A MODEL FOR SUPERCONDUCTING THIN-FILMS HAVING VARIABLE THICKNESS

A two-dimensional macroscopic model for superconductivity in thin films having variable thickness is derived through an averaging process across the him thickness. The resulting model is similar to the well-known Ginzburg-Landau equations for homogeneous, isotropic materials, except that a function that describes the variations in the thickness of the film now appears in the coefficients of the differential equations. Some results about solutions of the variable thickness model are then given, including existence of solutions and boundedness of the order parameter. It is also shown that the model is consistent in the sense that solutions obtained from the new model are an appropriate limit of a sequence of averages of solutions of the three-dimensional Ginzburg-Landau model as the thickness of the him tends to zero. An application of the variable thickness thin film model to flux pinning is then provided. In particular, the results of numerical calculations are given that show that the vortex-like structures that are present in certain superconductors are attracted to relatively thin regions in a material sample. Finally, extensions of the model to other settings are discussed.