Introduction

In many geometrically frustrated magnets, a weak perturbation can potentially induce new competing ground states, leading to an extremely rich phase diagram. A particularly important but less explored case is that of the frustrated system with defects, i.e., nonmagnetic impurities, which are ubiquitous in real-world materials. Combined with the geometrical frustration, impurities can introduce several nontrivial effects in a wide variety of frustrated magnets. For example, extensive search for a genuine quantum spin liquid (QSL) state in frustrated magnets has been consistently plagued by the impurity issues, e.g., herbertsmithite and YbMgGaO₄ as the S = 1/2 kagome^1–3 and triangular^4–6 QSL candidates, respectively. It has been theoretically pointed out that impurities themselves can induce continuum-like excitations in geometrically frustrated magnets, which is difficult to distinguish from the genuine signature of a QSL state. Other examples are a rich phase diagram due to impurities in a triangular lattice antiferromagnet⁷ (TLAF), an impurity-induced spin-glass state in the frustrated spinel^8–10 and spin ice^11,12 systems, suppression of the QSL state due to impurities¹³, and a disorder-induced classical^14,15 and quantum¹⁶ QSL state, to name a few.

One of the salient features resulting from a nontrivial interplay between the geometrical frustration and disorder is a significant modification of spins’ directions near impurities. In frustrated magnets, a vacancy gives rise to a partial relief of local frustration around it, which leads to a sizable reorientation of spins proximate to the impurity site (Fig. 1a). In a coplanar magnetic ground state with a well-defined easy plane, the reorientation only consists of in-plane components. The reorientation at the six nearest sites points toward the opposite direction to the effective impurity moment, leading to a screening of the effective impurity moment. Subsequently, spins at the next-nearest sites are affected by the nearest sites’ reorientation, leading again to the canting toward the direction opposite to the former reorientation (See Fig. S1 for demonstration of the spatial structure of the spin reorientation). Such a spatially-correlated pattern of the spin reorientation acts as a partial screening of the effective impurity moments, resulting in an uncompensated fractional impurity moment¹⁷, among other phenomena.

Fig. 1

Formation of a spin texture in 2D-TLAF h-Y(Mn,Al)O₃.

a Local spin canting induced by a vacancy in a triangular lattice antiferromagnet (blue arrows) and original 120^◦ magnetic order (gray arrows). b Distance dependence of the canting angle $$ \mid \mathrm{\delta }\mathrm{\Theta }\left(r\right)\mid $$ in the spin texture around a vacancy determined from the numerical simulations. Black (Blue) dots denote the simulation with D₂ = 0 (−0.02) meV, repectively. The red line is a guided plot of the 1/r³ behavior. c Doping dependence of the transition temperature T_N. The inset denotes the magnetic susceptibility of YMn_1-xAl_xO₃ as a function of temperature, from which the transition temperatures (T_N) were determined. d Two-dimensional reciprocal space of YMnO₃ with the labels of high-symmetric points and the high-symmetric lines used in Fig. 2. Color plot shows magnetic Bragg peaks of h-YMn_0.85Al_0.15O₃ measured by neutron scattering.

Moreover, the spins’ canting angle decays algebraically to the distance from the vacancy^17–19 (Fig. 1b), resulting in a formation of an extended spin object around the vacancy called spin texture^17,20,21. This impurity-induced texture can affect the spin dynamics and the ground state of the original system, which determines the thermal and spin transport properties of the material. Therefore, it may have a broad applicability to several fields in magnetism, not to mention its importance as a new way of generating large spin objects analogous to a skyrmion. However, despite such outstanding interests, there has been no experimental demonstration of the impurity-induced spin textures as of yet.

h-YMn_1-xAl_xO₃ is a unique model system to study dilution effects in a frustrated magnet, where non-magnetic Al³⁺ ions are doped into the Mn³⁺ triangular lattice^22,23. Pure h-YMnO₃ has noncollinear 120° magnetic order below T_N = 74 K due to the geometrical frustration. The spin dynamics of h-YMnO₃ have been well established by the previous studies^24–31, which used a simple model Hamiltonian (Eq. 1) with some additional terms from magnon-magnon/phonon coupling. Most importantly, increasing Al concentration x up to 0.2 does not change the crystalline symmetry but only reduces the antiferromagnetic transition temperature (Fig. 1c) and the magnetic moments of Mn³⁺ gradually, enabling a systematic study of the dilution effect on the magnetism of a triangular lattice antiferromagnet (TLAF)²³. While a large single crystal of h-YMn_1-xAl_xO₃ suited for inelastic neutron scattering (INS) is available²³, previous studies on this material focused only on the crystal structure and its bulk properties^23,32,33.

