Introduction

Topological matter phases beyond the Landau paradigm have promoted the emergence and development of topological physics and materials. Marvelous topological phases exist in real and momentum spaces1,2,3,4,5,6,7,8. Among them, magnetic skyrmions have been extensively studied for their intriguing physics and potential applications in spintronic devices9,10,11. They often exhibit skyrmion lattices (SkLs) assembled by skyrmion quasi-particles with long-range order, as theoretically and experimentally affirmed in various magnets3,4,12,13. Magnetic SkLs exhibit responses to external stimuli distinct from those of atomic crystal14,15,16, along with novel emergent properties, e.g., topological magnon band structure17, topological Nernst effect18, and novel microwave processing schemes in reconfigurable quasi-particle arrays19,20,21. As analogies to magnetic skyrmions, polar skyrmions in ferroelectric superlattices and multilayers22,23,24,25,26,27,28,29 have garnered widespread attention due to similar topology but distinct properties, such as negative capacitance30 and giant electric field-induced second harmonic generation31. However, they have yet to be experimentally demonstrated to form long-range ordered polar SkLs27, despite theoretical predictions suggesting the formation of hexagonal polar SkLs (H-SkLs)32,33. Constructing long-range order of polar skyrmions is critical to the functional properties of devices based on polar topological textures. Researches on the collective dynamics of polar vortex lattices34,35,36 suggest potential applications of polar topological textures in metamaterials and optoelectronic devices. The interaction between SkLs and external fields will significantly differ depending on the lattice type, and thus exhibiting distinct properties. Owing to the smaller coordination number of the tetragonal lattice compared to the hexagonal lattice, the anisotropy of the tetragonal lattice should be stronger. The macroscopic properties of materials are directly related to this anisotropy. The potential of polar topological textures for application of nonlinear optics has been confirmed31. SkLs with different lattice type may also exhibit distinct dispersion characteristics when used as optical gratings or photonic crystals. Due to the coupling of strain and polarization, SkLs typically have a periodic distribution strain field, which may be used as periodic pre-strained substrates or acoustic filters. If the lattice types of SkLs are reconfigurable, the optical and electromechanical performance of devices based on the SkL may be able to dynamically control. Therefore, exploration and a deterministic control of long-range ordered polar SkLs are not only physically intriguing but also practically significant.

The flexoelectric effect, electric polarization response induced by strain gradients, is ubiquitous in common solids37. Due to the small flexoelectric coefficients, this effect is often neglected in bulk materials. However, in thin films, strain gradients are typically several orders of magnitude larger than that of bulk materials38,39. Therefore, this effect becomes significant and may even dominate in such systems. In addition, polarization defects like domain walls, vortices and skyrmions, typically of nanoscale size and with large strain and polarization gradients, should be susceptible to the flexoelectric effect. Indeed, previous studies have shown the flexoelectric effect can introduce orientation anisotropy of non-Ising domain walls relative to the crystal axis40,41. This anisotropy is expected to influence the long-range order of polar skyrmions. Interesting questions hence arise: Is it possible to stabilize other types of SkLs besides H-SkLs? How does flexoelectricity play its role in stabilization and transformation of polar SkLs? Can we gain deterministic control of polar SkLs via flexoelectricity?

In this work, phase-field simulations were performed to explore the critical roles of flexoelectricity playing in the formation and transformation of polar SkLs. It shows that diverse polar SkL patterns can emerge in the ferroelectric PbTiO3 (PTO) thin films as schematically shown in Fig. 1. The structure of single polar skyrmion bubble at the top, middle, and bottom (001) planes of the thin film are schematically presented on the right of Fig. 1a. This structure is consistent with previous reports7,32,42. The diverse polar SkL patterns include tetragonal-SkL (T-SkL) and hexagonal-SkLs (H-SkLs) in various orientations (denoted as d1H-SkL, d2H-SkL, v1H-SkL and v2H-SkL). Specifically, d1H-SkL and d2H-SkL denote H-SkLs with one of the basis vectors parallel to the <110> axes, while v1H-SkL and v2H-SkL denote H-SkLs with one of the basis vectors parallel to the [100] and [010] axes, respectively. Stabilization of these emergent SkL states is summarized by phase diagrams and is attributed to the material anisotropy modified by the flexoelectric effect. Interestingly, we further found that the hexagonal-SkLs can be rotated by applying strain gradient or in-plane (IP) electric field to the films, based on the modified IP anisotropy. Besides, an abnormal nonreciprocal bending response of tetragonal-SkL is also induced by the flexoelectric effect. Our results thus demonstrate the critical role of flexoelectricity in discovery, design, and manipulation of emergent crystals assembled from polar skyrmions.

