Josephson coupling through ferromagnetic heterojunctions with noncollinear magnetizations

We study the Josephson effect in clean heterojunctions that consist of superconductors connected through two metallic ferromagnets with insulating interfaces. We solve the scattering problem based on the Bogoliubov-de Gennes equation for arbitrary relative orientation of in-plane magnetizations, transparency of interfaces, and mismatch of Fermi wave vectors. Both spin-singlet and -triplet superconducting correlations are taken into account, and the Josephson current is calculated as a function of thickness of ferromagnetic layers and of the angle $\ensuremath{\alpha}$ between their magnetizations. For relatively weak ferromagnets, we find that the critical Josephson current ${I}_{c}$ is a monotonic function of $\ensuremath{\alpha}$ when the junction is far enough from $0\text{\ensuremath{-}}\ensuremath{\pi}$ transitions. No substantial impact of long-range spin-triplet superconducting correlations on the Josephson current has been found in the clean limit. For stronger ferromagnets, variation of $\ensuremath{\alpha}$ induces switching between 0 and $\ensuremath{\pi}$ states and ${I}_{c}(\ensuremath{\alpha})$ is a nonmonotonic function, displaying characteristic dips at the transitions. This nonmonotonicity is due to decreasing influence of the exchange potential with increasing misorientation of magnetizations. We suggest that the critical current and $0\text{\ensuremath{-}}\ensuremath{\pi}$ transitions can be controlled by varying the angle between magnetizations.