Magnetic dynamics in NiTiO3 honeycomb antiferromagnet using neutron scattering

TL;DR

This study investigates the magnetic structure and excitations of NiTiO3, revealing anisotropic Heisenberg behavior, weak frustration, and temperature-dependent magnetic dynamics using neutron scattering.

Contribution

It provides detailed insights into the magnetic interactions and excitations in NiTiO3, highlighting its anisotropic Heisenberg characteristics and weak frustration.

Findings

Dispersive acoustic and nearly dispersionless optical bands at ~3.7 meV observed.

Anisotropy gap of 0.95 meV identified.

Critical exponent indicates behavior between Heisenberg and 2D Ising models.

Abstract

The ilmenite NiTiO3 consists of a buckled honeycomb lattice, with the Ni spins aligned ferromagnetically in-plane and antiferromagnetically out-of-plane. Using neutron spectroscopy, the magnetic structure and the dynamics were investigated as a function of temperature. Dispersive acoustic bands and nearly dispersionless optical bands at ~3.7 meV are described by a highly anisotropy Heisenberg model with stronger antiferromagnetic (AFM) out-of-plane, weaker ferromagnetic (FM) in-plane interactions and an anisotropy gap of 0.95 meV. The order parameter yields a critical exponent between the Heisenberg and two-dimensional Ising models, consistent with highly anisotropic Heisenberg systems. The frustration parameter ~ 2 supports a weakly frustrated system.

Tables

Table 1. Table 1: Exchange interaction constants and anisotropy value used in the Heisenberg spin Hamiltonian for NiTiO 3 , with the corresponding distances between Ni atoms (d) corresponding to the Js depicted in Fig. 5(a) and the number of bonds corresponding to the each exchange interaction (n) for an atom.

	$J$1	$J$2	$J$3	$J$4	$J$5	$J$6	$D$
strength (meV)	-0.12	0.24	-0.08	0.25	-0.07	0.23	0.12
d(Å)	2.94	4.12	5.03	5.43	5.83	5.92	-
n	3	1	6	6	3	3	-

Table 2. Table S1: Exchange interactions for different the J models and corresponding R values.

	1J	2Js	3Js	4Js	5Js	6Js
J1	-0.55	-0.50	-0.50	-0.28	-0.17	-0.12
J2	–	0.16	0.31	0.35	0.35	0.24
J3	–	–	-0.15	-0.13	-0.10	-0.08
J4	–	–	–	0.25	0.25	0.25
J5	–	–	–	–	-0.13	-0.07
J6	–	–	–	–	–	0.23
D	0.20	0.20	0.21	0.20	0.20	0.12
R	131.4	121.8	116.8	35.6	33.0	13.3

Equations

$$H=\sum _{⟨i,j⟩}J\left(S_i^x{S}_j^x+S_i^y{S}_j^y+S_i^z{S}_j^z\right)+D\sum _i{\left(S_i^z\right)}^2$$

$$R=\sqrt{\frac{1}{n_E}\cdot \sum _{i,q}\frac{1}{{\sigma }_{i,q}^2}{\left(E_{i,q}^{\text{sim}}-E_{i,q}^{\text{meas}}\right)}^2}$$

Full text

I Introduction

Interest in low-dimensional magnetic materials (LDMs) has been growing in part due to their widespread application in magnetic memory devices and magnetic sensorsSoumyanarayanan et al. (2016); Miao et al. (2018); Fert (2008); Wolf et al. (2001); Felser et al. (2007), and also because they are prone to phenomena such as topological and, frustrated magnetism, and multiferroic behaviorsTokura et al. (2019); Šmejkal et al. (2018); Bishop et al. (2012); Hu and Kan (2019); Dey et al. (2021). Specifically, the halides CrX3 (X = Cl, Br, I), sulfides APS3 (A = Mn, Fe, Co, Ni) and ilmenites ATiO3 consist of layered honeycomb magnetic structures where their reduced dimensionality can lead to Dzyaloshinskii–Moriya interactions (DMI), quantum hall effect, spin Nernst effect, and Dirac magnons to mention a few  Schneeloch et al. (2022); Chen et al. (2023); Ederer and Fennie (2008); Niu et al. (2017); Haldane (1988); Go et al. (2024); McClarty (2022); Yuan et al. (2020a). For instance, gapped and gapless Dirac magnons were elucidated in CrI3, CrCl3 and CoTiO3 using inelastic neutron scattering Schneeloch et al. (2024, 2022); Rathnayaka et al. (2024).

