Spatially reconfigurable antiferromagnetic states in topologically rich free-standing nanomembranes

Antiferromagnets hosting real-space topological textures are promising platforms to model fundamental ultrafast phenomena and explore spintronics. However, they have only been epitaxially fabricated on specific symmetry-matched substrates, thereby preserving their intrinsic magneto-crystalline order. This curtails their integration with dissimilar supports, restricting the scope of fundamental and applied investigations. Here we circumvent this limitation by designing detachable crystalline antiferromagnetic nanomembranes of α-Fe2O3. First, we show—via transmission-based antiferromagnetic vector mapping—that flat nanomembranes host a spin-reorientation transition and rich topological phenomenology. Second, we exploit their extreme flexibility to demonstrate the reconfiguration of antiferromagnetic states across three-dimensional membrane folds resulting from flexure-induced strains. Finally, we combine these developments using a controlled manipulator to realize the strain-driven non-thermal generation of topological textures at room temperature. The integration of such free-standing antiferromagnetic layers with flat/curved nanostructures could enable spin texture designs via magnetoelastic/geometric effects in the quasi-static and dynamical regimes, opening new explorations into curvilinear antiferromagnetism and unconventional computing.


Main Text
Topological textures in antiferromagnetic (AFM) materials are whirling structures with spins oppositely aligned between ferromagnetic (FM) sublattices.2][3][4][5] In fact, some topological AFM textures are predicted to exhibit spintronic analogues of relativistic physics, where their speed limit is set by the magnon group velocity. 2,6[9] Central to nucleating and controlling topological textures are various magneto-crystalline interactions, namely anisotropy, exchange, and relativistic spin-orbit torques, which are intrinsic to specific antiferromagnets.1][12] This markedly restricts their utility and flexibility in comparison to typical FM-based, topological texture hosting metallic heterostructures, which are polycrystalline and can be grown simply by sputter-deposition. 1,8Therefore, further exploration and exploitation of topological AFM textures necessitates the development of crystalline AFM layers that can be readily integrated with dissimilar supports, for example, silicon or even non-crystalline flexible substrates.
To this effect, we drew inspiration from recent developments in crystalline quantum materials membranes, which are freestanding crystals of macroscopic lateral dimensions with a thickness of ~1-100 nm. 14- 16These membranes are a relatively new form of crystalline matter occupying an intermediate position between bulk and 2D materials, whilst having properties that are distinct from both.Generally, crystalline membranes have bulk-like magnetic/electronic properties, but, similar to 2D materials, are quite flexible, and can therefore withstand extreme deformations without undergoing fracture. 17,18They can also be transferred post-growth to any desirable host, including silicon 15,19 or flexible supports, 20,21 enhancing the ability to stack and twist complex heterostructures. 22re, we demonstrate the design and fabrication of high-quality AFM nanomembranes that preserve the all-important magneto-crystalline interactions post delamination.To image the local AFM order, we developed a scanning transmission X-ray microscopy (STXM)-based Néel vector reconstruction technique and demonstrated that our detached membranes host a multi-chiral family of topological AFM textures, analogous to the Kibble-Zurek-like phenomenology previously observed in attached epitaxial films. 10Moreover, we provide a striking demonstration that the flexibility of our membranes can be exploited to drive a local reconfiguration of the AFM properties and topological textures across membrane 'folds'.We present a set of mechanical models, which demonstrate that our observations are consistent with the magneto-structural effects expected from flexure-based strains.Our results pave the way for the development of AFM spintronics platforms exploiting membrane tunability via geometry and strain.

Membrane design and fabrication
We fabricated high-quality freestanding membranes of (001)-oriented, Rh-doped α-Fe2O3 (referred to hereafter as α-Fe2O3) using the selective water-etching technique 14,21,23 (see Methods) on epitaxial heterostructures grown by pulsed laser deposition. 10,24Achieving high-quality α-Fe2O3 was found to depend critically on the choice of substrate and intermediate (buffer) layers to reduce inter-layer lattice mismatch.Due to the trigonal symmetry of α-Fe2O3 (space group 3 ̅ ), we chose single-crystalline (001)-oriented α-Al2O3 and (111)-oriented SrTiO3 (STO) substrates as the growth templates, and (111)-oriented Sr3Al2O6 (SAO) as the water-soluble sacrificial layer. 14,21,23For all samples, water-etching of SAO resulted in freestanding oxide membranes, which were shifted to the desired support via either direct or indirect transfer, see Figure 1a.The former involves direct scooping of the floated membrane onto the support, whereas the latter requires spincoating of a temporary organic support to hold the delaminated membrane before its final transfer.We have used both approaches for different experiments throughout this work.
