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

The competition between kinematic, relativistic, and Coulomb interactions has spurred intense experimental and theoretical investigations in iridium-based oxides. Their electronic structure is mostly understood in terms of the spin-orbital coupled effective $J$ state (${J}_{\mathrm{eff}}$). However, the role of the Ir charge state in shaping the strength of effective spin-orbit coupling and defining the stability of the ${J}_{\mathrm{eff}}$ ground state has not been thoroughly explored. We argue here that the iridium-ruthenium triple perovskites, ${\mathrm{Ba}}₃{M\text{RuIrO}}₉$ ($M=\mathrm{Li}$, Mg, and In), are of particular interest in this regard. Using ab initio theory, we show here that the nominal charge states of Ir can be tuned from $+6$ ($5{d}³$) to $+4$ ($5{d}⁵$) by choosing nonmagnetic $M$ ions as $\mathrm{Li}\phantom{\rule{0.16em}{0ex}}(+1), \mathrm{Mg}(+2)$, and $\mathrm{In}\phantom{\rule{0.16em}{0ex}}(+3)$, while the Ru ions always remain in nominal $+5$ ($4{d}³$) charge state. This variation modulates the influence of the spin-orbit coupling (SOC), which is found to be negligible in ${\mathrm{Ba}}₃{\mathrm{LiRuIrO}}₉$, moderate in ${\mathrm{Ba}}₃{\mathrm{MgRuIrO}}₉$, and determining in ${\mathrm{Ba}}₃{\mathrm{InRuIrO}}₉$. Our analysis classifies ${\mathrm{Ba}}₃{\mathrm{LiRuIrO}}₉$ as a correlation-driven insulator arising from Hubbard interactions and exchange-split ${t}_2g$ states, ${\mathrm{Ba}}₃{\mathrm{MgRuIrO}}₉$ as a SOC and correlation-driven insulator that does not conform to the commonly expected $J=0$ ground state and ${\mathrm{Ba}}₃{\mathrm{InRuIrO}}₉$ as ${J}_{\mathrm{eff}}=1/2$ Mott-Hubbard insulator. In the data reported here, the correlational electronic structure theory results in sizable magnetic moments of both Ru and Ir atoms in these systems and atomistic spin-dynamics simulations capture the experimental N\'eel temperature for ${\mathrm{Ba}}₃{\mathrm{LiRuIrO}}₉$ and ${\mathrm{Ba}}₃{\mathrm{MgRuIrO}}₉$ and provide evidence for a phase transition for ${\mathrm{Ba}}₃{\mathrm{InRuIrO}}₉$ when T $\ensuremath{\rightarrow}$ 0 K, to a multivalley magnetic state with strong magnetic frustration. The theory identifies that strong SOC in ${\mathrm{Ba}}₃{\mathrm{InRuIrO}}₉$ induces bond-dependent magnetic couplings with significant Dzyaloshinskii-Moriya interaction, strong symmetric anisotropic exchange, and finite in-plane single-ion anisotropy. The realization of such strong anisotropic interactions helps to stabilize a particularly complex energy landscape of ${\mathrm{Ba}}₃{\mathrm{InRuIrO}}₉$, which opens up for exotic magnetic quantum phases such as quantum spin liquid. Thus, by comparing the electronic structure and magnetism of isostructural iridates with different Ir-charge states, we provided a theoretical framework to demonstrate the structural and electronic conditions that drive the deviations from conventional magnetic ordering, facilitating the emergence of exotic quantum phases.