Highly anisotropic exchange interactions of $j_{\mathrm{eff}}=\frac{1}{2}$ iridium moments on the fcc lattice in ${\mathrm{La}}_{2}B{\mathrm{IrO}}_6 (B=\mathrm{Mg},\mathrm{Zn})$

We have performed inelastic neutron scattering (INS) experiments to investigate the magnetic excitations in the weakly distorted face-centered-cubic (fcc) iridate double perovskites ${\mathrm{La}}_2{\mathrm{ZnIrO}}_6$ and ${\mathrm{La}}_2{\mathrm{MgIrO}}_6$, which are characterized by A-type antiferromagnetic ground states. The powder inelastic neutron scattering data on these geometrically frustrated $j_{\mathrm{eff}}=\frac{1}{2}$ Mott insulators provide clear evidence for gapped spin-wave excitations with very weak dispersion. The INS results and thermodynamic data on these materials can be reproduced by conventional Heisenberg-Ising models with significant uniaxial Ising anisotropy and sizeable second-neighbor ferromagnetic interactions. Such a uniaxial Ising exchange interaction is symmetry forbidden on the ideal fcc lattice, so that it can only arise from the weak crystal distortions away from the ideal fcc limit. This may suggest that even weak distortions in $j_{\mathrm{eff}}=\frac{1}{2}$ Mott insulators might lead to strong exchange anisotropies. More tantalizingly, however, we find an alternative viable explanation of the INS results in terms of spin models with a dominant Kitaev interaction. In contrast to the uniaxial Ising exchange, the highly directional Kitaev interaction is a type of exchange anisotropy which is symmetry allowed even on the ideal fcc lattice. The Kitaev model has a magnon gap induced by quantum order by disorder, while weak anisotropies of the Kitaev couplings generated by the symmetry lowering due to lattice distortions can pin the order and enhance the magnon gap. Our findings highlight how even conventional magnetic orders in heavy transition metal oxides may be driven by highly directional exchange interactions rooted in strong spin-orbit coupling.

Figure 1

Color contour plots of the coarse energy resolution HYSPEC data for (a) ${\mathrm{La}}_2{\mathrm{MgIrO}}_6$ and (b) ${\mathrm{La}}_2{\mathrm{ZnIrO}}_6$ with $E_i=15$ meV and $T=1.5$ K. Inelastic modes with weak dispersion are clearly observed for both materials. The color contour plots with $E_i=15$ meV and $T=20$ K shown in (c) and (d) indicate that these modes disappear above the respective $T_N$'s of 12 K and 7.5 K for the Mg and Zn systems, which suggests that they have a magnetic origin. (e) and (f) Similar color contour plots for ${\mathrm{La}}_2{\mathrm{MgIrO}}_6$ and ${\mathrm{La}}_2{\mathrm{ZnIrO}}_6$ at $T=1.5$ K, but with better energy resolution arising from the choice of $E_i=7.5$ meV. The same excitations are visible in these inelastic spectra. Note that the lowest-$Q$ regions in the color contour plots show no intensity; this issue results from a background oversubtraction of the direct beam, which is a consequence of the strong neutron absorption of Ir in the samples. (g) and (h) show constant-$\hslash \omega$ cuts through the $T=1.5$ K fine-resolution data sets with an integration range of [0.6, 1] meV. The lack of increased intensity near the magnetic zone center $Q=0.79{\AA }^{-1}$ for each system indicates that the magnetic excitations are fully gapped.

Figure 2

(a) and (b) Constant-$Q$ cuts through the HYSPEC data shown in Figs. 1 and 1, integrated over $Q=[0.6,1]{\AA }^{-1}$. These cuts clearly show that the inelastic modes are gapped in each case. Gaussian fits are superimposed on the data, which were used to determine the peak positions. For ${\mathrm{La}}_2{\mathrm{MgIrO}}_6$, the mode is centered at $\hslash \omega =2.57\left(4\right)$ meV, and for ${\mathrm{La}}_2{\mathrm{ZnIrO}}_6$, it is centered at $2.09\left(3\right)$ meV. (c) and (d) HB-3 constant-$Q$ scans with $Q=0.79{\AA }^{-1}$ for ${\mathrm{La}}_2{\mathrm{MgIrO}}_6$ and ${\mathrm{La}}_2{\mathrm{ZnIrO}}_6$ at selected temperatures. The spin gaps close around $T_N$ for each compound, indicating that the modes have a spin wave origin. Note that 1 mcu (monitor count unit) $\approx$ 10 000 and 11 000 monitor counts for the Mg and Zn data, respectively.

