Probing Short-Range Correlations in the van der Waals Magnet CrSBr by Small-Angle Neutron Scattering

The layered metamagnet CrSBr offers a rich interplay between magnetic, optical and electrical properties that can be extended down to the two-dimensional (2D) limit. Despite the extensive research regarding the long-range magnetic order in magnetic van der Waals materials, short-range correlations have been loosely investigated. By using Small-Angle Neutron Scattering (SANS) we show the formation of short-range magnetic regions in CrSBr with correlation lengths that increase upon cooling up to ca. 3 nm at the antiferromagnetic ordering temperature (TN ~ 140 K). Interestingly, these ferromagnetic correlations start developing below 200 K, i.e., well above TN. Below TN, these correlations rapidly decrease and are negligible at low-temperatures. The experimental results are well-reproduced by an effective spin Hamiltonian, which pinpoints that the short-range correlations in CrSBr are intrinsic to the monolayer limit, and discard the appearance of any frustrated phase in CrSBr at low-temperatures within our experimental window between 2 and 200 nm. Overall, our results are compatible with a spin freezing scenario of the magnetic fluctuations in CrSBr and highlight SANS as a powerful technique for characterizing the rich physical phenomenology beyond the long-range order paradigm offered by van der Waals magnets.


Introduction
[13] .The structural features of these layered vdW materials -which typically shows strong exchange interactions within the layers but very weak interlayer interactions-provide a unique situation in which the emergence of long-range magnetic order is strongly dependent on the spin dimensionality. [14] this regard, the role of short-range correlations coupling electronic and magnetic degrees of freedom is fundamental for understanding the properties of these materials.For instance, shortrange correlations have been related to an enhancement of the thermoelectric properties, [15] the appearance of quantum phase transitions [16] and even to the origin of high-temperature superconductivity. [17]However, most of the experimental efforts regarding van der Waals magnets have focused on the long-range magnetic ordered phase, being the quantification of the short-range correlations relegated to a secondary place, likely due to the lack of proper experimental techniques able to its quantification. [14][23][24][25][26] This material is formed by ferromagnetic layers that couple antiferromagnetically (Figure 1.a) undergoing a long-range magnetic ordering at TN ~ 140 K. [21] In addition, upon the application of moderate magnetic fields, it is possible to reorient the spin of the layers (at 10 K: 0.6 T, 1 T and 2 T for fields applied along the b, a and c axes, respectively). [12][30][31] However, due to the emergence of long-range order at lower temperatures, it was not possible to quantify these short-range correlations.Here, we employ Small-Angle Neutron Scattering (SANS) in order to determine the correlation length of the short-range fluctuations, as well as its temperature dependence.[38][39] Our results show the appearance of short-range correlations below ca.200 K, characterized by correlations lengths up to ca. 3 nm at TN, and highlight SANS as a powerful technique for characterizing vdW magnets.

Results and discussion
Crystals of CrSBr are grown by a solid-state reaction (see Methods).CrSBr crystallizes in an orthorhombic space group, characterized by α = β = γ = 90°, a = 3.512 Å, b = 4.762 Å and c = 7.962 Å, [12] being c related to the distance between the layers (Figure 1   the SANS signal by subtracting the structural component (in our case, the spectra at 300 K in the high-temperature paramagnetic phase). [40]The complete Q-range is presented in the Supplementary Figure 2.b.While lowering the temperature, the SANS magnetic contribution is already detectable at 200 K, i.e., 60 K above TN, exhibiting its maximum at TN and starting to decrease upon further cooling down (Figure 3.a).An example of the SANS spectrum is shown in Figure 3.b.We determine the correlation length, ξ, based on an Ornstein-Zernike analysis, following a well stablished phenomenological method in SANS as previously reported [41][42][43] (see Methods).From the fit (Figure 3.b), ξ is estimated to be in the order of 3 nm at 140 K.
By performing similar fittings at different temperatures (Supplementary Section 2), the thermal dependence of ξ and the intensity scaling (related to the volume fraction of the correlated regions, assuming that the net magnetic moment is constant) is determined (Figure Below 200 K, IOZ(0) increases and reaches its maximum at TN. Below TN, IOZ(0) decreases rapidly (by a factor of 3) in the 80 K -140 K range and, finally, is suppressed below 40 K (thus, a short-range frustrated magnetic state in CrSBr, as it could be speculated based on the magnetic exchange interactions, [23] is not likely to occur at very low-temperatures).Overall, the observed magnetic behavior is in agreement with a spin-freezing scenario occurring in CrSBr, as previously suggested by magnetization and muon experiments, [23] where only long-range interactions become relevant at low-temperatures and, therefore, are not detected in our SANS signal.In contrast with previous observations, where a slowing down of the magnetic fluctuations was observed below ca. 100 K, we observe a rapid suppression of these short-range correlations just below TN, becoming negligible below ca.40 K. Summarizing, we observe correlated regions with a net magnetic moment well above the ordering temperature, that increase while approaching the ordering temperature and are characterized by a correlation length in the order of 3 nm at TN.By cooling below TN, these magnetic fluctuations decrease, as observed by the diminishment of the intensity scaling, IOZ(0).As a secondary fingerprint of the short-range correlations, we consider the role of applying an external magnetic field.In Figure 4.a, we show the magnetic contribution to the SANS signal at TN upon the application of different external magnetic fields, observing a suppression of the signal as the field is increased.By performing the same analysis as discussed above, we determine ξ and intensity scaling (Figure 4.b), which diminish as the applied field is increased.A similar field dependence is observed at 150 K (Supplementary Section 2).On the contrary, no field dependence in the region between 0.02-0.3Å -1 is observed at 10 K (Supplementary Section 2), in agreement with a scenario with only long-range order at low temperatures and, therefore, not sensitive by SANS.For modelling the short-range correlations in bulk CrSBr, we employ the following effective spin Hamiltonian:

