Main

Topological patterns in magnetic systems (for example, skyrmions) have garnered enormous interest in condensed-matter physics, as well as for potential applications in spintronics or skyrmionics4,5,6,7, including the design of racetrack memories8,9. This raises the question of whether analogous arrangements of electrical dipoles (that is, polar skyrmions) can be realized. (PbTiO3)n/(SrTiO3)n heterostructures provide an intriguing opportunity to test for this possibility, since researchers have already demonstrated the formation of emergent polar topologies, such as chiral vortices1,3, ferroelectric chiral and achiral domain walls10,11 or ferroelectric bubble domains12. Theoretical predictions suggest that there is considerable potential for obtaining bubble-shaped nano-domains13 and skyrmion-like topological structures in ferroelectric materials driven by the interplay of elastic, electrostatic and gradient energies14,15,16,17. By recognizing the crucial role of lattice-mismatch strain, we demonstrate the formation of chiral, polar-skyrmion bubbles with a skyrmion number of +1 using a combination of real-space imaging, second-principles ab initio calculations and phase-field modelling.

Heterostructures of [(PbTiO3)n/(SrTiO3)n]m (n = 12–20, m = 1–8) were fabricated on TiO2-terminated single-crystalline SrTiO3 (001) substrates by reflection high-energy electron diffraction (RHEED)-assisted pulsed-laser deposition (Methods)1. Three-dimensional reciprocal space mapping (RSM) of the trilayer (m = 1) and superlattice (m = 8) structures (Extended Data Fig. 1) about the 003 diffraction condition shows satellite peaks about the film peak along the in-plane directions (side lobes, Extended Data Fig. 1a, c), corresponding to an ordered phase with a periodicity of about 8 nm. The intensity distribution shows a ring with a preference for alignment along the cubic directions of the substrate (Extended Data Fig. 1b, d). The appearance of such a textured ring in reciprocal space can arise from ferroelectric nanodomains18. The RSM indicates consistency between the trilayer and superlattice on SrTiO3, where the ring size and its in-plane widths (Extended Data Fig. 1g, h) are nearly the same, but very distinct from periodic vortex nanodomains observed previously2 on DyScO3 (Extended Data Fig. 1e, f). To clarify the source of this diffraction pattern, we present real-space imaging results obtained by transmission electron microscopy (TEM).

Cross-sectional dark-field TEM for trilayer (m = 1) (Fig. 1a) and superlattice (m = 8) (Fig. 1b) structures reveals intensity modulations with a length scale of about 8 nm, suggesting the formation of anti-parallel polar regions. Examining the corresponding planar-section images, taken using scanning transmission electron microscopy (STEM), trilayers exhibit a mixture of circular (about 8 nm diameter) and elongated features (Fig. 1c), whereas the superlattices predominantly exhibit arrays of circular features (about 8 nm diameter; Fig. 1d, Extended Data Fig. 2). Certainly, the topology of these domain walls is distinct, in that they are broken up into many regions of shorter-range order along multiple in-plane directions, leading to distinct circular and elongated textures (Fig. 1, Extended Data Figs. 2, 3), which were not observed in previously studied vortex arrays1 (Extended Data Fig. 3). Consistent with the X-ray RSM studies, fast Fourier transforms (FFT) of the images reveal side lobes with a fourfold symmetry (insets, Fig. 1c, d). It is noteworthy that the in-plane modulation of about 8 nm is identical to that observed in RSM studies. Such features closely resemble the results of phase-field simulations (Extended Data Fig. 4a–d), where the blue markers represent in-plane polarization (Pip), whereas the white regions have out-of-plane polarization (Pop). Here, we see the formation of these features in plan view, where circular regions of Pop are surrounded by Pip borders with Néel-like rotation of the polarization.

Fig. 1: Observation of ordered polar structure.
figure 1

a, b, Cross-sectional dark-field TEM images of a (SrTiO3)16/(PbTiO3)16/(SrTiO3)16 trilayer (a) and a [(SrTiO3)16/(PbTiO3)16]8 superlattice (b), revealing a regular in-plane modulation of about 8 nm. c, d, Planar-view dark-field STEM imaging shows the widespread occurrence of nanometre-size round and elongated features in a (SrTiO3)4/(PbTiO3)11/(SrTiO3)11 trilayer (c) and only circular features in a [(SrTiO3)16/(PbTiO3)16]8 superlattice (d) along the [100] and [010] directions. The STEM studies were repeated in a minimum of 10 separate samples and the observations were repeatable. Insets, FFT of the images in c and d show a ring-like distribution with stronger intensities along the cubic directions—the same feature seen in RSM studies.

