Tilted and Twisted Magnetic Moments in the Kitaev Magnet $α$-RuCl$_3$

Abstract

The layered honeycomb magnet $α$-RuCl$_3$ has attracted intense scrutiny as a prime candidate for realizing the Kitaev quantum spin liquid, yet a consensus on its microscopic Hamiltonian remains elusive due to the material's extreme sensitivity to structural details. Here, we report a comprehensive reexamination of the low-temperature crystallographic and magnetic structures of high-quality $α$-RuCl$_3$ single crystals using unpolarized and polarized neutron diffraction. We confirm a sharp, first-order structural phase transition to the rhombohedral $R\bar{3}$ space group with a pronounced thermal hysteresis. Crucially, using both spherical and longitudinal neutron polarization analysis, we determine the 3D orientation of the ordered magnetic moment without the ambiguity typically arising from domain distributions. We find that the Ru$^{3+}$ magnetic moments in the zigzag phase are tilted by $15.7^\circ$ out of the hexagonal plane and, remarkably, exhibit an additional in-plane twist of $-13.8^\circ$. This "tilted and twisted" geometry differentiates the ground state from the previously reported models based on unpolarized neutron diffraction or resonant elastic X-ray scattering (REXS) analysis.

Figures (4)

• Figure 1: Structures and phase transition of $\alpha$-RuCl$_{3}$. (a) View of a single honeycomb layer along the $c$-axis. The light-blue line indicates the mirror plane present in the single layer but broken in the $R\bar{3}$ stacking. The labels (1--3) indicate the stacking vector positions. Crystal structure model was plotted by Vesta Momma2011. (b) 3D view of the unit cell. (c, d) Temperature dependence of the integrated intensities for Bragg peaks (1,1,4) and (1,1,9). The (1,1,4) reflection is allowed only in the monoclinic ($C2/m$) phase, while (1,1,9) is specific to the rhombohedral ($R\bar{3}$) phase. The clean hysteresis and lack of residual intensity at base temperature indicate a complete structural transition. Insets show rocking curves at 20 K and 200 K. (e) Magnetic susceptibility $\chi(T)$ with $B \parallel ab$, showing the structural hysteresis around 150 K and the magnetic transition at low temperature. (f) Comparison of observed ($|F_{\mathrm{obs}}|^2$) and calculated ($|F_{\mathrm{cal}}|^2$) neutron diffraction structure factors for the $R\bar{3}$ model at 2 K.
• Figure 2: Spherical neutron polarimetry (SNP) analysis at $T=1.5$ K. $\text{SF}_{ij}$ and $\text{NSF}_{ij}$ denote the spin-flip and non-spin-flip scattering cross-sections, respectively, where the subscripts $i, j \in \{x, y, z\}$ indicate the polarization directions of the incident and scattered neutrons. (a) Rocking curves ($\omega$-scans) of key magnetic reflections measured in the $\text{SF}_{xx}$ channel. (b)-(f) $Q$ scans across the $(-0.5, 0, -1)$ reflection, resolved into various SF and NSF polarization channels. The local coordinate system is defined with $x \parallel \mathbf{Q}$ and $z$ perpendicular to the scattering plane (vertical). The absence of peak intensity in the $\text{NSF}_{xx}$ channel (b) confirms the purely magnetic origin of this reflection. Furthermore, the substantial intensity in the off-diagonal channels, such as $\text{SF}_{yz}$ (e) and $\text{SF}_{yx}$ (c), directly probes the cross-terms of the magnetic interaction vector. This provides the tight constraints necessary to uniquely determine the tilted and twisted moment direction.
• Figure 3: Determination of the magnetic moment direction by the SNP method. (a) Three zigzag magnetic configuration domains in the obverse hexagonal structure. Their magnetic configurations and propagation directions are marked. The right panel shows their corresponding ordered positions in reciprocal space. In our SNP scattering geometry, we exclusively access one domain, as the other two domains point out of the scattering plane. (b) Scattering geometry and definition of the $xyz$ coordinate system used for the SNP experiment. The moment direction is defined by the tilt angle $\theta$ (out of the $ab$ plane) and the twist angle $\beta$ (in-plane rotation). (c) Comparison of observed (open circles) and calculated (solid bars) polarization matrix elements $P_{ij}$. The sequence of $P_{ij}$ for each reflection is plotted as $P_{xx}$, $P_{xy}$, $P_{xz}$, $P_{yx}$, $P_{yy}$, $P_{yz}$, $P_{zx}$, $P_{zy}$, $P_{zz}$. (d) The determined zigzag magnetic structure. The best fit yields moment orientation angles of $\theta = 15.7(1.1)^\circ$ and $\beta = -13.8(1.5)^\circ$, indicating a "tilted (out-of-plane) and twisted (in-plane)" geometry. A twist of $\beta = 0^\circ$ confines the moment to the hexagonal $ac$ plane when the antiparallel bond of the zigzag order points along $b^\star$, and negative $\beta$ value indicates a clockwise rotation away from the hexagonal $ac$ plane.
• Figure 4: Differentiation of magnetic structure models via XYZ-polarization analysis. (a)--(d) $Q$-scans along $K$ and $L$ directions for magnetic reflections $(0, 0.5, L)$ with $L=1, 2, 4, 5$. Data were collected in the $x, y, z$ spin-flip ($\text{SF}_{xx}$, $\text{SF}_{yy}$, $\text{SF}_{zz}$) and $x$ non-spin-flip ($\text{NSF}_{xx}$) channels. For each scan direction, the isolated magnetic intensity components are obtained via the subtractions $M_{\perp z}^2 \sim \text{SF}_{xx} - \text{SF}_{zz}$ and $M_{\perp y}^2 \sim \text{SF}_{xx} - \text{SF}_{yy}$. (e) The ratio of magnetic intensity components $M_{\perp z}^2 / M_{\perp y}^2$ as a function of $L$. Symbols represent experimental values derived from the fits in (a)--(d). The curves represent calculated ratios for different magnetic moment orientations: blue ($\theta = 35^\circ, \beta = 0^\circ$) Cao2016, orange ($\theta = 32^\circ, \beta = 0^\circ$) Sears2020, and yellow ($\theta = 24.53^\circ, \beta = 0^\circ$) obtained by forcing $\beta = 0^\circ$ in the SNP fit. The solid magenta line indicates the best-fit structure determined from the full SNP analysis ($\theta \approx 15.7^\circ, \beta \approx -13.8^\circ$), with the shaded region representing the uncertainty band. The data clearly exclude the $\beta=0^\circ$ models and confirm the twisted moment structure.