Topological phases in polar oxide nanostructures

Javier Junquera, Yousra Nahas, Sergei Prokhorenko, Laurent Bellaiche, Jorge Íñiguez, Darrell G. Schlom, Long-Qing Chen, Sayeef Salahuddin, David A. Muller, Lane W. Martin, and R. Ramesh

Rev. Mod. Phys. 95, 025001 – Published 20 April 2023

Abstract

The past decade has witnessed dramatic progress related to various aspects of emergent topological polar textures in oxide nanostructures displaying vortices, skyrmions, merons, hopfions, dipolar waves, or labyrinthine domains, among others. For a long time, these nontrivial structures (the electric counterparts of the exotic spin textures) were not expected due to the high energy cost associated with the dipolar anisotropy: the smooth and continuous evolution of the local polarization to produce topologically protected structures would result in a large elastic energy penalty. However, it was discovered that the delicate balance and intricate interplay between the electric, elastic, and gradient energies can be altered in low-dimensional forms of ferroelectric oxide nanostructures. These can be tuned to manipulate order parameters in ways once considered impossible. This review addresses the historical context that provided the fertile background for the dawning of the polar topological era. This has been possible thanks to a fruitful, positive feedback between theory and experiment: advances in materials synthesis and preparation (with a control at the atomic scale) and characterization have come together with great progress in theoretical modeling of ferroelectrics at larger length and timescales. An in-depth scientific description to formalize and generalize the prediction, observation, and probing of exotic, novel, and emergent states of matter is provided. Extensive discussions of the fundamental physics of such polar textures, a primer explaining the basic topological concepts, an explanation of the modern theoretical and computational methodologies that enable the design and study of such structures, what it takes to achieve deterministic, on-demand control of order-parameter topologies through atomically precise synthesis, the range of characterization methods that are key to probing these structures, and their thermodynamic field-driven (temperature-driven, stress-driven, etc.) susceptibilities are included. The new emergent states of matter join together with exotic functional properties (such as chirality, negative capacitance, and coexistence of phases) that, along with their small size and ultrafast dynamical response, make them potential candidates in multifunctional devices. Finally, some open questions and challenges for the future are presented, underlining the interesting future that is anticipated in this field.

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Received 1 August 2021

DOI:https://doi.org/10.1103/RevModPhys.95.025001

Physics Subject Headings (PhySH)

Magnetic nanoparticles Topological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Javier Junquera

Departamento de Ciencias de la Tierra y Física de la Materia Condensada, Universidad de Cantabria, Avenida de los Castros s/n, 39005 Santander, Spain

Yousra Nahas, Sergei Prokhorenko , and Laurent Bellaiche

Physics Department and Institute for Nanoscience and Engineering, University of Arkansas, Fayetteville, Arkansas 72701, USA

Jorge Íñiguez

Materials Research and Technology Department, Luxembourg Institute of Science and Technology, 5 avenue des Hauts-Fourneaux, L-4362 Esch/Alzette, Luxembourg and Department of Physics and Materials Science, University of Luxembourg, 41 Rue du Brill, L-4422 Belvaux, Luxembourg

Darrell G. Schlom

Department of Materials Science and Engineering, Cornell University, Ithaca, New York 14853, USA, Kavli Institute at Cornell for Nanoscale Science, Ithaca, New York 14853, USA, and Leibniz-Institut für Kristallzüchtung, Max-Born-Straße 2, 12489 Berlin, Germany

Long-Qing Chen

Department of Materials Science and Engineering, Materials Research Institute, The Pennsylvania State University, University Park, Pennsylvania 16802, USA

Sayeef Salahuddin

Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, California 94720, USA and Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA

David A. Muller

School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA and Kavli Institute at Cornell for Nanoscale Science, Ithaca, New York 14853, USA

Lane W. Martin

Department of Materials Science and Engineering, University of California, Berkeley, Berkeley, California 94720, USA and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

R. Ramesh ^*

Department of Physics and Department of Materials Science and Engineering, University of California, Berkeley, Berkeley, California 94720, USA and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

^*Present address: Rice University, Houston, Texas 77005, USA.

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Vol. 95, Iss. 2 — April - June 2023

