General critical states in type-II superconductors

The magnetic flux dynamics of type-II superconductors within the critical state regime is posed in a generalized framework by using a variational theory supported by well-established physical principles. The equivalence between the variational statement and more conventional treatments based on the solution of the differential Maxwell equations together with appropriate conductivity laws is shown. Advantages of the variational method are emphasized, focusing on its numerical performance that allows us to explore a number of physical scenarios. In particular, we present the extension of the so-called double critical state model to three-dimensional (3D) configurations in which only flux transport (T states), cutting (C states), or both mechanisms ($CT$ states) occur. The theory is applied to several problems. First, we show the features of the transition from T to $CT$ states. Second, we give a generalized expression for the flux cutting threshold in 3D and show its relevance in the slab geometry. In addition, several models that allow us to treat flux depinning and cutting mechanisms are compared. Finally, the longitudinal transport problem (current is applied parallel to the external magnetic field) is analyzed both under T and $CT$ conditions. The complex interaction between shielding and transport is solved.