Introduction

The advancements in the research of antiferromagnetic (AFM) spintronics have opened new avenues for spin-based devices1,2,3. Generally, collinear antiferromagnets are utilized as the elementary basis for spintronic functionalities4,5,6. However, recent examinations of intriguing and unexpected physical phenomena in noncollinear antiferromagnets have expanded the scope of potential materials for exploiting AFM spintronics2,7,8,9. In particular, a large anomalous Hall effect was observed in noncollinear antiferromagnets, despite the vanishingly small magnitude of magnetization7,10. The Hall effect originates from the Berry curvature associated with topologically nontrivial spin textures11,12,13. Although its experimental observation is challenging, recent magnetic imaging techniques such as single spin relaxometry and scanning thermal gradient microscopy can be used for imaging noncollinear AFM textures and domain structures14,15,16.

The ability to control and detect AFM memory states is imperative for AFM spintronics, and accordingly, magnetocrystalline anisotropy has been exploited as a generic foundation for manipulating AFM states17,18. As controlled anisotropy provides an exceptional opportunity for extensive spintronic applications17,19, anisotropic magnetoresistance (AMR) has been adopted to detect the various resistive states associated with crystal axes20,21,22,23. Nonetheless, in several cases, an intricate stacking geometry with additional reference layers is required to achieve unified spintronic functionality24,25.

Helimagnets exhibit a prototypical noncollinear spin structure in which the spin direction is spatially rotated in the plane, whereas the rotation axis is parallel to the propagation direction26. In principle, helimagnets are noncollinear antiferromagnets that correspond to the zero net moment inherent in rotating spins. Thus, helimagnets offer the same advantages as antiferromagnets, such as the absence of a stray field and the features of intrinsically fast spin dynamics27,28,29. Despite these merits, a crucial problem with noncollinear AFM spintronics is the establishment of controllable factors and mechanisms for the anisotropy of noncollinear spin configurations. Specifically, the lack of a comprehensive understanding of AMR affects its control and application in spintronic devices. In this research, we demonstrate that the helix-to-fan transition induces a sign reversal of the AMR in a helical antiferromagnet of EuCo2As2 (ECA). Moreover, we verified the electrically discernible dual magnetic phases producing the alterable sign of the AFM memory state through both experimental and theoretical methods. In addition, we identified diverse AFM memory states relevant to the development of the spin structure under magnetic fields, which facilitates AFM spintronics based on noncollinear antiferromagnets.

Materials and methods

Sample preparation

The ECA single crystals were grown following the flux method with Sn flux30. Eu (99.9%, Alfa Aesar), Co (99.5%, Alfa Aesar), As (99.999%, Sigma Aldrich), and Sn (99.995%, Alfa Aesar) were mixed at a 1.05:2:2:15 molar ratio of Eu:Co:As:Sn. The mixture was placed in an alumina crucible sealed in an air-evacuated quartz tube. The quartz tube was placed in a high-temperature furnace at 1050 °C for 20 h and thereafter gradually cooled to 600 °C at a rate of 3.75 °C/h. At 600 °C, the incorporation of the remaining Sn flux in the crystals was prevented by using a centrifuge composed of stainless steel, while the quartz tube was cooled to room temperature. Ultimately, we obtained crystals with typical dimensions of 1.5 × 1.5 × 0.1 mm3.

Scanning transmission electron microscopy (STEM) measurement

We prepared ECA samples with a cutting plane perpendicular to the a-axis utilizing a dual-beam focused ion beam system (Helios 650, FEI). Along the cutting plane, the images presented a well-discernible atomic structure. To minimize any damage to the sample, the acceleration voltage conditions were gradually reduced from 30 to 2 keV. In addition, dark-field images were obtained using STEM (JEM-ARM200F, JEOL Ltd, Japan) at 200 keV with a Cs-corrector (CESCOR, CEOS GmbH, Germany) and a cold field-emission gun. The size of the electron probe was 83 pm, and the range of the high-angle annular dark-field detector angle was varied from 90 to 370 mrad.