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Second-principles ab initio calculations19 were used to examine the microscopic details of these polarization textures. With these calculations, we show that they are analogous to magnetic skyrmions and have a mathematically definable and characteristic skyrmion number (Methods). These calculations show that the local electric dipoles continuously rotate both in- and out-of-plane at the PbTiO3/SrTiO3 boundary (Fig. 2a, b). At the top and bottom of the PbTiO3 layer, the electric dipoles display an in-plane polarization connecting the up and down polar regions with diverging and converging polarization, respectively (Fig. 2c, e). Superimposed on this hedgehog-like, cylindrically symmetric polar texture, we observe the condensation of a curling polarization pattern in the middle x–y plane of the PbTiO3 layer (Fig. 2b, d), which is also indicated from phase-field simulations at 0 K and 298 K (Extended Data Fig. 4e–h).

Fig. 2: Second-principles calculations of the skyrmion structure in a PbTiO3–SrTiO3 superlattice.
figure 2

a, Right, a cross-section of the polarization texture predicted by second-principles calculations. Bloch walls appear along both the out-of-plane (left) and in-plane directions. b, Repetition of the skyrmion structure in the x and y directions, similar to the one observed experimentally (Fig. 1d). The polarization diverges at the top of the structure, rotating smoothly from +z to –z polarization. At the domain wall of the structure, the polarization component rotates in plane, resembling a Bloch skyrmion. c, Hedgehog-like skyrmion at the top interface between SrTiO3 and PbTiO3. d, Bloch skyrmion at the central plane in PbTiO3. e, Hedgehog-like skyrmion at the bottom interface. The up and down domains are represented by white and grey regions, respectively. f–h, The corresponding Pontryagin densities. The arrows represent the normalized electric dipole moments in the x–y plane.

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Analysis of the polarization texture at the top and bottom PbTiO3/SrTiO3 interfaces (Fig. 2c, e) reveals hedgehog-like (Néel) skyrmion structures, whereas in the middle PbTiO3 layer (Fig. 2d) a ‘combed hedgehog’ (Bloch) skyrmion is observed. In addition, we identify a polarization texture where the local order parameter has a null rotational component, but a positive divergence (Fig. 2c) at the top plane. The reverse happens at the bottom plane (Fig. 2e), with an inward-pointing local polarization profile (negative divergence). At the middle layer, the in-plane component of the polarization displays only a component parallel to the domain wall, as in conventional Bloch skyrmions (Fig. 2d). From this, we see that the three-dimensional polar-skyrmion bubble can be described as an evolution of two-dimensional skyrmions along the film normal (from top to bottom in a single PbTiO3 layer, going from a hedgehog to a vortex to a hedgehog structure). This complex polar evolution—including the formation of the in-plane Bloch component—arises from the presence of a strong depolarization field. Despite this evolution, these structures are topologically equivalent because they can be transformed into each other by a continuous deformation20. The class of these topological structures can be indexed by a skyrmion number, defined as14

$${N}_{{\rm{s}}{\rm{k}}}=\frac{1}{4\pi }\int \int {\boldsymbol{u}}\cdot (\frac{{\rm{\partial }}{\boldsymbol{u}}}{{\rm{\partial }}x}\times \frac{{\rm{\partial }}{\boldsymbol{u}}}{{\rm{\partial }}y}){\rm{d}}x{\rm{d}}y$$
(1)

where u denotes the normalized local dipole moment, and the surface integral is taken over the corresponding (001) plane. The integrand of eq. (1), the Pontryagin density, is also plotted in Fig. 2f–h. For this system, Nsk = +1 is an integer that is invariant for every plane (Methods). It is also interesting and important to note that even the elongated polar regions observed in the TEM images (Fig. 1c, Extended Data Fig. 5) manifest a skyrmion number of +1, indicating that they have the same topology.

