Probing atomic-scale symmetry breaking by rotationally invariant machine learning of multidimensional electron scattering

www.nature.com/npjcompumats ARTICLE OPEN Probing atomic-scale symmetry breaking by rotationally invariant machine learning of multidimensional electron scattering 1234567890():,; Mark P. Oxley 1 , Maxim Ziatdinov 1,2 ✉ , Ondrej Dyck 1 , Andrew R. Lupini 1 , Rama Vasudevan 1 and Sergei V. Kalinin 1✉ The 4D scanning transmission electron microscopy (STEM) method maps the structure and functionality of solids on the atomic scale, yielding information-rich data sets describing the interatomic electric and magnetic fields, structural and electronic order parameters, and other symmetry breaking distortions. A critical bottleneck is the dearth of analytical tools that can reduce complex 4D-STEM data to physically relevant descriptors. We propose an approach for the systematic exploration of 4D-STEM data using rotationally invariant variational autoencoders (rrVAE), which disentangle the general rotation of the object from other latent representations. The implementation of purely rotational rrVAE is discussed as are applications to simulated data for graphene and zincblende structures. The rrVAE analysis of experimental 4D-STEM data of defects in graphene is illustrated and compared to the classical center-of-mass analysis. This approach is universal for probing symmetry-breaking phenomena in complex systems and can be implemented for a broad range of diffraction methods. npj Computational Materials (2021)7:65 ; https://doi.org/10.1038/s41524-021-00527-3 INTRODUCTION Functionalities of materials including ferroics1,2, superconductors3, and charge density wave systems4 are governed by the physics of symmetry breaking phenomena. In systems with long-range discrete translation symmetries, these behaviors are readily amenable to neutron and X-ray scattering, providing insight into the minute details of atomic structure, electronic density distribution, and elastic and inelastic vibrational properties5,6. In these systems, the long-range periodicity allows integrating the behaviors over multiple unit cells. Similar approaches can be extended to ordered 2D systems such as surfaces and interfaces, as accessed via low-energy electron diffraction or surface X-ray methods7,8. However, this approach offers only limited applicability to materials such as nanoscale phase-separated oxides, ferroelectric relaxors, and morphotropic phase boundary systems, incommensurate charge- and spin density wave systems, and, more generally, systems with non-uniform ground states. Similarly, the local mechanisms describing the interplay between chemical disorder, including both lattice-preserving substitution and lattice breaking structural defects, and physical functionalities are often unknown. In all these cases, the lack of long-range translational symmetry limits the applicability of classical scattering techniques and requires the development of methods for probing correlated disorders. At the same time, the last several years have seen an exponential growth of atomic-scale electron diffraction in scanning transmission electron microscopy (4D-STEM). The fast electrons in the electron probe are deflected by the electric field within the crystal. Negatively charged electrons are attracted to positively charged nuclei, which are screened by the surrounding electrons, meaning they contain sub-atomic scale components. This variation is most clearly seen in diffraction space, where the center-of-mass (COM) of the convergent beam electron diffraction (CBED) pattern is deflected toward the nuclei. Practically, the atomically sized focused electron beam is used to collect the local (2D) diffraction patterns over a dense spatial grid of (2D) points, producing the 4D-STEM data sets. A unique aspect of this method is that the size of the probe can be below the distance between the scatterers, resulting in very complex local diffraction patterns and encoding minute details of the local scattering potential. Originally, 4D-STEM in its modern form was proposed by Rodenburg as an approach to achieve high spatial resolution9,10, enabling a practical embodiment of the ptychographic idea of Hoppe11,12. However, there were two main difficulties that prevented the widespread adoption of these methods. First, a practical problem was that CCD cameras were not fast or sensitive enough to keep up with the speed of the STEM probe, resulting in long acquisition times creating sample damage and stability problems. The second main problem was that the data sets were too large for existing computer infrastructure and the amount of computation required made it prohibitively expensive. Both of these difficulties have been addressed over the last 4–5 years. Modern computers and their associated storage and datahandling capabilities have improved dramatically in accordance with the well-known Moore’s law. Electron detection capabilities have grown both evolutionarily with incremental improvements in conventional designs and revolutionarily with the advent of direct-electron detectors13–16. Methods other than ptychography have been developed to analyze scanning nanodiffraction data. The position averaged CBED (PACBED) approach has been used primarily to determine specimen thickness17. PACBED has recently been enhanced by the application of deep convolution neural networks to automatically analyze the data sets. Differential phase contrast (DPC) in the STEM was originally proposed in the early 1970s18 and was recently implemented using segmented detectors19. The development of high-speed electron detectors has allowed DPC-STEM 1 Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN, USA. 2Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA. ✉email: ; Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences M.P. Oxley et al. 1234567890():,; 2 to be readily applied. By determining the deflection of the COM of the CBED pattern as a function of probe position, insight can be gained about the local charge densities and fields20 or alternatively the electron scattering potential21. Despite these initial advances and the well-recognized promise of 4D-STEM for the sub-atomic scale exploration of materials properties, progress has been stymied by a lack of analysis tools to convert the 4D-STEM data sets into physically relevant parameters. The vast majority of the work presently relies on using a simple COM. Alternatively, a number of approaches using linear unsupervised dimensionality reduction methods such as principal component analysis (PCA) and non-negative matrix factorization (NNMF) and clustering techniques have been explored and recently have become part of open-source platforms. The applicability of linear separation methods for the analysis of 4D-STEM data sets is limited, stemming from the intrinsic symmetries of the atomic lattice. Linear unmixing methods such as PCA will separate Ronchigram (...truncated)