Magnetic and transport property measurements

The dependence of magnetization on temperature and magnetic field was determined with the magnetic fields along the a- and c-axes using a vibrating sample magnetometer module in a physical property measurement system (PPMS, Quantum Design, Inc.). The electric transport measurements were performed using the conventional four-probe method in the PPMS. The AMR was measured by rotating the magnetic field in the ac plane in a PPMS equipped with a single-axis rotator.

Theoretical calculations

The easy-plane anisotropic spin model can be expressed as

$$\begin{array}{l}{{{\mathcal{H}}}}/N = J_1 {\sum \limits_{i = 1}^5} {\vec S_i \cdot \vec S_{i + 1}} + J_2 {\sum \limits_{i = 1}^5} {\vec S_i \cdot \vec S_{i + 2} - g\mu _B\vec H} \cdot {\sum \limits_{i = 1}^5} {\vec S_i}\\ + K_\theta {\sum \limits_{i = 1}^5} {cos^2\theta _i-K_5S\frac{{\left( {1 + \sin ^4\theta } \right)}}{2} {\sum \limits_{i = 1}^5} {\cos 5\varphi _i} } \end{array}$$

where N denotes the number of Eu2+ moments in a single layer; J1 and J2 represent the AFM coupling strength between the Eu2+ moments of the two nearest layers and the next-nearest layers, respectively; S = 7/2 for Eu2+ ions; g = 2; and Kθ denotes the magnetocrystalline anisotropy constant31. We considered the helical spin structure of the ECA as commensurate (k = 0.8) for the convenience of calculation and thus included only five layers with the periodic boundary condition. The first and second terms indicate the competing exchange interactions for an AFM helical state. The third term designates the Zeeman energy, where the magnetic field \(\vec H\) is situated on the ac plane, forming an angle θ with the c-axis. The fourth term denotes the easy-plane magnetocrystalline anisotropy energy that favors the planar spin orientation. The fifth term reflects that we are working with a three-dimensional system, considering the ferromagnetic interactions within a given layer as a mean-field term. For the helical AFM state, where φ = \(\frac{4}{5}\pi\), i.e., J2 = 0.31J1, we estimated the following parameters by fitting the theoretical results of anisotropic magnetization to the experimental data: \(g\mu _BH_{{{\mathrm{m}}}}/J_1S = 1.18\), \(K_\theta = 0.35J_1S^2\), and \(K_5 = 0.022J_1S\), where Hm indicates the occurrence of the helix-to-fan transition. Using Hm = 4.7 T for the ECA at 2 K, J1S2 and Kθ can be evaluated as 5.45 × 106 and 1.96 × 106 J/m3, respectively.

Results

Structure and properties of helical ECA antiferromagnets

A two-dimensional (2D) layered ECA helimagnet forms a body-centered tetragonal structure (I4/mmm space group)30. The crystal exhibits a strong 2D nature that enables it to be mechanically exfoliated. By composition, it consists of two Co2As2 layers on opposite sides of a unit cell, separated by a magnetic Eu layer, as depicted in Fig. 1a32. As crystal quality is an essential aspect for investigating distinguished anisotropy, it was reviewed using the single-crystal X-ray diffraction technique. The analyses revealed the crystals as a single phase with high quality (refer to Supplementary S1). The magnetic moments of the Eu2+ ions (S = 7/2 and L = 0) were helically ordered, whereas those of the Co ions were paramagnetic and not relevant to the magnetic ordering30,33. The direction of the net moment in the Eu layer rotates in the ab plane and propagates along the c-axis (Fig. 1a). In general, the pitch of the helix was incommensurate with the lattice parameter, and thus, no two layers exhibited the same directions of the net moments34. In the ECA, a slightly incommensurate helimagnetic order with a propagation vector k = (0, 0, 0.79) has been observed via neutron diffraction32, which indicates that the AFM interaction between the adjacent layers is perturbed by an additional AFM interaction acting between the second-neighbor layers31. Accordingly, STEM was utilized to visualize the structural units of alternately arranged Eu and Co2As2 layers (Fig. 1b). The STEM image with lower magnification indicates that all the layers were regularly aligned. The lattice constants were obtained using fast Fourier transformation (FFT) from the STEM data with a = 0.402 nm and c = 1.150 nm, similar to the results of previous reports30,32.

