Flexoelectrically Induced Polar Topology in Twisted SrTiO₃ Membranes

ABSTRACT

The recent realization of freestanding membranes of perovskite oxides has enabled their deterministic mechanical assembly into twisted homo bilayers. Twisted oxide interfaces feature a non‐homogeneous pattern of shear strains determined by the atomic registry through the strong ionic‐covalent bonding of this family of compounds. Twisted ferroelectrics open an unprecedented opportunity to tailor topological polar landscapes in a way determined by the flexoelectric coupling of polarization to the large strain gradients developing with the modulation of the moiré coincidence pattern. Yet, flexoelectricity is a universal phenomenon which may render polar landscapes in non‐ferroelectric materials. Here we report a polar topology in twisted membranes of SrTiO₃ (STO), a paraelectric centrosymmetric compound. The polar landscape is generated by the flexoelectrically induced polarization driven by the strain modulation triggered by twisting, as also supported by machine‐learned force fields based on first‐principles calculations. We further show that the strain and polarization patterns in top and bottom layers are correlated in a way which breaks inversion and mirror symmetries as dictated by the emergent chirality imposed by twisting. This finding opens exciting opportunities for the exploration of novel physical effects and functionalities.

Twisted SrTiO₃ bilayers host polar vortices of flexoelectric origin, revealed through combined experiment and theory. By reconstructing polarization from the toroidal moment of strain gradients, the work establishes a 3D chiral state with broken inversion and mirror symmetries. Vortex handedness and strain‐gradient polarity are shown to be deterministically controlled by the twist angle, in agreement with theory.

1. Introduction

When two crystalline lattices are stacked with a mismatch angle, a larger pattern emerges, the so‐called moiré pattern, with a periodicity determined by the twist angle. It imposes a secondary lattice potential which has been shown to deeply modify the electronic structure of 2D van der Waals layers, yielding exciting electronic ground states [1, 2]. 2D layered materials are prone to exfoliation, stacking and twisting at any material combination and free choice of angle, and mechanically assembling twisted layers has prompted an exciting new strategy, named ‘twistronics’, to tailor novel properties and responses.

The recently developed epitaxial growth of transition metal oxides on soluble sacrificial buffer layers with (pseudo) perovskite structure has enabled the mechanical assembly of freestanding oxide membranes [3, 4], extending ‘twistronics’ to a new family of oxide materials with strong ionic and covalent bonds [5, 6, 7, 8, 9, 10]. Transition metal oxides with perovskite structure host a rich variety of robust ferroic orders, some of them stable at room temperature, which may feature exciting ferroic topologies driven by profound bonding modification at the twisted interface. For example, in twisted ferroelectric membranes, the inhomogeneous pattern of shear strains developing at the interface gives rise to an array of polar vortices and antivortices via flexoelectric coupling of the ferroelectric polarization to shear gradients [10].

Flexoelectricity is a universal electrical polarization response to inhomogeneous mechanical strain [11, 12] commonly observed in insulators and more rarely in metals. On general grounds, the flexoelectric coefficients are expected to be small, yet it has been theoretically predicted that they are enhanced at the nanoscale [13] or in systems with large values of the static permittivity [11, 14, 15], enabling potentially functional polarization switching by externally imposed strain gradients [16, 17].

Here, we investigate the nucleation of flexoelectrically induced polar topologies in twisted membranes of non‐ferroelectric SrTiO₃ (STO) oxide. STO is a quantum paraelectric whose low‐temperature ferroelectric polar order is suppressed by quantum fluctuations [18, 19]. It remains centrosymmetric across the cubic to tetragonal transition at 105 K driven by an antiferrodistortive instability; it also displays diverging permittivity at low temperatures, although large values of the permittivity are already observed at room temperature (ε∼300) [19]. The absence of ferroelectricity combined with relatively high dielectric permittivity makes STO one of the better studied flexoelectric materials [20, 21, 22]. Moreover, despite the non‐ferroelectric ground state, STO is prone to become ferroelectric under epitaxial mismatch strain [23, 24, 25, 26], externally applied uniaxial strain [27, 28], cation [29] or isotope substitution [30], and defect engineering [31]. Interestingly, the tendency of strained STO membranes to display ferroelectricity together with the enhanced flexoelectricity expected due to the dominant scaling of strain gradient elasticity in the 2D limit [32, 33] makes twisted STO membranes a natural system to display flexoelectrically induced polar landscapes. The incipient STO ferroelectricity together with the strong flexoelectric fields developing in twisted membranes by the nanoscale modulation of shear strains combine to nucleate a polar topology of alternating vortices and antivortices. It is worth noting that vortex‐antivortex arrays have been reported before in twisted STO bilayers [34] and their flexoelectric origin has been suggested from simulations [35].

Here we show experimental and theoretical evidence of the connection of the vortex pattern to the inhomogeneous field of strain gradients induced by twisting in the individual layers. We also show that at moiré lattice sites of maximal coincidence of the atomic registry of both layers, vortices and the field of strain gradients at the top and bottom layers are correlated and have opposite handedness triggered by twisting. Moreover, we experimentally demonstrate that, as recently predicted theoretically [36], vortex chirality is also determined by twisting, such that at a given site of the moiré lattice the circulation of the polarization field is inverted when the sign of the twist angle is reverted.

2. Results and Discussion

2.1. Strain and Polarization Analysis of Twisted SrTiO₃ Bilayers

9–16 nm thick STO layers epitaxially grown on 10 nm thick LSMO sacrificial buffer layers deposited on (001) STO substrates were delaminated and mechanically assembled to form twisted bilayer homojunctions with twist angles comprised between 2.5 and 10° (see Methods section). The twisted stack was transferred to a porous Si₃N₄ membrane support grid for planar view observation in an aberration‐corrected scanning transmission electron microscope (STEM). Atomic resolution high angle annular dark field (HAADF) images of the (individual) top and bottom layers were acquired exploiting the relative narrow focus depth (5–8 nm) obtained in aberration‐corrected instruments, by using large‐angle illumination conditions [37]. Supplementary Figure S1 shows images of a 30 nm STO twisted bilayer focusing at the top surface of the stack (defocus = 0 nm), at the interface (defocus = −15 nm) and at the bottom surface (defocus = −30 nm). The image focusing on the interface shows the characteristic moiré contrast produced by the superposition of the twisted atomic registries. Notice that the atomic columns of the bottom layer appear rotated with respect to the top layer. The moiré contrast was observed over large areas as illustrated by the low magnification microscopy image of Figure S2.

