Multistep magnetization switching in orthogonally twisted ferromagnetic monolayers

The advent of twist engineering in two-dimensional crystals enables the design of van der Waals heterostructures with emergent properties. In the case of magnets, this approach can afford artificial antiferromagnets with tailored spin arrangements. Here we fabricate an orthogonally twisted bilayer by twisting two CrSBr ferromagnetic monolayers with an easy-axis in-plane spin anisotropy by 90°. The magnetotransport properties reveal multistep magnetization switching with a magnetic hysteresis opening, which is absent in the pristine case. By tuning the magnetic field, we modulate the remanent state and coercivity and select between hysteretic and non-hysteretic magnetoresistance scenarios. This complexity pinpoints spin anisotropy as a key aspect in twisted magnetic superlattices. Our results highlight control over the magnetic properties in van der Waals heterostructures, leading to a variety of field-induced phenomena and opening a fruitful playground for creating desired magnetic symmetries and manipulating non-collinear magnetic configurations.

Metamagnets and their field-induced phase transitions offer a plethora of counterintuitive phenomenology, as already quoted by Kramers, 1 with a direct competition between magnetic anisotropy, exchange, and dipolar energies. 2In absence of magnetic field, these materials show zero net magnetization that suddenly increases until its saturation -thus, resembling a ferromagnet-above a certain magnetic field threshold. 1A good example of an A-type metamagnet is offered by the layered vdW semiconductor CrSBr.The spins in every single layer (ab plane) couple ferromagnetically between them (TC ~150 K), pointing along the easy b axis, whereas the layers couple between them antiferromagnetically (TN ~ 140 K). 3 By applying a magnetic field, it is possible to flip the layers' magnetization in a parallel fashion via a spin reversal and to induce a spin reorientation along the magnetic field direction.][6][7][8][9][10][11][12] In bulk, the saturation fields at 2 K are 0.6 T, 1 T and 2 T for the easy (b), intermediate (a) and hard (c) magnetic axis, respectively. 10This vdW material can be thinned down to the monolayer limit and integrated into electronic nano-devices.3,14 The monolayer limit is characterized by the absence of MR for fields applied along the easy axis and small and positive MR for fields applied along the intermediate and hard axis. 10,13he ability for isolating, manipulating and twisting 2D crystals adds a new degree of control in vdW heterostructures, affording emergent new properties, like superconductivity in twisted bilayer graphene. 15As far as the 2D magnetic materials are concerned, twisting is much less explored.Still, it has allowed the creation of new magnetic ground states.0][21][22] However, no 2D twisted-magnets have been incorporated into electronic devices so far, remaining the magneto-transport effects in twisted-magnets fully unexplored.
Here, we twist by ca.90 degrees two CrSBr ferromagnetic monolayers, thus forming an orthogonally-twisted bilayer.In analogy with the artificial antiferromagnets reported in synthetic spintronics -where the magnetic properties are tailored by growing multilayers of different antiferromagnets, in contrast with crystalline bulk antiferromagnets-, 23 this twisted heterostructure can be envisaged as an artificial antiferromagnetic bilayer.5][26][27] Note that, in stark contrast with CrI3, where the spins are out-of-plane, in CrSBr the spins are in-plane pointing along the easy magnetic b-axis, with an intermediate a-axis -also in-plane-but with a hard magnetic c-axis -out-of-plane direction-.This orthogonal configuration yields to an intriguing spin scenario where several terms might compete with an applied magnetic field as the Zeeman split energy, the inter-layer magnetic interactions (which favors an antiparallel orientation between the layers) and the local spin anisotropy at each CrSBr layer (which are perpendicular at the twisted configuration).This case is different from the common Moiré patterns in twisted bilayers, where a modification of the band structure is reached by twisting small angles. 15 example of an orthogonally-twisted-CrSBr heterostructure is shown in Fig. 1a-b.In this vertical geometry, the MR can be rationalized within a spin-valve picture, considering a twocurrent channel model: when the magnetization of both layers is antiparallel (parallel), there is a higher (lower) resistance across the heterostructure. 