3.c). ξ (blue dots in
The first term describes the bilinear exchange interactions, where Jij is an isotropic exchange parameter up to seven neighbor's order in the layers [44] (extracted from inelastic neutron scattering experiments) and up to second neighbor's order between the layers (taken from first-principles calculations). [45]The second term takes into account the antisymmetric anisotropic Dzyaloshinskii-Moriya interaction (DMI) that are allowed for some bonds of the CrsBr based on its symmetry (following the first-principles calculations). [45]The third term reflects the triaxial on-site anisotropy with the values adopted from microwave absorption spectroscopy measurements with Ai being a diagonal matrix . [46]The last term accounts for magnetic dipole-dipole interactions.For every temperature and magnetic field, we calculate the spin-spin correlation function from the result of the Monte-Carlo sampling as implemented in the VAMPIRE computational package. [47]The calculated correlation function for bulk exhibits an exponential decay (Supplementary Section 3) and is separated into the interlayer and intralayer contribution due to the relatively small exchange between the layers of CrSBr (in the range of 10 -3 meV) as compared to the intralayer ones (in the range of 1 meV), indicating that the 2D intralayer short-range correlations are the main ones in this material. [44]The intralayer term dominates the exponential decay at each temperature while the interlayer contribution remains close to zero for all temperatures and exhibits little or none temperature and field dependence, thus pointing towards the appearance of long-range interlayer antiferromagnetic

Conclusion
In this work we have probed the role of short-range correlations in the van der Waals magnet CrSBr by Small-Angle Neutron Scattering experiments.Our experimental results quantify the short-range magnetic fluctuations in CrSBr, which are characterized by a correlation length in the order of 3 nm at the ordering temperature, TN, and confirm the antiferromagnetic ordering below 140 K as well as the absence of frustrated magnetic states at low-temperatures.These correlations exhibit an interesting thermal dependence since they are already present below 200 K -that is, well above TN -, they exhibit a maximum at TN and, then, decrease rapidly while cooling down, being absent at low-temperatures.In accordance, the application of an external field suppresses these fluctuations.In addition, these experimental observations are well reproduced by a theoretical model based on an effective spin Hamiltonian, highlighting that the appearance of short-range correlations are intrinsic to the monolayer limit.
Overall, our results are in accordance with a spin-freezing scenario in CrSBr, where the magnetic fluctuations cease while cooling down, and highlight SANS as an optimal technique for characterizing the rich physical phenomenology occurring in van der Waals magnets beyond the conventional long-range order picture.

Methods
Crystal growth: Crystals of CrSBr are grown by solid-state techniques, as previously reported by some of us. [12]The crystal structure is verified by powder and single-crystal X-ray diffraction together with the elemental composition by energy-dispersive X-ray spectroscopy (EDS).

