Topological polar structures in ferroelectric oxide films

The topic of topological polar structures may originate from the observation of magnetic flux-closures in confined nanometer/micrometer scale ferromagnetic materials,^1,2 where the anisotropy for a ferromagnet is generally small, and thus, the magnetostatic energy of the confined ferromagnet could be reduced by forming flux-closures or a vortex [Fig. 1(a), Ref. 3]. In addition, by further considering the Dzyaloshinskii–Moriya (DM) interactions^4–6 in magnets (which tend to swing spins), many unusual spin textures such as helical spin order,⁷ skyrmion^8–10 and antiskyrmion,¹¹ magnetic bobber,¹² and so on were observed by theoretical and experimental methods. In particular, special transformation behaviors between these spin topologies were further identified,¹³ which shed more light on further exploration of related emergent electromagnetic properties of these spin structures. Based on these novel spin textures, high-density memory units and logic elements, such as “racetrack memory” devices,^14–16 can be envisioned in the future.

It should be emphasized that by considering the low anisotropy of a ferromagnet and the rigidity of spins, the internal characteristic length scales of magnets (for example, the widths of magnet domain walls) are generally in the ranges of several tens of nanometers to approximately 100 nm.^3,6 This aspect indicates that the dimensions of the special spin structures should also be comparable to the internal characteristic length scales (several tens of nanometers to approximately 100 nm), which is illuminative for understanding the basic behaviors and potential capabilities of these spin topologies, such as the potential memory densities and external field dynamics based on these spin textures.

In the counterpart of a magnet, ferroelectric materials with similar dipole textures have received great interest, not only for the curiosity for basic condensed matter physics, but also for potential applications as new electric devices, like ultra-high density memories.^17–23 Since the ferroelectrics are highly anisotropic in structure than magnets,⁶ the internal characteristic length scales in ferroelectrics are much smaller than that of magnets.³ Consequently, ferroelectric domain wall widths are unusually in the range of several nanometers, or even nearly one or two unit-cells,⁶ which are nearly 1/10 of the ferromagnetic domain walls. Thus, the polar topological structures in ferroelectrics should be much smaller as well than that in ferromagnets (if they do exist), which stimulate the thinking on potential applications in ultra-high density ferroelectric memory devices.¹⁷ Here, the dimensions and geometric shapes of ferroelectrics are also important for considering the stabilization of polar topological structures.^17–27 Based on these preconditions, in 2004, Naumov et al. predicted polar vortex structures in low-dimensional ferroelectric PbZr_0.5Ti_0.5O₃ nanorods,¹⁷ which aroused the investigations on novel topological polar structures in ferroelectrics [Fig. 1(b)].

However, unlike the ferromagnetic materials, the anisotropy energies or strain energy costs in potential topological polar structures are much larger than that in ferromaget,⁶ which is a main obstacle for introducing dipole rotation and further topological polar structures in ferroelectric nano-structures.^26–29 It was almost a general consensus before ca. 2014 that it is the high anisotropy energies that stop the formation of potential polar topologies in ferroelectric materials. Nevertheless, by including an interface depolarization field and strain/electric field gradient-related energies,^17–23 nano-scale continuous polarization rotations were observed in PbZr_0.2Ti_0.8O₃ and BiFeO₃ films^28,29 [Figs. 1(c) and 1(d)], which opened the experimental exploration on novel topological polar structures in ferroelectrics. Here, two critical factors should be emphasized for the experimental exploration of topological polar structures in ferroelectric oxides: (1) advanced microscopy technologies with high spatial resolutions, which are based on atomic force microscopy (AFM) and aberration-corrected (scanning) transmission electron microscopy [(S)TEM] and enable the characterizations of oxide films/interfaces/superlattices on the nanometer and, most importantly, the sub-Ångström and atomic level [Figs. 1(c) and 1(d)]. Here, the AFM-based methods are very important for the new identifications of polar topologies in organic ferroelectric materials.^30–32 For example, a novel type of toroidal topological texture was reported in poly(vinylidene fluoride-ran-trifluoroethylene) [P(VDF-TrFE)] polymer showing capability of manipulating a terahertz wave on a mesoscopic scale.³⁰ Vortex−antivortex-related domain structures were observed in [4-fluoro-quinuclidinium]ReO₄ ([4-F-Q]ReO₄) and lead iodide perovskite ferroelectrics.^31,32 These results have useful implications for future exploration of novel emergent phenomena in organic ferroelectrics. (2) The controlled growth and preparations of the target oxide films/nanostructures^{14–16,24–29} were specified in several review articles.^33,34