Fig. 1: Diverse polar SkLs in PTO thin films.
figure 1

a Three-dimensional OP polarization distribution of T-SkL in the film, simulated at the condition of E[001] = –1.25 MV/cm and f12 = 5 V, f11 = f44 = 0 V. The configurations of individual polar skyrmion bubble at the top, middle, and bottom (001) planes of the film are schematically shown on the right. b Polarization distributions of T-SkL formed near the top plane of the film at the condition of E[001] = –1.25 MV/cm and f11 = 3 V, f12 = f44 = 0 V. c–g Schematics of different polar SkL patterns predicted in PTO thin films. All the polarization vectors are normalized, and the color represents Pz, sharing the color bar in b. fij are the flexocoupling coefficients and can be found in Methods, and E[001] is the out-of-plane electric field.

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Results

Formation of diverse polar patterns

To gain an essential picture, we first consider the evolution of polarization of the PTO thin film system excluding the flexoelectric effect and including the flexoelectric effect but with only f11 taken to be nonzero (3 V), as shown in Fig. 2a, b, respectively. Without the flexoelectric effect (Fig. 2a), labyrinthine state is formed in the PTO thin film under zero out-of-plane (OP) electric field along E[001]. As E[001] increases to –0.5 MV/cm, the labyrinthine state evolves into d1H-SkL state. As E[001] increases to –1.25 MV/cm, the polar skyrmions shrink and the SkL becomes imperfect with defects emerged (hereafter, we denote such an imperfect H-SkL state as iH-SkL state). The structural factor (which is computed via two-dimensional Fourier transform of (P3 – <P3>)) of the labyrinthine state implies a strong orientational order at room temperature, similar to that revealed by previous work43. At high enough temperature, the labyrinthine state will transforms into stripe state or vortex lattice state, characterized by a single wavevector q (single-q state)44. In contrast, the structural factor of the H-SkL state exhibits a six-fold rotational symmetry, which is a triple-q state (detailed explanations for the formation of H-SkL can be found in Supplementary Discussion II). As defects appear in the iH-SkL state, the structural factor no longer retains an obvious six-fold rotational symmetry. The size shrinkage of the skyrmions is accompanied by the expansion of the OP polarization region separating the skyrmions, as clearly reflected by evolution of the Pontryagin density. For case with f11 = 3 V and f12 = f44 = 0 V (Fig. 2b), the labyrinthine state exhibits stripes with nearly 90° angle due to the flexoelectric effect at zero E[001]. At E[001] of –0.5 MV/cm, the labyrinthine state breaks into elongated polar skyrmions along [100] or [010] axes instead of forming H-SkL state. When E[001] reaches –1.25 MV/cm, a double-q state (denoted as T-SkL) emerges (explanations for the formation of T-SkL can be found in Supplementary Discussion II and III). From Supplementary Fig. S1, as f11 increases, the IP flexoelectric field \({E}_{\text{IP}}^{\text{flexo}}\) localizing around the skyrmions (i.e., near their edges) would strengthens, consequently strengthening the interaction between skyrmions. When the IP flexoelectric field is small, its effect is mainly limited to the vicinity of each skyrmion. As the IP flexoelectric field increases, an overlapping effect emerges, and the skyrmion interaction becomes more significant. As previously discussed, the flexoelectric effect introduces an orientation anisotropy of ferroelectric domain walls relative to the crystal axis40,41. Evolution of the skyrmion pattern and the IP flexoelectric field \({E}_{\text{IP}}^{\text{flexo}}\) reflects such a flexoelectricity-modified anisotropy also exists in complex polar systems like skyrmion systems. With the enhancement of system anisotropy along the crystal axis, the SkL transforms from a hexagonal lattice to a tetragonal lattice.

Fig. 2: Evolution of polar states in PTO thin films under an OP electric field E[001].
figure 2

Field-dependent polarization configuration, structural factor and Pontryagin density near the top plane of the PTO thin films simulated at the condition of a f11 = f12 = f44 = 0 V, b f11 = 3 V and f12 = f44 = 0 V, respectively. Structural factors for these polar states are computed via two-dimensional Fourier Transform of (P3 – <P3>), with <P3> being the average OP polarization of the PTO thin films.

