Electric field control of ferroelectric polarization

To study a single FE nanoparticle morphology and its vortex structure, we designed a composite consisting of BTO nanoparticles and carbon nanoparticles dispersed within a non-ferroelectric polymeric dielectric matrix (Fig. 1, Supplementary Fig. 1 and the Methods section). Laboratory X-ray powder diffraction (Supplementary Fig. 2) confirms the crystalline nature of the BTO nanoparticles. In BCDI experiments the sample is illuminated with focused coherent X-ray beam (Fig. 1). A random orientation of the BTO nanoparticles along with the experimental geometry allows us to isolate and record the (111) Bragg reflections from a single BTO particle on an area detector. By applying an electric field in cycles and monitoring changes in diffraction pattern, we can differentiate BTO nanoparticles acting as nanocapacitor (Supplementary Fig. 3) from the particles which are electrically insulated in dielectric matrix (Supplementary Fig. 4). By iteratively inverting the coherent X-ray diffraction data, we obtain information on the three-dimensional (3D) Bragg-electron density distribution along with ionic displacement fields (Fig. 2b) with 20 nm spatial resolution (Supplementary Fig. 5). Supplementary Note 1 for more information.

Fig. 1 Experimental scheme of Bragg coherent diffraction imaging. Incident coherent X-ray beam is scattered by a nanoparticle embedded in conducting non-polarizing polymer with attached electrodes. Constructive interference patterns are recorded during application of an external electric field on the particle. Recorded high-resolution Bragg-peak diffraction carries information on the electron density and atomic displacement variations, allowing to reconstruct the complex process of defect evolution and monitoring of vortex. Scale bar corresponds to 0.1 Å−1 Full size image

Fig. 2 Correlations between Bragg coherent diffraction measurements and phase-field simulations. a A blue isosurface shows the reconstructed particle shape (amplitude) with green planes marking the locations of 2D cuts through the volume for the extracted planes in d. b Under zero electric field (initial state), the 3D projection of displacement field along the [111] direction is mapped onto the surface of the particle. c Slices through the particle volume at cut planes of 30, 60, 100, and 145 nm showing the inhomogeneity of the displacement and dynamics under external electric field. d Phase-field simulations for similar cut planes of 30, 60, 100, and 145 nm support the interpretation of experimental results. Scale bars correspond to 60 nm Full size image

Figure 2a shows the isosurface rendering of the particle’s shape, and Fig. 2b shows the [111] projection of the 3D displacement field u 111 (r) onto the isosurface at the virgin electric field state E 1 = 0 kV cm−1. We further compare reconstructions of ionic displacements with phase-field simulations within the nanoparticle when exposed to 223 kV cm−1 (E 2 ) and at remnant (E 3 = 0 kV cm−1) (Fig. 2d) states. The electric field magnitude estimates are obtained from the phase-field simulation based on the convergence of the model to the experimental results. The total elastic strain ϵ ij can be estimated from u 111 (r) by ϵ ij = 1/2 (∂u i 111 /∂x j + ∂u j 111 /∂x i ). The symmetric nature of the strain relative to the bulk structure enables coupling to the polarization and is given by

$$\epsilon _{ij}^o{\rm{ = }}{Q_{ijkl}}{P_k}{P_l}$$ (1)

where Q ijkl is the electrostrictive tensor, ϵ o ij is the spontaneous strain, and the projections of the ferroelectric polarization along the x, y, and z directions are given by P k and P l . The displacement field in Fig. 2b is indicative of a topological polar vortex in the nanoparticle given as isosurfaces in Fig. 3a–c. The electrostrictive coefficients can therefore be obtained as a fitting parameter by using Equation 1. The elastic coefficients used in the fit are consistent with phase-field simulations (Supplementary Note 2 and Supplementary Table 1). Since the displacements u 111 (r) are relatively small compared to the particle size, we assume that u 111 (r) scales with the local polarization, which can be obtained by a summation of Born effective charges35, 36. The formation of the vortex in the nanoparticle is a result of the competition between the elastic energy, the electrostatic energy, the gradient energy and boundary conditions, including nanoparticle shape, and surface facets (Supplementary Fig. 6). The elastic energy, which arises from the fact that BaTiO 3 nanoparticle is constrained by the non-ferroelectric polymer matrix, drives the nanoparticle to adopt to a mixture of in-plane P x , P y and out-of-plane P z polarizations. The second major contribution is the electrostatic energy induced by the built-in electric fields. If the interface between the ferroelectric nanoparticle and the non-ferroelectric matrix is charge free or has a very small charge density, that is, \(

abla \cdot {\bf{P}} \approx 0\), the polarizations tend to align parallel to the interface. Finally, the gradient energy tends to change the direction and magnitude of the polarization. The three energy contributions lead to the topological structure observed in the nanoparticle (Fig. 3a).

