Revealing the three-dimensional arrangement of polar topology in nanoparticles

Supplementary Data Codes

The data and source codes for the paper [C.J., J.L., H.J., J.O., H.B., K.-J.G., J.S., S.-Y.C., S.P., L.B., and Y.Y., "Revealing the three-dimensional arrangement of polar topology in nanoparticles", Nat. Commun. 15. 3887, (2024)] are posted below.

Overview

In the early 2000s, low dimensional ferroelectric systems were predicted to have topologically nontrivial polar structures, such as vortices or skyrmions, depending on mechanical or electrical boundary conditions. A few variants of these structures have been experimentally observed in thin film model systems, where they are engineered by balancing electrostatic charge and elastic distortion energies. However, the measurement and classification of topological textures for general ferroelectric nanostructures have remained elusive, as it requires mapping the local polarization at the atomic scale in three dimensions. Here we unveil topological polar structures in ferroelectric BaTiO3 nanoparticles via atomic electron tomography, which enables us to reconstruct the full three-dimensional arrangement of cation atoms at an individual atom level. Our three-dimensional polarization maps reveal clear topological orderings, along with evidence of size-dependent topological transitions from a single vortex structure to multiple vortices, consistent with theoretical predictions. The discovery of the predicted topological polar ordering in nanoscale ferroelectrics, independent of epitaxial strain, widens the research perspective and offers potential for practical applications utilizing contact-free switchable toroidal moments.

System Requirements

1. All software dependencies and operating systems (including version numbers)

Software: MATLAB R2021a – academic use

Operating systems: CentOS Linux 7

2. Versions the software has been tested on

MATLAB R2021a – academic use

3. Any required non-standard hardware

There are no non-standard hardware

Repositary Contents

1. Raw Experiment Data

Folder : 1_Raw_data

This folder contains raw experimental projections before denoising as well as their corresponding angles for Particle 1 and Particle 2.

File name

: ImageData.mat : raw experimental projections

: Angles.mat : corresponding angles

2. Experiment Data

Folder: 2_Measured_data

This folder contains experimental projections after denoising and alignment as well as their corresponding angles for Particle 1 and Particle 2.

File name

: projections.mat : experimental projections after denoising and alignment

: zeroprojections.mat : zero-degree projections at the beginning, in the middle, and at the end of the experiment 

: angles.mat : corresponding angles

3. Reconstructed 3D Volume

Folder: 3_Final_reconstruction_volume

This folder contains the 3D volume (before and after orientation) of the BaTiO3 nanoparticles (Particle 1 and Particle 2) reconstructed from GENFIRE algorithm.

The original GENFIRE URL is below.

(https://www.physics.ucla.edu/research/imaging/dataSoftware.html)

File name

: volume_Particle1.mat : Reconstructed 3D volume of Particle 1

: volume_Particle2.mat : Reconstructed 3D volume of Particle 2

: reorientedvol_Particle1.mat : Reoriented volume along crystallographic direction of Particle 1

: reorientedvol_Particle2.mat : Reoriented volume along crystallographic direction of Particle 2

: Up_Particle1.mat : Rotation matrix for rotating along crystallographic direction of Particle 1

: Up_Particle2.mat : Rotation matrix for rotating along crystallographic direction of Particle 2

4. Experimental Atomic Structure

Folder : 4_Final_atomic_structures

This folder contains the final 3D atomic model and chemical species (i.e. type 1 for Ti atom, type 2 for Ba atom) of the BaTiO3 nanoparticles.

File name

: Final_atomic_model_Particle1.mat : Final 3D atomic model and chemical species of Particle 1 (after orientation along crystallographic direction)

: Final_atomic_model_Particle2.mat : Final 3D atomic model and chemical species of Particle 2 (after orientation along crystallographic direction)

5. Data analysis

Folder : 5_Data_analysis

This folder contains the analysis of creating Ti atomic displacement field and its topological analysis.

1. Run the code Main_1_Ti_atomic_displacement_field.m to compute Ti atomic displacement fields for all Ti atoms surrounded by 8 nearest neighbor Ba atoms based on bcc fitting and calculate tetragonality map (c/a ratio) by using local tetragonal fitting.

2. Run the code Main_2_Obtain_Topological_Charge.m to conduct topological analysis (i.e. helicity, winding number, pontryagin charge density, skyrmion number, monopole analysis and hopf invariants.

If you use any of the above data or source codes in your publications and/or presentations, our paper should be properly cited: C.J., J.L., H.J., J.O., H.B., K.-J.G., J.S., S.-Y.C., S.P., L.B., and Y.Y., "Revealing the three-dimensional arrangement of polar topology in nanoparticles", Nat. Commun. 15. 3887, (2024)If you have any questions regarding the above data or source codes, please contact Yongsoo Yang, Department of Physics, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, Korea. Email: [e-mail]