# Abrikosov vortex

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Vortices in a 200-nm-thick YBCO film imaged by scanning SQUID microscopy[1]

In superconductivity, an Abrikosov vortex (also called a fluxon) is a vortex of supercurrent in a type-II superconductor theoretically predicted by Alexei Abrikosov in 1957.[2] Abrikosov vortices occur generically in the Ginzburg–Landau theory of superconductivity, and can be explicitly demonstrated as solutions to that theory in a general mathematical setting, viz. as vortices in complex line bundles on Riemannian manifolds.

## Overview

The supercurrent circulates around the normal (i.e. non-superconducting) core of the vortex. The core has a size ${\displaystyle \sim \xi }$ — the superconducting coherence length (parameter of a Ginzburg-Landau theory). The supercurrents decay on the distance about ${\displaystyle \lambda }$ (London penetration depth) from the core. Note that in type-II superconductors ${\displaystyle \lambda >\xi /{\sqrt {2}}}$. The circulating supercurrents induce magnetic fields with the total flux equal to a single flux quantum ${\displaystyle \Phi _{0}}$. Therefore, an Abrikosov vortex is often called a fluxon.

The magnetic field distribution of a single vortex far from its core can be described by

${\displaystyle B(r)={\frac {\Phi _{0}}{2\pi \lambda ^{2}}}K_{0}\left({\frac {r}{\lambda }}\right)\approx {\sqrt {\frac {\lambda }{r}}}\exp \left(-{\frac {r}{\lambda }}\right),}$

where ${\displaystyle K_{0}(z)}$ is a zeroth-order Bessel function. Note that, according to the above formula, at ${\displaystyle r\to 0}$ the magnetic field ${\displaystyle B(r)\propto \ln(\lambda /r)}$, i.e. logarithmically diverges. In reality, for ${\displaystyle r\lesssim \xi }$ the field is simply given by

${\displaystyle B(0)\approx {\frac {\Phi _{0}}{2\pi \lambda ^{2}}}\ln \kappa ,}$

where κ = λ/ξ is known as the Ginzburg-Landau parameter, which must be ${\displaystyle \kappa >1/{\sqrt {2}}}$ in type-II superconductors.

Abrikosov vortices can be trapped in a type-II superconductor by chance, on defects, etc. Even if initially type-II superconductor contains no vortices, and one applies a magnetic field ${\displaystyle H}$ larger than the lower critical field ${\displaystyle H_{c1}}$ (but smaller than the upper critical field ${\displaystyle H_{c2}}$), the field penetrates into superconductor in terms of Abrikosov vortices. Each vortex carries one thread of magnetic field with the flux ${\displaystyle \Phi _{0}}$. Abrikosov vortices form a lattice, usually triangular, with the average vortex density (flux density) approximately equal to the externally applied magnetic field. As with other lattices, defects may form as dislocations.

## Abrikosov vortex and proximity effect

Here is shown, that a quantum vortex with a well-defined core can exist in a rather thick normal metal, proximized with a superconductor [3].

## References

1. ^ Wells, Frederick S.; Pan, Alexey V.; Wang, X. Renshaw; Fedoseev, Sergey A.; Hilgenkamp, Hans (2015). "Analysis of low-field isotropic vortex glass containing vortex groups in YBa2Cu3O7−x thin films visualized by scanning SQUID microscopy". Scientific Reports. 5: 8677. arXiv:1807.06746. Bibcode:2015NatSR...5E8677W. doi:10.1038/srep08677. PMC 4345321. PMID 25728772.
2. ^ Abrikosov, A. A. (1957). "The magnetic properties of superconducting alloys". Journal of Physics and Chemistry of Solids. 2 (3): 199–208. Bibcode:1957JPCS....2..199A. doi:10.1016/0022-3697(57)90083-5.
3. ^ Stolyarov, Vasily S.; Cren, Tristan; Brun, Christophe; Golovchanskiy, Igor A.; Skryabina, Olga V.; Kasatonov, Daniil I.; Khapaev, Mikhail M.; Kupriyanov, Mikhail Yu.; Golubov, Alexander A.; Roditchev, Dimitri (11 June 2018). "Expansion of a superconducting vortex core into a diffusive metal". Nature Communications. 9 (1): 2277. doi:10.1038/s41467-018-04582-1.