A new era in ferroelectrics

Topological structures in ferroic materials have drawn great interest in recent years due to the richness of the underlying physics and the potential for applications in next generation electronics. Recent advances in atomically precise thin-film materials synthesis and characterization of structural/physical phenomena at unprecedented length/energy/time scales have enabled us to study exotic phases and their associated physics [Rößler et al., Nature 442, 797 (2006); S. Das, Nature 568, 368 (2019); Yadav et al., Nature 530, 198 (2016); and Stoica et al., Nat. Mater. 18, 377 (2019)].14 It is appropriate that, in the second century of ferroelectrics, some dramatic discoveries are propelling the field into directions heretofore unimaginable. In this review, we explore the recent progress in ferroelectric-oxide superlattices in which researchers can control structure and physical properties through the application of epitaxial strain, layer thickness, temperature, electric field, etc. We provide a discussion of exotic topological structures (e.g., closure domains, vortices, polar skyrmions, and other exotic phases) and associated functionalities in ferroelectric/paraelectric superlattices. We conclude with a brief overview of and prospects for how the field may evolve in the coming years. © 2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0034914., s


INTRODUCTION
In the hundred years since the original discovery of ferroelectricity in Rochelle salt, 5 numerous significant discoveries and technological manifestations of these materials have been demonstrated. 6 While it is essentially impossible to cover all of these discoveries in a single review, we attempt to put into perspective some striking new discoveries within the family of well-known perovskite ferroelectrics, driven by the confluence of three scientific advances in the field. The first is the ability of materials synthesis approaches to control and tailor artificially designed materials at the atomic scale, as exemplified by the creation of epitaxial superlattices, 7 the second is the dramatic advancements in ab initio and mesoscale phase-field computational approaches, 8,9 and the third is the emergence of a host of probes using x-rays, electrons, and neutrons and near field techniques that are now routinely enabling us to study physical phenomena at the atomic scale at unprecedented length/energy/time scales. In parallel, there has been a revolution developing that is exploring the possibility of using ferroelectrics as a key element in beyond Moore's law electronics, serving the dual role of a memory element and enabling logic operations. 10 Another key manifestation of ferroelectrics is in the form of magnetoelectric multiferroics, exemplified by the ferroelectric BiFeO 3 . While we urge the interested reader to explore recent reviews on this subject, 11 this review is focused on emergent topologies in ferroelectrics. These recent discoveries The classical thinking (i.e., ferroelectrics vs ferromagnets) had been that ferroelectrics represent the extreme case where the dipolar interaction is of a lower energy scale than the anisotropy (which comes about directly due to the strong dipole-lattice coupling) that manifests as a strong lattice distortion below the transition temperature (e.g., the prototypical ferroelectric PbTiO 3 has a strong tetragonal distortion of about 5%). In contrast, ferromagnets are exemplified by a much stronger exchange energy than the anisotropy energy (except for ferromagnets which contain rare earth ions with inherently large spin-orbit coupling). As an immediate manifestation, the domain-wall width (which is directly related to the tradeoff between these two energies) is small in ferroelectrics (a few unit cells wide) since anisotropy dominates. In contrast, domain-wall widths in classical ferromagnets (e.g., iron and cobalt) are much larger than the unit-cell dimensions since exchange dominates. As a direct consequence of the strong role played by anisotropy energy in ferroelectrics, rotations of the dipoles away from the crystallographic easy axes (e.g., the [001] in tetragonal PbTiO 3 ) have always been thought to be challenging. Work in the past decade (particularly in the last five years), however, has shown that this is likely not the case. 2,3 When appropriate boundary conditions 4 are imposed onto the ferroelectric crystal, it is forced to take up dipolar arrangements that exhibit exotic topologies that are typically not favored in the bulk. This forms the backbone for our discussion in this review.
