Importance of anisotropic exchange interactions in honeycomb iridates: Minimal model for zigzag antiferromagnetic order in ${\mathrm{Na}}_2{\mathrm{IrO}}_3$

In this work, we investigate the microscopic nature of the magnetism in honeycomb iridium-based systems by performing a systematic study of how the effective magnetic interactions in these compounds depend on various electronic microscopic parameters. We show that the minimal model describing the magnetism in ${\mathrm{A}}_2{\mathrm{IrO}}_3$ includes both isotropic and anisotropic Kitaev-type spin-exchange interactions between nearest and next-nearest neighbor Ir ions, and that the magnitude of the Kitaev interaction between next-nearest neighbor Ir magnetic moments is comparable with nearest neighbor interactions. We also find that, while the Heisenberg and the Kitaev interactions between nearest neighbors are correspondingly antiferro- and ferromagnetic, they both change sign for the next-nearest neighbors. Using classical Monte Carlo simulations we examine the magnetic phase diagram of the derived super-exchange model. We find that the zigzag-type antiferromagnetic order occupies a large part of the phase diagram of this model and, for the ferromagnetic next-nearest neighbor Heisenberg interaction relevant for ${\mathrm{Na}}_2{\mathrm{IrO}}_3$, it can be stabilized at small and even at zero third nearest neighbor coupling. Our results suggest that a natural physical origin of the zigzag phase experimentally observed in ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ is due to the interplay of the Kitaev anisotropic interactions between nearest and next-nearest neighbors.

Figure 1

(a) Schematic representation of the ${\mathrm{A}}_2{\mathrm{IrO}}_3$ structure. $x-,y-$, and $z-$ n.n. Ir-Ir bonds are shown by red, green, and blue solid lines. $\tilde{x}-,\tilde{y}-$, and $\tilde{z}-$ second n.n. Ir-Ir bonds are shown by red, green, and blue dotted lines. Thick magenta lines represent Ir-O-Na-O-Ir second n.n. super-exchange paths. (b) Undistorted $90{}^{\circ }$ Ir-O-Ir bond. Local axes for ${\mathrm{Ir}}^{4+}$ ions on A and B sublattices are the same as the global axes. Two possible super-exchange paths via upper or lower oxygen are shown.

Figure 2

The matrix elements of the tensor ${\Xi }_1^{\alpha \beta }$ on the $z$ bond as functions of trigonal crystal field $\Delta$, and Hund's coupling $J_H$. The diagonal n. n. exchange couplings $J_1^x,J_1^y,J_1^z$ in meV (shown by blue, green, and red lines, respectively) plotted as functions of (a) $\Delta$ (in eV) and (e) $J_H$ (in eV). The n. n. Kitaev interaction $K_1$ and the n. n. isotropic exchange $J_1$ (shown by brown and green lines, respectively) plotted as functions of (b) $\Delta$ (in eV) and (f) $J_H$ (in eV). The DM-type antisymmetric off-diagonal interactions $D_1^x,D_1^y,D_1^z$ in meV (shown by blue, green, and red lines, respectively) plotted as functions of (c) $\Delta$ (in eV) and (g) $J_H$ (in eV). The symmetric off-diagonal interactions ${\Gamma }_1^x,{\Gamma }_1^y,{\Gamma }_1^z$ in meV (shown by blue, green, and red lines, respectively) plotted as functions of (d) $\Delta$ (in eV) and (h) $J_H$ (in eV). For (a)–(d) and (e)–(h) plots we put $J_H=0.3$ eV and $\Delta =0.1$ eV, respectively. Other microscopic parameters of the model are considered to be $U_2=1.8$ eV, $\lambda =0.4$ eV, $t_{1o}=230$ meV, $t_d=67$ meV, and $t_{2o}=95$ meV.

Figure 3

The matrix elements of the tensor ${\Xi }_2^{\alpha \beta }$ on the $z$ bond as functions of trigonal crystal field $\Delta$, and Hund's coupling $J_H$. The diagonal n. n. exchange couplings $J_2^x,J_2^y,J_2^z$ in meV (shown by blue, green, and red lines, respectively) plotted as functions of (a) $\Delta$ (in eV) and (e) $J_H$ (in eV). The second neighbor Kitaev interactions $K_2$ and $K_2^{\prime }$, as well as the second neighbor isotropic exchange $J_2$ in meV (shown by brown, orange, and green lines, respectively) plotted as functions of (b) $\Delta$ (in eV) and (f) $J_H$ (in eV). The DM-type antisymmetric off-diagonal interactions $D_2^x,D_2^y,D_2^z$ in meV (shown by blue, green, and red lines, respectively) plotted as functions of (c) $\Delta$ (in eV) and (g) $J_H$ (in eV). The symmetric off-diagonal interactions ${\Gamma }_2^x,{\Gamma }_2^y,{\Gamma }_2^z$ in meV (shown by blue, green, and red lines, respectively) plotted as functions of (d) $\Delta$ (in eV) and (h)$J_H$ (in eV). For (a)–(d) and (e)–(h) plots we put $J_H=0.3$ eV and $\Delta =0.1$ eV, respectively. Other microscopic parameters of the model are considered to be $U_2=1.8$ eV, $\lambda =0.4$ eV, $t_{1o}=230$ meV, $t_d=67$ meV, and $t_{2o}=95$ meV.

Figure 4

Schematic representation of the effective super-exchange model for ${\mathrm{Na}}_2{\mathrm{IrO}}_3$. Color coding is the same as in Fig. 1. $X,Y$, and $Z t_{2g}$ electronic orbitals, participating in the super-exchange, are shown by red, green, and blue small circles.

Figure 5

Phase diagrams of the effective model (21) obtained with the classical Monte Carlo simulations at low temperature $T=0.1J_1$ for (a) second neighbor Kitaev interaction equal to $K_2=0$, (b) second neighbor Kitaev interaction $K_2=-2J_2$. The simulation is done for $J_1=3$ meV and $K_1=-17$ meV. The blue, rose, green, white, cyan, and emerald regions show the ferromagnetic (FM), the stripy, the zigzag, the incommensurate $3\mathbf{Q}$ spiral, the $120{}^{\circ }$ structure, and the intermediate state, respectively. (c) The structure factors obtained as a Fourier transform of a snapshot of a given configuration for each of these magnetic phases. Sharp peaks appear at the corresponding ordering wave vector.