Pearl vortex

In superconductivity, a Pearl vortex is a vortex of supercurrent in a thin film of type-II superconductor, first described in 1964 by Judea Pearl. A Pearl vortex is similar to Abrikosov vortex except for its magnetic field profile which, due to the dominant air-metal interface, diverges sharply as 1/$r$ at short distances from the center, and decays slowly, like 1/$r^{2}$ at long distances. Abrikosov's vortices, in comparison, have very short range interaction and diverge as $\log(1/r)$ near the center.

Derivation

In Pearl's thesis , he uses the London equations to derive the magnetic response of a thin superconducting film in the Meissner state. For a film where the thickness is on the order of the superconducting coherence length or smaller, the ability to screen magnetic field is geometrically suppressed. Whereas in a bulk superconductor the characteristic length scale over which magnetic field can penetrate is the London penetration depth $\lambda$, in a thin film this is increased to the Pearl length $\Lambda _{P}=2\lambda ^{2}/d$. This occurs because in a thin film, inductive coupling through free space plays a stronger role in magnetic field penetration.

This suppressed screening plays a role in film dynamics far beyond vortex dynamics. In most models, including Ginzburg-Landau theory, this can be accounted for by substituting $\Lambda _{P}$ instead of $\lambda$ Because the London equations assume a film in the Meissner state, Ginzburg-Landau theory is a more natural choice for studying vortex dynamics. Studying vortices in Ginzburg-Landau theory with a magnetic penetration depth of $\lambda$ yields Abrikosov vortices, while using a magnetic penetration depth of $\Lambda _{P}$ gives the dynamics of Pearl vortices.

Consequences

Because the magnetic penetration depth of Pearl vortices is a function of both geometry and material properties, their existence implies that in sufficiently thin films the modified Ginzburg-Landau parameter $\kappa =\Lambda _{P}/\xi$ may become greater than $1/{\sqrt {2}}$ even in films with Type-I superconductor behavior in the bulk. In other words, type-I superconducting thin films can host Pearl vortices, when normally in the bulk they transition directly from the Meissner state to the normal state with applied magnetic field .

Additionally, the long interaction length of Pearl vortices enable the Berezinskii-Kosterlitz-Thouless transition (BKT) to occur in superconducting thin films. The short interaction length of Abrikosov vortices was identified as insufficient to support a BKT transition. However, Beasley, Mooij, and Orlando showed that Pearl vortices could theoretically enable a BKT transition in thin film superconductors.

Measuring Pearl vortices

A transport current flowing through a superconducting film may cause these vortices to move with a constant velocity $v$ proportional to, and perpendicular to the transport current. Because of their proximity to the surface, and their sharp field divergence at their centers, Pearl's vortices can actually be seen by a scanning SQUID microscope. The characteristic length governing the distribution of the magnetic field around the vortex center is given by the ratio $\Lambda =2\lambda ^{2}$/$d$, also known as "Pearl length," where $d$ is the film thickness and $\lambda$ is London penetration depth. Because this ratio can reach macroscopic dimensions (~1 mm) by making the film sufficiently thin, it can be measured relatively easy and used to estimate the density of superconducting electrons.

At distances shorter than the Pearl's length, vortices behave like a Coulomb gas (1/$r$ repulsive force).