Multiband $d-p$ model and self-doping in the electronic structure of ${\text{Ba}}_2{\text{IrO}}_4$

We introduce and investigate the multiband $d-p$ model describing a ${\mathrm{IrO}}_4$ layer (such as realized in ${\mathrm{Ba}}_2{\mathrm{IrO}}_4$) where all 34 orbitals per unit cell are partly occupied, i.e., $t_{2g}$ and $e_g$ orbitals at iridium and $2p$ orbitals at oxygen ions. The model takes into account anisotropic iridium-oxygen $d-p$ and oxygen-oxygen $p-p$ hopping processes, crystal-field splittings, spin-orbit coupling, and the on-site Coulomb interactions, both at iridium and at oxygen ions. We show that the predictions based on assumed idealized ionic configuration (with $n_0=5+4\times 6=29$ electrons per ${\mathrm{IrO}}_4$ unit) do not explain well the independent ab initio data and the experimental data for ${\mathrm{Ba}}_2{\mathrm{IrO}}_4$. Instead we find that the total electron density in the $d-p$ states is smaller, $n=29-x<n_0$ ($x>0$). When we fix $x=1$, the predictions for the $d-p$ model become more realistic and weakly insulating antiferromagnetic ground state with the moments lying within ${\mathrm{IrO}}_2$ planes along (110) direction is found, in agreement with experiment and ab initio data. We also show that (i) holes delocalize over the oxygen orbitals and the electron density at iridium ions is enhanced; hence (ii) their $e_g$ orbitals are occupied by more than one electron and have to be included in the multiband $d-p$ model describing iridates.

Figure 1

Artist's view of the $5d$ orbital energies split by the elements $\{D_1,D_2,D_3\}$ at Ir ions in the absence of spin-orbit coupling ($\zeta =0$) (left), and the $d-p$ splitting $\mathrm{\Delta }$ Eq. (2.3) between the $\mathrm{Ir}(5d$) and $\mathrm{O}(2p$) orbitals (right).

Figure 2

Antiferromagnetic spin alignment along (1,1,0) direction in a $4\times 4{\mathrm{IrO}}_4$ cluster. The arrows represent magnetic moments within an ${\mathrm{IrO}}_2$ plane with the structure of ${\mathrm{CuO}}_2$ plane in cuprates (apical oxygens are not shown), with red arrows for iridium ions, and blue arrows for in-plane oxygen ions; the spin magnitudes are arbitrary and only the directions are relevant. Vertical and horizontal lines are only guides to the eye—they indicate ${\mathrm{IrO}}_4$ units for a better visibility of individual symmetry and unit cells within the cluster.

Figure 3

Electron occupation and spins in the multiband model for increasing $\mathrm{\Delta }\equiv -{\varepsilon }_p$ obtained in the formal ionic model without self-doping ($x=0$). The investigated ground state is insulating with AF order aligned along (1,1,0) direction. No significant orbital order was detected. Legend: data for Ir ions are represented by red lines; data for O ions are represented by blue lines; data points show electron occupation numbers in $t_{2g}$ orbitals (pluses), all $5d$ orbitals ($\times$), $2p$ orbitals at in-plane oxygens (circles), and $2p$ orbitals at apical oxygens (diamonds).

Figure 4

Electron occupations and magnetic moments in the multiband model for increasing $\mathrm{\Delta }\equiv -{\varepsilon }_p$, obtained in the realistic model with self-doping $x=1$. The investigated ground state is weakly insulating with AF order aligned along (1,1,0) direction. No significant orbital order was detected. Legend: data for Ir ions are represented by red lines; data for O ions are represented by blue lines; data points show electron occupation numbers in $t_{2g}$ orbitals (pluses), all $5d$ orbitals ($\times$), $2p$ orbitals at in-plane oxygens (circles), and $2p$ orbitals at apical oxygens (diamonds).