Noncoplanar and Counterrotating Incommensurate Magnetic Order Stabilized by Kitaev Interactions in $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$

Materials that realize Kitaev spin models with bond-dependent anisotropic interactions have long been searched for, as the resulting frustration effects are predicted to stabilize novel forms of magnetic order or quantum spin liquids. Here, we explore the magnetism of $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$, which has the topology of a three-dimensional Kitaev lattice of interconnected Ir honeycombs. Using magnetic resonant x-ray diffraction, we find a complex, yet highly symmetric incommensurate magnetic structure with noncoplanar and counterrotating Ir moments. We propose a minimal Kitaev-Heisenberg Hamiltonian that naturally accounts for all key features of the observed magnetic structure. Our results provide strong evidence that $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ realizes a spin Hamiltonian with dominant Kitaev interactions.

Figure 1

Magnetic Bragg peak at $(0,0,16)+\mathit{q}$. (a) Scans along orthogonal directions in reciprocal space [filled (open) symbols at base temperature (above $T_N$)]. Dashed lines are fits to a Gaussian shape. (b) Temperature dependence of the integrated magnetic peak intensity (solid line is guide to the eye, temperature values are corrected for beam heating effects, see text). Inset: specific heat data showing an anomaly at the onset of magnetic order. (c) Scans with a polarizer in the scattered beam: the magnetic signal is present only in the $\sigma \text{−}{\pi }^{\prime }$ channel (filled circles) and disappears in the $\sigma \text{−}{\sigma }^{\prime }$ (open circles) dominated by charge scattering (intensity scaled by $1/10$). (d) Energy scan through the magnetic peak (blue squares) and a structural Bragg peak (0,0,16) (dotted line, scaled by $1/10^4$) as well as the fluorescence scan (solid line).

Figure 2

(a) Scan along the ($h,0,24$) direction observing structural peaks at $h=0,4$ (intensity scaled by $1/10^4$ for clarity), a multiple-scattering signal centered at $h=2$, and magnetic peaks at $h=0+q$, $2\pm q$, $4-q$. Solid red line is the calculated magnetic scattering intensity [21] for the magnetic structure model depicted in Fig. 4. (b) ($h0l$) reciprocal plane with filled circles, diamonds, and red crosses indicating positions of structural peaks, measured magnetic peaks, and the absence of peaks, respectively. Lattice points are also labeled by the magnetic basis vectors that have finite structure factor for magnetic peaks at satellite $\pm \mathit{q}$ positions.

Figure 3

Integrated intensity as a function of azimuth for three magnetic Bragg peaks (a) pure $A_x$, (b) pure $F_y$, and (c) mixed $F_z{A}_x$. Top diagram illustrates the scattering geometry. Data points (filled circles) are integrated peak intensities from rocking curve scans corrected for absorption and Lorentz factor. Thick (red) lines show fits that include all contributions to the MRXD structure factor [16, 21] from the magnetic structure model $i(A,-A{)}_x$, $-i(F,-F{)}_y$, $(F,F{)}_z$, depicted in Fig. 4. Blue and green curves in (a) and (c) illustrate that other phase combinations of basis vectors are ruled out.

Figure 4

Projection of the magnetic structure on the ($ac$) plane showing 3 unit cells along the horizontal propagation direction $a$. Light and dark blue arrows show the moments on the Ir and ${\mathrm{Ir}}^{\prime }$ sublattices, with sites 1–4 and $1^{\prime }–4^{\prime }$, respectively. Curly arrows on the left side illustrate counterrotating magnetic order between consecutive sites along $c$. In unit cell 2, light ($-\varphi$) and dark ($+\varphi$) shaded elliptical envelopes emphasize the confinement of the moments to alternate planes obtained from the ($ac$) plane by a rotation by $\mp \varphi$ around $c$. In unit cell 3, the color of bonds indicates the anisotropy axis of the Kitaev exchanges in Eq. (1), with $\eta =\mathsf{x},\mathsf{y},\mathsf{z}$ for black/green/red bonds, where $\widehat{\mathsf{x}}=(\widehat{\mathit{a}}+\widehat{\mathit{c}})/\sqrt{2}$, $\widehat{\mathsf{y}}=(\widehat{\mathit{a}}-\widehat{\mathit{c}})/\sqrt{2}$ and $\widehat{\mathsf{z}}=\widehat{\mathit{b}}$ [15]).