Theory of the lower critical magnetic field for a two-dimensional superconducting film in a non-uniform field

We consider the first appearance of vortices in a two-dimensional (2-D) superconducting film exposed to a non-uniform magnetic field, \(\mathbf{B_a}\), 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 applied perpendicular field, \(B_0\), at the origin exceeds the external critical field: \(B_{c1}^0 = \frac{4\sqrt{2}\Lambda}{R}\frac{\Phi_0}{4\pi \Lambda^2}ln\left(\frac{\Lambda}{\xi}\right)\), where \(\frac{\Phi_0}{4\pi\Lambda^2}ln\left(\frac{\Lambda}{\xi}\right) \equiv B_{c1}^{2D}\) is the intrinsic critical field in 2-D, \(\Lambda \equiv 2\lambda^2/t\) the 2-D penetration depth introduced by Pearl, and \(\lambda\) is the bulk penetration depth. The prefactor, \(4\sqrt{2}\Lambda/R\), is calculated in the strong-screening regime, \(\Lambda/R \ll 1\). \(R\) is the radial distance at which the applied perpendicular field, \(B_{a,z}(\rho)\), 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.