Introduction

Two-dimensional (2D) van der Waals (vdW) magnets are composed of elemental magnetic multilayers and provide a unique platform for studying the fundamental aspects of magnetism in terms of reduced dimensionality and a favorable foundation for attaining atomically thin spin-processing functionalities1,2,3,4,5,6,7,8,9. At finite temperatures, the long-range magnetic order is in principle prohibited in 2D isotropic magnets without the presence of magnetocrystalline anisotropy (MCA), which breaks the ideal Heisenberg symmetry10. The intrinsic MCA, induced by a sizable spin-orbit interaction, plays a crucial role in preserving the magnetic order and preventing random spin orientations driven by thermal fluctuations11,12,13,14. Furthermore, indirect exchange interactions between the 2D magnetic layers can be facilitated through the vdW gap15,16,17,18. Consequently, significant efforts have been dedicated toward enhancing and controlling anisotropy to improve the magnetization stability for various spintronic device applications, especially at higher temperatures14,19,20. Therefore, understanding the intimate connection between the magnetic order and MCA provides a fundamental basis for investigating diverse 2D magnetic characteristics.

The spin‒flop transition is a field-induced spin reorientation phenomenon and occasionally observed in antiferromagnets. Because this causes significant changes in magnetically anisotropic properties through a phase conversion, the characterization of the spin‒flop transition provides valuable insights into the impact of MCA on antiferromagnetism21,22,23,24. The MCA energies in vdW antiferromagnets have been theoretically estimated to be in the sub-meV range, which is comparable to the typical magnitude of antiferromagnetic exchange interactions25,26,27. However, investigating the spin‒flop transition in the presence of weak MCA energy has proven challenging mainly due to the lack of suitable candidate materials for these investigations. vdW antiferromagnets MPS3 (M: Fe, Ni, and Mn) exhibit antiferromagnetic order in the Néel (MnPS3) or zigzag types (FePS3 and NiPS3) configurations28,29. These spin arrangements result from intralayer antiferromagnetic couplings, typically characterized by strong exchange interactions that lead to high spin‒flop fields. For instance, the spin‒flop transition in FePS3 occurs above 35 T with pronounced magnetic hysteresis at T = 4 K30; thus, it is unsuitable for investigating the weak limit of MCA.

In this study, we focus on the vdW interlayer antiferromagnet CrPS4 to investigate the weak limit of thermally evolved MCA energy. The spin‒flop transition could be sustained up to T = 33 K, just below TN, and this was verified through the presence of a pronounced peak in the magnetic torque measurements. By adopting the easy-axis spin model, we could estimate the MCA energy, which was only 0.12% of the interlayer antiferromagnetic exchange interaction. Furthermore, we established various spin textures in the presence of rotating magnetic fields, and these spin textures were strongly correlated with the angular dependence of magnetic torque data. This research highlights the essential role of MCA in antiferromagnetism and extends our understanding to previously undetected limits. The results contribute to the expanding field of 2D magnetism and its potential applications in spintronics and other emerging technologies.

Materials and methods

Crystal growth

Plate-like single crystals of CrPS4 were synthesized using the chemical vapor transport method with iodine as the chemical agent31,32,33. Stoichiometric ratios of Cr, P, and S powders (1:1:4) were thoroughly mixed. The mixture was then placed at one end of a quartz tube, which was subsequently vacuum-sealed and loaded into a horizontal two-zone furnace. The hot zone of the furnace was maintained at 680 °C, while the cold zone was kept at 600 °C for one week. Then, the furnace was turned off and allowed to cool to room temperature. Shiny crystals a few millimeters in length formed in the cold zone of the quartz tube. Any residual iodine on the crystal surfaces was removed by washing with ethanol.

Scanning transmission electron microscopy (STEM)

The CrPS4 crystals used for the STEM measurements were carefully prepared based on a cutting plane perpendicular to the b axis using a dual-beam focused ion beam system (Helios 650, FEI). To ensure minimal specimen damage, the acceleration voltage conditions were progressively reduced from 30 to 2 keV. Dark-field images were acquired using a STEM (JEM-ARM200F, JEOL Ltd., Japan) operating at 200 keV with a Cs-corrector (CESCOR, CEOS GmbH, Germany) and a cold field emission gun. During scanning, an electron probe with a size of 83 pm was employed, and the high-angle annular dark-field detector angle was varied within the range of 90–370 mrad.