In particular, the ilmenite CoTiO3 exhibits strong spin-orbit coupling that yields an effective spin S = $\frac{1}{2}$, XXZ-type magnetic interactions and Dirac-type magnon dispersions. On the other hand, isostructural systems such as MnTiO3 and FeTiO3 exhibit three-dimensional (3D) magnetic behaviors with relatively strong interplane coupling, unlike in CoTiO3, that exhibits magnetic behavior with weaker interplane interactions Goodenough and Stickler (1967); Osmond (1964); Shirane et al. (1959); Kato et al. (1983); Hwang et al. (2021); Yuan et al. (2020a, b, 2024); Elliot et al. (2021); Dey et al. (2021); Rathnayaka et al. (2024); Newnham et al. (1964); Yamaguchi et al. (1986); Charilaou et al. (2012). Included in the ilmenite family is NiTiO3 that shares a similar magnetic structure as in CoTiO3. However, NiTiO3 exhibits very different magnetic dynamics, absent of Dirac magnon nodes and topological properties. Earlier studies on the dielectric properties showed a frequency response that is significant at high temperatures Acharya and Choudhary (2015); Bamzai et al. (2011); Zhang et al. (2015) but little is known of the magnetic properties of NiTiO3 and is the focus of the present work.

The NiTiO3 crystal structure is rhombohedral with the R$\overline{3}$ symmetry and the unit cell shown in Fig. 1(a) Goodenough and Stickler (1967); Osmond (1964). The unit cell consists of alternating layers of corner-sharing NiO6 and TiO6 polyhedra in a trigonal prismatic coordination. The layers form a honeycomb-like structure within the $ab$-basal plane, as shown in Fig. 1(b). NiTiO3 orders antiferromagnetically with a Neel transition temperature (TN), 23 K, with spin $S$=1, and an effective orbital angular momentum in the ground state that is quenched, unlike in some other ilmenites Goodenough and Stickler (1967). The spin structure of the system is A-type, as illustrated in the inset of Fig. 2(b). The spins are aligned along the $ab$-plane, with in-plane FM interactions and out-of-plane AFM interactions, similar to the spin structure observed in CoTiO3  Goodenough and Stickler (1967). The propagation wave vector, k=(0, 0, 3/2) and the reported magnetic moment per Ni atom is 1.46(1) $\mu_{B}$ Dey et al. (2021, 2020); Xin et al. (2014). The in-plane FM interactions follow the Goodenough-Kanamori rule for Ni-O-Ni superexchange pathways, where the Ni-O-Ni bond angle is approximately 90∘ Goodenough (1955); Goodenough and Stickler (1967); Kanamori (1959). In contrast, the out-of-plane AFM interactions follow a Ni-O-O-Ni interaction path, which is weaker compared to the in-plane interactions Goodenough and Stickler (1967); Dey et al. (2021).

Previous studies on NiTiO3 reported field-dependent thermal expansion and magnetostrictive effects, measured via dilatometry and neutron diffraction Dey et al. (2021); Harada et al. (2016); Dey et al. (2020). These measurements show an anomaly at TN mostly associated with magneto-elastic coupling. Magnetic and phonon entropies obtained from heat capacity and dielectric data indicated the presence of spin-phonon interactions.