Figure 3

Theoretical powder-averaged $S(Q,\omega )$ for the Heisenberg-Ising models around the A-II AFM ground state considered in the text within linear spin-wave theory with nearest-neighbor AFM Heisenberg $J_1=1$, various second neighbor Heisenberg exchange $J_2$, and varying degrees of uniaxial anisotropies ${\lambda }_{1,2}$ in the first and second neighbor exchanges. We set $J_{1,2}^y=J_{1,2}^z=J_{1,2}$ and $J_{1,2}^x=(1+{\lambda }_{1,2})J_{1,2}$. (a)–(d) Models with only first neighbor exchange, with $J_2=0$. The corresponding frustration parameters $f_H$ are also shown, and range from $f_H\approx 9$ for the isotropic case with ${\lambda }_1=0$ to $f_H\approx 2$ for extreme anisotropy ${\lambda }_1\gg 1$. (e)–(h) Heisenberg-Ising models with AFM second neighbor exchange, $J_2>0$, and varying degrees of anisotropy. (i)–(l) Heisenberg-Ising models with FM second neighbor exchange, $J_2<0$, and varying degrees of anisotropy. We have incorporated a small energy broadening in all panels to account for the finite instrumental energy resolution of HYSPEC.

Figure 4

Theoretical powder-averaged $S(Q,\omega )$ for the Kitaev models around the A-II AFM state for indicated spatial anisotropy ${\lambda }_K$. (a)–(e) Kitaev model within linear spin-wave theory for the ideal fcc limit (${\lambda }_K=0$, no spin gap) and various anisotropies $\left({\lambda }_K\le 4\right)$. (f)–(j) Kitaev models incorporating magnon interactions which induce an order-by-disorder spin gap even for ${\lambda }_K=0$, and a slight enhancement of the spin gap for ${\lambda }_K>0$. The corresponding frustration parameters $f_K$ are also shown; they range from $f_K\approx 2$ for the ideal fcc lattice to smaller values with increasing ${\lambda }_K$. We have incorporated a small energy broadening in all panels to account for the finite instrumental energy resolution of HYSPEC.

Figure 5

Illustrative plots of $S(Q,\omega )$ for the Heisenberg-Ising models and Kitaev-Heisenberg models. We have incorporated the ${\mathrm{Ir}}^{4+}$ form factor, and also used the instrumental energy resolution and the kinematic cutoff corresponding to the HYSPEC instrument with a neutron incident energy $E_i=7.5$ meV. (a) Heisenberg-Ising model for ${\mathrm{La}}_2{\mathrm{MgIrO}}_6$ with $J_1=0.5$ meV, a second-neighbor FM Heisenberg coupling $J_2=-0.3J_1$, and uniaxial anisotropies ${\lambda }_1={\lambda }_2=1$. (b) Heisenberg-Ising model for ${\mathrm{La}}_2{\mathrm{ZnIrO}}_6$, with $J_1=0.2$ meV, $J_2=-J_1$, and ${\lambda }_1={\lambda }_2=1$. (c) Kitaev-dominant model for ${\mathrm{La}}_2{\mathrm{MgIrO}}_6$ with $J_K=1.7$ meV, anisotropy ${\lambda }_K=0.2J_K$, and $J_2=0.2J_K$. (d) Kitaev-dominant model for ${\mathrm{La}}_2{\mathrm{ZnIrO}}_6$ with $J_K=0.7$ meV, anisotropy ${\lambda }_K=0.2J_K$, and second-neighbor FM Heisenberg exchange $J_2=-0.2J_K$.