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To comprehensively reveal the influences of the flexoelectricity on polar SkL, we calculated the f11 vs. E[001] and f12 vs. f11 “phase diagrams”, as illustrated in Fig. 3a, b. Up to eight polar states are confirmed and their polarization distributions are shown in Fig. 3c. Here, “C” indicates the uniform downward polarization state. “H-L” and “T-L” represent the labyrinthine states that does not and does spontaneously form 90° angle stripes, respectively. “TH-L” also denotes the labyrinthine state, but with the angle close to but not equal to 90°. “TH-SkL” is a polar SkL with mixed features of both hexagonal and tetragonal lattices. Due to the possible influence of the geometry of the system on the results45, we performed calculations with different IP model sizes. The results show that the size of the skyrmions varies with the change of the IP model size, but this numerical effect related to the IP periodic boundary condition does not have a significant impact on our main conclusions (Supplementary Discussion I). From Fig. 3a, with the increase of f11, the TH-L state emerges as the transitional state between the H-L and T-L states, and the TH-SkL state emerges as the transitional state between the H-SkL and T-SkL states. It reflects that the structural transformation of SkL from hexagonal lattice to tetragonal lattice is a gradual process occurring with the enhance of the flexoelectric effect. The structural transformation is attributed to the material anisotropy modified by the flexoelectric effect. In addition, under stronger flexoelectric effect, the threshold OP electric fields among the states of labyrinthine, SkL and C states increase.

Fig. 3: The simulated topological polar states in PTO thin films.
figure 3

a f11 vs. E[001] and b f12 vs. f11 “phase diagrams” of the polar states. f12 and f44 are both set to 0 V in a. f44 and E[001] are set to 0 V and –1.25 MV/cm in b, respectively. Eight polar patterns near the top (001) planes of the thin film are shown in c. Due to the degeneracy of d1H-SkL (v1H-SkL) and d2H-SkL (v2H-SkL), the regions marked with d1H-SkL (v1H-SkL) in the phase diagrams should also stabilize d2H-SkL (v2H-SkL). The conditions for each state in c are: H-L (fij = 0 V and E[001] = 0 MV/cm), TH-L (f11 = 2 V, f12 = f44 = 0 V and E[001] = 0 MV/cm), T-L (f11 = 3 V, f12 = f44 = 0 V and E[001] = 0 MV/cm), TH-SkL (f11 = 3 V, f12 = f44 = 0 V and E[001] = –1.00 MV/cm), iH-SkL (fij = 0 V and E[001] = –1.25 MV/cm), v1H-SkL (f11 = 1 V, f12 = f44 = 0 V and E[001] = –1.25 MV/cm), d1H-SkL (f11 = f12 = 4 V, f44 = 0 V and E[001] = –1.25 MV/cm), and T-SkL (f11 = 3 V, f12 = f44 = 0 V and E[001] = –1.25 MV/cm).

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From the “phase diagram” shown in Fig. 3b, calculated at the condition of E[001] = –1.25 MV/cm, one can infer how the anisotropy of the flexocoupling tensor affects the polar SkL. Along the diagonal, the flexocoupling tensor is isotropic (f11 = f12 ≠ 0, f44 = 0), satisfying the equality f11 – f12 = f44. Therefore, the PTO thin film systems only have the intrinsic anisotropy along the diagonal induced by biaxial misfit strain and stabilize the d1H-SkL state. At the off-diagonal region (f11 ≠ f12 ≠ 0, f44 = 0), the flexocoupling tensor becomes in cubic symmetry. The systems exhibit stronger IP biaxial anisotropy along the [100] and [010] axes when further deviating the diagonal, leading to a gradual transformation from d1H-SkL state into T-SkL state. Consequently, the symmetry of the flexocoupling tensor is directly related to the formation of T-SkL. When only f12 ≠ 0, similar conclusions can be obtained (Supplementary Fig. S2a). However, the IP easy axis directions induced by f11 and f12 are orthogonal, thus being isotropic at the diagonal in Fig. 3b. The polar patterns corresponding to the “phase diagrams” in Fig. 3 and those of the f12 vs. E[001] “phase diagram”, are presented in Supplementary Fig. S3.

