Spin supercurrent in Josephson contacts with noncollinear ferromagnets

We present a theoretical study of the Josephson coupling of two s-wave superconductors which are connected through a diffusive contact consisting of noncollinear ferromagnetic domains. First, we consider a contact with two domains with magnetization vectors misoriented by an angle $\theta$. Using the quantum circuit theory, we find that in addition to the charge supercurrent, a spin supercurrent, which is even in $\phi$ and odd in $\theta$, with a spin polarization normal to the magnetization vectors flows between the domains. Furthermore, with asymmetric insulating barriers at the interfaces of the junction, the system may experience an antiferromagnetic-ferromagnetic phase transition for $\phi=\pi$. Secondly, we discuss the spin supercurrent in an extended magnetic texture with multiple domainwalls. We find the position-dependent spin supercurrent. The magnitude of the spin supercurrent strongly depends on the phase difference between the superconductors and the number of domain walls. Our results demonstrate the possibility to couple the superconducting phase to the magnetization dynamics.