Fig. 3 Three-dimensional nanodomain and vortex dynamics in BTO nanoparticle. Isosurface of the spontaneously formed nanodomain arrangements in BTO nanoparticle as obtained from Bragg Coherent Diffraction Imaging. Evolution of the spontaneous polarization distribution a at field E 1 (0 kV cm−1), b at field E 2 (223 kV cm−1), and c at field E 3 (0 kV cm−1) shows that the in-plane components of P 111 are always arranged in a flux-closure (vortex) manner in the virgin state. d Phase-field simulations confirm a mixture of Tetragonal and Monoclinic (T+M) structural phases that accounts for the vortex (clockwise) structure at zero electric field E 1 (0 kV cm−1). e The curl of the axial polarization is characterized by an electric toroidal moment (\({\bf{T}} = 1/{T_0}{\int} {\left( {{\bf{r}} \times {\bf{p}}} \right)} \,{\rm{d}}{V_{{\rm{cell}}}}\), where T 0 is the toroidal moment without field at E 1 , V cell is the volume of the cell located at position r, and p is the local dipole) from phase-field simulations. The disappearance of the toroidal moment indicates a vortex-to-polarization transformation in the nanoparticle37. Under field E 2 (223 kV cm−1), the vortex core is off-centered with a predominant Monoclinic phase. As we decrease the field to remnant E 3 (0 kV cm−1), the vortex core returns to the center of the particle. Scale bars correspond to 60 nm Full size image

Structural phase transition and vortex core transformation path

An external electric field E in the [111] direction was applied to study the influence of E on the domain structure. The recorded Bragg coherent diffraction (Fig. 1 and Supplementary Fig. 3) shows evidence of structural phase coexistence within the nanoparticle and its transformation under fields E 1 = 0 kV cm−1, E 2 = 223 kV cm−1, and E 3 = 0 kV cm−1. Slices of the reconstructed displacements (Fig. 2d) and polarization (Fig. 3a–c) at different depths within the nanoparticle for a given values of E show a heterogeneous distribution of the domains. At the intersection of the domain walls, we observe the formation of a polar vortex (Fig. 3d). The transformation path of the vortex37 is accompanied by a structural phase transition from coexisting tetragonal (T) and monoclinic (M) phases at E 1 to a predominant M structural phase at E 2 . For bulk BTO crystal at room temperature without an external electric field, the stable domain structure is T phase. In bulk, the structure of the domain wall is usually neglected because of its very small thickness (about 5–20 nm). In such cases, the reported thickness of 90° domain wall is ~14 nm. However, in our nanoparticle with size of 160 nm, our simulations indicate that the domain wall is essentially a cross-over region with width of ~10–20 nm and is monoclinic in structure. The signature of this monoclinic cross-over region in reciprocal space is given by the splitting of the Bragg peaks as shown in our measured coherent X-ray diffraction patterns (Fig. 1 and Supplementary Fig. 3). This splitting becomes centrosymmetric at electric field E 2 (223 kV cm−1) since the entire particle becomes predominantly monoclinic (Supplementary Note 3 and Supplementary Fig. 11). The simulated 2D polar distribution in the central slice is plotted in Fig. 3d with color representing P 111 . Figure 3d for field E 1 = 0 kV cm−1 shows the coexistence of T and M phases with a vortex core at the center. For E 2 = 223 kV cm−1, the field induced T to M phase transition leads to an M phase with the vortex core displaced to the edge of the particle. Finally, when the field is decreased to zero (E 3 = 0 kV cm−1), the core returns to the center of the particle but the polarization domain morphology is not the same as that for E 1 = 0 kV cm−1. This difference reflects the non-linearity of the polarization response. We observe an average polarization of 0.18 C m−2 at E 3 , comparable to bulk spontaneous values38 of 0.23 C m−2. This allows us to postulate that the non-linear behavior of the polarization response is truly governed by the local vortex structure and behavior.

