Current and field pattern in rectangular and inhomogeneous superconductors

The penetration and exit of magnetic flux in type-II superconductors is investigated for the realistic situation where a transverse magnetic field is applied to a square or rectangular plate or film. In rectangular specimens the pattern of the sheet current and of the density of the perpendicular flux has some common features with the one-dimensional distributions in circular disks or long strips. Other features, however, are characteristic for the rectangular shape, e.g., the starlike pattern of the penetrating flux and, in the fully penetrated critical state, the discontinuity lines at which the current stream lines perform sharp bends and at which the perpendicular magnetic field ${\mathit{H}}_{\mathit{z}}$(x,y) exhibits sharp ridges. These typical features have to be calculated from a genuine two-dimensional theory. Such a theory based on a highly nonlinear current-voltage law is outlined. The field patterns obtained by this general theory are compared with patterns observed magneto-optically at the surface of square and rectangular single crystals or films of high-${\mathit{T}}_{\mathit{c}}$ superconductors with homogeneous and inhomogeneous critical-current distribution. It is shown that the analysis of the current-discontinuity lines is essential to understand the flux dynamics in superconductors. In samples with inhomogeneous critical current density ${\mathit{j}}_{\mathit{c}}$(r), a strong concentration of flux motion and electric field can occur along the lines where ${\mathit{j}}_{\mathit{c}}$ changes abruptly. This may trigger flux jumps.