Effect of geometry on the critical currents of thin films

We consider experimentally and theoretically the effect of the thickness on the critical current density ${\mathit{J}}_{\mathit{c}}$ of superconducting films. In order to eliminate possible contributions from intrinsic pinning, our measurements of ${\mathit{J}}_{\mathit{c}}$(cphi) as a function of the amplitude and orientation of the magnetic field H with respect to the film plane were performed on isotropic Nb-Ti films having thicknesses d ranging from λ/4 to 4λ, where λ is the London penetration depth, and H⊥J. The angular dependent ${\mathit{J}}_{\mathit{c}}$(cphi) has a sharp peak for H parallel to the film surface, similar to that observed for high-${\mathit{T}}_{\mathit{c}}$ films. The amplitude of the peak increases as d decreases and reaches 20–30 % of the depairing current density (${\mathit{J}}_{\mathit{d}}$) for the λ/2 film. The ratio of ${\mathit{J}}_{\mathit{c}}$ values for parallel (${\mathit{J}}_{\mathit{c}\mathrm{\parallel }}$) and perpendicular (${\mathit{J}}_{\mathit{c}\mathrm{\perp }}$) film orientation increases as d decreases, so that ${\mathit{J}}_{\mathit{c}\mathrm{\parallel }}$≪${\mathit{J}}_{\mathit{c}\mathrm{\perp }}$ for the 4λ film and ${\mathit{J}}_{\mathit{c}\mathrm{\parallel }}$≫${\mathit{J}}_{\mathit{c}\mathrm{\perp }}$ for the λ/4 film, the crossover occurring at d≊2λ. A proposed interpretation of these results is based on our calculations of the vortex behavior in thin (d≪λ) films, which give analytical formulas for the field distribution around a fluxon, the lower critical field, ${\mathit{H}}_{\mathit{c}1}$, the surface barrier, and the vortex-vortex interaction potential.