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# Abrikosov vortex

In superconductivity, an Abrikosov vortex is a vortex of supercurrent in a type-II superconductor. The supercurrent circulates around the normal (i.e. non-superconducting) core of the vortex. The core has a size ˜ξ — the superconducting coherence length (parameter of a Ginzburg-Landau theory). The supercurrents decay on the distance about λ (London penetration depth) from the core. Note that in type-II superconductors λ > ξ. The circulating supercurrents induce magnetic field with the total flux equal to a single flux quantum Φ0. 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 $B(r) = \frac{\Phi_0}{2\pi\lambda^2}K_0\left(\frac{r}{\lambda}\right) \approx \sqrt{\frac{\lambda}{r}} \exp\left(-\frac{r}{\lambda}\right),$

where K0(z) is the so-called 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 first critical field Hc1 (but smaller than the second critical field Hc2), the field penetrates into superconductor in terms of Abrikosov vortices. Each vortex carries one thread of magnetic field with the flux Φ0. Abrikosov vortices form a lattice (usually triangular, may be with defects/dislocations) with the average vortex density (flux density) approximately equal to the externally applied magnetic field.