We also explore the effect of the flexocoupling coefficient f44 on the formation of SkL. As shown in Supplementary Fig. S4a, b, when only f44 ≠ 0, although the flexocoupling tensor is cubic symmetric, T-SkL cannot be stabilized. However, when f11 or f12 is also nonzero, f44 can influence the lattice type of SkL via affecting the symmetry of the flexocoupling tensor. As shown in Supplementary Fig. S4c, where the flexocoupling tensor is isotropic (f11 = f44 = 3 V, f12 = 0 V), the H-SkL state is stabilized. Conversely, the T-SkL state emerges for the case (f11 = 3 V, f44 = f12 = 0 V). In Supplementary Fig. S4d, the values of the flexocoupling coefficients (f12 = f44 = 3 V, f11 = 0 V) brings a strong enough cubic symmetry to the system, leading to the formation of T-SkL state. However, the case (f12 = 3 V, f11 = f44 = 0 V) stabilizes TH-SkL. In addition, f44 changes the threshold OP electric field for the transformation of diverse polar states. The above results together demonstrate the lattice type of polar SkL is intimately related to the symmetry of the flexocoupling tensor.

In-plane rotation of hexagonal polar skyrmion lattices

The macroscopic properties of polar SkL may depend on the lattice type. Deterministic control of the lattice type of SkL is therefore meaningful to regulating their functional properties. Our simulation shows that appropriately large values of f11 and f12 can cause the orientation of the H-SkL to rotate (see details in the “phase diagram” plotted in Supplementary Fig. S3b). In the following, we aim to investigate the influences of field-induced uniaxial IP anisotropy on the orientation of the H-SkL. Owing to the strain gradients are associated with the flexoelectric effect, we first consider modifying the IP easy axis by bending the PTO thin films. Two pure bending operations along the [010] axis (denoted as \({{\rm{B}}}_{1}^{+}\) and \({{\rm{B}}}_{1}^{-}\) for downward and upward bending, respectively) were applied to the films and the corresponding strain distributions (ε11) are illustrated in Fig. 4a. Simulation results show that, when annealing the PTO thin films (under either \({{\rm{B}}}_{1}^{+}\) or \({{\rm{B}}}_{1}^{-}\) bending strains) from paraelectric phase at the condition (E[001] = –1.25 MV/cm, f11 = f12 = 4 V, and f44 = 0 V) intermediate states with stripe-like polarization patterns would first appear (Fig. 4b, c) before the films finally evolve into v1H-SkL state (Fig. 4d). It is noteworthy that the emergent time points of the stripe-like intermediate states are different for \({{\rm{B}}}_{1}^{+}\) and \({{\rm{B}}}_{1}^{-}\) bending operations (see the number of simulation steps N shown in Fig. 4b, c). This is related to the different effective OP electric field due to the different bending-induced OP flexoelectric field. However, the OP flexoelectric field is not the decisive factor causing the rotation of H-SkL during the bending-annealing process. The rotation of H-SkL can still occur after the flexoelectric effect is switched off (Supplementary Fig. S5). This is due to the fact that the bending strain (gradient) mainly affects the OP flexoelectric field, while the latter has no significant effect on the IP anisotropy of the system. The rotation of H-SkL is driven by the electrostriction effect: since the electrostriction coefficients of PTO have the relation Q11 > –Q12 > 0, the two bending operations along [010] axis introduce an IP easy axis parallel to the [100] direction (see details in Supplementary Discussion IV).

Fig. 4: In-plane rotation of H-SkL under bending.
figure 4

a Distributions of strain in the thin films along the [001] axis (z axis) generated by bending operations (\({{\rm{B}}}_{1}^{+}\) and \({{\rm{B}}}_{1}^{-}\)). The strain gradient \({\varepsilon }_{\mathrm{11,3}}^{\text{b}}\) generated by different bending operations is marked in a. b, c Stripe-like intermediate states appear in the PTO thin films during bending-annealing processes. N denotes the number of simulation steps. d, e H-SkLs with different orientations emerged after bending-annealing processes. In the simulations, E[001] = –1.25 MV/cm and f11 = f12 = 4 V, f44 = 0 V are set.

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Note also that the formation of v1H-SkL state in PTO thin films under bending is contrast with those without bending operations, which are found to form d1H-SkL state at the same condition (E[001] = –1.25 MV/cm, f11 = f12 = 4 V, and f44 = 0 V) as previously shown in Fig. 3. Moreover, v2H-SkL state instead of v1H-SkL state would emerge in PTO thin films under bending operations along [100] axis (denoted as \({{\rm{B}}}_{2}^{+}\) and \({{\rm{B}}}_{2}^{-}\) for downward and upward bending, respectively), as shown in Fig. 4e. For this case, the two bending operations induce an IP easy axis along the [010] axis to the PTO thin film. The results of Fig. 4 therefore imply that bending annealing could serve as an effective strategy to regulate the lattice of polar skyrmions.

