Introduction

Recent advancements in computational materials science, high-quality thin film growth techniques, and state-of-the-art characterization tools have led to the discovery of various polar topological phases1,2,3,4,5,6,7,8,9,10. One notable example is the polar skyrmion, a swirling electrical dipole texture, which has been observed at room temperature in (PbTiO3)n/(SrTiO3)n oxide superlattices (known as the PTO/STO superlattice)7,11,12,13. It is now well recognized that the topological structures usually form due to the complex interplay between strain, depolarization field, interfacial coupling, and polar gradient14. These topological phases exhibit exotic properties, such as negative capacitance11,15,16, chirality12,17,18,19,20,21, subterahertz collective modes22,23, and electric field tunable second harmonic generation responses24. They have been envisioned for practical applications in nanoelectronic devices25,26.

Specifically, skyrmions have shown great potential for applications in innovative data storage devices using the race-track memory concept27,28,29. In the existing race-track memory model, a single skyrmion is assigned to represent a binary bit, which could increase the risk of data loss due to the changes in the distance between skyrmions. Therefore, researchers are exploring higher-order topological structures formed with skyrmions, such as skyrmion bags, sacks, or bundles that could offer the potential to encode more data while preserving the properties of skyrmions30,31,32,33,34. A skyrmion bag typically consists of an outer boundary formed by a single skyrmion, along with several inner skyrmion/antiskyrmions inside the large outer skyrmion that formed the bag. Skyrmions or antiskyrmions with opposite polarity can be filled in the bag, thereby representing various types of bag configurations with distinct topological charges.

The formation of stable skyrmion bags has been widely predicted and investigated in chiral magnets with Dzyaloshinskii–Moriya interaction (DMI), as well as in liquid crystals31,32. Theoretical calculations and micromagnetic simulations have been performed to further understand the parameters responsible for stabilizing these skyrmion bags in magnetic thin films31,32,34,35,36,37,38. Although polar skyrmions have been extensively studied in the ferroelectric system, a room-temperature polar skyrmion bag has yet to be discovered, partially due to the absence of a mechanism similar to DMI. Herein, we demonstrate the creation, evolution, and manipulation of room-temperature polar skyrmion bags in a ferroelectric/dielectric (PTO/STO) superlattice by applying an electric pulse via a charged tip, in addition to the mechanism induced by DMI. A voltage phase diagram was further constructed to establish design principles for creating and manipulating the polar skyrmion bag.

Results

Structure and polar skyrmion of the as-grown thin film

The model system used in this study was the PTO/STO superlattice on a SrTiO3 (001) substrate. Previous studies have shown that polar skyrmions can be observed in this material at room temperature7. The superlattice of [10 uc PTO/2 uc STO]20 ([(PTO)10/(STO)2]20) with total thickness of 100 nm was fabricated on 5 nm LSMO, and [16 uc PTO/16 uc STO]16 ([(PTO)16/(STO)16]16) with total thickness of 200 nm was fabricated on 5 nm SrRuO3 on TiO2-terminated single-crystalline SrTiO3 (001) substrates by reflection high-energy electron diffraction (RHEED)-assisted pulsed-laser deposition (details in “Methods”). The θ-2θ X-ray diffraction (XRD) patterns reveal the high-quality epitaxial and periodic nature of the superlattice sample (Fig. 1a). Reciprocal space mapping (RSM) (Figs. 1b and  S1a) at about the (002)-diffraction condition shows the several superlattice peaks, further confirming the periodic structure of the superlattice. Additionally, satellite peaks along the in-plane directions (side lobes, marked by arrows in Figs. 1b, and  S1b), correspond to an ordered skyrmion phase with an in-plane modulation period ~25–30 nm for [(PTO)10/(STO)2]20 superlattice and 10–12 nm for ([(PTO)16/(STO)16]16 superlattice. The in situ transmission electron microscopy (TEM) observations were carried out on FEI Titan G2 80–200 ChemiSTEM in STEM mode (details in “Methods”). The planar view image of the pristine sample shows the formation of a disordered bubble state (Fig. 1c)7. The full 3-D structure and the corresponding polarization vector mappings extracted from TEM characterizations are further provided in Figs. S1c and  1d, which confirms the presence of polar skyrmion bubbles with hedge-hoge-like feature in the planar view and vortex-like structure in the cross-section view, consistent with the previous reports7,13. Dual AC resonance tracking piezo-force microscopy (DART-PFM) scanning (details in “Methods”) reveals skyrmion-like features, which are distinctly evident in both the vertical amplitude (Left, Fig. 1e) and phase (Right, Fig. 1e) images (few of them are marked with white circles). The darker areas within the circles possess a distinct polarization compared to their surroundings, attributed to the intrinsic swirling polarizations in skyrmions. The conducting atomic force microscopy (cAFM) characterizations are further performed, as shown in Fig. 1f, which demonstrates that the formation of polar skyrmions leads to higher local conductions, desired for potential applications in electronic devices. Such an enhanced local conduction for the core of the polar skyrmions has also been observed in previous literature25, which could be attributed to the band bending due to the local swirling polarization, as well as the bound charge induced by the local head-to-head polarization arrangement. Notably, the size of the region with enhanced conduction is approximately 2 nm, which corresponds to the characteristic length scale of the core of a polar skyrmion. Phase-field simulations are simultaneously performed to unveil the polar structure of the as-grown PTO/STO superlattice system (details in “Methods”, the model setup is provided in Fig. 1g). The calculated equilibrium polar texture under zero applied voltage shows skyrmion bubbles with a size of 10–20 nm (Fig. 1h), in good agreement with the experimental observations and previous reports7,11,13,29. It is interesting to note that the polar skyrmions are elongated oval structure rather than circular in shape, as shown in both the TEM images (Fig. 1c) and the simulation polar mapping (Fig. 1h). The observed size of the skyrmion is smaller in STEM measurement than that measured in PFM imaging, likely due to the resolution limitations of the PFM technique, which are inherently constrained by the tip size used during the measurements (Fig. S2)39.

