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 \ensuremath{\lambda}/4 to 4\ensuremath{\lambda}, where \ensuremath{\lambda} is the London penetration depth, and H\ensuremath{\perp}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 \ensuremath{\lambda}/2 film. The ratio of ${\mathit{J}}_{\mathit{c}}$ values for parallel (${\mathit{J}}_{\mathit{c}\mathrm{\ensuremath{\parallel}}}$) and perpendicular (${\mathit{J}}_{\mathit{c}\mathrm{\ensuremath{\perp}}}$) film orientation increases as d decreases, so that ${\mathit{J}}_{\mathit{c}\mathrm{\ensuremath{\parallel}}}$\ensuremath{\ll}${\mathit{J}}_{\mathit{c}\mathrm{\ensuremath{\perp}}}$ for the 4\ensuremath{\lambda} film and ${\mathit{J}}_{\mathit{c}\mathrm{\ensuremath{\parallel}}}$\ensuremath{\gg}${\mathit{J}}_{\mathit{c}\mathrm{\ensuremath{\perp}}}$ for the \ensuremath{\lambda}/4 film, the crossover occurring at d\ensuremath{\approxeq}2\ensuremath{\lambda}. A proposed interpretation of these results is based on our calculations of the vortex behavior in thin (d\ensuremath{\ll}\ensuremath{\lambda}) 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.The film geometry gives rise to a significantly enhanced surface barrier and ${\mathit{H}}_{\mathit{c}1}$, a marked decrease of the range of the intervortex repulsion (to d instead of \ensuremath{\lambda}), and noncentral, position-dependent forces between vortices. These results are employed to evaluate the bulk and surface contributions to ${\mathit{J}}_{\mathit{c}\mathrm{\ensuremath{\parallel}}}$(d), both being shown to increase as d decreases. The bulk component of ${\mathit{J}}_{\mathit{c}\mathrm{\ensuremath{\parallel}}}$ exhibits a 1/${\mathit{d}}^{2}$ dependence at dg${\mathit{d}}_{\mathit{c}}$ due to the decrease of the tilt elastic modulus ${\mathit{c}}_{44}$(d) of the vortex structure, a crossover from the collective to a single-vortex regime of pinning occurring below a critical thickness at d${\mathit{d}}_{\mathit{c}}$. The surface magnetic pinning gives the main contribution to ${\mathit{J}}_{\mathit{c}}$ for our \ensuremath{\lambda}/2 and \ensuremath{\lambda}/4 films, leading to ${\mathit{J}}_{\mathit{c}\mathrm{\ensuremath{\parallel}}}$, which increases as 1/d and becomes of order ${\mathit{J}}_{\mathit{d}}$ at H\ensuremath{\approxeq}${\mathit{H}}_{\mathit{c}1}$. These calculations show that the ratio ${\mathit{J}}_{\mathit{c}\mathrm{\ensuremath{\parallel}}}$(d)/${\mathit{J}}_{\mathit{c}\mathrm{\ensuremath{\perp}}}$(d) increases as d decreases, with the ${\mathit{J}}_{\mathit{c}\mathrm{\ensuremath{\parallel}}}$(d)/${\mathit{J}}_{\mathit{c}\mathrm{\ensuremath{\perp}}}$(d) value being much less than unity at d\ensuremath{\gg}\ensuremath{\lambda} and much larger than unity at d\ensuremath{\ll}\ensuremath{\lambda}. 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.