SANS measurements:
SANS experiments are performed at the Larmor instrument (ISIS neutron and muon source, United Kingdom) on 115 mg of CrSBr powder sample.The coherence length of the neutron beam is tens of micrometers, but at least higher than 57 nm, as experimentally probed. [42]The incident neutron beam scatters in the sample in a transmission geometry, being the diffuse magnetic scattering recorded in a 2D detector (sample-detector distance is 4 m, with wavelengths from 0.9 to 13 Å).The signal is integrated over each azimuthal angle around the center of diffraction, with an experimental detection in the 0.004-0.7 Å −1 Q range, being Q the wavevector.SANS spectra are recorded as a function of temperature (10 K -300 K) and magnetic field (0 T -2 T), being the magnetic field applied perpendicular to the incident neutron beam.No significant thermal dependence of the SANS pattern is expected unless some structural or magnetic inhomogeneities in the range between 2-200 nm -that is, within our experimental sensitivity-start developing. [41]The neutron data reduction (including conversion of the Time of Flight and corrections for accounting the background scattering and transmission) is performed using the Mantid software. [48]Data fits are done with SasView application (http://www.sasview.org/,using dQ Data instrumental smearing).The data is fitted considering two different approaches.In the first approach (Supplementary Section 2.a), we fit the spectra at 300 K (high-temperature paramagnetic phase) to a power law, being , where IP is a Porod scale term (a particularly common dependence which main contributions are the divergence of the incident neutron beam as well as the Porod scattering from interfaces such as powder grain surfaces and grain boundaries, applicable to our powder measurement), [35] n is an exponent and B is a Q-independent background constant.Then, we employ Computational details: Spin-spin correlation function is calculated from the result of the Monte-Carlo sampling as implemented in the VAMPIRE computational package. [47]The computation is performed considering a 10x10x10 (nm) sample with open boundary conditions.
The lateral size of the simulated sample is chosen in a balance of covering the measured range of the correlation length and the computational cost of the simulation.10 5 Monte-Carlo steps are computed for each temperature and field with first 3•10 4 reserved for the thermalization.
From the microscopic spin distribution, the spin-spin correlation function (  ,   ) = 〈    〉 − 〈  〉〈  〉 is computed for each temperature, where 〈. . .〉 denotes the canonical ensemble average.Next, the distance-dependent function (  −   ) is computed by averaging over the pairs with the same distances.Finally, the correlation length is computed assuming the exponential decay of the spin-spin correlation function over distance.In particular, we consider the exponential fit A•e -r/ξ for the intralayer part of the correlation function, where A is the amplitude, r the distance and ξ the correlation length.In order to account for the surface effects only the pairs with the distance smaller than 5nm are considered for the fit.For constructing the spin Hamiltonian, the intralayer isotropic exchange parameters were extracted from inelastic neutron scattering experiments. [44]The interlayer isotropic and intralayer DMI parameters are taken from first-principles calculations. [45]The on-site anisotropy values are adopted from the results of the microwave absorption spectroscopy measurements. [46]We remark that the notation of the spin Hamiltonians differs between sources; thus, the values are converted during the construction of the Hamiltonian (see Supplementary Table 1 for details).After the construction of the Hamiltonian all parameters are scaled by a factor 1.5 in order to match the experimental value of TN.In the calculations, the magnetic alignment of the layers does not follow the sequence expected for a pure A-type antiferromagnet.Still, the total balance of the spin-up and spin-down layers is preserved (thus out of 13 layers calculated in Figure 5.c, 6 are spin-up and 7 are spin-down).This result is not unexpected, as the interlayer exchange parameters are ca.1000 times smaller than the intralayer ones.The intralayer correlation length saturates at the value ~2 nm below the phase transition, which is comparable with the result of the measurements.However, it is less stable in the vicinity of the phase transition.This suggests that the description of the equilibrium properties near the phase transition with the Monte-Carlo simulations may require considerably more iteration steps, which is not feasible from the computational point of view for the bulk. [49]For the monolayer case, we start the Monte-Carlo sampling from 20 different random initial spin configurations and estimate the mean value and standard deviation of the correlation length for each temperature.

ToC figure
The correlation length of the layered magnetic semiconductor CrSBr is measured by Small-Angle Neutron Scattering.The ferromagnetic correlations start developing below 200 K, i.e., well above TN (140 K), with a length of ca. 3 nm at TN.The experimental results are wellreproduced by an effective spin Hamiltonian, which pinpoints that the short-range correlations are intrinsic to the monolayer limit.

Data analysis.
The data is fitted considering two different approaches.In the first approach (Supplementary Section 2.a), we fit the spectra at 300 K (high-temperature paramagnetic phase) to a power law, being I T=300K (Q) = I P,300K Q 4−n,300K + B 300K , where IP is a Porod scale term, n is an exponent and B is a Q-independent background constant.Then, we employ the relationship I T≠300K (Q) = .a).SANS experiments are performed on a CrSBr powder sample in a transmission geometry (see Methods).As sketched in Figure 1.b, the incident neutron beam scatters in the sample and the diffuse magnetic scattering is recorded in a 2D detector.All 2D neutron data are shown in the Supplementary Section 1.Then, the SANS pattern is obtained by integrating over each azimuthal angle around the center of diffraction, with an experimental detection in the 0.004-0.7 Å −1 Q range, being Q the wavevector.Magnetic fields are applied perpendicular to the incident neutron beam.