We found that direct growth of α-Fe2O3|SAO on (001)-oriented α-Al2O3 substrates (sample type-A) results in oriented polycrystalline samples due to the large lattice mismatch between various layers in the stack, see Supplementary-S1.Film quality improves notably when α-Fe2O3|SAO are grown on (111)-oriented STO substrates (sample type-B) due to the significantly lower mismatch between SAO and STO (in this orientation in bulk,  SAO /4 ~ 5.60 Å,  STO ~ 5.51 Å), although the resulting α-Fe2O3 itself remains quite defective.To improve the sample quality further, we added an intermediate buffer consisting of an ultra-thin STO (111) layer followed by a thicker LaAlO3 (LAO) (111) layer between SAO and α-Fe2O3 (sample type-C), see Methods and Supplementary-S1.Here, LAO acts as a good buffer as it has a slightly smaller lattice constant ( LAO ~ 5.35 Å), reducing the mismatch with α-Fe2O3 ( Fe 2 O 3 ~ 5.03 Å), whilst being structurally close to both STO and SAO. 23Moreover, the ultra-thin STO increases the overall crystallinity 23 and is found to be critical in aiding the delamination of the overlayers in our buffered heterostructures.The addition of the buffer layers results in freestanding AFM membranes with much larger crack-free areas compared with the unbuffered counterparts, see Figure 1b and Supplementary-S1.Lastly, the sample quality and yield of these membranes are significantly superior to hematite layers prepared via chemical exfoliation. 25e quality and orientation of our buffered α-Fe2O3 crystal membranes were ascertained by X-ray diffraction (XRD) and selected-area electron diffraction (SAED), see Figure 1c-f, and Supplementary-S1.A unique feature of buffered α-Fe2O3 membranes is the formation of a moiré pattern evident in the reciprocal space as satellite peaks in SAED.This is expected to be a 'mismatch' moiré pattern, 26,27 which results from electron beam interference through the slightly mismatched lattices of α-Fe2O3 and the buffer layers.This picture is validated by our diffraction simulation, which closely reproduces the experimental pattern (Figure 1f).The resulting periodic perturbation at the α-Fe2O3-buffer interface does not appear to affect the magnetic properties of α-Fe2O3, as the length scales we study in magnetometry and X-ray microscopy, and the membrane thickness are significantly larger than those of the mismatch pattern.

Magnetic transition in membranes
The reliable generation of topological textures in α-Fe2O3 requires the presence of a spin reorientation (Morin) phase transition, which mimics the Kibble-Zurek phenomenology. 10 At the Morin transition temperature  M , the anisotropy undergoes a sign reversal, 10,24 causing spins to flip from out-of-plane (OOP) to in-plane (IP) configurations.The presence of a distinct Morin transition in the proximity of room temperature was confirmed both by SQUID magnetometry 10,24 and by dichroic X-ray spectroscopy, see Supplementary-S1.This is in sharp contrast to chemically exfoliated hematene membranes, which do not display any Morin transition. 25Crucially, the transition in our detached membranes is qualitatively similar to those reported in attached epitaxial films, 10,24 despite the former being more defective than the latter, with transitions in buffered α-Fe2O3 being particularly sharp, see Supplementary-S1.We conclude that our water-etched membranes are good freestanding platforms to seek out real-space topological AFM order.

Nanoscale mapping of the AFM order
To image the local AFM textures, we performed scanning transmission X-ray microscopy (STXM) in X-ray magnetic linear dichroism (XMLD) modality (see Methods)an element-specific spectro-microscopy technique, with a large depth of focus, that enables unambiguous identification of the AFM contrast.In XMLD-STXM, a beam of Fe L3-edge X-rays is focussed onto the AFM membranes at normal incidence, whilst changes in absorption are monitored in transmission by a point detector (Figure 2a).In this geometry, the X-ray polarisation (Linear Horizontal -LH) is in the basal plane of the α-Fe2O3 membranes, and IP and OOP AFM orientations are clearly distinguished as they contribute different XMLD contrast signals. 10Moreover, by varying the sample azimuth using an in situ rotation stage, we changed the relative orientation of the X-ray polarization and IP Néel order systematically, enabling the nanoscale reconstruction of the AFM order. 10,28,29kin to our previous work with XMLD photoemission microscopy (PEEM), 10,28 the XMLD contrast can resolve the IP AFM directions but cannot distinguish the absolute sign of the AFM order.Nevertheless, we can clearly identify topological textures in our membranes from these reconstructions.