Fig. 1: Structure and observation of polar skyrmions in PTO/STO superlattice.
Fig. 1: Structure and observation of polar skyrmions in PTO/STO superlattice.
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a XRD θ–2θ scan around (001)-reflection of the [(PTO)10/(STO)2]20 superlattice showing main superlattice film (SL0) peak along with several satellite peaks (SL1, SL−1, SL−2, and so), indicating the well periodic and high quality of the superlattice. b Reciprocal space mapping (RSM) around the (002)-reflection confirms again the well-periodic structure of the superlattice. Additionally, the appearance of side lobes along the in-plane direction (marked by white arrows) indicates the presence of the polar skyrmion phase in the samples. c Planar-view HAADF-STEM image of the [(PTO)16/(STO)16]16 superlattice shows the polar skyrmions. d Planar and cross-sectional polarization mapping from high resolution TEM images. e PFM images in DART mode, showing amplitude (left) and phase (right), highlighting the presence of polar skyrmions (marked by white circle). f cAFM images of the initial pristine sample, showing a higher conduction for the polar skyrmions. g Schematics of the model PTO/STO superlattice system with a charged probe on the top STO layer in the Phase-field simulations. Reprinted from Matter, 5, 3, Zhou et al.40 local manipulation and topological phase transitions of polar skyrmions, 1031–1041, Copyright (2022), with permission from Elsevier. h Simulated in-plane polar texture of the as-grown [(PTO)16/(STO)16]8 superlattice using phase-field simulations. Here, \(\left|{{{\rm{P}}}}_{{\mathrm{in}}}\right|\) is the magntitude of the in-plane polarization, defined by \(\left|{{\rm{P}}}_{{\mathrm{in}}}\right|=\sqrt{{P}_{{{\rm{x}}}}^{2}+{P}_{{{\rm{y}}}}^{2}}\).

Creation of the polar skyrmion bag

The concept of a polar skyrmion bag emerges from a pocket created within skyrmion walls, which contains skyrmions with opposite polarity to the walls. To create the skyrmion bag, an external electric field is applied through a charged tip to the skyrmion state (Fig. 2 and details in “Methods”). The schematics of the experimental setups (including the PFM and in situ STEM) are given in Fig. S3. The kinetic evolution of the in-plane polar pattern on the top PTO layer is simulated by the Phase-field simulations (Fig. 2a–f), the applied electric potential through a PFM tip is modeled by a Lorentz distribution, as plotted in Fig. S3a. The skyrmions underneath the probe region expand after 100 timesteps (Fig. 2a) due to the external out-of-plane electric field, consistent with previous reports11,40. Skyrmions near the edge of the tip become asymmetric due to the large in-plane field, resulting in greater in-plane polarization on the side near the tip. Consequently, the skyrmions near the edge of the tip are interconnected, effectively acting as a “fence” to separate the inner region near the tip from the outer region far away from the tip (Fig. 2b). Subsequently, after 10,000 timesteps (Fig. 2c), the skyrmions near the tip edge are displaced by the in-plane electric field, forming a ring-like pattern. Small skyrmions with opposite polarizations are observed upon melting of the stripes. After the applied field is removed, the asymmetric skyrmions recover to a circular pattern, and nucleation seeds gradually expand to a regular skyrmion, filling the region underneath the probe (Fig. 2d–f). Eventually, a grapefruit-like structure is formed, where 13 skyrmions are enclosed by a circular coat like pattern.