Magnetic property measurements

The magnetic susceptibility and isothermal magnetization measurements were performed using a vibrating sample magnetometer option in a physical property measurement system (PPMS, Quantum Design, Inc.). The measurements were conducted at temperatures in the range of T = 2–200 K with magnetic fields applied along the b and c axes of the crystals in the range of H = − 9-9 T. To measure the magnetic torque depending on the magnetic field and angular orientation, a calibrated cantilever chip (P109A, Quantum Design, Inc.) was attached to a single-axis rotator in the PPMS. The magnetic torque is measured under the condition of an exceptionally low noise level (1 × 10−9 N·m), with a Wheatstone bridge circuit to the cantilever chip.

Easy-axis anisotropic spin Hamiltonian

The spin Hamiltonian for the A-type antiferromagnetic order of CrPS4 is given by the following:

$${\mathcal{H}}{\mathscr{/}}N=J\mathop{\sum }\limits_{i=1}^{2}{\vec{S}}_{i}\,\cdot\, {\vec{S}}_{i+1}-g{\mu }_{B}\vec{H}\,\cdot\,\mathop{\sum }\limits_{i=1}^{2}{\vec{S}}_{i}+K\mathop{\sum }\limits_{i=1}^{2}{\sin }^{2}{\theta }_{i},$$

where \(N\) denotes the number of Cr3+ moments in a single layer, \({\vec{S}}_{i}\) is the Cr3+ moment in the i-th layer, \(g\) = 2, and \({\mu }_{B}\) is the Bohr magneton. The first term corresponds to the antiferromagnetic interactions, where J denotes the coupling strength (\(J\) > 0) between the Cr3+ moments in two neighboring layers. The system exhibits a pattern of alternating net magnetic moments along the c axis due to interlayer antiferromagnetic exchange interactions, as shown in Fig. 1a. Employing a periodic boundary condition with the summation of i up to 2 ensures sufficient accuracy, considering the identical nature of the 3rd and 1st moments. The second term denotes the Zeeman energy. The third term reflects the easy c-axis MCA, and here, \(K\) is the MCA energy. The angle \({\theta }_{i}\) is the polar angle of the Cr3+ moment in the i-th layer on the bc plane (\({\theta }_{i}=\) 0° for the c-axis and \({\theta }_{i}=\) 90° for the b-axis).

Fig. 1: Structure and magnetic properties of antiferromagnetic CrPS4.
figure 1

a Crystallographic structure of CrPS4, viewed from a axis. Cr3+ ions composed of CrS6 octahedra form quasi-1D chains along the b axis. The red arrow on each Cr3+ ion corresponds to the individual spin direction, displaying A-type antiferromagnetic order. The large red arrow adjacent to a layer represents net magnetic moments alternately repeating along the c axis. b Structure viewed from c axis. The green, purple, and yellow spheres represent the Cr3+, P5+, and S2- ions, respectively. The dotted rectangle denotes crystallographic unit cell. c Dark-field images collected by scanning transmission electron microscopy (STEM) in the ac plane. STEM image with lower magnification illustrates the regular alignment of all layers. d Temperature (T) dependence of the magnetic susceptibility for H//c (χc) and H//b (χb), measured at H = 0.1 T. The vertical dotted gray line indicates the Néel T, TN = 38 K. e Solid curve of the isothermal magnetization for H//c (Mc), measured at T = 2 K. Calculated Mc is plotted as the dotted curve. The red inverted triangle marks occurrence of spin-flop transition, Hflop = 0.9 T. The inset shows a magnified view of Mc around Hflop. The dotted gray line is extrapolated from the linear slope above Hflop. The schematics illustrate the angular variation of the Cr3+ magnetic moments in the consecutive layers upon increasing H along the c axis (Hc). f Solid curve of the isothermal magnetization for H//b (Mb), measured at T = 2 K. The calculated Mb is shown as the dotted curve. the schematics illustrate gradual canting of the Cr3+ moments upon increasing H along the b axis (Hb).