In this study we performed elastic and inelastic neutron scattering measurements on powder and single crystals. NiTiO3 has a magnon spectrum with a maximum energy cutoff of approximately 3.7 meV. Magnons show strong damping with increasing temperature while the inelastic intensity persist up to twice TN due to short-range magnetic fluctuations persisting above the magnetic transition. The magnetic order parameter was used to calculate the critical exponent yielding a value of $\beta$ = 0.213(2), which indicates an anisotropic Heisenberg behavior. Spin-wave calculations yielded a Heisenberg-like Hamiltonian with six exchange interactions (Js). From the single-crystal inelastic neutron scattering, two magnon bands were discerned: a quasi-flat optic-like band and a dispersive acoustic-like band. An anisotropy gap of about 0.95 meV is observed at the Brillouin zone (BZ) center, which can be explained by single-ion anisotropy. The calculated frustration parameter of $\simeq$ 2 indicates that the system is weakly frustrated.

II Experiment and simulations

The NiTiO3 sample was prepared using solid-state reaction method. NiO and TiO2 powders were thoroughly mixed in a 1:1 molar ratio and reacted at 1350*∘C in a tube furnace under argon atmosphere for 48 hours. Powder samples obtained from the solid-state reaction were used as the starting material for the single-crystal growth. The powder was ground using a mechanical mortar and pestle for 24 hours to achieve fine consistency. The powder was then pressed into long, thin rods (approximately 15 cm in length and 1 cm in diameter) using a hydrostatic press. The rods were subsequently annealed at 1350∘*C for 12 hours in a tube furnace. Large single crystals were grown in the floating zone furnace. The resulting crystal, measuring 10 cm in length and 8 mm in diameter, was cut into two parts. The quality of the single crystal was confirmed by Laue diffraction, reflective light polarization, and single crystal X-ray diffraction. Magnetic susceptibility measurements were performed using a Physical Property Measurement System (PPMS).

The powder neutron scattering experiments were carried out using the VISION spectrometer at the Spallation Neutron Source (SNS) and the HB-2A powder diffractometer at the High Flux Isotope Reactor (HFIR) of Oak Ridge National Laboratory. A 10 g of powder sample was used for the VISION experiment, where elastic and inelastic data were acquired from 5 to 300 K, with a fixed final neutron energy of 3.5 meV. Data were collected over a finite momentum ($Q$) window: detector positions correspond to the low-$Q$ (LQ) path at 45*∘* and the high-$Q$ (HQ) path at 135*∘. For the HB-2A measurements, a 10 g powder sample was loaded into a vanadium can, and experiments were performed at 5, 200, 295, 400, 500, and 700 K using a Ge(115) monochromator and a neutron beam of 1.54 Å wavelength. To ensure consistent temperature control within the HB-2A chamber, the sample environment was maintained under vacuum above 295 K and with helium exchange gas below 295 K. Refinement of the diffraction data confirmed the presence of a single NiTiO3* phase at 5 K.

Single crystal neutron scattering measurements were performed using the Cold Neutron Triple Axis Spectrometer (CTAX) at HFIR. The first crystal weighing 8 g was aligned along the (HHL) plane, while a second crystal weighing 5 g, was aligned along the (H0L) plane. For the measurements, the final neutron energy was fixed at 4.8 meV and the energy resolution at the elastic line is 0.25 meV. A cooled Be filter was placed after the sample to reduce higher-order neutron contaminations. The magnon dispersions were simulated using SpinW and SU(N)NY  Toth and Lake (2015); git .

III Results and Discussion

The neutron powder diffraction measurements carried out at HB-2A did not show a structural transition in the 5 - 700 K temperature range. The HB-2A instrument can access $Q$ values as low as 0.45 Å*-1* with the selected wavelength for these measurements. Fig. 2(a) is a plot of analyzed neutron scattering data at 5 K using the Rietveld refinement. Analysis was performed using FullProf and SARAh-Representational Analysis to determine the nuclear and magnetic structure Rodriguez-Carvajal (1990); Wills (2000). The refinement of room-temperature data confirms the high quality of the sample, which consists solely of the NiTiO3 phase with an R$\overline{3}$ space group and no detectable impurity phases. In the diffraction pattern, nuclear peaks are marked by green ticks, while magnetic peaks are labeled. The main (0 0 3/2) magnetic Bragg peak appears at 0.6725 Å*-1* (2$\theta$ = 9.05*∘). The analysis confirmed that the magnetic k-vector for this system is (0, 0, 3/2). First, the calculation was performed allowing the magnetic moment to have an in-plane component ($M_{X}$). Refinement using both the in-plane ($M_{X}$) and out-of-plane ($M_{Z}$) components resulted in a better fit to certain magnetic peaks compared to refinement using only the in-plane component and the results of the two refinements are shown in Supplementary Fig. S1 sup . The results shows a possible small out-of-plane magnetic moment, with a resultant in-plane magnetic moment per Ni atom of $M$ = 1.245(1) $\mu$B. The magnetic space group is PS*$\overline{1}$ (#2.7), consistent with previous reports  Dey et al. (2020). The resulting magnetic structure is shown in the inset of Fig. 2(a). This space group allows moments both in the plane and out of the plane. The direction of the spin in the ab-plane was chosen to be along the $a$–axis, however, the exact direction can not be determined due to lack of directional information in the powder data.

