Emergent chirality in a polar meron to skyrmion phase transition
Yu-Tsun Shao, Sujit Das, Zijian Hong, Ruijuan Xu, Swathi Chandrika, Fernando Gómez-Ortiz, Pablo García-Fernández, Long-Qing Chen, Harold Y. Hwang, Javier Junquera, Lane W. Martin, Ramamoorthy Ramesh, David A. Muller
EEmergent chirality in a polar meron to skyrmion phase transition
Yu-Tsun Shao , Sujit Das , Zijian Hong , Ruijuan Xu , Swathi Chandrika , Fernando Gómez-Ortiz , Pablo García-Fernández , Long-Qing Chen , Harold Y. Hwang , Javier Junquera , Lane W. Martin , Ramamoorthy Ramesh & David A. Muller Department of Applied and Engineering Physics, Cornell University, Ithaca, New York, USA. Department of Materials Science and Engineering, University of California, Berkeley, CA, USA. Materials Research Institute and Department of Materials Science and Engineering, The Pennsylvania State University, State University Park, PA, USA Laboratory of Dielectric Materials, School of Materials Science and Engineering, Zhejiang University, Hangzhou 310027, China Department of Applied Physics, Stanford University, Stanford, CA, USA. Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CAs, USA. Departamento de Ciencias de la Tierra y Física de la Materia Condensada, Universidad de Cantabria, Cantabria Campus Internacional, Avenida de los Castros s/n, 39005 Santander, Spain Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA. Department of Physics, University of California, Berkeley, CA, USA. Kavli Institute at Cornell for Nanoscale Science, Ithaca, New York, USA. (Email: [email protected]) bstract
Polar skyrmions are theoretically predicted to emerge resulting from the interplay of elastic, electrostatic and gradient energies, in contrast to the key role of the anti-symmetric Dzyalozhinskii-Moriya interaction in magnetic skyrmions. With the discovery of topologically stable polar skyrmions reported by Das et al., (Nature 568, 368, 2019), it is of both fundamental and practical interest to understand the microscopic nature and the possibility of temperature- and strain-driven phase transitions in ensembles of such polar skyrmions. Here, we explore the emergence of a two-dimensional, tetratic lattice of merons (with topological charge of +1/2) from a skyrmion state (topological charge of +1) upon varying the temperature and elastic boundary conditions in [(PbTiO ) /(SrTiO ) ] lifted-off membranes. Such a topological phase transition is accompanied by a change in chirality, e.g. from left-handed to zero-net chirality, as measured by four-dimensional scanning transmission electron microscopy (4D-STEM). We show how 4D-STEM provides a robust measure of the local polarization simultaneously with the strain state at sub-nm resolution, while directly revealing the origins of chirality in each skyrmion. Using this, we demonstrate strain as a crucial order parameter to drive isotropic-to-anisotropic structural transitions of chiral polar skyrmions to non-chiral merons, validated with X-ray reciprocal space mapping and theoretical phase-field simulations. These results provide the first illustration of systematic control of rich variety of topological dipole textures by altering the mechanical boundary conditions, which may offer a promising way to control their functionalities in ferroelectric nanodevices using the local and spatial distribution of chirality and order for potential applications. ntroduction A structural transition in materials involves the rearrangement of a periodic array of motifs in response to external stimuli, which governs the materials’ functional properties. Such a transition is not limited to atoms, but also anticipated in lattices consisting of unconventional quasi-particles such as skyrmions or merons(8–11). The recent discovery of polar skyrmions in ferroelectric-oxide superlattices provides a framework for exploring topology and exotic physical phenomena in condensed matter physics with a focus on polar order(7, 12). Polar skyrmion bubbles consist of three-dimensional (3D) electric dipole textures which, plane by plane, are characterized by a topological charge of +1, or the skyrmion number 𝑁 sk = 14𝜋 ∬ 𝑑 𝑟𝑛 → ⋅ (∂𝑛 → ∂𝑥 × ∂𝑛 → ∂𝑦) where 𝑛 → is the normalized local dipole moment. At the central xy -plane, a polar skyrmion exhibits an out-of-plane polarization ( P op ) at the core, antiparallel P op outside the boundary, and, most importantly, a Bloch-component(13) of in-plane polarization ( P ip ) at the periphery that ascribes chirality to the skyrmion.