Purely even harmonic Josephson current due to crossed pair transmission across strongly spin-polarized materials

Abstract We revisit the problem of the second harmonic generation in the current-phase relation across ferromagnetic bilayers placed between BCS superconductors. In particular, we consider a strongly spin-polarized metallic ferromagnet coupled to two superconducting leads via thin spin-active (left) and non-spin-active (right) insulating layers. The system is examined in the framework of the quasiclassical Green’s function formalism, both in the ballistic (Eilenberger) and the diffusive (Usadel) limit. Strong spin polarization allows for neglecting short-range mixed-spin correlations, and the Josephson supercurrent across the ferromagnet is fully mediated by long-range equal-spin triplet correlations. Considering the ballistic junction, we derive a general analytic formula for the Josephson current-phase relation that contains only even harmonics. Using a diagrammatic technique, we interpret the aforementioned formula in terms of multiple Andreev processes occurring at the right (non-spin-active) interface. In particular, we show that the lowest-order Andreev processes are responsible for the effective conversion of two spin-singlet Cooper pairs in the superconductor into one ↑↑ and one ↓↓ pair in the ferromagnet. The self-consistent numerical treatment of the diffusive junction, typically more relevant in experiments, confirms this picture quantitatively.