Polarization Vortices in a Ferromagnetic Metal via Twistronics

Polarization Vortices in a Ferromagnetic Metal via Twistronics | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Letter Polarization Vortices in a Ferromagnetic Metal via Twistronics Gustau Catalan, Yingzhuo Lun, Xinxin Hu, Qi Ren, Umair Saeed, and 13 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6656512/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted You are reading this latest preprint version Abstract Recent advances in moiré engineering have revealed new pathways for manipulating lattice distortions and electronic properties in low-dimensional materials. Here, we demonstrate that twisted-stacking can induce polarization vortices in metallic SrRuO 3 membranes, despite the presence of free charge screening. These polarization vortices are correlated with moiré-periodic flexoelectricity induced by shear strain gradients, and exhibit a pronounced dependence on the twist angle. In addition, multiferroic behavior emerges below the ferromagnetic Curie temperature of the films, whereby polarization and ferromagnetism coexist and appear to compete, showing opposite twist-angle dependencies of their respective magnitudes. Density functional theory calculations provide insights into the microscopic origin of these observations. Our findings extend the scope of topological polarization design beyond dielectric materials and into metals. Physical sciences/Materials science/Condensed-matter physics/Ferroelectrics and multiferroics Physical sciences/Physics/Condensed-matter physics/Ferroelectrics and multiferroics Physical sciences/Physics/Electronics, photonics and device physics/Electronic and spintronic devices Physical sciences/Materials science/Nanoscale materials/Two-dimensional materials Figures Figure 1 Figure 2 Figure 3 Figure 4 Highlights Twistronics enables the emergence of polarization vortices and multiferroic characteristics in metals. Introduction Since the discovery of superconductivity in twisted bilayer graphene 1 , the field of so-called “twistronics” 2,3 has blossomed. The key idea behind twistronics is that stacking two thin crystalline layers rotated with respect to each other generates a moiré interference pattern in the electronic density of states at the interface, which in turn modifies the electronic properties of the bilayer system. Twistronics was initially studied in the context of two-dimensional (2D) materials 4 , which can be readily exfoliated and stacked. The development of methods for growing epitaxial films of complex oxides and separating them from the substrate via dissolution of a sacrificial layer 5–7 allows these materials to partake in the twistronic game. A seminal result in this context has been the discovery that two layers of BaTiO 3 twisted with respect to each other display polarization vortices 8 . The proposed explanation is that the torsional strain introduced at the interface between the twisted layers is not homogeneous, being minimal at the sites where the atomic columns of the two layers coincide across the interface, and maximal where equivalent atomic columns of the two layers are furthest apart. This causes torsional strain gradients and therefore gradient-induced polarization, known as flexoelectricity 9,10 . Flexoelectricity is allowed in materials of any symmetry 11 , and twist-induced polar vortices have also been reported in SrTiO 3 12 , which is cubic and non-polar in the absence of strain (although strain can make it ferroelectric 13 ). Since there are not strain gradients without strain, it can be difficult to separate how much of the polarization vortices is due to gradient-induced flexoelectricity, and how much to strain-induced ferroelectricity or piezoelectricity. While piezoelectricity has been ruled out from the explanation of vortices based on geometrical considerations, the fact remains that there is a strong polarization to start with that may be affected both by strain and by the depolarization fields expected whenever polarization is disrupted. It is therefore highly desirable to examine whether polar vortices can be induced in a material where piezoelectricity and/or depolarization fields can be fully excluded. Importantly, if the origin of polar vortices is truly flexoelectric, we no longer need to limit ourselves to examining insulating materials in our on-going search for chiral polarization 14,15 : metals can also be flexoelectric 16–18 . Therefore, a tantalizing possibility emerges of inducing polarization vortices in a metal, where the depolarization field engineering approach typically used in ferroelectrics is just not available. In this work, we have fabricated twisted-bilayer (t-BL) stacks of SrRuO 3 (SRO), a well-known oxide metal whose flexoelectricity has been recently reported 17 , and we have combined atomic-scale imaging with theoretical calculations to examine the polar displacement pattern. The results show that the Ru cation is shifted with respect to the Sr lattice, forming periodic arrays of vortices and antivortices like those reported in insulating BaTiO 3 and SrTiO 3 8,12 , displaying a significant twist-dependence. Moreover, we observed that the twisted-stacking induced polarization has functional consequences on the ferromagnetic and metallic properties, indicating multiferroic characteristics in metallic membranes. Film growth and structural characterization Epitaxial thin films of SRO (10.7 nm) were grown by pulsed-laser deposition (PLD) on (001)-oriented SrTiO 3 (STO) substrates with water-soluble Sr 3 Al 2 O 6 (SAO) buffer layer. The films were attached onto a polydimethylsiloxane (PDMS) stamp and immersed in deionized water to dissolve the SAO layer. Upon removal of SAO and separation of SRO from the substrates, the freestanding single-layer (SL) SRO membranes were transferred onto silicon oxide or TEM supports. The t-BL SRO was assembled by stacking a second membrane, released from the same epitaxial film and transferred onto the first one with a prescribed rotational angle. To remove potential surface adsorbates at the interface and/or hydrogen intercalation 19 , the transferred samples were annealed at 330 ℃ for 8 hours in an oxygen atmosphere before measurements. The schematic of fabrication process is provided in Fig. 1a, and further details for film growth and transfer are described in Methods. The crystallinity of the initial epitaxial films and subsequently transferred membranes was confirmed by X-ray diffraction (XRD) 2θ-ω scans (Fig. 1b). The fully epitaxial strained state of the as-grown heterostructures was confirmed using reciprocal space mapping (Fig. 1c), revealing that the SRO and SAO layers share the same in-plane lattice parameter as the STO substrate. The diffraction peak of the transferred membrane (Fig. 1b), however, is shifted, suggesting a relaxation of the lattice upon epitaxial strain release. The morphologies of the as-grown film and the transferred bilayer membrane (Fig. 1d and Supplementary Fig. S1) show smooth and flat surfaces with roughness of 0.25 nm and 0.38 nm, respectively. The bilayer interface was examined using cross-sectional scanning transmission electron microscopy with high-angle annular dark-field (STEM-HAADF) imaging (Supplementary Fig. S2). A high-quality interface with minimal interlayer distance of 4.97 Å is observed in the twisted bilayer. Polarization vortices in twisted-bilayer SrRuO3 We used high-resolution STEM-HAADF imaging to visualize the in-plane atomic structure of the metallic membranes and determine the eventual presence of polarization vortices (see Methods). Bulk SRO crystallizes in a centrosymmetric perovskite structure with orthorhombic symmetry (Pbnm) at room temperature 20 . We found that, even when the size reduces to the nanometer scale, the freestanding SL SRO membrane retains its orthorhombic characteristics, as evidenced by the presence of ½ superlattice spots in the selected area electron diffraction (SAED) pattern (Supplementary Fig. S3). To quantify polarization, we measured the off-center displacement of Ru ( δ Ru ) with respect to its four adjacent Sr atoms. The Ru displacement extracted from the STEM-HAADF image of the SL SRO (Figs. 2a, e) is both random and minimal (1.3 pm on average), indicating that the freestanding SRO remains nonpolar as its bulk counterpart. The t-BL SRO exhibits a different behaviour. Upon focusing the electron beam at the bilayer interface, a continuous and well-ordered moiré was found in low-magnification image (Supplementary Fig. S4), suggesting a homogeneous interfacial contact. Subsequent high-resolution moiré images measured from the 3.0 ο , 4.8 ο , and 7.3 o twisted bilayers are shown in Figs. 2b-d, displaying a decreasing moiré periodicity with increasing twist angle. The twist angle was determined from the relative rotation between the two distinct sets of diffraction spots observed in the fast Fourier transform (FFT) of the moiré images (Supplementary Fig. S5). By shifting the focus away from the interface to the upper surface, we selectively imaged the lattice structure of the top layer (Supplementary Fig. S6). Prior to this, we conducted depth-dependent STEM imaging and analyzed the corresponding variations in FFT patterns. The results confirmed that the diffraction spots from the bottom lattice completely vanish as the focus shifts to the upper surface (Supplementary Fig. S7), thus excluding geometric interference artifacts from the bottom layer in the images obtained at top layer. The Ru displacement maps of the top layers (Figs. 2f-h), calculated based on the lattice structure observed at STEM-HAADF images (Supplementary Fig. S6), reveal a periodic arrangement of alternating vortices and antivortices, indicating that chiral polarization has emerged in the metallic membranes. While chiral polarization is becoming increasingly frequent among ferroelectrics, where it can be driven either by depolarization field 21,22 or torsional strain gradients 8 , it is unprecedented in SRO or any other metal. The toroidal moment was quantified as , where δ i is the adjacent Ru displacement vector located at position r i , and N = 24 is the number of adjacent cells, and then superimposed onto the Ru displacement maps to visualize the