Figure 1a,d, and g shows, respectively, HAADF planar view images of the interface moiré patterns of 10, 6 and 2.5° twisted bilayers fabricated by clockwise rotation of the top layer respect to the bottom. Sites of maximal atom‐on‐atom coincidence: AA sites with Ti on Ti or Sr on Sr and AB sites with Sr on Ti or Ti on Sr are identified with blue and red circles, respectively. Interface images acquired exploiting the technique's shallow depth of focus (∼5 nm) feature distinct moiré patterns. The moiré periodicity Λ increases systematically as the twist angle θ is reduced (images of Figure 1 are taken at comparable magnification), consistent with the geometric relation Λ ≈ a/tan θ, where a is the lattice constant of STO. Using geometric phase analysis, GPA [38], which supplies maps of relative atomic displacements between sites obtained from Fourier analysis of atomic resolution images, we obtained the pattern of symmetric shear strains (${\epsilon }_{xy}=\frac{1}{2}(\frac{\partial u_x}{\partial y}+\frac{\partial u_y}{\partial x}$)). See Figure 1b,e, and h for the 10, 6 and 2.5° twisted bilayers respectively displaying shear strain contour plots obtained from the GPA analysis of images focusing at the top surface. Notice that, as in our previous work [10] on twisted BaTiO₃ layers, strain is modulated with the periodicity of the moiré lattice. Strain is zero at AA and AB sites, as expected from the maximal atom‐on‐atom coincidence, and in between there are sites of maximum shear strain, which can be either positive or negative. Maximum values of the shear strain in the range 1%–2 % are obtained for the 3 samples, similar to those obtained previously in twisted BaTiO₃ bilayers. These values should be considered as an upper bound as GPA may introduce artifacts related to the larger (polar) Ti displacements (see below).

FIGURE 1.

Flexoelectric polar vortex in twisted STO bilayers. Planar view STEM‐HAADF image of the moiré pattern at the interface of a 10° clockwise (CW) twisted STO bilayer (defocus = −12 nm) (a), a 6°CW twisted STO bilayer (9 nm thick layers) (defocus = −9 nm) (d) and for a 2.5°CW twisted STO bilayer (16 nm thick layers) (defocus = −16 nm) (g). Blue (red) circles mark AA (AB) sites of maximal atom‐on‐atom coincidence (see text). b GPA Shear Strain (ε_xy ) maps measured at the surface of the top STO layer (defocus = 0 nm) for the 10°CW (b), for the 6°CW (e), and 2.5°CW (h), twisted STO bilayers showing a periodic strain modulation with the same periodicity as the moiré pattern. c Ti─Ti displacement map ( d _Ti − d _Ti ) after subtracting the homogeneous component (black arrows amplified by a x50 factor for clarity) of the top layer STO superimposed on the contour plot of the polarization toroidal moment (Q) for the 10°CW (c), for the 6°CW (f), and 2.5°CW (i), twisted STO bilayers. Blue counterclockwise (CCW) and red clockwise (CW) polar vortices sit in AA and AB sites respectively.

A rotational polarization texture was measured from the off‐centering of the Ti cations from the centrosymmetric position of the square lattice determined by the Sr atoms. In‐plane polar displacements in the range of 0.03 to 0.08 Å were obtained yielding a polarization (0.06 and 0.13 Cm⁻²) whirling around AA and AB sites (of zero shear strain). Superimposed to the curling polar texture we found a homogenous polarization component with polarization values ranging between 0.02 − 0.07 C m ⁻² (corresponding to a Ti atom off‐centering ranging between 0.01–0.04 Å) and directed along 〈001〉 directions. The polar texture can be better analyzed by subtracting the average polarization ( P − 〈 P 〉). See arrow plots on Figure 1c,f,i for the 10, 6 and 2.5° twisted bilayers respectively. The polar topology was characterized using the toroidal moment of the polarization field defined as $\mathit{Q}=\frac{1}{2N}{\sum }_1^N{\mathit{r}}_{\mathit{i}}\times {\mathit{P}}_{\mathit{i}}$, where P_i is the local dipole moment located at r_i and N is the number of dipoles (cells). Contour (false color) plots of the toroidal moment are displayed on Figure 1c,f,i for the 10, 6 and 2.5° twisted bilayers, evidencing that polarization vortices wind around AA (counterclockwise (CCW) vortices, blue spots in the figures) and AB (clockwise (CW) vortices, red spots) position with opposite handedness. A similar polar topology has been reported previously from ptychography experiments [34], yet the connection of the vortex pattern to the inhomogeneous shear strain induced by twisting in the individual layers to establish its flexoelectric origin has not been demonstrated yet.

A notable trend in Figure 1 is that both the shear‐strain amplitude and the Ti─Ti displacements are larger in the 6° bilayer (Figure 1e,f) than in the 10° sample (Figure 1b,c). This difference does not arise from the twist angle itself, but from variations in layer thickness across the samples. As shown in [10], the shear strain decays as it propagates from the interface toward the surface; consequently, thicker layers exhibit systematically smaller surface strains. This is borne out by the three CW‐twisted bilayers in Figure 1—10° (12 nm layers, panel b), 6° (9 nm, panel e), and 2.5° (16 nm, panel h)—where the thickest samples display the weakest strain signatures.

A similar thickness dependence is expected for the Ti displacements, although in this case, reduced strain gradients at smaller twist angles—arising from the larger moiré periodicity—also play a role (discussed below). We note in particular the reduced scale of the toroidal‐moment maps in panels c and i, consistent with the suppressed distortions in the thicker bilayers.