Fig. 1: Structure and magnetic susceptibility of ECA.
figure 1

a Crystal and spin structures of ECA. Red, orange, and light purple spheres denote Eu, Co, and As atoms, respectively. The red arrow on each Eu atom indicates the individual spin direction. Competing exchange interactions form the helical AFM order along the c-axis with k = (0, 0, 0.79). b Dark-field STEM images in the bc plane. Inset: diffraction pattern obtained from FFT. c Temperature (T) dependence of magnetic susceptibility χ = M/H, measured upon warming at H = 0.1 T after zero-field cooling for the a- and c-axes. Vertical gray line specifies Néel T, TN = 46 K. d H–T phase diagram for H // a. Red dots distinguish the phase boundary between helix and fan phases, estimated by isothermal magnetization. Green dots indicate the Néel T at each H.

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The helical AFM order emerges at TN = 46 K, as evident from the temperature (T) dependence of the magnetic susceptibility defined by the magnetization (M) divided by the magnetic field (H), χ = M/H, measured at H = 0.1 T (Fig. 1c). The χ curves below TN for H along the a- and c-axes (Ha and Hc) indicate an anisotropic nature. The rapid decrease in χ for Ha below TN is consistent with the alignment of the magnetic moments of Eu ions along the ab plane, as previously verified via neutron diffraction and nuclear magnetic resonance experiments32,33. We observed metallic behavior with a distinct anomaly at TN, as shown by the variations in resistivity with T (refer to Supplementary Fig. S2). Therefore, an indirect Ruderman–Kittel–Kasuya–Yosida (RKKY) exchange interaction between the Eu2+ spins mediated by the spins of conduction electrons can be expected35. The phase diagram of the T and Ha dependences of the magnetic properties is illustrated in Fig. 1d, which clarifies the phase boundary between the helix and fan structures.

Electrically distinguishable dual magnetic phases and magnetoresistance anisotropy

In an antiferromagnet, an adequately strong H along a magnetic easy-axis often generates a magnetic phase transition through spin reorientation, such as a spin-flop or spin-flip transition30,36. Phase conversion occurs with marked anomalies in the physical properties and stabilizes the flopped or flipped state. We observed a similar magnetic phase transition in the helimagnetic ECA. At 2 K, Ma (M along the a-axis) abruptly increases at Hm = 4.7 T, thereby indicating a helix-to-fan transition30. The similarity of this transition to spin-flops is signified by the extrapolation of the linear slope above Hm that merges at the origin (Fig. 2a). Hm was determined from the peak in the derivative of Ma. As displayed in the inset of Fig. 2a, slight magnetic hysteresis is observed to arise from the first-order characteristic of this transition (refer to Supplementary Information S2). Across the phase transition, a fan structure emerges in which the net magnetic moments spatially oscillate along the propagation vector30,31,35. In contrast, Mc (M along the c-axis) at 2 K displays a linear increase associated with the gradual canting of the net moments (Fig. 2c). The slope of Ma is smaller than that of Mc below Hm and becomes larger in magnetic fields greater than Hm. Therefore, the value of Ma exceeds Mc at Hm.

