Coexistence of the Meissner and vortex states on a nanoscale superconducting spherical shell

We show that on superconducting spherical nanoshells, the coexistence of the Meissner state with a variety of vortex patterns drives the phase transition to higher magnetic fields. The spherical geometry leads to a Magnus-Lorentz force pushing the nucleating vortices and antivortices toward the poles, overcoming local pinning centers, preventing vortex-antivortex recombination, and leading to the appearance of a Meissner belt around the sphere equator. In sufficiently small and thin spherical shells paramagnetic vortex states can be stable, enabling spatial separation of freely moving shells with different radii and vorticity in an inhomogeneous external magnetic field.

Figure 1

(Color online) The Gibbs free energy of a vortex-antivortex pair, displaced away from the poles down to a latitude $\theta$, is shown for different values of the applied magnetic field. The origin of the metastability barrier, which develops with increasing the applied magnetic field, is a Magnus-Lorentz $F_M$ force pushing vortices toward the poles, as illustrated in the inset. This force arises from a differential velocity field across the vortex due to the interplay between the supercurrent of the vortex, proportional to the phase gradient $\nabla \phi$, and the screening supercurrent induced by the magnetic field $H$.

Figure 2

(Color online) The phase diagram for a thin spherical shell is compared to that of a flat thin disk. The shaded regions show the values of magnetic field and radius where the superconducting (SC) state is supported. The solid lines correspond to the boundaries between the thermodynamically stable normal and superconducting states. The dashed lines correspond to the boundaries between the thermodynamically stable Meissner and vortex states.

Figure 3

(Color online) Boundaries of the region, where paramagnetic states with $L=1$ on a spherical shell of radius $\mathcal{R}$ and thickness $\mathcal{W}$ can be thermodynamically stable. Inset: magnetic fields, which correspond to the equilibrium positions of thin spherical shells with vorticity $L=1$ and $L=2$ in an inhomogeneous magnetic field, as a function of the shell radius.

Figure 4

(Color online) Flux hysteresis in a superconducting nanoshell. The Gibbs free-energy difference between the normal and superconducting states is plotted as a function of the magnetic field. The solid curve shows the thermodynamically stable state. The dashed (dotted) curve corresponds to a slow increase (decrease) in the applied magnetic field. Inset: on the left, a vortex pattern that arises at high magnetic field $\left(H=20\right)$ when a ringlike vortex in a nanoshell with $\mathcal{R}/\xi =11.5$ and $\mathcal{W}\mathcal{R}⪡{\lambda }^2$ breaks up into separate vortices. The color scale indicates the Cooper pair density from blue (dark gray background) (high) to red (gray patches) (low). The supercurrents are indicated by the arrow field. On the right, a cloud pattern observed at the north pole of Saturn (Ref. 17).

Figure 5

(Color online) Four snapshots of the dynamical process of vortices entering a superconducting nanoshell with $\mathcal{R}/\xi =5.66$ and $\mathcal{W}\mathcal{R}/{\lambda }^2=0.02$ at $H=4.15$. The color scale indicates the Cooper pair density from blue (dark gray background) (high) to red (gray patches) (low). The supercurrents are indicated by the arrow field. (a) A redistribution of the Cooper pair density is already present in the Meissner state. (b) The vortices nucleate at a depression of the Cooper pair density at the equator, (c) separate into two counter-rotating two-dimensional vortices, and (d) proceed toward the poles.