spatial arrangement of polar vortices (Figs. 2f-h). The periodic array of negative toroidal moments reveals that the AA- stacked (Sr-Sr and Ru-Ru sites) regions host clockwise vortices, while anticlockwise vortices form near the AB- stacked (Sr-Ru sites) regions. These features in polarization vortices can be robustly observed in the different bilayers even with a high twist angle of 10.4 o (Supplementary Fig. S8) but were entirely absent in the SL SRO (Fig. 2e). As the twist angle increases, the periodicity of polar vortices in different bilayers becomes smaller, and always coincides with the moiré periodicity, consistent with the twistronic origin of the polarization. The magnitudes of the toroidal moment and the polarization also decrease as the twisting increases. Figure 2i shows that the Ru displacement exhibits an inverse dependence on the twist angle, being more significant at the smallest twist angle. The maximum displacement reaches ~20 pm (8.3 pm on average) for the 3.0 o twisted bilayer, which is comparable to reported values for polarized SRO 17,23,24 . To exclude the possibility that the observed polarization vortices arise from imaging artifacts associated with the moiré pattern, we performed the same measurements on unannealed samples. The idea is that a weaker interlayer coupling at the unannealed interface should make it difficult to distort the lattice and polarize it. Indeed, the unannealed samples remain non-polar, with a Ru displacement map (Supplementary Fig. S9) like that of SL SRO, despite the presence of a moiré pattern at the interface. The observed polarization vortices are therefore not a geometric interference artifact, since the moiré exists both in the annealed and unannealed bilayers. Since electrostatic interactions are expected to be screened by the free charges in metallic systems, whereas strain (gradient) cannot be screened, the polar vortices of the SRO bilayers consistent with a mechanical rather than electrostatic origin, lending support to the flexoelectric hypothesis 8,25 : the periodic misalignment of the atoms in the top and bottom layers generates shear torsional strain gradients and thus flexoelectric fields, which can serve as pseudo-electric fields, manipulating polarization 26,27 and driving the formation of polar vortices. The strain-gradient origin is further supported by the geometric phase analysis (GPA) (see Supplementary Text I), showing that the shear strain gradients are distributed in a strip network of alternating signs, with the polarization components pointing in the same direction within each strip region (Supplementary Fig. S10). A flexoelectric coupling in metals is evidenced as the polarization systematically switches towards the opposite direction upon the reversal of the strain gradients. The flexoelectric fields reconstructed by shear strain gradients (Figs. 3j-l) significantly correlate with the observed polar vortices maps, providing direct supports of a flexoelectric origin for the polar vortices. This electromechanical coupling accounts for the observed twist angle-dependent polarization, where the larger polarization at small angle is attributed to the more significant lattice distortion (shear strain) and its localized inhomogeneity (strain gradients). DFT calculation for polarized t-BL SrRuO3 We then used first-principles density functional theory (DFT) to simulate the polarization in t-BL SRO (see Methods). Commensurate bilayer supercell structure was constructed by stacking two individual unit-cell-thick layers with a relatively high twist angle of 18.92 o . The smaller moiré periodicity with fewer atoms in a high-angle model improves computational efficiency. The established twisted bilayer model, depicted in Fig. 3a, exhibits a characteristic moiré pattern arising from alternating AA- and AB-stacked sequences. After fully relaxing structure, we computed the off-center displacement of Ru atoms relative to the surrounding oxygen octahedra in each unit cell. The corresponding maps (Figs. 3b-c) reveal the formation of polarization vortices in the top and bottom layers, with opposite chirality in each layer. In the top layer, clockwise vortices predominantly form at AA-stacked regions and rough anticlockwise ones at AB-stacked regions, consistent with our experimental observations. Similar polarization configuration can be robustly reproduced in bilayers with higher twist angles of 22.62 o and 28.07 o (Supplementary Fig. S13). The calculated Ru displacement decreases with twist angle (Fig. 3d), in agreement with experimental results. Since the calculations adopted unit-cell-thick SRO layers, the lattice near the bilayer interface is expected to be subjected to larger strain, the magnitude of calculated polarization is thereby larger than the experimental one that measured at the upper surface. To demonstrate the crucial role of twisted stacking, we conducted the same calculations for 0 o -stacked bilayers with different translational shifts; irrespective of AA- or AB-stacked bilayers, SRO maintains a centrosymmetric structure without any polarization (Supplementary Fig. S14). Additionally, we calculated the electronic density of states (DOS) of the polarized twisted bilayers (Fig. 3e and Supplementary Fig. S15). All twisted bilayers exhibit nonzero density of states at the Fermi level, primarily contributed by Ru d -orbital electrons, confirming their metallic nature. The overlap of the Ru d and O p states indicates a strong Ru-O orbital hybridization. Overall, our calculations provide solid theoretical evidence that topological polarization can be physically compatible with metallicity in twisted bilayers. Functional consequences of polar vortices Having established the presence of polarization vortices in t-BL SROs, the question arises: do these vortices have any tangible impact on the material's functional properties? A consideration is that bulk SRO is a paramagnetic metal at room temperature, but it undergoes a transition to a ferromagnetic state below ~160 K 28 (~150K for epitaxial films 29 ). This ferromagnetic ordering manifests as a distinct change in the electrical resistance 30,31 . Given that polarization, conductivity, and magnetism are all closely linked to the Ru atoms and their surrounding oxygen octahedra, coupling among these properties may be possible. With this in mind, we measured the temperature-dependent magnetization ( M - T ) curve and ferromagnetic hysteresis loop ( M - H ) for SL and t-BL SROs (see Methods). Figures 4a-b show that the t-BL SROs exhibit ferromagnetism with higher coercive field at low temperature. We examined the variation in ferromagnetic transition temperature and magnetic moment, which show strong dependencies on the twist angle. The Curie temperature T C (inset of Fig. 4a) and the saturation magnetization M s (hysteresis loops in Fig. 4b) both decrease as the twist angle is reduced. The twist-dependent T C and M s are summarized and compared with δ Ru in Fig. 4c, showing that the polarization magnitude and the ferromagnetism display opposite trends with respect to the twist angle, suggesting a competition between the two types of ordering. We also measured the temperature-dependent resistance curves (Supplementary Fig. S16), which show that all samples exhibit typical metallic conductivity. The kink in resistance due to the ferromagnetic transition, highlighted in Fig. 4d, show the same twist angle-dependence. These results reveal coexistence and interaction of polar vortices and ferromagnetic ordering in correlated metallic systems, indicating that SRO becomes a “multiferroic metal” under the twisted stacking. The apparent competition between polar and magnetic orderings echoes the known difficulty of stabilizing ferroelectricity and ferromagnetism within the same material 32,33 . To understand this multiferroic competition, we constructed an orthorhombic SRO structure and modulated polarization by manually off-centering Ru cations, thus approximating the polarization observed in t-BL SROs. The magnetic exchange parameter of the Heisenberg model was calculated based on DFT total energies of ferromagnetic and anti-ferromagnetic phases (see Methods). The results shown in Fig. 4e indeed reveal a gradual decrease in the exchange parameter with increasing polarization, consistent with the observed lower magnetization. The reduced exchange interaction is attributed to the decreasing of the mean Ru-O-Ru bond angle due to the Ru displacement along the polar direction. This off-centering reduces the overlap between Ru 4d and O 2p orbitals, thereby weakening the Ru–O hybridization. The Curie temperature estimated based on exchange parameter also decreases with increasing polarization, in good agreement with the experimental observations (Fig. 4f). Conclusion The discovery of polarization vortices in metallic membranes expands the field of polar topology 34,35 into metals and strongly correlated electron systems 36,37 . Our findings demonstrate that moiré engineering can induce polarization vortices in a metallic system, and, since SRO is also ferromagnetic, the result is a multiferroic metal. The results indicate an anticorrelation and therefore a possible competition between the two orders. Beyond SRO, our results highlight the broader implications of flexoelectricity as a universal property, suggesting that similar polarization phenomena could emerge in other twisted bilayer metallic systems. The presence of polarization within a metal introduces new degrees of freedom for spintronic applications, mediated by flexo-Rashba effect 18 , opening new avenues for designing materials with tunable electronic, magnetic, and topological properties, and exploring potential polarization-related behaviour in metallic systems. Methods Epitaxial film growth All epitaxial films were synthesized using pulsed-laser deposition (PLD) with a KrF excimer laser (248 nm, COMPex 102, Lambda Physik). The SrRuO 3 (SRO) thin films were grown on (100)-oriented SrTiO 3 (STO) substrates with an 8.8 nm-thick Sr 3 Al 2 O 6 (SAO) sacrificial layer to enable the subsequent membrane release. Prior to deposition, the chamber was evacuated to a base pressure of 1.0 × 10 -4 mTorr. Stoichiometric ceramic targets were used to ensure the