Bragg Interferometry of Moiré Superlattices: From Geometric Phase Principles to Atomic Reconstruction

The emergence of moir\'e superlattices formed by twisting and stacking two-dimensional materials has created a need for characterization techniques capable of mapping sub-angstrom atomic reconstruction across micron-scale fields of view. This review surveys a suite of methodologies developed to extract geometric phases in electron microscopy and X-ray spectroscopy, with a primary focus on the interference of overlapping Bragg reflections in the dark field. We provide a comparative analysis of some established techniques, including geometric phase analysis, converged beam electron diffraction holography, and various ptychographic paradigms, culminating in a discussion of Bragg interferometry for measuring interlayer displacement fields and strain in moir\'e materials. We demonstrate the utility of Bragg interferometry through case studies of twisted bilayer and trilayer graphene as well as transition metal dichalcogenide moir\'e systems. Special attention is given to the unique advantages of this dark-field interferometric method for probing buried interfaces and encapsulated heterostructures, as well as the inherent challenges of interpreting incomplete phase information in the presence of dynamical scattering. By examining the physical principles underlying these approaches, this review highlights the conceptual similarities and practical trade-offs involved in high-resolution structural mapping of materials in which the interplay between structural relaxation and electronic behavior defines a scientific frontier at the nexus of modern condensed matter physics, nanomaterials engineering, and interfacial chemistry.