Electronic and magnetic properties of iridium ilmenites $A{\mathrm{IrO}}_3$ $(A=\mathrm{Mg}, \mathrm{Zn}, \mathrm{and} \mathrm{Mn})$

We theoretically investigate the electronic band structures and magnetic properties of ilmenites with edge-sharing ${\mathrm{IrO}}_6$ honeycomb layers, $A{\mathrm{IrO}}_3$ with $A= \mathrm{Mg}, \mathrm{Zn}, \mathrm{and} \mathrm{Mn}$, in comparison with a collinear antiferromagnet ${\mathrm{MnTiO}}_3$. The compounds with $A= \mathrm{Mg} \mathrm{and} \mathrm{Zn}$ were recently reported in Y. Haraguchi et al., Phys. Rev. Mater. 2, 054411 (2018), while ${\mathrm{MnIrO}}_3$ has not been synthesized yet but the honeycomb stacking structure was elaborated in a superlattice with ${\mathrm{MnTiO}}_3$ in K. Miura et al., Commun. Mater. 1, 55 (2020). We find that, in contrast to ${\mathrm{MnTiO}}_3$, where an energy gap opens in the Ti $3d$ bands by antiferromagnetic ordering of the high-spin $S=5/2$ moments, ${\mathrm{MgIrO}}_3$ and ${\mathrm{ZnIrO}}_3$ have a gap in the Ir $5d$ bands under the influence of both spin-orbit coupling and electron correlation. Their electronic structures are similar to those in the spin-orbit coupled Mott insulators with the $j_{\mathrm{eff}}=1/2$ pseudospin degree of freedom, as found in monoclinic $A_2{\mathrm{IrO}}_3$ with $A=$ Na and Li, which have been studied as candidates for the Kitaev spin liquid. Indeed, we find that the effective exchange interactions between the $j_{\mathrm{eff}}=1/2$ pseudospins are dominated by the Kitaev-type bond-dependent interaction and the symmetric off-diagonal interactions. On the other hand, for ${\mathrm{MnIrO}}_3$, we show that the local lattice structure is largely deformed, and both Mn $3d$ and Ir $5d$ bands appear near the Fermi level in a complicated manner, which makes the electronic and magnetic properties qualitatively different from ${\mathrm{MgIrO}}_3$ and ${\mathrm{ZnIrO}}_3$. Our results indicate that the ${\mathrm{IrO}}_6$ honeycomb network in the ilmenites $A{\mathrm{IrO}}_3$ with $A=$ Mg and Zn would offer a good platform for exotic magnetism by the spin-orbital entangled moments like the Kitaev spin liquid.

Figure 1

Lattice structure of ilmenite $A{B\mathrm{O}}_3$: (a) bird's-eye view and (b) projection from the $c$ axis. The orange and yellow octahedra denote ${A\mathrm{O}}_6$ and ${B\mathrm{O}}_6$, each of which form a two-dimensional honeycomb network with edge sharing. The adjacent ${A\mathrm{O}}_6$ and ${B\mathrm{O}}_6$ honeycomb layers are stacked with face and corner sharing. The black lines in (a) denote the conventional unit cell with the lattice constants, $a$ and $c$. (c) The first Brillouin zone. The red lines denote the symmetric lines used in the plots of the band structures in Sec. 3 and Appendices pp1 and pp2.

Figure 2

(a) The energy gap and (b) the antiferromagnetic moment at the Mn site in the $c$-AFM state in ${\mathrm{MnTiO}}_3$ as functions of the Coulomb repulsion at the Mn site, $U_{\mathrm{Mn}}$, obtained by the $\mathrm{LDA}+\mathrm{SOC}+U$ calculations.

Figure 3

Electronic band structure of ${\mathrm{MnTiO}}_3$ obtained by the $\mathrm{LDA}+\mathrm{SOC}+U$ calculations for the $c$-AFM state at $U_{\mathrm{Mn}}=6$ eV. The right panels display the projected DOS for the relevant orbitals in each ion. The Fermi level is set to zero.