Fig. 2: Formation of polar skyrmion bag from skyrmion under applied potential with a charged tip.
Fig. 2: Formation of polar skyrmion bag from skyrmion under applied potential with a charged tip.
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a–c The evolution kinetics from phase-field simulations (potential −4.5 V, diameter 40 nm) after 100, 200, and 10,000 timesteps, respectively, showing the expansion of skyrmions under the probe and their transformation into the “coat” of the skyrmion bag, forming nucleation seeds due to the blocking effect of outside skyrmions and mutual squeezing. d–f The relaxation kinetics after the field is removed, depicted after 100, 200, and 10000 timesteps, respectively, illustrate the recovery process where asymmetric skyrmions revert to a circular pattern, and nucleation seeds gradually expand into a regular skyrmion. g 3-D vector plot of the polar skyrmion bag. The positive/negative polarizations are shown in green/yellow. h cAFM image after poled with a charged tip under −10 V, showing the formation of two skyrmion bags. i, j Experimental observation of a polar skyrmion bag with PFM: an applied voltage of −15 V for approximately 20 msec induces the formation of a skyrmion bag, outlined by the white dashed ring. The skyrmions located outside the bag (indicated by a white circle) exhibit polarization opposite to those inside the bag (indicated by an orange circle). k Planar-view HAADF-STEM images showing the transitions from skyrmions and skyrmions bags after the bias loading. l The enlarged views of the skyrmions bags marked in (k).

The Pontryagin density of the topological feature of the grapefruit-like structure as calculated by phase-field simulation is depicted in Fig. S4a. Each inner skyrmion has a negative Pontryagin density, resulting in a net topological charge of −1. Conversely, the circular boundary exhibits alternating positive and negative Pontryagin density with an opposite topological charge of +1, similar to outside skyrmions. This implies that the inner skyrmions possess opposite polarity with outside skyrmions and the circular “coat”. The 3-dimensional polar vector arrangement of the polar skyrmion bag is shown in Fig. 2g, while the 3-D isosurface structure (Fig. S4b) reveals that the skyrmion bundles extend throughout the entire PTO layer in the z-direction, akin to the magnetic skyrmion bag32,34.

To experimentally harness a skyrmion bag, the cAFM measurements is performed to unveil the presence of polar skyrmion bags (Fig. 2h), it can be seen that after poled through a charged tip at −10 V, at least one ring formed, which enclosed multiple skyrmions inside (marked with white dashed line). The skyrmion bag wall shows a higher conduction comparable to the polar skyrmions. The PFM characterizations are further performed, with the PFM tip positioned over the designated area (indicated by a cross in Fig. 1d), and a bias voltage of −15 V was applied through the PFM tip of pulse duration 20 ms, resulting in the formation of a skyrmion bag within the [(PTO)10/(STO)2]20 superlattice. This phenomenon was observed in the PFM amplitude and phase images (Fig. 2i, j). The dashed white boundary highlights the perimeter of the skyrmion bag, which is visible in the PFM amplitude image as a darker contrast (Fig. 2i). The creation of the skyrmion bag structure is attributed to the merging of skyrmion walls, which interconnect to form a closed circular boundary. This boundary confines multiple skyrmions with distinct polarities within the bag.

The PFM images provide a detailed visualization of the skyrmion bag structure, aligning well with theoretical predictions. Several individual skyrmions inside and outside the bag are marked with orange and white circles, respectively. Additionally, it is also discovered that a skyrmion bag can form under lower voltages if the voltage is applied for a longer duration. For instance, applying a voltage of −6 V for 30 s can induce the formation of a polar skyrmion bag, as demonstrated in Fig. S5a, b. The PFM phase image and the butterfly loops (Fig. S5c–f) clearly reveals the opposite polarities of skyrmions inside and outside the skyrmion bag. Skyrmions located outside the bag exhibit a darker phase contrast, while those within the bag display a brighter phase contrast. This difference in polarity highlights the spatial variation in local polarization.

