Quasiclassical theory of superconductivity: A multiple-interface geometry

A method is suggested that allows one to study multiple coherent reflection /transmissions by partially transparent interfaces (e.g., in multilayer mes oscopic structures or grain boundaries in high T-c's), in the framework of the quasiclassical theory of superconductivity. It is argued that in the pr esence of interfaces, a straight-line trajectory transforms to a simple con nected one-dimensional tree (graph) with knots, i.e., the points where the interface scattering events occur and pieces of the trajectories are couple d. For the two-component trajectory "wave function" which factorizes the Go r'kov matrix Green's function, a linear boundary condition on the knot is f ormulated for an arbitrary interface, specular or diffusive (in the many ch annel model). From the new boundary condition, we derive (i) the excitation scattering amplitude for the multichannel Andreev/ordinary reflection/tran smission processes; (ii) the boundary conditions for the Riccati equation; (iii) the transfer matrix which couples the trajectory Green's function bef ore and after the interface scattering. To show the usage of the method, th e cases of a him separated from a hulk superconductor by a partially transp arent interface, and a SIS' sandwich with finite thickness layers, are cons idered. The electric current response to the vector potential (the superflu id density rho(s)) with the pi phase difference in S and S' is calculated f or the sandwich. It is shown that the model is very sensitive to imperfecti on of the SS' interface: the low temperature response being paramagnetic (r ho(s) < 0) in the ideal system case, changes its sign and becomes diamagnet ic (rho(s) > 0) when the probability of reflection is as low as a few perce nt.