2.2. Handedness of the Polar Vortices

We now show that the handedness of the polar vortices in the top layer at a given site of the moiré lattice is determined by the twist angle (CW or CCW) of the top layer respect to the bottom layer. Figure 2 (top panels) shows the effect of an 8°CCW rotation of the top layer. Figure 2a displays a map of the shear strain, illustrating that AA sites have negative hot spots of shear strain at their right and left neighboring sites and positive hot spots of shear strain at the top and the bottom neighboring sites. Figure 2b shows maps of the polarization field (illustrated with black arrows) and a contour color plot of its toroidal moment with CW (red) vortices sitting at AA sites. The close correlation of polarization and strain maps indicate the flexoelectric origin of the vortex‐ antivortex array. In fact, polarization maps can be retrieved from strain maps assuming flexoelectric coupling where polarization components δP_x  and δP_y  follow from the strain gradients according to the following relations $\delta P_x\approx {\mu }_{xyxy}^{\mathrm{eff}}\frac{\partial {\epsilon }_{xy}}{\partial y}$ and $\delta P_y\approx {\mu }_{xyxy}^{\mathrm{eff}}\frac{\partial {\epsilon }_{xy}}{\partial x}$, where ${\mu }_{xyxy}^{\mathrm{eff}}$ is an effective flexoelectric coefficient. Figure 2c shows a polarization arrow map obtained from the strain gradients and a contour plot of its toroidal moment $\mathit{Q}=\frac{1}{2}\mathit{r}\times \mathit{P}=\frac{1}{2}{\mu }_{xyxy}^{\mathrm{eff}}(r_x\frac{\partial {\epsilon }_{xy}}{\partial x}-r_y\frac{\partial {\epsilon }_{xy}}{\partial y})$. Notice that the polarization maps derived from Ti displacements appear noisier than the strain maps. Strain and polarization are obtained independently: strain is extracted from the relative displacements of Sr atoms, whereas polarization is determined from the Ti off‐centering within each unit cell. Because strain is imposed by atomic registry and bonding across the twisted interface, its magnitude (∼2% at hotspots) is comparable to that reported for BaTiO₃ bilayers in [10]. The situation differs for polarization. In BaTiO₃, strain gradients act on robust ferroelectric moments, producing Ti displacements of ∼0.15 Å, whereas in STO the induced off‐centering is much smaller (0.03–0.06 Å), yielding correspondingly weaker polar responses. Moreover, GPA‐based strain retrieval benefits from FFT filtering, while polarization requires direct measurement of these minute Ti shifts within the Sr cages, making it inherently more susceptible to noise. Although polarization can be reconstructed from strain gradients via the flexoelectric coupling—yielding cleaner maps—the independently measured Ti‐based and strain‐derived polar textures show good agreement, as reflected in the toroidal‐moment contour plots.

FIGURE 2.

Chirality of polar vortices. Shear Strain (ε_xy ) map measured at the top surface of an 8°CCW twisted STO bilayer (with 15 nm thick layers) (a) and for a 10°CW twisted STO bilayer (with 12 nm thick layers) (d). Blue (red) circles indicate AA (AB) sites. Ti─Ti displacement map ( d _Ti − d _Ti ) after subtracting the homogeneous component (black arrows amplified by a x50 factor for clarity) of the top layer superimposed on the corresponding toroidal moment (Q) of the polarization of the 8°CCW twisted STO bilayer (b) and for a 10°CW twisted STO bilayer (e). Arrow plot of the polarization field retrieved from the strain gradients (flexoelectric field) at the top layer and contour plot of its toroidal moment for the 8°CCW twisted STO bilayer (c) and for a 10°CW twisted STO bilayer (f). Notice the inversion of the chirality of the flexoelectric and polarization fields at AA (AB) sites for the CW with respect to the CCW twisted bilayer.

The close correspondence to the polarization maps of Figure 2b directly determined from Ti atom off‐centering attests the flexoelectric origin of the polarization. Bottom panels of Figure 2 show the effect of a 10°CW rotation of the top layer respect to the bottom (data taken from the same bilayer as the 10°CW data of Figure 1). The shear strain maps of Figure 2d show that, in this case, AA sites have positive hot spots of shear strain at their right and left neighboring sites and negative hot spots of shear strain at the top and the bottom neighboring sites. On the other hand, polarization maps shown in Figure 2e show that CCW (blue) vortices sit now at AA sites (marked with blue circles). Figure 2f shows polarization maps retrieved from strain gradients according to flexoelectric coupling closely confirming this scenario. This demonstrates that the chirality of the polar vortices at a given site of the moiré lattice is determined by the twist angle, and chirality is reversed when the twist angle is inverted. This result can be intuitively understood within the flexoelectric description of the induced polarization as inverting the twist angle changes sign of the flexoelectric field induced in the individual layers.

A central aspect of the flexoelectric scenario is the distinction between the strain‐gradient field and the resulting “flexoelectric field,” i.e., the polarization induced by those gradients. The two are linked by a flexoelectric coupling term in the free‐energy density,

$$F=-\sum _i\frac{1}{2}{\mu }_{xyxy}^{\mathrm{eff}}\left(d_x\left(i\right)\frac{\partial {\epsilon }_{xy}}{\partial y}\left(i\right)+d_y\left(i\right)\frac{\partial {\epsilon }_{xy}}{\partial x}\left(i\right)\right)$$ (1)