Fig. 2(b) is a plot of the lattice constants $a$ and $c$ obtained from Rietveld refinement, based on the elastic neutron scattering data from VISION from 5 to 300 K. AFM magnetic Bragg peaks appear below the magnetic transition, $T$N $\approx$ 23 K. Above 70 K, $a$ and $c$ gradually increase with temperature. Below that, the $c$-lattice constant shows minimal change until 23 K, after which it drops. A negative thermal expansion (NTE) below 50 K has been previously reported for the $a$ lattice constant; however, our data does not support this observation. The drop at 23 K of the lattice parameters supports the magneto-elastic coupling previously demonstrated through dilatometry and heat capacity measurements Dey et al. (2020); Harada et al. (2016).

The magnon density of states (DOS) along the LQ and HQ trajectories at 5 K are compared to the calculated phonon DOS at 0 K (Figs. 3(a) and (b)). The neutron data contains both magnon and phonon contributions where the LQ trajectory has a higher magnetic contribution, whereas the HQ trajectory has less magnon and more phonon contribution due to the magnetic form factor of Ni2+ and the fact that the HQ path reaches higher Q transfers. This distinction helps in identifying the phonon contribution in the data. A comparison of the DOS in Figs. 3(a) and (b) indicates that the phonon contribution is negligible below 10 meV. Note also that the prominent peak that is seen at  12 meV is an instrument artifact due to $\lambda$/2 higher-order reflections from the analyzer.

Shown in Figs. 3(c) and (d) are plots of the temperature dependence of the $\chi^{\prime\prime}$(Q,$\omega$) along the LQ and HQ trajectories from 5 to 300 K. Unlike what was previously observed in CoTiO3, only a single sharp peak is present with a maximum magnon energy of about 4 meV. A small peak can be seen at 5 meV in the low-Q data which most likely is an instrument artifact previously observed in other measurements on VISION Tao et al. (2023). The temperature changes of the magnon intensity are significant between 5 K and TN as seen in Figs. 3(c) and (d). The integrated area under the central peak and its center of mass (COM) are plotted as a function of temperature and shown in Fig. 4(a). It can be seen that, the area under the peak decreases sharply with TN and shifts down in energy due to softening. Above TN, the intensity persists up to 50 K, well above TN. The persistence of inelastic intensity above TN supports the existence of short-range magnetic correlations at temperatures as high as 2TN. Similar behavior has been reported in other AFM honeycomb systems such as CrCl3 and CoTiO3, where strong in-plane FM correlations above TN are responsible for short range ordering McGuire et al. (2017); Rathnayaka et al. (2024). Furthermore, the sharp change in the COM with increasing temperature reflects the sharp softening of the magnon energy as the system warms up. Thermal fluctuations and spin disorder contribute to the magnons damping.