chemical composition of films. The SAO layer was deposited at 750 ℃ under an oxygen pressure of 1 mTorr, with a laser fluence of 1.60 J/cm 2 . The SRO layer was subsequently grown at 700 ℃ under an oxygen pressure of 100 mTorr O 2 , with a laser fluence of 1.28 J/cm 2 . The laser repetition rate was maintained at 1 Hz throughout the deposition process. After deposition, the films were cooled down to room temperature at 5 ℃/min under an oxygen pressure of 10 Torr. Twisted bilayer membrane transfer To obtain freestanding SRO membranes, the epitaxial films were adhered to a polydimethylsiloxane (PDMS) stamp and immersed in deionized water to dissolve the SAO sacrificial layer. The complete dissolution of SAO led to the detachment of the film from the substrate. The released membrane, attached to the PDMS stamp, was then transferred onto a target substrate using a transfer platform that allows to control the rotation of the substrate with a precision of 0.1 o . Half of the membrane on the PDMS was brought into contact with the substrate, and the temperature was then increased to 40 ℃ to remove residual water. The PDMS was subsequently peeled off slowly, ensuring an intact transfer of the membrane. To make a twisted-bilayer sample, the substrate with the transferred bottom layer was rotated to a predefined angle using a micrometer probe in the transfer system. The remaining half of the membrane on the PDMS was then transferred onto the bottom one in the same manner, thereby obtaining twisted-bilayer samples. To enhance interlayer interaction and eliminate possible adsorbates at the interface, the samples were annealed at 330 ℃ for 8 hours under oxygen flow rate of 50 sccm. X-ray diffraction (XRD) and surface characterizations High-resolution 2𝜃-𝜔 scans and reciprocal space mappings (RSMs) were conducted using a Panalytical X’pert Pro diffractometer with a Cu K𝛼 1 (1.540598Å) source. The system featured a hybrid two-bounce primary monochromator on the incident beam side and a PIXcel line detector for high-resolution data acquisition. The surface morphology was characterized using MFP-3D Asylum AFM (Asylum Research -Oxford Instruments). Tapping mode was performed to minimize surface damage using a soft tip. TEM, SAED, and STEM characterizations The low-magnification HRTEM images and SAED patterns were acquired using a FEI Tecnai F20 TEM at 200 kV. High-quality cross-sectional samples were fabricated using the focused ion beam (FIB) technique at a Thermo Fisher Helios 5 UX dual-beam system. The STEM experiments were conducted using a double aberration-corrected Spectra 300 operated at 300 kV. Depth-sectioning STEM-HAADF was performed by acquiring a series of atomic-resolution images at different defocus depth 8 , allowing for discrimination between the top layer and the bilayer interface. STEM-HAADF images were acquired with a probe convergence semi-angle of 27.3 mrad while the HAADF detector collection semiangle was set as 63–200 mrad. Polarization vortices and strain analysis The atomic positions of Ru and Sr were firstly extracted from Wiener-filtered STEM-HAADF images using the Python package Atomap 38 . After locating the Ru and Sr columns, the off-center displacement of each unit cell was calculated as the deviation of the Ru column from the mass center of the four adjacent Sr columns. The mean Ru displacement value is subtracted in the mappings to improve the visibility of the polarization vortex array. Strain analysis was performed using the Geometrical Phase Analysis (GPA) 39 software package (HREM Research) in Digital Micrograph. Electrical transport and magnetization measurements Temperature-dependent resistance measurements were performed in an Advanced Research Systems cryostat using a Keithley 2614B SourceMeter. The SL and t-BL membranes were transferred onto an insulating oxidized silicon substrate, and the Pt electrode arrays were sputtered through a square array-mask placed on the sample surface, defining 100 μm electrodes separated by 20 μm spacing. The electrodes were contacted via wire-bonding to a leadless chip carrier (Kyocera) electrically connected to the sourcementer by a chip-carrier socket, and thermally connected to the cold finger of the cryostat. The resistance was measured using two-point contact mode, as the temperature gradually varied at a rate of 1 K/min. Ferromagnetic hysteresis loops and temperature-dependent magnetization field-cooling curves were measured using a vibrating sample magnetometer (VSM, Quantum Design MPMS-3). The SL and t-BL membranes were transferred onto sapphire substrates. The external magnetic field was applied parallel to the membrane plane. DFT calculation for polar vortex DFT calculations were performed using the Vienna Ab Initio Simulation Package (VASP) 40,41 . The projector-augmented wave (PAW) method 42 was employed, with the Perdew–Burke–Ernzerhof (PBE) 43 generalized gradient approximation (GGA) 44 used as the exchange-correlation functional. The cubic structure was adopted to construct bilayer supercell structures, following the method proposed by Lee et al. 45 . The electronic wave functions were expanded using a plane-wave basis set with a kinetic energy cutoff of 600 eV and the 12×12×1 k-point mesh was used to sample the Brillouin zone. The monolayer structure was built based on the relaxed bulk SRO and was fully relaxed. The equilibrium lattice constant of cubic monolayer was calculated to be 3.781 Å, and the same value was used for all bilayer structures. The 12×12×1 k-mesh was used for monolayer and untwisted bilayer structures, whereas Γ-point sampling was adopted for the twisted structures. Structural relaxations were performed until the Hellmann-Feynman forces on each atom were reduced below 0.01 eV/Å. Van der Waals interactions between adjacent layers were accounted for using the Grimme D3 method 46 . To prevent spurious interlayer interactions, a vacuum region exceeding 15 Å was introduced in the supercell. DFT calculation for magnetism The magnetic properties were investigated based on an orthorhombic structure with manually off-centering Ru along the c -axis direction ( i.e. [001]-orientation), which lies within the membrane plane in as-grown film. In DFT calculations, a plane-wave basis set with a kinetic energy cutoff of 500 eV was used to expand the electronic wave functions, and an 8 × 8 × 6 k-mesh was employed to sample the Brillouin zone. The on-site Coulomb interaction parameters were set to U = 3 eV and J = 0.3 eV for Ru 4d orbitals. The Curie temperature was estimated using the Heisenberg model and the self-consistent Gaussian approximation (SCGA) method 47 . Specifically, the magnetic exchange parameter ( J ) of Heisenberg model was derived from the DFT calculated total energies of ferromagnetic and G-type antiferromagnetic phases 48,49 , as shown in Supplementary Fig. S17, , (1) where E AFM ( E FM ) is the total energy of the ferromagnetic and antiferromagnetic phases, N mag = 4 is the number of magnetic Ru ions in the unit cell, and N nn = 6 is the number of nearest-neighbour Ru ions of each Ru ion. The Curie temperature was estimated using SCGA method 47 as , (2) θ C = 0.719 is a structure dependent parameter 48 . Declarations Acknowledgments The authors acknowledge support by the National Key Projects for Research and Development of China (Grant No. 2021YFA1400300), the National Natural Science Foundation of China (Grant Nos. 12172047, 12402183, 12374080), the Beijing Natural Science Foundation (Grant No. 1244057). We also acknowledge ICN2 and ALBA (JEMCA) for providing key facilities and technical guidance. ICN2 is supported by the Severo Ochoa program from Spanish MCIN / AEI (Grant No. CEX2021-001214-S), the European Regional Development Fund from European Union (Grant No. IU16-014206 (METCAM-FIB)), and the CERCA Programme, Generalitat de Catalunya. ICN2 is a founding member of e-DREAM. J.A. and X.H. acknowledge funding from Generalitat de Catalunya (Grant No. 2021SGR00457). This study is part of the Advanced Materials programme and was supported by MCIN with funding from European Union NextGenerationEU (PRTR-C17.I1) and by Generalitat de Catalunya (In-CAEM Project). We acknowledge support from CSIC Interdisciplinary Thematic Platform (PTI+) on Quantum Technologies (PTI-QTEP+). This work has been funded by the European Commission – NextGenerationEU (Regulation EU 2020/2094), through CSIC's Quantum Technologies Platform (QTEP). Y.L. acknowledges support from the Severo Ochoa Seed Funding program (Grant CEX2021-001214-S/MICIU/AEI/10.13039/501100011033). D.P. acknowledges the “Consolidación Investigadora” grant CNS2024-154522 and grant PID2022-140589NB-I00, funded by MCIN/AEI/10.13039/501100011033. This work is in the framework of the Universitat Autonoma de Barcelona Materials Science PhD program. X.H. acknowledges PhD scholarship support from the China Scholarship Council (CSC) (Grant No. 202304910019). Q.R. acknowledges support from the BIT Research and Innovation Promoting Project Graduate (Grant No. 2023YCXZ002). U.S. acknowledges the funding for this project by Grant PID2019-108573GB-C21 (FOx-Me) funded by the Spanish Ministry of Science and Innovation (DOI: 10.13039/501100011033). I.P.