Charge-state dependent spin-orbit coupling and quantum phase transitions in Ir-Ru oxides

The competition between kinematic, relativistic and Coulombic interactions in iridium-based oxides has spurred intense experimental and theoretical investigations regarding the electronic structure and magnetism. We argue here that the Iridium-Ruthenium triple perovskites, Ba$_3$MRuIrO$_9$ (M = Li, Mg and In), are of particular interest in this regard. We show here, using ab-initio theory, that the nominal charge states of Ir can be tuned from +6 to +4 by choosing non-magnetic 'M' ions as Li (+1), Mg(+2) and In (+3). This variation modulates the influence of the spin-orbit coupling (SOC) which is found here to be negligible in Ba$_3$LiRuIrO$_9$, moderate in Ba$_3$MgRuIrO$_9$ and determining in Ba$_3$InRuIrO$_9$. Our analysis classifies Ba$_3$LiRuIrO$_9$ as a band-insulator, Ba$_3$MgRuIrO$_9$ as a SOC and correlation driven insulator and Ba$_3$InRuIrO$_9$ as $J_{\rm eff} = 1/2$ Mott-Hubbard insulator. As reported here, correlated electronic structure theory results in sizeable magnetic moments of both Ru and Ir atoms in these systems and atomistic spin-dynamics simulations capture the experimental N\'eel temperature for Ba$_3$LiRuIrO$_9$ and Ba$_3$MgRuIrO$_9$ and provide evidence for a phase transition for Ba$_3$InRuIrO$_9$ when T $\to$ 0 K, to a multi-valley magnetic state with strong magnetic frustration. The theory identifies the presence of Kitaev interaction among the iridium atoms in Ba$_3$InRuIrO$_9$. The realization of such strong anisotropic interactions helps to stabilize a particularly complex energy landscape of Ba$_3$InRuIrO$_9$, that opens up for exotic magnetic quantum phases.