Evolution across the Morin transition
XMLD-STXM reveals that our α-Fe2O3 membranes host magnetic textures remarkably similar to those seen in attached films (Figure 2b and Supplementary-S1). 10 For  <  M , we observe large OOP AFM domains (purple) separated by antiphase domain walls (ADWs) with IP AFM order (yellow/orange).As the system is warmed, the ADWs widen and small IP islands nucleate and progressively increase in size.At ~ M , the ADW length-scale diverges, as anisotropy approaches zero, resulting in a complex distribution of AFM domains hosting nearly equal fractions of IP and OOP regions.At  >  M , IP regions enlarge and become dominant, whilst OOP regions shrink dramatically.Nonetheless, we still observe several OOP regions across the sample.
To determine the topological character of the AFM textures in our membranes we constructed Néel vector-maps for  >  M .We used red-green-blue colours to denote IP domains with spin directions at 120° from each other, as expected based on the underlying trigonal symmetry. 28Based on previous work in attached films, 10 we expect topological textures to be associated with small OOP 'cores'.Although such small topological cores are usually difficult to detect with LH-polarized X-rays, we observed some larger OOP 'bubbles', not all of which are associated with a whirling texture, and are therefore likely to be topologically trivial and produced by pinning of the OOP phase at local defects.More importantly, we were able to observe many topological AFM textures, including AFM merons and antimerons, see Figure 2c.Individual AFM (anti)merons can be characterised by an AFM winding number ±1 (depending on whether the texture whirls along or opposite to the azimuth angle), and an AFM topological charge ±1/2 (depending on the product of the winding number and the core polarisation). 4,9,10,28Moreover, (anti)merons can combine locally to form pairs, which may have a net AFM topological charge of ±1 (bimerons) or 0 (topologically trivial pairs), depending on the relative core polarisation of the (anti)merons.It should be noted that bimerons and topologically trivial pairs cannot be distinguished by XMLD techniques. 10The observation of a multi-chiral topological AFM family unequivocally confirms that our membranes harbour the Kibble-Zurek phenomenology originally discovered in attached films, 10 despite the larger concentration of structural defects.Noteworthily, topological states are observed in the absence of any high spin-orbit heavy-metal overlayer, indicating that the creation and stabilization of topological order are intrinsic to α-Fe2O3, and do not rely on interfacial interactions present in typical FM skyrmionic systems. 1,5,8,9clear difference between membranes and attached films is that AFM textures in the former are more strongly pinned than in the latter, so that texture patterns are reproduced almost identically even after performing multiple thermal cycles across  M (Supplementary-S2).We hypothesise that texture pinning results from highly localised alteration of the magnetic properties due to an increased density of point and extended defects in the membranes.This reasoning is supported by previous studies performed extensively in other topological systems.[30][31][32][33] We also performed in situ imaging with magnetic fields and found the AFM state to remain largely unperturbed (Supplementary-S2), indicating that our topological textures are much more robust compared to counterparts observed in synthetic AFMs. 5

Flexure-driven three-dimensional texture reconfiguration
5][36] We find that our buffered α-Fe2O3 membranes are not brittle, as one expects of ceramic-like oxides, but are very flexible and can develop 'folds'.An example is illustrated in Figure 3, where the fold has a maximum curvature of ~ 3×10 5 m -1 .In extreme scenarios, we even observe complete 180° 'folded-over' membranes, see Supplementary-S3.Large-area buffered membranes (type-C) are particularly remarkable, as they can hold complex strain distributions without undergoing fracture.
To study magneto-structural effects, we imaged naturally flexed regions across the membrane folds that emerged serendipitously upon direct transfer.Their shape was confirmed through confocal microscopy, which maps the height profile of the membrane (Figure 3b and Methods).Moreover, the slopes of the flexed region appear darker in STXM images (Figure 3c) because the signal scales inversely with the effective sample thickness  eff ~ / cos  ( -actual thickness,  -deviation angle from horizontal).
