A Scattering method to the equilibrium spin current in a ferromagnet junction

We extended McMillan's Green's function method to study the equilibrium spin current (ESC) in a ferromagnet/ferromagnet (FM/FM) tunnelling junction, in which the magnetic moments in both FM electrodes are not collinear. The single-electron Green's function of the junction system is directly constructed from the elements of the scattering matrix which can be obtained by matching wavefunctions at boundaries. The ESC is found to be determined only by the Andreev-type reflection amplitudes as in the Josephson effect. The obtained expression of ESC is an exact result and at the strong barrier limit gives the same explanation for the origin of ESC as the linear response theory, that is, ESC comes from the exchange coupling between the magnetic moments of the two FM electrodes, ${\mathbf{J}}\sim{\mathbf{h}}_{l}\times{\mathbf{h}}_{r}$. In the weak barrier region, ESC cannot form spontaneously in a noncollinear FM/FM junction when there is no tunneling barrier between the two FM electrodes.