-H. acknowledges funding from AGAUR-FI scholarship (Grant No. 2023FI-00268) Joan Oró of the Secretariat of Universities of the Generalitat of Catalonia and the European SocialPlus Fund. Calculations were performed using resources of the Supercomputer Centre in Chongqing. The authors acknowledge the use of the Spectra 300 microscope, provided under ALBA Synchrotron proposal (Grant No. 20240320035). The authors gratefully acknowledge the assistance of Francesco Salutari, Athique Ahmed Ali and Francisco Javier Belarre Triviño in the preparation of cross-sectional FIB samples, and Bernat Bozzo in the magnetic measurements. Note: after writing this manuscript, we have become aware of another work on SrRuO 3 membranes (Alejandro Martin Merodio et al., group of Jacobo Santamaría at the Universidad Complutense de Madrid, presented at QUOROM, 22/05/25). So far they only report transport properties, but we expect their membranes will also display vortices. Author contributions Experimental measurements were mainly performed at ICN2 and ALBA (JEMCA), and theoretical calculations were carried out at BIT. All institutions contributed substantially to the development and completion of this work. G.C., J.H., and J.A. designed and supervised the project. Y.L. conceived the original idea for this work. Y.L. fabricated the thin films and transferred the samples with the help from U.S. and J.M.C.R.. X.H., J.A., B.M., K.G., and I.P. conducted STEM, SAED and TEM experiments. X.H. and Y.L. performed data analysis on the polar vortices, strain gradients, and flexoelectric fields. Q.R., G.T., and J.H. carried out the atomistic simulations and analysis. Y.L. conducted the ferromagnetism and resistance measurements under the guidance of D.P. and X.W.. Y.L. performed the surface morphology measurements. J.P. and J.S. conducted the XRD measurements. Y.M. and Q.L. provided assistance with the experiments. Y.L., X.H. and G.C. wrote the manuscript with the help from all authors. All authors contributed to the discussion of the results and the revision of the manuscript. Competing interests The authors declare that they have no competing interests. Data and materials availability All data are available in the manuscript or the supplementary information. References Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556 , 43–50 (2018). Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist. Nat. Mater. 19 , 1265–1275 (2020). Pryds, N., Park, D.-S., Jespersen, T. S. & Yun, S. Twisted oxide membranes: A perspective. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6656512","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Letter","associatedPublications":[],"authors":[{"id":481921844,"identity":"445d7c2e-1e47-4566-aba1-4eab41d2dad9","order_by":0,"name":"Gustau 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(BIT)","correspondingAuthor":false,"prefix":"","firstName":"Gang","middleName":"","lastName":"Tang","suffix":""},{"id":481921858,"identity":"225ae6e9-6003-4cbb-b541-2a6a2ea669d7","order_by":14,"name":"David Pesquera","email":"","orcid":"https://orcid.org/0000-0003-0681-3371","institution":"Catalan Institute of Nanoscience and Nanotechnology","correspondingAuthor":false,"prefix":"","firstName":"David","middleName":"","lastName":"Pesquera","suffix":""},{"id":481921859,"identity":"06c05684-da66-4c24-9393-faaf17845154","order_by":15,"name":"Xueyun Wang","email":"","orcid":"https://orcid.org/0000-0001-5264-9539","institution":"Beijing Institute of Technology","correspondingAuthor":false,"prefix":"","firstName":"Xueyun","middleName":"","lastName":"Wang","suffix":""},{"id":481921860,"identity":"2dd4bc53-4994-4da3-8c64-361e41964d11","order_by":16,"name":"Jiawang Hong","email":"","orcid":"https://orcid.org/0000-0002-9915-8072","institution":"Beijing Institute of Technology, Zhuhai Beijing Institute of Technology(BIT)","correspondingAuthor":false,"prefix":"","firstName":"Jiawang","middleName":"","lastName":"Hong","suffix":""},{"id":481921861,"identity":"6f1636b4-bcb0-43d1-a100-d4f816c1b8f6","order_by":17,"name":"Jordi Arbiol","email":"","orcid":"https://orcid.org/0000-0002-0695-1726","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Jordi","middleName":"","lastName":"Arbiol","suffix":""}],"badges":[],"createdAt":"2025-05-13 14:33:26","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6656512/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6656512/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":86242995,"identity":"6b4232fb-14c0-4bbb-976c-48f6e2096167","added_by":"auto","created_at":"2025-07-08 11:04:50","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":280433,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eTransfer and characterization of twisted-bilayer SrRuO\u003c/strong\u003e\u003csub\u003e\u003cstrong\u003e3\u003c/strong\u003e\u003c/sub\u003e\u003cstrong\u003e membranes. a\u003c/strong\u003e, Schematic illustration of the transfer process of twisted bilayer membranes. \u003cstrong\u003eb\u003c/strong\u003e, X-ray diffraction 2θ-ω scans of the as-grown epitaxial film and transferred membrane (a.u., arbitrary units). \u003cstrong\u003ec\u003c/strong\u003e, Reciprocal space mapping of the as-grown film around the (−103) diffraction peak. \u003cstrong\u003ed\u003c/strong\u003e, Surface morphology of the transferred t-BL SRO membrane. The inset shows the line profile, indicating a thickness of 10.7 nm for a single layer.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-6656512/v1/d4444e58839b82b95f1a3f9c.png"},{"id":86242993,"identity":"e7695d31-6c6d-43bc-ba96-dcf936061b5b","added_by":"auto","created_at":"2025-07-08 11:04:50","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":1206603,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eAtomic-scale imaging of polar vortices in t-BL SROs. a\u003c/strong\u003e, Planar-view STEM-HAADF image of the freestanding SL SRO. \u003cstrong\u003eb-d\u003c/strong\u003e, Moiré images focusing on the interface in the 3.0\u003csup\u003eο\u003c/sup\u003e, 4.8\u003csup\u003eο\u003c/sup\u003e, 7.3\u003csup\u003eο\u003c/sup\u003e twisted bilayers, respectively. \u003cstrong\u003ee\u003c/strong\u003e, Ru displacement vector maps superimposed with their toroidal moment Q(δ\u003csub\u003eRu\u003c/sub\u003e) for the SL SRO. \u003cstrong\u003ef-h\u003c/strong\u003e, Same Ru displacement analyses for the top layer in the 3.0\u003csup\u003eο\u003c/sup\u003e, 4.8\u003csup\u003eο\u003c/sup\u003e, 7.3\u003csup\u003eο\u003c/sup\u003e twisted bilayers, respectively. The Ru displacements in \u003cstrong\u003ee\u003c/strong\u003e, \u003cstrong\u003ef\u003c/strong\u003e,\u003cstrong\u003e g\u003c/strong\u003e, and \u003cstrong\u003eh \u003c/strong\u003eare amplified by factors of 30, 30, 50, 100 for clarity, respectively. \u003cstrong\u003ei\u003c/strong\u003e, Histograms comparing the magnitude of Ru displacement of SL and t-BL SROs. \u003cstrong\u003ej-l\u003c/strong\u003e, Normalized flexoelectric field vector maps reconstructed by shear strain gradients, superimposed with their toroidal moment Q(E\u003csub\u003eflexo\u003c/sub\u003e). The yellow and purple circles in \u003cstrong\u003eb-d\u003c/strong\u003e, \u003cstrong\u003ef-h\u003c/strong\u003e, and \u003cstrong\u003ej-l \u003c/strong\u003eindicate the AA- and AB-stacked regions, respectively.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-6656512/v1/05b76012579bdfc593ec765a.png"},{"id":86243933,"identity":"bb776b27-9ca3-4981-9eac-263855c2c1e2","added_by":"auto","created_at":"2025-07-08 11:12:50","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":292542,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eDFT calculation for polar vortices in metallic SRO. a\u003c/strong\u003e, Planar view of constructed 2x2 supercell for the 18.92\u003csup\u003eo\u003c/sup\u003e twisted bilayer. The black and red dashed squares denote the supercell and unit cell of SRO, respectively. \u003cstrong\u003eb-c\u003c/strong\u003e,\u0026nbsp; In-plane Ru displacement maps of the top (red) and bottom (blue) layers, respectively. All displacements are amplified by a factor of 25 for clarity. The AA- and AB-stacked regions are marked with yellow and purple circles, respectively. \u003cstrong\u003ed\u003c/strong\u003e, Maximum in-plane Ru displacements as functions of the twist angle, extracted from the calculations. \u003cstrong\u003ee\u003c/strong\u003e, Electronic density of states of the polarized twisted bilayer. The Fermi level is indicated by a dashed line.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-6656512/v1/814becf439d24d80c02fb82f.png"},{"id":86243934,"identity":"df92c7f8-8ae4-4cae-bdf2-a4580c40eb6d","added_by":"auto","created_at":"2025-07-08 11:12:50","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":297841,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eMagnetic and electrical transport characterizations of polarized t-BL SrRuO\u003c/strong\u003e\u003csub\u003e\u003cstrong\u003e3\u003c/strong\u003e\u003c/sub\u003e\u003cstrong\u003e. a\u003c/strong\u003e, Temperature-dependent magnetization field-cooling curves of SL and BL SROs measured under an in-plane magnetic field of 100 Oe. The inset highlights the variations in ferromagnetic transition temperature.\u003cstrong\u003e b, \u003c/strong\u003eIn-plane ferromagnetic loops measured at 10 K.\u003cstrong\u003e c\u003c/strong\u003e, Twist-angle dependencies of ferromagnetic transition temperature (\u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e), saturation magnetization (\u003cem\u003eM\u003c/em\u003e\u003csub\u003es\u003c/sub\u003e) and Ru displacement (\u003cem\u003eδ\u003c/em\u003e\u003csub\u003eRu\u003c/sub\u003e). The dashed lines indicate the \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e and \u003cem\u003eM\u003c/em\u003e\u003csub\u003es\u003c/sub\u003e of SL SRO. The mean values of \u003cem\u003eδ\u003c/em\u003e\u003csub\u003eRu\u003c/sub\u003e are extracted from the Gaussian fitting in Fig. 2i.\u003cstrong\u003e d\u003c/strong\u003e, Normalized temperature-dependent resistance curves. The vertical dashed lines indicate the kinks in resistance, which is determined by linearly fitting the local raw data on the right and left sides. \u003cstrong\u003ee\u003c/strong\u003e, Magnetic exchange parameter (\u003cem\u003eJ\u003c/em\u003e) and mean Ru-O-Ru bond angle as a function of Ru displacement. \u003cstrong\u003ef\u003c/strong\u003e, Comparison of \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e from DFT calculations and experiments. The experimental values were obtained via linear interpolation based on the mean \u003cem\u003eδ\u003c/em\u003e\u003csub\u003eRu\u003c/sub\u003e and \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e across samples with varying twist angles.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-6656512/v1/96e4116da9446c1fb2e4eb1d.png"},{"id":86244099,"identity":"dc61ef94-fc48-456b-b73d-ec64410e3302","added_by":"auto","created_at":"2025-07-08 11:20:52","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2753361,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6656512/v1/926cc34f-c0a1-4b9a-a290-6b3e4c2fafe1.pdf"},{"id":86242998,"identity":"0ab33bfe-2050-483e-999a-8595fe160251","added_by":"auto","created_at":"2025-07-08 11:04:51","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":32040740,"visible":true,"origin":"","legend":"SUPPLEMENTARY INFORMATION","description":"","filename":"SupplementaryInformation.