To elucidate the atomic structure of the skyrmion bubble, we carried out atomic-scale polarization mapping using a displacement vector-mapping algorithm on both plan view and cross-sectional high-angle annular dark-field (HAADF)-STEM images, along with four-dimensional STEM (4D-STEM) with an electron microscope pixel array detector (EMPAD)21 (Fig. 3, Extended Data Fig. 6; details in Methods)22. Titanium-displacement vector mapping analysis of the plane-view HAADF-STEM image (Extended Data Fig. 6a) reveals the microscopic displacement field around a single skyrmion bubble (Extended Data Fig. 6b). The vector map (Fig. 3a) corresponding to this image of a single skyrmion bubble (white circle, Extended Data Fig. 6a) shows a region where the reversed titanium displacement is converging from the edge to the centre, reminiscent of a hedgehog-like skyrmion structure23. In cross-sectional HAADF-STEM images (Fig. 3c), vector displacement mapping clearly shows a cylindrical polar region with anti-parallel (up–down) polarization. The polarization vector rotates at the boundaries near the PbTiO3/SrTiO3 interfaces, consistent with the divergence of polarization observed in the plane-view images. The combined plane-view and cross-sectional vector displacement mapping reveals a hedgehog-like structure at the top of the PbTiO3 layer, consistent with the second-principles calculations (Fig. 2).

Fig. 3: Observation of polar-skyrmion structure.
figure 3

a, b, Reversed Ti-displacement vector map (top) based on the atomically resolved plane-view HAADF-STEM image (bottom) of a single skyrmion bubble (marked by a white circle in Extended Data Fig. 6a), showing the hedgehog-like skyrmion structure. The sketch of the superlattice in b is overlaid with the planar-view dark-field TEM image and gives a top view of the superlattice. c, Ti-displacement vector map (front) based on the atomically resolved cross-sectional HAADF-STEM image (back), showing a cylindrical domain with anti-parallel (up–down) polarization. The sketch in b is overlaid with the cross-sectional dark-field TEM image and shows the cross-sectional view of the superlattice. d, e, The 4D-STEM image of a [(PbTiO3)16/(SrTiO3)16]8 superlattice gives the ADF image (d) and maps of polar order using the probability current flow (e), which were reconstructed from the same 4D dataset. f, g, Multislice simulations of the beam propagation through the model structure from Fig. 2 show the ADF image (f) and the probability current flow (g), which were analysed using the same process as the experimental data. The signals are not simple projections, but weighted by electron beam channelling towards the middle of the skyrmion bubble, where the polarization exhibits a Bloch-like character.

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In 4D-STEM, the full momentum distribution is collected at every scan position, and in polar materials the asymmetry in scattering intensity between conjugate diffracted beams reflects the crystal polarity (Extended Data Fig. 7; details in Methods)24. The first moments of the beams give a probability current-flow vector map that is proportional to the in-plane polar order at each beam position. The constant of proportionality is depth-dependent as a result of electron channelling changing the shape of the beam. As an example, the annular dark-field (ADF) image (Fig. 3d) and the in-plane probability current flow (p) of a skyrmion bubble (Fig. 3e) from a [(PbTiO3)16/(SrTiO3)16]8 superlattice are shown. To understand the depth dependence of the plane-view images, multiple-scattering calculations of electron beam propagation through the model skyrmion bubble (Fig. 2) were performed using the multislice method25 to produce the simulated ADF (Fig. 3f) and p (Fig. 3g) maps. These simulations show that the electron scattering signal in 4D-STEM is largely weighted towards the Bloch-like component in the middle of the skyrmion bubble, rather than being a simple uniformly depth-weighted projection. The scalar ADF signal is sensitive to in-plane displacements of the atom columns leading to a strong dechannelling signal from the waist of the skyrmion bubble structure. The probability current flow captures the polar order from the middle and some of the upper portion of the skyrmion bubble, showing a Bloch-like character in both simulation and experiment.

These data, collectively, point to the formation of a polar-skyrmion structure with hedgehog-like components, where the polarization direction rotates smoothly from up to down and from the centre to the edge of the skyrmion; along with a Bloch-like, in-plane component, consistent with the theoretical predictions. The diverging and converging polarization rotation at the top and bottom of the polar structure resembles the spin configuration of Néel skyrmions in ferromagnets26. Associated with the realization of such a polar-skyrmion bubble, we anticipated the emergence of interesting properties, such as chirality and a non-zero skyrmion number.

To probe this potential chirality, resonant soft-X-ray diffraction (RSXD) studies were performed on a [(PbTiO3)16/(SrTiO3)16]8 superlattice (Fig. 4, Extended Data Figs. 8, 9; Methods). According to the calculated structure (Fig. 2), an imaginary line along any direction within the central plane of each PbTiO3 layer crosses only Bloch domain walls, so that along that line the local polarization exhibits a helical rotation. This helical polarization texture (embedded within the surrounding skyrmions) is virtually identical to the helical polarization configuration that emerges in polar-vortex arrays3 (Extended Data Fig. 8) and should therefore result in a similar dichroic signal from the chiral electronic structure.