Since the scales of internal characteristic length, such as ferroelectric domain walls and domain wall–interface interactions that are important for generating rotations of dipoles, are much smaller than that of magnets,⁶ instruments for the observation of topological polar structures are thus essential. With the development of aberration correction in electron microscopes,³⁵ the TEM- and STEM-based imaging provides great opportunity to directly visualize the single electric dipole and ferroelectric domain walls, where the common ferroelectric domain wall width was identified to be nearly within 2–3 unit cells^36–38 (smaller than 2 nm). These studies suggest that the scale of topological polar structures should be in the range of nanometers. By using the advanced TEM/STEM technologies, the continuous polarization rotations were imaged in PbZr_0.2Ti_0.8O₃ and BiFeO₃ films,^28,29 and genuine flux-closure polar structures in tensile strained multilayer PbTiO₃/SrTiO₃ films were observed [Refs. 39–43, Figs. 2(a)–2(c)]. The finding of flux-closure provides a new similarity between magnet and ferroelectrics, which implies that there is plenty of room to explore the dipole topologies in ferroelectric or polar materials corresponding to that in ferromagnetic materials.^1,2 Since the first observation of flux-closures,³⁹ several topological polar structures in ferroelectric superlattice/film systems have been then observed, such as chiral polar vortex in tensile-strained PbTiO₃/SrTiO₃ superlattices and BiFeO₃/GdScO₃ multilayers^44–48 [Fig. 2(c)], chiral polar skyrmion lattice in PbTiO₃/SrTiO₃ superlattices⁴⁹ [Fig. 2(c)], polar meron lattice in tensile-strained single PbTiO₃ layer films⁵⁰ [Fig. 2(c)], polar vortex/center-convergent/divergent polar domains in BiFeO₃, and nanometer bubble domains in ultrathin PbZr_0.2Ti_0.8O₃/SrTiO₃/PbZr_0.2Ti_0.8O₃ films.^51–58 The lateral dimensions are within ∼15 to ∼40 nm for the flux-closures, ∼4 to ∼10 nm for vortices, ∼7 nm for skyrmions, and ∼4 to ∼10 nm for merons. In each of the above explorations, the aberration-corrected STEM imaging together with related quantitative analysis has played critical roles. So far, aberration-corrected STEM imaging is unique in exploring ferroelectric dipoles and their patterns at an atomic resolution.

In this aspect, the experimental observation of topological polar structures could be described as re-discovery, since they might be present in the previous studies but were invisible without the tool of atomic mapping. One of the examples is the periodic 180° domains observed in ultrathin PbTiO₃ films and PbTiO₃/SrTiO₃ superlattices grown on the SrTiO₃ substrate.^59–62 According to the present knowledge on the formation of topological polar structures, some kind of polarization rotations or even special polar structures may evolve in these systems, since polarization rotations are common for 180° domain walls connecting with other interfaces, which is finally identified to be crucial for stabilizing the topological structure of polar skyrmions and merons in ultrathin PbTiO₃ films and PbTiO₃/SrTiO₃ superlattices.^49,50 However, without direct atomic-scale imaging, it was difficult to reveal the polar details in the nanometer-scale ferroelectrics.

In almost all the topological polar structures, the interface depolarization fields play critical roles for generating dipole rotations,⁶ which is similar to the way of forming spin vortex in magnets.^1–3 Moreover, the shapes and dimensions are also important for mediating the topological polar structures.^1,2,6,63 Detailed studies were specified in several review articles.^14–16,63

Here, we emphasize the external strains introduced by the substrates on which the ferroelectric films are grown. Precise preparation technology enables the epitaxial growth of different perovskite oxides, and it works even for the systems with large lattice mismatches^64–68 (for instance, >2% mismatches). Thus, large external strains, which are generally inaccessible for bulk perovskite oxides, can be routinely used to modulate structures and properties of perovskite oxide films/superlattices.⁶⁷ The strains and polarizations inside ferroelectrics are strongly coupled with each other,⁶ which prevent dipole rotations and thus the further formations of topological polar structures.⁶ Thus, external strains from a substrate, which can be stabilized via epitaxial growth, can thus be used to aid dipole rotations and thus facilitate the formations of topological polar structures. This is true in the finding of polar flux-closure^39–42 (on GdScO₃ substrate, tensile strain of ∼1.7%), vortex^44,45 (on DyScO₃ substrate, tensile strain of ∼1.3%), and meron⁵⁰ (on SmScO₃ substrate, tensile strain of ∼2.2%) in ultrathin PbTiO₃ layers.