In this work, we report the full spin dynamics of h-YMn_1-xAl_xO₃ studied by INS and model calculations. Our INS data of h-YMn_1-xAl_xO₃ reveal the presence of magnon damping with clear momentum dependence, in accordance with the theoretical calculation incorporating the spin texture. Our result provides the first experimental evidence of the impurity-induced spin texture, and prompt a fundamentally new understanding of the impurity effects in a wide variety of non-collinear magnetic systems.

Results and discussion

Figure 2a–c show the energy-momentum (Q) slices from the INS data of 0%, 10%, and 15% Al-doped h-YMnO₃ single crystals along the high-symmetric lines (see Fig. 1d), which demonstrate the influence of the Al doping. In contrast to the clear INS spectra of the pristine h-YMnO₃, the INS spectra of the Al-doped samples show broad magnon signals as expected for magnetic systems with impurities. Interestingly, however, the observed energy linewidth broadening is not uniform over the Brillouin zone but Q-dependent. A stack of constant Q-cuts along the B − C ([K − K 0] direction) in Fig. 3a–c supports this statement: near the magnetic zone center (C point), the linewidth of the magnon peaks is hardly changed with doping, whereas the magnon peaks near the zone boundary (B point) undergo a drastic broadening. For a more transparent demonstration of the Q-dependent energy linewidth broadening, see Supplementary Fig. S5. For a deeper understanding of the Q-dependent linewidth broadening, we analyzed the half-width at half-maximum (HWHM) of the magnon peaks (see Methods) over the full Brillouin zone (Fig. 3d–f), which was averaged for all magnon modes at each Q-point. We note that there is no clear Q-dependence of the HWHM in the pure h-YMnO₃ (Fig. 3d). In h-Y(Mn,Al)O₃, however, the HWHM steadily increases when the magnon branches get closer to the zone boundary, and reaches its maximum near the B point (equivalent to the M point in the reciprocal space of a triangular lattice). Another noticeable feature is the Q-dependent magnon energy renormalization, manifested in the increasingly downward shift of the magnon dispersion along the A-B direction (Fig. 2a–c). Such Q-dependent energy linewidth broadening and renormalization of magnons may imply the presence of something beyond a simple dilution effect, as a point-like scattering potential made by an impurity is known to induce a Q-independent behavior of magnons generally^21,34,35.

Fig. 2

Magnetic excitation spectra of h-YMn_1-xAl_xO₃.

a–c INS spectra of a h-YMnO₃ (ISIS), b h-YMn_0.9Al_0.1O₃ (J-PARC) and c h-YMn_0.85Al_0.15O₃ (J-PARC) measured at 5 K with the incident neutron energy of E_i = 30 meV. The scattering intensity of the data was integrated over the c^∗-axis. d–f Theoretical INS cross-section of d h-YMnO₃, e h-YMn_0.9Al_0.1O₃, and f h-YMn_0.85Al_0.15O₃, which include the instrumental resolution convolution as well as the data integration effect over the c^∗-axis (see Methods).

Fig. 3

Q-dependent magnon linewidth broadening due to doping.

a–c Constant Q-cuts (colored circles) and theoretically calculated INS cross-sections (solid black lines) of a h-YMnO₃, b h-YMn_0.9Al_0.1O₃, and c h-YMn_0.85Al_0.15O₃ at various Q points along the [K, -K, 0] direction (the C-B direction). Both the experimental data and the theoretical calculation results are rendered with the intensity integration along the c^∗-axis. d–f Fitted intrinsic HWHM (Γ(Q)) of the magnon modes in d h-YMnO₃, e h-YMn_0.9Al_0.1O₃, and f h-YMn_0.85Al_0.15O₃ over the full Brillouin zone, where the instrumental resolution effects are excluded (see Methods).