To confirm the universality of this phenomenon, additional experiments were conducted on the [(PTO)16/(STO)16]16 superlattice, with further details provided in Fig. S6. These results substantiate the reproducibility and robustness of skyrmion bag formation across different superlattice configurations. In situ STEM is further performed to validate the formation of the skyrmion bag, with the electrical bias of 20 V applied for ~30 s between a tungsten tip and the Au conductive layer (~5 nm) on the film top. It can be seen that after applying the electric pulse, a skyrmion bag structure is formed, with a clear boundary that encloses multiple skyrmions (Fig. 2k). The radius of the skyrmion bag is ~20 nm. The zoom-in image of the polar skyrmion bag is given in Fig. 2l, an oval-like skyrmion bag structure can be identified, with six polar skyrmions inside. It should be noted that in the phase-field simulations and STEM images, the skyrmions are closely packed within the skyrmion bag. In contrast, the PFM images show only a few skyrmions present in the skyrmion bag. This discrepancy is primarily due to the differences in tip radius and the relatively low resolution of the PFM technique. Nevertheless, the PFM and STEM characterizations clearly confirm the formation of polar skyrmion bag with electric potential applied through a charged tip. Moreover, the concept of polar skyrmion bag has been introduced by Sun et al.39 recently, as a circular structure that encloses several skyrmions inside, where the polarity inside and outside the circle remains the same. Notably, the skyrmion bags created in this study exhibit a different polarity inside the bag compared to the outside. This resembles the skyrmion bag observed in a magnetic system, which represents a distinctly different structure from that of Sun et al.39.

Erasure and recovery of skyrmion bag

It is also crucial to erase and recover the skyrmion bag for memory application. We explore ways to destruct and recover the skyrmion bag state by manipulating the sign of the applied potential (Fig. 3). Our phase-field simulations reveal that for the skyrmion bag state formed after poling at −4.5 V (Fig. 3a), while an applied potential of +4 V will switch the polarity of the inner skyrmions and break the “coat” region into several asymmetric walls (Fig. 3b). The kinetics of the “coat” destruction process is illustrated in Fig. S7. The melting of the skyrmion bag wall can be observed when +4 V is applied, accompanied by the shrinking of the inner skyrmions. Upon removal of the applied electric field, the asymmetric skyrmion walls evolve into long vortex stripes. While the nucleation seeds continue to grow, eventually forming a regular skyrmion pattern (Figs. 3c and  S7f–j). The reconstruction of the skyrmion bag from this regular skyrmion pattern can be achieved by utilizing a repeatable −4 V pulse (Fig. 3d), indicating the excellent reliability and durability of this process.

Fig. 3: Erasing and recovering the skyrmion bag state.
Fig. 3: Erasing and recovering the skyrmion bag state.
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From phase-field simulations: a Planar views of the initial polarization distribution after applying −4 V to pack the skyrmions and form a stable skyrmion bag when the field is removed. b–c Applying +4 V could unpack the skyrmion bag and form skyrmions. d Reversible transition from skyrmions to skyrmion bag with another −4 V pulse. From PFM experiments: e Initial skyrmion structure. f Formation of skyrmion bag after applying −6 V. g Erasing the skyrmion bag after applying 6 V. h Recreation of polar skyrmion bag after applying −6 V again.

To experimentally demonstrate the reversible formation of a skyrmion bag, PFM images were first captured for a pristine region of the [(PTO)10/(STO)2]20 superlattice, which contains skyrmions (some skyrmions are marked by white circles in Fig. 3e). To create a skyrmion bag, a bias voltage of −6 V was applied for ~30 s, resulting in the formation of the skyrmion bag (Fig. 3f). To erase the skyrmion bag, the PFM tip was moved to the central region, and a reversed bias voltage of +6 V was applied for ~30 s. The resultant PFM images show the skyrmion bag boundary disappearing, with the skyrmions reverting to their original state (Fig. 3g). To recreate the skyrmion bag, the PFM tip was again positioned at the central region, and a −6 V bias was applied for ~30 s. The PFM images clearly demonstrate the reformation of the skyrmion bag (Fig. 3h), confirming the reproducibility of its formation and erasure. A similar experiment was conducted on the [(PTO)16/(STO)16]16 superlattice, which also exhibited consistent erasure and recreation of the skyrmion bag (details provided in Fig. S8).