A comparison of the integrated intensity of the (0, 0, 3) nuclear Bragg peak with the (0, 0, 3/2) magnetic peak is shown in Fig. 4(b). The data were obtained from the single crystal neutron scattering measurement. The nuclear peak intensity remains unchanged within the temperature range shown while the magnetic peak intensity decreases, as expected, and eventually vanishing at 22.5(1) K, which corresponds to $T$N. The behavior of the magnetic peak can be described by Landau’s theory of phase transition using, $I$=A($T$N-$T$)2β, with $\beta$ as the critical exponent Landau et al. (1980). The critical exponent $\beta$, is found to be 0.213(2) from fitting the intensity. This value falls between a 3D Heisenberg model ($\beta$ = 0.368) and a 2D Ising model ($\beta$ = 0.125) Landau et al. (1980); Gebara and Hasiak (2021); Ahlberg (2012) and is close to reported values for MnTiO3, MnPS3 and CrI3, which are layered and highly anisotropic Heisenberg systemsHwang et al. (2021); Wildes et al. (2006); Liu and Petrovic (2018).

In Figs. 5(a) and (b), the $\chi^{\prime\prime}$ is compared data at 5 K with the calculated spectrum at 0 K along the HQ and LQ paths of the VISION instrument. The calculated HQ and LQ $\chi^{\prime\prime}(Q,\omega)$ were obtained from modeling the magnon dispersions using linear spin wave theory and the powder-averaged to yield the magnon DOS. The interaction pairs used for the exchange model are shown in Fig. 5(c). The Hamiltonian for the linear spin wave calculation is:

[TABLE]

where J is the exchange interaction constant, and $D$ is the single-ion anisotropy (SIA) constant. For the calculation, we used $S=1$ and k = (0, 0, 3/2), along with the magnetic form factor of Ni2+. Starting with nearest-neighbor in-plane and out-of-plane interactions, six exchange constants ($J_{1}$–$J_{6}$) were included in the spin Hamiltonian. $J_{1}$, $J_{3}$, and $J_{5}$ correspond to in-plane nearest-neighbor FM interactions, while $J_{2}$, $J_{4}$, and $J_{6}$ represent out-of-plane nearest-neighbor AFM interactions. The weighted residual factor was calculated by introducing different numbers of J’s into the system, as detailed in the Supplementary section II sup . The six J values listed in Table I yielded the lowest weighted residual factor, and the corresponding calculation is shown in supplementary figure S3 (f).

Fig. 5(b) shows the powder-averaged magnon dispersion calculated at 0 K. The DOS was obtained by integrating the intensities along the LQ and HQ paths, as shown in the figure. The simulation accounted for the energy resolution of the instrument, with the $Q$ resolution set to 0.2 Å*-1*. The model reproduces key features of the experimental data as shown in Figs. 5(c) and (d), including the maximum in $\chi^{\prime\prime}(Q,\omega)$ and the energy dependence of the DOS. However, the experimental data exhibit a broader distribution than the simulation, likely due to several factors. At low temperatures, dynamical interactions between spin waves, such as magnon-magnon interactions, are not included in linear spin wave theory, potentially leading to additional broadening of the magnon peaks Dyson (1956). Additionally, the calculations assume 0 K, while the experimental data were collected at 5 K. This small temperature difference could allow for thermal spin fluctuations, further contributing to peak broadening. This effect may be particularly relevant given that $J_{1}$ is extremely small ($\sim 1$ K) and $J_{2}$ is on the order of 3 K.

Magnon dispersions along several high-symmetry directions were collected on CTAX using the single crystal (Fig.6). Well-defined magnetic excitations are evident in the data, attributed to the good quality of the sample. Generally, honeycomb AFM systems exhibit four magnetic dispersion bands Liu and Finkel’Stein (2022); Choe et al. (2024); Huang et al. (2022); Brehm et al. (2024). The four bands correspond to four magnon modes associated with the two magnetic sublattices from the two magnetic atoms, with each sublattice contributing to in-phase and out-of-phase modes. The data presented in Fig. 6 exhibits a quasi-flat optic-like band at approximately 3.7 meV and a low-energy acoustic-like dispersive band below 3.7 meV. The bands are most likely degenerate and are not resolved within the resolution of CTAX. Fig. 6(a) shows a dispersion along the direction perpendicular to the layers, indicative of a stronger inter-layer coupling which is consistent with the values of J2. The plots in Figs. 6(d)-(k) illustrate that at the high-symmetry K point, the lower dispersive band touches the high-energy quasi flat band.