Topology is a branch of mathematics that studies the properties of geometric objects that are preserved under continuous deformations -such as under contraction, expansion, rotation, etc. 12 In recent years, topological states, for example, spin/dipolar textures, have become an intense topic of study in condensed-matter physics and materials science, 1 and the concept of topology 13,14 has been woven into condensed-matter physics. 15,16 Topology can arise in either real-space arrangements of the order parameter (i.e., dipole moment, spins, etc.) or in reciprocal space arising from the topology of the band structure, leading to topological insulators, semimetals, and superconductors. 17 Focusing on real-space topologies, textures such vortices and skyrmions of ferroic order have attracted attention in the past decade due to their exotic nature and the consequent diversity of interesting physics and properties that hold promise for future technologies. [18][19][20] The formation of domains in ferroic materials such as ferroelastics, ferromagnets, or ferroelectrics is due to the thermodynamic driving force to reduce the depolarization/demagnetization/strain fields that occur at surfaces or interfaces. Regions of the uniform order parameter are separated by domain walls, which can be thought of as a two-dimensional (2D) topological defect. The three archetypes of domain walls are classified by either their axis of rigid rotation (Néel and Bloch type) or the scaling of the moment across the wall (Ising type). [21][22][23] Despite their chemical and phase homogeneity, even basic domain walls approximating these archetypes will exhibit emergent properties due to the broken symmetry and strong spin, lattice, and charge coupling of the respective ferroic order. [24][25][26][27][28] Formation of complex topologies, however, has, until recently, been the purview of ferromagnetic materials whose relative isotropy, rigid spin moments, and exchange energy produce rotational Néel and Bloch walls as well as complex textures such as helices, vortices, and skyrmions. 29 Furthermore, in ferromagnets, the formation of spin textures requires symmetry breaking either in the form of an interface breaking inversion symmetry or in the bulk through the anti-symmetric Dzyalozhinski-Moriya coupling. Three-dimensional (3D) topological defects with different geometric shapes can appear in ferroic materials under specific conditions. Typical topological defects include vortices, skyrmions, merons, etc. (Fig. 1). 4,[30][31][32][33][34] Although these topological defects may decay over time in the bulk, they are surprisingly stable in nanoscale systems. At the same time, they are responsive to external fields (mechanical, thermal, electric, and magnetic), which may lead to exotic physics such as chirality reversal, 35 ratchet effects, 36 negative electrocaloric effects, 37 negative capacitance effects, 38 etc.
Kittel's work (dating to 1949) 39 in magnetic materials outlined the formation of different magnetization patterns depending on the competition of various energies associated with the exchange and crystallographic anisotropy as well as electrostatic/magnetostatic energy arising from depolarization/demagnetization fields. 40 A fluxclosure-like domain is predicted when anisotropy energy dominates, whereas, a vortex-like structure is developed when exchange interactions between adjacent magnetic moments dominate over anisotropy. It is easy to find such arrangements of magnetic dipoles when the material is reduced to small sizes, and this has led to experimental observations of flux-closure domains and vortices. 12,[41][42][43][44][45][46][47] More complex topological structures can be developed in ferroic materials -i.e., those with a spontaneous reversible ordering, for example, magnetic skyrmions with swirling arrangements of magnetic dipoles, as described by Mermin [ Fig. 1(e)]. 12,45,48,49 Magnetic flux-closure domains were first observed in permalloy plates and disks. 50 When permalloy films are reduced to smaller sizes to form circular disks (of ∼1 μm diameter), the magnetic moments become arranged into continuous circular loops, forming vortex structures instead of flux-closure domains. 39 Complex magnetic topological patterns like a periodic array of magnetic skyrmions have also been recently observed in (Fe 0.5 Co 0.5 )Si. 47 Finally, one key attribute that underlies the formation of such spin textures is the existence of inversion symmetry breaking, for example, through an interfacial Rashba effect 51 or an anti-symmetric Dzyalozhinski-Moriya interaction.
