Impact of spin-orbit coupling on the magnetism of Sr$_{3}$MIrO$_{6}$ (M $=$ Ni, Co)

Recently, Iridates have recently drawn considerable attention due to their significant spin-orbit coupling (SOC) effect and possibly exotic properties [1]. In this work, we demonstrate, using density functional calculations, that the SOC of Ir$^{4+}$ ions plays an essential role in determining the antiferromagnetism of hexagonal spin-chain materials Sr$_{3}$MIrO$_{6}$ (M$=$Ni, Co) by tuning the crystal-field level sequence and altering the Ir-M inter-orbital interactions. Owing to the SOC effect, the single t$_{\mathrm{2g}}$ hole of the Ir$^{4+}$ ion resides on the e'$_{\mathrm{g}}$ upper branch and gives rise to the Ir$^{4+}$-M$^{2+}$ antiferromagnetic coupling. In absence of the SOC, however, the single t$_{\mathrm{2g}}$ hole would occupy the crystal-field a$_{\mathrm{1g}}$ singlet instead, which would mediate an unreal ferromagnetic exchange This work clarifies the nature and the origin of the intra-chain Ising antiferromagnetism of Sr$_{3}$MIrO$_{6}$(M $=$ Ni, Co) [2].\\[4pt] [1] B. J. Kim, et al., Phys. Rev. Lett. 101, 076402 (2008)\\[0pt] [2] X. Ou and H. Wu, Sci. Rep. 4, 4609 (2014); Phys. Rev. B 89, 035138 (2014).