Four-dimensional scanning transmission electron microscopy (4D--STEM) is a modern operating mode of a transmission electron microscope in which a focused electron probe is rastered across the sample and the diffraction pattern is recorded at each position.The resulting diffraction patterns can be analyzed to obtain a wealth of local structural information, such as deformation or strain, changes in symmetry or lattice distortions, orientation of a crystal lattice, as well as to measure electric and magnetic fields. More advanced analyses, i.e. ptychography, can also extract structural information at a spatial resolution finer than the size of the electron probe.

Several challenges arise in realizing these measurements: First, the sheer number of diffraction patterns recorded in a 4D-STEM experiment leads to computational challenges and puts demands on the complexity of the algorithms used to recover the structural information. Second, experimental considerations often strictly limit the number of electrons in each of the diffraction patterns, which can be mitigated through robust analysis approaches or by de-noising that takes advantage of the high dimensionality of the data. Most critically, all of the structural measurements described above are effectively trivial in the limit of thin and weakly scattering materials but become rather challenging when analyzing diffraction from a thick sample where multiple scattering effects are present.

In this work, we will explore several means to mitigate these challenges. First, to handle the large quantities of data and the low number of electrons recorded by modern detectors operated at their full speed, we will show a hyperspectral denoising method based on total variation denoising and show its application to 4D--STEM datasets.

The bulk of this work, however, will focus on the latter challenge: dynamical scattering. In 4D--STEM measurements of local strain or deformation, dynamical scattering causes unwanted contrast inside of diffraction disks which hinders accurate determination of the lattice. To mitigate this, we demonstrate a method for imprinting known contrast into the diffraction disks to improve the precision of the measured lattice. In measurements of the local orientation of the crystal, multiple scattering causes the diffraction disk intensities to vary in a highly nonlinear way as the crystal tilts, and as a function of the thickness of the crystal. We present a hybrid pattern-matching and simulation-matching algorithm for precisely determining both the orientation and thickness of a crystalline sample from 4D--STEM measurements. Finally, many polar structures of technological interest exist only under exacting electrical and mechanical boundary conditions and so can only be studied as a thick and heterogeneous sample. To measure polarization structures in such materials, we construct a dynamical scattering model for the system and demonstrate an optimization procedure which recovers local polar order from large-area scans of a thick multilayer sample.