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Results

Anisotropic magnetic properties and theoretical fittings

CrPS4 exhibits a monoclinic crystal structure belonging to the C2 space group, with lattice constants a = 1.086 nm, b = 0.725 nm, c = 0.614 nm, and β = 91.88°31. Each chromium ion (Cr3+, S = 3/2) is coordinated by six sulfur ions (S2-), thus forming a slightly distorted CrS6 octahedron, as depicted in Fig. 1a. These octahedra align along the b axis, thereby creating quasi-1D chains. Phosphorus ions (P5+) connect these chains, forming a 2D rectangular lattice, as shown in Fig. 1b. Close-packed S2- forms separate layers, resulting in significant vdW gaps, which facilitate easy exfoliation. This further leads to the formation of thin flakes or even monolayers34,35. The structural units are also visualized by STEM, and the vdW gaps that separate these units along the cutting plane are clearly resolved (Fig. 1c). X-ray diffraction was performed, and the distinct 112.5° arrangement of the Cr atoms within the ab plane (Supplementary Figs. S1 and S2) confirmed the crystallographic axes of the CrPS4 crystals.

Recent studies employing powder neutron diffraction have confirmed that the magnetic Cr3+ ions in CrPS4 exhibit an A-type antiferromagnetic order12,31,36; this result is consistent with previous density functional theory calculations37. As shown in Fig. 1a, the net magnetic moments of the adjacent layers of Cr3+ ions are aligned in the opposite direction along the c axis, thus yielding a magnetic unit cell that is twice the size of the crystallographic unit cell. The Cr3+ spins in the spin‒flop state are known to lie in the bc plane, and this is associated with the presence of the quasi-1D chains of the CrS6 octahedra along the b axis (Fig. 1a)31. Therefore, the magnetic properties were characterized by applying magnetic fields along the c axis (Hc) and b axis (Hb).

The magnetic susceptibility (χ = M/H) was measured at H = 0.1 T after zero-field cooling for orientations parallel to the c axis (χc) and b axis (χb) (Fig. 1d). Anomalies in both χc and χb indicate the onset of an antiferromagnetic order at a transition temperature (TN) of 38 K. TN was additionally estimated through molecular-field theory (see Supplementary S2). The high rate of decrease in χc below TN is consistent with the A-type antiferromagnetic arrangement of Cr3+ moments.

Isothermal magnetization (M) measurements at T = 2 K reveal the magnetically anisotropic behavior of CrPS4, as displayed in Figs. 1e, f. The slight linear slope of Mc at low Hc values can be attributed to fractional flips of Cr3+ spins that were initially aligned in a direction opposite to that of Hc. A small but sudden increase at Hflop = 0.9 T indicates a spin‒flop transition, wherein the moments of Cr3+ ions are partially polarized, and this can be identified through the extrapolation of a linear slope above Hflop intersecting at the origin (inset of Fig. 1e)38,39. Furthermore, the increase in Hc induces additional canting of the flopped moments, which produces the saturated state at Hsat. ≈ 8.1 T. The schematics in Fig. 1e illustrate the evolution of the Cr3+ magnetic moments across the spin‒flop transition. In contrast, Mb exhibits a linear increase due to the gradual canting of Cr3+ magnetic moments toward the Hb direction, and the magnetic moment saturates at approximately 8.4 T (Fig. 1f).

