repulsion.

Figures (7)

• Figure 1: Sketch of a superconducting tubular bottleneck. A thin-walled diffusive superconducting tube has a variation of its radius along the axial direction $z$. A geometrical defect or bottleneck is created between sections of different radii $R_-$ and $R_+$. In the presence of a longitudinal magnetic field $B_z$, the phase of the superconductor order parameter $\Delta(\bm{r})$ acquires a radius-dependent integer number $n$ of windings, called fluxoids, so ${\textrm{arg}}(\Delta)=n\varphi \mod 2\pi$. At the bottleneck, the fluxoid is forced to change abruptly from $n_-$ to $n_+$, giving rise to $|\delta n|=|n_+-n_-|$ fluxoid solitons in $\Delta(\bm{r})$ that are free to move along the azimuthal direction $\varphi$ for cylindrically symmetric systems.
• Figure 2: Fluxoid solitons in short tubular bottlenecks. Pairing modulus $|\Delta|$ (a) and phase ${\rm{arg}}(\Delta)$ (b), versus cylindrical coordinates $\varphi$ and $z$ (normalized to the superconducting coherence length $\xi_0$), for a bottleneck of length $L_b=3\xi_0$ represented by the solid red line in (a). Temperature is $T=0.25 T_C^0$, and $B_z$ field is such that the dimensionless flux is $\Phi_-/\Phi_0=0.25$ on the $R_-$ section, and $\Phi_+/\Phi_0=1$ on the $R_+$ section. The dashed red line shows the position at which the local fluxoid $n(z)$ jumps by one. The total fluxoid mismatch is $\delta n=1$, so one soliton emerges at the bottleneck, close to the dashed line [black shadow in (a)]. The associated supercurrent density $\bm{J}$ [white arrows in (b)] forms a vortex around the soliton. (c,d) are like (a,b) but for larger $R_\pm/\xi_0$ and the same flux. (e,f) are similar to (c,d) but with an integer flux on both sides of the bottleneck, which makes currents vanish asymptotically. (g,h) are like (c,d) but with $\delta n=2$ solitons.
• Figure 3: Behavior of the soliton with system parameters. The soliton profile, $\textrm{min}_\varphi(|\Delta|)$, along $z$ is plotted from bottom to top for increasing values of temperature (a), radius $R_+$ (b) and magnetic field (c). All other parameters are fixed in each case, and correspond to the bottleneck of Fig. \ref{['fig2']}(a,b). Panels (d,e,f) show the asymptotic $|\Delta_\pm|$ far from the bottleneck as a function of the same parameters.
• Figure 4: Fluxoid solitons in tapered tubes. Four tapered nanowire geometries with increasing number of solitons are represented from top to bottom, with the same plotting conventions as in Fig. \ref{['fig2']}. Unlike in the short bottleneck case, solitons are spaced along the tapered section ($z$ direction), with their equilibrium positions governed by their mutual repulsion. In the case of a narrow tube, (a,b), the destructive LP effect transforms solitons into strips of near-zero $|\Delta|$ across a finite $z$ interval.
• Figure 5: Longitudinal location of a soliton in the abrupt-bottleneck toy model of Appendix \ref{['ap:analyticalsoliton']}. Anomalous superconducting Green's functions to the left and right sections of the bottleneck step versus longitudinal coordinate $z$. The soliton (green dot) appears on the left side of the bottleneck (at $z_0\approx-0.17\xi_0$), because the pair amplitudes on the left are weaker than on the right for the chosen flux. Parameters: $R_+ = 2R_- = 0.4\xi_0$, $\Phi_-/\Phi_0 = 0.5$, $\Delta_+ = \Delta_- \equiv \Delta$ and $\omega = 2\Delta$.