Equilibrium spin transport in the ferromagnet/ferromagnet tunnelling junction

We study theoretically the equilibrium spin current (ESC) flowing through a ferromagnet/insulator barrier/ferromagnet (FM/I/FM) tunnelling junction in a lattice model. By using the Keldysh Green's function technique, we obtained an exact result of ESC based on the model. At the low transmission barrier limit, the exact solution can approximate the well-known linear response result that ESC comes from the exchange coupling between two magnetizations, ${\bf J}_{s}\sim {\bf h}_{L}\times {\bf h}_{R}$; whereas at the case of zero barrier between two FMs the ESC will vanish, which is attributed to a quantum mechanics effect that the electron reflection at the barrier (interface) is crucial for ESC formation in the junction. An analytic relation between ESC and the system conductance was found at the case of the weak magnetization and strong barrier