Distribution of the magnetic field and current density in   superconducting films of finite thickness
  A one-dimensional equation describing the distribution of the effective vector potential $\bar A(y)$ across a film width, which holds for thin ($d<\lambda$) and thick ($d>\lambda$) films alike, is derived based on the analysis of a 2D Maxwell-Londons equation for superconducting films in a perpendicular magnetic field. The validity of this equation for a finite-thickness film is verified by a numerical analysis. An approximation dependence $\bar A(y)$, finite (with all of its derivatives) across the entire film width, is found for films, being in the Meissner state. The flux-entry field is evaluated for a film of arbitrary thickness. An approximation expression is obtained for the distribution of the sheet current density in the mixed state of a pin-free superconducting film with an edge barrier. The latter approximation allows to estimate magnetic field concentration factor at the film edge as a function of external magnetic field and geometrical parameters of the sample. 
D. Yu. Vodolazov, I.L.Maksimov