Cubic Double Perovskites Host Noncoplanar Spin Textures

Magnetic materials with noncoplanar magnetic structures can show unusual physical properties driven by nontrivial topology. Topologically-active states are often multi-q structures, which are challenging to stabilize in models and to identify in materials. Here, we use inelastic neutron-scattering experiments to show that the insulating double perovskites Ba2YRuO6 and Ba2LuRuO6 host a noncoplanar 3-q structure on the face-centered cubic lattice. Quantitative analysis of our neutron-scattering data reveals that these 3-q states are stabilized by biquadratic interactions. Our study identifies double perovskites as a highly promising class of materials to realize topological magnetism, elucidates the stabilization mechanism of the 3-q state in these materials, and establishes neutron spectroscopy on powder samples as a valuable technique to distinguish multi-q from single-q states, facilitating the discovery of topologically-nontrivial magnetic materials.

Most magnetic materials order with simple magnetic structures in which spins are collinear or coplanar.Noncoplanar magnetic structures are relatively rare, but are of great current interest, because they can exhibit topological character and exotic physical properties [1,2].For example, the finite scalar spin chirality of noncoplanar spin textures can generate a topological magneto-optical effect [3] and anomalous quantum Hall effect [4,5], even in the absence of spinorbit coupling.Topologically-nontrivial spin textures are typically multi-q structures, which superpose magnetic modulations with symmetry-related wavevectors q [2].Multi-q spin textures with long-wavelength modulations, such as skyrmion and hedgehog crystals, are well-studied as hosts of topologydriven phenomena [6][7][8].In this context, multi-q antiferromagnets are increasingly important [9], because they offer higher densities of topological objects with the potential to generate stronger physical responses [10].
To probe the relationships between spin structure, interactions, topology, and physical response, it is crucial to identify real materials that host noncoplanar spin textures.This has proved a challenging task, for three main reasons.First, it is necessary to identify noncoplanar spin textures that are robust to subleading effects such as magnetic anisotropies, spinlattice coupling [11,12], fluctuations [13][14][15][16], and anisotropic interactions [17], which usually favor collinear states.Second, most noncoplanar states are found in metals, such as USb [18,19] and γ-Mn alloys [20][21][22][23][24][25], and are often stable only under an applied magnetic field [6,26].On the one hand, itinerant electrons can support the generation of physical responses; on the other hand, modeling the magnetic interactions of metals presents fundamental challenges [27][28][29][30][31][32], such that insulators are often more suitable as model materials.Third, powder neutron-diffraction measurements play a central role in solving magnetic structures, but suffer from a "multi-q problem": Such measurements are generally unable to distinguish 1-q from multi-q structures [33].Therefore, multi-q spin textures are challenging to stabilize in models, and to identify in real materials.
Here, we identify the Mott-insulating double perovskites Ba 2 YRuO 6 and Ba 2 LuRuO 6 [34][35][36][37] as prototypical examples of noncoplanar 3-q magnetism on the face-centered cubic (FCC) lattice in zero magnetic field.We obtain evidence for 3-q magnetism from a spin-wave analysis of neutron spectroscopy data.By optimizing the magnetic structure and interactions simultaneously against our data, we show that the 3q structure is stabilized by biquadratic interactions within an antiferromagnetic Heisenberg-Kitaev model.Our study experimentally establishes that noncoplanar multi-q states are stabilized in frustrated FCC antiferromagnets, identifies cubic double perovskites as model materials to realize this behavior, and identifies guiding principles to facilitate design of materials with noncoplanar states.
(multi-q) structures associated with Type I antiferromagnetism.A remarkable property of the FCC lattice is that 1-q, 2-q, and 3-q structures are energetically degenerate for all bilinear interactions for which Type I ordering is stable [39,40].Moreover, uniaxial anisotropy (∼S 2 z ) and antisymmetric exchange terms are forbidden by Fm 3m symmetries, and quartic anisotropy (∼S 4 x + S 4 y + S 4 z ) is forbidden for S = 3/2 operators in a cubic environment.Consequently, interactions that would usually favor collinear magnetic structures are inactive in the S = 3/2 FCC antiferromagnet.This remarkable property potentially allows noncollinear structures to appear.
