Theory Of The Lower Critical Magnetic Field For A Two-Dimensional Superconducting Film In A Nonuniform Field

We consider the appearance of vortices in a two-dimensional (2D) superconducting film exposed to a nonuniform magnetic field B-a produced by a nearby coil. The film has "infinite" radius R-f and thickness t about equal to the coherence length xi. The coil is approximated as a point dipole. We find that the first vortex-bearing state to appear has both a vortex and an antivortex. The Gibbs free energy of this state is lower than the vortex-free state when the maximum applied perpendicular field, i.e., the applied field B-0 at the origin, exceeds the external critical field B-c1(0) = 4 root 2 Lambda/R Phi(0)/4 pi Lambda(2) ln(Lambda/xi), where Phi 0/4 pi Lambda(2) ln(Lambda/xi) = B-c1(2D) is the intrinsic critical field in 2D, Lambda = 2 lambda(2)/t is the 2D penetration depth introduced by Pearl, and lambda is the bulk penetration depth. The prefactor 4 root 2 Lambda/R is calculated in the strong-screening regime Lambda/R << 1. R is the radial distance at which the applied perpendicular field B-a,B- z (rho) changes sign. In the lab, the onset of vortex effects generally occurs at a field much higher than B-c1(0), indicating that vortices are inhibited by the vortex-antivortex unbinding barrier, or by pinning. DOI: 10.1103/PhysRevB.87.064503