(14). A fundamental question pertaining to the spatial arrangement of the skyrmions is the degree of long-range order, if any, amongst the skyrmions, in terms of both the orientational order as well as translational order(11, 15). The emergence of long range order, or conversely the disappearance of a possible long range order in the polar skyrmion lattice driven by temperature can provide a heretofore unexplored possibility of a melting phase transition in such a topologically protected two-dimensional (2D) array of polar skyrmions (akin to the well-known Kosterlitz-Thouless transition(11, 16)). Although phase field models had predicted the possibility of forming a long-range ordered skyrmion lattice for certain values of mismatch strain with the substrate(7), experimentally this is yet to be demonstrated. Given that the magnitude and sign (compressive vs. ensile) of the strain in the superlattice is a critical component of such long-range order, we sought to manipulate this by lifting off the superlattice from the substrate(17–19). In doing so, we are able to study the ground state of the skyrmions without any interference from substrate constraint. On such a free-standing membrane, we then imposed different elastic boundary conditions by varying the temperature to manipulate the degree of long-range order in the skyrmion lattice. Here, we report the direct observation of sequential structural transformations for the polar skyrmions: from stripe-shaped to circular disordered skyrmions bubbles, to a tetratic-ordered meron lattice in lifted-off [(PbTiO ) /(SrTiO ) ] superlattice membranes through integrated experimental measurements and theoretical phase-field calculations. To image the topological polar textures with sub-nm resolution, we developed an approach for the analysis of Kikuchi bands recorded by scanning convergent beam electron diffraction (SCBED) along with an electron microscopy pixel array detector (EMPAD)(20). We also demonstrate, for the first time, a direct experimental determination of chirality of the polar skyrmions using SCBED and dynamical diffraction analysis. Our observations reveal the emergence of a square meron lattice with N sk =+1/2 from the disordered skyrmion phase with a N sk =+1, in which the chirality changed from left-handed to zero-net chirality, respectively. Results and Discussion
To examine the local polarization distribution, we performed SCBED experiments on plan-view, lifted-off samples to image the in-plane Bloch components (details in Methods). Briefly, SCBED works by using an EMPAD detector which records the 2D electron diffraction pattern over a 2D grid of probe positions, resulting in 4D datasets (Fig. 1a, details in Methods)(21–23). As a result of dynamical diffraction effects, the charge redistribution associated with ferroelectric olarization leads to the violation of Friedel’s law(24, 25). Thus, the polarization field within the top PbTiO layer (Fig. S1) can be measured quantitatively from intensity differences of polarity-sensitive Kikuchi bands(26, 27) (Fig. 1B & D) or Bragg reflections(28, 29) (Fig. S2). We employed these Kikuchi bands for polarity mapping as they are less sensitive to artifacts such as disinclination strain or crystal mistilts(30, 31). For example, Fig. 1C shows a high-angle annular dark-field (HAADF) image of the skyrmions from plan-view, where the out-of-plane polarization ( P op ) is separated by domain walls with circular or elongated features. Figure 1E shows that the in-plane polarization ( P ip ) map of Bloch-like rotation can be reconstructed from SCBED dataset. The white arrows and colors in Fig. 1E denote the direction of P ip , whereas the saturation represents the vector magnitude. The dark color indicates the P op regions. Figure 1. Imaging in-plane polarization textures. ( A ) Schematic of the plan-view SCBED imaging technique on the [(PbTiO ) /(SrTiO ) ] superlattice which uses a scanning electron probe and pixelated array detector, where a diffraction pattern was recorded at each probe position. The local polarization direction can be determined by observing the difference of diffracted intensities of Friedel pairs, 𝐼 𝐺⃑ and 𝐼 −𝐺⃑ . Representative CBED patterns taken from ( B ) top and ( D ) ottom of a skyrmion, where the polarity-sensitive Kikuchi bands in the thermal diffuse scattering are selected for determining polarization, as marked by pink and yellow boxes. For clarity of display, the Kikuchi bands intensity were weighted by k , where k ⃑⃑ denotes the scattering vector from the transmitted spot. ( C ) Plan-view dark field STEM image of a (SrTiO ) /(PbTiO ) /(SrTiO ) trilayer shows nanometer-size round and elongated features. ( E ) Polarization map reconstructed from the SCBED dataset showing the in-plane Bloch components of polar-skyrmions, acquired from the nearby region of (C). The color map represents the in-plane polarization direction at each point. Upon heating from 223 K to 373 K, we observed successive structural transitions in the skyrmion ensemble, from striped to circular-shaped polar skyrmions to a tetratic-ordered lattice. Figure 2a shows the polarization configuration at 223 K, which consists of elongated stripes of ~100 nm in length. In analogy with in-plane cuts ( 𝑄 𝑥 − 𝑄 𝑦 ) from X-ray reciprocal space maps (RSM), the fast Fourier transform (FFT) patterns of HAADF images indicate the in-plane ordering of polar textures (Fig. S3). For example, two peaks were found in the FFT pattern (inset, Fig. 2A) consistent with the stripe features. When heated to 298 K, the stripes deformed into a circular shape (Fig. 2B & 2E) of ~10 nm in diameter. The arrangements of circular skyrmions appears to be random, as indicated by the halo in the FFT pattern (inset, Fig. 2B). With further increase of temperature to 373 K, the random skyrmion arrangements were replaced by an ordered tetratic arrangement (Fig. 2C), confirmed by the four peaks in the corresponding FFT. To elucidate the details of various polar textures, we performed SCBED experiments at several temperatures, in Fig. 2D-F we show the polarization maps reconstructed from SCBED datasets. For the stripe- (223 K) and disordered, circular-shaped (298 K) skyrmions, the polarization maps (Fig. 2D & 2E) indicate that the maximum P ip is observed at the periphery of the skyrmions, whereas the minima (almost zero) are at the core and outside the boundaries. Upon heating from 298 K to 373 K, the disordered skyrmions transformed into a square lattice (Figs. 2C & 2F). Figs. 2G-I show the corresponding phase-field simulations of polar textures at different temperatures and under various train conditions, in which the average effective in-plane lattice constants are obtained from experimental measurements (Fig. S4). These simulations demonstrate a systematic change from the labyrinthine skyrmions at low temperature (223K) to an ordered tetratic structure at higher temperatures (373K), driven not by a pure thermal effect but by the changes in the in-plane strain state originated from lattice thermal expansion. Figure 2. Variations of polar textures with temperature.
Plan-view dark field STEM imaging of [(SrTiO ) /(PbTiO ) ] superlattice acquired with temperatures at ( A ) 223 K, ( B ) 298 K, and ( C ) 373 K. Insets, fast Fourier transform (FFT) of the images in A-C showing different types of rdering. Polarization maps reconstructed from the SCBED dataset of superlattice at ( D ) 223 K, ( E ) 298 K, and ( F ) 373 K, showing the in-plane Bloch components of polar-skyrmions. Figures D-F are acquired from nearby regions of A-C. The corresponding phase-field simulations with temperatures and in-plane lattice parameters ( a,b ) of ( G ) 223 K, a =3.875 Å, b =3.885 Å; ( H ) 298 K, a = b =3.905 Å; and ( I ) 373 K, a = b =3.899 Å. The color wheel hue (saturation) corresponds to the direction (magnitude) of the in-plane component of the ferroelectric polarization. To compare the difference of polar textures, regions in Fig. 3A-B (yellow box) were selected to show the P ip polarization maps (Figs. 3C-D). The white arrows and colors in Fig. 3C-D denote the magnitude and the direction of P ip , whereas the saturation represents the vector magnitude. The dark color indicates the P op regions at skyrmion cores and outside the boundary, which are separated by Bloch domain walls ( P ip ) consistent with cross-section data (Fig. S5). The P op at the skyrmion cores points positively towards the growth direction ([001], or + z ) and are antiparallel to P op outside of the boundary, labeled as green dots (+ z ) and red crosses (- z ), respectively (Fig. 3E). The Bloch components ( P ip ) exhibit a continuous rotation of the local polarization vector forming a closed loop, as illustrated in a map of the curl of the polarization vector field (∇ × P ) [ ] (Fig. 3A). Both elongated and circular skyrmions exhibit P ip having clockwise (CW) rotation at the periphery yielding a vorticity of +1. Combining P ip and P op information, we confirm that both elongated and circular skyrmions manifest with a skyrmion number of N sk =+1. Figure 3. Real-space observations of disordered polar skyrmions and a square lattice of merons.