10,28,29The field-dependence of the MR at 10 K is presented in Fig. 1c for in-plane magnetic fields aligned along the easy-axis of one of the layers (in this case, the top layer; α = β = 0º).Starting at high negative fields (red curve in Fig. 1c), the MR is negative and field independent down to -1 T; then, it increases until a maximum positive MR is observed at ca. +0.16 T. Above this field, it decreases again until reaching a saturation value above +1 T. This value coincides with that observed for the spin reorientation along the intermediate magnetic axis, a, thus suggesting that this is determined by the spin anisotropy.Reversing the magnetic field yields to a symmetrical curve that exhibits the maximum in MR at ca. -0.16 T (blue curve in Fig. 1c).These two curves cross at zero field (ZF) showing a hysteretic behavior when the field modulus is kept below ca.0.32 T. For an easier visualization of the hysteresis, we present as a top panel in Fig. 1c the increment value, defined as ΔX = X+B→-B -X-B→+B, where X states either for the resistance (R) or the MR while decreasing (+B→-B; blue curve in Fig. 1c) or increasing (-B→+B; red curve in Fig. 1c) the external magnetic field (B).Then, non-zero ΔX values indicate a hysteretic effect.As well, a zoom of the hysteretic region is presented in Fig. 1d, showing several resistance drops and plateaus and two lower limiting MR branches (with positive -redand negative -blueslopes) crossing at ZF.No relevant influence of the field sweeping rate is observed (Supplementary Figure 1).For a better comparative with the orthogonally-twisted bilayer, we show the corresponding MR behavior for pristine monolayer and bilayer CrSBr in Fig. 1e-g. 10In the pristine case, the spin reversal takes place via a spin-flip for fields applied along the easy-magnetic axis and a spin-canting process for fields along the intermediate-and hard-magnetic axis. 7,8,10,13 qualitative understanding of the MR behavior of the orthogonally-twisted bilayer upon the application of a magnetic field along the easy-(intermediate-) magnetic axis of the top (bottom) monolayer (Fig. 1c) is as follows: at high negative fields (region from -3 T to -1 T) the magnetization of both layers is parallel (φ = 0°, where φ is the angle formed between the magnetization of the top and bottom layer) and yielding to a state of low resistance according with a spin-valve picture.Below -1T the anisotropy is able to progressively reorient the magnetization of the bottom layer from its intermediate magnetic axis towards its easy-magnetic axis, while that of the top layer stays unchanged since the field is applied along its easy magnetic axis.As a consequence, from -1 T to 0 T an increase of the resistance is observed in agreement with the progressive increase of φ.In fact, at zero-field, the magnetization of both layers would be orthogonal (φ = 90°), assuming negligible inter-layer interactions.Upon the application of positive fields, the magnetization of the bottom layer continues the canting process (φ > 90°), tending to adopt an antiparallel configuration to satisfy the antiferromagnetic coupling, thus increasing the resistance to a maximum value at 0.16 T. At this point, the top layer flips its magnetization to be oriented along the positive magnetic field and φ decreases (φ < 90°), thus yielding to a big drop of the resistance.Further magnetic fields tend to continue canting the magnetization of the bottom layer, thus decreasing φ and, therefore, the resistance.Above 1 T (range from 1 T to 3 T), the magnetization of the top and bottom layers is parallel (φ = 0°) and the lower resistance state is observed.Decreasing magnetic fields yields to a symmetric configuration but observing the MR peak at negative fields and, consequently, yielding to a hysteretic effect (a detailed view of the process is presented in the Supplementary Figure 2).This scenario, which is possible thanks to the in-plane magnetic anisotropy of CrSBr, cannot be observed in twisted CrI3, since it exhibits an out-of-plane anisotropy.This behavior is in sharp contrast with that of the pristine bilayer, which shows a single maximum of MR at ZF, as a result of the antiparallel orientation between the two layers, and no hysteretic effects (Fig. 1e-g). 