$J_{eff}$ states in a quasi one dimensional antiferromagnetic spin chain hexagonal Iridates Sr$_3$MIrO$_6$ (M=Mg, Zn, Cd): an $ab-initio$ comparative perspective

We employ first-principles density-functional theory, to perform a comparative investigation of the effect of the spin-orbit coupling (SOC) on the electronic and magnetic properties of three experimentally synthesized and characterized hexagonal perovskites Sr$_3$MIrO$_6$(M=Mg, Zn, Cd). The electronic structure calculations show that in all the compounds, Ir is the only magnetically active site in +4[5$d^5$] configuration, whereas M$^{+2}$ (M=Cd, Zn, Mg), remains in nonmagnetic states with Cd/Zn and Mg featuring $d^{10}$ and $d^{0}$ electronic configurations, respectively. The insulating gap could be opened by switching on the correlation parameter $U$ for Sr$_3$CdrO$_6$ and Sr$_3$ZnIrO$_6$ which qualifies it to be a correlated Mott insulator. However, in the case of Sr$_3$MgIrO$_6$ both $U$ and antiferromagnetic ordering is not enough and the gap could only be opened by including the SOC which classifies it to fall under the category of a typical SOC Mott insulator. The $j_{eff}$ states are visualized from the orbital projected band structure. The magnetism is studied from the point of view of exchange interactions and magnetocrystalline anisotropy in the presence of the SOC. We also present the comparative analysis of the renormalized impact of SOC on the three compounds, which shows that all the three compounds fall under the $intermediate$ coupling regime, where Sr$_3$MgIrO$_6$ is comparatively closer to the atomic $j_{eff}=\frac{1}{2}$ picture from the others.