There are still much work need to be done before considering the utility of topological polar structures. First, revealing the dipole alignments inside these topological polar structures is critical, such as the dipoles inside a single vortex or skyrmion. Resonant soft x-ray diffraction method based on synchrotron was successfully applied to reveal electric dipoles inside the topological polar structures.^45,49 Strong circular dichroism signals indicate clear chirality of polar vortex arrays and polar skyrmion lattice in PbTiO₃/SrTiO₃ superlattices, which might inspire the future development of controllable chiral devices and memory units.⁴⁵ 

Second, the dynamics of these topological polar structures under different external fields will be very important for manipulating their potentials. Sub-picosecond optical pulses were identified to be able to create a novel supercrystal phase in PbTiO₃/SrTiO₃ superlattices containing a₁/a₂ domains and polar vortices,⁶⁹ which is stimulating for manipulating topological polar structures with emergent structural and electronic responses. Moreover, electric fields, high energy electron beams, and mechanical stimulations could trigger novel phase interconversions between vortex/flux closure phases and ferroelectric phases,^70–72 which allow constructions of new electromechanical and nonlinear optical electronic devices on the basis of topological polar structures.

In addition, while direct controlling of topological polar structures as high density memories is still challenging, topologically confined ferroelectric domain walls in special polar structures show practical potentials.^51–54 Switching of center-type domains in BiFeO₃ nano-structures alters the conductivity of topologically confined domain walls by several orders of magnitude,^{51,52,54–56} which holds promise for non-volatile memory with low energy consumptions (Figs. 3 and 4).

The studies on topological polar structures in oxide films are just starting. Many fundamental aspects are still unclear for future applications of these structures as potential electronic devices. In particular, we note that new theoretical studies on the interactions between two polar displacements have confirmed DM-like interactions in ferroelectrics and antiferroelectrics, which may stimulate researches on novel polar topologies.⁷³ We propose three topics that might accumulate substantial achievements in the near future. They are the 3D dipole structures on the atomic scale, flexoelectric properties, the mechanical-related properties, and strain accumulating/relaxation mechanisms of the topological polar structures.

As mentioned above, TEM-based methods with the capability of atomic imaging are essential for direct revealing topological polar structures in oxide films. However, these methods mainly provide the 2D atom projections of the topological polar structures, which is far from enough for understanding the 3D dipole structures of a single polar structure. Nevertheless, recent progress on atomic electron tomography, based on the advanced TEM instrument, allows the atomic level 3D reconstruction of a single nanometer scale particle.⁷⁴ Moreover, the developments of new electron microscopy hardware and technologies, such as electron microscope pixel-array detector and full-field ptychography methods, have pushed the resolution of a TEM to over 0.04 nm.⁷⁵ These new microscopy technologies might thus be useful for investigating the atomic level 3D reconstruction of dipole structures inside the topological polar structures.

Nevertheless, big challenges could be expected here. For instance, how to isolate and stabilize a single vortex or skyrmion and locate them in an atomic electron tomography experiment? Pre-explorations might be conducted on the interacting areas between the domain walls and the interfaces, such as the 109° wall in BiFeO₃ and the BiFeO₃/GdScO₃ interface [Figs. 5(a) and 5(b)], where vortex-like polar structure emerges. Future studies will be expected to focus on the 3D chemical information⁷⁶ [such as Ti valences inside vortex cores, Figs. 5(c) and 5(d)] and 3D dipole structures of ferroelectric domain walls⁷⁷ [Figs. 5(e) and 5(f)]. These details are not only critical for understanding the chirality and chemical properties of polar topologies but also important for the designing of information processing and memory devices.

Flexoelectric effect refers electric polarizations that are induced by strain gradient. Flexoelectricity is an important concept for new device designs, where piezoelectric effects might occur with no piezoelectric materials.^78–82 This is a promising research area since all dielectric materials could intrinsically exhibit flexoelectric effects.⁷⁸ However, the coupling between electrical and mechanical properties is generally negligible because large strain gradients (commonly not obtainable for oxide materials) are required to insure a detectable flexoelectric response. Even though the large strain gradient of about 10⁴/m and enhanced piezoelectricity were achieved in mechanically buckled micrometer PbZr_0.52Ti_0.48O₃ ribbons,⁸³ this strain gradient is actually not large enough for generating polarizations comparable to that of the spontaneous polarization of PbZr_0.52Ti_0.48O₃ because the flexoelectric coefficients of perovskite are generally in the orders of 10⁻⁹–10⁻⁸ Cm⁻¹ (Refs. 84 and 85).