The spin wave calculations were performed using the J parameters listed in Table I, and the calculated dispersions are directly compared with the experimental data, as shown in Fig. 6. The 0.95 meV gap observed at the (0, 0, 3) BZ center in Figs. 6(a) and 6(i) is the SIA along the z-axis, with the anisotropy value estimated to be approximately 0.12 meV. In the out-of-plane directions, as shown in Figs. 6(a) and 6(b), the calculations show only two bands, while along other directions, three bands are visible in the calculation. Without the anisotropy, only the flat band and a single low-energy dispersive band is present in the calculation; however, the presence of the anisotropy lifts the degeneracy of the lower dispersive band, resulting in two distinct bands, while the flat band remains unaffected. The band arising from the anisotropy is only present in the in-plane measurements because the intensity of the band resulted from anisotropy decreases near the (0, 0, 3/2) magnetic Bragg point and is not observed in out-of-plane measurements. The lifted degeneracy from the anisotropy cannot be seen in the data due to the limited resolution of the measurement.

Quasi flat magnon bands are characterized by highly localized spin waves resulting from strongly correlated spins, often exhibiting zero group velocity. Similar bands observed in layered kagome and triangular lattices where the spins have geometric induced frustration associated with DMI and SIA  Chisnell et al. (2015); Riberolles et al. (2024); Seifert and Savary (2022). The bands in the NiTiO3 should not be due to DMI because the collinear spin structure does not support DMI. The honeycomb lattice is not generally considered frustrated, but frustration can arise due to competing nearest-neighbor and next-nearest-neighbor magnetic interactions among hexagonally arranged magnetic ions Regnault and Rossat-Mignod (1990); Smirnova et al. (2009); Seibel et al. (2013). The frustration parameter ($f=|\Theta_{CW}|/T_{N}$) is an indicator of the degree of frustration in a system. Here the $\Theta_{CW}$ is the Curie-Weiss temperature(= -43.48 $\pm$ 0.32 K) of the NiTiO3 sample and the corresponding calculations are shown in Supplementary section III sup . For an ideal, non-frustrated system, $f$ would be close to 1. Several honeycomb systems with small $f$ values, such as BaNi2X2O8 (X = V, P, As) and CrPSe3 exhibit weak frustration Klyushina et al. (2017); Baithi et al. (2023). For NiTiO3, the calculated $f\approx 2$, indicates weak frustration and this frustration could arise from competition between two channels of exchange coupling such as from the AFM interlayer and FM intra-layer coupling. Although there is no frustration for the in-plane FM interaction, long-range 2-dimensional FM order is not allowed due to Mermin-Wagner theorem Chern . Therefore, the magnetic phase transition has to be a 3-dimensional order, and the establishment of a 3D order through inter-layer coupling is weakly frustrated, which could lead to the $f\approx 2$.

A recent paper on NiTiO3 provided with exchange interaction constants closer to our model Kikuchi et al. (2025). However, in their model, nearest neighbor inter-plane interaction (J2) was absent which is surprising given that this is a layered AFM system. Also, Ref. Kikuchi et al. (2025) claims that Dirac magnon band crossing is observed without confirming the topological nature of the crossing. It is equally likely that the bands are approaching but not actually touching or crossing. Even if they cross, it does not mean that the crossing is topological. In our data we cannot confirm the presence of Dirac-like magnon band crossing. The dispersion is quite different from CoTiO3 where Dirac magnon crossing points were suggested. In the present case, accidental band touching at high symmetry K points is possible and does not imply Dirac crossing.