Fig. 2: Magnetization and magnetoresistance anisotropies.
figure 2

a Isothermal magnetization along the a-axis (Ma) at T = 2 K. Experimental and calculated data are plotted as solid and dotted curves, respectively. The vertical gray line denotes the occurrence of the helix-to-fan transition. Hm = 4.7 T. Inset: magnified view of Ma to elucidate the magnetic hysteretic behavior. b Isothermal magnetoresistance, MR (%), along the a-axis (MRa) at T = 2 K. c Isothermal magnetization along the c-axis (Mc) at T = 2 K. Inset: schematics of canted helical-spin structures at Hc, in which the beginnings of arrows in layers indicate net magnetic moments placed together at a single point. d Isothermal magnetoresistance, MR (%), along the c-axis (MRc) at T = 2 K. e Estimated spin configurations of helix states formed below the phase transition (H < Hm). f Estimated spin configurations of fan states formed above the transition (H > Hm). The red arrow indicates the net magnetic moment in each Eu layer numbered from 1 to 5 along the c-axis.

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An easy-plane anisotropic spin model was adopted to investigate the evolution of the helix-to-fan phase. The model Hamiltonian comprises the competing exchange interactions, Zeeman energy, magnetocrystalline anisotropy, and mean-field terms (refer to Methods and Supplementary Information S2 for details). Additionally, the commensurate helical spin structure was considered for convenience of calculation. The relative angle of the two moments between the most proximate layers at zero H was \(\varphi = \frac{4}{5}\pi\), corresponding to j2 = 0.31J1, because the relative angle of the two spins for a helical order can be expressed as \(\varphi = \cos ^{ - 1}\left( { - \frac{{J_1}}{{4J_2}}} \right)\)31,34. As the planar spin rotation is explicitly disintegrated with the application of Ha, the order parameter of the helical state can be expressed as \(\mathop {\sum }\nolimits_{i = 1}^5 \cos 5\varphi _i\). Thus, the ferromagnetic interactions within a given layer can be treated by adding a mean-field term. The estimated Ma and Mc values obtained from the experimental data are represented as dotted curves in Fig. 2a, c. The well-matched fitting in the absence of a fourfold rotational magnetocrystalline anisotropy in the ab plane implies the formation of helical spins that are independent of the planar crystalline axes in the ECA. Moreover, the theoretical estimation directly generated the spin configurations pertaining to the helix and fan phases, as illustrated in Fig. 2e, f. In the helix phase, the orientations of the net moments continually turned in the H direction as Ha increased. Greater than Hm, spin reorientation occurred in such a manner that the moments acting farther from the H direction tended to be perpendicular. A further increase in Ha generated an additional canting of the magnetic moments.

The influence of anisotropic M on transport was examined based on the magnetoresistance, \(MR = \frac{{R\left( H \right) - R\left( 0 \right)}}{{R\left( 0 \right)}}\), for both the a- and c-axes (MRa and MRc). The increase in Ma with increasing H below Hm (Fig. 2a) indicates a partial and gradual alignment of the magnetic moments in the H direction. The average net magnetic moment along the H direction disrupts the equally distributed angles between the moments in the layers (Fig. 2e). The spin configuration away from the helix state increases the resistance, i.e., a positive MRa (Fig. 2b). With a further increase in H, MRa exhibits an abrupt reduction with the maximum slope at Hm and becomes negative. The spin-reorientation-driven switching from positive to negative MRa is consistent with the crossing behavior of Ma. In the high-H regime, MRa decreases faster with the additional canting of the moments toward the flipped state (Fig. 2f). In contrast, MRc decreases with increasing H (Fig. 2d). This intimate correlation between the M and MR plots suggests that the magnetic order governs the magnetotransport and its anisotropy. As T increases, Hm decreases, and the shape of the anomaly is broadened, as displayed in the anisotropic M and MR plots at various T values in Supplementary Fig. S6.