Figure 1 .
Figure 1.CrSBr crystal structure and Small-Angle Neutron Scattering (SANS) experimental configuration.a) Crystal structure of the layered vdW magnet CrSBr.The chromium, sulphur and bromine atoms are represented as cyan, yellow and pink balls, respectively.In the ordered state, the spins within each single layer couple ferromagneticallyspins represented as red arrows-pointing along the b-axis.The ferromagnetic layers couple antiferromagnetically among them along the c-axis.b) SANS experiment schematics in a

Figure 2 .
Figure 2. Antiferromagnetic order in CrSBr probed by Small-Angle Neutron Scattering (SANS).a) SANS pattern at selected temperatures (the whole dataset is presented in the Supplementary Figure 1.a).At low temperatures, a Bragg peak is observed at Q = 0.407 Å -1 ~ 2c as a consequence of the antiferromagnetic order.b) Thermal dependence of the SANS intensity at Q = 0.407 Å -1 , where an enhancement of the signal is observed below the Néel temperature (TN ~ 140 K).c) SANS spectra at T = 10 K.The peak at Q = 0.407 Å -1 vanishes at high applied magnetics fields due to the spin-reorientation of the layers.

Figure 3 .
c) increases below 225 K up to a maximum around TN. Below TN, ξ diminishes and, at temperatures lower than 40 K, the magnetic signal in the studied region decreases very quickly due to the reduction in the volume fraction of the correlated region.This fact is better observed in the intensity scale (red squares in Figure 3.c).At high temperatures, IOZ(0) is almost zero, indicating that the magnetic contribution to the SANS signal is negligible.

Figure 3 .
Figure 3. Thermal dependence of the short-range correlations in CrSBr probed by SANS.a) Thermal dependence of the magnetic contribution of the SANS signal after removing the structural component (IT = 300 K). b) SANS signal at T = 140 K fitted following an Ornstein-Zernike law, yielding to an estimate of the volume fraction, IOZ, and correlation length, ξ (see text for details).c) Thermal dependence of the volume fraction and correlation length.The Néel temperature is marked as a grey dashed line.The complete set of fittings is presented in the Supplementary Section 2.

Figure 4 .
Figure 4.-Field dependence of the short-range correlations in CrSBr at T = 140 K. a) Magnetic contribution of the SANS signal at different applied magnetic fields after removing the structural component (IT = 300 K). b) Field dependence of the volume fraction and correlation length.The complete set of fittings is presented in the Supplementary Section 2.
interactions only below TN.By fitting the correlation function (see Methods), we obtain the computed correlation length as a function of temperature (Figure 5.a) and magnetic field (Figure 5.b), reproducing well our experimental observations.To further illustrate the key role of the intralayer interactions, we compute as well the thermal dependency of the correlation length for a monolayer (blue squares in Figure 5.a).Compared to the bulk case (red dots in

Figure 5 .Figure 5 .
Figure 5.a), the bulk thermal dependency of the correlation length mimics the behavior of the monolayer, reflecting the major role of the intralayer interactions and highlighting that the short-range correlations in CrSBr arise from the monolayer.
the relationship I T≠300K (Q) = I OZ (0) 1+(ξQ) 2 + I P,300K Q 4−n,300K + B , where IOZ(0) is the Ornstein-Zernike intensity scaling and ξ is the correlation length.In the second approach (Supplementary Section 2.b), we consider I(Q) − I 300K (We note that both approaches are compatible between them, although the second one yields to a correlation length with larger error bars at low temperatures, that arise from an overparametrized fitting due to the absence of correlations within our experimental window range resolution.Despite being ξ constant below TN, the absence of correlations is accounted by the suppression of IOZ(0).Fits shown in the main text are obtained with the first approach.All fitted data following both approaches are shown in the Supplementary Section 2.
n,300K + B, where IOZ(0) is the Ornstein-Zernike intensity scaling and ξ is the correlation length.In the second approach (Supplementary Section 2.b), we consider I(Q) − I 300K (We note that both approaches are compatible between them, although the second one yields to a correlation length with larger error bars at low temperatures, that arise from an overparametrized fitting due to the absence of correlations within our experimental window range resolution.Despite being ξ constant below TN, the absence of correlations is accounted by the suppression of IOZ(0).Fits shown in the main text are obtained with the first approach.a. I(T) analysis.Supplementary Figure 4.-Fitted SANS signal at different temperatures (see Methods).