Generalized proximity effect model in superconducting bi- and trilayer films

This article presents a general model for calculating the density of states and the Cooper pair potential in proximity-coupled superconducting bi- and trilayer films. It is valid for any kind of bilayer S1-S2, whatever the quality of the materials S1 and S2, the quality of the S1-S2 interface, and the layer thicknesses. The trilayer model is valid for a thin S3 layer, whereas the other two layers have arbitrary thicknesses. Although the equations of the dirty limit are used, it is argued that the model stays valid in clean bi-and trilayer films. The typical example of superconducting tunnel junctions is used to show that existing models, which apply to very thin or very thick layers or to perfectly transparent S1-S2 interfaces, are too restrictive to apply to an arbitrary bilayer. The new model is applied to practical junctions, with layer thicknesses intermediate between the “thick” and the “thin” approximation.