Fig. 4: RSXD circular dichroism.
figure 4

a, Hard-X-ray reciprocal space mapping about the SrTiO3 (002) peak for a [(PbTiO3)16/(SrTiO3)16]8 superlattice shows superlattice peaks along the Qz direction, corresponding to an out-of-plane periodicity of about 12 nm, and satellite peaks along the Qy direction, corresponding to an in-plane periodicity of about 8 nm. b, An image from the charge-coupled device (CCD) used to collect RSXD data, with the resonant diffraction peak clearly visible. The satellite peaks accessible by RSXD are near (Qx, Qy, Qz) = (0, 0, ≪2π/c), and the satellite peak shown is at around Qy = 0.072 Å−1 and Qz = 0.141 Å−1, corresponding to the periodic helical polarization texture that forms within the Bloch skyrmions of the central PbTiO3 layers. c, d, Spectra from a satellite peak for right- (red) and left-circularly (blue) polarized light (c) and the same spectra with the background fluorescence subtracted (d). The peaks corresponding to the L3 and L2 transitions to the t2g and eg states are labelled. e, The XCD difference spectrum (right-circular minus left-circular X-ray absorption spectrum) shows a clear circular dichroism peak at the titanium L3 t2g edge. f, XCD measured at a constant energy (457.4 eV) near resonance as the lateral-scattering vector Qy is varied. The dichroism is always positive for +Qy and negative for −Qy. The reversal of XCD as the Q-vector is inverted is a tell-tale signature of chirality. a.u., arbitrary units.

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A hard-X-ray RSM study about the 002-diffraction condition (Fig. 4a) shows the presence of two sets of ordering. First, there are regular peaks along the out-of-plane direction, related to the out-of-plane periodicity of the superlattice (about 12 nm). Second, there are satellite peaks in the perpendicular or in-plane direction, corresponding to the in-plane ordering of the polar structures (about 8 nm). RSXD studies were performed at satellite peaks near the reciprocal lattice vector components (Qx, Qy, Qz) = (0, 0, ≪2π/c) (one such peak is shown in Fig. 4b; Methods). Because the X-ray energy is tuned through the titanium L3 and L2 absorption edges, the intensity of the satellite diffraction peaks is sensitive to periodic distortions, specifically chiral arrangements of the anisotropic titanium orbitals. By measuring the spectra with both right- and left-circularly polarized light (Fig. 4c), subtracting the background fluorescence (Fig. 4d), and taking the difference between the two spectra (Fig. 4e), the X-ray circular dichroism (XCD) can be measured. The clear dichroism at the L3 edge (corresponding to the t2g states)3 is indicative of chiral structures. We note that circular dichroism is observed for both the superlattice and trilayer structures (Fig. 4, Extended Data Fig. 9). Theoretical reports have shown that a likely explanation for the nonmagnetic circular dichroism is a chiral configuration of charge quadrupole moments27. This chiral configuration is composed of a helical rotation of titanium t2g orbitals, which possess a quadrupole moment and are related to the ferroelectric distortions of TiO6 octahedra. The XCD observed here is consistent with the presence of Bloch domain walls in the PbTiO3 layers.

Additionally, resonant scattering intensity and circular dichroism were measured as a function of the lateral scattering vector (Fig. 4f, Extended Data Fig. 10a). The polarization-averaged scattering intensity exhibits a clear asymmetry under reversal of the scattering vector (Extended Data Fig. 10b). The presence of the Bloch component will lead to a breaking of Friedel’s law in resonant diffraction experiments because of the broken inversion symmetry along the in-plane directions. Therefore, the observed asymmetry in the intensity of mirrored diffraction peaks provides strong evidence for the Bloch component of polarization in the skyrmion bubbles. Furthermore, as the lateral scattering vector Qy is reversed, the sign of the XCD also reverses, which is a telltale signature of chirality3,27. The simplest explanation for both the asymmetric diffraction intensity and the measured circular dichroism is a Bloch component of polarization, because it is the simplest structure to exhibit broken inversion symmetry as well as chirality. This, along with the atomic-scale images and computational studies, confirms the presence of a polar-skyrmion bubble structure.

