Vortices in superconductors
The superconducting state of some material is characterized by a vanishing (DC-) resistivity ρ(T) and by the complete repulsion of magnetic flux, even if a magnetic field is applied before a superconductor is cooled below the critical temperature Tc (Meissner and Ochsenfeld 1933). The existence of the Meissner effect shows that the superconducting state is a thermodynamic state which does not depend on previous history.
Until 1986, about 30 metallic elements and more than thousand alloys and compounds were known to change into the superconducting state below a critical temperature Tc, allowing them to transport an electrical current without losses up to a critical current density jc. All critical temperatures of these superconductors were below 23 K (about -250°C), today they are known as conventional or low temperature superconductors (TTSL). Discovery of superconductivity in Ba-La-Cu-O-compounds by Alex Müller and Georg Bednorz in 1986 (Nobel price 1987) led to the discovery of many high temperature superconductors (HTSL), some showing a critical temperature more than 100 K above those of conventional superconductors. The highest Tc-values under normal pressure (138 K) were measured in Hg-Tl-Ba-Ca-Cu-O-compounds . Some of the more than 50 known HTSL have critical temperatures above the boiling point of liquid nitrogen (77 K or -196 °C), which allows for new technical applications involving superconductivity because cooling with liquid nitrogen is much cheaper (and a lot easier) than cooling with liquid nitrogen used to cool conventional superconductors.
Some superconductors (type-II superconductors) do not exhibit a complete expulsion of flux if an external field is larger than a critical field. In this case they allow the applied field to partly penetrate the material, the magnetization of the specimen depends on its magnetic history in a complicated way. An explanation of this partial flux penetration was given by A. Abrikosov in a pioneering work  on a periodic solution of the phenomenological Ginzburg-Landau theory . Flux lines (or vortices) thread the specimen, each carrying a single quantum of magnetic flux Φ0=h/2e, at the center of a flux line the superconducting order Parameter Ψ(r) vanishes (in other words: the core of a flux line is normal conducting).
A current density j flowing through the vortex lattice perpendicular to the magnetic field leads to a lorentz force fL = j x B on each vortex. This force tends to move the vortex liquid in the direction perpendicular to current and field. If the vortices start to flow into this direction, work is done and there is energy dissipation, the vortex motion directly leads to a finite resistance. In the mixed state, superconductors only have zero resistance when the vortices are pinned and unable to move.
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 W. Buckel, R. Kleiner, Supraleitung: Grundlagen und Anwendungen, Wiley-VCH, 2004
 P. Dai et al., Physica C 243:201-206 (1995)