To explain the observed features, we have modeled the spin Hamiltonian of a diluted triangular lattice with randomly distributed vacancies (see Methods for details). We assumed that Al doping does not change the parameters of the spin Hamiltonian, which is plausible considering the simple linear relation between x and T_N (Fig. 1c)³⁶. The spin dynamics of pure h-YMnO₃ can be described by the following spin Hamiltonian suggested by the previous study:²⁴

First, we tested the ground state modification by a single impurity (see Methods) and compared it with our previous theoretical work results. Figure 1b shows the canting angle of spins $$ \mid \mathrm{\delta }\mathrm{\Theta }\left(r\right)\mid $$ as a function of the distance from the impurity site (r) on a logarithmic scale, both with and without the easy-axis anisotropy D₂. For the directions of the spin canting, see Supplementary Fig. S1. Without the easy-axis anisotropy (D₂=0), the canting angle follows asymptotically the algebraic decay law, 1/r³, indicated by the guide line¹⁷. The long-range spin texture is formed with the canting angle depending on the distance r as well as on the sublattice number⁷. At large distances (r > 30), the numerical data are affected by finite-size effects. Turning on D₂, the canting angle decreases faster with distance, as expected for anisotropic models, but the spin texture remains almost intact at intermediate distances.

Using the relaxed spin configuration, including the spin texture and Eq. 1, we calculated the INS cross-section of YMn_1-xAl_xO₃ at the level of linear spin-wave theory (LSWT); see Methods for the details. Figures 2d–f and 3a–c show the results of the calculations convoluted with the instrumental resolution for x=0, 0.1, and 0.15, which are in excellent agreement with our INS data (for the results without the resolution convolution, see Supplementary Figs. S7 and S8). These results suggest that despite the simplicity of LSWT, our model calculation has successfully captured the unusual features observed in the data: the Q-dependent energy linewidth broadening and the increase of the downward shift of the magnon dispersion along the A-B direction. Note that the discrepancy between the magnon dispersion along A-B in Figs. 2a and 2d (x=0) is due to the effect of magnon-phonon coupling present in h-YMnO₃ (ref. ²⁴), which was not considered in our model calculation.

Such a Q-dependent magnon lifetime in a diluted noncollinear magnet has already been suggested by a previous theoretical study²¹, which argued that the spin texture gives rise to Q-dependent magnon scattering. According to ref. ²¹, the impurity-induced spin texture generates a spatially-dispersed effective magnetic field around an impurity (unlike a point-like potential made by the impurity itself), which acts as a potential for the magnon Umklapp scattering. Importantly, this scattering makes the scattering rate strongly Q-dependent as exactly found in our experiments. As the effect mentioned above is already embedded in our model calculations, the Q-dependent behavior observed in both the data and the calculations would be from the magnons’ scattering on the spin textures. Note that similar Q-dependent magnon scattering was also suggested in the system with skyrmions, where the effective magnetic field from its topological texture acts as a scattering potential³⁷. To verify that the observed behavior is unique characteristics of a frustrated magnet, we contrast it with a diluted non-frustrated square-lattice antiferromagnet’s spin dynamics. As shown in Supplementary Fig. S9, impurities in a square lattice create a flat localized mode, consistent with the previous INS study on a perovskite fluoride K(Co,Mn)F₃ (ref. ³⁸). However, one does not see any noticeable Q-dependent energy linewidth broadening or renormalization when increasing doping. Such distinctive difference comes from the absence of geometrical frustration (and, therefore, the absence of spin textures) in the square lattice antiferromagnet as opposed to the TLAF. Further theoretical analysis is needed for a comprehensive understanding of the specifics of the observed Q-dependence, such as why the most drastic changes in the spectrum due to dilution occur in the vicinity of the B point, which we leave for further studies.

To explicitly examine the spin textures’ role in the spin dynamics of h-Y(Mn,Al)O₃, we have also calculated the INS cross-sections of a diluted triangular lattice without the feedback of the vacancy, i.e., without the spin texture. To perform such calculation, we artificially forced the spins to retain the 120˚ magnetic order for the calculation, which is similar to the approach used in ref. ²¹. It amounts to neglecting the feedback of the impurity onto the host spins, so that no texture is created. This allows us to separate scattering effects of the conventional dilution from the effects associated with extended textures. Figure 4a, b show the calculated magnon spectra of 10% Al-doped h-YMnO₃ with and without the spin texture. While there exist a couple of slight differences when comparing the results in Fig. 4a and b over the full Brillouin zone, we found a particularly large difference in the intensity of the 5 meV mode near the C point. Complementary calculations without the spin texture confirmed that the 5 meV mode becomes strongly suppressed due to the non-magnetic impurities (Supplementary Fig. S10). In comparison, there is no noticeable suppression of the 5 meV mode in our experimental data and the theoretical calculation with the spin texture (Fig. 4c). This result implies that the in-plane spin texture formation is a key factor in retaining the 5 meV mode’s stability against the non-magnetic impurity.