In summary, we report the observation of polar-skyrmion bubbles in a confined PbTiO3 layer. This is the first observation of this intriguing electric skyrmion structure, which is composed of a superposition of Néel and Bloch textures, in any ferroic material. This structure resembles vortex rings previously described in the context of topological semimetals28, superfluids29 and quantum chromodynamics30, and three-dimensional skyrmions and solitons that have been predicted and observed in ultra-cold gases (that is, Bose–Einstein condensates)31,32. Such a three-dimensional structure deviates from the ordinary two-dimensional skyrmions observed in magnetic systems. Although three-dimensional, tube-like skyrmions or three-dimensional skyrmion lattices exist33, they are different from this new polar-skyrmion bubble phase. The stabilization of such non-uniform polarization topology results in highly enhanced susceptibilities and provides a pathway for engineering functionalities previously inaccessible in these materials. Furthermore, if ferroelectric skyrmions can be moved with an electric field, they could be employed to create novel device architectures.

Methods

Sample preparation using RHEED-assisted pulsed-laser deposition

n-SrTiO3/n-PbTiO3/n-SrTiO3 trilayers (n is the number of monolayers) and [(PbTiO3)n/(SrTiO3)n]m superlattices were synthesized on TiO2-terminated single-crystalline SrTiO3 (001) substrates via RHEED-assisted pulsed-laser deposition (KrF laser). The growth temperature and oxygen pressure for the bottom SrTiO3 layer were 700 °C and 50 mtorr, respectively. The PbTiO3 and the top SrTiO3 layer were grown at 610 °C in 100 mtorr oxygen pressure. For all materials, the laser fluence was 1.5 J cm−2 with a repetition rate of 10 Hz. RHEED was used during the deposition to ensure the maintenance of a layer-by-layer growth mode1 for both PbTiO3 and SrTiO3. The specular RHEED spot was used to monitor the RHEED oscillations. After deposition, the heterostructures were annealed for 10 min in oxygen pressure of 50 torr to promote full oxidation and then cooled to room temperature at the same oxygen pressure.

Structural analysis

Laboratory-based X-ray diffraction

Structural characterization of the heterostructures was carried out using a Panalytical X’Pert Pro X-ray Diffraction (XRD) system with Cu-Kα radiation (wavelength λ = 1.5405 Å). The high crystalline quality of the films and the smooth nature of the interfaces were confirmed from X-ray reflectivity and θ–2θ symmetric XRD scans around the 002 reflection plane, which showed strong superlattice peaks and Laue oscillations (not shown).

Synchrotron XRD

To obtain a comprehensive picture of the crystal structure of the trilayers and superlattices, as well as information on the in-plane and out-of-plane ordering, further structural characterization was done using synchrotron-based XRD. The synchrotron XRD studies were performed at the Sector 33-BM-C and Sector 7-ID-C beamline of the Advanced Photon Source at Argonne National Laboratory, USA. The high flux from the synchrotron X-ray source delivered at this beamline permits the detection of the weak diffracted intensities arising from the lattice modulations associated with the polar-skyrmion bubbles present in the trilayers and superlattices (Extended Data Fig. 1). A double crystal monochromator, in conjunction with two mirrors, was used to deliver a highly monochromatic beam with negligible contamination from higher harmonics. Moreover, the excellent angular accuracy of the Huber 4-circle diffractometer at Sector 33-BM-C and the Newport 6-circle diffractometer (20 keV) at Sector 7-ID-C allowed us to determine the orientation of our crystals very reliably and to obtain three-dimensional (3D) RSMs with high accuracy. Lastly, the availability of the PILATUS 100K pixel detector was essential for the efficient acquisition of 3D RSMs.

STEM

The plane-view samples of the SrTiO3/PbTiO3/SrTiO3 trilayers and superlattices for the STEM experiments were prepared by gluing a 2.5 mm × 2.5 mm film on a 3-mm molybdenum grid. The plane-view samples were then grinded, dimpled and ion-milled. Cross-sectional samples for the dark-field TEM and STEM experiments were prepared by slicing, gluing, grinding, dimpling and ion milling. A Gatan PIPS II was used for the final ion milling. Before ion milling, the samples were dimpled down to less than 20 μm. The final ion-milling voltage was 0.5 kV to reduce ion-beam damage. HAADF-STEM images were recorded by using a Cs-corrected TEAM1 FEI Titan microscope working at 300 kV. A HAADF detector acquiring ‘Z-contrast’ images was used to record the HAADF-STEM images. The beam convergence angle was 17 mrad. The diffraction contrast image (Fig. 1a) was recorded using a TitanX microscope (FEI) working at 300 kV.