IV Conclusion

In conclusion, from neutron scattering measurements on single crystal and powder samples, observed a highly anisotropic spin state in NiTiO3. Unlike other ATiO3 ilmenites, the exchange interactions in NiTiO3 indicate relatively stronger inter-plane interactions compared to in-plane interactions ($J$1,$J$3,$J$5 $<$ $J$2,$J$4,$J$6), consistent with previous models of strong $J$AFM and weak $J$FM in Refs. Goodenough and Stickler (1967); Dey et al. (2020). However, among the ATiO3 family, NiTiO3 exhibits the weakest overall exchange interactions. The spin Hamiltonian is Heisenberg in nature, with SIA along the z-axis. It is possible that the frustration arises from competing nearest-neighbor FM interactions within the honeycomb layer. The weak in-plane Js, comparable in magnitude to the SIA of the system, that results in a complex spin arrangement and contributes to the observed frustration. In contrast, other ATiO3 ilmenites do not exhibit such frustration, likely due to the anisotropies presents in the systems are not comparable with exchange interactions as in NiTiO3.

NiTiO3 presents a case where competing interactions prevent the system from settling into a simple unfrustrated FM ground state because in-plane J1 is comparable to out-of-plane J2. Thus it is possible that J1 nearest neighbor is competing with J2, next nearest neighbor. If J2 is strong, then it can disrupt the FM order by promoting configurations that are AFM. In this system, the SIA is very weak and allows the spins to align in-plane. This is because the energy gain from the exchange interaction dominates over the penalty associated with aligning against the anisotropy.

V Acknowledgements

The authors would like to acknowledge discussions with Yang Yang and Gia-wei Chern. They also thank Armando di Biase for assistance with the measurements. The work at the University of Virginia is supported by the Department of Energy, Grant number DE-FG02-01ER45927. This research used resources at the High Flux Isotope Reactor and Spallation Neutron Source, both DOE Office of Science User Facilities operated by the Oak Ridge National Laboratory. The beam time was allocated to VISION on proposal number IPTS-31947.1, HB-2A on proposal number IPTS-31669.1, CTAX on proposal number IPTS-32124.1.

VI Determination of the Magnetic Moments

In this section, refinement was performed on the HB-2A data at 5 K to investigate the presence of a possible out-of-plane magnetic moment in the system. Figure S1(a) shows the refinement using only an in-plane magnetic moment ($M_{X}$), while Fig. S1(b) presents the refinement including both in-plane ($M_{X}$) and out-of-plane ($M_{Z}$) components. The refinement incorporating both components yields a slightly better fit compared to the one using only $M_{X}$. In particular, the magnetic peak at 36.5∘ shows a noticeable improvement in fitting when both $M_{X}$ and $M_{Z}$ are included.

VII Fitting of the linear spin wave model to the data.

In this section, calculations were performed to assess the quality of the different J models with different J combinations. Fig. S2 shows the fitting of the Q-cut data using Gaussian fits, and Fig. S3 presents the extracted peak centers along with their corresponding experimental errors and the calculated dispersions from the linear spin wave theory. The goodness of fit (R-value) calculations were performed using fitspec method in the SpinW with the following equation:

[TABLE]

Here, $E^{\text{meas}}_{i,q}$ represents the energy obtained from the peak fitting of the experimental data, $E^{\text{sim}}_{i,q}$ represents the energy obtained from the calculation, and ${\sigma_{i}^{2}}$ is the standard deviation obtained from the Gaussian fit of the experimental Q-cuts. ${i,q}$ indexing the spin wave mode and momentum respectively.

In this analysis, the experimental data were fitted using linear spin wave theory, starting with a single J-value and progressively adding up to six J-values. Using the model and the data, R-values were calculated for each model. The corresponding R-values were calculated and are presented in Table I. The lower R-value, corresponding to the model with six J-values, confirms the accuracy and reliability of this model.

VIII Determination of the Curie–Weiss Temperature

Magnetic susceptibility measurements were performed on the single-crystal NTO sample using a PPMS in the temperature range of 5–300 K. Fig.S4(a) shows the magnetization per Ni atom, calculated from the susceptibility data with temperature. The data exhibit AFM behavior in both FC warming and cooling conditions. Fig.S4(b) displays the inverse magnetization (1/M) from the FC cooling data. The data were fitted using the Curie–Weiss law, M = C/(T-$\theta$CW), where C is the Curie constant. The fit yields a $\theta_{CW}$ value of -43.48$\pm$0.32 K , indicating dominant antiferromagnetic interactions in the system.