It was heretofore thought that formation of such textures in ferroelectrics would be difficult, which exhibit large crystalline anisotropy (because dipole-lattice coupling is much stronger than spin-lattice coupling). However, the emergence of sophisticated computational methods 52,53 has triggered an intellectual renaissance in this field. Such theoretical studies in ferroelectric nanostructures first showed enormous potential in formation of new electric-dipole arrangements including those with a toroidal moment (which is nonzero for a curling configuration of electric dipoles) in BaTiO 3 and PbZr 0.5 Ti 0.5 O 3 . 54,55 Ferroelectric domain walls do exhibit rotational components 56 including a pronounced Néel character in the vicinity of the domain-wall interface terminations where minimization of stray field energies leads to domain broadening 57 and formation of rotational flux-closure structures 58,59 including closed loops. 3,60 In the absence of an equivalent mechanism for a proper ferroelectric, non-collinear moments such as chiral host structures or spin-orbit interactions, 61 vortices, and "hedgehog" skyrmions are the most promising parity structures that might be obtained through these rotational-ferroelectric transitions. Experimentally, it was shown that ferroelectric flux-closure-domain topologies can emerge, similar to those in a ferromagnet, by juxtaposing the dimensionality of layer thickness, strain, and electrostatic energies. 55 Such flux-closure domain structures in ferroelectrics were first observed in PbZr 0.2 Ti 0.8 O 3 layers and later in a PbTiO 3 layer. 58,60 Despite this interest, the first experimental realization of continuously rotating polar topological configurations (polar vortices) and swirling polar topological configurations (polar skyrmions) in (PbTiO 3 )n/(SrTiO 3 )n heterostructures was demonstrated only recently. 2,3 This required the convergence of precise materials synthesis, oxide superlattices in this case, in which thermodynamic boundary conditions are imposed on a ferroelectric phase and directly studied with state of the art material characterization tools. 3 With this as the background, this review focuses on the advances and breakthroughs over the past decade (particularly the past five years) in the growth of ferroelectric model systems and their topological structures. We focus on polar vortices, polar skyrmions, and the associated functionalities recently observed in ferroelectric epitaxial heterostructures. Finally, we conclude this review with an outlook for this burgeoning field.

Growth of (PbTiO 3 )m /(SrTiO 3 )n as a model system
Bringing together advances in reflection high-energy electron diffraction (RHEED) controlled layer-by-layer growth, the similarity of in-plane lattice parameters of PbTiO 3 and SrTiO 3 , and advanced microscopy techniques, superlattices are ideal model systems to study the effects of the competition between various components of the ferroelectric Hamiltonian, in a manner that is not present in the bulk form. Short-period (PbTiO 3 )m/(SrTiO 3 )n interfaces also lead to the emergence of improper ferroelectricity due to antiferrodistortive/ferroelectric coupling, driven by interfacial oxygen octahedral rotations. [62][63][64] It has also been shown that in (PbTiO 3 )m/(SrTiO 3 )n (m, n < 9 unit cells) superlattices, PbTiO 3 has a local negative dielectric constant which imbues a large overall, positive permittivity to the superlattice. 65,66 At relatively large length scales (m, n > 20 unit cell), these superlattices form classical flux-closure domain patterns in order to reduce the depolarization fields. 57,59 At intermediate length scales (n and m = 10-20 unit cells), however, theoretical studies suggested that such superlattices could potentially support the formation of exotic topological structures (vortices and skyrmions) as a result of the competition of different energies (i.e., elastic, depolarization, and gradient). 53,[67][68][69] What this work has demonstrated is that careful control of material heterostructures and the local competition of energies can provide a potential pathway to manipulate the structural, polar, and physical nature of the mesoscale structures that emerge. 53,[70][71][72] On the synthesis side, the most important recognition came with the realization that having excess of lead in the target is beneficial to the layer-by-layer growth of the PbTiO 3 layers in the superlattices. As a direct consequence, such a growth mode can be maintained up to even 200 nm of total thickness. One of RHEEDcontrolled growths of (SrTiO 3 ) 6 /(PbTiO 3 ) 6 superlattices ( Fig. 2) reveals RHEED oscillations that are present throughout the growth of the 100-nm-thick superlattice, indicating that both the PbTiO 3 and SrTiO 3 layers are grown in a layer-by-layer fashion, which, in turn, enables controlled growth of the superlattices with different period-thicknesses while achieving the same total thickness. 3 Thus, it should not be surprising that atomically sharp chemical interfaces can be observed in cross-sectional, STEM images [ Fig. 2(b)], and atomic-scale high-resolution STEM (HR-STEM) confirms sharp and coherent interfaces (b). As we will discuss later, such chemically sharp interfaces (i.e., with a minimum of interfacial interdiffusion) are critical to impose other thermodynamic boundary conditions on the superlattice. With such atomically ideal superlattices on hand, one can ask the following: what are the consequences of the imposition of such electrostatic and elastic boundary conditions on the PbTiO 3 layer?

