Charge-transfer model for the electronic structure of layered ruthenates

Motivated by the earlier experimental results and ab initio studies on the electronic structure of layered ruthenates (${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ and ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$) we introduce and investigate the multiband $d\text{−}p$ charge transfer model describing a single ${\mathrm{RuO}}_4$ layer, similar to the charge transfer model for a single ${\mathrm{CuO}}_2$ plane including apical oxygen orbitals in high $T_c$ cuprates. The present model takes into account nearest-neighbor anisotropic ruthenium-oxygen $d\text{−}p$ and oxygen-oxygen $p\text{−}p$ hopping elements, crystal-field splittings, and spin-orbit coupling. The intraorbital Coulomb repulsion and Hund's exchange are defined not only at ruthenium but also at oxygen ions. Our results demonstrate that the ${\mathrm{RuO}}_4$ layer cannot be regarded to be a pure ruthenium $t_{2g}$ system. We examine a different scenario in which ruthenium $e_g$ orbitals are partly occupied and highlight the significance of oxygen orbitals. We point out that the predictions of an idealized model based on ionic configuration (with $n_0=4+4\times 6=28$ electrons per ${\mathrm{RuO}}_4$ unit) do not agree with the experimental facts for ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ which support our finding that the electron number in the $d\text{−}p$ states is significantly smaller. In fact, we find the electron occupation of $d$ and $p$ orbitals for a single ${\mathrm{RuO}}_4$ unit $n=28-x$, being smaller by at least 1–1.5 electrons from that in the ionic model and corresponding to self-doping with $x\simeq 1.5$.

Figure 1

A fragment of the studied ${\mathrm{RuO}}_4$ cluster with four Ru ions shown by dark (red) big spheres and surrounding them O ions shown by smaller gray (dark blue) spheres. The calculations in Sec. 5 were performed for a $4\times 4{\mathrm{RuO}}_4$ periodic supercell.

Figure 2

Electron densities in the multiband model (2.1) for increasing $\Delta \left(\Delta \equiv -{\varepsilon }_p\right)$ obtained in the formal ionic model (with $n_0=28$ electrons per ${\mathrm{RuO}}_4$ unit): at Ru ions (solid lines) and at O ions (dashed lines). Data points show electron densities within $t_{2g}$ orbitals ($n_{t2g}$, circles), all $4d$ orbitals ($n_{4d}$, triangles), $2p$ orbitals at in-plane oxygens $(n_{p\parallel },+)$, and $2p$ orbitals at apical oxygens ($n_{p\perp }$, $\times$). The obtained ground state is ferromagnetic.

Figure 3

Total electron densities in the realistic $d\text{−}p$ multiband model based on ab initio calculations for increasing $\Delta \left(\Delta =-{\varepsilon }_p\right)$ for $\mathrm{Ru}(t_{2g})$ orbitals (solid line, circles), $\mathrm{Ru}(4d)$ orbitals (solid line, triangles), and $\mathrm{O}(2p)$ orbitals (dashed lines). There are $(4\times 6+2.5)$ electrons per a single ${\mathrm{RuO}}_4$ unit, corresponding to self-doping by $x=1.5$ electrons. The ground state is nonmagnetic for $\Delta >1.0$ eV, and weakly ferromagnetic for $\Delta \le 1.0$ eV. The change of magnetic order is responsible for the change of the slope of the lines.