Exchange interactions in $d^5$ Kitaev materials: From ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ to $\alpha \text{−}{\mathrm{RuCl}}_3$

We present an analytical study of the exchange interactions between pseudospin one-half $d^5$ ions in honeycomb lattices with edge-shared octahedra. Various exchange channels involving Hubbard $U$, charge-transfer excitations, and cyclic exchange are considered. Hoppings within $t_{2g}$ orbitals as well as between $t_{2g}$ and $e_g$ orbitals are included. Special attention is paid to the trigonal crystal field $\mathrm{\Delta }$ effects on the exchange parameters. The obtained exchange Hamiltonian is dominated by ferromagnetic Kitaev interaction $K$ within a wide range of $\mathrm{\Delta }$. It is found that a parameter region close to the charge-transfer insulator regime and with a small $\mathrm{\Delta }$ is most promising to realize the Kitaev spin liquid phase. Two representative honeycomb materials ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ and $\alpha \text{−}{\mathrm{RuCl}}_3$ are discussed based on our theory. We have found that both materials share dominant ferromagnetic $K$ and positive nondiagonal $\mathrm{\Gamma }$ values. However, their Heisenberg $J$ terms have opposite signs: AFM $J>0$ in ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ and FM $J<0$ in $\alpha \text{−}{\mathrm{RuCl}}_3$. This brings different magnetic fluctuations and results in their different magnetization behaviors and spin excitation spectra. Proximity to FM state due to the large FM $J$ is emphasized in $\alpha \text{−}{\mathrm{RuCl}}_3$. The differences between the exchange couplings of these two materials originate from the opposite $\mathrm{\Delta }$ values, indicating that the crystal field can serve as an efficient control parameter to tune the magnetic properties of $d^5$ spin-orbit Mott insulators.

Figure 1

(a) Top view of the honeycomb plane. NN bonds hosting $x$-, $y$-, and $z$-type Ising couplings are shown in blue, green, and red colors, respectively. The hexagonal lattice coordinates ($X,Y,Z$) and octahedral axes ($x,y,z$) are indicated. (b) Level structure of $t_{2g}$ manifold in a hole representation. Positive $\mathrm{\Delta }>0$ corresponds to a compression of octahedra along $Z$-axis within a point-charge model. (c) Wave-function shapes of pseudospin $|\widetilde{\frac{1}{2}}〉$ state in cubic case ($\mathrm{\Delta }=0$) and two opposite limits of trigonal crystal field $\mathrm{\Delta }$. Red (blue) color indicates the spin up (down) polarization of the hole. ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ and ${\mathrm{RuCl}}_3$ have opposite signs of $\mathrm{\Delta }$, thus the shapes of their ground state wave functions should be different and result in the distinct exchange interactions and magnetic spectra (see text).

Figure 2

(a) The 90${}^{\circ }$ bonding geometry between the neighboring magnetic ions ${\mathrm{M}}_i$ and ${\mathrm{M}}_j$. Sketches of the considered (b) intersite $U$ process where two holes meet at the transition metal ion site, (c) charge-transfer process where two holes are created at the same ligand ion, and (d) cyclic exchange where two holes are created on different ligand ions and do not meet each other.

Figure 3

(a) Schematic of $p$ and transition-metal ion $d(t_{2g},e_g$) energy levels. $t_{pd\pi }(t_{pd\sigma }$) is the hopping integral between $p$ and $t_{2g}(e_g$) orbitals. The corresponding $pd$ charge-transfer gap ${\mathrm{\Delta }}_{pd}$ and cubic splitting $10Dq$ are indicated. Sketch of different hopping processes between $d$ orbitals along $z$-type NN-bond: (b) indirect hopping $t$ between $t_{2g}$ orbitals, (c) direct hopping $t^{\prime }$ between $c=d_{xy}$ orbitals, and (d) indirect hopping processes between $t_{2g}$ and $e_g$ orbitals. Notice that there is a prefactor ($-1/2$) of the overlap between $e_g$ orbital and lower ligand $p_x$ orbital in (d).