Despite the relatively large isotropic spin state within the octahedral environment of the Cr3+ ions, the slight preference for the Cr3+ moment orientation along the c axis and the occurrence of a small spin‒flop transition at low Hc indicate the existence of a marginal MCA. To obtain further insights into the essential characteristics of the spin‒flop transition within the context of a weak MCA, we employed an easy-axis anisotropic spin model, as described in the Methods section40,41. Based on a systematic analysis of the total magnetic energy as Hc increases, we conducted a precise fitting of the theoretical model to the experimental Mc at 2 K. This enabled the determination of the ratio between the exchange coupling (\(J{S}^{2}\)) and Zeeman energy (\(g{\mu }_{B}{HS}\)), yielding a value of \(g{\mu }_{B}{H}_{{\rm{flop}}}S/J{S}^{2}=\) 0.43 and \(J{S}^{2}=\) 0.24 meV. Furthermore, we performed an additional fitting to the experimental Mb at 2 K, yielding a ratio between the MCA energy (\(K\)) and \(J{S}^{2}\), \(K/J{S}^{2}=\,\)2.26% with \(K=\) 0.0054 meV. Notably, this result differed from \(K/J{S}^{2}=\,\)27.27%, as estimated for another vdW A-type antiferromagnet, MnBi2Te4 (see Supplementary Fig. S3).

Probing spin‒flop transition by magnetic torques

The measurement of magnetic torque per unit volume, denoted as τ = M × H, serves as an ideal approach to investigate the anisotropic magnetic properties and the occurrence of spin‒flop transitions, as shown in Figs. 2a and 2b. The anisotropic signals in the H-dependent τ were observed at angles θ close to 0° and 90°, where θ represents the deviation of H from the c axis (θ = 0° corresponds to the c axis, and θ = 90° corresponds to the b axis) (see Supplementary Fig. S4). In contrast to the small step observed in Mc at Hflop, the τ at T = 2 K and θ = –2° exhibits a distinct and sharper peak at Hflop (Fig. 2a). With a further increase in H, τ gradually decreases and changes its sign at H1 = 4.2 T. The increase in the rotation of θ to θ = –4° enhances the peak height at Hflop. By changing the sign of θ that corresponds to the opposite angle between the net M and H, τ is fully reversed. Conversely, the τ at angles close to 90° demonstrates a minor and broad variation as H increases, followed by a change in the negative values and crossing zero at H2 = 5.0 T (Fig. 2b). The anisotropic characteristics of CrPS4 are evident in the behavior of τ despite the relatively small value of the MCA energy.

Fig. 2: Anisotropy of the magnetic torques at 2 K.
figure 2

a H dependence of the magnetic torque, τ, measured at θ = −4°, −2°, 2°, and 4° for T = 2 K. The red and gray inverted triangles indicate occurrence of spin-flops at Hflop and crossing of zero τ value at H1, respectively. b H dependence of τ, measured at θ = 85°, 89°, 91°, and 95° for T = 2 K. The gray inverted triangle designates crossing of zero τ value at H2. c Calculated H dependence of τ, at θ = −4°, −2°, 2°, and 4°. The schematics display angle variation of the magnetic moments upon increasing H at θ = 4°. d Calculated H dependence of τ, for θ = 85°, 89°, 91°, and 95°. The schematics illustrate the angle variation of the magnetic moments upon increasing H at θ = 85°. e Angular variation of the net magnetization as H is increased at θ = 4°. y axis is plotted on logarithmic scale. f Angular variation of the net magnetization as H is increased at θ = 85°.

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By employing a uniaxial anisotropic spin Hamiltonian and explicitly estimating the net Cr3+-moment orientation relative to H in each layer, we successfully reproduced the anisotropic H-dependent τ at angles θ close to 0° and 90°, as illustrated in Figs. 2c and 2d. For θ = 4° and H below Hflop, the net Cr3+ moment in the lower layer (S2 moment) is slightly canted due to the weak Hb component, while the net Cr3+ moment in the upper layer (S1 moment) tends to align predominantly along the Hc direction (Fig. 2c). This larger rotation of the S2 moment results in an angle of net M below 90°, and this yields negative values of τ (Fig. 2e). The relative angle between M and H gradually decreases up to Hflop, but τ continues to increase due to the high rate of increase in H. When H surpasses Hflop, the angle of net M rapidly decreases but is larger than θ = 4°, thereby sustaining negative values of τ. For H above H1, the strong Hc component predominantly aligns the two net moments of both layers closer to the Hc direction. Since the influence of the Hb component on the moments is not significant, the angle of net M is situated below θ = 4°, causing a sign reversal in τ. Under H for θ = 85°, the angle of net M becomes larger than θ due to the configuration of the S1 moment (Figs. 2d and 2f). This arrangement causes the generation of negative values for τ. At H greater than H2, the Hc component causes the S1 moment to be less tilted toward the Hb direction, generating a change in the sign of τ to positive values. The T development of anisotropic H-dependent τ is shown in Supplementary Fig. S5.