To identify candidate systems for 3-q spin textures among the diverse magnetic ground states of double perovskites [42][43][44][45][46][47][48][49][50], we consider two criteria: Type I antiferromagnetic ordering, and strictly cubic symmetry below the magnetic ordering temperature, T N .The second criterion is key because 3-q structures have cubic symmetry, while 1-q and 2-q structures have tetragonal or orthorhombic symmetry that could drive a crystallographic distortion via spin-lattice coupling [Figure 1(a)].We investigate Ba 2 YRuO 6 and Ba 2 LuRuO 6 because they are chemically well-ordered and show no evidence for low-temperature deviations from cubic symmetry [34,36].Moreover, recent first-principles calculations predict that their magnetic structures might not be collinear [51], in apparent contradiction with interpretations of previous experiments [34].
We prepared ∼8 g polycrystalline samples of Ba 2 YRuO 6 and Ba 2 LuRuO 6 by solid-state reaction [52].Rietveld refinement revealed stoichiometric samples with minor Lu 2 O 3 (1.94wt.%) or Y 2 O 3 (0.65 wt.%) impurities.The magnetic ordering temperature T N ≈ 37 K is the same for both samples, and is suppressed compared to the Weiss temperature θ ∼ −500 K, indicating strong magnetic frustration [36].We performed inelastic neutron-scattering measurements on the SEQUOIA instrument at ORNL [53] using incident neutron energies E i = 62 and 11.8 meV, yielding elastic energy resolutions δ ins ≈ 1.68 and 0.27 meV, respectively.Figure 1(b) shows magnetic Rietveld refinements to our elastic neutron-scattering data, measured with E i = 11.8 meV at T ≈ 5 K. Applying the q = [1, 0, 0] propagation vector to Fm 3m crystal symmetry generates two magnetic irreducible representations (irreps), notated mX + 3 and mX + 5 [54,55].These irreps can be distinguished by their magnetic Bragg profiles.The mX + 5 irrep agrees well with our elastic-scattering data for both materials; we obtain ordered magnetic moment lengths of 2.56(2) and 2.43(2) µ B per Ru for Ba 2 YRuO 6 and Ba 2 LuRuO 6 , respectively, from Rietveld refinement.Since the magnetic form factor for Ru 5+ is not known, we tested several 4d magnetic form factors [56]; while this choice does not qualitatively affect our results, the form factor for Zr + (isoelectronic with Ru 5+ ) yields optimal agreement with our data and is used throughout.In contrast to the mX + 5 irrep, the mX + 3 irrep strongly disagrees with our data, as it yields zero intensity for the strong (100) magnetic Bragg peak.This can be understood intuitively for a collinear 1-q structure, because neutrons are only sensitive to spin components perpendicular to the scattering wavevector, and the mX 3+ irrep has S q while the mX 5+ irrep has S ⊥ q [Figure 1(a)].A previous elas- tic neutron-scattering study of Ba 2 YRuO 6 and Ba 2 LuRuO 6 considered only collinear 1-q structures [34], but could not rule out multi-q structures, due to the multi-q problem.
To overcome the multi-q problem, we consider the energy dependence of our neutron-scattering data [57]., from which we extract the average energy E ph and width σ ph of this phonon band via Gaussian fits for each material.The energy overlap of magnon and phonon modes suggests that spin-lattice coupling may be significant, which we consider further below.
Our starting point for modeling the magnetic scattering is the Heisenberg-Kitaev model, Eq. ( 1).For all models, we require J > 0 and K > 0 to stabilize mX + 5 ordering.We consider three additional interactions in turn.First, the symmetric off-diagonal interaction is the only additional bilinear nearest-neighbor interaction allowed by symmetry.Second, the Heisenberg next-nearest neighbor interaction H 2 = J 2 ∑ i, j S i • S j has been invoked for Ba 2 YRuO 6 [37]; we require J 2 ≤ 0 to stabilize Type I ordering.Third, the nearest-neighbor biquadratic coupling H bq = J bq ∑ i, j (S i • S j ) 2 has been invoked in densityfunctional-theory calculations for 4d double perovskites due to their increased electron hopping relative to 3d analogs [51].For J bq = 0, the classical energy of 1-q, 2-q, and 3-q structures is equal for all K, Γ, and J 2 that stabilize Type I ordering.Nonzero J bq removes this degeneracy, and stabilizes R wp (%) Table I.Refined values of magnetic interaction parameters for the J-K-J bq model and 3-q structure.Uncertainties indicate 1σ statistical confidence intervals.