Phase diagram of topological charge of skyrmion bag (Qsk) under different applied voltage

The effect of the magnitude of the applied potential is further investigated through phase-field simulations (Fig. 4). The final states depicted here were obtained after poling for 10,000 timesteps followed by relaxation at zero field for another 10,000 timesteps. When the applied potential is small, e.g., −1 V, the polar skyrmion states can be recovered upon removing the field (Fig. 4a). With a higher magnitude of the applied potential (−2 V), we observe the merging of polar skyrmions to form curved stripes (Fig. 4b). Moreover, further increasing the applied potential can lead to the formation of skyrmion bag (Fig. 4c, d), where the number of enclosed skyrmions increases with the applied potential. This can be understood since smaller skyrmion nuclei can formed under a higher field, breaking the long stripes through the Raleigh-Plateau mechanism, as has been reported previously41. As the applied potential increases to higher than −5 V (Fig. 4e, f), the skyrmions inside the bag start to merge into stripes, resulting in a decreasing of net topological charge. The electric potential phase diagram shows three regions (Fig. 4g): when the applied potential is small, the full skyrmion state can be preserved; under higher electric field, the skyrmions merge to form curved stripes; when the applied potential is >3 V, skyrmion bag state can be stabilized even when the field is removed. Notably, the net topological charge of the skyrmion bag initially increases with the increasing applied potential, reaching a maximum of 13 at −4.5 V, which gradually decreases to 4 at −8 V. The decrease in the net topological charge inside the bag can be attributed to the merging of the newly-formed skyrmions inside the bag.

Fig. 4: Topological state and topological charge vs electric potential phase diagram of skyrmion bag.
Fig. 4: Topological state and topological charge vs electric potential phase diagram of skyrmion bag.
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From phase-field simulations: a–f Planar views of the polarization distribution after electrical potential with −1 V, −2 V, −3 V, −3.5 V, −5.5 V, and −8 V, respectively. The boundary appears when potential is below −3 V, and inner skyrmions begin to melt into strips when voltage is below −5 V. g Absolute value of topological charge as a function of the absolute value of voltage (electrical potential), demonstrating the variation in topological charge under different voltages. From PFM measurements: h PFM phase images reveal the experimental observation of skyrmion bags containing varying numbers of skyrmions and stripe domains within the bag, depending on the magnitude of the applied bias voltage for same duration of 30 s. i Schematics of the energy landscape for the formation of high-order topologies. j Evolution of the total energy after the application and removal of the external field (−4.5 V) from phase-field simulations, showing that the skyrmion bag and skyrmion have similar total energy, and the external electric field could help overcome the energy barrier between the polar skyrmion and polar skyrmion bag.

The electric potential manipulation of the skyrmion bag state is further studied. Figure 4h presents the experimental observation of skyrmion bags containing varying numbers of skyrmions within the bag, depending on the magnitude of the applied voltage bias. This observation highlights the potential to stabilize different topological charge states, showcasing the feasibility of utilizing skyrmion bags in multistate memory applications. To illustrate the formation mechanism for the high-order topologies, a schematic diagram of the energy landscape is further plotted in Fig. 4i. It can be seen that the system has multiple stable energy wells with similar energy, where the polar skyrmion is the closest to the paraelectric phase, which forms spontaneously upon cooling. Meanwhile, to form skyrmion bags from the stable skyrmion state, one has to overcome the barriers with an externally applied electric field. The evolution of the total energy under an applied tip electric field of −4.5 V is further given in Fig. 4j, demonstrating that the external field provides an energy of ~1.7 MJ/m3 to overcome the barrier between skyrmion and skyrmion bag (details of each energy component is plotted in Fig. S9). After the field is removed, the system relaxes to a skyrmion bag state, which has a similar yet slightly lower energy as compared to the initial skyrmion state (−2.33 MJ/m3 for skyrmion vs. −2.48 MJ/m3 for skyrmion bag). This study demonstrates that the number of skyrmions within a skyrmion bag can be dynamically controlled in real-time by tuning the applied voltage, offering a promising avenue for real-time reconfigurable topological systems.