The curl of in-plane polarization ( 𝛻⃑⃑ × 𝑃⃑⃑ ) [001] showing the rotation directions of the structures at temperatures of ( A ) 298 K and ( B ) 373 K. ( C and D ) Enlarged in-plane polarization mapping from the yellow box region of A and B, respectively exhibiting the skyrmion texture at 298 K and local ordered meron textures at 373 K. ( E and F ) Details of (C and D), where vortices (clockwise: green, counterclockwise: blue) and antivortices (red) are labeled. The dots in circles represent polarization pointing out of the page, while the cross points into the page. he most striking observation is the appearance of an ordered structure at 373 K (Figs. 3D & 3F), which represents a square meron lattice with N sk =+1/2. Fig. 3F clearly demonstrates these periodic arrays, which also indicates that the maximum P ip polarization is observed at the periphery, whereas the minimum (almost zero) is at the core. Three types of core regions were observed, which we labeled the cores of P ip having CW rotation (green), counterclockwise rotation (CCW; blue), and antivortices (red). The dot in the circle indicates the P op pointed out of the page (along the growth direction), while the cross indicates P op pointing into the page. From a cross-section polarization map (Fig. S5), the P op at the cores of vortices (vorticity of +1) and antivortices (vorticity of -1) are plausibly antiparallel. On the other hand, P op appears at vanishing points of P ip , implying the P op at vortex cores are not fully surrounded by P op of the opposite direction (Fig. S6). From this, we can compute the skyrmion number(10): 𝑁 sk = 14𝜋 ∬ 𝑑 𝑟𝑛 → ⋅ (∂𝑛 → ∂𝑥 × ∂𝑛 → ∂𝑦) = 12 𝑣 · (𝑃 𝑜𝑝𝑐 − 𝑃 𝑜𝑝𝑝 ), where 𝑣 represents the vorticity, 𝑃 𝑜𝑝𝑐 the value of the out of plane polarization at the core and 𝑃 𝑜𝑝𝑝 the value of the out of plane polarization at the periphery. We deduce the vortices (blue and green dots) to be merons with a topological number of 𝑁 𝑆𝑘 = , and the antivortices as merons(32) with 𝑁 𝑆𝑘 = (−1) · (−1 − 0) = . While SCBED works well for mapping P ip polarizations, we have so far only speculated about the corresponding P op at skyrmion cores based on the cross-section data (Fig. S5). To test our hypothesis, it is necessary to experimentally determine the chirality of the 3D polar-vector field for each skyrmion, which poses challenges for projection techniques such as TEM. Fortunately, we can overcome this problem by utilizing the dynamical diffraction effects in higher-order Laue zone (HOLZ) reflections, which was established to retrieve 3D structural information uch as handedness of chiral crystals(33, 34). In this study, we specifically examine the intensity differences of chirality sensitive Bijvoet pairs(35), such as (671)/(6 ̅
71) and (771)/ (7 ̅
71) (yellow box, Fig. 4A).
Figure 4. Handedness determination of the chiral polar textures using SCBED. ( A ) A representative experimental diffraction pattern acquired at 373 K (which shows meron square lattice), incidence of ~6.2° away from [001] zone axis, tilted along one of the mirror planes. ( B ) Map of normalized intensity difference between (771) and (7 ̅
71) reflections reconstructed from the SCBED dataset. The positive (negative) regions indicate the polar textures having left-handed (right-handed) chirality. ( C ) Intensity line profiles of HOLZ reflections from selected regions labeled in (B), displaying the intensity difference between two pairs of reflections: (671) and (6 ̅ ̅ ̅
71) is stronger than that of (671) at this exact incident beam direction, while other pairs of reflections remain approximately symmetrical (Fig. 4C, black curve). With this in mind, we can determine the chirality of an individual skyrmion by comparing intensity variations of Bijvoet pairs in a SCBED dataset. At 298 K, we carefully selected regions with minimal crystal mis-tilts and determined that both elongated and circular skyrmions are left-handed (Fig. S7). An ordered tetratic lattice appears upon heating to 373 K, as shown in the (771)/(
71) intensity ratio map (Fig. 4B). In this region, two types of chiral structures are identified, labeled as ̅
71) is weaker than (671) from the diffraction pattern averaged over skyrmion type ) /(SrTiO ) ] lifted-off membranes is caused by the thermal stress upon heating/cooling. In our experiment, the oxide membranes are ~4 × higher in thermal expansion coefficients than the SiN x TEM grid to which they are attached(39, 40). As a consequence, the lifted-off membrane is under compressive strain at 373 K and tensile strain at 223 K, with additional local bending. In case of SCBED experiments, we deliberately searched for flat regions to avoid artefacts in measuring P ip . A detailed SCBED analysis of averaged in-plane lattice parameters shows that the lifted-off membrane is locally more rectangular at 223 K than at 373 K (Fig. S4). To decouple the effects of strain versus temperature on the phase transition, controlled phase-field simulations were carried out at controlled strain boundary conditions at various temperatures (Fig. S8). For a fixed in-plane lattice constant, simulation results at various temperatures showed a similar disordered skyrmion phase, indicating that strain is playing an important role in the skyrmion ordering. Furthermore, the change in anisotropy of in-plane lattice