Nevertheless, for the topological polar structures in perovskite oxides, both strain gradient and polar gradient are involved.^39–50 Most importantly, the strains and polarizations in these structures change in the scales as short as nanometers, which would introduce very large strain or polarization gradients (polarization gradients could introduce strains in dielectric materials, referred to as the converse flexoelectric effect). For instance, the strain gradients in the flux-closures of PbTiO₃ are as large as 10^6–7/m [Figs. 6(a)–6(d)], which may lead to considerable flexoelectric responses.^39,43,86 Giant strain gradient as 10⁶/m in the BiFeO₃ lattice could also be stabilized, which would induce a large built-in electric field of several MV/m and will enhance solar absorption properties.⁸⁵ 

It is worthwhile to add that the flexoelectric coefficients and flexoelectric coupling strength of perovskite oxides are still under debate.⁸⁴ In some cases, so far even the sign of a specific flexoelectric coefficient cannot be exactly determined.^87,88 Intriguingly, by using machine learning and phase-field simulations, the flexoelectric coefficients for both PbTiO₃ and SrTiO₃ were determined on the basis of atomic images for the PbTiO₃/SrTiO₃ superlattice containing the vortex phase [Ref. 87, Fig. 6(e)]. Moreover, by combinations of atomic-scale imaging, first-principles calculations, and the Landau–Ginzburg–Devonshire (LGD) theory, converse flexoelectricity and corresponding magnitude and sign of some flexoelectric coefficients were specified for ferroelectric PbTiO₃ [Ref. 88; Figs. 6(f)–6(h)]. These results suggest a nontrivial role of flexoelectric response corresponding to the emergent topological polar structures in perovskite oxides, which will attract enormous attentions to conduct further explorations on these topics.

As mentioned above,^64–68,89 the strain engineering of ferroic films and superlattices is crucial for mediating topological polar structures. Understanding these mechanically related properties of perovskite oxides are important for future engineering of strains in the ferroic oxide materials, which is useful for tuning ferroic transitions, bandgaps, carrier mobility, and superconductivity of functional materials.^67,89 For perovskite oxide materials, future studies should focus on these open questions: how elastic strains far beyond the elastic limit accumulate in perovskite oxide films and/or superlattices? How the misfit strains relax? What are the origination mechanisms for misfit dislocations and is there any way to prevent them so that the elastic strain engineering strategy can be employed? We see that the studies on strain relaxation mechanisms and the origin of misfit dislocations for the perovskite oxide films were mainly based on the knowledge of semiconductor films as Si or GaAs materials.^90–93 However, the crystal structures, bonding type, and mechanical properties of the semiconductor materials (mainly covalent bond) are largely different from that of oxide materials (ionic or mixed bonds), which should possess totally different defect behaviors, such as the nucleation, gliding, and interactions of dislocations. Thus, the detailed formation mechanisms (nucleation, gliding, and interactions) of misfit dislocations might be remarkably different from that of the semiconductor films, which should be paid more attentions in the future. These mechanical properties are crucial for guiding how to engineer strains and/or how to modulate physical properties of perovskite oxides.⁸⁹ 

The topological polar structures show novel responses, dynamic behaviors, and phase transitions under optical, mechanical, and electric field stimulations, which hold promise for future information processing and storage applications. We propose three topics that should be inspiring in the future for investigations of novel polar structures in oxide materials: visualizing the 3D dipole structures at the atomic scale, exploring flexoelectric properties, and clarifying strain accumulation and relaxation mechanisms. We hope that these open questions will stimulate further studies on how to stabilize and manipulate the novel topological phases and, consequently, pave some ways toward the applications of topological polar structures in the development of electronic devices.

This work was supported by the National Natural Science Foundation of China (NNSFC) (Nos. 51922100, 51971223, and 51671194), the Key Research Program of Frontier Sciences CAS (No. QYZDJ-SSW-JSC010), the Scientific Instrument Developing Project of CAS (No. YJKYYQ20200066), and the Youth Innovation Promotion Association of CAS (No. Y202048).

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