Figure 4

Exchange parameters $K$ (red), $J$ (black), $\mathrm{\Gamma }$ (blue), and ${\mathrm{\Gamma }}^{\prime }$ (grey dashed) in units of $t^2/U$ as a function of trigonal field parameter $\delta =2\mathrm{\Delta }/\lambda$ for different hopping processes in $t_{2g}\text{−}{t}_{2g}$ (top) and $t_{2g}\text{−}{e}_g$ (bottom) channels. Other microscopic parameters are $t^{\prime }/t=0.5, U/{\mathrm{\Delta }}_{pd}=0.5, 10Dq/U=1, J_H/U=0.15, U_p/U=0.6, J_H^p/U_p=0.4$, and $t_{pd\sigma }/t_{pd\pi }=2$.

Figure 5

Exchange parameters $K$ (red), $J$ (black), $\mathrm{\Gamma }$ (blue), and ${\mathrm{\Gamma }}^{\prime }$ (grey dashed) as a function of (a)-(b) $\delta =2\mathrm{\Delta }/\lambda$, (c) $t^{\prime }/t$, and (d) $U/{\mathrm{\Delta }}_{pd}$. Other microscopic parameters $10Dq/U=1, J_H/U=0.15, U_p/U=0.6, J_H^p/U_p=0.4$, and $t_{pd\sigma }/t_{pd\pi }=2$ are fixed. (e) The framed areas indicate the parameter subspace of $t^{\prime }/t$ and $U/{\mathrm{\Delta }}_{pd}$, where the non-Kitaev interaction terms are by an order of value smaller than the FM Kitaev coupling, i.e., $\sqrt{J^2+{\mathrm{\Gamma }}^2+{\mathrm{\Gamma }}^{\prime 2}}<0.1\left|K\right|$. The enclosed area of dominant Kitaev interaction varies as function of $\delta$ and is largest near the cubic limit.

Figure 6

Exchange parameters $K$ (red), $J$ (black), $\mathrm{\Gamma }$ (blue), and ${\mathrm{\Gamma }}^{\prime }$ (grey) as a function of $\delta =2\mathrm{\Delta }/\lambda$, calculated for (a) ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ and (b) ${\mathrm{RuCl}}_3$. For ${\mathrm{Na}}_2{\mathrm{IrO}}_3$, we use $10Dq=3.3$ eV, $U=1.35$ eV, $J_H=0.25$ eV, and ${\mathrm{\Delta }}_{pd}=3.3$ eV values from Ref. [43]. $U_p=4$ eV and Hund's $J_p=1.6$ eV for O-$p$ orbitals are taken from Ref. [38], and we set $t^{\prime }=0.6t$. For ${\mathrm{RuCl}}_3$, we use $10Dq=2.4$ eV, $J_H=0.34$ eV, ${\mathrm{\Delta }}_{pd}=5.5$ eV, and $U=2.5$ eV [22]. $U_p=1.5$ eV and Hund's $J_p=0.7$ eV are used for Cl-$p$ orbitals, and $t^{\prime }=0.5t$. The shaded intervals cover the $\delta$ values suggested by the experimental data on ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ [44, 45] and ${\mathrm{RuCl}}_3$ [22, 55].

Figure 7

(a) Sketch of the magnetic structure for zigzag order. Open and closed circles represent spins with opposite directions. (b) Brillouin zones of the honeycomb (inner hexagon) and the completed triangular lattice (outer hexagon). Expected spin excitation spectrum using linear spin wave theory (LSWT) and the exact diagonalization (ED) with (c) ($K,J,\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime },J_3$)=($-1,0.44,0.3,-0.15,0.1\left)\right|K|$ for ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ and (d) ($K,J,\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime },J_3$)= ($-1,-0.31,0.56,0.15,0.22$) $\left|K\right|$ for ${\mathrm{RuCl}}_3$, along the ${\mathrm{\Gamma }}^{\prime \prime }\text{−}Y\text{−}\mathrm{\Gamma }\text{−}K\text{−}X$ path of the Brillouin zone. Plotted in (c,d) is the trace of the spin susceptibility tensor. The LSWT spectra are averaged over three possible zigzag pattern directions. The ED data are a combination of the results for hexagonal 24-site and 32-site clusters.