Quantitative comparison of long-range electric field measurements using off-axis electron holography and 4D-STEM via differential phase contrast

Phase contrast techniques in the transmission electron microscope (TEM), such as off-axis electron holography (OAEH) and four-dimensional scanning TEM (4D-STEM), are widely utilized for mapping electromagnetic fields both within and surrounding nanoscale materials. In this study, the two techniques are used to measure long-range electrostatic potentials and electric fields generated by electrically-biased colinear conducting needles. The results are compared between the two techniques and with a theoretical model. The experimental measurements obtained using OAEH and 4D-STEM via differential phase contrast reveal discrepancies in the magnitudes and distributions of the electric fields surrounding the needles. A comparison of both approaches with a theoretical model reveals that the discrepancy results from perturbation of the reference wave in OAEH by the highly extended electric field outside the needles, leading to an underestimate of the electrostatic potential when using OAEH. In contrast, the 4D-STEM measurements are more directly interpretable. We provide a theoretical background for the OAEH results, which fully explains and supports the findings.