parameters, as seen in the SCBED strain analysis (inset, Fig. S4D), is also consistent with FFT patterns for stripes and square lattice (Figs. 2A and 2C). Thus, we anticipate the occurrence of a long-range ordered meron lattice at room temperature in (PbTiO ) n /(SrTiO ) n superlattices grown on a compressive substrate such as LaAlO . In summary, we report the observation of a topological phase transition sequence in polar skyrmions, whereby tuning the temperature and strain boundary conditions, the anisotropic stripe phase deforms into an isotropic disordered circular phase, and finally transforms into an nisotropic ordered phase. This is the first observation of such transitions in a dipolar topological structure, in which the skyrmions deform in shape and a square lattice of merons appears, thereby preserving the topological charge of the system. The chiralities for each phase were also determined experimentally at nm-scale, for the first time. We hope that the microscopic observation of such a topological phase transition will stimulate further work to explore the macroscopic manifestation of the changes in topology with strain, which clearly is the critical external stimulus (in contrast to magnetic systems, where the Dzyalozhinskii-Moriya coupling plays a key role). Finally, our findings imply that such dipolar textures are a fertile ground for exploring new phases and topology, and with possible applications in nanoscale ferroelectric logic and storage devices. Acknowledgments
The authors acknowledge fruitful discussions with Prof. Jian-Min Zuo, Prof. Kin Fai Mak, Dr. Shengwei Jiang, and Zui Tao. Funding was primarily provided by the Department of Defense, Air Force Office of Scientific Research under award FA9550-18-1-0480. The electron microscopy studies were performed at the Cornell Center for Materials Research, a National Science Foundation (NSF) Materials Research Science and Engineering Centers program (DMR-1719875). The Cornell FEI Titan Themis 300 was acquired through NSF-MRI-1429155, with additional support from Cornell University, the Weill Institute and the Kavli Institute at Cornell. The authors thank M. Thomas, J. G. Grazul, M. Silvestry Ramos, K. Spoth for technical support and careful maintenance of the instruments. The materials synthesis work is supported by the uantum Materials program from the Office of Basic Energy Sciences, US Department of Energy (DE-AC02-05CH11231). The membrane lift-off techniques were developed with support from
US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under contract number DE-AC02-76SF00515. The phase-field simulation work is supported as part of the Computational Materials Sciences Program funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0020145. F.G.O., P.G.F., and J. J. acknowledge financial support from the Spanish Ministry of Science, Innovation and Universities through the grant No. PGC2018-096955-B-C41. L.W.M. acknowledge support from the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-SC-0012375 for the development and study of ferroic heterostructures.
METHODS
Sample preparation using RHEED-assisted pulsed-laser deposition.
The epitaxial lift off sacrificial layer of 16 nm Sr CaAl O with a 2.4 nm SrTiO capping layer was synthesized on single-crystalline SrTiO (001) substrates via reflection high-energy electron diffraction (RHEED) – assisted pulsed laser deposition. The growth of the Sr CaAl O layer was carried out in a dynamic argon pressure of 4×10 −6 Torr, at a growth temperature of 710 °C, a laser fluence of 1.35 J/cm , and a repetition rate of 1 Hz, using a 4.8 mm imaged laser spot. The growth of the SrTiO layer was conducted in dynamic oxygen pressure of −6 torr, at a growth temperature of 710 °C, a laser fluence of 0.9 J/cm , and a repetition rate of 1 Hz, using a 3.0 mm imaged laser spot. The heterostructure was then cooled down to room temperature at the growth ressure. Subsequent to this, n -SrTiO / n -PbTiO / n -SrTiO trilayers ( n - is the number of monolayers ; n =16 ) and [(PbTiO ) /(SrTiO ) ] superlattices were synthesized ex-situ on this template via RHEED-assisted pulsed-laser deposition (KrF laser). The PbTiO and the SrTiO were grown at 610 °C in 100 mTorr oxygen pressure. For all materials, the laser fluence was 1.5 J/cm with a repetition rate of 10 Hz. RHEED was used during the deposition to ensure the maintenance of a layer-by-layer growth mode for both the PbTiO and SrTiO . The specular RHEED spot was used to monitor the RHEED oscillations. After deposition, the heterostructures were annealed for 10 minutes in 50 Torr oxygen pressure to promote full oxidation and then cooled down to room temperature at that oxygen pressure. Membrane lift-off and transfer
The heterostructure was first spin-coated with a polymer support of 500 nm thick polymethyl methacrylate (PMMA) film and placed in deionized water at room temperature until the sacrificial Sr CaAl O layer was fully dissolved. The PMMA coated film was then released from the substrate and transferred onto the TEM gird (NH050D2, Norcada Inc.). Finally, the PMMA layer was dissolved and removed from the membrane in acetone, as schematically shown in Fig. S8. The membrane sample surface was further cleaned in ozone at 180 ℃ for 10 mins. X-ray structural analysis
Laboratory-based X-ray diffraction.