Fig. 4

A role of spin textures in the spin dynamics of h-YMn_1-xAl_xO₃.

Theoretical spin-wave spectra of h-YMn_0.9Al_0.1O₃ a with vacancies and spin textures, and b with vacancies but without spin textures. c A constant Q-cut at the C point, which demonstrates the suppression of the 5 meV mode in the calculation results without spin texture. Black and orange points are the INS data of h-YMn_0.9Al_0.1O₃ and h-YMn_0.95Al_0.05O₃, respectively. Blue and Red solid lines are the constant-Q cuts of a and b. Note that this figure’s calculation results do not include the effect of data integration along the c^∗-axis.

To further understand the origin of the significant difference at the C point near 5 meV between Fig. 4a and b, we analyzed the eigenvector of the 5 meV magnon mode at the C point in pure h-YMnO₃ (Supplementary Fig. S10). This exercise allows us to examine why it becomes particularly susceptible to the vacancy at the C point as a perturbation. As expected, an out-of-plane motion is dominant for the spin precession of the 5 meV mode, in accordance with the fact that this mode is gapped by the easy-plane anisotropy D_1. However, we also found some finite in-plane precession components in its eigenvector. Notably, the in-plane precession of the six spins nearest to a specific site for the 5 meV magnon mode is almost identical to the effect of the spin canting of six spins due to the vacancy (see Supplementary Fig. S10). In other words, the 5 meV eigenmode will undergo significant energy (eigenvalue) variation due to the vacancy, indicating its sensitivity to the vacancy as a perturbation. Moreover, since the spin precession is synchronized over the triangular lattice due to a zero magnon wave-vector at the C point, the effects of this sensitivity will be amplified, leading to an ill-defined spectrum of the 5 meV mode seen in Fig. 4b. While further confirmation of whether such a minor portion of the spin precession can result in the significant change is required, these results imply that spin textures may play an important role in explaining the diluted noncollinear magnets’ magnetic excitations.

Although our INS data of YMn_1-xAl_xO₃ together with the model calculations have provided some valuable insight about the spin texture, we would like to note that such an approach would be somewhat close to the indirect examination. Therefore, further measurements to directly detect the spin texture in real space will be of great help to deeper understanding of it. For instance, small angle neutron scattering (SANS) may give further hidden information about its spatial correlation, which is the key feature of the spin texture.

In summary, our work presents a unique experimental study on the energy and momentum-resolved spin dynamics of a diluted frustrated magnet. It contributes to answering the critical fundamental problem of the nontrivial impurity effects in frustrated magnets. Furthermore, our results provide the first experimental confirmation of the impurity-induced spin textures, which have been long-advocated theoretically. We demonstrate that generating a spin texture can be easily achieved using frustrated noncollinear magnets. It may also be conceivable that this giant spin texture can be manipulated and so used as potential applications as done for skyrmion.

Methods

Magnon linewidth analysis

To estimate the Q-dependence of magnon energy linewidth, we performed magnon peak fittings at the Q points within the full Brillouin zone. We used a Lorentzian function to fit magnon peaks, while a Gaussian function fitted the incoherent quasi-elastic signal. Instrumental resolution effects were removed from the fitted HWHM, assuming that the following relation holds:

where HWHM_instrument was derived from the profile shown in Supplementary Fig. S4. Note that for the doped samples, HWHM_intrinsic is much larger than HWHM_instrument at most Q points ($$ {\mathrm{HWHM}}_{\mathrm{intrinsic}}\cong 10{\mathrm{HWHM}}_{\mathrm{instrument}}$$), which guarantees the validity of Eq. 2. As there is more than one magnon peak at a certain Q point, HWHM values of the magnon peaks were averaged at each Q point, the results of which are displayed in Figs. 3d–f.