Weak limit of magnetocrystalline anisotropy

In the context of the anisotropic spin model, the magnetic energies of the spin‒flop state and A-type antiferromagnetic state in CrPS4 can be described as follows: \({E}_{{\rm{flop}}}=2J{S}^{2}\cos 2\theta -2g{\mu }_{B}{HS}\cos \theta +2K{\sin }^{2}\theta\) and \({E}_{{\rm{AFM}}}=-2J{S}^{2}\), respectively (see Supplementary Fig. S6). The optimal canting angle of the Cr3+ moments in the spin‒flop state, \(\cos {\theta }_{{\rm{flop}}}=g{\mu }_{B}{HS}/2\left(2J{S}^{2}-K\right)\), can be derived from \(\partial {E}_{{\rm{flop}}}/\partial \theta =0\). Consequently, the condition that \({E}_{{\rm{flop}}}\left({\theta }_{{\rm{flop}}}\right)={E}_{{\rm{AFM}}}\) at Hflop leads to \({H}_{{\rm{flop}}}=\frac{2}{g{\mu }_{B}S}{\left[K\left(2J{S}^{2}-K\right)\right]}^{1/2}\). This result indicates that the spin‒flop transition disappears as the MCA energy, \(K\), approaches zero. Therefore, the thermal evolution of the intrinsic \(K\) can be examined by observing the spin‒flop transition. This often results in peculiar variations in anisotropic magnetic characteristics through phase transformation.

In the plot of the thermally evolved magnetization curve (Mc) (Fig. 3a), the small step-like feature at Hflop and T = 2 K progressively shifts to a lower Hc and continuously decreases as T increases to T = 30 K. At T = 35 K, slightly below TN, Hflop is not observed, which is consistent with previous findings13,31,36. Additionally, Hsat. also moves to a lower Hc, and its shape becomes less pronounced as T increases. The nonzero slope after Hsat., which is more evident at higher T, are attributed to thermal fluctuations inhibiting the saturation of Mc. In Fig. 3b, the initial magnetization (Mb) monotonically increases without any magnetic transition and follows a similar behavior to that of Hsat. for Mc. The identical shapes of Mc and Mb at T = 35 K indicate a magnetically isotropic aspect as the MCA energy fades.

Fig. 3: Weak limit of magnetocrystalline anisotropy.
figure 3

a Measured isothermal Mc at T = 2, 10, 20, 30, 35 and 45 K. The Mc data are shifted vertically for clear visualization. The marked scale denotes 1 μB/f.u. The inverted triangles indicate occurrence of Hflop at each T. b Measured isothermal Mb at T = 2, 10, 20, 30, 35 and 45 K. c Measured (solid curves) and calculated (dotted curves) isothermal Mc in the low Hc regime at T = 30, 31, 32 and 33 K. d Hc derivative of the measured Mc at T = 30, 31, 32, and 33 K. The inverted triangles denote occurrence of spin-flop transitions, Hflop = 0.32, 0.28, 0.24 and 0.18 T for T = 30, 31, 32 and 33 K, respectively. e Measured (closed circles) and calculated (dotted curves) H dependence of the magnetic torque, τ, taken at θ = –4° for T = 30, 31, 32, 33 and 34 K. f Dark green and light green circles plotting the T dependence of the interlayer exchange coupling energy (JS2) and MCA energy (\({\rm{K}}\)) divided by JS2, respectively.