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Images

Figure 1
(a) Elastic band wrapped around an infinite cylinder. Green (lower), blue (middle), and orange (upper) bands wind around the cylinder one, two, and three times, respectively. (b) Examples of homotopy equivalent band placements that wind twice around the cylinder. These configurations can be continuously transformed into one other and belong to the same homotopy class. (c) The green (lower) band does not wrap the cylinder and belongs to a class of trivial loops that can be contracted to a point. Bands that wind the cylinder clockwise and counterclockwise [blue (dark gray) arrows] cannot be continuously matched. Their respective classes are labeled with winding numbers of opposite signs. (d) Any loop on a sphere can be continuously contracted to a point. The sequence of contracting loops is shown with blue (dark gray) lines. (e) Loops on the surface of a torus are characterized by a combination of winding numbers in the poloidal (orange solid arrow) and toroidal (blue dashed arrow) directions.
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Figure 2
Order-parameter spaces $X$ for (a) the Ising, (b) $X Y$ , and (c) Heisenberg models. The arrows indicate examples of orientations of the spin vector. The locus of all possible spin tip positions constitute the order-parameter space. The relevant topological defects in three dimensions for the Ising, $X Y$ -, and Heisenberg models are (d) domain walls (e) vortex lines, and (f) hedgehogs or Bloch points. The Heisenberg model also allows for various topological solitons in $D = 1$ , 2, and 3 dimensions. (g) Néel skyrmion, which is an example of such a soliton. This can be considered the stereographic projection of (f) taken from the south pole.
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Figure 3
A domain wall in (a) three-dimensional ( $D = 3$ ), (b) two-dimensional ( $D = 2$ ), and (c) one-dimensional ( $D = 1$ ) spaces is a planar, line, and point defect, respectively. The core dimension $d$ of a domain wall is always equal to $D - 1$ .
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Figure 4
(a) Ising, Bloch, and Néel domain-wall structures. (b) Schematic illustration of fourfold and threefold domain-wall vertices. (c) Splitting of a vortexlike domain-wall vertex [blue (dark gray) circle at the left] into two threefold vertices [yellow (light gray) circles at the right]. The domain wall shown by the dashed-orange line carries $w = + 1 / 2$ half vortices at its end points.
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Figure 5
(a) Two-dimensional center-divergent vortex. The arrows represent the 2D polarization vectors $P$ and are colored according to their $x$ Cartesian component. The black circle is a loop surrounding the defect core. Moving counterclockwise along this loop from point 1 to point 5 generates a counterclockwise rotation of polarization, as shown in (b). Consequently, the tip of the polarization vector draws a loop wrapping around $P = 0$ . The winding number $w$ of configuration (a) is equal to 1 since one full turn around the vortex core yields a 360° polarization rotation. (c)–(e) Topologically equivalent vortex configurations with $w = 1$ . (f) Antivortex configuration with $w = - 1$ .
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Figure 6
(a) Néel and (b) Bloch skyrmions. The arrows represent spins and are colored according to their out-of-plane Cartesian component. The topological charge of a skyrmion (the skyrmion number) is computed by mapping the spin values within a disk containing the skyrmion core to the order-parameter space. (c)–(e) Schematic of the skyrmion number calculation for the Néel skyrmion texture. Increasing the diameter of the mapped area [bottom images in (c)–(e)] eventually creates a closed surface [top images in (c)–(e)] that fully covers the order-parameter space $S^{2}$ one time.
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Figure 7
Schematic structures of (a) a polar bubble, (b) polar bubble skyrmion, and (c) polar hopfion. Lines indicate the polarization flux lines, while arrows show the orientation of the local dipoles. The in-plane polar vortex of a bubble skyrmion is indicated by an orange (light gray) horizontal curled arrow in (b). (d) Linking of two polarization flux lines (shown in solid blue and dashed red) on a toroidal surface. (c) From [227].
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Figure 8
Table of the relevant topological structures in two-dimensional (the $D = 2$ , top row) and three-dimensional (the $D = 3$ , bottom row) systems with the circle-equivalent (left $X = S^{1}$ column) and sphere-equivalent (right $X = S^{2}$ column) order-parameter spaces. Structures belonging to the families of topological defects and solitons are found below the pink and blue labeled boxes, respectively. The most frequently encountered meron structures are shown below the gray labeled boxes. For vortices, skyrmions, bubbles, bubble skyrmions, and hopfions the structure schematics are complemented with indications of physical properties (chirality $h$ , as well as the divergence and curl of the polarization field). Merons and skyrmion tubes (antiskyrmion tubes; not shown) have the same physical properties as the corresponding skyrmion (antiskyrmion) structures, while vortex lines (antivortex lines; not shown) inherit their properties from 2D vortices (antivortices). For the antivortices the local value of the curl is not zero, but it vanishes when the surface integral is taken (the circulation over circular contours surrounding the core vanishes). $⋆$ As discussed in Sec. 3b4, polar bubbles can be split into two half-bubble structures featuring $N = \pm 1$ layers using a cylindrical Ising domain wall characterized by $N = 0$ . Schematic hopfion illustration from [227].
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Figure 9