10,13inally, we consider the in-plane angular dependence (Supplementary Figure 3).All the curves exhibit the same general trend discussed above but with different coercivity fields and ΔX values.The field orientation hence allows for a fine tuning and control of the hysteretic parameters.Note the asymmetry between 0º and 90º, suggesting that the underlying spin dynamics are dominated by one of the layers -as discussed later, it is due to the larger stray fields at the twisted layers-.We note that, for fields applied along directions different that the easy-magnetic axis, the reversal mechanism can be more complex since both layers can be canting, thus motivating future magnetic imaging experiments in these CrSBr twisted layers.Regarding magnetic fields applied along the hard-magnetic axis c (out-of-plane direction), a hysteretic behavior is manifested as well, but with a significantly broader maximum of MR (Supplementary Figure 4).In this case, the MR curves are saturating for fields above 2 T, which, as for the in-plane case, coincides with the field needed to reorient the spins along the magnetic field direction (c in the present case; Fig. 1e-g).Similar results are observed in different orthogonally-twisted bilayer CrSBr heterostructures, underlying the robustness of the observed phenomenology (Supplementary Figure 5), although the exact switching magnetic values differ between the different devices, probably due to slightly different twisting angles.Next, we consider both the field and temperature dependence of the MR (Fig. 2).We observe that the behavior resembles that reported for the pristine bilayer. 10Upon cooling down the system, a negative MR starts developing below 200 K due to the onset of short-range interactions within the layers.Then MR reaches a broad plateau at ca. 150 K, near Tc, and below 100 K it increases again (Fig. 2a).However, some differences with the pristine bilayer are observed.First, in the pristine bilayer a minimum in MR, instead of a plateau, is observed at 150 K, followed at 100 K by a decrease.Second, a hysteretic behavior is observed from temperatures below TN (Fig. 2b-d), increasing the coercive field and ΔMR upon cooling down, while no hysteresis is observed in the pristine bilayer.Similar trends are observed for fields applied along different directions (Supplementary Figure 6).To further explore the irreversibility of the observed hysteresis in Fig. 1, we perform a series of first-order reversal curves (FORC).The FORC analysis lies behind the Preisach model. 30,31We increment sequentially the maximum applied magnetic field (Bmax) in steps of 20 mT, after an initial saturation at negative fields (sequence: -0.6 T → +Bmax → -0.6 T).Selected curves are shown in Fig. 3a (see the Supplementary Video 1 for the whole data set).For sweeping fields below ca.0.1 T (|Bmax| = 0.06 T in Fig. 3a), the resistance increases/decreases upon increasing/decreasing B following the behavior already observed in Fig. 1.d when sweeping from negative fields (limiting branch with positive slope).No hysteresis is observed for this loop, being the MR curve symmetric (ΔMR = 0 at ZF).A more interesting scenario is offered when this field threshold is overcome (|Bmax| = 0.16 T in Fig. 3a).In this case, the resistance increases (red curve) upon increasing B, as before, until a sharp drop occurs at ca. 0.1 T.Then, upon decreasing B (blue curve), the resistance decreases but with a smaller slope until a second drop is observed at ca. -0.1 T, when it returns to the initial path (limiting branch with positive slope).This behavior results in the emergence of an asymmetric hysteresis (ΔMR ≠ 0 at ZF).Similar asymmetric curves with successive drops in the resistance, giving rise to steps and plateaus at well-defined magnetic fields, are observed upon increasing the maximum sweeping magnetic field value (|Bmax| = 0.18 T in Fig. 3, Supplementary Video 1 and Supplementary Figure 7).Interestingly, each step observed for positive fields is characterized by a different slope while returning to ZF.This slope decreases until a saturation field is reached (0.32 T in the present case).For B > 0.32 T, the limiting branch with negative slope is reached and the hysteresis loop becomes fully symmetric with respect to the R axis (|Bmax| = 0.50 T in Fig. 3a).Interestingly, when coming from positive saturated fields (sequence: +0.6 T → -Bmax → +0.6 T in Fig. 3b and Supplementary Video 1), the same phenomenology is observed but reversing the modulus of the switching fields (mirror image with respect to the R axis).