In order to obtain a comprehensive picture of the crystal structure of superlattices, as well as information on the in-plane and out-of-plane ordering, was carried out using a Panalytical X’Pert Pro X-ray Diffraction (XRD) diffractometer with Cu- K α radiation (λ = 1.5405 Å). The high rystalline quality of the films, the smooth nature of the interfaces and the skyrmion ordering, was confirmed from reciprocal space mapping of the as-grown and lifted-off superlattice. Phase-field simulations
Phase-field simulations were performed to study equilibrium polar structures of (PTO) /(STO) superlattice under different temperatures and strain conditions. The spontaneous polarization vector 𝑃⃑⃑ is used as the primary order parameter. The temporal evolution of
𝑃⃑⃑ is governed by the time dependent Ginsburg-Landau equation, i.e., 𝜕𝑃 𝑖 𝜕𝑡 = −𝐿 𝛿𝐹𝛿𝑃 𝑖 (𝑖 = 1,3) Where L is the kinetic coefficient, t is the evolution time. The total free energy F can be obtained by integrating the contributions from the mechanical, electrical, Landau chemical and polar gradient energies, 𝐹 = ∫(𝑓
𝐸𝑙𝑎𝑠𝑡𝑖𝑐 + 𝑓
𝐸𝑙𝑒𝑐𝑡𝑟𝑖𝑐 + 𝑓
𝐿𝑎𝑛𝑑𝑎𝑢 + 𝑓
𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 )𝑑𝑉
Detailed expressions of the energy terms, materials parameters as well as the numerical simulation procedure is described in the published literature [1-5]. The simulation system is discretized into a three-dimensional grid of 200 x y z, with x= y= z=0.4 nm. A periodic boundary condition is used along the in-plane dimensions, while a superposition method is applied in the out-of-plane direction [6]. In the out-of-plane direction, the thickness of the substrate, film and air are set as 30 z, 288 z and 32 z, respectively; while the film is comprised of periodic stacking of 16 z of PTO layers and 16 z of STO layers. A closed-circuit, electric boundary condition is assumed where the electric potential is fixed to zero at the top and bottom of the film surface [2]. When a thin film boundary condition is applied where the stress on the top of the thin film is zero, nd the displacement at the bottom of the substrate sufficiently far away from the film/substrate interface is set to zero [3]. An iteration-perturbation method is adopted to account for the inhomogeneity in the elastic constants of PTO and STO [7]. To determine the local strain state, the reference pseudocubic lattice constants for STO (PTO) at 223K, 300 K and 373 K are set as 3.901 Å (3.953 Å), 3.905 Å (3.957 Å) and 3.909 Å (3.961 Å), respectively. The average effective substrate lattice constants are taken from experimental measurements, which are set as a =3.875 Å, b =3.885 Å for 223 K, a = b =3.905 Å for 300 K and a = b =3.899 Å for 373 K. The large compressive strains are due to local bending when the lift-off membrane is heated/cooled down. A background dielectric constant of 40 is used [8-9]. Random noise with a magnitude of 0.0001 C/m is added to the system as the initial polarization distribution of the system. The polar structures are shown in Figs. 2G-I & S8. At 223K, due to the anisotropic bending of the membrane during cooling, elongated skyrmion stripes, or bimerons, are formed along X -axis. Meanwhile, at room temperature, a disordered skyrmion lattice is observed, consistent with a previous report [10]. When the system is further heated up to 373K, a locally ordered square skyrmion lattice is observed. The formation of this locally ordered square lattice is attributed to the large local compressive strain that generated due to the bending of the membrane during heating. Interestingly, some dislocation-like features are also observed, which could locally disturb the ordered structure, consistent with experimental observations. STEM
The plan-view samples of the [(PbTiO ) /(SrTiO ) ] superlattices were lifted-off and transferred to TEM grids. Cross-sectional TEM specimens were prepared on the same plan-view samples, using a FEI Strata 400 focused ion beam (FIB) with a final milling step of 2 keV to reduce damage. The initial sample surface was protected from ion-beam damage by depositing carbon and latinum layers prior to milling. The cross-sectional TEM specimen has a thickness of ~25 nm as determined by CBED analysis.