(a) Unwinding of a vortex in a two-dimensional order-parameter field. In the order-parameter space, a vortex (top left panel) and a perfectly ordered state (top right panel) corresponds to a closed loop (bottom left panel) and a point (bottom right panel), respectively. The transformation between the two states is therefore impossible without making a cut. Such a “surgery” with a consequent contraction of the circular arch creates a midpoint state (middle panels) with a characteristic head-to-head 180° domain wall. (b) Allowing out-of-plane rotations of the order parameter (top subpanels) opens the door to smoothly eliminate a vortex. The order-parameter space representation of such a transformation looks like a contraction of a loop on a sphere (bottom subpanels).
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Figure 10
(a) For $T < T_{BKT}$ tightly bound pairs of vortices [orange (light gray) circles] and antivortices [blue (dark gray) circles] are formed. (b) For $T > T_{BKT}$ the vortex-antivortex pairs unbind.
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Figure 11
Formation of nanodomains in ferroelectric thin films under open-circuit boundary conditions. (a) Uniformly polarized ferroelectric of thickness $d$ . The normal component of the polarization gives rise to surface polarization charges $σ_{bound} = P \cdot \hat{n}$ , where $\hat{n}$ is the unitary vector perpendicular to the surface. These polarization charges generate a depolarization field $E_{dep}$ . (b) One way to screen the depolarization energy is the formation of stripe domains: regions with different orientations of spontaneous polarization. $w$ stands for the width of the domain, which according to Landau-Kittel’s law scales as $w \propto d^{1 / 2}$ . (c) To further screen the stray field $E_{st}$ , 90° domain walls are formed at the near-surface shell, yielding flux-closure domains or (d) vortices depending on the thickness $d$ . In the vortices the polarization continuously rotates, in contrast to the flux-closure domains, where at some regions 180° domain walls between the up and down domains are present.
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Figure 12
Different regimes for the polarization coupling in a ferroelectric-dielectric interface. For a sufficiently thin ratio of the dielectric layer with respect to the ferroelectric (left panels) and under compressive epitaxial strain, the dielectric is expected to polarize with nearly the same spontaneous out-of-plane polarization as the ferroelectric layer, resulting in a roughly homogeneous polarization throughout the structure. The two layers are said to be electrostatically coupled. Under tensile epitaxial strain, the polarization might rotate to in plane. For a thick enough layer of the dielectric, the polarization is confined within the ferroelectric layer, breaking into domains to minimize the electrostatic energy (right panels). The two layers are said to be electrostatically decoupled. Depending on the thickness of the ferroelectric, the local polarization can rotate continuously to impose a vanishing divergence of the polarization (vortices) or a 180° domain might appear at the central layers of the ferroelectric (flux-closure domains).
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Figure 13
Phase diagram and total energy density for the $({PbTiO}_{3})_{n} / ({SrTiO}_{3})_{n}$ superlattice grown on the ${DyScO}_{3}$ substrate, as calculated using the phase-field simulations and verified experimentally. Top left inset: simulation and planar STEM result of an in-plane view of the $a_{1} / a_{2}$ twin-domain structure for $n = 6$ . Middle left and right insets: vortex structures for $n = 10$ from simulation and experimental TEM mapping, respectively. Bottom left and right insets: cross sections of flux-closure structure for $n = 50$ from phase-field simulation and experimental TEM vector mapping, respectively. SIM, simulation; TEM, transmission electron microscopy. From [154].
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Figure 14
Phase diagram of a $({PbTiO}_{3})_{16} / ({SrTiO}_{3})_{16}$ superlattice under short-circuit electrical boundary conditions obtained from phase-field simulations, as described in Sec. I.B.3 of the Supplemental Material ([472]). For a particular value of the in-plane lattice constant, three independent calculations were carried out, starting with a random configuration of the local polarization. After a minimization of the energy, the three calculations converged to a similar ground state. The “ $a$ -twin” phase corresponds to the $a_{1} / a_{2}$ domain structure. In the top $x$ axis, the lattice constants of some of the available commercial substrates are represented. LSAT, $({LaAlO}_{3})_{0.29} − ({SrAl}_{0.5} {Ta}_{0.5} O_{3})_{0.71}$ ; STO, ${SrTiO}_{3}$ ; SAGT, ${Sr}_{1.04} {Al}_{0.12} {Ga}_{0.35} {Ta}_{0.50} O_{3}$ ; DSO, ${DyScO}_{3}$ ; GSO, ${GdScO}_{3}$ . Courtesy of Z. Hong.
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Figure 15
Schematic sketch of the effect of the antisymmetric Dzyaloshinskii-Moriya interaction in the alignment of neighboring spins. The order parameter rotates around an axis parallel to the Dzyaloshinskii-Moriya vector $D_{12}$ . (a) If $D_{12}$ is parallel to the relative position between the two neighboring spins $R_{12}$ , then the appearance of chiral helices and Bloch-like skyrmions, such as the one shown at the bottom, are favored. (b) If $D_{12}$ is perpendicular to $R_{12}$ , then a nonchiral spin texture and a Néel-like skyrmion are supported.
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Figure 16