1-q ordering for J bq < 0 and 3-q ordering for J bq > 0 [Figure 3(a)].Importantly, since single-ion anisotropies are forbidden for S = 3/2 in a cubic environment, biquadratic exchange is the only physically-plausible mechanism that can remove the degeneracy of 1-q and 3-q structures.We performed extensive fits to our inelastic neutronscattering data to optimize the magnetic structure and interactions simultaneously.For each structure associated with the mX + 5 irrep (1-q, 2-q, or 3-q), we optimized three spin Hamiltonian parameters (J, K, and either Γ, J 2 , or J bq ) against the broadband inelastic data shown in Figure 4(a) and the energy dependence near the (100) magnetic Bragg position shown in Figure 4(b).The powder-averaged magnon spectrum was calculated within a renormalized linear spin-wave theory [58] using the SpinW program [59].The renormalization factor, which takes into account higher-order corrections in the 1/S expansion, is strictly necessary to extract a correct value of J bq , since the unrenormalized spin-wave theory would lead to a value of J bq that is 2.25 times smaller than the correct value [60].The parameter values were optimized to minimize the sum of squared residuals using nonlinear least-squares refinement [52].We calculated the energy-dependent broadening of the magnon spectrum as δ (E) = δ ins (E) + Ae −(E−E ph ) 2 /2δ 2 ph , where δ (E) is the overall Gaussian energy width, δ ins (E) is the instrumental resolution, and A is a refined parameter that phenomenologically accounts for magnon broadening due to coupling with phonons at E ∼ E ph .
Figure 4(a) compares our broadband inelastic data (E i = 62 meV) with the best fit for each of the 1-q, 2-q, and 3-q Figure 4. (a) Broadband inelastic neutron-scattering data (E i = 62 meV) and optimal spin-wave fits for different magnetic structures, showing (left to right) experimental data, 1-q fit, 2-q fit, and 3-q fit.(b) Low-energy inelastic neutron-scattering data (E i = 11.8 meV) and 3-q model calculations, showing (left to right) a cut at Q = 0.7450 ± 0.0175 Å −1 comparing experimental data (black circles) and spin-wave fit (red lines), experimental data as a Q-E slice, and spin-wave calculation.
structures.The data show two V-shaped features centered at ≈ 0.85 and ≈ 1.70 Å −1 , with a sharp cutoff of magnetic signal for energies above ∼14 meV.For both materials, these characteristics are best reproduced by the 3-q structure, while the 1-q structure disagrees with our experimental data.These observations are confirmed by the goodness-of-fit metric R wp , shown in Figure 3(b).For both materials and for every interaction model we considered, the 3-q structure yields better agreement with our data than the 1-q or 2-q structures.Notably, the goodness-of-fit is more sensitive to the structure than the precise magnetic interactions; indeed, the main differences between 1-q and 3-q spectra are apparent for Heisenberg exchange only [52].The global best fit is for the 3-q structure and J, K, and J bq interactions with the refined values given in Table I.The refined values of A indicate significant magnon broadening, which is larger for Ba 2 LuRuO 6 and is likely due to magnon-phonon coupling.Importantly, for both materials, the biquadratic term is significant, with J bq /J ∼ 0.06.Hence, our key results are that only the 3-q spin texture agrees well with our neutron data, and this state is stabilized by biquadratic interactions in Ba 2 YRuO 6 and Ba 2 LuRuO 6 .
