Long-range magnetic interaction and frustration in double perovskite Sr$_{2}$NiIrO$_{6}$

Sr$_{2}$NiIrO$_{6}$ would be a ferromagnetic (FM) insulator in terms of the common superexchange mechanism between the first nearest neighboring (1NN) magnetic ions Ni$^{2+}$ ($t_{2g}^{6}e_{g}^{2}$) and Ir$^{6+}$ ($t_{2g}^{3}$). However, the observed antiferromagnetic (AF) order questions this viewpoint. In this work, we present first-principles calculations and find that while the 1NN Ni$^{2+}$-Ir$^{6+}$ exchange is indeed FM, the 2NN and 3NN couplings in the fcc Ir (and Ni) sublattice are AF. Moreover, the 2NN AF Ir-Ir coupling turns out to be even stronger than the 1NN FM Ni-Ir coupling, thus giving rise to a magnetic frustration. Sr$_{2}$NiIrO$_{6}$ hence becomes a distorted low-temperature antiferromagnet. Naturally, a very similar magnetic property in Sr$_{2}$ZnIrO$_{6}$ can be explained by the frustrated AF coupling in the fcc Ir$^{6+}$ sublattice. This work highlights the long-range magnetic interaction of the delocalized $5d$ electrons, and also addresses why the spin-orbit coupling is ineffective here.