(a) Sketch showing how the combination of (left images) the DMI (which favors a 90° rotation between neighboring magnetic moments $i$ and $j$ ) and (center images) the Heisenberg interaction (which favors, depending on the coupling constant $J_{i j}$ , 0° or 180° angles between neighboring magnetic moments $i$ and $j$ , such as in the antiferromagnetic configuration) produces weak ferromagnetism in (right images) an antiferromagnetic crystal. $m_{i}$ and $m_{j}$ are the localized magnetic moments in neighboring atoms. The straight arrow represents the spin and the curled arrow indicates the angular momentum of an electron on a given site, highlighting the fact that the Dzyaloshinskii-Moriya vector $D_{i j}$ stems from spin-orbit coupling. $m_{T}$ stands for the total magnetization of the corresponding configuration. (b) Sketches showing different distortions in a ${A B O}_{3}$ perovskite structure. The trilinear coupling between oxygen octahedra rotations and tilts (left image) with two antiferroelectric (center left image) or ferroelectric modes (center right image) leads to the appearance of noncollinear (anti)ferroelectricity (right image). $A$ , $B$ , and O atoms are represented by green, blue, and red balls, respectively. The arrows represent the atomic displacements from the reference centrosymmetric positions, $u_{i}$ . The electric DMI $D_{i j}^{'} \cdot (u_{i} \times u_{j})$ is formally equivalent to the magnetic counterpart $D_{i j} \cdot (m_{i} \times m_{j})$ , but the $D_{i j}^{'}$ vector depends on the octahedral tilts, which play the role of the spin-orbit coupling. From [166], and [464].
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Figure 17
(a) Selected area electron diffraction (SAED) patterns of the $n = 18$ ${PbTiO}_{3} / {SrTiO}_{3}$ superlattice grown on ${DyScO}_{3}$ showing the existence of superlattice reflections for both the (a) $00 l$ and (b) $0 l 0$ parent reflections. (c) Synchrotron scattering-based reciprocal-space map of the same superlattice as in (a) and (b) that captures the scattering from the superlattice period as well as the vortex lattice, which has a periodicity of $\sim 13 nm$ . Adapted from [448].
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Figure 18
A dark-field image obtained from a cross section of the ${PbTiO}_{3} / {SrTiO}_{3}$ superlattice using the $(002)_{pc}$ reflection that shows periodic intensity modulation corresponding to the location of the vortices in the ${PbTiO}_{3}$ layer. From [448].
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Figure 19
Observation of clockwise and counterclockwise vortex pair structures. (a) Cross-section HRSTEM image with an overlay of the polar displacement vectors ( $P_{PD}$ ), indicated by yellow (light gray) arrows for a $({SrTiO}_{3})_{10} / ({PbTiO}_{3})_{10}$ superlattice, showing that an array of clockwise-counterclockwise vortex pairs is present in each ${PbTiO}_{3}$ layer. (b) A magnified image of a single clockwise-counterclockwise vortex pair, showing the full density of data points (one for each atom) and the continuous rotation of the polarization state within such pairs. (c) Polarization vectors from a phase-field simulation of the same $({SrTiO}_{3})_{10} / ({PbTiO}_{3})_{10}$ superlattice, which predicts pairs that closely match the experimental observations. (d) Orbital angular momentum transferred to the electron beam. (a)–(c) From [448]. (d) From [279].
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Figure 20
3D visualization of the vortex array in the ${PbTiO}_{3} / {SrTiO}_{3}$ superlattice. The front of the left side is the vector map from a HAADF-STEM image, while the top is a dark-field image showing the vortex “tubes.” The front of the right side is the phase-field simulation, and the side view gives the 3D perspective. From [448].
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Figure 21
(a) Phase-field cross-section simulation of a mixed vortex-ferroelectric phase ensemble showing how the vortex phase accommodates the in-plane head-to-head or tail-to-tail domains. (b) Dark-field image of an $n = 8$ ${SrTiO}_{3} / {PbTiO}_{3}$ superlattice that also exhibits the mixed phase; the ferroelectric phase is identified by the yellow dashed arrows. Adapted from [153].
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Figure 22
(a) Lateral piezoresponse image for a ${SrTiO}_{3} / {PbTiO}_{3}$ superlattice with $n = 16$ revealing stripelike order in which alternating stripes exhibit high (checkered white and black) and low or zero [brown (gray)] piezoresponse with a periodicity of $\sim 300 nm$ along the $[010]_{pc}$ . (b) Variations in the surface topography. (c) Changes in the lateral piezoresponse amplitude. (d) Line trace across the topography (green dashed line) and lateral piezoresponse images (red dashed line) revealing opposite and dramatic modulations in surface topography and lateral piezoresponse between the $a_{1} / a_{2}$ and vortex phases. These measurements in combination with RSM x-ray diffraction studies confirm that the vortex phase corresponds to protruding features in surface topography with low piezoresponse, while the $a_{1} / a_{2}$ phase is recessed in topography with large lateral piezoresponse. (e) 3D phase-field calculated image of the mixed vortex-ferroelectric $a_{1} / a_{2}$ phase ensemble. Adapted from [77].
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Figure 23
Lateral piezoresponse amplitude images for an $n = 16$ superlattice. (a) As-grown mixed-phase structure. (b) Conversion to a uniformly low-piezoresponse state corresponding to the vortex phase following application of a positive 15 V bias. (c) Reversal of portions of region back into a mixed-phase structure following application of a negative 15 V bias. (d) These results are confirmed by nanodiffraction studies showing the intensity of the $a_{1} / a_{2}$ $002_{pc}$ diffraction peak. (e) Ti-absorption edge, x-ray linear-dichroism-based photoemission electron microscopy image of the region in (c) showing the difference in XLD between the vortex and ferroelectric phases. (f) SHG polar plots obtained from regions with the as-grown mixed-phase structure [solid orange (light gray)], the electrically poled pure vortex structure [solid blue (dark gray)], and that of the volume normalized signal from the $a_{1} / a_{2}$ phase (red dashed ellipse) revealing a large SHG intensity change after transitioning to the vortex phase. Adapted from [77].
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Figure 24
