A diagrammatic approach to triplet supercurrents in ferromagnetic Josephson junctions

We present a numerical study of ferromagnetic Josephson junctions with one intermediate layer and spin active interfaces in the ballistic regime, comparing the behaviour of one, two and three dimensional junctions. We find that in two and three dimensional junctions the geometries which lack spin flip scattering at one of the interfaces have a critical current that decays to zero, as we increase the magnetization of the intermediate layer towards the half metal limit. As known, magnetization direction inhomogeneity produces a long range supercurrent with width variation, even when spin flip scattering exists in one of the interfaces only, although of lower characteristic decay length, compared to junctions with spin flip scattering at both interfaces. In the case of a single spin-flip interface and large thickness of the junction, we find a dominant second harmonic in the current-phase relation, in agreement with earlier work. We apply a diagrammatic approach, which explains the strong second harmonic and the long range property of the supercurrent. We find that in the three dimensional junction case the main contribution to both the long range supercurrent and the second harmonic in thick Josephson junctions with one spin flip interface comes from diagrammatic terms that correspond to scattering loops inside the junction which are subject to anomalous Andreev reflections on the spin flip interface and normal Andreev reflections at the non spin flip interface, thus exhibiting the triplet nature of the phenomenon. We also find the emergence of the strong first harmonic for small variation of the magnetization geometry and comment on the effect of normal interface scattering and temperature on the second harmonic.