Our model provides insight into the mechanism of gap opening [35].Figure 4(b) compares our low-energy inelastic data (E i = 11.8 meV) with the 3-q magnon spectrum for the optimal J-K-J bq model [Table I].This calculation reproduces the observed ≈ 2.8 meV gap, unlike the J-K-J 2 model that yields the next-best R wp [52].Since single-ion anisotropies are forbidden here, the mechanism of gap opening is subtle.If K = 0, there is no gap, because the energy of the Heisenberg and biquadratic terms is unchanged by global spin rotations.For K > 0, whether a gap opens depends on both structure and interactions.If the structure is 1-q with J bq < 0, the classical energy is unchanged by global spin rotations in the plane perpendicular to q.In this case, there is no gap at the linear spin-wave level; a gap is generated only by magnon interactions in the quantum (S = 1/2) limit [61].By contrast, if the structure is 3-q with J bq > 0, a gap is present at the linear spinwave level, because J bq > 0 and K > 0 together favor 111 spin alignment.Since Ba 2 YRuO 6 and Ba 2 LuRuO 6 are not in the quantum limit, the experimental observation of a gap supports the presence of biquadratic and Kitaev interactions in a 3-q structure.We have shown that the magnetic ground states of Ba 2 YRuO 6 and Ba 2 LuRuO 6 are noncoplanar 3-q structures stabilized by biquadratic interactions.Macroscopic topological physical responses may be generated synthesizing thin films of these materials with [111] strain [62].Our experimental results strikingly confirm recent first-principles predictions [51].The positive sign of J bq suggests that the effect of inter-site electron hopping outweighs spin-lattice coupling, since the latter would give a negative contribution to J bq [11,12].Crucially, we quantify the magnetic interactions that stabilize the noncoplanar state, in contrast to other proposed 3-q structures in NiS 2 [63][64][65], MnTe 2 [66], and UO 2 [67][68][69][70], where the relevant interactions are not yet well understood.Our work provides several guiding principles to facilitate the identification of multi-q spin textures.First, the near-degeneracy of 1-q and multi-q structures on the FCC lattice makes double perovskites enticing systems.In candidate materials, the crystal symmetry should be higher than a 1-q model would imply.Second, magnets that are not deep inside the Mott-insulating regime are expected to have larger J bq and, consequently, more robust 3-q orderings.This criterion hints that cubic Ba 2 YOsO 6 [71,72] may also host a 3-q state, due to its extended Os 5d orbitals, potentially offering a route to investigate the effect of increased electron hopping.For small J bq , we anticipate a thermally-induced transition from 3-q to 1-q ordering, since thermal fluctuations fa-vor collinear states.Third, quartic single-ion anisotropy may play a role in FCC magnets with S > 3/2; in particular, easy-111 axis anisotropy should favor 3-q ordering.Finally, our key methodological insight is that refining the magnetic structure and interactions simultaneously enables 1-q and multi-q structures to be distinguished on the FCC lattice, even when single-crystal samples are not available.
This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division.This research used resources at the Spallation Neutron Source, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory.

Figure 2 .
Figure 2. Broadband inelastic neutron-scattering data (E i = 62 meV) measured at T = 5 K for Ba 2 YRuO 6 (upper panels) and Ba 2 LuRuO 6 (lower panels), showing (a) intensity as a color plot, and (b) energy dependence integrated over 4.0 ≤ Q ≤ 4.5 Å −1 , where experimental data are shown as black circles, and Gaussian fits to the ∼14 meV phonon band as red lines.
Figure 2(a) shows our inelastic data measured with E i = 62 meV at T ≈ 5 K.A structured inelastic signal appears at T < T N for small scattering wavevectors, Q 2 Å −1 , which we identify as magnon scattering.The top of the magnetic band overlaps with an intense phonon signal for Q 2 Å −1 .Figure 2(b) shows the scattering intensity integrated over 4.0 ≤ Q ≤ 4.5 Å −1

Figure 3 .
Figure 3. (a) Schematic phase diagram showing the magnetic ground states of the J-K-J bq model.(b) Goodness-of-fit metric R wp for candidate magnetic structures and interaction models of Ba 2 YRuO 6 (upper graph) and Ba 2 LuRuO 6 (lower graph).The graphs show R wp for refinements of the Heisenberg-Kitaev (J-K) model including a third refined parameter Γ (red bars), J 2 (blue bars), or J bq (green bars); note that the 2-q structure is stable only for J bq = 0.