Therefore, this magnetic system is formed by two ground states that are degenerated at ZF but that evolve with opposite MR slopes in presence of B. Thus, an initial saturation at negative fields leads to a state defined by the MR branch with positive slope (Fig. 3a).This state is sketched as a set of blue circles in the Fig. 3. Conversely, when coming from positive fields, a different state is obtained (set of red circles in Fig. 3) leading to the MR branch with negative slope.For B values within the range ±0.32 T the system evolves hysteretically and selectively towards one of these two ground states and only for higher |B| values a change of ground state is possible.This allows to select at will the ground state of the system.Furthermore, in the hysteretic region such evolution takes place through successive steps at specific fields that may be associated to intermediate states.This multistep phenomenology can be related with the Preisach model.Thus, starting from one of the two MR branches, each one of the resistance drops observed in the hysteresis curves is associated to the switch of an individual hysteron, leading to each one of the intermediate states postulated above.In applied terms, every one of these switches could be potentially employed as a bit of information.This is schematically sketched in Fig. 3 by sweeping red/blue bytes.Importantly, there is also magnetic memory at ZF since we can select between hysteretic and non-hysteretic MR scenarios depending on the initial ground state of the system.In the Supplementary Figure 8, we consider different magnetic field sweep protocols and, for example, in the sequence ZF → + Bmax → ZF we observe hysteresis only after an initial saturation in negative magnetic fields.Therefore, the magnetic history allows us to control the appearance or not of hysteresis.To manifest the robustness of these results, we present in Supplementary Figure 9 the study for other orthogonally-twisted CrSBr bilayers.Overall, similar trends, although at different switching fields, are observed under different in-plane field orientations (Supplementary Figure 10) and temperatures (Supplementary Figure 11).Regarding the origin of the multistep magnetization switching, we can attribute it to the stabilization of different domain configurations and spin textures as unveiled by atomistic spin dynamic simulations.We have considered the case of a CrSBr-based orthogonally twisted bilayer, where the top monolayer is rotated 90 degrees with respect to the bottom one (inset in Fig. 4a).The size of the simulation system is 100 nm × 100 nm along x-and y-axes, with no periodic boundary conditions along the aforementioned directions, and a cell thickness along the -axis corresponding to two-unit cells to accommodate the two stacked monolayers (see Methods for details).In line with the experimentally-based measurement protocol, we apply a simulated field of varying strengths along the x-axis, i.e., along the easy (intermediate) magnetic axis of the bottom (top) layer.Note that, for an easier visualization of the results, the easy (intermediate) magnetic axis of the bottom (top) layer is rotated if compared with Fig. 1a-d.We then simulate the field-cooling from 200 K (above   ) to 0 K using spin dynamics techniques for 2D magnets 32- 34 .In this way, it is possible to follow microscopically the variation of the magnetic features at the final simulated state of the system for different field strengths.We evaluate the angle  between the magnetic moment vector /  (where   is the volumetric saturation magnetization) of the top monolayer and the x direction, as it provides a strong descriptor of the spin orientations at the layers.Interestingly, we have observed that when low fields are applied (0.01 − 0.095T), the magnetization of the top monolayer is canted from its easy b-axis towards its intermediate a-axis (Fig. 4a,b).If we increase the magnitude of the magnetic field to 0.10 − 0.14 T (Fig. 4a,c), we observe the appearance of non-collinear spin configurations in the form of hybrid domain walls (Bloch-type) in the top monolayer.Intriguingly, this type of magnetic configurations, for this range of field strengths, only occurs if chiral spin-interactions like Dzyaloshinskii-Moriya interactions (DMI) are considered in the simulations, 35 causing a magnetic frustration due to the competing contributions.This could lead to the appearance of more complex non-collinear spin distributions for larger systems.When the applied field is increased further to 0.18 − 0.30 T (Fig. 4a,d), the Zeeman-like contribution will overpower the internal fields causing the magnetization of the top monolayer to align along the magnetic field direction, that is, along its intermediate magnetic axis, a.The illustration of each spin phase at specific field magnitudes is displayed in Fig. 4e-g with the snapshots extracted from the simulations in Supplementary Figure 12.Supplementary Movies S2-S4 show the entire evolution of the dynamics at 0.06 T, 0.10 T and 0.20 T, respectively.For instance, we observed that at 0.10 T (Fig. 4e and Supplementary Movie S3) the domain wall profile of the top layer flips from the +y toy direction and the spins at the centre are along the applied field direction, x, parallel to the bottom monolayer.We observed that these different spin textures are not present on the pristine bilayer CrSBr as expected, since both layers have their easy-axes along the same direction.Moreover, we have applied temperature to the system (5 K) and the simulated results remain consistent despite of the thermal fluctuations and noise (Supplementary Figure 13).It is worth mentioning that as one of the layers is twisted (e.g., top layer), the dipolar fields   generated at that layer become larger