where d_x (i),d _x(i) are the components of the local dipole moment in cell (i) [36]. Because of this coupling, the topologies of the polarization field and the strain‐gradient field differ markedly. Their respective toroidal moments take the forms $\mathit{Q}=\frac{1}{2}\mathit{r}\times \mathit{P}=\frac{1}{2}{\mu }_{xyxy}^{\mathrm{eff}}(r_x\frac{\partial {\epsilon }_{xy}}{\partial x}-r_y\frac{\partial {\epsilon }_{xy}}{\partial y})$ and for the field of strain gradients ${\mathit{Q}}_{\mathit{fg}}=\frac{1}{2}\mathit{r}\times \nabla {\epsilon }_{xy}=\frac{1}{2}(r_x\frac{\partial {\epsilon }_{xy}}{\partial y}-r_y\frac{\partial {\epsilon }_{xy}}{\partial \mathrm{x}})$. These expressions account for the contrasting winding patterns observed experimentally. As shown in Figure S3 for 8°CCW (Figure S3a) and for 10°CW twist angle (Figure S3b), the polarization field hosts vortices (winding +1) at AA and AB sites and antivortices (−1) in between, whereas the strain‐gradient field displays the opposite topology: antivortices at AA and AB sites and hedgehog‐like textures (winding +1) at the strain maxima. This behaviour, demonstrated for both 8°CCW and 10°CW bilayers, is in excellent agreement with the analytical predictions of [36]. Notice that reversing the twist angle flips the polarity of the strain field at AA sites. To further characterize the polar topology we have also retrieved the antitoroidal moment D recently introduced [36] to characterize the antivortices. The electric dipole vortex is characterized by the toroidal moment of the vortex pattern $Q=\frac{1}{2N}{\sum }_1^N(r_{xi}{P}_{yi}-r_{yi}{P}_{xi})$. A similar moment should exist for the antivortex dipole structure, the antitoroidal moment, D, introduced in reference [36], which can be retrieved from a vortex by substituting Px by −Px to obtain $D=\frac{1}{2N}{\sum }_1^N(r_{xi}{P}_{yi}+r_{yi}{P}_{xi})$. Supplementary Figure S4 shows a comparison of toroidal and antitoroidal moments for the polar landscapes of Figure 2 corresponding to CW and CCW twisted bilayers. Notice that in agreement with the findings of reference [34] the antitoroidal moment of the antivortex and the toroidal moment of the vortex are phase shifted but have precisely the same magnitude, which hints toward the strong coupling and interplay between these two different topological defects. It can be shown that Q and D are respectively proportional to the averaged flexoelectric energy given by Equation 1 in quadrants defined around vortices and antivortices which further supports the flexoelectric origin of polar topology.

Finally, once we have demonstrated that the experimentally determined polar topology can be consistently accounted for by the flexoelectric coupling of polarization to strain gradients, we can quantitatively estimate the flexoelectric coefficient. Polarization values of 0.06 − 0.13 C m ⁻² can be estimated from polarization maps, which, together with (maximum) values of the strain gradients of the order of 4  ×  10⁷ m ⁻¹obtained from experimental strain maps, yield the effective flexoelectric coupling as:

$${\mu }_{xyxy}^{\mathrm{eff}}\approx \delta P_x{\left(\frac{\mathrm{\Delta }{\epsilon }_{xy}}{\mathrm{\Delta }y}\right)}^{-1}\approx \frac{0.06-0.13C{m}^{-2}}{4\times {10}^7{m}^{-1}}\approx 1-3nC{m}^{-1},$$

These values are well in the range (2.4 − 6.1 nC m ⁻¹) of the (absolute) values reported from experiments bending single crystals [20]. Furthermore, the flexoelectric coefficients experimentally estimated from strain and vortex patterns of CW and CCW bilayers are always positive as opposed to the case of the experiments on STO single crystals, which yield positive or negative values depending on measurement direction and crystal orientation.

2.3. Correlation Between Polar Vortices and Fields of Strain Gradients in Top and Bottom Layers

The possibility of obtaining atomic resolution images of top and bottom layers together with the moiré pattern at the interface allows examining the correlation between strain and polarization textures of both layers. We have used a beam‐induced defect (see dark spot at the upper right corner of the image of the moiré lattice in Figure 3a) to align patterns of the top and the bottom layers. Figure 3 shows strain patterns and toroidal moments of the polarization and strain gradient fields obtained at the surface of the top layer (panels 3b, 3c and 3d) and in the bottom layer (panels 3e, 3f and 3 g). Notice that the pattern of shear strains of the top layer corresponds to a CCW rotation of the top layer with positive hot spots of shear strain at the top and the bottom neighbors of AA sites and negative hot spots at right and left neighboring sites of AA positions. Conversely, at the bottom layer, the strain pattern has changed (to resemble the CW twist of Figure 2) with positive hot spots of shear strain at neighboring left and right sides of AA positions and negative shear strain at top and bottom neighbors of AA sites. Polarization and gradients of shear strain in the top layer show CW (red) vortices sitting in AA sites while in the bottom layer, it is CCW (blue) vortices that sit in AA sites. It becomes clear then that top and bottom layers display polarization fields (and fields of strain gradients) with opposite handedness. The emergent chirality imposed in the bilayer by twisting has an imprint in the polar and strain textures which have themselves the broken inversion and mirror symmetries.

FIGURE 3.

Correlation of polar vortex chirality in top and bottom layers. a STEM‐HAADF planar view image of the moiré pattern at the interface of an 8° (CCW) twisted STO bilayer (defocus = −15 nm). Shear strain (ε_xy ) map at the top STO layer (defocus = 0 nm) (b) and at the bottom surface of the bottom STO layer (defocus = −30 nm) (e). Ti─Ti displacement map ( d _Ti − 〈 d _Ti 〉) after subtracting the homogeneous component (black arrows amplified by a x50 factor for visibility) superimposed to the contour plot of its toroidal moment (Q) map at the top STO layer (defocus = 0 nm) (c) and at the bottom surface of the bottom STO layer (defocus = −30 nm) (f). Arrow plot of the polarization field retrieved from the field of shear strain gradients (flexoelectric field) and contour plot of its toroidal moment at the top STO layer (defocus = 0 nm) (d) and at the bottom surface of the bottom STO layer (defocus = −30 nm) (g). The maps of the bottom layer are rotated respect to the top layer. Notice the inversion of the chirality of the flexoelectric and polarization fields at AA (AB) sites of top and bottom layers.