Discussion

In conclusion, this study provides a comprehensive investigation into the formation, manipulation, and temporal dynamics of polar skyrmion bags within the (PbTiO3)/(SrTiO3) superlattices, integrating experimental observations (PFM and STEM) with theoretical calculations. The ability to create, manipulate, and dissolve skyrmion bags with various adjustable topological charges provides valuable insights into their potential applications in nonvolatile, multistate high-density memory devices42. Here we explicitly link the integer topological charge Q to an electrical readout, enabling multilevel cells in which each stable Q corresponds to a distinct resistance/conductance state. In particular, we demonstrate that both the skyrmions within the bag and the wall of the skyrmion bag exhibit higher current density compared to the surrounding regions. These current-density “hot spots” originate from the local polarization texture and associated domain-wall conduction, so changing Q modifies the fraction and topology of conducting regions. In practical devices, planar electrodes or a ferroelectric tunnel junction could map Q (topological charge) → R (resistance) for compact multilevel readout without resolving individual skyrmions. Additionally, having multiple skyrmions within a single bag significantly reduces the risk of data loss, thereby enhancing the reliability of these memory devices. As shown in Fig. S10, the skyrmions within the bag are significantly protected by the bag’s wall when subjected to high external stimuli. Even though the skyrmion bag may decompose into clusters of skyrmions, these skyrmions remain more stable than regular skyrmions under high voltage. This robustness further supports Q-based encoding for reliable multistate memory. The consistency of our experimental results across multiple regions and the phase-field simulations bolsters the credibility of our findings. Looking forward, previous work has demonstrated the current-driven motion of the magnetic skyrmion bag/bundles due to topological protection30. In the ferroelectric system, it has been shown that the motion of skyrmions may be possible under an external field29, yet it still remains a challenging and exciting area to move the skyrmion bag collectively. Ultimately, this research contributes to both the fundamental understanding and practical utilization of polar skyrmion bags under external stimuli such as electric voltage, paving the way for advancements in multifunctional device architectures and the development of emerging technologies.

Methods

Materials synthesis

[(PbTiO3)16/(SrTiO3)16]16 superlattices (total 200 nm thickness) and [(PbTiO3)10/(SrTiO3)2]20 superlattices (total 100 nm thickness) were synthesized on TiO2-terminated single-crystalline SrTiO3 (001) substrates via RHEED (reflection high energy electron diffraction)-assisted pulsed-laser deposition (KrF laser). The growth temperature for the bottom SrRuO3, bottom La0.7Sr0.3MnO3 and PbTiO3/SrTiO3 layers were set as 700 °C, 650 °C, and 610 °C, respectively. While the oxygen pressure was fixed to 0.067 mbar, 0.067 mbar, and 0.133 mbar for the growth of bottom SrRuO3, bottom La0.7Sr0.3MnO3, and PbTiO3/SrTiO3 layers. During the whole synthesize process, the laser fluence was 1.1 J/cm2 with a repetition rate of 10 Hz. RHEED was used during the deposition to ensure a layer-by-layer growth mode for both PbTiO3 and SrTiO3, the RHEED oscillations were monitored by the specular RHEED spot. After deposition, the film was annealed for 10 min and then cooled to room temperature with oxygen pressure fixed to 66 mbar to promote full oxidation.

Structural and surface morphology characterization

XRD and RSM measurements were carried out using a Bruker D8 discover diffractometer with Cu-Kα radiation (λ = 1.54056 Å). The 2θ−ω XRD and RSM have been performed along (002) plane of the substrate.

In-situ STEM

The in situ experiments were carried out on FEI Titan G2 80–200 ChemiSTEM (operated at 200 kV, 30 mrad convergence semi-angle, 46–200 mrad angle collection semi-angle) in STEM mode. The surface of the bulk sample was coated with a Au layer using sputter coater. The sample was then cut into lamellas with the widest faces perpendicular to the [001] direction using Focused Ion Beam (FEI Quanta 3D FEG) for observation. The substrate was completely removed while the Au layer was partially retained as a bottom conductive electrode layer. In consideration of that potential slight damage caused by ion beam, we minished parameters, including voltage and electric current of ion beam down to 2 kV/27 pA to minimize the surface damage. The out-of-plane bias in in situ STEM experiments was performed by using the electrical holder system from Zeptools. An electrical bias was applied between a tungsten tip and the bottom electrode Au layer. The Au layer is connected to the holder ground. The diameter of probe tip is ~100 nm. During the whole in situ experiments, the electron beam was only turned on during the acquisition, to minimize the potential impacts of electron-beam irradiation.