Flexure effects are immediately apparent from the images collected through the Morin transition.We define  M FF as the Morin transition in the far-field (flat) region of the membrane away from the fold.All the data in Figure 3 were collected for a type-C membrane with the α-Fe2O3-layer facing upward, and the buffer layer lying underneath (for characterization details see Methods).Above  M FF (Figure 3d), both the far-field regions and the peak of the fold exhibit an IP AFM matrix hosting several OOP cores, as expected.However, narrow bands near the base of the fold host a much large fraction of OOP AFM regions.Upon cooling to T ~  M FF (Figure 3e) the far-field regions exhibit mixed IP and OOP contrast, the base of the fold has a robust OOP matrix with clear AFM ADWs, whilst the top of the fold surprisingly remains in the IP state.Finally, at temperatures well below  M FF , all regions are in the OOP state interspersed with ADWs (Figure 3f).This remarkable evolution, consistent with the magnetic anisotropy changing rapidly through the fold, is in stark contrast with the behaviour in flat membranes, corresponding to Morin temperatures significantly raised near the base and lowered at the peak.
To further explore these effects, we imaged a buffered membrane in the reversed configuration, with the buffer layer facing upward, and the α-Fe2O3-layer is underneath, see Supplementary-S4.Here, the fold has the opposite effect with respect to the trend in Figure 3: the OOP state is stabilised on top of the fold (consistent with a raising of  M ), while the topologically-rich IP state is stabilised at the base (lowering of  M ).Finally, to confirm the presence of topological textures on folds, we also performed vector-mapping and found a clear instance of a meron-antimeron pair near the base.This suggests that structural reconfiguration could enable new form of spatial control over AFM topology.
The overall thermal evolution across different folded membranes can be summarised as follows: flexure alters the magnetic anisotropy in opposite directions near the peak and the base, such that the respective local  M is either suppressed or elevated relative to  M FF , but with the overall sign of the effect being reversed depending on whether α-Fe2O3 is the top or the bottom layer.Hence, we observe in Figure 3 that  M peak <  M FF <  M base , whereas in Supplementary-S4 we find  M base <  M FF <  M peak .These results indicate that magnetostructural effects can be used to spatially design AFM states outside their typical thermal stability window.

Strain and Anisotropy Modelling
Magnetic anisotropy in α-Fe2O3 results from a delicate competition between dipolar and on-site interactions that are sensitive to structural variations. 24,34,37In particular, epitaxial studies 34 revealed that uniform compressive and tensile strains applied via substrate clamping raise and lower  M in α-Fe2O3, respectively.Consequently, we seek to explain our experimental observations through flexure-induced strain.
To develop this simple insight, we estimated the strain distribution across our fold numerically through a finite-element mechanical model of a buffered α-Fe2O3 membrane (see Methods), whose flexed region closely reproduces the profile determined by confocal microscopy.We find that flexure results in sizable uniaxial inplane tensile and compressive strains,   , distributed along the membrane thickness (-direction), such that the neutral (unstrained) line 17 is located close to the middle of the buffered membrane.Due to the presence of the buffer, which itself accommodates some strain, the net strain 〈  〉 ,F averaged across the α-Fe2O3 layer is actually non-zero.Moreover, 〈  〉 ,F strength varies gradually across the length of the fold, changing sign near the point where the curvature is zero, see Figure 4a.Finally, this model predicts that reversing the buffered membrane should also reverse the sign of the strain distribution (Figure 4b).
To provide a semi-quantitative estimate of the flexure-induced variation of the magnetic anisotropy across the fold, we calculated the local  M (Figure 4c) by combining the thickness-averaged strain profile from our mechanical model with the strain dependence of  M determined in literature. 34A caveat of this analysis is that the strains in Ref 34 were substrate-induced and biaxial, whereas the flexure-induced strains here are primarily uniaxial.In buffered membranes with the α-Fe2O3-side facing up, we find that the net compressive strain at the base of the fold (〈  〉 ,F base < 0) and the net tensile strain at the peak (〈  〉 ,F peak > 0) lead to an increase and decrease, respectively, of the local  M by ~10%.This is consistent with our experimental observations in Figure 3. Furthermore, the sign of this effect is reversed for the flipped buffered membrane, while its magnitude remains approximately the same, consistent with our results in Supplementary-S4.