Our work differs fundamentally from previous studies. Unlike in the work of Sha et al. [34], where the very small focus depth of the ptychography experiment requires two images to geometrically retrieve the moiré pattern of the interface from their superposition, in our case we take advantage of the larger focus depth (several nanometers) to directly image the moiré pattern at the cost of layer resolution. A deterministic transfer protocol enables systematic control of the twist angle, allowing us to track the evolution of strain and vortex textures across multiple bilayers. By mapping strain gradients—absent in earlier work—and correlating them with the polar landscape, we establish compelling experimental evidence for a flexoelectric origin of the vortices. Crucially, independent measurements of the top and bottom layers reveal a 3D chiral state, in which vortex handedness and strain‐gradient sign invert across the bilayer and reverse with the twist sense. These findings demonstrate twist‐controlled chirality in STO and define a previously unrecognized mechanism for engineering polar textures in oxide moirés.

2.4. Analysis with First‐Principles Machine‐Learned Force Fields

We run atomistic calculations to study the correlation between strain gradients and polarization vortices in STO and, thus, ascertain whether they exist together and can be explained via the flexoelectric couplings described above. Our approach—which focuses on the polarization‐strain correlations and does not treat explicitly the interface between the twisted layers—follows closely the one adopted in our previous work [10]. For the simulations we use a Gaussian kernel‐based machine‐learned force field based on density functional theory (DFT) calculations (see methods).

We use the model to run constrained structural optimizations in simulation boxes of 8 × 8 × 2 perovskite unit cells (under periodic boundary conditions in the three directions of space), which is roughly the in‐plane periodicity of the moiré patterns observed experimentally at a 10° rotation angle. We allow the lattice parameters to fully relax. We perform two types of simulations.

In the first type, we start from cubic STO, and we impose in‐plane Ti displacements as indicated in Figure 4a, mimicking the dipole pattern around AA and AB sites. When we relax this structure (keeping only the relative in‐plane Ti─O distances fixed), we find that a shear strain modulation appears, as computed from the Sr cation displacements. The strain texture we obtain from the calculations closely resembles that of the experiments. We do this calculation for different amplitudes of the initial Ti displacements, and we see a roughly linear relation between the Ti displacements and the maximum shear strain obtained. For the largest Ti displacements considered (0.056 Å, similar in magnitude to those experimentally observed, 0.03–0.06 Å) the largest Sr displacements are of 0.050 Å. This procedure allows us to estimate the inhomogeneous strain pattern that accompanies the dipole vortex observed in the experiments.

FIGURE 4.

Theoretical calculations by first principles. a Dipole and shear strain map of a SrTiO₃ layer from our atomistic calculations. We show an in‐plane 2 × 2 repetition of one layer from our 8 × 8 × 2 simulation box (the other layer is virtually identical). Dashed lines indicate the limits of our simulation box. The arrows indicate the displacements of the Ti atoms with respect to the oxygen octahedra centers. Solid grey lines follow the Sr sublattice (with a Sr atom at each intersection), and we have magnified the distortion from a perfect square Sr sublattice by a factor of 50 for visualization purposes. Imposing and fixing the vortical dipole pattern on an unstrained cell results in the appearance of the shear strain displayed. Similarly, imposing and fixing the Sr sublattice distortions (and thus the shear strain map) on a SrTiO₃ layer with no local dipoles results in the vortical dipole texture displayed upon structural optimization. Blue (red) circles indicate AA (AB) sites. b Sketch of the SrTiO₃ layer, showing regions of approximately constant shear strain either positive (red) or negative (blue) separated by regions of zero strain corresponding to AA and AB sites discussed in the text. The black arrows stand for the polarization induced by the flexoelectric effect; these arrows are consistent with a negative flexoelectric coefficient ${\mu }_{xyxy}^{\mathrm{eff}}<0$, and they correspond to the vortices and antivortices observed experimentally. Note that the flexoelectric polarization can be intuitively understood if at any given lattice point (shared by four cells, with four associated cell angles), we always find an arrow pointing toward the cell with the largest (>90°) angle.

In the second type of simulation we perform, we also start from cubic STO, but we now impose shear strain modulations, via in‐plane Sr displacements, as obtained from the output of the first type of calculation. Upon relaxing this structure (while fixing only the in‐plane relative positions of the Sr atoms, to mimic the effect of the moiré‐induced shear pattern), we observe that Ti─O displacements develop, forming a dipole vortex, essentially identical to the one imposed in the first type of calculations (Figure 4a). This shows that the moiré‐induced shear maps induce vortical dipole textures, and that the occurrence of one such distortion (electric dipoles or inhomogeneous shears) forces the occurrence of the other. Overall, these observations indicate that the shear gradients induce vortical dipole patterns around the less‐strained areas and vice versa. These calculations also show that, given a shear pattern qualitative identical to the one experimentally observed and compatible with the moiré period, the material naturally develops electric dipoles, showing no indication of an alternative order than might compete with (and preclude) this vortical state.

Our model correctly identifies the tetragonal phase of STO with oxygen octahedra tilts as the ground state. The structural relaxations described above, however, preserve the symmetry of the initially imposed shear or dipolar pattern. As a result, octahedral tilt modes, although present in bulk SrTiO₃, are not activated in these calculations. This enables us to isolate the strain–polarization coupling without interference from competing tilt instabilities. Moreover, since the experiments are performed at room temperature, well above the antiferrodistortive transition of bulk SrTiO₃, where octahedral tilts are absent, the computed structures are suitable for comparison with the experimental observations.

A relevant question is the possibility of the emergence of an outofplane polarization. Although the present experiment, done of planar view samples, is not sensitive to atomic displacements in the out of plane direction, our atomistic simulations show no out‐of‐plane polarization. However, note that the atomistic simulations were carried out in bulk‐like conditions, with no explicit treatment of the interface or the surfaces, which would require machine‐learned models that accurately describe bond breaking at interfaces as well as open surfaces. This is a cumbersome and challenging task that falls beyond the scope of the present manuscript.