We further consider modifying the IP easy axis by applying an IP electric field to the PTO thin films, to explore the effects of such a field-induced IP anisotropy on the orientation of the H-SkL state. Typical simulation results at various IP electric fields are depicted in Supplementary Fig. S6. In the simulations, the flexoelectric effect is switched off to reveal the pure effects of IP electric field. The PTO thin film evolves into v2H-SkL state after a sufficiently long period of evolution without applying an IP electric field (Supplementary Fig. S6a). In contrast, v1H-SkL and v2H-SkL can be stabilized when the IP electric field is applied along the [100] and [010] axis, as shown in Fig. S6b, c, respectively. The stripe-like intermediate states, similar to those found during bending-annealing process in Fig. 4b, c, appear before the final formation of skyrmion lattice states. When an IP electric field is applied along [110] or [\(\bar{1}\)10] axis, d2H-SkL or d1H-SkL can also be stabilized (Supplementary Fig. S6d, e). These results together with Fig. 4 and Supplementary Fig. S5 indicate the strong impacts of IP anisotropy on the nucleation of SkL, leading to the rotation of H-SkL. Although the rotation of H-SkL under bending or IP field application is not closely related to the flexoelectric effect, the inherent mechanisms of both are similar, that is, by modifying the anisotropy of the system.

Nonreciprocal bending response of tetragonal polar skyrmion lattices

Previous simulations show that bending operations need to be combined with additional annealing to trigger rotation of H-SkL, and flexoelectric effect does not play a crucial role in such a rotation. However, the situation of T-SkL is found to be quite different. Particularly, a nonreciprocal bending response of T-SkL under bending operations is observed and flexoelectric effect plays an important role, as shown in Fig. 5. Similar to previous simulations, two pure bending operations along the [010] axis (denoted as \({{\rm{B}}}_{1}^{+}\) and \({{\rm{B}}}_{1}^{-}\) for downward and upward bending, respectively) were applied to the films. T-SkL is formed at the condition (E[001] = –1.25 MV/cm and f11 = 3 V, f44 = f12 = 0 V). As schematically shown in Fig. 5a, when under \({{\rm{B}}}_{1}^{+}\) bending, the initial T-SkL transforms into v1H-SkL, and the T-SkL state recovers upon removal of bending strain. However, such a SkL transformation does not occur for case of \({{\rm{B}}}_{1}^{-}\) bending. This nonreciprocal electromechanical phenomenon can be attributed to the inconsistency between favorite polar structures induced by the flexoelectric effect and the structures of polar skyrmion bubbles near the top and bottom (001) planes (more detailed discussion can be found in Supplementary Discussion V). As discussed by previous researches40, flexoelectric effect tends to stabilize Ising-Néel-like domain walls. Specifically, head-head or tail-tail domain wall can be induced when f11 is greater than or less than 0, as respectively shown in Fig. 5b. For a polar skyrmion bubble (with a topological charge Q being +1 or –1), its polar structures at the top and bottom planes are different: one is of central divergence and the other is of central convergence. This inconsistency leads to skyrmion bubbles no longer possessing inversion symmetry along the [001] axis, and forming asymmetric distributions of flexoelectric field along [001] axis, as illustrated in Fig. 5c, d. The IP distributions of polarization and flexoelectric field in the PTO films under the two bending operations are also depicted in Fig. S7. The asymmetry coming from the coupling of flexoelectricity and topology of the polar skyrmion results in the nonreciprocal response of T-SkL state to bending operations. Variations in energy landscape of the PTO film upon bending operations are schematically shown in Fig. 5e, f, respectively. With the simulated condition (E[001] = –1.25 MV/cm, f11 = 3 V, Q = +1), T-SkL state is stable, and H-SkL state is unstable. Under \({{\rm{B}}}_{1}^{+}\) bending, T-SkL state becomes unstable and spontaneously transforms into H-SkL state. After removing the \({{\rm{B}}}_{1}^{+}\) bending, the energy landscape returns to initial shape, and H-SkL state transforms back to T-SkL state. In contrast, the \({{\rm{B}}}_{1}^{-}\) bending operation does not significantly affect the energy landscape of the system, thus it does not change the lattice type of SkL.