HAADF-STEM images were recorded by using a Cs-corrected
FEI Titan operated at 300 keV, with beam semi-convergence angle of 21.4 mrad and beam current of 30 pA.
SCBED for polarization mapping.
We performed scanning convergent beam electron diffraction (SCBED) experiments using an electron microscopy pixel array detector (EMPAD), where the 2D electron diffraction pattern was recorded over a 2D grid of real space probe positions, resulting in 4D datasets. Experimental data was acquired using a FEI Titan operated at 300 keV with 15 pA beam current, 2.45 mrad semi-convergence angle, having a probe of ~8 Å FWHM (full-width at half-maximum). A double-tilt liquid-nitrogen-cooled Gatan specimen holder was used for temperature-dependent studies ranging from 95 K to 373 K. The CBED patterns were captured by the EMPAD with exposure time set to 1 ms per frame, for which a 256 × 256 scan can be recorded in under 2 minutes. Due to dynamical diffraction effects, the charge redistribution associated with polarization leads to the breakdown of Friedel’s law. From the collected CBED patterns, the polarization direction in the plan-view samples is reconstructed by calculating the center-of-mass in polarity sensitive Kikuchi bands along the cubic directions. We employ the Kikuchi bands as a more robust means to extract polarity information against internal crystal mis-tilts, which often occurs in ferroic oxides due to disinclination strain. In addition, due to electron channeling effects, the polarity information obtained from Kikuchi bands at this experimental condition is mostly arising from the topmost PbTiO layer, which overcomes the problem of overlapping signals from PbTiO multilayers projected along the plan-view geometry (Fig. S1). For SrTiO /PbTiO /SrTiO trilayer samples, riedel pairs of Bragg reflections were used, such as (300)/(
00) and (030)/(0
0) for x and y components of polarization, respectively (Fig. S2). By matching with dynamical diffraction simulations, we can unambiguously determine the polarization directions in real space.
SCBED for chirality determination.
In order to excite the higher order Laue zone (HOLZ) reflections, the plan-view samples were deliberately tilted ~6.2° away from the [001] zone axis, along one of the mirror planes (Fig. S7). We then perform SCBED experiments exactly at this diffraction condition at various temperatures. Within a SCBED dataset, we carefully selected local regions with minimal tilt and thickness variations. By using dynamical diffraction simulations as the reference, the chirality can thus be determined by comparing the intensity asymmetry of Bijvoet pairs, such as (671)/( 6 ̅
71) and (771)/(7 ̅ SCBED for strain analysis.
We perform exit wave power cepstrum (EWPC) analysis on SCBED datasets to look at changes in lattice parameters. The EWPC works by a discrete Fourier transform of the logarithm of a CBED pattern, resulting units in real-space. Figure S5 shows an EWPC pattern, in which the peaks correspond to projected Pb-Pb inter-atomic distances. Thus, the change in mean projected, in-plane lattice parameters can be measured by comparing the peak distances in EWPC patterns. Sub-picometer precision can be achieved by sub-pixel peaking fitting using the algorithm described previously(41). For the sake of self-consistency, SCBED datasets for temperature-dependent strain analysis were acquired using the exact same TEM optics, within one experimental session. ynamical diffraction simulation.
The CBED simulations were carried out using the µSTEM software(42), with neutral atomic scattering factors of Waasmaier & Kirfel(43). The atomic coordinates were taken from results of 2 nd -principles simulations for a right-handed polar skyrmion. 25×25 diffraction patterns with a 3.2-Å scan step size were simulated at 300-keV beam energy and with 2.45-mrad semi-convergence angle. To simulate Kikuchi bands, thermal diffuse scattering effect was included with the frozen-phonon approximation. Supplementary Figures
Figure S1. Electron beam channeling along a column of Sr atoms in SrTiO . The intensity
𝐼(𝑧) of ( A ) (300) Bragg reflection and ( C ) high-angle ADF signal (40-100 mrad) for a 300-keV electron beam centered on the Sr-site as a function of depth z into the crystal. The focused probe ith semi-convergence angle of 2.45 mrad is similar to that used in the EMPAD experiments to separate diffraction disks. ( B ) The derivative of the (300) diffracted intensity, showing several peaks of signal are generated at 6 nm, 22 nm, 32 nm into the SrTiO , corresponding to the points where 1 st , 2 nd , and 3 rd PbTiO layers would begin in the multilayer structure. ( D ) From derivative 𝑑𝐼/𝑑𝑧 , the signal is channeling most efficiently at 6 nm. The yellow box shows the thickness of 16 unit cells of SrTiO . By changing the collection angle, the Kikuchi bands are more suitable than (300) reflections for retrieving polarization information from a single (top) PbTiO layer in the repeated cell in the 16
16 multilayer structure.