Determination and mapping of the polar atomic displacements was performed on the atomic-scale HAADF-STEM images by using the column offsets of each titanium and lead atom, obtained from the position fitting of the titanium and lead atoms1. The atom positions were determined by fitting them as two-dimensional Gaussian peaks using Matlab. The titanium displacement in each PbTiO3 unit cell was calculated as a vector between each titanium atom and the centre of mass of its four nearest lead neighbours. The displacement of the titanium in each unit cell was opposite to the polarization direction of the PbTiO3. The visualization of the two-dimensional polar displacement vectors was carried out using Matlab. Random noise in the displacement vector maps was reduced using a weighted smoothing length of 1.2 nm.

4D-STEM

We performed 4D-STEM using an EMPAD21. The full momentum distribution—that is, the electron diffraction pattern—was collected at every scan position by the EMPAD. In our experiments, we used a beam energy of 300 kV and a semi-convergence angle of 1.9 mrad to ensure that the diffracted beams did not overlap, corresponding to a 6.3-Å-wide beam on an aberration-corrected Thermo Fisher Themis-Titan STEM system in nanobeam diffraction mode. From the diffraction patterns collected on the EMPAD, we reconstructed the low-angle ADF image and the probability current flow (\(\left\langle p\right\rangle \)) in x and y giving the vector components of polar order34. We compared our experimental results to multislice simulations of the electron beam propagation through the crystal. For the low-order diffraction beams used to reconstruct \(\left\langle p\right\rangle \), the simulations show that the largest signal comes from the middle of the skyrmion and is not a simple projection throughout the beam path. This holds for both the low-angle ADF and \(\left\langle p\right\rangle \) signals. The higher-angle signals show stronger dechannelling (they are generated by electrons scattered closer to the nucleus, where the scattering probability is much higher). Extended Data Fig. 7 shows the HAADF signal for a 17.2-mrad convergence-angle probe used for atomic-resolution imaging, which shows its highest scattering rate midway through the SrTiO3 layer and is largely depleted before the PbTiO3 is reached. By contrast, when the probe is narrowed to 1.7 mrad, close to the EMPAD experimental conditions, the probe is still well focused on the atom columns as it leaves the SrTiO3 layer and produces a larger signal from the PbTiO3.

Phase-field modelling

Phase-field simulations were performed to simulate the polar structure of PbTiO3/SrTiO3 heterostructures on a STO substrate. The spontaneous polarization vectors (Pi, i = {1, 3}) were selected as the order parameters, governed by the time-dependent Ginzburg–Landau equations35:

$$\frac{{\rm{\partial }}{{\boldsymbol{P}}}_{i}}{{\rm{\partial }}t}=-L\frac{{\rm{\delta }}F}{{\rm{\delta }}{{\boldsymbol{P}}}_{i}}$$
(1)

where L and t are the kinetic coefficient and the time, respectively. The total free energy F can be expressed as the volume integration of the individual energy densities, namely, the Landau chemical, mechanical, electrostatic and polar gradient energy densities:

$$F=\int \left({f}_{{\rm{Landau}}}+{f}_{{\rm{elastic}}}+{f}_{{\rm{electric}}}+{f}_{{\rm{gradient}}}\right){\rm{d}}V$$
(2)

Detailed expressions of the energy densities, the numerical treatments, the phase-field equations and the parameters can be found in previous reports36,37,38,39,40. A three-dimensional mesh of 200 × 200 × 350 was used, with a grid spacing of 0.4 nm. Periodic boundary conditions were assumed in the two in-plane dimensions, whereas a superposition scheme41 was applied in the thickness dimension. From the bottom to the top, the thickness of the substrate, thin film and air was set to be 30, 280 and 32 grids, respectively, whereas 16 unit cells of PbTiO3 and 16 unit cells of SrTiO3 layers were periodically stacked to form the superlattice thin film. A short-circuit electric boundary condition36 was used, where the electric potential at the top of the film and at the film/substrate interface was zero. The mechanical-boundary condition was set such that the stress on the film top was zero, whereas the displacement was set to be zero at the bottom of the substrate, sufficiently far from the substrate/film interface35. A wide range of strain conditions were simulated with the substrate lattice constant ranging from 3.875 Å to 3.99 Å, with an interval of 0.01 Å. Random noise with a magnitude of 0.0001 C m−2 was used as the initial condition for the evolution of polarization to simulate the thermal fluctuation during the annealing process.