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6656512/v1/82d2a162b361b77ce1685299.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Polarization Vortices in a Ferromagnetic Metal via Twistronics","fulltext":[{"header":"Highlights","content":"\u003cp\u003eTwistronics enables the emergence of polarization vortices and multiferroic characteristics in metals.\u003c/p\u003e"},{"header":"Introduction","content":"\u003cp\u003eSince the discovery of superconductivity in twisted bilayer graphene\u003csup\u003e1\u003c/sup\u003e, the field of so-called “twistronics”\u003csup\u003e2,3\u003c/sup\u003e has blossomed. The key idea behind twistronics is that stacking two thin crystalline layers rotated with respect to each other generates a moiré interference pattern in the electronic density of states at the interface, which in turn modifies the electronic properties of the bilayer system. Twistronics was initially studied in the context of two-dimensional (2D) materials\u003csup\u003e4\u003c/sup\u003e, which can be readily exfoliated and stacked. The development of methods for growing epitaxial films of complex oxides and separating them from the substrate \u003cem\u003evia\u003c/em\u003e dissolution of a sacrificial layer\u003csup\u003e5–7\u003c/sup\u003e allows these materials to partake in the twistronic game. A seminal result in this context has been the discovery that two layers of BaTiO\u003csub\u003e3\u003c/sub\u003e twisted with respect to each other display polarization vortices\u003csup\u003e8\u003c/sup\u003e. The proposed explanation is that the torsional strain introduced at the interface between the twisted layers is not homogeneous, being minimal at the sites where the atomic columns of the two layers coincide across the interface, and maximal where equivalent atomic columns of the two layers are furthest apart. This causes torsional strain gradients and therefore gradient-induced polarization, known as flexoelectricity\u003csup\u003e9,10\u003c/sup\u003e.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFlexoelectricity is allowed in materials of any symmetry\u003csup\u003e11\u003c/sup\u003e, and twist-induced polar vortices have also been reported in SrTiO\u003csub\u003e3\u003c/sub\u003e\u003csup\u003e12\u003c/sup\u003e, which is cubic and non-polar in the absence of strain (although strain can make it ferroelectric\u003csup\u003e13\u003c/sup\u003e). Since there are not strain gradients without strain, it can be difficult to separate how much of the polarization vortices is due to gradient-induced flexoelectricity, and how much to strain-induced ferroelectricity or piezoelectricity. While piezoelectricity has been ruled out from the explanation of vortices based on geometrical considerations, the fact remains that there is a strong polarization to start with that may be affected both by strain and by the depolarization fields expected whenever polarization is disrupted. It is therefore highly desirable to examine whether polar vortices can be induced in a material where piezoelectricity and/or depolarization fields can be fully excluded. Importantly, if the origin of polar vortices is truly flexoelectric, we no longer need to limit ourselves to examining insulating materials in our on-going search for chiral polarization\u003csup\u003e14,15\u003c/sup\u003e: metals can also be flexoelectric\u003csup\u003e16–18\u003c/sup\u003e. Therefore, a tantalizing possibility emerges of inducing polarization vortices in a metal, where the depolarization field engineering approach typically used in ferroelectrics is just not available.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIn this work, we have fabricated twisted-bilayer (t-BL) stacks of SrRuO\u003csub\u003e3\u003c/sub\u003e (SRO), a well-known oxide metal whose flexoelectricity has been recently reported\u003csup\u003e17\u003c/sup\u003e, and we have combined atomic-scale imaging with theoretical calculations to examine the polar displacement pattern. The results show that the Ru cation is shifted with respect to the Sr lattice, forming periodic arrays of vortices and antivortices like those reported in insulating BaTiO\u003csub\u003e3\u003c/sub\u003e and SrTiO\u003csub\u003e3\u003c/sub\u003e\u003csup\u003e8,12\u003c/sup\u003e, displaying a significant twist-dependence. Moreover, we observed that the twisted-stacking induced polarization has functional consequences on the ferromagnetic and metallic properties, indicating multiferroic characteristics in metallic membranes.\u003c/p\u003e\n"},{"header":"Film growth and structural characterization","content":"\u003cp\u003eEpitaxial thin films of SRO (10.7 nm) were grown by pulsed-laser deposition (PLD) on (001)-oriented SrTiO\u003csub\u003e3\u003c/sub\u003e (STO) substrates with water-soluble Sr\u003csub\u003e3\u003c/sub\u003eAl\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e6\u0026nbsp;\u003c/sub\u003e(SAO) buffer layer. The films were attached onto a polydimethylsiloxane (PDMS) stamp and immersed in deionized water to dissolve the SAO layer. Upon removal of SAO and separation of SRO from the substrates, the freestanding single-layer (SL) SRO membranes were transferred onto silicon oxide or TEM supports. The t-BL SRO was assembled by stacking a second membrane, released from the same epitaxial film and transferred onto the first one with a prescribed rotational angle. To remove potential surface adsorbates at the interface and/or hydrogen intercalation\u003csup\u003e19\u003c/sup\u003e, the transferred samples were annealed at 330\u0026nbsp;℃\u0026nbsp;for 8 hours in an oxygen atmosphere before measurements.\u0026nbsp;The schematic of fabrication process\u0026nbsp;is provided in Fig. 1a,\u0026nbsp;and further details\u0026nbsp;for film growth and transfer are\u0026nbsp;described\u0026nbsp;in Methods.\u003c/p\u003e\u003cp\u003eThe crystallinity of the initial epitaxial films and subsequently transferred membranes was confirmed by X-ray diffraction (XRD) 2θ-ω scans (Fig. 1b). The fully epitaxial strained state of the as-grown heterostructures was confirmed using reciprocal space mapping (Fig. 1c), revealing that the SRO and SAO layers share the same in-plane lattice parameter as the STO substrate. The diffraction peak of the transferred membrane (Fig. 1b), however, is shifted, suggesting a relaxation of the lattice upon epitaxial strain release. The morphologies of the as-grown film and the transferred bilayer membrane (Fig. 1d and Supplementary Fig. S1) show smooth and flat surfaces with roughness of 0.25 nm and 0.38 nm, respectively. The bilayer interface was examined using cross-sectional scanning transmission electron microscopy with high-angle annular dark-field (STEM-HAADF) imaging (Supplementary Fig. S2). A high-quality interface with minimal interlayer distance of 4.97 Å is observed in the twisted bilayer.\u003c/p\u003e"},{"header":"Polarization vortices in twisted-bilayer SrRuO3","content":"\u003cp\u003eWe used high-resolution STEM-HAADF imaging to visualize the in-plane atomic structure of the metallic membranes and determine the eventual presence of polarization vortices (see Methods). Bulk SRO crystallizes in a centrosymmetric perovskite structure with orthorhombic symmetry (Pbnm) at room temperature\u003csup\u003e20\u003c/sup\u003e. We found that, even when the size reduces to the nanometer scale,\u0026nbsp;the freestanding SL SRO membrane\u0026nbsp;retains its orthorhombic characteristics, as\u0026nbsp;evidenced by the presence of\u0026nbsp;½ superlattice spots in the\u0026nbsp;selected area electron diffraction (SAED) pattern (Supplementary\u0026nbsp;Fig. S3).\u0026nbsp;To quantify polarization, we measured the off-center displacement of Ru (\u003cem\u003eδ\u003c/em\u003e\u003csub\u003eRu\u003c/sub\u003e) with respect to its four adjacent Sr atoms. The Ru displacement extracted from the STEM-HAADF image of the SL SRO (Figs. 2a, e) is both random and minimal (1.3 pm on average),\u0026nbsp;indicating that the freestanding SRO remains nonpolar\u0026nbsp;as its bulk counterpart.\u003c/p\u003e\u003cp\u003eThe t-BL SRO exhibits a different behaviour. Upon focusing the electron beam at the bilayer interface, a continuous and well-ordered moiré was found in\u0026nbsp;low-magnification image\u0026nbsp;(Supplementary Fig. S4), suggesting a homogeneous interfacial\u0026nbsp;contact.\u0026nbsp;Subsequent\u0026nbsp;high-resolution\u0026nbsp;moiré images measured from the\u0026nbsp;3.0\u003csup\u003eο\u003c/sup\u003e, 4.8\u003csup\u003eο\u003c/sup\u003e, and\u0026nbsp;7.3\u003csup\u003eo\u003c/sup\u003e twisted bilayers are shown in Figs. 2b-d,\u0026nbsp;displaying\u0026nbsp;a decreasing\u0026nbsp;moiré\u0026nbsp;periodicity\u0026nbsp;with increasing twist angle. The twist angle was determined from the relative rotation between the two distinct sets of diffraction spots observed in the fast Fourier transform (FFT) of the moiré images (Supplementary Fig. S5). By shifting the focus away from\u0026nbsp;the interface to the upper surface, we\u0026nbsp;selectively imaged\u0026nbsp;the lattice structure of the top layer (Supplementary Fig. S6). Prior to this, we conducted depth-dependent STEM imaging and analyzed the\u0026nbsp;corresponding variations in FFT patterns. The results confirmed that the diffraction spots from the bottom lattice completely vanish\u0026nbsp;as the focus shifts to the upper surface\u0026nbsp;(Supplementary Fig. S7), thus\u0026nbsp;excluding\u0026nbsp;geometric\u0026nbsp;interference\u0026nbsp;artifacts from the bottom layer in the images obtained at top layer.