Spin Josephson effect in ferromagnet/ferromagnet tunnel junctions
We consider the tunnel spin current between two ferromagnetic metals from a perspective similar to the one used in superconductor/superconductor tunnel junctions. We use fundamental arguments to derive a Josephson-like spin tunnel current $I_J^{\rm spin}\propto\sin(θ_1-θ_2)$. Here the phases are associated with the planar contribution to the magnetization, $ \sim e^{iθ}$. The crucial step in our analysis is the fact that the $z$-component of the spin is canonically conjugate to the phase of the planar contribution: $[θ,S^z]=i$. This is analogous to the commutation relation $[ϕ,N]=i$ in superconductors, where $ϕ$ is the phase associated to the superconducting order parameter and $N$ is the Cooper pair number operator. We briefly discuss the experimental consequences of our theoretical analysis.