Second-principles calculations

For computational feasibility, we focused on a representative [(SrTiO3)4/(PbTiO3)10] superlattice periodically repeated in space (20 × 20 × 14 elemental perovskite cells in a periodically repeated simulation box), except in those simulations used to compare with the 4D-STEM experiments, where a (SrTiO3)16/(PbTiO3)16 superlattice was used. The search for a stable skyrmion structure was carried out in a two-step process. First, starting from a monodomain, downward-pointing ferroelectric configuration in the PbTiO3 layer, a cylindrical domain with the opposite polarization was defined. This polydomain structure was used as the initial seed for the structural relaxation at zero temperature.

The second-principles simulations were performed using the same methodology presented in previous works42,43, as implemented in the scale-up package19,42. The interactions inside the PbTiO3 or SrTiO3 were described using the potentials for the bulk materials42. These potentials give qualitatively correct descriptions for the lattice dynamical properties and structural phase transitions of both materials. Then, we treated the interactions in the interface between PbTiO3 and SrTiO3, assuming a simple numerical average for the interactions of the ions pairs touching or crossing the interface. The main effect of the stacking is electrostatic, where long-range dipole–dipole interactions are governed by a bare electronic dielectric constant ε∞, which is taken as a weighted average of the first-principles results for bulk PbTiO3 (8.5ε0, where ε0, is the vacuum permittivity) and SrTiO3 (6.2ε0), with weights reflecting the composition of the superlattice. To preserve the electrostatic energy within each material as close as possible to that of the bulk parent compound, the Born effective charge tensors of the inner atoms were rescaled by \(\sqrt{{\varepsilon }_{\infty }}/{\varepsilon }_{\infty }^{{{\rm{ABO}}}_{3}}\) (where ‘ABO3’ stands for PbTiO3 or SrTiO3, depending on the layer).

The second-principles parameters of both materials were fitted from density functional theory imposing a hydrostatic pressure of −11.2 GPa to counter underestimation of the local density approximation of the cubic-lattice constant that was taken as the reference structure. We impose an epitaxial constrain assuming in-plane lattice constants of a = b = 3.901 Å forming an angle of γ = 90°, which correspond to a SrTiO3 (001)-oriented substrate.

Following this strategy, we are able to construct models for superlattices with arbitrary n-stacking. For the simulations, we used periodically repeated supercells that contained 20 × 20 elemental perovskite units in plane and one full superlattice period in the out-of-plane direction. To solve the models, we used standard Monte Carlo and Langevin molecular-dynamics methods. At low temperature, only random configurations along the Markov chains that produced a decrease in energy were accepted, following this procedure as an equivalent to structural relaxations to find the ground state.

Finally, the local polarizations were obtained within a linear approximation by computing the product of the atomic displacements measured from the ideal cubic structure with the Born effective charge tensor divided by the volume of the unit cell. The skyrmion number, Nsk, was obtained from the local dipoles by computing the normalized polarization field defined along each slice of 20 × 20 × 1 elemental perovskite units in the plane of our supercell, and solving equation (1) by applying the strategy presented in ref. 44, obtaining a skyrmion number that corresponds to an integer number.

