Vortex-vortex interaction in thin superconducting films

The interaction between Pearl vortices in thin superconducting films is revisited. For infinitely extended films this problem was solved by Judea Pearl who obtained the sheet current J(r) around the vortex and the force Φ0J(r) on a second vortex with magnetic flux Φ0 and at distance r in terms of the two rarely used Bessel functions S1 and N1. It is shown that the interaction potential V(r) and force -V'(r) can be approximated with high precision in the entire range of r by a simple logarithm. This expression directly shows the correct limits V(r)=(Φ02/μ0) ln (2.27Λ/r)/(2πΛ) for r<<Λ and V(r)=Φ02/(μ0 πr) for r>>Λ. Here Λ=λ2/d is the effective penetration depth, λ the London depth, and d < λ the film thickness. The effect of finite film size on the vortex interaction is discussed. The interaction now depends not only on the distance r but on both vortex positions and on the film shape. It is shown how the vortex interaction in finite films of any shape and size can be computed.