(a) A two-phase mixture of in-plane ferroelectric-ferroelastic $a_{1} / a_{2}$ domains (FE) and polar vortex ( $V$ ) is converted to a single 3D supercrystal ( $S$ ) phase by subpicosecond optical pulses in a ${PbTiO}_{3} / {SrTiO}_{3}$ superlattice. The $S$ phase contains ferroelectric, ferroelastic, and polar-vortex subregions ordered in three dimensions. Thermal annealing reverses this transition. The arrows in (a) indicate the local polar displacements obtained from a phase-field model. The red and blue color contrasts illustrate the up and down $z$ components of the polarization. The white regions correspond to in-plane polarization. (b) Diffraction along the $K$ - $L$ ( $q_{y} - q_{z}$ ) plane near the $004_{pc}$ peak for the mixed-phase ( $FE + V$ ) pristine sample shows evidence of order only along the $z$ direction, with distinct peaks due to the FE and $V$ phases, as indicated by the horizontal lines. (c) Superlattice peaks near $002_{pc}$ (pseudocubic notation) showing two distinct diffraction peaks, corresponding to the $V$ and FE phases in the pristine sample, that transform into a single uniform $S$ phase with single-shot optical excitation above a certain threshold energy density. a.u., arbitrary units. From [383].
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Figure 25
Electron microscopy characterization of the supercrystal phase. (a) Cross-section HAADF-STEM image of the $({PbTiO}_{3})_{27} / ({SrRuO}_{3})_{5}$ superlattice showing the full superlattice structure with well-defined periodicity. Scale bar, 100 nm. (b) Higher magnification image revealing a complex periodic arrangement of horizontal and vertical flux-closure domains. (c) Out-of-plane strain ( $ε_{z z}$ ) with respect to the substrate ( ${DyScO}_{3}$ , $c = 3.946 Å$ ) extracted using geometric phase analysis (GPA). (d) In-plane strain ( $ε_{x x}$ ) extracted using GPA. Scale bars in (b)–(d), 30 nm. (e) Laterally compressed HAADF-STEM image that has been Fourier filtered to retain only the out-of-plane periodicity. White curves superposed over atomic planes near the ${PbTiO}_{3} - {SrRuO}_{3}$ interfaces are included as guides for the eye to highlight the large bending of the lattice. ${PbTiO}_{3}$ layers are found to exhibit periodic expansion and contraction along the out-of-plane direction with opposite deformation in neighboring ${PbTiO}_{3}$ layers, while the ${SrRuO}_{3}$ layers bend to accommodate this distortion. (f) 2D sketch of the domain pattern deduced from the TEM measurements. (The absolute directions of the polarization may be reversed in the experimental image.) (g) 3D sketch of the overall domain pattern deduced from XRD, PFM, and TEM studies. From [134].
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Figure 26
(a) Fourier spectra (blue) of calculated time-dependent response of polar vortices on terahertz-field excitation. The spectrum of the terahertz pulse is shown as a broad pink background. The collective modes of polar vortices (orange arrows) are shown as a separate set of modes with respect to the known superlattice acoustic modes (black arrow) and the soft mode of ${PbTiO}_{3}$ at room temperature. The vortexon mode shifts to higher frequency (red peak; note that this is not the calculated peak but rather a schematic peak to show the temperature dependence) as the sample temperature increases. To the right is a schematic of the terahertz-pump and x-ray-diffraction-probe experiment using an x-ray free-electron laser (FEL). The colored stripes on the $({PbTiO}_{3})_{16} / ({SrTiO}_{3})_{16}$ superlattice film represent in-plane vortex orders with opposing polarization vorticities. (b) Emergence and evolution of the vortexon (atomic displacement vortices, purple circles) during its oscillation period $τ$ , overlaid with the static polarization vortices (magenta circles). $+$ and $-$ are signs for the vortexon vorticities, which reverse dynamically. Right panel: enlarged view of the region of the dashed box with the calculated static polarization (magenta arrows) and lead-cation displacement (purple arrows) in each unit cell of the vortexon mode at $t = τ / 4$ . (c),(d) Snapshots of the two types of coherent deformation modes that are observed in the vortexons at two different frequencies: (c) 0.328 and (d) 0.08 THz. From [204].
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Figure 27
Theoretical predictions of the formation of polar skyrmions from second-principles simulations in (a) ${BaTiO}_{3} / {SrTiO}_{3}$ nanocomposite, (b) skyrmion tubes written in nanocolumns of ${PbTiO}_{3}$ , and (c) polar bubble skyrmions in ${PbTiO}_{3} / {SrTiO}_{3}$ superlattices. (d) Planar section HRTEM image showing the locally ordered arrays of skyrmions. (a) From [265]. (b) From [292]. (c),(d) From [82].
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Figure 28
(a) Reversed Ti-displacement vector map (top image) based on the atomically resolved plane-view HAADF-STEM image (bottom image) of a single skyrmion bubble showing the Néel-like skyrmion structure. The sketch of the superlattice in the central top panel is overlaid with the planar-view dark-field TEM image and gives a top view of the superlattice. (b) Ti-displacement vector map (front image) based on the atomically resolved cross-section HAADF-STEM image (back image) showing a cylindrical domain with antiparallel (up-down) polarization. The sketch in the central top panel is overlaid with the cross-section dark-field TEM image and shows the cross-section view of the superlattice. (c),(d) 4D STEM image of a $[({PbTiO}_{3})_{16} / ({SrTiO}_{3})_{16}]_{8}$ superlattice giving the (c) ADF image and (d) maps of polar order using the probability current flow, which were reconstructed from the same 4D dataset. (e),(f) Multislice simulations of the beam propagation through the model structure from Fig. 27 showing (e) the ADF image and (f) the probability current flow, which were analyzed using the same process as the experimental data. The signals are not simple projections but are instead weighted by electron-beam channeling toward the middle of the skyrmion bubble, where the polarization exhibits a Bloch-like character. (g) Néel-like skyrmion at the top interface between ${SrTiO}_{3}$ and ${PbTiO}_{3}$ . (i) Bloch-like skyrmion at the central plane in ${PbTiO}_{3}$ . The up and down domains are represented as white and gray regions, respectively. (h),(j) Corresponding Pontryagin densities. The arrows represent the normalized electric dipole moments in the $x − y$ plane. From [82].
