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.^[1] 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/${\displaystyle r}$ at short distances from the center, and decays slowly, like 1/${\displaystyle r^{2}}$ at long distances. Abrikosov's vortices, in comparison, have very short range interaction and diverge as ${\displaystyle \log(1/r)}$ near the center.

Derivation

In Pearl's thesis,^[2] 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 ${\displaystyle \lambda }$, in a thin film this is increased to the Pearl length ${\displaystyle \Lambda _{P}=2\lambda ^{2}/d}$. 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 ${\displaystyle \Lambda _{P}}$ instead of ${\displaystyle \lambda }$ 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 ${\displaystyle \lambda }$ yields Abrikosov vortices, while using a magnetic penetration depth of ${\displaystyle \Lambda _{P}}$ 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 ${\displaystyle \kappa =\Lambda _{P}/\xi }$ may become greater than ${\displaystyle 1/{\sqrt {2}}}$ 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.^[3]

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 ^[4] 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 ${\displaystyle v}$ proportional to, and perpendicular to the transport current.^[5] 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.^[6][7][8] The characteristic length governing the distribution of the magnetic field around the vortex center is given by the ratio ${\displaystyle \Lambda =2\lambda ^{2}}$/${\displaystyle d}$, also known as "Pearl length," where ${\displaystyle d}$ is the film thickness and ${\displaystyle \lambda }$ is London penetration depth.^[9] 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.^[8]

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