Background

As analogies to magnetic skyrmions, polar skyrmions in ferroelectric superlattices and multilayers have garnered widespread attention for their non-trivial topology and novel properties like negative capacitance and nonlinear optical effect. So far, they have yet to be experimentally demonstrated to form long-range ordered polar skyrmion lattices (SkLs), despite theoretical predictions suggesting the formation of hexagonal polar SkLs (H-SkLs). 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 lattices 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. Therefore, exploration and a deterministic control of long-range ordered polar SkLs are not only physically intriguing but also practically necessary. 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 is known to induce oriented anisotropy of domain walls relative to the crystal axis. This anisotropy is expected to influence the long-range order of polar skyrmions. An interesting question hence arises: How does flexoelectricity play its role in stabilization and transformation of polar SkLs? Can we gain deterministic control of polar SkLs via flexoelectricity? Here in this work, based on phase-field simulations, we report the critical roles of flexoelectricity playing in the stabilization and transformation of polar SkLs. Different polar SkL patterns can emerge in the ferroelectric PbTiO₃ (PTO) thin films, including tetragonal-SkL (T-SkL), and H-SkLs with diverse orientations. These emergent SkL states are attributed to the material anisotropy modified by the flexoelectric effect. Interestingly, we further found that the H-SkLs can be rotated by applying strain gradient or in-plane electric field to the films. Moreover, a nonreciprocal bending response of T-SkL is also induced by the flexoelectric effect. Our results provide useful guidelines for the implementation of polar skyrmion lattices in experiments.

Formation of diverse polar patterns

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. 1a and b, where f_ij (f₁₁, f₁₂ and f₄₄) are the flexocoupling coefficients in the reduced form according to the Voigt notation. Up to eight polar states are confirmed and their polarization distributions are shown in Fig. 1c. The structural transformation from H-SkL to T-SkL with the increase of f₁₁ is attributed to the material anisotropy modified by the flexoelectric effect. From the “phase diagram” shown in Fig. 1b, one can infer how the anisotropy of the flexocoupling tensor affects the polar SkL. Along the diagonal, the flexocoupling tensor is isotropic, satisfying the equality f₁₁ – f₁₂ = f₄₄. Thus, 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, the flexocoupling tensor becomes cubic symmetry. The systems exhibit stronger in-plan 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 lattice type of polar SkL is intimately related to the symmetry of the flexocoupling tensor.

Deterministic manipulation of 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. We further found that, when annealing the PTO thin films (under either B₁⁺ or B₁-;bending strains shown in Fig. 2a or with the application of in-plane electric field) from paraelectric phase, H-SkL can be rotated. The rotations of H-SkLs indicate the strong impacts of in-plane anisotropy introduced by the bending operation or the in-plane field on the nucleation of SkL. Although the rotation of H-SkL is not closely related to the flexoelectric effect, the inherent mechanisms of both are similar, that is, by modifying the anisotropy of the system.

Different from H-SkL, a nonreciprocal bending response of T-SkL under bending operations is observed and the flexoelectric effect plays an important role, as shown in Fig. 3. When under B₁⁺ 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 B₁^- 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. This inconsistency leads to skyrmion bubbles no longer possessing inversion symmetry along the [001] axis. It is this asymmetry coming from the coupling of flexoelectricity and topology of the polar skyrmion which results in the nonreciprocal response of T-SkL state to bending operations.

Outlook

Our work demonstrates 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. Furthermore, the rotation of hexagonal-SkLs can be realized by applying strain gradient or in-plane electric field to the films. 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.