relative to the untwisted layer (e.g., bottom layer), which conditionate the response of the system (Supplementary Figure 14).That is, one of the layers becomes more dominant than the other inducing the appearance of some of the MR effects discussed before in the measurements.Indeed, the variations of stray-fields with the applied field follow those observed in the spin textures with the formation of canting fields and domain walls (Supplementary Figure 12).This suggests that the dynamic evolution of the magnetisation with the external magnetic field follows a Barkhausen-like effect trend 36 with a series of sudden changes in the size and orientation of the magnetic domains.In our case, however, since the top layer is twisted with respect to the underlying layer, a systematic flip of the spins with the field is possible until saturation is reached.The smaller fields to saturate the system in the simulations (~0.2 T) relative to the measurements (~ 1 T) may be due to variations of the magnetic parameters used, 37,38 but the overall picture is well described and in sound agreement with the measurements.The field is initially applied along one of the easy-axis of the layers (dark arrows), and as its magnitude increases M/Ms changes accordingly to be aligned with B (see inset).Three magnetic phases can be stabilised with the applied field: spin canting (the spins are aligned but at an angle between the anisotropy easy-axis direction of the top layer and the applied field direction), domain wall (part of the spins of the top layer orient along of the field, and another with the bottom layer underneath which induced the formation of domain walls) and homogeneous (both layers have their spins aligned with the field).Three values of the field (0.06 T, 0.1 T, 0. In conclusion, we have shown that twisting engineering of magnetic 2D materials is a fruitful platform for the emergence of new correlated phases in artificial metamagnets, as exemplified here by the appearance of multistep spin switching accompanied by hysteretic MR effects in orthogonally-twisted bilayer CrSBr.These field-induced features can be controlled by playing with the modulus and direction of the applied magnetic field, being absent in pristine CrSBr mono-and bi-layers.Overall, our results pinpoint twisted bilayer CrSBr as a versatile and rich platform for controlling and addressing the magnetic information on 2D magnets -of special relevance in areas such as spintronics or magnonics 39 -, as well as for motivating a new playground for fundamental studies.In particular, this orthogonally-twisted bilayer CrSBr may offer a promising route for the creation and manipulation of non-colinear magnetic textures, like vortices or topologically protected skyrmions and merons. 20,21On the other hand, the controlled stacking of 2D magnetic monolayers under defined angles opens new avenues to increase the magnetic symmetry in the plane, thereby reducing the anisotropy energy.Of special interest is to reach the crossover from easy-axis to easy-plane anisotropy, since easy-plane (XY) systems 40 are predicted to host dissipationless spin transport. 41,42n particular, device 1 is formed by a top (bottom) CrSBr monolayer of 77.2 µm 2 (53.3 µm 2 ), with an overlap area of 9.3 µm 2 and a twisted angle of 92.5º.Device 2 is formed by a top (bottom) CrSBr monolayer of 190.1 µm 2 (117.3 µm 2 ), with an overlap area of 15.9 µm 2 and a twisted angle of 89.3º.Device 3 is formed by a top (bottom) CrSBr monolayer of 206.6 µm 2 (121.1 µm 2 ), with an overlap area of 7.9 µm 2 and a twisted angle of 87.0º.

Magneto-transport measurements:
Electrical measurements were performed in a Quantum Design PPMS-9 cryostat with a 4-probe geometry, where a DC current was passed by the outer leads and the DC voltage drop was measured in the inner ones.DC voltages and DC currents were measured (MFLI from Zurich Instruments) using an external resistance of 1 MΩ, i.e., a resistance much larger than the sample.Temperature sweeps were performed at 1 K•min −1 , field sweeps at 200 Oe/s, rotation sweeps at 3 °/s and the current bias was 1 µA, unless otherwise explicitly specified.Magneto-resistance (MR) is defined as: MR = 100[R(B) -R(0)]/R(0), where B is the external magnetic field and R(0) is the resistance at zero field in the symmetric case (see text).

Atomistic spin dynamic simulations:
Our simulations were performed using atomistic spin dynamics simulation techniques [32][33][34][35][43][44][45][46][47] as implemented in the VAMPIRE software package. 47 Theenergetics of the system is described by the spin Hamiltonian: where   and   are unit vectors describing the local spin directions on Cr sites.The first term is the symmetric Heisenberg exchange and   is the exchange tensor between Cr sites, being ,  = , , .The first input is the Heisenberg exchange, which in CrSBr has seven intra-layer exchange terms ( 1−7 ) occurring between atoms within the same monolayer and two inter-layer terms ( 1 and  2 ) occurring from one monolayer to another.The value of the second intra-layer nearest neighbour exchange ( 2 ) was taken from Wang, H. et al. 38 and was used as a reference to define the magnitude of the   elements, which are outlined below.In order to obtain satisfactory predictions of the critical temperature of CrSBr-based systems, the relative ratios between exchange parameters have been taken from Bo, X. et al. 48For the inter-layer interactions,  1 and  2 , we have used the values for the unrotated bilayer due to the absence of dramatic changes in the intermonolayer distances.The distances for the  1 only differ by about 5.44% and the average deviation in the  2 interactions is only 2.66%.As commented below, variations of these magnitudes do not change the results.
