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The 4D scanning transmission electron microscopy (STEM) method has enabledmapping of the structure and functionality of solids on the atomic scale,yielding information-rich data sets containing information on the interatomicelectric and magnetic fields, structural and electronic order parameters, andother symmetry breaking distortions. A critical bottleneck on the pathwaytoward harnessing 4D-STEM for materials exploration is the dearth of analyticaltools that can reduce complex 4D-STEM data sets to physically relevantdescriptors. Classical machine learning (ML) methods such as principalcomponent analysis and other linear unmixing techniques are limited by thepresence of multiple point-group symmetric variants, where diffractograms fromeach rotationally equivalent position will form its own component. Thislimitation even holds for more complex ML methods, such as convolutional neuralnetworks. Here, we propose and implement an approach for the systematicexploration of symmetry breaking phenomena from 4D-STEM data sets usingrotationally invariant variational autoencoders (rrVAE), which is designed todisentangle the general rotation of the object from other latentrepresentations. The implementation of purely rotational rrVAE is discussed asare applications to simulated data for graphene and zincblende structures thatillustrate the effect of site symmetry breaking. Finally, the rrVAE analysis of4D-STEM data of vacancies in graphene is illustrated and compared to theclassical center-of-mass (COM) analysis. This approach is universal for probingof symmetry breaking phenomena in complex systems and can be implemented for abroad range of diffraction methods exploring the 2D diffraction space of thesystem, including X-ray ptychography, electron backscatter diffraction (EBSD),and more complex methods.

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