As a final test of our magneto-structural model, we investigate flexed regions in unbuffered samples (type-B).Absence of the buffer layer should cause the neutral strain line in a folded membrane to lie approximately in the middle of the α-Fe2O3 layer, so that 〈  〉 ,F ~ 0. This would markedly suppress the flexure-driven anisotropy changes, in stark contrast to what is expected in buffered membranes.This is indeed confirmed by our experimental results illustrated in Supplementary-S5.We conclude that our model effectively explains the AFM state reconfiguration observed across the folded membranes.

Discussion
Using a powerful transmission-based AFM vector-mapping technique, we demonstrated that highquality α-Fe2O3 membranes host topological AFM states, which are reconfigurable via flexure-induced strain in three-dimensional folded structures.
At a fundamental level, our results establish that strain modulation is a novel and powerful tool to design and manipulate topological AFM textures, adding a completely new vista of reconfigurable magnetic topology to the burgeoning research landscape built on exploiting membranes of quantum materials to generate exotic states. 17,18,21Our results also pave the way for the exploration of static and dynamical AFM evolution [38][39][40][41] triggered by in situ electric, magnetic, optical, or structural perturbations. 1,9For example, we envisage electrically triggering topological reconfiguration and dynamics via localised piezoelectric control.Moreover, by integrating extremely flexible AFM membranes/ribbons onto carefully designed three-dimensional nanostructures, it may become possible to induce novel symmetry-breaking exchange or anisotropy interactions, e.g. through curvilinear geometric 35,36 and magnetoelastic 42,43 effects, thereby enabling the design of spatiallyvarying magnetic states, 44 or the realization of hitherto undiscovered chiral textures. 45,46n the applied front, the development of substrate-free AFM membranes that preserve magnetocrystalline properties and topological order addresses a major roadblock inhibiting the integration of crystalline AFM materials into established spintronics platforms.Specifically, complex and dense topological AFM fabrics are expected to possess fast non-linear dynamics, 2,47 which could open explorations into AFM-based siliconcompatible ultra-fast reservoir computing 7,48 or dense AFM memory-in-logic arrays in three dimensions. 49,50

Methods
Membrane growth and fabrication: Throughout this work, we have studied Rh-doped α-Fe2O3 (α-Fe1.97Rh0.03O3)membranes.Rh-doping was used to elevate the Morin transition temperature beyond what is typically achievable in undoped counterparts, as discussed in the literature. 10,24Membrane layers were grown either on (111)-oriented SrTiO3 or (001)-oriented α-Al2O3 substrates from CrysTec, using a pulsed laser deposition setup fitted with a KrF excimer laser.Firstly, the growth of the water-soluble Sr3Al2O6 layer was performed at 950 °C, in a pure oxygen atmosphere of 1 mTorr, and a laser repetition rate of 2 Hz.Deposition of the buffer SrTiO3 layer (thickness ~ unit cells) was then performed at 850 °C, 10 mTorr, and 2 Hz.This was followed by the LaAlO3 layer (~10 nm) grown at the same temperature and repetition rate, with an oxygen pressure of 1 mTorr.Subsequently, the α-Fe2O3 layer (~30 nm) was deposited at 700 °C, 2 mTorr, and 3 Hz.Finally, the samples were gradually cooled in a high-oxygen-pressure environment to minimise oxygen vacancies formed during the growth.
To delaminate the membranes, 14 samples were placed in high-purity deionised water at room temperature to dissolve the SAO layer.In the case of direct transfer (see Figure 1a), the membranes gradually floated to the top of the water surface, following which they were scooped out using the desired support (Si, Si3N4, etc.).The presence of non-uniform forces in the scooping process can result in the serendipitous formation of folded regions.These flexed geometrical structures are held in place due to van der Waals interactions with the underlying Si3N4 support.Alternatively, in the case of indirect transfer, a temporary support consisting of Poly(methyl methacrylate) (PMMA) was spin-coated on the top of the sample, which was then held by a flexible tape.The entire stack was then placed in deionised water.After delamination, the membrane was carefully moved to the final support (Si, Si3N4, etc.) using a transfer stage that held onto the tape.Lastly, the PMMA layer and the tape were removed from the top surface of the membrane through a room-temperature acetone wash.The indirect process enables targeted and controlled transfer of membranes with a much higher yield compared to its direct counterpart.