The second term is the anti-symmetric exchange or DMIwhich stabilizes topological states, where   is the DMI vector.Due to the absence of inversion symmetry between interacting Cr-based atoms, 49 we have included the reported anti-symmetric contributions with DMI unit vectors parallel to the -th (mediating  3 ) and -th (mediating  1 ) axes, whose values are given, respectively, by  1 = 0.07 meV and  3 = 0.18 meV. 48e third term is the on-site anisotropy energy, which is made up of two uniaxial terms, where the relative values of the anisotropy constants,   = 8.06 meV and   = 31.53meV, govern the intermediate -th and easy -th axes of the system. 38It is important to note that previously introduced single-ion anisotropies are not, theoretically, the only ones that should contribute to the overall magnetocrystalline anisotropy of the system.The larger spin-orbit coupling of the Br atoms compared to those of Cr points to the existence of in-basal-plane-based exchange anisotropy terms at the previously defined spin Hamiltonian. 50However, in the computational characterization of the system for the twisted bilayer we have chosen not to include them, despite the fact that it has been reported that they share the same order of magnitude as the on-site anisotropy contributions.This is because these second-ion terms can induce the -th axis to be the easiest one in the system. 38Moreover, the single-ion contributions are enough to unravel the main features observed experimentally.It is worth noting that, due to the rotation process, the easy-axis of the top monolayer is directed along the -th spatial direction and the intermediate one the -th axis (orthogonally directed with respect to the untwisted bottom monolayer).
The fifth term is the Zeeman energy, where  represents the externally applied magnetic field and   is the atomic magnetic moment, to which the value   = 2.88 B has been assigned in consonance with the bulk scenario, 37 being  B the Bohr magneton.
The final term is the long-range dipole-dipole interaction, ℋ  , which can be expressed as: where |  | is the distance between site i to j.
We also calculated the inter-layer exchange field as: being the magnetisation of the bottom and top layer represented by  bottom and bottom , respectively.Taking into account that there are four nearest-neighbours, mediated by  1 , and one next-nearest-neighbour, mediated by  2 , interactions, we can estimate a maximum exchange field of ~0.15 T if we assume that  bottom and  top are fully parallel.This magnitude is much smaller than the dipolar fields induced by the twisted layer (Supplementary Figure 14), and suggests that variations of the order of 5-10% in exchange interactions will not affect the results in case the rotation might play a role.This correlates with potential variations due to the interlayer distance between Cr sites as commented above.

Figure 1 .
Figure 1.-Magnetic field dependence of the magneto-resistance (MR) in orthogonally-twisted bilayer CrSBr.a, Optical image of a vertical van der Waals heterostructure consisting of twisted CrSBr monolayers (black dashed lines) in between few-layers graphene (blue dashed lines).Different insulating h-BN layers (green dashed lines) are employed both for avoiding shortcuts and protecting the heterostructure.Red arrows indicate the easy-magnetic axis (b) of every CrSBr monolayer, being the intermediate-magnetic axis (a) perpendicular to it.The hard-magnetic axis (c) corresponds to the out-of-plane direction.Scale bar: 5 µm.b, Schematic view of the heterostructure (not to scale), highlighting the twisted CrSBr monolayers (pink, yellow and cyan balls correspond to bromine, sulfur and chromium atoms, respectively; red arrows represent the spin lying along the easy-magnetic axis, assuming negligible inter-layer magnetic interactions) placed in between few-layers graphene or NbSe2 thin layers (blue color) on top of pre-patterned electrodes (gold color) together with a sketch of the electrical measurement configuration.c-d, Field-dependence of the resistance and MR (bottom panel) as well as its increment (top panel), defined as ΔX = X+B→-B -X-B→+B, where X states either for the resistance or the MR at T = 10 K and θ = φ = 0º.Sweeping up (down) trace is depicted in red (blue).Red/blue arrows indicate the sweeping direction of the magnetic field.Black arrows sketch the relative configuration of both layers' magnetization.MR is defined as MR (%) = 100•[R(B) -R(0)]/R(0).e-g, Reference experiments on pristine monolayer and bilayer based on our previous work, 10 including the corresponding sketches (panel e) and field dependence of the MR for fields applied along the easy (b), intermediate (a) and hard (c) magnetic axes for pristine monolayer (panel f) and bilayer (panel g) at T = 10 K.

