Current density inhomogeneity throughout the thickness of   superconducting films and its effect on their irreversible magnetic   properties
  We calculate the distribution of the current density $j$ in superconducting films along the direction of an external field applied perpendicular to the film plane. Our analysis reveals that in the presence of bulk pinning $j$ is inhomogeneous on a length scale of order the inter vortex distance. This inhomogeneity is significantly enhanced in the presence of surface pinning. We introduce new critical state model, which takes into account the current density variations throughout the film thickness, and show how these variations give rise to the experimentally observed thickness dependence of $% j$ and magnetic relaxation rate. 
R. Prozorov, E. B. Sonin, E. Sheriff, A. Shaulov and Y.Yeshurun