Fig. 5: Nonreciprocal bending response of T-SkL induced by the flexoelectricity.
figure 5

a Schematic illustration of transformation between T-SkL and v1H-SkL under bending operations. b Ising-Néel-like domain walls induced by flexoelectric effect, and the polarization configurations at the top and bottom (001) planes of the polar skyrmion bubbles with opposite topological charge Q. c Flexoelectric field at the (010) plane of the PTO film along the black line in d. d Flexoelectric field near the top and bottom (001) planes of the PTO film. e, f Schematics of the variations in energy landscape of the PTO film upon bending operations. In the simulations, E[001] = –1.25 MV/cm and f11 = 3 V, f44 = f12 = 0 V are set.

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To have deeper insight into the nonreciprocal response of T-SkL to bending and the role of flexoelectricity, we further conducted simulations at the condition (E[001] = +1.25 MV/cm and f11 = +3 V, f44 = f12 = 0 V) and at the condition (E[001] = –1.25 MV/cm and f11 = –3 V, f44 = f12 = 0 V). Simulation results at the two conditions are shown in Supplementary Figs. S8 and S9, respectively. For the case previously presented in Fig. 5 (E[001] = –1.25 MV/cm and f11 = 3 V, f44 = f12 = 0 V), the topological charge Q of each polar skyrmion is +1, and f11 is greater than 0, i.e., Q × f11 > 0. The \({{\rm{B}}}_{1}^{+}\) bending operation can control the lattice type of SkL. For the case shown in Supplementary Fig. S8, Q = –1 and f11 = +3 V, and for the case shown in Supplementary Fig. S9, Q = +1 and f11 = –3 V, i.e., for both cases Q × f11 < 0. For these two cases, the \({{\rm{B}}}_{1}^{-}\) bending operation instead of \({{\rm{B}}}_{1}^{+}\) bending operation cause the transformation of T-SkL to H-SkL. These results clearly confirm that the nonreciprocal electromechanical response of T-SkL to bending operations arises from the coupling between flexoelectricity and polar topology.

Discussion

Abundant and novel properties can emerge in long-range ordered quasi-particle crystals. The rotation and distortion14,15,46,47, along with structural transformation48 of magnetic SkLs have been widely reported. They can stabilize either hexagonal3,4 or tetragonal12,13 SkLs and melt from lattice states to liquid-like states16. As analogies to magnetic skyrmions, polar skyrmions have garnered widespread attention for their non-trivial topology and novel properties like negative capacitance30 and nonlinear optical effect31. However, so far very few works have focused on the formation of polar SkLs32,33. Polar SkLs with long-rang order may carry exotic collective properties which are critical for realization of devices based on polar topology. Construction of polar SkLs with diverse lattice types and achieving deterministic manipulation are prerequisites for applications. Hence, our work is committed to reveal the important factors for forming different polar SkLs and their control and provide guidelines for the realization of polar SkLs in experiments.

In summary, based on phase-field simulations, we demonstrate the critical roles of flexoelectricity playing in the stabilization and transformation of polar SkLs. Different polar SkL patterns can emerge in the ferroelectric thin films, including tetragonal-SkL, and hexagonal-SkLs with diverse orientations. Stabilization of these polar SkL states is summarized by phase diagrams and is attributed to the material anisotropy modified by the flexoelectric effect. Furthermore, the rotation of hexagonal-SkLs can be realized by applying strain gradient or IP electric field to the films, based on the modified IP anisotropy. Moreover, a novel nonreciprocal bending response of tetragonal-SkL induced by the flexoelectric effect is presented. The coupling of the flexoelectricity and polar topology not only significantly influences the configuration of polar topology but also raises new challenges for its manipulation. Comprehensive understandings of these periodic polar structures are the basis of development and design of metamaterials and optoelectronic devices based on polar topological textures. Our results demonstrate the critical role of flexoelectricity and other factors like bending strain and external electric field in discovery, design, and manipulation of emergent crystals assembled from polar topological textures. While this study focuses on the influences of flexoelectricity on polar topological textures, the impacts of polar topological textures on the macroscopic flexoelectric response of ferroelectrics are also worth further exploration.

