Attractive vortex interaction and the intermediate-mixed state of superconductors

The magnetic vortices in superconductors usually repel each other. Several cases are discussed when the vortex interaction has an attractive tail and thus a minimum, leading to vortex clusters and chains. Decoration pictures then typically look like in the intermediate state of type-I superconductors, showing lamellae or islands of Meissner state or surrounded by Meissner state, but with the normal regions filled with Abrikosov vortices that are typical for type-II superconductors in the mixed state. Such intermediate-mixed state was observed and investigated in detail in pure Nb, TaN and other materials 40 years ago; last year it was possibly also observed in MgB$_2$, where it was called "a totally new state" and ascribed to the existence of two superconducting electron bands, one of type-I and one of type-II. The complicated electronic structure of MgB$_2$ and its consequences for superconductivity and vortices are discussed. It is shown that for the real superconductor MgB$_2$ which possesses a single transition temperature, the assumption of two independent order parameters with separate penetration depths and separate coherence lengths is unphysical.