Abrikosov vortex



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. 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), $ |undefined 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.