Piezo-force microscopy

PFM imaging was performed by using the Oxford Asylum research MFP-3D origin+ instrument. PFM images were taken using Budget sensor Pt/Ir coated AFM tip having spring constant, resonate frequency and tip radius of ~3 nN/nm, ~75 KHz, and 25 nm, respectively. All the imaging was performed in dual AC resonance tracking (DART) mode having an ac amplitude of 0.7–1 V. Tip movement was controlled via software and a specific region was chosen to perform the scan. For applying the bias voltage, we keep the tip at particular position for certain amount of time, after that voltage was removed and required area scanned.

Phase field simulation

Phase-field simulations were performed for the [(PbTiO3)16/(SrTiO3)16] superlattices on a SrTiO3-(001) substrate by solving the time-dependent Ginzburg–Landau equations43,44,45,46:

$$\frac{d\vec{{\mathbf{P}}}}{{dt}}=-L\frac{{{\rm{\delta }}}{{\rm{F}}}(\vec{{{\mathbf{P}}}},\nabla \vec{{{\mathbf{P}}}})}{{{\rm{\delta }}}\vec{{\mathbf{P}}}}$$
(1)

where \(\vec{P}\), t, and L are the spontaneous polarization vector, evolution timestep, and kinetic coefficient, respectively. The total free energy F is fundamentally defined by the volume integral of distinct energy densities, including elastic, electric, Landau, and gradient energy densities,

$$F(\vec{P},\nabla \vec{P})=\int ({f}_{{elas}}+{f}_{{elec}}+{f}_{{Land}}+{f}_{{grad}}){dV}$$
(2)

The Landau chemical energy can be expressed as:

$${f}_{{\mbox{Land}}}= {\alpha }_{1}\left({{P}_{1}}^{2}+{{P}_{2}}^{2}+{{P}_{3}}^{2}\right)+{\alpha }_{11}\left({{P}_{1}}^{4}+{{P}_{2}}^{4}+{{P}_{3}}^{4}\right) \\ +{\alpha }_{12}\left({{P}_{1}}^{2}{{P}_{2}}^{2}+{{P}_{1}}^{2}{{P}_{3}}^{2}+{{P}_{2}}^{2}{{P}_{3}}^{2}\right)+{\alpha }_{111}({{P}_{1}}^{6}+{{P}_{2}}^{6}+{{P}_{3}}^{6}) \\ +{\alpha }_{112}\left[{{P}_{1}}^{2}\left({{P}_{2}}^{4}+{{P}_{3}}^{4}\right)+{{P}_{2}}^{2}\left({{P}_{1}}^{4}+{{P}_{3}}^{4}\right)+{{P}_{3}}^{2}({{P}_{1}}^{4}+{{P}_{2}}^{4})\right] \\ +{\alpha }_{123}{{P}_{1}}^{2}{{P}_{2}}^{2}{{P}_{3}}^{2}$$
(3)

In this study, the Landau polynomial is expanded to the sixth order, with the coefficients taken from previous study47,48. The expressions, parameters, and numerical treatment of other energy densities can be found in supplementary information and previous literatures14,40,41. A three-dimension mesh is built with the size of 320 × 320 × 350, where each grid represents 1 unit cell (0.4 nm). From the bottom to the top, the substrate, film, and air layers contain 30-, 272-, and 48-unit cells. The pseudocubic lattice parameters for PbTiO3 and SrTiO3 were taken as 3.9547 Å and 3.905 Å, respectively, to determine the lattice mismatch. The normalized timestep is set as 0.01 in this study, corresponding to 10–100 fs per simulation timestep. To compute the energy density, we sum the total free energy across each grid point within the simulation system and then normalize it by the simulation volume. The Pontryagin density q is calculated by \(q=\frac{1}{4\pi }\vec{{{\bf{P}}}}.\left(\frac{\partial \vec{{{\bf{P}}}}}{\partial x}\times \frac{\partial \vec{{{\boldsymbol{P}}}}}{\partial y}\right)\), the topological charge is calculated by the surface integral of the Pontryagin density \(Q=\int {qdxdy}\).

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.