Abrikosov vortex

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.