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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, $ξ$. 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: $B_{c1}^0 = \frac{4\sqrt{2}Λ}{R}\frac{Φ_0}{4πΛ^2}ln\left(\fracΛξ\right)$, where $\frac{Φ_0}{4πΛ^2}ln\left(\fracΛξ\right) \equiv B_{c1}^{2D}$ is the intrinsic critical field in 2-D, $Λ\equiv 2λ^2/t$ the 2-D penetration depth introduced by Pearl, and $λ$ is the bulk penetration depth. The prefactor, $4\sqrt{2}Λ/R$, is calculated in the strong-screening regime, $Λ/R \ll 1$. $R$ is the radial distance at which the applied perpendicular field, $B_{a,z}(ρ)$, 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.