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Figure 29
(a) In-plane cross section of the simulated dipolar structure of the hexagonal bubble lattice wherein each bubble is topologically equivalent to a Néel skyrmion. (b) Schematic illustration of the equilibrium structure of a spherical polar bubble averaged over thermal fluctuations (c) Calculated dependence of the free energy on the polar bubble size indicative of a constrained nucleation process. $R_{c}$ and $R_{eq}$ denote the critical and equilibrium radii. (d) Field-induced sublimation of polar bubbles and formation of the bubble gas. The distribution of the out-of-plane polarization displays increasing bias field magnitudes from left to right. From [266].
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Figure 30
(a) Temperature-electric-field phase diagram of a $Pb ({Zr}_{0.4} {Ti}_{0.6}) O_{3}$ ultrathin film. Phases I and II correspond to (b1) connected and (b2) disconnected labyrinthine patterns, respectively, while phases III and IV denote (b3) the mixed bimeron-skyrmion phase and (b4) the bubble phase, respectively. Light gray [red (dark gray)] dipoles are oriented along the $[001]$ ( $[00 \bar{1}]$ ) pseudocubic direction. Phase V corresponds to a homogeneously polarized state. The dashed line separating phases II and III marks the spinodal-like boundary, while the solid line separating phases IV and V marks the binodal-like boundary. (c1),(c2) Experimental PFM amplitude images of PZT films with a one-unit-cell ${SrTiO}_{3}$ spacer and labyrinthine and bubble morphologies, respectively. (d)–(k) Elementary and composite topological defects within the predicted dipolar modulated phases in $Pb ({Zr}_{0.4} {Ti}_{0.6}) O_{3}$ ultrathin film, namely, a threefold junction or (d) an antimeron, (e) meron, (f) saddle or fourfold junction, (g) the handle or meron-antimeron pair, (h) dislocation, (i) the bimeron or meron-meron pair, (j) the polar bubble, and (k) the target skyrmion pattern. (l)–(n) Numerical onsets of the disconnection processes of domains via the removal of a fourfold junction or saddle defect and of a threefold junction or antimeron, as well as the cleavage of an elongated bimeron into two bubbles. From [266].
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Figure 31
Key features defining the chiral character of polar vortices. Straight arrows point along the direction of the local polarization. Clockwise (counterclockwise) vortices are represented as black (dark gray) [red (light-gray)] curled arrows. Circles represent the direction of the axial component of the polarization, with the sign indicated by a cross or dot. The original dipolar configuration in every panel is surrounded by a orange box. Three orthogonal reflections are then examined for each of the configurations. The mirrors are perpendicular to (left) $[100]_{pc}$ , (upper) $[001]_{pc}$ , and (bottom) $[010]_{pc}$ . If the reflections can be mapped onto each other but not to the original one by means of rotations and translations, then the structure is chiral in three dimensions. If not, the configuration is achiral. (a) Vortices without an axial component of polarization (achiral in three dimensions). (b) Vortices coexist with antiparallel axial components of the polarization in neighboring vortices (chiral; the handedness is indicated in the original configuration and in the reflection in the left panel). (c) Vortices coexist with parallel axial components of the polarization in neighboring vortices. The sizes of the up and down domains differ (achiral). (d) Buckled vortices (nonzero offset between its centers) coexist with parallel axial components of the polarization in neighboring vortices. This favors a net polarization along the $[100]_{pc}$ direction, as represented by the horizontal double arrow. The structure is achiral; the mirror images can be superimposed on the original one by a rigid translation of half a unit cell (indicated by the horizontal orange arrow at the center of the upper mirror). (e) Combination of the cases sketched in (d) and (e). This structure is chiral since the handedness of one of the vortices is larger than the other, as indicated by the size of the hands.
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Figure 32
Different experimental techniques to characterize chirality in topologically nontrivial polar systems. (a) Schematic illustrating the experimental setup for the RSXD studies in polar vortices in ${PbTiO}_{3} / {SrTiO}_{3}$ superlattices. The data on the top right are actual images of the specular beam and various higher-order satellite peaks that can be observed. (b) Linecut of the scattered intensity vs lateral momentum transfer using right- (red) and left-circularly (blue) polarized x rays for an $n = 16$ superlattice. The difference in intensity between the two helicities is shown in green, displaying a significant dichroism at the vortex peaks, indicative of the chiral nature of the polar-vortex phase. (c) Map of XCD intensity at $q_{lateral} = + q_{pair}$ across an $n = 14$ sample. Regions of positive (negative) XCD indicate where chiral polar arrays have positive (negative) helicity. (d) Hard-x-ray reciprocal-space mapping about the ${SrTiO}_{3}$ $(002)$ peak for a $[({PbTiO}_{3})_{16} / ({SrTiO}_{3})_{16}]_{8}$ superlattice shows superlattice peaks along the $Q_{z}$ direction, corresponding to an out-of-plane periodicity of about 12 nm, and satellite peaks along the $Q_{y}$ direction, corresponding to an in-plane periodicity of about 8 nm. (e) Image from the CCD used to collect RSXD data, where the resonant diffraction peak is clearly visible. (f),(g) Spectra from a satellite peak for right- (red) and left-circularly (blue) polarized light. (h) The XCD difference spectrum (right-circular minus left-circular x-ray absorption spectrum) shows a clear circular dichroism peak at the titanium $L_{3}$ $t_{2 g}$ edge. (i) SHG images taken with right-circularly and (j) left-circularly polarized excitation in ${PbTiO}_{3} / {SrTiO}_{3}$ trilayers. (k) SHG CD calculated from the images shown in (i) and (j). (a)–(c) From [359]. (d)–(h) From [83]. (i)–(k) From [22].