\u0026nbsp;\u003c/p\u003e\u003cp\u003eThe Ru displacement maps of the top layers (Figs. 2f-h), calculated based on the lattice structure observed at STEM-HAADF images (Supplementary Fig. S6), reveal a periodic arrangement of alternating vortices and antivortices, indicating that chiral polarization has emerged in the metallic membranes. While chiral polarization is becoming increasingly frequent among ferroelectrics, where it can be driven either by depolarization field\u003csup\u003e21,22\u003c/sup\u003e or torsional strain gradients\u003csup\u003e8\u003c/sup\u003e, it\u0026nbsp;is unprecedented in SRO or any other metal. The toroidal moment was quantified as\u0026nbsp;\u003cimg width=\"128\" height=\"40\" src=\"data:image/wmf;base64,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\" alt=\"image\"\u003e, where\u0026nbsp;\u003cstrong\u003eδ\u003c/strong\u003e\u003cem\u003e\u003csub\u003ei\u003c/sub\u003e\u003c/em\u003e is the adjacent Ru displacement vector located at position \u003cstrong\u003er\u003c/strong\u003e\u003cem\u003e\u003csub\u003ei\u003c/sub\u003e\u003c/em\u003e, and \u003cem\u003eN\u003c/em\u003e = 24 is the number of adjacent cells, and then superimposed onto the Ru displacement maps to visualize the spatial arrangement of polar vortices (Figs. 2f-h). The\u0026nbsp;periodic array of negative toroidal moments\u0026nbsp;reveals that\u0026nbsp;the AA- stacked (Sr-Sr and Ru-Ru sites) regions host clockwise vortices, while\u0026nbsp;anticlockwise vortices form near the AB- stacked\u0026nbsp;(Sr-Ru sites)\u0026nbsp;regions.\u0026nbsp;These features in polarization vortices can be robustly observed in\u0026nbsp;the different\u0026nbsp;bilayers even with a high twist angle of 10.4\u003csup\u003eo\u003c/sup\u003e (Supplementary Fig. S8) but\u0026nbsp;were entirely absent\u0026nbsp;in the\u0026nbsp;SL SRO\u0026nbsp;(Fig. 2e). As the twist angle increases, the periodicity\u0026nbsp;of\u0026nbsp;polar vortices in different bilayers becomes smaller, and always coincides with the moiré periodicity,\u0026nbsp;consistent with\u0026nbsp;the twistronic origin of the polarization.\u003c/p\u003e\u003cp\u003eThe magnitudes of the toroidal moment and the polarization also decrease as the twisting increases. Figure 2i shows that the Ru displacement exhibits an inverse dependence on the twist angle, being more significant at the smallest twist angle. The maximum displacement reaches\u0026nbsp;~20 pm (8.3 pm on average) for the 3.0\u003csup\u003eo\u003c/sup\u003e twisted bilayer, which is comparable to reported values for polarized SRO\u003csup\u003e17,23,24\u003c/sup\u003e. To exclude the possibility that the observed polarization vortices arise from imaging artifacts associated with the moiré pattern, we performed the same measurements on unannealed samples. The idea is that a weaker interlayer coupling at the unannealed interface should make it difficult to distort the lattice and polarize it. Indeed, the unannealed samples remain non-polar, with a Ru displacement map (Supplementary Fig. S9) like that of SL SRO, despite the presence of a moiré pattern at the interface. The observed polarization vortices are therefore not a geometric interference artifact, since the moiré exists both in the annealed and unannealed bilayers.\u003c/p\u003e\u003cp\u003eSince electrostatic interactions are expected to be screened by the free charges in metallic systems, whereas strain (gradient) cannot be screened, the \u0026nbsp;polar vortices of the SRO bilayers consistent with a mechanical rather than electrostatic origin, lending support to the flexoelectric hypothesis\u003csup\u003e8,25\u003c/sup\u003e:\u0026nbsp;the periodic misalignment of the atoms in the top and bottom layers generates shear torsional strain gradients\u0026nbsp;and thus flexoelectric fields, which can serve as pseudo-electric fields, manipulating polarization\u003csup\u003e26,27\u003c/sup\u003e and driving the formation of polar vortices. The strain-gradient origin is further supported by the geometric phase analysis (GPA) (see Supplementary Text I), showing that the shear strain gradients are distributed in a strip network of alternating signs, with the polarization components pointing in the same direction within each strip region (Supplementary Fig. S10). A flexoelectric coupling in metals is evidenced as the polarization systematically switches towards the opposite direction upon the reversal of the strain gradients. The flexoelectric fields reconstructed by shear strain gradients (Figs. 3j-l) significantly correlate with the observed polar vortices maps, providing direct supports of a flexoelectric origin for the polar vortices. This electromechanical coupling accounts for the observed twist angle-dependent polarization, where the larger polarization at small angle is attributed to the more significant lattice distortion (shear strain) and its localized inhomogeneity (strain gradients).\u0026nbsp;\u003c/p\u003e"},{"header":"DFT calculation for polarized t-BL SrRuO3","content":"\u003cp\u003eWe then used first-principles density functional theory (DFT) to simulate the polarization in t-BL SRO (see Methods). Commensurate bilayer supercell structure was constructed by stacking two individual unit-cell-thick layers with a relatively high twist angle of 18.92\u003csup\u003eo\u003c/sup\u003e. The smaller moiré periodicity with fewer atoms in a high-angle model improves computational efficiency. The established twisted bilayer model, depicted in Fig. 3a, exhibits a characteristic moiré pattern arising from alternating AA- and AB-stacked sequences. After fully relaxing structure, we computed the off-center displacement of Ru atoms relative to the surrounding oxygen octahedra in each unit cell. The corresponding maps (Figs. 3b-c) reveal the formation of polarization vortices in the top and bottom layers, with opposite chirality in each layer. In the top layer, clockwise vortices predominantly form at AA-stacked regions and rough anticlockwise ones at AB-stacked regions, consistent with our experimental observations. Similar polarization configuration can be robustly reproduced in bilayers with higher twist angles of 22.62\u003csup\u003eo\u003c/sup\u003e and 28.07\u003csup\u003eo\u003c/sup\u003e (Supplementary Fig. S13). The calculated Ru displacement decreases with twist angle (Fig. 3d), in agreement with experimental results. Since the calculations adopted unit-cell-thick SRO layers, the lattice near the bilayer interface is expected to be subjected to larger strain, the magnitude of calculated polarization is thereby larger than the experimental one that measured at the upper surface. To demonstrate the crucial role of twisted stacking, we conducted the same calculations for 0\u003csup\u003eo\u003c/sup\u003e-stacked bilayers with different translational shifts; irrespective of AA- or AB-stacked bilayers, SRO maintains a centrosymmetric structure without any polarization (Supplementary Fig. S14). Additionally, we calculated the electronic density of states (DOS) of the polarized twisted bilayers (Fig. 3e and Supplementary Fig. S15). All twisted bilayers exhibit nonzero density of states at the Fermi level, primarily contributed by Ru \u003cem\u003ed\u003c/em\u003e-orbital electrons, confirming their metallic nature. The overlap of the Ru \u003cem\u003ed\u003c/em\u003e and O \u003cem\u003ep\u003c/em\u003e states indicates a strong Ru-O orbital hybridization. Overall, our calculations provide solid theoretical evidence that topological polarization can be physically compatible with metallicity in twisted bilayers.\u003c/p\u003e"},{"header":"Functional consequences of polar vortices","content":"\u003cp\u003eHaving established the presence of polarization vortices in t-BL SROs, the question arises: do these vortices have any tangible impact on the material's functional properties? A consideration is that bulk SRO is a paramagnetic metal at room temperature, but it undergoes a transition to a ferromagnetic state below ~160 K\u003csup\u003e28\u003c/sup\u003e (~150K for epitaxial films\u003csup\u003e29\u003c/sup\u003e). This ferromagnetic ordering\u0026nbsp;manifests\u0026nbsp;as a\u0026nbsp;distinct\u0026nbsp;change in the electrical\u0026nbsp;resistance\u003csup\u003e30,31\u003c/sup\u003e. Given that polarization, conductivity, and magnetism are all closely linked to the Ru atoms and their surrounding oxygen octahedra, coupling among these properties may be possible.\u003c/p\u003e\u003cp\u003eWith this in mind, we measured the temperature-dependent magnetization (\u003cem\u003eM\u003c/em\u003e-\u003cem\u003eT\u003c/em\u003e) curve and ferromagnetic hysteresis loop (\u003cem\u003eM\u003c/em\u003e-\u003cem\u003eH\u003c/em\u003e) for SL and t-BL SROs (see Methods). Figures 4a-b show that the t-BL SROs exhibit ferromagnetism with higher coercive field at low temperature. We examined the variation in ferromagnetic transition temperature and magnetic moment, which show strong dependencies on the twist angle. The Curie temperature \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e (inset of Fig. 4a) and the saturation magnetization \u003cem\u003eM\u003c/em\u003e\u003csub\u003es\u003c/sub\u003e (hysteresis loops in Fig. 4b) both decrease as the twist angle is reduced. The twist-dependent \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e and \u003cem\u003eM\u003c/em\u003e\u003csub\u003es\u003c/sub\u003e are summarized and compared with\u0026nbsp;\u003cem\u003eδ\u003c/em\u003e\u003csub\u003eRu\u003c/sub\u003e in Fig. 4c, showing that the polarization magnitude and the ferromagnetism display opposite trends with respect to the twist angle, suggesting a competition between the two types of ordering. We also measured the temperature-dependent resistance curves (Supplementary Fig. S16), which show that all samples exhibit typical metallic conductivity. The kink in resistance due to the ferromagnetic transition, highlighted in Fig. 4d, show the same twist angle-dependence. These results reveal coexistence and interaction of polar vortices and ferromagnetic ordering in correlated metallic systems, indicating that SRO becomes a “multiferroic metal” under the twisted stacking.\u003c/p\u003e\u003cp\u003eThe apparent competition between polar and magnetic orderings echoes the known difficulty of stabilizing ferroelectricity and ferromagnetism within the same material\u003csup\u003e32,33\u003c/sup\u003e. To understand this multiferroic competition, we constructed an orthorhombic SRO structure and modulated polarization by manually off-centering Ru cations, thus approximating the polarization observed in t-BL SROs. The magnetic exchange parameter of the Heisenberg model was calculated based on DFT total energies of ferromagnetic and anti-ferromagnetic phases (see Methods). The results shown in Fig. 4e indeed reveal a gradual decrease in the exchange parameter with increasing polarization, consistent with the observed lower magnetization. The reduced exchange interaction is attributed to the decreasing of the mean Ru-O-Ru bond angle due to the Ru displacement along the polar direction. This off-centering reduces the overlap between Ru 4d and O 2p orbitals, thereby weakening the Ru–O hybridization. The Curie temperature estimated based on exchange parameter also decreases with increasing polarization, in good agreement with the experimental observations (Fig. 4f).