Spin Josephson effects in Exchange coupled Anti-ferromagnets

The energy of exchange coupled antiferromagnetic insulators (AFMIs) is a periodic function of the relative in-plane orientation of the N\'{e}el vector fields. We show that this leads to oscillations in the relative magnetization of exchange coupled AFMIs separated by a thin metallic barrier. These oscillations pump a spin current ($I_{S}$) through the metallic spacer that is proportional to the rate of change of the relative in-plane orientation of the N\'{e}el vector fields. By considering spin-transfer torque induced by a spin chemical potential ($V_{S}$) at one of the interfaces, we predict non-Ohmic $I_{S}$-$V_{S}$ characteristics of AFMI exchange coupled hetero-structures, which leads to a non-local voltage across a spin-orbit coupled metallic spacer.