Methods

Phase-field model

To investigate the polar states in the PTO thin films grown on the substrates, the phase-field method is applied49. A three-dimensional spontaneous polarization vector P = (P1, P2, P3) is chosen as the order parameter. The temporal-spatial evolution of polarizations is captured by the time-dependent Ginzburg-Landau (TDGL) equation as

$$\begin{array}{c}\displaystyle\frac{\partial {P}_{i}({\bf{r}},t)}{\partial t}=-M\displaystyle\frac{\delta F}{\delta {P}_{i}({\bf{r}},t)}\end{array},(i=1,2,3)$$
(1)

where M is the kinetic coefficient, along with r and t being the spatial coordinate vector and the time variable, respectively. The total free energy F, based on Landau–Ginzburg–Devonshire theory, can be expressed as the sum of the volume integrals over the entire space for Landau fLand, elastic felas, electrostatic felec, gradient fgrad, and flexoelectric energy densities fflex, as well with the surface energy density fsurf integrated over the surface,

$$F=\iiint ({f}_{{\rm{Land}}}+{f}_{{\rm{elas}}}+{f}_{{\rm{elec}}}+{f}_{{\rm{grad}}}+{f}_{{\rm{flex}}}){\rm{d}}V+\iint {f}_{{\rm{surf}}}{\rm{d}}S$$
(2)

For PTO, the Landau energy density can be expressed as,

$${f}_{{\rm{Land}}}={a}_{1}\sum _{i}{P}_{i}^{2}+{a}_{11}\sum _{i}{P}_{i}^{4}+{a}_{12}\sum _{i > j}{P}_{i}^{2}{P}_{j}^{2}+{a}_{111}\sum _{i}{P}_{i}^{6}+{a}_{112}\sum _{i\ne j}{P}_{i}^{4}{P}_{j}^{2}+{a}_{123}\prod _{i}{P}_{i}^{2}$$
(3)

where \({a}_{i},{a}_{ij}\,{\rm{and}}\,{a}_{ijk}\)(with i, j, k =1, 2, 3) are the expansion coefficients, and a1 is linearly dependent on temperature and obeys the Curie-Weiss law.

Based on the microscopic elasticity theory50, the elastic energy density is given by,

$$\begin{array}{c}{f}_{{\rm{elas}}}=\frac{1}{2}{C}_{ijkl}{e}_{ij}{e}_{kl}=\frac{1}{2}{C}_{ijkl}({\varepsilon }_{ij}-{\varepsilon }_{ij}^{0})({\varepsilon }_{kl}-{\varepsilon }_{kl}^{0})\end{array}$$
(4)

where \({C}_{ijkl},{e}_{ij}({\bf{r}}),{\varepsilon }_{ij}({\bf{r}})\,{\rm{and}}\,{\varepsilon }_{ij}^{0}({\bf{r}})\) are the components of elastic stiffness tensor, elastic strain, total strain, and eigenstrain, respectively. Via electrostriction effect, eigenstrain depends on spontaneous polarization as \({\varepsilon }_{ij}^{0}={Q}_{ijkl}{P}_{k}{P}_{l}\), with \({Q}_{ijkl}\) being electrostrictive coefficients.

The electrostatic energy density is related to the spontaneous polarization P(r), and can be expressed as

$${f}_{{\rm{elec}}}=-{P}_{i}{E}_{i}-\frac{1}{2}{\varepsilon }_{{\rm{b}}}{E}_{i}^{2}$$
(5)

where \({\varepsilon }_{{\rm{b}}}\) is the background dielectric constant51,52,53, and Ei is the components of the total electric field E. The total electric field E is the sum of depolarization field Ed and external electric field Eext acting on the ferroelectric films. The depolarization field Ed can be expressed as Ed = βESC + (1 – β)EOC, with β being a screening factor ranging from 0 to 1, along with ESC and EOC being the depolarization field at ideal short-circuit (SC) and open-circuit (OC) boundary condition54, respectively.

The gradient energy density caused by non-uniform polarization can be written as

$$\begin{array}{c}{f}_{{\rm{grad}}}=\,\displaystyle\frac{1}{2}{g}_{ijkl}{P}_{i,j}{P}_{k,l}\end{array}$$
(6)

with \({g}_{ijkl}\) being the gradient energy coefficients, and the subscript “,i” represents the spatial derivative along the i direction.

The surface energy density attributed by lattice relaxation near the surface of the thin films55,56,57, is expressed as

$${f}_{{\rm{surf}}}=\frac{{D}_{1}^{S}{P}_{1}^{2}}{2{\delta }_{1}^{{\rm{eff}}}}+\frac{{D}_{2}^{S}{P}_{2}^{2}}{2{\delta }_{2}^{{\rm{eff}}}}+\frac{{D}_{3}^{S}{P}_{3}^{2}}{2{\delta }_{3}^{{\rm{eff}}}}$$
(7)

with \({\delta }_{i}^{{\rm{eff}}}\) being the extrapolation lengths, and \({D}_{i}^{S}\) being the coefficients related to \({g}_{ijkl}\) and surface orientation, respectively.