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To investigate the decrease in the MCA energy as T increases, the Mc in the low Hc regime was measured for T = 30–33 K (Fig. 3c). Fitting the spin model to the measured Mc data produces the theoretical Mc, represented by the dotted curves in Fig. 3c. The feature of Mc at Hflop, which is already weakened for T = 30 K, decreases further as T increases, and Mc exhibits nearly linear behavior at T = 33 K. This trend is also evident in the Hc derivative of Mc, shown in Fig. 3d; here, the peak indicating Hc is consistently reduced and broadens with a slight increase in T. The diminished spin‒flop transition is more pronounced in the measured and calculated H-dependent τ at θ = –4° (Fig. 3e). The broad peak, which results in the occurrence of spin‒flops, completely disappeared at T = 34 K.

Fitting the experimental data over the entire T range leads to an understanding of the impact of thermal softening on \(J{S}^{2}\) and the MCA energy. As TN is relatively low in CrPS4, \(J{S}^{2}\) decreases at a relatively slow rate; this shows a decrease of 15.4% at T = 33 K compared to that at T = 2 K (Fig. 3f). Conversely, the MCA energy decreases at a considerably higher rate, with \(K/J{S}^{2}\) remaining only at 0.12% at T = 33 K. For a higher T and reduced antiferromagnetic coupling strength, the Zeeman energy overcomes the coupling strength at lower Hc, and this contributes to the decline in Hflop. Additionally, as the MCA energy decreases, the tendency of magnetic moments to align along the easy c-axis significantly weakens. This leads to an increased angle between two magnetic moments in adjacent layers in the spin‒flop state and further diminishes the step-like feature at Hflop. The increased angle between magnetic moments also shows a reduction in the Zeeman energy that triggers the spin‒flop transition; this result is consistent with the shift of Hflop to lower Hc. Moreover, approaching the weak limit of MCA energy suppresses the emergence of the spin‒flop transition. The theoretical fitting parameters for each T are summarized in Supplementary Table S1. The calculated angles of the Cr3+ spins and the Mc data for T = 2 K are also listed in Supplementary Table S2.

Visualizing spin states directly connected to torque data in rotating magnetic fields

In particular, the τ magnetometer has significantly advanced and has enabled accurate measurements of angular-dependent magnetic characteristics. The occurrence of magnetic τ stems from the relative angular disparity between M and H, which arises from specific axial anisotropy associated with the MCA. This confirms the suitability of the method for examining the intricate evolution of anisotropic magnetic properties across the spin‒flop transition in CrPS4. We conducted angle-dependent measurements of τ with various H values at T = 2 K and constructed the 3D contour plots in Fig. 4a; the results reveal a distinct characteristic of τ exhibiting a gradual reversal above Hflop. Furthermore, we performed precise calculations of the minimum total magnetic energy with respect to the uniaxial anisotropic spin Hamiltonian while systematically rotating a specific H. This further enable the determination of the value of τ corresponding to the magnetic moment direction at a given angle. The resulting trend of the τ variations closely resembles the experimental outcomes, as shown in Fig. 4b. This reversal behavior observed in angle-dependent τ highlights the significant role of the MCA, despite its relatively small magnitude.

Fig. 4: Measured and calculated angle dependence of magnetic torques at 2 K.
figure 4

a 3D contour plot of the angular dependent τ data measured at different H’s for T = 2 K. b 3D contour plot established from the angular dependent τ data calculated by its fitting to the experimental data. c Calculated angular dependent τ in presence of rotating H = 0.5 T in the bc plane. Schematics describe the configurations of the net magnetic moments in the upper (S1) and lower (S2) layers with respect to the H directions. The inset shows the directions of τ at θ = 45° and 135°, where a positive τ value at θ = 45° corresponds to +a axis, and a negative τ value at θ = 135° corresponds to –a axis. d–f Calculated angular dependent τ and corresponding net-moment orientations at H = 2, 4.8, and 9 T, respectively.