2.5. Value and Sign of the Flexoelectric Coefficient

Our simulations thus support the “flexoelectric picture” that emerges from our experimental results and explains the formation of the electric texture in our STO layers. Our atomistic calculations also allow us to estimate the effective flexoelectric coefficient ${\mu }_{xyxy}^{\mathrm{eff}}$ from finite differences. If we consider that the maximum shear strain (peaking around 0.4%, red regions in Figure 4a) is 4 perovskite unit cells away (∼15.6 Å) from the dipole vortex centre (yellow regions in the Figure 4a), then the relevant change in shear strain per unit length is about $\frac{\mathrm{\Delta }{\epsilon }_{xy}}{\mathrm{\Delta }y}\approx \frac{0.004}{15.6\AA }\approx 2.6·{10}^{-4}{\AA }^{-1}=2.6·{10}^6{m}^{-1}.$ If we take the nominal charge of Ti (Z_Ti =   + 4|e|) to compute the atomic dipole from the maximum Ti─O displacement (along x) around the vortex (d _{Ti − O} =   − 0.056 Å) and the average volume per formula unit (V  =  59.3 ų), we get that in the region of vortical dipoles the polarization can be estimated to be $\delta P_x\approx \frac{Z_{Ti}·d_{Ti-O}}{V}\approx -3.76·{10}^{-3}e{\AA }^{-2}\approx -0.06C{m}^{-2}$. These give an effective flexoelectric coefficient of ${\mu }_{xyxy}^{\mathrm{eff}}=\delta P_x{(\frac{\mathrm{\Delta }{\epsilon }_{xy}}{\mathrm{\Delta }y})}^{-1}\approx -0.06C{m}^{-2}{(2.6·{10}^6{m}^{-1})}^{-1}\approx -23nC{m}^{-1}$ substantially different to the values estimated from the experiments (1 − 3 nC m ⁻¹). Nonetheless, this value matches (in both magnitude and sign) a previous DFT‐based result reported in the literature for cubic STO [‐19.392 nC m ⁻¹] [39]. Furthermore, simulations and experiments disagree on an important qualitative feature, namely, the sign of the effect. While this discrepancy may seem problematic, we should note that there are good reasons that could explain the difference. First and foremost, our simulations do not treat the twisted interface explicitly, as we assume that the effect of the layer misalignment is essentially captured by the resulting shear strain modulation. This is a serious approximation; the actual interfacial atoms are subject to chemical environments that may greatly differ from those of bulk STO, which will affect their energy landscape and, in particular, the interaction between strain and polarization. Thus, while flexoelectricity is driven by a universal symmetry‐breaking effect, the actual coupling may strongly depend on the structural details, which our simulations greatly simplify. (It would be tempting to think that, at the experimental level, we will have a surface‐dependent contribution to the flexoelectric effect [22], which we are surely missing in our bulk‐like theoretical simulations. However, such a surface contribution only affects flexoelectric couplings where a polar direction is normal to said surface, which is not the case here.) Second, we must note that our estimates of the flexoelectric coupling involve large strain gradients, at the theoretical level (∼10⁶ m⁻¹), and especially at the experimental one (∼10⁷ m⁻¹). As regards our theoretical estimate, the agreement with the DFT literature on STO [39] seems to suggest we are within the linear regime. Then, since the experimentally observed strain gradients are more than 10 times bigger, this suggests that the effective flexoelectric coefficient estimated from our experimental results may contain significant non‐linear contributions, which would thus complicate a direct comparison with our theoretical value. Beyond these points, we can also note the difficulties associated to an experimental quantification of flexoelectric effects pointed out previously [21], related to the challenge of modelling the internal distribution of strains in bent single crystals, and uncertainties arising from the inter‐sample variation. Moreover, it has to be taken into account that GPA‐determined strains may be overestimated as a result of the effect of inhomogeneous atomic displacements (Sr vs. Ti) on FFT‐filtered methodology [40].

To further investigate the connection between shear pattern and electric vortices, we have carried out molecular dynamics simulations to explore the evolution of the dipole texture with temperature (see Methods). In particular, we use an 8×8×8 simulation box and several temperatures (10, 50, 100, 200, 300, 400, and 500 K). We define a normalized vortical order parameter such that an amplitude of 1 corresponds to the Ti─O displacements of the dipole vortex obtained upon structural relaxation (in absence of oxygen octahedra tilts) in an 8 × 8 × 1 STO layer. We then monitor the vortical order as a function of temperature by projecting the Ti─O displacements of each molecular dynamics step onto the reference vortical pattern for the 8 layers of STO in the 8 × 8 × 8 simulation box. We find that the dipole vortex order persists, with a decreasing average amplitude, as temperature increases (Figure S5), yielding a non‐negligible dipolar vortical order even at room temperature. Notably, the helicity of the vortical order (i.e., the sign of the order parameter in Figure S5) does not change upon heating, suggesting that the flexoelectric coupling connecting vortical order and inhomogeneous shears does not change sign upon heating.

Molecular dynamics simulations also reveal that octahedral rotations characteristic of bulk SrTiO₃ emerge at low temperatures, vanishing almost completely at 100 K. Importantly, these tilts coexist with the dipole vortex at low temperatures, but they do not affect the occurrence of the dipole vortex or the sign of the underlying flexoelectric coupling.

A last comment regards the appearance of a small uniform ferroelectric polarization component (ranging between 0.02 − 0.07 C m ⁻²). Although we cannot entirely exclude the possibility that inhomogeneous strains arising from membrane bending over the TEM grid induce a uniform polarization via flexoelectric coupling, alternative mechanisms may also contribute. Recent work has shown that STO can host polar nanoregions associated with point‐defect (stoichiometry) fluctuations, which act as seeds of the ferroelectricity [41, 42] that emerges under strain or when the layer thickness approaches the characteristic size of these regions [31]. Stoichiometric defects and localized charge, recently shown to occur due to the bonding between the incommensurate atomic registry of the twisted layers, [43] may be the precursors of the uniform polarization in our case. One possibility is that the nucleation of the polar flexoelectrically‐driven ferroelectric triggers a homogeneous polarization component. Further work is required to clarify if, beyond the rotational topology created by the inhomogeneous strains, the bonding between the twisted layers also plays a role to yield a uniform polarization component.