Polar vortices
The first surprise from HR-STEM studies came when the atomic-resolution images were analyzed using vector-mapping algorithms. In contrast to what was expected, researchers observed rotating patterns in dark-field transmission electron microscopy (DF-TEM) images [ Fig. 3(a)] comprising a long-range, periodic array of intensity modulations along both the in-and out-of-the plane directions of the superlattice with a spacing of ∼10 nm. At higher resolution [ Fig. 3(b)], vector mapping of the atomic polar displacement (P PD ) 44 in a cross-sectional HR-STEM image enabled researchers to measure the local non-centrosymmetry of the lattice. The vector map of these polar displacements indicates the formation of long-range arrays of clockwise (CW) and counter clockwise (CCW) vortex pairs in each PbTiO 3 layer. A single CW-CCW vortex pair is shown in the magnified image [ Fig. 3(c)] and reveals the full density of data points (one for each atom) and the continuous rotation of the polarization state within the vortices. The lateral periodicity of this pair is ∼10 nm (and scales with the superlattice periodicity). Such pairs of CW-CCW vortices have been confirmed by both second-principles calculations 8 and phase-field simulations. 73 The 3D structure of the CW-CCW array of a vortex in the (PbTiO 3 ) 10

Polar skyrmions
Skyrmions were originally proposed by Skyrme in the 1960s to explain the stability of hadrons as quantized topological defects in three-dimensional (3D) nonlinear sigma models. 75

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where n is the unit vector pointing to the magnetization/polarization direction. This topological charge is the in-plane integral of the Pontryagin density. 12 Further interplay of elastic, electrostatic, and gradient energies in the (PbTiO 3 )n/(SrTiO 3 )n heterostructures provides an intriguing opportunity to produce such complex electric skyrmions (analogous to magnetic skyrmions). Importantly, it was discovered that (PbTiO 3 )n/(SrTiO 3 )n heterostructures have to be grown under a slight compressive strain, for instance, on SrTiO 3 (001) substrates, to tip the balance of energies in the right manner to induce these new structures. 2 Looking down at the surface of a (PbTiO 3 ) 16 /(SrTiO 3 ) 16 superlattice, plan-view STEM images (Fig. 4, middle panel of the top row) reveal long-range ordered arrays of circular features with a size of 8 nm-9 nm, suggesting that this polar order extends through the film over many hundreds of nanometers length scale. Lowresolution, cross-sectional DF-TEM imaging (Fig. 4, blue shaded area in the middle panel of the top row), in turn, revealed a pseudo-long-range periodic array of intensity modulations along both the in-and out-of-plane directions of the superlattice. Atomicscale polarization mapping using a displacement vector-mapping algorithm was used on both plan-view and cross-sectional highangle annular dark-field (HAADF)-STEM images to extract the local polarization structures [Figs. 4(a) and 4(b)]. 58 The reverse titanium-displacement vector mapping [ Fig. 4(a), top] based on the high-resolution plan-view HAADF-STEM image [ Fig. 4(a), bottom] reveals a single skyrmion bubble. The reversed titanium displacement is diverging from the edge to the center, corresponding to a hedgehog-like skyrmion structure. 50 To complement this, vector displacement mapping of cross-sectional HAADF-STEM imaging [ Fig. 4(b)] clearly shows a cylindrical polar region with antiparallel (up-down) polarization. The polarization vector rotates at the boundaries near the PbTiO 3 -SrTiO 3 interfaces, consistent with the divergence of polarization observed in plan-view images. The combined plan-view and cross-sectional vector displacement mapping reveals a hedgehog-like structure at the top of the PbTiO 3 layer. Along this line, four-dimensional STEM (4D-STEM) with an electron microscope pixel array detector (EMPAD) 77 provides information about the center PbTiO 3 layers. In 4D-STEM, the full momentum distribution -that is, the electron diffraction pattern -can be collected at every scan position by the EMPAD. From the diffraction patterns collected on the EMPAD, the reconstructed low-angle annular dark-field (ADF) image  Fig. 4(e)]. 78 This calculation shows the three-dimensional view of a single skyrmion where the polar skyrmions consist of a diverging and converging hedgehog skyrmion pair at the top and bottom, respectively, unlike magnetic skyrmions, 12 and at the center layer, there is a Bloch-type skyrmion structure. The calculated topological charge (Q) is found to be +1 in every case, which reveals that they are topologically equivalent, i.e., one can transform to another by continuous deformation.