Volumetric morphology and evolution of the vortex core

The sensitivity of BCDI to whole volume (Supplementary Fig. 7) allows us to not only track the evolution of competing polarization states under an electric field but also to identify boundaries (domain walls) separating these states and their role in polarization and vortex evolution (Fig. 3d). The vortex core forms a nanorod in the 3D nanoparticle (Fig. 4a, f). Since the diameter of this nanorod falls within the limit of our spatial resolution of 20 nm (as given by the phase retrieval transfer function in Supplementary Fig. 5), we used the variance of the displacement to study the 3D morphology of the vortex core as a 30 nm thick intrinsic paraelectric nanorod (Fig. 4a, b). The variance analysis also allows us to confirm that the vortex core coincides with regions of intersecting domain walls (Supplementary Fig. 8). The mobility of the domain walls under an applied electric field translates to a transformation of the vortex core. To estimate the number of domain walls that intersect to form the vortex core, we count the number of zeros in the angular dependence of the displacement field around the vortex (Fig. 4e). The two zeros of the displacement in Fig. 4e (blue curve) indicate the presence of one nanorod and hence only one vortex core within the FE nanoparticle. In the absence of an external electric field, the nanorod is in the center of the particle as shown in Fig. 4b. When the field is increased to 223 kV cm−1 (E 2 ), the nanorod rotates (Fig. 4a) in the plane as the vortex core moves to the edge of the particle as confirmed by our phase-field simulation results in Figs 2d and 3d. As predicted by our simulations in Fig. 3e, if we continue to increase the field beyond 270 kV cm−1, the vortex core will finally disappear and the nanorod can be thought of as being erased.

Fig. 4 Identification of the domain wall and 3D rendering of nanorod as a defect within the nanoparticle. a Variance of displacement in the nanoparticle under the maximum electric field E 2 (223 kV cm−1), calculated in the vicinity of the domain wall. b Variance of displacement in the nanoparticle under the electric field E 3 (0 kV cm−1) calculated in the vicinity of the domain wall. c Map of displacement values and magnification of the boxed region for the nanoparticle at the slice shown as green plane in f for the state at field E 2 (223 kV cm−1). d Displacement values as a function of position of lineplots in c. e Angular dependence of the displacement field and of the gradient of the displacement measured along the defect line in f with error bars indicating standard deviation over the slices in the particle. f Rendering of the particle under field E 2 shows the 3D nanorod as a defect line whose 2D cross-section corresponds to the vortex core. Scale bars correspond to 60 nm Full size image

Since the summation of Born effective charges allows us to scale the displacements with the local polarization within the particle, Eq. 1 allows us to use the zeros of the displacement gradients (red curve in Fig. 4e) to estimate the nature and number of domain walls and hence FE multiple states for any given slide and component of the polarization within our resolution limit. The regions where the displacement gradient changes sign indicates transition across the domain wall (Fig. 4c–e). Using this analysis, the orientation and behavior of the polarization vector across the domain wall (Supplementary Fig. 9), and from the locations of the first and second zeros in Fig. 4e, we determine a 173 ± 10° domain wall parallel to the spontaneous polarization of the adjacent domains. This is in good agreement with the predicted value of 180° for Bloch walls (Supplementary Note 4 for more information). To compare with our experimental tracking of the vortex structure under an external electric field, our phase-field model in Fig. 3e predicts that at fields above 270 kV cm−1, the toroidal moment disappears leaving the axial polarization as the only non-zero order parameter. This disappearance indicates a vortex-to-polarization transformation10, 37 in the nanoparticle making it useful for NRAM applications. However, for other potentially new applications such as tunable optical behavior, electrically controllable chirality of the toroidal moment is essential (Fig. 5).

Fig. 5 Electrically controllable chirality of the toroidal moment. Projections of the toroidal moment of the ferroelectric displacement (polarization), T x (r), T y (r), and T z (r) when the particle is subjected to a cyclic external electric field: E 1 (0 kV cm−1), E 2 (223 kV cm−1) and back to remnant E 3 (0 kV cm−1). a Shows the projection T x (r), b T y (r), and c T z (r) under field E 1 (0 kV cm−1). At the maximum field of E 2 (223 kV cm−1) the projections T x (r), T y (r), and T z (r) are shown in d, e and f respectively. When the field is returned to E 3 (0 kV cm−1) the projections T x (r), T y (r), and T z (r) are shown in g, h and i respectively. Each projection of the moment can be seen as a new ferroelectric phase within a single FE particle with electrically controllable chirality. The displayed view of the particle is in the plane perpendicular to the vortex core. For more views please see the Supplementary Movie 1. Scale bars correspond to 60 nm Full size image

Chirality of the three-dimensional toroidal moment