## A model for variable thickness superconducting thin films

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S. J. Chapman, Qiang Du & M. D. Gunzburger

A model for superconductivity in thin films having variable
thickness is derived through an averaging process across the
film. When the film is of uniform thickness the model is
identical to a model for superconducting cylinders as the
Ginzburg-Landau parameter tends to infinity. This means that all
superconducting materials, whether type I or type II in bulk,
behave as type-II superconductors when made into sufficiently
thin films. When the film is of non-uniform thickness the
variations in thickness appear as spatially varying coefficients
in the thin-film differential equations. After providing a formal
derivation of the model, some results about solutions of the
variable thickness model are given. In particular, it is
shown 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
film tends to zero. An application of the variable thickness
thin film model to flux pinning is then provided. In particular,
the results of a numerical calculation are given that show that
the vortex-like structures present in superconductors are
attracted to relatively thin regions.