Chirality of polar skyrmions and vortices
Chirality is a geometrical property that is ubiquitous in nature. When an object is not superimposable on its mirror image, i.e., imparting a handedness or helicity, we call it a "chiral object." Examples include amino acids and sugars in our body and naturally occurring crystals such as quartz. In the case of (PbTiO 3 )n/(SrTiO 3 )n superlattices, one does not, a priori, expect them to be chiral since the constituent layers are by themselves not chiral. When assembled into such superlattices and polar textures begin to form, however, they are found to exhibit chirality. One way to look for chirality in condensed-matter systems is to measure the interaction with circularly polarized electromagnetic radiation, i.e., circular dichroism (XCD). 35 In the case of (PbTiO 3 )n/(SrTiO 3 )n superlattices, recent resonant soft x-ray diffraction studies have shown that the polar vortices in such superlattices do indeed exhibit strong circular dichroism 2,3 (Fig. 5), which is also reminiscent of magnetic-topological structures. 2,35 In hard x-ray reciprocal space mapping (RSM) studies, lateral satellites appear due to the regular spacing of the skyrmions [ Fig. 5(a)] and vortex structures [ Fig. 5(b)], where the spacing is dx_pair (∼8 nm for skyrmions and ∼10 nm for vortices). These satellites occur at q-vectors equal to ±n×qx_pair, where qx_pair = 2π/dx_pair. By tuning the energy of the x-rays through the titanium L 3,2 edge (where electronic transitions from the 2p to 3d levels occur), absorption spectra are collected at q-values corresponding to the satellite peaks. At the titanium L 3,2 resonance, effects from the electronic structure (specifically the d-orbital configuration) are strongly enhanced. If the spiraling polar distortions that create the skyrmion or vortex phase are chiral, this will manifest as a non-zero circular dichroism since chiral objects interact differently with left-and right-circularly polarized light. 35

Negative capacitance state in polar vortices
The phenomenon of negative capacitance has garnered enormous interest due to its exotic physics as well as promise in applications in next-generation electronics [79][80][81][82] specifically to overcome the "Boltzmann tyranny" in electronics. 84 Classical Landau-Devonshire-Ginzburg theory of ferroelectrics describes the ferroelectric state through a doubly degenerate free-energy landscape 6 between total free energy G and electric displacement D; the shape of the double-well, free-energy landscape points to the existence of a range of polarizations where the curvature of the free energy (i.e., the second derivative of the free energy with respect to the electric displacement, which is also the permittivity of the material permittivity ε ∝ ∂ 2 G ∂ 2 D < 0) is negative [hence negative capacitance emerges; Figs. 6(a) and 6(b)]. It should be noted that the entire volume of the material cannot exhibit such a negative permittivity since in that case, it would spontaneously evolve to a state where the curvature of the free energy landscape is positive. Such negative permittivity states can, however, be stabilized in portions of the material or out of equilibrium (i.e., in the temporal regime 83 ). A direct measurement of the spatially resolved, steady-state negative capacitance is rare in nature and was first observed microscopically in a vortex structure of (PbTiO 3 )n/(SrTiO 3 )n superlattices. 38 The existence of negative permittivity at the vortex core and the potential-energy landscape across the vortex structure can be directly probed via 4D-STEM.