Materials characterization:
The structural quality and crystallinity of the samples were determined by XRD involving 2 −  scans, rocking curves ( scan),  scans, and pole figure measurements, see Figure 1 and Supplementary-S1.Measurements were performed for both as-deposited films on crystalline substrates and membranes transferred onto Si substrates.The structural phase of α-Fe2O3 was further confirmed through Raman spectroscopy, performed using a Jobin Yvon Horiba LabRAM Evolution Spectrometer in reflection geometry (514.5 nm laser).Transmission electron microscope based SAED experiments were carried out using a JEM-ARM200F JEOL equipped with a cold field emission gun, operated at 200 kV.To ensure electron transparency, the AFM membranes were mounted on commercial 30-nm Si3N4 holders fabricated on top of Si substrates from Agar Scientific Ltd.Magnetic characterization was performed using a Quantum Design MPMS SQUID system on field-cooled samples under a 5000 Oe field during warming and cooling measurement scans.Detached membranes were supported on Kapton tape for magnetometry.Although there is generally good correspondence between magnetometry and STXM imaging, in some cases, the temperature dependence may have a minor difference, most likely due to small strain variation introduced during the corresponding sample preparation steps.The shape and height profile of membrane folds was studied through confocal microscopy using the Sensofar S Neox metrology tool.
STXM imaging: Fe L3-edge resonant STXM imaging was performed at the PolLux X07DA endstation of the Swiss Light Source. 51Images were obtained by recording the transmission of normally incident X-rays, polarised linearly (XMLD-STXM) or circularly (XMCD-STXM), and focused using a Fresnel zone plate (FZP).The outermost zone width of the FZP was selected, in tandem with the size of the monochromator exit slits of the beamline, resulting in a spatial resolution of about 40-50 nm.As the focusing efficiency of a diffractive optical element is not unitary, an order-sorting aperture (OSA), combined with a centre stop fabricated on the FZP, was employed to guarantee that only the focused X-rays illuminate the sample.The FZP and OSA are indicated in Figure 2a.An image is then obtained by scanning the sample with a piezoelectric scanner and recording the transmitted X-ray intensity for each point in the image.Typically, the field of view of our images had a square or rectangular shape.
For all X-ray transmission experiments, the AFM membranes were mounted on commercial 100-nm Si3N4 holders fabricated on top of Si substrates from Silson Ltd.The membrane temperature was controlled using a thin Au/Ti heater coil, lithographically fabricated directly onto the Si3N4/Si holders.Passing a current through the heater coil leads to resistive dissipation, thereby heating the sample.The resistance of the heater is calibrated and can be simultaneously measured to monitor the sample temperature.The Au/Ti heaters can be seen in Supplementary-S3,S4.For in situ field studies, magnetic fields were applied in the plane of the samples (i.e., in the crystal basal planes) using a rotatable permanent magnet setup, which could apply fields up to ~ 120 mT, see Supplementary-S2.
XMLD-STXM imaging was performed by collecting a pair of images, using linear horizontal X-ray polarization at two photon energies near the Fe L-edge (around ~ 710 eV), E1 and E2, chosen to provide maximal AFM contrast in our samples.The XMLD energy contrast was then calculated from Δ = (  1 , −   2 , )/(  1 , +   2 , ).Based on crystal symmetry analysis, 10 it can be shown that the LH XMLD intensity varies as  =   +   cos 2 , where  is the relative angle between the linearly polarised X-ray electric field and the magnetization, allowing us to map out the local AFM order.This immediately reveals that XMLD imaging is unable to distinguish IP AFM orientations separated by 180°.Likewise, it is also not possible to resolve the direction of OOP AFM orientation. 10,28As the predominant AFM textures vary with temperature, the intensity range from the energy contrast also changes.Hence, we have used the same colour scheme (purple to yellow/orange for OOP to IP) with different energy contrast limits to aid the visualization across the transition. 10Next, XMCD-STXM images presented in Supplementary-S1 were acquired by taking a set of data with both right (RCP) and left circularly polarised (LCP) X-rays at a fixed energy   .The XMCD contrast was calculated as δ = (   , −    , )/(   , +    , ).Owing to the absence of strong ferromagnetic textures in the sample, XMCD-STXM images showed negligible contrast.The data reduction was performed using a custom-built MatLab tool (available on the public repository https://gitlab.psi.ch/microspectro-public/). Spatially averaged X-ray absorption spectra were obtained at the Fe L-edges from the transmitted signal by scanning a straight line that straddled across regions both on and off the membrane, see Supplementary-S1.The signal measured outside the membrane was used as the reference signal for the normalization of the spectra acquired on the α-Fe2O3.