3. Summary and Conclusions

We have found a polar topology in twisted membranes of STO, a non‐ferroelectric centrosymmetric compound, which expands the opportunities offered by the growing field of twisted oxide membranes. During the review process of this article a paper appeared in Advanced Materials reporting polar vortices in twisted STO membranes as well [44]. Yet our paper differs in establishing theoretically and experimentally the flexoelectric origin, in assessing the chirality and establishing its link with the twist angle. The polar landscape, an ordered array of vortices and antivortices with the symmetry of the moiré lattice, is generated by the flexoelectrically induced polarization driven by the strain modulation triggered by twisting, as also supported by machine‐learned force fields based on first‐principles calculations. We have demonstrated that vortex handedness is determined by the twist angle and it is reversed when the twist angle is inverted. The strain and polarization patterns in top and bottom layers are correlated in a way which breaks inversion and mirror symmetries showing that the emergent chirality imposed by twisting unlocks a chiral degree of freedom in the polar texture. These results indicate that twisting membranes may offer new avenues to engineer polar topologies with tailored chiral properties in non‐ferroelectric materials which may offer exciting opportunities for the exploration of novel physical effects and functionalities.

4. Experimental Methods

4.1. Preparation of Freestanding Samples

High‐pressure oxygen RF sputtering was used to grow high‐quality epitaxial SrTiO₃ (STO)/La_0.7Sr_0.3MnO₃ (LSMO) (sacrificial layer) heterostructures on (001) single‐crystalline STO substrates. STO thin films with thickness in the range of 9–16 nm, and LSMO sacrificial layers (6–10 nm thick), were grown at 900°C substrate temperature in a 3.2 mbar pure oxygen atmosphere, with a 60 min 800°C post‐annealing at 900 mbar. To release the STO layer, the bilayer structure was first coated with a polypropylene carbonate (PPC) solution (15 g of PPC per 100 mL of anisole) by spin coating (at 7500 rpm for 30 s) and thermal curing for 10 min at 80°C and was subsequently attached onto a commercial polydimethylsiloxane (PDMS) support. The LSMO buffer layer was selectively etched by immersion in a potassium iodide (KI) and hydrochloric acid (HCl) solution (2.49 g KI + 5 mL HCl (37%) + 105 mL H₂O) for an average time lapse of two days. The membrane supported by the polymer stamp was washed by immersion in deionized water. The resulting freestanding STO membrane was then transferred onto a Si₃N₄ membrane grid for STEM imaging using an in‐house made transfer set up [10] at 130°C. Prior to transfer, the Si₃N₄ grid was baked at 300°C for 30 min to remove surface contaminants and enhance adhesion. The Si₃N₄ grid is kept at 130°C during the membrane transfer, which improves adhesion and assists the release of the STO membrane from the PDMS support by melting the PPC coating. Residual PPC was removed by an additional thermal treatment at 300°C for 30 min in air. The second STO layer in the twisted stack was manually aligned with the first layer using membrane edges (aligned with the substrate´s crystalline directions) as references. This procedure sets twist angles deterministically with an accuracy better than 0.5°. Note that a more precise value of the final twist angle can be obtained ex‐post from the shift between fast Fourier transform (FFT) spots of electron microscopy images (Figure S2b).

4.2. Scanning Transmission Electron Microscopy

Planar view experimental high‐angle annular dark field (HAADF) images of the twisted stacks were obtained using a JEOL JEM ARM200cF aberration‐corrected microscope, installed at the Centro Nacional de Microscopía Electrónica (ICTS CNME‐UCM). For large‐angle illumination conditions, the microscope was operated at 200 kV using a probe forming semi angle of 30 mrad. To increase signal to noise ratio for atom position mapping, each raw data image consists of 30 sequential 1024 × 1024 pixel images acquired with an exposure time of 1 µs/pixel for a total acquisition time of 32 s. The stack was then aligned, summed and resized, and the analysis was performed in the resulting images.

4.3. Analysis of Strain and Polarization in Twisted Bilayers

Nanometre‐thick (9–16 nm) STO layers delaminated to form twisted bilayer homojunctions with deterministic twist angles (2.5°–10°) were transferred onto holey Si₃N₄ membranes for electron microscopy imaging. To study the structural properties of individual layers of the twisted bilayers, we performed a STEM‐HAADF experiment (see the previous section). Focusing on the entrance surface of the stack (defocus  =  0 nm), we observed the typical perovskite structure corresponding to the top STO layer (Figure S1a). The moiré contrast was shown by changing the defocus to reach the interface of the twisted bilayer (defocus  =  − Thickness) (Figure S1b). Further increasing defocus brings the bottom layer into focus, which appears rotated by the twist angle of the bilayer (Figure S1c). Twisted bilayer interfaces exhibit characteristic moiré patterns over large sample areas attesting the high‐quality of the membranes (Figure S2a). The twist angle can be determined from the FFT image (Figure S2b) as previously stated and is close to the nominal rotation value of the films during the deterministic transfer process. We analyzed the strain and polarization on averaged sequential STEM HAADF images of the different twisted STO bilayer stacks acquired focusing on the entrance (defocus  =  0 nm) and exit (defocus = −30 nm) surfaces of the stack. The strain and polarization maps extracted from top‐surface images thus capture how interfacial distortions propagate through the upper layer. To probe the lower layer, the focal plane is shifted to the bottom surface (focus depth = –bilayer thickness), where the measured strain and polar textures reflect the decay of the interfacial fields through the lower layer. Strain analysis was carried out using the Geometric Phase Analysis (GPA) [38] tool in the FRWRtools plugin (www.physics.hu‐berlin.de/en/sem/software/software_frwrtools) for DigitalMicrograph. Strain maps quantified using GPA were independently verified via Sr–Sr displacements, yielding qualitatively consistent results.