RSXD

RSXD experiments were performed at the scattering endstation of beamline 4.0.2 at the Advanced Light Source, Lawrence Berkeley National Laboratory. The elliptically polarizing undulator at beamline 4.0.2 allows both the energy and polarization of X-rays to be tuned. The spot size of the X-ray beam used was about 100 μm in diameter. Experiments were performed on a [(PbTiO3)16/(SrTiO3)16]8 superlattice sample as well as a trilayer (SrTiO3)16 /(PbTiO3)16/(SrTiO3)16 sample. This technique has been used previously to demonstrate the chiral nature of both magnetic and non-magnetic structures, including magnetic domain walls45 and polar vortices3, by using the dichroism effects that arise from the magnetic exchange scattering and anisotropic tensor susceptibility scattering, respectively. All data were collected in vacuum, using a 1,024 × 1,024-pixel CCD. The out-of-plane ordering of the superlattice gives rise to specular ‘superlattice peaks’, that is, modulations in the intensity of the specular reflection versus the out-of-plane scattering vector that correspond to the period of the superlattice. In proximity to the third-order superlattice reflection, the lateral satellite intensity from the in-plane skyrmion bubble modulation (for example, Fig. 4b) was measured. At the determined Bragg condition, spectra were collected using both right- and left-circularly polarized X-rays tuned through the Ti L3 and L2 absorption edges (445–476 eV), where the diffractometer and sample positions were adjusted to account for the changes in Bragg angle with energy. Additionally, ‘dark images’ were recorded with no X-rays present. All data were normalized to the incoming beam intensity, and the dark images were subtracted from the diffraction data to remove background from detector dark current. The spectra were then normalized to the pre-edge region near 450–455 eV. Subtraction of the normalized detector images (right-handed minus left-handed) shows that the dichroism is localized at the diffraction peak. An example is shown (Extended Data Fig. 9d, Extended Data Fig. 10c) at the L3 t2g resonance (457.2 eV) for the trilayer and superlattice, respectively. Spectra were then extracted from several locations on the CCD (Extended Data Fig. 9d), one at the diffraction peak (Fig. 4c, Extended Data Fig. 9a) and two nearby in background regions. To correct for nonlinear background intensity, the average of the background regions was weighted slightly with a 44:56 ratio. This weighted average was subtracted from the spectra at the satellite peak. The resulting spectra (Fig. 4d, Extended Data Fig. 9b) show the resonant diffraction intensity for right- and left-circularly polarized light. The L3 t2g peak intensity was normalized to 1 in the average spectra (that is, the average of the right- and left-circularly polarized spectra), and the right- and left-circularly polarized spectra were adjusted by the same normalization factor (Fig. 4d, Extended Data Fig. 9b). Then, the difference spectrum was calculated as right- minus left-circularly polarized datasets (Fig. 4e, Extended Data Fig. 9c). The resulting spectra give the XCD. For the superlattice and trilayer samples, the XCD at resonance is found to be ±63% and ±61% of the polarization-averaged t2g peak height. The skyrmion bubble diffraction peaks for both trilayer and superlattice samples exhibit strong XCD, indicative of the chiral nature of the phase. Q-dependent scans were taken at a constant energy near resonance (457.4 eV) while varying the lateral scattering vector Qy. All data were normalized to the incoming beam intensity, and dark images were subtracted from the diffraction data to remove background from detector dark current. The diffraction intensity for right- and left-circularly polarized light and the polarization-averaged intensity were measured (Extended Data Fig. 10a, b). The XCD was calculated at the satellite peak position as the difference in intensity between right- and left-circularly polarized light.

In addition to their observation in the spectra for right- and left-circularly polarized light, these differences can also be visualized using images from the CCD. For a superlattice sample at energies near the L3 t2g peak, detector images (Extended Data Fig. 10c) show that the diffraction intensity is considerably enhanced for right-circularly polarized light but barely visible for left-circularly polarized light. The intensity difference between CCD images for right- and left-circularly polarized light shows that the XCD is localized at the diffraction peak on the detector, indicating that the dichroism is coming from the periodic chiral structures in the superlattice.

The qualitative terminology of skyrmions

There is a general consensus on the phenomenological difference between the topological structure of skyrmions and bubbles46,47. We have compared this with magnetic skyrmions and (chiral) bubbles. The key point to “distinguish magnetic skyrmions from general concepts of bubble-like spin textures” is the fact that “the key antisymmetric chiral Dzyaloshinskii–Moriya interaction (DMI) that stabilizes magnetic skyrmions in chiral magnets gives rise to a uniform spin topology of the involved spin textures over the entire system.”26,46,47 On the other hand, in magnetic bubble-like spin textures “the spin chiralities of neighboring magnetic bubbles are random.” In other words, magnetic skyrmions necessarily display a topological charge and arise from the DMI, which gives rise to large regions with well defined chirality. On the other hand, bubbles appear from magnetic dipolar interactions and may (chiral bubbles) or may not (bubbles) individually display a topological charge/chirality. Even if they display chirality, however, they do not lead to large regions with a preferred handedness.

The objects observed in this work arise from interactions that are quite different from their magnetic equivalents, and classifying them according to the predominance of DMI or magnetic dipolar interactions does not seem reasonable. It should be noted, however, that the fundamental outcome of the DMI—namely, breaking inversion symmetry—happens naturally in ferroelectrics. Conversely, as shown in the main text and Figs. 2, 3, our polar structures display a well defined topological charge and chirality individually. Such a chirality is manifested in a macroscopic measurement through XCD observed in resonant X-ray scattering studies (Fig. 4), and the chirality is macroscopically maintained over large length scales (millimetres), which has been observed in spatially resolved resonant scattering studies. This macroscopic ordering of the polar texture is closely related with the notion of magnetic skyrmions in the literature. Thus, we believe that denoting these structures as polar skyrmions is a sensible choice.