Quasiclassical theory of superconductivity: A multiple-interface geometry

Abstract

A method is suggested that allows one to study multiple coherent reflection/transmissions by partially transparent interfaces (e.g., in multilayer mesoscopic structures or grain boundaries in high Tc’s), in the framework of the quasiclassical theory of superconductivity. It is argued that in the presence of interfaces, a straight-line trajectory transforms to a simple connected one-dimensional tree (graph) with knots, i.e., the points where the interface scattering events occur and pieces of the trajectories are coupled. For the two-component trajectory “wave function” which factorizes the Gor’kov matrix Green’s function, a linear boundary condition on the knot is formulated for an arbitrary interface, specular or diffusive (in the many channel model). From the new boundary condition, we derive (i) the excitation scattering amplitude for the multichannel Andreev/ordinary reflection/transmission processes; (ii) the boundary conditions for the Riccati equation; (iii) the transfer matrix which couples the trajectory Green’s function before and after the interface scattering. To show the usage of the method, the cases of a film separated from a bulk superconductor by a partially transparent interface, and a SIS′ sandwich with finite thickness layers, are considered. The electric current response to the vector potential (the superfluid density ρs) with the π phase difference in S and S′ is calculated for the sandwich. It is shown that the model is very sensitive to imperfection of the SS′ interface: the low temperature response being paramagnetic (ρs<0) in the ideal system case, changes its sign and becomes diamagnetic (ρs>0) when the probability of reflection is as low as a few percent.

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