To theoretically probe the magnetotransport property, we propose that the interlayer hopping amplitude can be expressed by \(t_{i,i + 1} = t_0 + t_s\left|<{\hat n_i{{{\mathrm{|}}}}\hat n_{i + 1}>} \right|\) (i = 1–5) with the periodic boundary condition, where t0 and ts indicate the spin-independent and spin-dependent parts, respectively. Here, \(\left| {<\hat n_i{{{\mathrm{|}}}}\hat n_{i + 1}>} \right|\)denotes the overlap integral between the two spinors. Each spinor aligns with the direction of the net magnetic moment in each layer. The overlapping integral is given by \(\cos \frac{\gamma }{2}\), where γ denotes the relative angle between the two spinors. For the MR calculations, the conductance (σ) of the system was assumed to be proportional to the geometric mean of multiple hopping amplitudes through the layers as \(\sigma \propto \left( {\mathop {\prod }\nolimits_{i = 1}^5 t_{i,i + 1}} \right)^{\frac{1}{5}}\), where the geometric mean effectively presumes the average of the product \(\mathop {\prod }\nolimits_{i = 1}^5 t_{i,i + 1}.\) According to the definition of MR and R = 1/σ, MR is proportional to σ(σ) − σ(H). Upon setting t0/ts = 0.2, the highly anisotropic trend between MRa and MRc is moderately described by the purely spintronic consideration of σ (Fig. 3a, b). This is because of the L = 0 condition for the ground state of the Eu2+ ions34. The MR response to an applied Ha prominently distinguished two distinct magnetic phases. In the helix phase, the increased γ for certain adjacent layers at Ha below Hm contributes more to σ as it reduces, thereby inducing positive MRa. For small Ha, one can explicitly prove that σ decreases with Ha. In this regime, an interlayer hopping amplitude can be approximately expressed as \(t_{i,i + 1} \cong \bar te^{ - \alpha {\it{\epsilon }}_i/2 - \beta {\it{\epsilon }}_i^2/4}\), where \({\it{\epsilon }}_i = \varphi _{i + 1} - \varphi _i - \frac{{4\pi }}{5}\) represents the small deviation of the relative spin orientation φi+1−φi from the average value, \(\bar t = t_0 + t_scos\frac{{2\pi }}{5}\), \(\alpha = t_ssin\frac{{2\pi }}{5}/\bar t\), and \(\beta = t_s\left( {t_s + t_0cos\frac{{2\pi }}{5}} \right)/2\bar t^2\). As \(\mathop {\sum}\nolimits_{i = 1}^5 {{\it{\epsilon }}_i = 0}\),\({\upsigma} \propto \left( {\mathop {\prod }\nolimits_{i = 1}^5 t_{i,i + 1}} \right)^{\frac{1}{5}} = \bar te^{ - \beta \mathop {\sum}\nolimits_{i = 1}^5 {{\it{\epsilon }}_i^2/20} }\, < \, \bar t\). Therefore, σ reduces for small Ha. In the fan phase, \(\gamma\) continuously decreases with a further increase in Ha by approaching the completely spin-aligned state along the magnetic field direction, which enhances \(\sigma\) and diminishes MRa. Various AFM memory states relevant to the various spin configurations under an external Ha were theoretically identified, as depicted in Fig. 3a.

Fig. 3: Calculated magnetoresistance anisotropy.
figure 3

a Theoretically estimated isothermal magnetoresistance, MR, along the a-axis (MRa). Schematics of net magnetic moment configurations corresponding to multiple MRa states at various values of Ha. b Theoretically estimated isothermal magnetoresistance, MR, along the c-axis (MRc).