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These imaging experiments were carried out in cross-section samples and show the CW-CCW vortex pair [ Fig. 6(c)]. The long-range electric fields can be mapped out from the deflection of the electronbeam pattern due to the Lorentz force [ Fig. 6(d)]. 84 The spatial distribution of polarization (Pz) and electric field (Ez) are measured by focusing along the dashed horizontal line profile [Figs. 6(c)-6(e)], which goes through the vortex core (the magnitude of the measured polarization was calibrated using the polarization of PbTiO 3 , as shown in Ref. 85). With both the measured electric field and polarization, one can calculate the local potential energy of the system using dG = EdP. The estimated potential-energy (G) as a function of Pz across the vortex reveals the existence of a local potential-energy maxima ( ∂ 2 G ∂ 2 D < 0) at the location of the vortex core, where Pz is small [ Fig. 6(f)], thus illustrating that the negative capacitance exists in the CW-CCW vortex core. 3 Its intriguing that the existence of such regions of negative capacitance in the material actually leads to an enhancement in the effective dielectric permittivity that can then be probed even by macroscopic measurements. 86 Of greater interest is the fact that such an enhanced dielectric permittivity can be significantly changed with an electric field. This occurs through the destruction of the vortex/skyrmion state under the electric field, and the superlattice converts into a "normal" ferroelectric layer, thus representing a topological phase transition accompanied by a change in the topological charge from a finite value to zero at a finite field. Removal of the electric field restores the skyrmionic state.

SUMMARY AND OUTLOOK
The chiral-polar topologies and their associated complex phase mixtures suggest the presence of exotic physical responses in these ferroelectric superlattices. Along with gaining fundamental knowledge about them, researching and understanding vortices and skyrmions can lead to new technological applications, particularly those that take advantage of the negative capacitance that is embedded in the vortices/skyrmions. Specially, they could be used in skyrmionics -a new type of memory storage where data are encoded using the chirality of the individual skyrmion. 18,19 Magnetic skyrmions can be controlled by utilizing cross coupling between magnetic and electrical parameters, e.g., switching and controlling of chirality using electric or magnetic fields. The use of electric fields is preferable for reducing the power consumption associated with the switching. The control of chirality with the use of homogeneous dc fields can be achieved with skyrmions and/or vortices, which can couple to homogeneous electric/magnetic fields, and an electrically switchable chirality could be realized. In addition, the chiral domain walls with different handedness in solid-state ferroelectric materials have never been observed. In a confined ferroelectric, it would be interesting to demonstrate the polar vortex domains with antiphase boundaries as three-dimensional (3D) polarization textures which exhibit collective behavior, in the form of chiral domain walls. The prospect for control of chirality in both magnetic systems and polar skyrmions/vortices depends on the ability to easily switch their handedness, which will require coupling to a homogeneous electric field. [87][88][89] The ability to switch chirality with homogeneous magnetic fields 35 or ac electric currents 93 has been demonstrated for magnetic structures, and switching by electric fields is possible with, for example, the imposition of voltage-induced strain. 12 In polar skyrmion/vortex structures, their chiral nature 2,35 and toroidal coupling to electric fields 45 have already been established, and attempts to utilize this coupling to achieve electric-field control of chirality are underway. So far, studies demonstrate chirality switching in magnetic structures on the scale of hundreds of nanometers. Although theoretical predictions show that switching can occur in sub-100-nm electric and magnetic nanostructures, 60,63 these are difficult to realize experimentally. Thus, a significant challenge in the coming years will be the development of methods to write and read chirality on the nanoscale in such a way that it is useful for devices. The glassy behavior and extremely "soft" nature of the polarization and frequency dispersion pointing toward macroscale collective response of the vortices have been predicted in PbTiO 3 /SrTiO 3 heterostructures. In addition, strain-based routes to produce samples exhibiting a 100% vortex phase or polar skyrmion features are being explored; these can be used as model systems for in-depth studies of the physics and nature of these complex phases. 90 Imaging chiral domain walls at the nanoscale is possible with in situ techniques such as spin-polarized low energy electron microscopy (SPLEEM) 91 and x-ray magnetic circular dichroism microscopy (XMCD), 92 and chiral domain walls in ferroelectrics have been demonstrated using second-harmonic generation microscopy. 93 Moving forward, these or related techniques may inspire new mechanisms through which the deterministic writing and reading of nanoscale chiral structures can be achieved. In addition, electric field manipulation of spins has attracted interest due to their promise for new functionalities and low power device concepts. Using a dilute Fe-doped ferroelectric PbTiO 3 /SrTiO 3 as a model system, one can demonstrate electric field manipulation of spin directionality in individual spins. The intriguing potential for coupled effects between spin and polarization could also lead to new modalities of data storage. The topological phase transition and the production of steady-state negative capacitance and large field-tunable response of these emergent states could promise for electronic applications.