The flexoelectric energy density can be written in the form of Lifshitz invariant as56

$${f}_{{\rm{flex}}}=\frac{{f}_{ijkl}}{2}\left({\varepsilon }_{ij}\frac{\partial {P}_{k}}{\partial {x}_{l}}-{P}_{k}\frac{\partial {\varepsilon }_{ij}}{\partial {x}_{l}}\right)$$
(8)

with \({f}_{ijkl}\) being the bulk flexocoupling coefficients. Variation of the flexoelectric energy with polarization gives rise to flexoelectric field as \({E}_{i}^{{\rm{flexo}}}={f}_{ijkl}{\varepsilon }_{kl,j}.\) Note that the elastic stiffness, electrostrictive, gradient energy and flexocoupling coefficients are written in the reduced form according to the Voigt notation, e.g., f1111 = f11, f1122 = f12, f1212 = f44.

There are only three independent flexocoupling coefficients for materials belong to the cubic point group: the longitudinal coupling coefficient f11, the transversal coupling coefficient f12, and the shear coupling coefficient f44, respectively.

To obtain the elastic energy, the strain field should be calculated first by solving the mechanical equilibrium equation,

$$\begin{array}{c}{C}_{ijkl}{u}_{k,lj}({\bf{r}})={C}_{ijkl}{\varepsilon }_{kl,j}^{0}\end{array}({\bf{r}})-{f}_{ijkl}{P}_{k,lj}({\bf{r}})$$
(9)

The total strain can be written as\(\begin{array}{c}{\varepsilon }_{ij}=\tfrac{1}{2}(\end{array}{u}_{i,j}+{u}_{j,i})\), with u(r) = (u1, u2, u3) being the total displacement field. The mechanical boundary condition is the top surface of the thin films is stress-free while the bottom surface is clamped by the substrate .

The electric field is obtained by solving the electrostatic equilibrium equation

$$\begin{array}{c}{\varepsilon }_{{\rm{b}}}{\varphi }_{,ii}({\bf{r}})={{P}}_{i,i}\end{array}({\bf{r}})$$
(10)

under the screening boundary condition related to β, where \(\varphi ({\bf{r}})\) is the electric potential.

Numerical implement

The TDGL equations [Eq. (1)] are numerically solved by Euler iteration method, with the normalized time step \(\varDelta \tilde{t}\) taken to be 0.001. The simulated supercell of PTO thin film consists 128 × 128 × 12 grid points, with the mesh grid spacing ∆h being 0.4 nm. The mechanical and electrostatic equilibrium equations [Eqs. (9) and (10)] are solved by fast Fourier transform algorithm58,59,60. Periodic boundary conditions are applied in the IP directions. The thin films are subjected to a biaxial misfit strain (εm = –0.01) from the substrate. To apply the bending operations, a fixed strain distribution of each operation is first generated, and then is added to the total strain. The PTO thin films are under incomplete screening boundary conditions, with the screening factor β being 0.6. Random initial noises of polarizations are used for all simulations at room temperature, with a peak amplitude of 0.001 C/m2. The layer topological charge \(Q(z)=\iint q{\rm{d}}x{\rm{d}}y=\frac{1}{4\pi }\iint {{{{\hat{\bf{P}}}}}}\cdot ({\partial }_{x}{{{{\hat{\bf{P}}}}}}\times {\partial }_{y}{{{{\hat{\bf{P}}}}}}){\rm{d}}x{\rm{d}}y\) is calculated to characterize the polar skyrmion, with q and \({{\hat{\bf{P}}}}\) being the Pontryagin density and the normalized polarization vector. The structural factor is \(S(u,v)=\frac{1}{MN}\mathop{\sum }\nolimits_{x=0}^{M-1}\mathop{\sum }\nolimits_{y=0}^{N-1}({P}_{3}(x,y)-\overline{{P}_{3}})\exp [-I2\pi (\frac{xu}{M}+\frac{yv}{N})]\), where \(I=\sqrt{-1}\) and \(\overline{{P}_{3}}\) is the average OP polarization of the selected layer of the thin film, along with M and N are the lateral length and width of the simulation cell, respectively. Values of all the simulation parameters are listed in the Supplementary Table S1.