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Figure 33
Different paths to switch the helicity of the vortices when the chirality is due to the opposite direction of the axial polarization at the cores of neighboring counterrotating vortices. Straight arrows represent the direction of the local polarization in the $x − z$ plane, while the curled arrows indicate the sense of rotation of the vortices. Orange (light gray) and blue (dark gray) circles represent different directions of the axial component of the polarization. The crosses are the points where head-to-head and tail-to-tail domains are formed. From [22].
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Figure 34
Different mechanisms to switch the chirality of topologically nontrivial polar structures. (a) Distribution of the polarization and chirality for different homogeneous applied fields in a ferroelectric skyrmion. Left panel: hysteresis behavior of the polarization of the nanodot as a function of the applied field. The blue and red branches correspond to the up-down and down-up sweeps of the applied field. The numbers mark the different topological states of the polarization. From [400]. (b) Chirality switching of vortices by an electric field in ${PbTiO}_{3} / {SrTiO}_{3}$ superlattices. The blue and red in the pristine vortex array indicate the direction of the axial component of the polarization. The handedness can thus be determined for an individual vortex. From [55]. (c) Helicity (red, left axis) and polarization (blue, right axis) for a ${PbTiO}_{3} / {SrTiO}_{3}$ trilayers as a voltage is applied along $[100]_{pc}$ . Insets schematize the direct coupling between the sense of the buckling and the $[100]_{pc}$ component of the polarization. From [22].
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Figure 35
(a) Schematic of the classical double-well-energy landscape for a ferroelectric. $A$ and $B$ denote the stable polar states, and the regions shaded in red denote areas with negative curvature, i.e., negative permittivity. (b) Polarization vector map from a subregion of a ${PbTiO}_{3}$ layer embedded within a $({SrTiO}_{3})_{12} / ({PbTiO}_{3})_{12}$ superlattice as measured using STEM. (c) Local electric field in a ${PbTiO}_{3}$ layer corresponding to the region shown in (b). (d) Local potential-energy map in the same region as shown in (a) and (b). (e) Variation in the $z$ components of local polarization ( $P_{z}$ , red hexagons) and electric field ( $E_{z}$ , blue circles) along a horizontal line [indicated by the horizontal lines in (b) and (c) that pass through the core of the vortices]. (f) Local energy density estimated from the variation in $P_{z}$ and $E_{z}$ along the same line. Regions around the core (marked with arrows) have negative curvature $\partial^{2} G / \partial D^{2} < 0$ . (g) Second-principles calculation of the inverse of the local dielectric constant in the vortex structure (blue for negative and red for positive regions). (h) Phase-field calculation of a cross section of an array of skyrmions showing the regions of negative permittivity (identified by the dark blue rings). (i) Macroscopic dielectric permittivity as a function of electric field for vortices in $({SrTiO}_{3})_{16} / ({PbTiO}_{3})_{16}$ superlattices with different periodicities. (a)–(g) From [449]. (h)–(i) From [83].
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Figure 36
(a) Dipolar waves and (b) dipolar disclination observed by HAADF STEM in ${PbTiO}_{3} / {SrTiO}_{3}$ superlattices grown on a ${SrTiO}_{3}$ substrate. (c) Numerically simulated equilibrium dipolar waves and (d) metastable disclinations in a ${PbTiO}_{3}$ film under $- 0.6 %$ compressive strain show excellent reproduction of the experimental observations in (a) and (b), respectively. From [225].
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Figure 37
(a) Polarization vectors in 5 nm ${PbTiO}_{3} / {SmScO}_{3}$ ${(001)}_{pc}$ film, obtained from the phase-field simulation with an experiment-inspired lattice model. Convergent and divergent merons are marked with solid and dashed circles. (b),(c) 3D polarization vectors of the two merons marked 1 and 2 in (a), respectively. (d) In-plane polarization mapping exhibiting the local ordered meron textures in $[({PbTiO}_{3})_{16} / ({SrTiO}_{3})_{16}]_{8}$ lifted-off membranes at 373 K. (e) Details of (d) where vortices (clockwise, green; counterclockwise, blue) and antivortices (red) are labeled. The dots in the circles represent polarization pointing out of the page, while the cross points into the page. (f) Map of the local regions showing different handednees. The positive (negative) regions indicate the polar textures having left-handed (right-handed) chirality in the meron phase. (a)–(c) From [431]. (d)–(f) From [362].
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Figure 38
Some of the possible new directions that can emerge from fundamental phenomena in oxide superlattices in which we can juxtapose the various elements of the Hamiltonian for the system (as shown in the central box). The alignment of energy terms and control parameters on the horizontal scale is intended to show the ranges where each control parameter, within its variability, is effective in tuning the various energy terms in transition metal oxides. The box to the right shows the various fundamental physical properties that could be tuned, leading to the various physical phenomena that manifest in such heterostructures. These are described in the blue oval bubbles in the outer ring.
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Figure 39
