Even-odd effect for spin current through thin antiferromagnetic insulator

We theoretically study spin transport in a nonmagnetic metal-antiferromagnetic insulator (AFI)-ferromagnetic insulator (FI) system. Specifically, we consider a thin NiO layer oriented in (111) direction sandwiched between and coupled by exchange interaction to the metal and the FI. The preferred magnetic orientation in the NiO are ferromagnetically ordered planes parallel to the interfaces to the metal and the FI. These planes are antiferromagetically stacked on each other. Furthermore, the magnetic orientation in the FI is pinning the magnetic orientation of the neighboring NiO plane. As a consequence, the orientation of the NiO plane next to the metal depends on whether the number of planes in the NiO layer is even or odd. Furthermore, this determines the sign of the spin current through the insulator-metal interface generated by a thermal gradient (spin Seebeck effect). We compute the spin current in this scenario for clean interfaces and neglecting magnon decay and find that with increasing temperatures, the NiO thickness dependence of the spin current is vanishing except the mentioned even-odd effect. The reason for this lies in the nature of the magnonic eigenstates of the AFI-FI system yielding a normalization condition for their amplitudes next to the metal which are crucial for the spin current.