\u0026nbsp;\u003c/p\u003e"},{"header":"Conclusion","content":"\u003cp\u003eThe discovery of polarization vortices in metallic membranes expands the field of polar topology\u003csup\u003e34,35\u003c/sup\u003e into metals and strongly correlated electron systems\u003csup\u003e36,37\u003c/sup\u003e. Our findings demonstrate that moiré engineering can induce polarization vortices in a metallic system, and, since SRO is also ferromagnetic, the result is a multiferroic metal. The results indicate an anticorrelation and therefore a possible competition between the two orders. Beyond SRO, our results highlight the broader implications of flexoelectricity as a universal property, suggesting that similar polarization phenomena could emerge in other twisted bilayer metallic systems. The presence of polarization within a metal introduces new degrees of freedom for spintronic applications, mediated by flexo-Rashba effect\u003csup\u003e18\u003c/sup\u003e, opening\u0026nbsp;new avenues for designing materials with tunable electronic, magnetic, and topological properties, and exploring potential polarization-related behaviour in metallic systems.\u0026nbsp;\u003c/p\u003e\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n"},{"header":"Methods","content":"\u003cp\u003e\u003cstrong\u003eEpitaxial film growth\u003c/strong\u003e\u003c/p\u003e\u003cp\u003eAll epitaxial films were synthesized using pulsed-laser deposition (PLD) with a KrF excimer laser (248 nm, COMPex 102, Lambda Physik). The SrRuO\u003csub\u003e3\u003c/sub\u003e (SRO) thin films were grown on (100)-oriented SrTiO\u003csub\u003e3\u003c/sub\u003e (STO) substrates with an 8.8 nm-thick Sr\u003csub\u003e3\u003c/sub\u003eAl\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e6\u003c/sub\u003e (SAO) sacrificial layer to enable the subsequent membrane release. Prior to deposition, the chamber was evacuated to a base pressure of 1.0 × 10\u003csup\u003e-4\u003c/sup\u003e mTorr. Stoichiometric ceramic targets were used to ensure the chemical composition of films. The SAO layer was deposited at 750\u0026nbsp;℃\u0026nbsp;under an oxygen pressure of 1 mTorr, with a laser fluence of 1.60 J/cm\u003csup\u003e2\u003c/sup\u003e. The SRO layer was subsequently grown at 700\u0026nbsp;℃\u0026nbsp;under an oxygen pressure of 100 mTorr O\u003csub\u003e2\u003c/sub\u003e, with a laser fluence of 1.28 J/cm\u003csup\u003e2\u003c/sup\u003e. The laser repetition rate was maintained at 1 Hz throughout the deposition process. After deposition, the films were cooled down to room temperature at 5\u0026nbsp;℃/min under an oxygen pressure of 10 Torr.\u0026nbsp;\u003c/p\u003e\u003cp\u003e\u003cstrong\u003eTwisted bilayer membrane transfer\u003c/strong\u003e\u003c/p\u003e\u003cp\u003eTo obtain freestanding SRO membranes, the epitaxial films were adhered to a polydimethylsiloxane (PDMS) stamp and immersed in deionized water to dissolve the SAO sacrificial layer. The complete dissolution of SAO led to the detachment of the film from the substrate. The released membrane, attached to the PDMS stamp, was then transferred onto a target substrate using a transfer platform that allows to control the rotation of the substrate with a precision of 0.1\u003csup\u003eo\u003c/sup\u003e. Half of the membrane on the PDMS was brought into contact with the substrate, and the temperature was then increased to 40\u0026nbsp;℃\u0026nbsp;to remove residual water. The PDMS was subsequently peeled off slowly, ensuring an intact transfer of the membrane. To make a twisted-bilayer sample, the substrate with the transferred bottom layer was rotated to a predefined angle using a micrometer probe in the transfer system. The remaining half of the membrane on the PDMS was then transferred onto the bottom one in the same manner, thereby obtaining twisted-bilayer samples. To enhance interlayer interaction and eliminate possible adsorbates at the interface, the samples were annealed at 330\u0026nbsp;℃\u0026nbsp;for 8 hours under oxygen flow rate of 50 sccm.\u003c/p\u003e\u003cp\u003e\u003cstrong\u003eX-ray diffraction (XRD) and surface characterizations\u003c/strong\u003e\u003c/p\u003e\u003cp\u003eHigh-resolution 2𝜃-𝜔\u0026nbsp;scans and reciprocal space mappings (RSMs) were conducted using a Panalytical X’pert Pro diffractometer with a Cu K𝛼\u003csub\u003e1\u003c/sub\u003e (1.540598Å) source. The system featured a hybrid two-bounce primary monochromator on the incident beam side and a PIXcel line detector for high-resolution data acquisition. The surface morphology was characterized using MFP-3D Asylum AFM (Asylum Research -Oxford Instruments). Tapping mode was performed to minimize surface damage using a soft tip.\u003c/p\u003e\u003cp\u003e\u003cstrong\u003eTEM, SAED, and STEM\u003c/strong\u003e\u003cem\u003e\u0026nbsp;\u003c/em\u003e\u003cstrong\u003echaracterizations\u003c/strong\u003e\u003c/p\u003e\u003cp\u003eThe low-magnification HRTEM images and SAED patterns were acquired using a FEI Tecnai F20 TEM at 200 kV. High-quality cross-sectional samples were fabricated using the focused ion beam (FIB) technique at a Thermo Fisher Helios 5 UX dual-beam system. The STEM experiments were conducted using a double aberration-corrected Spectra 300 operated at 300 kV. Depth-sectioning STEM-HAADF was performed by acquiring a series of atomic-resolution images at different defocus depth\u003csup\u003e8\u003c/sup\u003e, allowing for discrimination between the top layer and the bilayer interface. STEM-HAADF images were acquired with a probe convergence semi-angle of 27.3 mrad while the HAADF detector collection semiangle was set as 63–200 mrad.\u0026nbsp;\u003c/p\u003e\u003cp\u003e\u003cstrong\u003ePolarization vortices and strain analysis\u003c/strong\u003e\u003c/p\u003e\u003cp\u003eThe atomic positions of Ru and Sr were firstly extracted from Wiener-filtered STEM-HAADF images using the Python package Atomap\u003csup\u003e38\u003c/sup\u003e. After locating the Ru and Sr columns, the off-center displacement of each unit cell was calculated as the deviation of the Ru column from the mass center of the four adjacent Sr columns. The mean Ru displacement value is subtracted in the mappings to improve the visibility of the polarization vortex array. Strain analysis was performed using the Geometrical Phase Analysis (GPA)\u003csup\u003e39\u003c/sup\u003e software package (HREM Research) in Digital Micrograph.\u0026nbsp;\u003c/p\u003e\u003cp\u003e\u003cstrong\u003eElectrical transport and magnetization measurements\u003c/strong\u003e\u003c/p\u003e\u003cp\u003eTemperature-dependent resistance measurements were performed in an Advanced Research Systems cryostat using a Keithley 2614B SourceMeter. The SL and t-BL membranes were transferred onto an insulating oxidized silicon substrate, and the Pt electrode arrays were sputtered through a square array-mask placed on the sample surface, defining 100 μm electrodes separated by 20 μm spacing. The electrodes were contacted via wire-bonding to a leadless chip carrier (Kyocera) electrically connected to the sourcementer by a chip-carrier socket, and thermally connected to the cold finger of the cryostat. The resistance was measured using two-point contact mode, as the temperature gradually varied at a rate of 1 K/min. Ferromagnetic hysteresis loops and temperature-dependent magnetization field-cooling curves were measured using a vibrating sample magnetometer (VSM, Quantum Design MPMS-3). The SL and t-BL membranes were transferred onto sapphire substrates. The external magnetic field was applied parallel to the membrane plane.\u003c/p\u003e\u003cp\u003e\u003cstrong\u003eDFT calculation for polar vortex\u003c/strong\u003e\u003c/p\u003e\u003cp\u003eDFT calculations were performed using the Vienna Ab Initio Simulation Package (VASP)\u003csup\u003e40,41\u003c/sup\u003e. The projector-augmented wave (PAW) method\u003csup\u003e42\u003c/sup\u003e was employed, with the Perdew–Burke–Ernzerhof (PBE)\u003csup\u003e43\u003c/sup\u003e generalized gradient approximation (GGA)\u003csup\u003e44\u003c/sup\u003e used as the exchange-correlation functional. The cubic structure was adopted to construct bilayer supercell structures, following the method proposed by Lee et al.\u003csup\u003e45\u003c/sup\u003e. The electronic wave functions were expanded using a plane-wave basis set with a kinetic energy cutoff of 600 eV and the 12×12×1 k-point mesh was used to sample the Brillouin zone. The monolayer structure was built based on the relaxed bulk SRO and was fully relaxed. The equilibrium lattice constant of cubic monolayer was calculated to be 3.781 Å, and the same value was used for all bilayer structures. The 12×12×1 k-mesh was used for monolayer and untwisted bilayer structures, whereas Γ-point sampling was adopted for the twisted structures. Structural relaxations were performed until the Hellmann-Feynman forces on each atom were reduced below 0.01 eV/Å. Van der Waals interactions between adjacent layers were accounted for using the Grimme D3 method\u003csup\u003e46\u003c/sup\u003e. To prevent spurious interlayer interactions, a vacuum region exceeding 15 Å was introduced in the supercell.\u003c/p\u003e\u003cp\u003e\u003cstrong\u003eDFT calculation for magnetism\u003c/strong\u003e\u003c/p\u003e\u003cp\u003eThe magnetic properties were investigated based on an orthorhombic structure with manually off-centering Ru along the \u003cem\u003ec\u003c/em\u003e-axis direction (\u003cem\u003ei.e.