Figure 2 .
Figure 2.-Field and temperature dependence of the MR in orthogonally-twisted bilayer CrSBr.a, Temperature dependence of the MR at saturated fields (B = 3 T).b-c, Field and temperature dependence of the MR while sweeping from negative (positive) to positive (negative) fields.d, Field and temperature dependence of ΔMR.MR is defined as MR (%) = 100•[R(B) -R(0)]/R(0), being R(0) the resistance obtained at zero field and ΔMR= MR+B→-B -MR-B→+B.θ = φ = 0º.

Figure 3 .
Figure 3.-Multistep magnetization switching with magnetic memory in orthogonally-twisted bilayer CrSBr.First-order reversal curves considering the sequence a) -0.6 T → +Bmax → -0.6 T and b) +0.6 T → -Bmax → +0.6 T at 10 K and θ = φ = 0 °.Bmax is incremented sequentially in steps of 20 mT and selected curves are shown (see Supplementary Video for the whole dataset).The saturated state at negative (positive) magnetic fields is schematically sketched as a set of blue (red) circles configuration, being every spin switch related to the change of one individual hysteron (squared hysteresis operator characterized by a coercive field and a field shift from zero) within the Preisach model.MR is defined as MR (%) = 100•[R(B) -R(0)]/R(0), being R(0) the resistance obtained at zero field in the symmetric case.

Figure 4 .
Figure 4.-Field-induced spin-textures in orthogonally-twisted bilayer CrSBr.a, After cooling under different applied field strengths (0.01 -0.3 T) applied along x -parallel (perpendicular) to easy (intermediate) magnetic axis of the bottom (top) layer-, we calculate the angle  between the magnetisation direction (M/Ms) of the top layer to the applied field B (blue arrow).The field is initially applied along one of the easy-axis of the layers (dark arrows), and as its magnitude increases M/Ms changes accordingly to be aligned with B (see inset).Three magnetic phases can be stabilised with the applied field: spin canting (the spins are aligned but at an angle between the anisotropy easy-axis direction of the top layer and the applied field direction), domain wall (part of the spins of the top layer orient along of the field, and another with the bottom layer underneath which induced the formation of domain walls) and homogeneous (both layers have their spins aligned with the field).Three values of the field (0.06 T, 0.1 T, 0.2 T) are highlighted with circles and further analysed in the following panels as an example.The crystallographic a-, b-and caxes for every monolayer are indicated.b-d, Projections of the magnetisation Mx, My and Mz at 0 K as a function of the position (nm) along the a-axis of the top layer at 0.06 T, 0.1 T and 0.2 T, respectively.e-g, Schematics of the spin configuration observed in the spin dynamics simulations (Supplementary Figure12) at 0.1 T (domain wall), 0.06 T (spin canting) and 0.2 T (homogeneous).
Figure 4.-Field-induced spin-textures in orthogonally-twisted bilayer CrSBr.a, After cooling under different applied field strengths (0.01 -0.3 T) applied along x -parallel (perpendicular) to easy (intermediate) magnetic axis of the bottom (top) layer-, we calculate the angle  between the magnetisation direction (M/Ms) of the top layer to the applied field B (blue arrow).The field is initially applied along one of the easy-axis of the layers (dark arrows), and as its magnitude increases M/Ms changes accordingly to be aligned with B (see inset).Three magnetic phases can be stabilised with the applied field: spin canting (the spins are aligned but at an angle between the anisotropy easy-axis direction of the top layer and the applied field direction), domain wall (part of the spins of the top layer orient along of the field, and another with the bottom layer underneath which induced the formation of domain walls) and homogeneous (both layers have their spins aligned with the field).Three values of the field (0.06 T, 0.1 T, 0.2 T) are highlighted with circles and further analysed in the following panels as an example.The crystallographic a-, b-and caxes for every monolayer are indicated.b-d, Projections of the magnetisation Mx, My and Mz at 0 K as a function of the position (nm) along the a-axis of the top layer at 0.06 T, 0.1 T and 0.2 T, respectively.e-g, Schematics of the spin configuration observed in the spin dynamics simulations (Supplementary Figure12) at 0.1 T (domain wall), 0.06 T (spin canting) and 0.2 T (homogeneous).