# Abrikosov vortex

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

## Overview

The solution is a combination of fluxon solution by Fritz London, combined with a concept of core of quantum vortex by Lars Onsager.

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

$B(r)={\frac {\Phi _{0}}{2\pi \lambda ^{2}}}K_{0}\left({\frac {r}{\lambda }}\right)\approx {\sqrt {\frac {\lambda }{r}}}\exp \left(-{\frac {r}{\lambda }}\right),$ where $K_{0}(z)$ is a zeroth-order Bessel function. Note that, according to the above formula, at $r\to 0$ the magnetic field $B(r)\propto \ln(\lambda /r)$ , i.e. logarithmically diverges. In reality, for $r\lesssim \xi$ the field is simply given by

$B(0)\approx {\frac {\Phi _{0}}{2\pi \lambda ^{2}}}\ln \kappa ,$ where κ = λ/ξ is known as the Ginzburg–Landau parameter, which must be $\kappa >1/{\sqrt {2}}$ 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 $H$ larger than the lower critical field $H_{c1}$ (but smaller than the upper critical field $H_{c2}$ ), 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 $\Phi _{0}$ . 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.