Figure S2. Imaging internal Bloch components of polar-skyrmions in the (SrTiO ) /(PbTiO ) /(SrTiO ) trilayer. ( A ) Polarization map reconstructed from the SCBED dataset showing the in-plane Bloch components of polar-skyrmions. The color map represents the in-plane polarization direction at each point. ( C ) Magnified skyrmion from the yellow box in (A). Ferroelectric polarization direction can be determined by observing the difference of diffracted intensities of Friedel pairs, 𝐼 𝐺⃑ and 𝐼 −𝐺⃑ . Representative CBED patterns taken from ( B ) top and ( D ) bottom of a skyrmion, where the polarity-sensitive <300> Bragg reflections are selected for determining polarization, as marked by pink and yellow circles. Fig. S3: Reciprocal space mapping of the [(PbTiO ) /(SrTiO ) ] superlattice lifted-off membrane. The white arrows indicate the satellite peaks associated with polar skyrmion modulation in the superlattice.
Figure S4. Temperature-dependent strain analysis of [(PbTiO ) /(SrTiO ) ] superlattice. ( A ) Representative CBED pattern shown in logarithmic scale. ( C ) Exit wave power cepstrum (EWPC) transformation of (A) shows peaks corresponding to projected Pb-Pb inter-atomic distances in real-space. Spots that correspond to the length of projected distances along (100) and (010) are selected for tracking changes in lattice parameters a and b , respectively. Histograms of lattice parameters along ( B ) a - and ( D ) b -axis, at temperatures of 223 K (blue) and 373 K (red), over regions of ~500 nm (>80,000 CBED patterns). The relative change in mean of a - and b -axis are ~0.2% and ~0.5%, respectively. Inset in (D) is an exaggerated cartoon of projected lattice parameters, indicating the film is more rectangular at 223 K than at 373 K. Figure S5. Out-of-plane polarization configurations of polar skyrmions. ( A ) Cross-sectional HAADF-STEM image of the (PbTiO )/(SrTiO ) superlattice. ( B ) Polarization configuration reconstructed from the SCBED dataset, where we can access the cross-section of both Neél (red box) and Bloch (blue box) components of polar skyrmions. The out-of-plane polarizations are separated by in-plane Bloch chiral domain walls (dark regions). Figure S6. Difference in topology between a skyrmion and a meron.
Details of polarization for ( A ) a skyrmion and ( B ) a meron, where vortices (clockwise: green, counter-clockwise: blue) and anti-vortices (red) are labeled. The dots in circles represent out-of-plane polarization pointing towards out of the page, while the cross points into the page. The red solid lines in (A) form a closed loop of P op (into the page) surrounding the core of the skyrmion with an antiparallel P op (out of the page). In contrast, the P op cannot form a closed loop around the core of the meron shown in (B), as indicated by dashed red lines. Figure S7. Handedness determination of the chiral polar skyrmions at room temperature. ( A ) Representative experimental (left) and simulated (right) CBED pattern at incidence of ~6.2° away from [001] zone axis, tilted along one of the mirror planes. ( B ) Virtual dark field image reconstructed from the SCBED dataset using (081) reflection. ( C ) Intensity line profiles taken from a row of HOLZ reflections from selected regions labeled in (b), displaying the intensity difference between two pairs of reflections: (671) and (6 ̅ ̅ Figure S8. Phase field simulations of polar skyrmion textures at different temperatures and under the same strain state.
The strain is imposed on the [(PbTiO )/(SrTiO )] superlattice by varying the substrate having in-plane lattice parameters ( a , b ) with a = b =3.905 Å, and at temperatures of ( A ) 223 K; ( B ) 298 K; and ( C ) 373 K. The color wheel hue (saturation) corresponds to the direction (magnitude) of the in-plane component of the ferroelectric polarization. Figure S9. Growth and transfer of PbTiO /SrTiO (PTO/STO) superlattice. Schematic of a superlattice with a SAO sacrificial buffer layer. The sacrificial SAO layer is dissolved in water to release the top oxide films with the mechanical support of PMMA. The freestanding film is then transferred onto the desired substrate or TEM grid.
SrTiO Scarified layer (SAO)PTO/STO SL Support: PDMS Transfer Grid
Release Transfer
Supportdetachment eference
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