Numerical computation of the flux-line-lattice structure of an inhomogeneous material in the London approximation

We present the basic equations that permit the efficient simulation of vortex structures in inhomogeneous materials whose superconducting behavior is described by the London equations. In analogy to homogeneous materials, we show that the energy of such structures is the sum of a self-interaction part, which depends on the local variation of the penetration depth, and a sum of pair interactions, which are the Greens functions of the London equation for an inhomogeneous material. This formalism permits simulations in which the exact, long-range London interactions between flux lines are used. In addition, the realistic modeling of defects by allowing the penetration depth to vary spatially is possible, and pinning, instead of being assumed, arises in a natural manner by local variations in Lorentz forces. The equations were solved for a variety of one-dimensional models of inhomogeneities, and physically reasonable flux-line lattices were found. A suggestion for the use of the formalism to study the dynamic properties of flux-line lattices is made.