Figure 4

(a) Energy measured from the $ab$-AFM state in ${\mathrm{MgIrO}}_3$ as a function of the Coulomb repulsion at the Ir site, $U_{\mathrm{Ir}}$, obtained by the $\mathrm{LDA}+\mathrm{SOC}+U$ calculations. (b) The energy gap and (c) the antiferromagnetic moment at the Ir site in the $ab$-AFM state.

Figure 5

The electronic band structure of ${\mathrm{MgIrO}}_3$ obtained by the LDA+SOC calculations ($U_{\mathrm{Ir}}=0$) for the paramagnetic metallic state. The black curves denote the LDA results and the red dashed ones represent the band dispersions obtained by tight-binding parameters estimated by the MLWFs. The right panels display the projected DOS for each orbital. The Fermi level is set to zero.

Figure 6

The electronic band structure of ${\mathrm{MgIrO}}_3$ obtained by the $\mathrm{LDA}+\mathrm{SOC}+U$ calculations for the $ab$-AFM state with $U_{\mathrm{Ir}}=3$ eV. The notations are common to those in Fig. 5.

Figure 7

Coupling constants for the nearest-neighbor pseudospins in ${\mathrm{MgIrO}}_3$ as functions of the Coulomb repulsion at the Ir site, $U_{\mathrm{Ir}}$; see Eq. (2). The Hund's-rule coupling $J_{\mathrm{H}}$ and the spin-orbit coupling coefficient $\lambda$ are set to $J_{\mathrm{H}}/U_{\mathrm{Ir}}=0.1$ and $\lambda =0.4$ eV, respectively. The data connected by the blue, red, green, and yellow lines represent the Heisenberg $J$, Kitaev $K$, and symmetric off-diagonal couplings $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}^{\prime }$, respectively; the dotted, dashed, and solid lines indicate the data obtained by taking $U_p=0.0$, 0.5, and 1.0 eV, respectively.

Figure 8

Coupling constants for the nearest-neighbor pseudospins in ${\mathrm{ZnIrO}}_3$ as functions of $U_{\mathrm{Ir}}$ with $J_{\mathrm{H}}/U_{\mathrm{Ir}}=0.1$ and $\lambda =0.4$ eV. The notations are common to those in Fig. 7.

Figure 9

The electronic band structure of ${\mathrm{MnIrO}}_3$ obtained by the $\mathrm{LDA}+\mathrm{SOC}+U$ calculations for the state with $c$-FM for Mn and $ab$-FM for Ir with $U_{\mathrm{Mn}}=6$ eV and $U_{\mathrm{Ir}}=3$ eV. The right panels display the projected DOS for each orbital. The Fermi level is set to zero.

Figure 10

The electronic band structure for the paramagnetic state of ${\mathrm{MgIrO}}_3$ obtained by using the HSE hybrid functional with the Hartree-Fock mixing parameter $\alpha =0.05$. The notations are common to those in Fig. 6.

Figure 11

(a) Energy measured from the $c$-AFM state in ${\mathrm{ZnIrO}}_3$ as a function of the Coulomb repulsion at the Ir site, $U_{\mathrm{Ir}}$, obtained by the $\mathrm{LDA}+\mathrm{SOC}+U$ calculations. (b) The energy gap and (c) the antiferromagnetic moment at the Ir site in the $c$-AFM state.

Figure 12

The electronic band structure of ${\mathrm{ZnIrO}}_3$ obtained by the LDA+SOC calculations ($U_{\mathrm{Ir}}=0$) for the paramagnetic metallic state. The notations are common to those in Fig. 5.

Figure 13

The electronic band structure of ${\mathrm{ZnIrO}}_3$ obtained by the $\mathrm{LDA}+\mathrm{SOC}+U$ calculations for the $c$-AFM state with $U_{\mathrm{Ir}}=3$ eV. The notations are common to those in Fig. 6.