Absolute scale off-axis electron holography of thin dichalcogenide crystals at atomic resolution

High-resolution transmission electron microscopy (HRTEM) is an enormously powerful technique for the investigation of material structures at atomic resolution. In addition, off-axis electron holography allows information to be obtained about electromagnetic fields inside and around the object. However, due to electron diffraction in the sample and subsequent electron optical imaging, the extraction of quantitative information from recorded images is by no means a trivial task, as information related to the object structure, electromagnetic fields and microscope properties are encoded in the recorded signal in a highly complex manner. The comparison of experimental data with accurate simulations, ideally on the same absolute scale, is a common approach in HRTEM to extract pure information about objects and fields. Prior to this work, absolute scale agreements had only been achieved manually in a few cases for HRTEM, but had not been demonstrated for off-axis electron holography. In this work, an automated optimization procedure is developed that enables the determination of unknown or only partially known experimental parameters directly from high-resolution electron wave functions measured using off-axis electron holography. The procedure is applied to the study of two-dimensional WSe$_{2}$, yielding one of the most precise local specimen orientation measurements that has been achieved in TEM for ultra-thin samples. Extensive tests on simulated data reveal that diffraction-related parameters, such as specimen tilt or absorption, can be determined unambiguously with extremely high accuracy and precision, even in the presence of realistic recording noise. In contrast, coherent aberration coefficients cannot be determined unambiguously from electron wavefunctions of periodic objects. By applying the procedure to a recorded off-axis electron hologram of five-layer-thick WSe$_{2}$, absolute scale agreement between experiment and simulation is achieved, which is limited primarily by the experimental recording noise. The automated procedure developed in this work is fast and computationally cheap. In comparison to previous manual optimizations, it is less prone to human error and bias. This work represents a significant advance for quantitative electron microscopy in general, as the procedure is not limited to off-axis electron holography, but can also be applied to HRTEM and other techniques