Bend Contour Electron Tomography (BCET): Quantitative strain and topography mapping
Freestanding thin films of quantum materials naturally develop sub-micrometer, nonuniform strain fields that strongly affect their electronic, magnetic, and structural properties in both equilibrium and nonequilibrium conditions. However, current methods to quantitatively resolve these mesoscopic features are primarily restricted to scanning probes, making it challenging for dynamical measurements such as single-shot imaging and femtosecond microscopy. Here, we present a new computational framework which we denote as bend contour electron tomography (BCET), which efficiently converts bend contours in transmission electron microscope images into quantitative two-dimensional maps of strain and topography. By iteratively minimizing a designed loss function between experimental and simulated bend contour images, BCET retrieves both surface morphology and in-plane strain tensor fields without requiring scanning or diffraction mapping. We applied BCET to freestanding SrTiO$_3$ thin films, demonstrating the successful reconstruction of the local strain distribution and curvature field with high fidelity. Our approach provides a quantitative framework for characterizing mesoscale structures in freestanding films using wide-field imaging, opening new avenues to investigate how spatial inhomogeneity governs phase transitions and nonequilibrium dynamics in two-dimensional quantum materials.