A diagrammatic approach for a clean multiferromagnetic Josephson junction

We consider a superconductor/multiferromagnet/superconductor (S/I/F1/. . ./Fn/I/S) ballistic junction with thin insulating layers in the interfaces. We develop a diagrammatic approach for the equation that determines the Andreev spectrum, by examining the closed loops in the intermediate ferromagnetic layers. It is expressed in terms of the S/F and F/F interface scattering amplitudes. A set of rules is determined. Several analytical formulas are obtained, and in particular we study the triple ferromagnetic layer case. We also obtain the maximum current from the scattering matrix, and show that it is consistent with the diagrammatic approach. A specific application is the asymmetric S/F/I/S junction. The diagrammatic approach is also checked with the matrix of scattering amplitudes method