The positions for the Sr and Ti sublattices were determined on the aligned image using an initial center of mass fitting and refining with 2D gaussian fittings [45], from which the relative displacement fields were calculated. The gaussian fitting windows were chosen to avoid intensity overlaps and minimize cross‐talk noise with neighboring atomic columns. As a precaution, the same fitting window was chosen for the analysis of all the images in this work. Polarization was calculated from the off‐centering of the B‐site Ti cations in the individual unit cells (relative displacement of the B‐site Ti cation from the centrosymmetric position, determined with the A‐site Sr cations within the same unit cell), and subtracting the average polarization ( P  − ⟨ P ⟩) value in the image. The toroidal moment associated to the polar displacements can be mapped by obtaining its local value at the different atomic positions [10]. Through this work, the local toroidal moment is calculated over the sum for all first and second neighbors, except for the 2.5° twisted STO bilayer, where the sum extends to the third neighbors to take into account the larger area of the plateaus of the moiré pattern.

4.4. First Principles Simulations Based on Machine‐Learned Force Fields

We used a Gaussian kernel‐based machine‐learned force field based on density functional theory (DFT) simulations. We build the model using the on‐the‐fly training algorithm as implemented in the Vienna ab initio simulation package VASP [46]. The training is carried out during a molecular dynamics simulation of 10⁴ steps with a timestep of 2 fs where the cell volume is allowed to evolve and the temperature is kept constant at 50 K. We set the Langevin damping parameter to 1 ps⁻¹ for the atomic degrees of freedom and 10 ps⁻¹ for the strain. We use a 2 × 2 × 2 perovskite unit cell simulation box and a 3 × 3 × 3 k‐point grid for Brillouin zone integrations. We employ a plane‐wave energy cut‐off of 500 eV for the basis set. The stopping criterion for the self‐consistent field calculations was set to 8 × 10⁻⁷ eV. For the exchange correlation functional, we use the Perdew–Burke–Ernzerhof implementation for solids (PBEsol) [47] of the generalized gradient approximation. We employ the plane augmented wave (PAW) [48, 49] method to represent the ionic cores as implemented in VASP, treating explicitly the following electronic orbitals: 4s, 4p and 5s for Sr; 3s, 3p, 3d and 4s for Ti; and 2s and 2p for O. Once the training is completed, we obtain a training set of 235 Density Functional Theory (DFT) calculations. The calculations are run in bulk conditions, with no vacuum. Direct DFT calculations of twisted bilayers (including vacuum) are essentially prohibitively expensive. This (unfortunately) falls beyond the scope of the present work.

To conclude, let us point out that we treat the atoms as classical objects in all our simulations, thus ignoring their quantum character and all potentially related effects. Note that this is the usual theoretical approach to investigate STO, which permits drawing conclusions relevant to the room‐temperature behavior of the material from standard and relatively light simulations. Here, one could note that bulk STO behaves as a quantum‐paraelectric at low temperatures, ferroelectric order being suppressed by quantum fluctuations that render anomalously large dielectric responses. Nevertheless, in our simulations we see no indication of a weak order (in particular, not a ferroelectric order) that could be suppressed by quantum effects. Hence, we have no reason to believe that a quantum‐mechanical treatment of the atoms is necessary in our simulated STO, subject to moiré‐like elastic modulations that drive the behavior away from that of the bulk compound.

4.5. Finite Temperature Vortex Texture: Molecular Dynamics Simulations

We impose the Sr displacements obtained in the structural relaxations above (which we assume to be temperature‐independent, as they mimic those imposed by the constant twist angle) and perform Langevin dynamics for 20 ps (with a timestep of 1 fs and a Langevin friction coefficient of 10 ps⁻¹ for all atoms) in a simulation box of 8 × 8 × 8 perovskite unit cells, which we found sufficient to obtain statistically meaningful thermodynamic averages. At each temperature, we isotropically scale the lattice parameters to follow the experimental volumetric thermal expansion coefficient of 3.23·10⁻⁵ K⁻¹ [50] and fix the lattice parameters during the dynamics.

Author Contributions

IT, VR prepared the samples with help and guidance of DSM, FG, FM, MGH and GSS. MC and GSS performed the electron microscopy. IT, GSS, VR, CL, JS analyzed the electron microscopy data. HA and JI did the theory analysis. IT, HA, CL, JI and JS wrote the manuscript with input and help from all authors.

Funding

Work at UCM supported by ERC Synergy METRIQS, EIC Pathfinder grant 101130224 “JOSEPHINE”, by Agencia Estatal de Investigación through PID2023‐148884OB‐I00, TED 2021–130196B‐C21, and by “(MAD2D‐CM)‐UCM” project funded by Comunidad de Madrid, by the Recovery, Transformation and Resilience Plan, and By NextGenerationEU from the European Union). Work at LIST Was Supported By the Luxembourg National Research Fund Through Grant C23/MS/17909853/BUBBLACED. G.S.‐S. Acknowledges Financial Support from Grants RYC2022‐038027‐I and CNS2024‐154548 Funded By MICIU/AEI/10.13039/501100011033. V.R. Was Supported By the Madrid Government (Comunidad De Madrid‐Spain) Under the Multiannual Agreement With Universidad Complutense De Madrid in the Line Research Incentive For Young PhDs, in the Context of the V PRICIT (Regional Programme of Research and Technological Innovation), and Grant CNS2024‐154507 Funded By MCIN/AEI/ 10.13039/ 50110001103. D. S‐M Acknowledges Support By Project SUPER ROTOX PR17/24‐31904 FUNDED BY CAM. I.T. Was Supported By UCM‐Santander Grant CT82/20‐CT83/20. Authors acknowledge the Use of STEM Systems of the Spanish National TEM Facility CNME ICTS Under ELECMI proposals ELC189‐2023 and ELC190‐2023.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Supporting File: adma72772‐sup‐0001‐SuppMat.pdf.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.