\u003c/em\u003e [001]-orientation), which lies within the membrane plane in as-grown film. In DFT calculations, a plane-wave basis set with a kinetic energy cutoff of 500 eV was used to expand the electronic wave functions, and an 8 × 8 × 6 k-mesh was employed to sample the Brillouin zone. The on-site Coulomb interaction parameters were set to U = 3 eV and J = 0.3 eV for Ru 4d orbitals. The Curie temperature was estimated using the Heisenberg model and the self-consistent Gaussian approximation (SCGA) method\u003csup\u003e47\u003c/sup\u003e. Specifically, the magnetic exchange parameter (\u003cem\u003eJ\u003c/em\u003e) of Heisenberg model was derived from the DFT calculated total energies of ferromagnetic and\u0026nbsp;G-type antiferromagnetic\u0026nbsp;phases\u003csup\u003e48,49\u003c/sup\u003e, as shown in Supplementary Fig. S17,\u003c/p\u003e\u003cp\u003e\u003cimg width=\"215\" height=\"31\" src=\"data:image/wmf;base64,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\" alt=\"image\"\u003e, \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;(1)\u003c/p\u003e\u003cp\u003ewhere\u0026nbsp;\u003cem\u003eE\u003c/em\u003e\u003csub\u003eAFM\u003c/sub\u003e (\u003cem\u003eE\u003c/em\u003e\u003csub\u003eFM\u003c/sub\u003e) is the total energy of the ferromagnetic and antiferromagnetic phases, \u003cem\u003eN\u003c/em\u003e\u003csub\u003emag\u003c/sub\u003e = 4 is the number of magnetic Ru ions in the unit cell, and \u003cem\u003eN\u003c/em\u003e\u003csub\u003enn\u003c/sub\u003e = 6 is the number of nearest-neighbour Ru ions of each Ru ion. The Curie temperature was estimated using SCGA method\u003csup\u003e47\u003c/sup\u003e as\u003c/p\u003e\u003cp\u003e\u003cimg width=\"117\" height=\"40\" src=\"data:image/wmf;base64,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\" alt=\"image\"\u003e, (2)\u003c/p\u003e\u003cp\u003e\u003cem\u003eθ\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e = 0.719 is a structure dependent parameter\u003csup\u003e48\u003c/sup\u003e.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgments\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors acknowledge support by the National Key Projects for Research and Development of China (Grant No. 2021YFA1400300), the National Natural Science Foundation of China (Grant Nos. 12172047, 12402183, 12374080), the Beijing Natural Science Foundation (Grant No. 1244057). We also acknowledge ICN2 and ALBA (JEMCA) for providing key facilities and technical guidance. ICN2 is supported by the Severo Ochoa program from Spanish MCIN / AEI (Grant No. CEX2021-001214-S), the European Regional Development Fund from European Union (Grant No. IU16-014206 (METCAM-FIB)), and the CERCA Programme, Generalitat de Catalunya. ICN2 is a founding member of e-DREAM. J.A. and X.H. acknowledge funding from Generalitat de Catalunya (Grant No. 2021SGR00457). This study is part of the Advanced Materials programme and was supported by MCIN with funding from European Union NextGenerationEU (PRTR-C17.I1) and by Generalitat de Catalunya (In-CAEM Project). We acknowledge support from CSIC Interdisciplinary Thematic Platform (PTI+) on Quantum Technologies (PTI-QTEP+). This work has been funded by the European Commission – NextGenerationEU (Regulation EU 2020/2094), through CSIC's Quantum Technologies Platform (QTEP). Y.L. acknowledges support from the Severo Ochoa Seed Funding program (Grant CEX2021-001214-S/MICIU/AEI/10.13039/501100011033). D.P. acknowledges the\u0026nbsp;“Consolidación Investigadora” grant CNS2024-154522\u0026nbsp;and grant PID2022-140589NB-I00,\u0026nbsp;funded by MCIN/AEI/10.13039/501100011033. This work is in the framework of the Universitat Autonoma de Barcelona Materials Science PhD program. X.H. acknowledges PhD\u0026nbsp;scholarship support from the China Scholarship Council (CSC) (Grant No. 202304910019). Q.R. acknowledges support from the BIT Research and Innovation Promoting Project Graduate (Grant No. 2023YCXZ002).\u0026nbsp;U.S.\u0026nbsp;acknowledges the funding for this project by Grant PID2019-108573GB-C21 (FOx-Me) funded by the Spanish Ministry of Science and Innovation (DOI: 10.13039/501100011033).\u0026nbsp;I.P.-H. acknowledges funding from AGAUR-FI scholarship (Grant No. 2023FI-00268) Joan Oró of the Secretariat of Universities of the Generalitat of Catalonia and the European SocialPlus Fund. Calculations were performed using resources of the Supercomputer Centre in Chongqing. The authors acknowledge the use of the Spectra 300 microscope, provided under ALBA Synchrotron proposal (Grant No. 20240320035). The authors gratefully acknowledge the assistance of Francesco Salutari, Athique Ahmed Ali and Francisco Javier Belarre Triviño in the preparation of cross-sectional FIB samples, and Bernat Bozzo in the magnetic measurements.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eNote: after writing this manuscript, we have become aware of another work on SrRuO\u003csub\u003e3\u003c/sub\u003e membranes (Alejandro Martin Merodio et al., group of Jacobo Santamaría at the Universidad Complutense de Madrid, presented at QUOROM, 22/05/25). So far they only report transport properties, but we expect their membranes will also display vortices.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eExperimental measurements were mainly performed at ICN2 and ALBA (JEMCA), and theoretical calculations were carried out at BIT. All institutions contributed substantially to the development and completion of this work. G.C., J.H., and J.A. designed and supervised the project. Y.L. conceived the original idea for this work. Y.L. fabricated the thin films and transferred the samples with the help from U.S. and J.M.C.R.. X.H., J.A., B.M., K.G., and I.P. conducted STEM, SAED and TEM experiments. X.H. and Y.L. performed data analysis on the polar vortices, strain gradients, and flexoelectric fields. Q.R., G.T., and J.H. carried out the atomistic simulations and analysis. Y.L. conducted the ferromagnetism and resistance measurements under the guidance of D.P. and X.W.. Y.L. performed the surface morphology measurements. J.P. and J.S. conducted the XRD measurements. Y.M. and Q.L. provided assistance with the experiments. Y.L., X.H. and G.C. wrote the manuscript with the help from all authors. All authors contributed to the discussion of the results and the revision of the manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare that they have no competing interests.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData and materials availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAll data are available in the manuscript or the supplementary information.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eCao, Y. \u003cem\u003eet al.\u003c/em\u003e Unconventional superconductivity in magic-angle graphene superlattices. \u003cem\u003eNature\u003c/em\u003e \u003cstrong\u003e556\u003c/strong\u003e, 43\u0026ndash;50 (2018).\u003c/li\u003e\n\u003cli\u003eAndrei, E. Y. \u0026amp; MacDonald, A. H. Graphene bilayers with a twist. \u003cem\u003eNat. 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Strain induced metal\u0026ndash;insulator transition of magnetic SrRuO\u003csub\u003e3\u003c/sub\u003e single layer in SrRuO\u003csub\u003e3\u003c/sub\u003e/SrTiO\u003csub\u003e3\u003c/sub\u003e superlattice. \u003cem\u003eAppl. Sci.\u003c/em\u003e \u003cstrong\u003e8\u003c/strong\u003e, 2151 (2018).\u003c/li\u003e\n\u003cli\u003eJiao, P., Liu, Y., Wang, X. \u0026amp; Chen, J. First principles investigation of Na doping effects on the structural, magnetic, and electronic properties in SrRuO\u003csub\u003e3\u003c/sub\u003e. \u003cem\u003eComput. Mater. Sci.\u003c/em\u003e\u003cstrong\u003e69\u003c/strong\u003e, 284\u0026ndash;288 (2013).\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nature-portfolio","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"","title":"Nature Portfolio","twitterHandle":"","acdcEnabled":false,"dfaEnabled":false,"editorialSystem":"ejp","reportingPortfolio":"","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-6656512/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6656512/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eRecent advances in moiré engineering have revealed new pathways for manipulating lattice distortions and electronic properties in low-dimensional materials. Here, we demonstrate that twisted-stacking can induce polarization vortices in metallic SrRuO\u003csub\u003e3\u003c/sub\u003e membranes, despite the presence of free charge screening. These polarization vortices are correlated with moiré-periodic flexoelectricity induced by shear strain gradients, and exhibit a pronounced dependence on the twist angle. In addition, multiferroic behavior emerges below the ferromagnetic Curie temperature of the films, whereby polarization and ferromagnetism coexist and appear to compete, showing opposite twist-angle dependencies of their respective magnitudes. Density functional theory calculations provide insights into the microscopic origin of these observations. Our findings extend the scope of topological polarization design beyond dielectric materials and into metals.\u003c/p\u003e","manuscriptTitle":"Polarization Vortices in a Ferromagnetic Metal via Twistronics","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-07-08 11:04:46","doi":"10.21203/rs.3.rs-6656512/v1","editorialEvents":[],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"nature-materials","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"nmat","sideBox":"Learn more about [Nature Materials](http://www.nature.com/nmat/)","snPcode":"","submissionUrl":"","title":"Nature Materials","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"ejp","reportingPortfolio":"Nature Research","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"77d8d621-bf88-40b3-8785-8c7a5786641b","owner":[],"postedDate":"July 8th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[{"id":51168219,"name":"Physical sciences/Materials science/Condensed-matter physics/Ferroelectrics and multiferroics"},{"id":51168220,"name":"Physical sciences/Physics/Condensed-matter physics/Ferroelectrics and multiferroics"},{"id":51168221,"name":"Physical sciences/Physics/Electronics, photonics and device physics/Electronic and spintronic devices"},{"id":51168222,"name":"Physical sciences/Materials science/Nanoscale materials/Two-dimensional materials"}],"tags":[],"updatedAt":"2026-03-23T10:32:53+00:00","versionOfRecord":[],"versionCreatedAt":"2025-07-08 11:04:46","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6656512","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6656512","identity":"rs-6656512","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}