Topological domain patterns in ferroelectric ${YMnO}_{3}$ . (a) TEM dark-field image taken using the $1 \bar{3} 1$ spot, showing six antiphase domains ( $α - β - γ - α - β - γ$ ) emerging from one central point. The $α$ , $β$ , and $γ$ antiphase domains correspond to the three options for the origin of trimerization. The red arrows in the insets indicate the anticipated crystallographic directions of antiphase boundaries. (b) Proposed cloverleaf configuration of six antiphase or ferroelectric domains. The presence of three types of antiphase domains ( $α$ , $β$ , and $γ$ ) and two types of 180° ferroelectric domains ( $+$ and $-$ ) results in the arrangement of six distinct domains meeting at one point. (c) Topography ( $6 \times 6 μ m^{2}$ ) and (d) conductive atomic force microscopy images simultaneously obtained in contact mode with a tip bias voltage of $- 2 V$ , demonstrating the presence of a nanometer-scale smooth surface and striking cloverleaf domains with conductive contrast. The bright (dark) conductive contrast corresponds to a domain with downward (upward) polarization. From [68].
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Figure 40
(a) Electron diffraction patterns of ${Fe}_{x} {TaS}_{2}$ with $x = 1 / 3$ . The ( $\sqrt{3} a \times \sqrt{3} a$ )-type (indicated as $S 4$ ) superlattice spots can be observed in addition to the fundamental spots. (b) Dark-field images taken using the superlattice spot $S 4$ . (c) Dark-field images of ${Fe}_{0.43} {TaS}_{2}$ taken using the superlattice spot $S 4$ . (d) Two-dimensional schematics of the $\sqrt{3} a \times \sqrt{3} a$ superstructures of intercalated Fe ions in ${Fe}_{1 / 3} {TaS}_{2}$ . The red and blue spheres depict Fe ions, and small (large) spheres represent the lower (upper) Fe layers. (e) Domain patterns with tensorial proper coloring (first step, dark and light; second step, red, blue, and green) in ${Fe}_{1 / 3} {TaS}_{2}$ . (f) Dark-field image taken under the so-called Friedel-pair-breaking condition exhibiting the presence of chiral domains without centrosymmetry. From [155].
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Figure 41
Manipulating energy landscapes in ferroelectrics. The schematic illustration shows how the energy landscape can be manipulated to create tunable barrier heights, multistates, and thresholdlike behavior.
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Figure 42
A weak beam dark-field-TEM image of a ${SrTiO}_{3} / {PbTiO}_{3} / {SrTiO}_{3}$ trilayer sample showing the formation of a quasiperiodic array of vortices (progressing from left to right). The magnified image on the right shows the formation of arrays of edge dislocation dipoles in the vortex lattice as well as the existence of antiphase boundaries. From [22].
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Figure 43
Topological defects at various length scales. (a) Dislocation dipole in a metal lattice, indicated as a $T$ and an inverted $T$ . From [441]. (b) Dislocation dipoles in the vortex lattice in a ${SrTiO}_{3} / {PbTiO}_{3} / {SrTiO}_{3}$ trilayer structure. From [22]. (c) Dislocation dipoles in a colloidal array. From [221]. (d) Dislocation dipoles in sand dunes. From [40]. Note the changes in the length scales of these patterns (from nm to cm), although the patterns are self-similar.
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Figure 44
(a) Two instances of the depleted Néel skyrmion lattice evolution at room temperature obtained from effective Hamiltonian molecular dynamics. Dark (bright) contrast correspond to positive (negative) out-of-plane polarization. The solid blue circles indicate a typical region where thermal skyrmion motion is observed. (b) Schematic illustration of the PFM tip-induced skyrmion “teleportation” in PZT-based heterostructures ([304]). Application of a local electric field abruptly transports a nearby Néel skyrmion to the position beneath the PFM tip.
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Figure 45
Moving ferroelectric topological textures with an electron beam. Scanning a subnanometer electron beam across an array of labyrinthine and bubble-shaped textures in a freestanding ${SrTiO}_{3} / {PbTiO}_{3} / {SrTiO}_{3}$ membrane enables the movement of portions of the texture. (a)–(d) Sequence of ADF-STEM images obtained over a period of 120 s showing the creation and annihilation of convex (lower blue circles) and concave disclinations (upper red circles) in the ferroelectric texture. The lower blue and upper red dots denote merons and antimerons with topological charges of $+ 1 / 2$ and $- 1 / 2$ , respectively. (e)–(h) Electric-field gradient control of polar merons and antimerons, a sequence of images in which the two-dimensional rastering of the electron beam moves the meron cluster. (i) Sum of differences between adjacent frames showing the trajectory of features tracking with the electron-beam parking positions. Adapted from [361].
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Figure 46
(a) Dielectric permittivity as a function of electric field (applied voltage divided by the thickness of the sample) for ${SrTiO}_{3} / {PbTiO}_{3}$ superlattices with different periodicities. (b) Measured capacitance of coplanar wave guide (CPW) transmission lines on the superlattice (upper orange line) and the ${SrTiO}_{3}$ substrate (black middle line). Finite-element simulations and the measurements of the bare ${SrTiO}_{3}$ substrate are used to analyze and isolate the portion of the total capacitance that is related to the permittivity of the superlattice (bottom blue line). (c) The real part of the in-plane dielectric permittivity (left-hand $y$ axis) and the loss tangent (right-hand $y$ axis) are determined from the capacitance and conductance measurements via a mapping function obtained from 2D finite-element modeling of the CPW structures. (d) Cole-Cole model of the in-plane complex permittivity of the superlattice from 100 MHz to 10 GHz. From [83].
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Figure 47
(a) EPR spectrum from the $[{Fe}^{+ 3} - {PbTiO}_{3}]$ . (b) Left image: schematic of the ${Fe}^{+ 3}$ -doped ${PbTiO}_{3}$ $[{Fe}^{+ 3} - {PbTiO}_{3}]$ in the tetragonal perovskite structure, with the green up arrow indicating the polarization direction and the horizontal blue sheet the easy plane of the spins in ${Fe}^{+ 3}$ . Right image: graphic showing that the easy plane of the spins rotates as the polarization direction is rotated by 90°. (c) Schematic indicating what happens to the crystal-field splitting and spin-orbit coupling when an impurity is placed within a polar vortex (or skyrmion).
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