The depth of focus of the FZP utilized for the experiments reported in this work is ~ 1 µm, meaning that the imaging of the folded membranes had to be performed in several steps, bringing different parts of the fold into focus.Finally, composite images of AFM textures across the folds were produced by 'stitching' together multiple images.
To study the effects of flexure-induced strain in buffered membranes, it was crucial to determine whether α-Fe2O3 was on the top or the bottom of the stack, as non-uniform forces introduced during the scooping process can flip the membrane.To make this determination, we performed depth sensitive local elemental mapping, which could be accomplished by either of the two following techniques: (i) energy dispersive spectroscopy in transmission electron microscopy, or (ii) STXM imaging performed in total electron yield detection geometry, see Supplementary-S3.The latter was performed using a channeltron detector biased to a voltage of 2.4 kV.Here, only the secondary electrons emitted by the first few monolayers at the surface of the membrane are detected, in contrast to a typical transmission measurement where the entire thickness of the sample is probed.This allows us to determine the orientation of the membrane, depending on whether chemical contrast is detected at the Fe L3-edge or the La M5-edge, respectively, as shown in Supplementary-S3.

Néel vector-maps:
The IP Néel vector-maps in Figure 2c and Supplementary-S4, which provide orientational information of the IP AFM order, were constructed by combining the energy-contrast XMLD-STXM images obtained at six azimuthal sample rotation angles about the crystallographic c-axis in the range -45° to +45° where the limits were set by in situ rotation stage, see Figure 2a.Theoretical and experimental details supporting this approach to Néel vector reconstruction can be found in our previous studies. 10,28,29For each pixel in the field of view, we fit the angular dependence of the XMLD (see the previous section) to extract the average spin direction.Owing to the trigonal symmetry and the weak basal-plane anisotropy of α-Fe2O3, we mapped the spin directions using a red-green-blue (R-G-B) colour scale, which indicates the IP directions of the AFM order.For easy identification, the local IP Néel vector direction is indicated using a thin white bar.Regions in these maps where the AFM axis was identified to be OOP were coloured white. 10delling and Simulation: A time-dependent finite element analysis was carried out to study the strain profile of folded membranes quantitatively.The simulated composite stack consisted of both the buffered membrane (30 nm α-Fe2O3/10 nm LAO) and the 100 nm thick Si3N4 support.3][54] The simulation starts with the composite stack lying flat at rest, to an impulsive upward external force is applied in the middle of the membrane causing a fold to emerge.Right after the removal of the external force, the two ends of the folded membrane are rigidly fixed to the Si3N4 support, with their separation distance corresponding to the experimental results from confocal microscopy (Figure 3b).Finally, both the membrane and the support deform and relax to their respective equilibrium states within ~1 µs simulation time.The resulting final state was found to closely reproduce the equilibrium state of a folded membrane held on the Si3N4 support.The only tuning parameter in the simulation is the length of the suspended composite membrane at rest, which is determined by matching the height of the folded membrane in the simulation to that in the experiment.
The SAED patterns were simulated by implementing the description of interfering diffraction patterns from two overlapping lattices, as developed in Refs. 26,27The spatial moiré pattern is obtained as the sum of the two lattice functions, and the diffraction pattern is the Fourier transform of this combined function.The in-plane lattice parameters were chosen to correspond to experimental values of about 5.10 Å for α-Fe2O3 and 5.51 Å for LAO.For simplicity, we did not include the ultra-thin STO layer in this simulation as it is much thinner than the LAO in our buffered membranes.for the two configurations given in (a,b), respectively.Strain driven modulation of the local  M was obtained from the model developed in the literature. 34Here,  M /  M FF larger and smaller than unity, refers to the elevation and suppression of the local Morin temperature and, therefore, the magnetic anisotropy, relative to the flat farfield regions.