Magnetic coupling of two-dimensional pancake vortex lattices in a finite stack of thin superconducting films with transport currents in the two outermost layers

We give a detailed study of the magnetic coupling between two-dimensional (2D) pancake vortices in a stack of N Josephson-decoupled superconducting thin films. The problem of a single pancake vortex in a finite stack of layers is first considered. We then investigate the magnetic interaction between 2D pancake lattices residing in different layers. It is assumed that all these 2D lattices have the same structure and orientation, although it is not required that they be in perfect registry. We derive an analytical solution for the coupling force on a pancake in a lattice arising from its interaction with a vortex lattice in another layer. As a direct application of this solution, we consider the case wherein a magnetic field is applied perpendicular to the layers and equal but oppositely directed surface current densities are introduced into the top and bottom layers, respectively. For weak currents, force-balanced configurations of pancake vortices are obtained. We then show the existence of a decoupling surface current density. Above this critical value, slippage occurs between 2D pancake lattices in different layers. This decoupling surface current density is then calculated for different magnetic fields and for different values of N.