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 Jc of superconducting films. In order to eliminate possible contributions from intrinsic pinning, our measurements of Jc(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 4{λ}, where {λ} is the London penetration depth, and H{⊥}J. The angular dependent Jc(cphi) has a sharp peak for H parallel to the film surface, similar to that observed for high-Tc films. The amplitude of the peak increases as d decreases and reaches 20--30 % of the depairing current density (Jd) for the {λ}/2 film. The ratio of Jc values for parallel (J_c∥) and perpendicular (J_c⊥) film orientation increases as d decreases, so that J_c∥{}J_c⊥ for the 4{λ} film and J_c∥{}J_c⊥ for the {λ}/4 film, the crossover occurring at d{}2{λ}. 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, Hc1, the surface barrier, and the vortex-vortex interaction potential.The film geometry gives rise to a significantly enhanced surface barrier and Hc1, a marked decrease of the range of the intervortex repulsion (to d instead of {λ}), and noncentral, position-dependent forces between vortices. These results are employed to evaluate the bulk and surface contributions to J_c∥(d), both being shown to increase as d decreases. The bulk component of J_c∥ exhibits a 1/d² dependence at d>dc due to the decrease of the tilt elastic modulus c₄₄(d) of the vortex structure, a crossover from the collective to a single-vortex regime of pinning occurring below a critical thickness at ddc. The surface magnetic pinning gives the main contribution to Jc for our {λ}/2 and {λ}/4 films, leading to J_c∥, which increases as 1/d and becomes of order Jd at H{}Hc1. These calculations show that the ratio J_c∥(d)/J_c⊥(d) increases as d decreases, with the J_c∥(d)/J_c⊥(d) value being much less than unity at d{}{λ} and much larger than unity at d{}{λ}. The results obtained indicate that the effect of the film geometry can be very important when interpreting the angular dependences of critical currents of thin films.