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Spin-reorientation-driven reversal of anisotropic magnetoresistance

A peculiar spintronic characteristic of a noncollinear AFM ECA is presented by angle-dependent magnetotransport. The results of the AMR, defined as \(\frac{{R\left( \theta \right) - R\left( 0 \right)}}{{R\left( 0 \right)}}\), are depicted in Fig. 4a, b. As indicated in the geometry of the AMR measurement in Fig. 4a, H is continually rotated perpendicular to the current, excluding the extrinsic Lorentzian MR effect. At H < Hm, ab-planar helix formation facilitates twofold rotational symmetry. The AMR is maximized at θ = 90° and 270° (Fig. 4a) owing to the positive value of MRa, which gradually increases with H (Fig. 2b). At H values slightly exceeding Hm, the AMR near θ = 90° and 270° starts to partially reverse, which is observed as a dip in Fig. 4a. A further increase in H results in a complete reversal of the AMR. The AMR contour map (Fig. 4b) demonstrates the sign-tunable AMR upon crossing Hm. The detailed influence of increasing T on the AMR is plotted in the contour maps of Supplementary Fig. S7. The overall trend of AMR development under the applied H was reproduced using theoretical calculations, as portrayed in Fig. 4c, d. The contrast emerging from the reversal behavior of the AMR effect reflects the intrinsic bulk properties and clarifies the different magnetotransport features between the helix and fan structures.

Fig. 4: Experimental and theoretical anisotropic magnetoresistance.
figure 4

a Plots of AMR (%) measured at T = 2 K by rotating H = 2, 4.4, 4.8, 5.2, 7, and 9 T in the ac plane with current along the b-axis, I // b. The geometry of AMR measurement is schematically presented. θ = 0° for the c-axis and θ = 90° for the a-axis. b Contour plots of AMR at T = 2 K. c Plots of calculated AMR data at H = 2, 4.4, 5.2, 6, 8, and 11 T. d Contour plots of AMR obtained from theoretical estimations.

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Discussion

In this study, we have demonstrated a mechanism to generate the sign-changing AMR phenomenon that originates from electrically distinctive dual magnetic phases separated by a magnetic phase transition. The magnetic transition emerges from the competition of diverse energy scales included in the model Hamiltonian. Specifically, the highly 2D characteristic of magnetocrystalline anisotropy is vital for determining the consistency between bulk measurements and theoretical calculations. In pursuance of the AMR effect predominantly caused by the magnetocrystalline anisotropy in AFM spintronics, the approach proposed in this study is applicable to a variety of antiferromagnets in which the distinct features of the anisotropic magnetotransport could be observed depending on the alteration of the magnetocrystalline anisotropy. The development of magnetic devices with the desired properties requires detectable macroscopic effects that are contingent on the variable magnetic states. In complex noncollinear antiferromagnets, more spintronic effects can exist. Therefore, the electrical access to various AFM memory states associated with the evolution of helical spin texture would provide an opportunity for realizing multilevel AFM memory devices.

Notably, the reversal phenomenon of AMR has seldom been reported. In a previous study on ferromagnetic La0.7Ca0.3MnO3 ultrathin films, a sign reversal of the AMR originated from the planar tensile strain that facilitated the rotation of the ferromagnetic easy axis, which is different from the present case37. The sign-tunable AMR effect in a single-phase noncollinear antiferromagnet is unique owing to its intrinsic origin from the electrically discernible magnetic phases. Additionally, a variety of topological states mediated by spin-orbit interactions yield exotic topological magnetism38,39,40. More recently, helical magnetism driven by Weyl-mediated RKKY interactions was observed in a Weyl semimetal, NdAlSi41. The recognition of versatile AFM memory states in a helimagnet offers valuable guidelines for investigating the intimate interplay between the electronic and magnetic topological properties and thus for implementing topological AFM spintronics.

In summary, we propose a new single-crystalline spintronic material in which the sign-tunable AMR effect reflects completely intrinsic bulk properties. These results are beneficial for the development of AFM spintronics, which has been driven by novel materials. Theoretically, the highly anisotropic 2D spin characteristic was highlighted as a core factor for anisotropic magnetotransport in a natural noncollinear AFM ECA. The scheme followed in this study is a particular type of spin-reorientation-driven mechanism to study the intriguing anisotropic properties, which can motivate further investigations of noncollinear AFM for extensive spintronic applications.