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Our extensive calculations based on the anisotropic spin model provide a visual representation of the precise orientations of the net Cr3+ moments during the rotation of H. These orientations directly correspond to the magnetic torque (τ) data, with theoretically obtained configurations of net magnetic moments in the upper (S1) and lower (S2) layers shown in Fig. 4c–f. The total magnetization, denoted as M = S1 + S2, resides in the bc plane with a relative angle to H, producing τ = M × H. At a weak H = 0.5 T, both the S1 and S2 moments exhibit different movements mainly due to the relative orientation of H, thus yielding a small periodic angle dependence of τ (Fig. 4c). For θ below 90°, while the S2 moment turns in the direction of the Hb component to some extent, the significant AFM exchange interaction causes the S1 moment to rotate slightly toward a negative angle. Consequently, the resulting value of τ is positive mainly because the configuration of the S1 moment is close to the direction of H. The sign of τ becomes negative for θ > 90° because the configuration of the S2 moment is adjacent to the direction of H. At H = 2 T, the spin‒flop state at θ = 0°, where the S1 and S2 moments are symmetrically arranged around H, transforms into another tilted configuration that incorporates the S1 moment close to the direction of H as θ increases, as shown in Fig. 4d. The impact of the Hc component on the S1 moment creates a positive value of τ for θ < 90°. Although the relative angle between the net moments and H becomes slightly smaller, a larger H generates a more pronounced τ signal with positive values. As θ crosses 90°, the τ state transforms from positive to negative because the alignment of the S1 moment moves farther away from the direction of H. With a rotation of H = 4.8 T (Fig. 4e), we observe a partially reversed behavior in τ near θ = 0 and 180°. The slight rotation of H causes the S2 moment to move closer to the direction of H, which aligns with the negative τ value observed at θ = 22.5°. With further rotation of H, the τ value switches to positive, corresponding to the orientation of the S1 moment close to H (the net-moment configuration is shown for θ = 67.5°). Similar variations in the sign of τ are repeated, and the sign changes occur twice as frequently with a complete rotation of H. At H = 9 T, the net moments become fully saturated and move collectively during the rotation of H (Fig. 4f). However, the orientations of the moments relative to H are still affected by their preference to align along the magnetic easy c-axis, leading to a complete reversal of the overall angle dependence of τ.

Discussion

The presence of an axial or planar MCA in an antiferromagnet significantly influences the angular-dependent magnetic τ and is suitable for observing the impact of anisotropy on the magnetic properties in vdW magnets. However, such an investigation has yet to be conducted for CrPS4. A prior study focused on the T dependence of isothermal magnetic τ on CrPS431. The analysis in the previous work employed an ordinary conventional phenomenological approach based on the angle derivative of the magnetic free energy, allowing fitting only for the low H quadratic portion of the isothermal τ below the spin‒flop field. Therefore, the interpretations were limited by the inherent difficulty in identifying the strength of the MCA and the configurations of relevant spin states. Moreover, validating magnetic τ variations through magnetic phase transitions remains limited.

In this comprehensive study, we performed a detailed investigation on the unexplored weak limit of the MCA in the vdW interlayer antiferromagnet CrPS4. Based on a combination of magnetic torque measurements and theoretical analysis, we demonstrated the occurrence of a spin‒flop transition even at an extremely small anisotropy, providing an essential result for a clear understanding of the magnetic stability at lower dimensionality. This finding is in distinctive contrast to previous research, which primarily focused on increasing the MCA in the vdW magnets to enhance the stability of the magnetic order parameter at higher temperatures or to achieve robust magnetic order down to a monolayer42,43,44,45.

To explore the intricate spin states in CrPS4, we devised an experimental approach based on rotating magnetic fields, and we complemented this approach with theoretical insights derived from the uniaxial anisotropy spin model. This methodology enables accessing and comprehending the spin directions in each magnetic layer. Notably, conventional experimental tools, such as neutron diffraction and resonant X-ray scattering, have technical challenges in detecting the continuously changing spin states due to limited field geometries. Our advanced approach overcomes this obstacle and provides a constructive means to study the magnetic properties of vdW interlayer antiferromagnets46,47,48.

The versatility of our approach is highlighted by the ability to flexibly modify the anisotropic spin Hamiltonian; thus, our approach is widely applicable for investigating other types of 2D antiferromagnetic materials. By understanding the subtle interactions between the MCA and magnetic order in these materials, we provide the groundwork for further advancements in the field of 2D magnetism. The findings of this study have great potential for the development of spintronic devices and the exploration of novel phenomena in the realm of low-dimensional magnetism.