Imaging and structure analysis of ferroelectric domains, domain walls, and vortices by scanning electron diffraction

High-energy electrons' sensitivity to local structure is leveraged in transmission electron microscopy (TEM) to study structure-property relations. Advances like high dynamic-range direct electron detectors (DEDs) in 4D-STEM significantly enhance information density, enabling spatially resolved diffraction imaging. However, deconvoluting the diverse phenomena contributing to electron scattering remains a challenge, often hindered by low signal-to-noise ratios and overlapping signals. Machine learning, particularly convolutional autoencoders (CAs), offers a promising avenue for disentangling these complex signals. This research utilizes 4D-STEM and a custom-designed CA to analyze the nanoscale structural features of Er(Mn,Ti)O₃, a uniaxial ferroelectric oxide. The goal is to statistically separate and interpret structural properties of ferroelectric domains, domain walls, and vortex structures, achieving nanoscale spatial resolution while improving structure-property correlations and enabling automated scientific experiments. The use of a CA with custom regularization strategies allows for the disentanglement of structural properties from other contributing factors, thereby increasing the accuracy and reliability of the results. This study aims to overcome the limitations of traditional analytical methods and provide a more powerful and efficient approach to the analysis of 4D-STEM data.

Numerous studies have explored the use of TEM and related techniques for studying ferroelectric materials. Advances in direct electron detectors and 4D-STEM have improved the resolution and information content of these measurements, but challenges remain in effectively analyzing the data. Previous works have demonstrated the application of machine learning to various microscopy tasks, including image segmentation and data reduction. However, the application of custom-designed CAs with bespoke regularization for the disentanglement of nanoscale structural features in ferroelectrics, particularly focusing on the separation of intrinsic structural information from extrinsic effects, remains a novel approach. The existing literature on the structural and electric properties of Er(Mn,Ti)O₃ provides a solid foundation for this study. The understanding of improper ferroelectricity, domain wall formation, and vortex structures in this material makes it an ideal model system for testing the efficacy of the proposed methodology. This study builds upon previous work by applying advanced machine learning techniques to extract more detailed information from the high-resolution data obtained through 4D-STEM.

The study employed 4D-STEM on Er(Mn,Ti)O₃ single crystals, grown using a pressurized floating-zone method. Lamellas, prepared using focused ion beam (FIB) milling, were analyzed. SED measurements were performed using a Jeol 2100F TEM at 200 kV with a Merlin 1S DED. A 2 nm electron beam with a convergence angle of 9 mrad raster-scanned a 256 x 256 grid with a 1.4 nm step size and a 50 ms dwell time. Data analysis involved a custom-designed CA in PyTorch. The CA architecture included an encoder, embedding layer, and decoder composed of ResNet blocks, convolutional layers, max-pooling layers, and upsampling layers. The model used a mean squared error (MSE) loss function, supplemented by L1 activity regularization to promote sparsity. For domain wall analysis, contrastive similarity and activation divergence regularizations were added to further improve disentanglement. The model was trained using ADAM optimization. Simulated diffraction patterns, generated using a Python multislice code, were compared to experimental data to validate the findings. The center-of-mass (COM) analysis of the diffraction patterns provided additional information about domain-dependent scattering. The combination of these techniques allowed for the separation and identification of scattering signatures from different structural features (domains, domain walls, and vortices) and extraneous contributions.

The COM analysis revealed a domain-dependent shift along the [001] axis, correlating with polarization orientation. The custom CA effectively disentangled features in the diffraction patterns, separating those associated with the ferroelectric domains, domain walls, and vortex textures from extrinsic contributions (e.g., thickness variations). One channel of the CA’s embedding showed sharp contrast between 180° domains, with variations in the 004 and 004 reflections. Comparison with simulated diffraction patterns confirmed that these intensity variations are linked to atomic displacements and polarization direction. The CA also identified distinct scattering signatures for head-to-head and tail-to-tail domain walls within vortex structures. These results demonstrate the sensitivity of the method to both the crystallographic structure and charge state of domain walls. The high spatial resolution achieved (limited by the 2 nm electron beam spot size) allowed for detailed mapping of these features within the ferroelectric material. The successful disentanglement of various features showcases the power of the CA for analyzing complex diffraction data and its ability to provide insights into subtle structural variations. The use of custom regularization techniques is key to enabling this detailed analysis and achieving high accuracy in the extraction of meaningful information.

This work presents a significant advancement in the analysis of 4D-STEM data for ferroelectric materials. The custom-designed CA successfully addresses the limitations of traditional analytical methods by disentangling complex scattering phenomena and isolating contributions from different structural features. The ability to separate intrinsic structural information from extrinsic effects significantly enhances the reliability and accuracy of nanoscale characterization. The successful identification and visualization of head-to-head and tail-to-tail domain walls highlights the method's sensitivity to both the crystallographic structure and electronic charge state. This approach has broad implications for studying other materials with complex nanoscale structures and opens new possibilities for understanding structure-property relationships. The development of this sophisticated machine learning approach combined with 4D-STEM provides a powerful tool for accelerating materials discovery and optimizing device performance.

This study demonstrates a novel and powerful method for imaging and characterizing ferroelectric domains, walls, and vortices at the nanoscale using 4D-STEM and a custom CA. The method successfully disentangles complex scattering signatures, allowing for precise mapping of structural features and their properties. This approach is highly versatile, capable of being adapted to other materials systems. Future research could explore the method's application to other ferroelectric materials, different types of domain walls, and other high-dimensional imaging modalities. The use of automation and increased imaging size would further enhance the efficiency and scope of this technique.

The current study focuses on a specific ferroelectric material, Er(Mn,Ti)O₃. While the method's principles are broadly applicable, the CA model needs to be retrained for different materials systems. The computational cost of training the CA can be significant, although optimized training strategies were employed in this research. The interpretation of the disentangled features may require additional knowledge and validation from complementary experimental techniques.