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Abrikosov vortex

## Summary

In superconductivity, a fluxon (also called an Abrikosov vortex or quantum vortex) is a vortex of supercurrent in a type-II superconductor, used by Alexei Abrikosov to explain magnetic behavior of type-II superconductors.[2] Abrikosov vortices occur generically in the Ginzburg–Landau theory of superconductivity.

## Overview

The solution is a combination of fluxon solution by Fritz London,[3][4] combined with a concept of core of quantum vortex by Lars Onsager.[5][6]

In the quantum vortex, 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 the same equation as in the London's fluxoid [3] [4]

${\displaystyle B(r)={\frac {\Phi _{0}}{2\pi \lambda ^{2}}}K_{0}\left({\frac {r}{\lambda }}\right)\approx {\sqrt {\frac {\lambda }{r}}}\exp \left(-{\frac {r}{\lambda }}\right),}$ [7]

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 obeys London's magnetic flux quantization and carries one quantum of magnetic flux ${\displaystyle \Phi _{0}}$ .[3][4] 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.