Abstract text (incl. figure legends and references)

Quantitative electric field mapping from mesoscopic scales down to sub-atomic distances is of fundamental interest with respect to materials science and device design. However, the direct probing of, e.g., polarisation-induced electric fields in a specimen via TEM is currently hampered by dynamical scattering. Consequently, quantifying intrinsic electric fields occurring in thicker specimens remains a challenge, for which a prerequisite is the quantification of electric fields in the absence of dynamical scattering, and obtaining consistency among different TEM methodologies. In this work, the stray electric fields produced by two biased gold needles (Fig. 1a) employing momentum-resolved STEM and electron holography (EH) have been studied in comprehensive simulations and were compared to experiments. Via finite element (FE) simulations, the 3D electric potential V was calculated (Fig. 1b). By projection along the z-direction, the projected potential Vpz is obtained (Fig. 1c). Subsequently, the projected electric field Ep is given by the gradient of Vpz (Fig. 1d) in x- or y-direction. This was done for different geometries, e.g., relative needle shifts and tip curvatures. For example, the line profiles in Fig 1e taken along the half axis show that a relative needle shift reduces and shifts the maximum significantly. The impact of further parameters such as needle shape and gap distance are also elucidated, as well as the impact of propagation through the large-scale electric potential for different acceleration voltages. The reliability of the simulation was examined by comparison with a STEM experiment in which Ep was measured using the centre of mass (COM) method where a respective line profile was extracted (Fig. 1f). Considering slightly deviating geometry parameters, i.e., the inaccurately known 3D tip shape, simulation and experiment are in good agreement. By experimentally probing Ep under alike conditions via EH, a discrepancy of about 50% between COM and EH became apparent. The EH geometry in Fig. 2a, however, suggests that the reconstructed phase is significantly affected by the perturbed reference wave effect (PRE), opposite to STEM where the field-free reference was recorded subsequently with unbiased needles. To numerically quantify the PRE in EH, Vpz gained from FE simulations (Fig. 2a) has been divided into two regions O and R of width w. In EH, the electron beam transmits both regions which are brought to interference via a biprism. Effectively, the measured projected electric potential in EH within the interference region is then given by the difference of Vpz in O and R. Ep is calculated from the gradient of the reconstructed phase, which would ideally equal Fig. 2b and yield the circled profile in Fig. 2c. However, the PRE causes the reconstructed Ep to be reduced by 50V as shown by the crosses in Fig. 2c which agree well with the EH experiment. Finally, Fig. 2d demonstrates that the experimentally measured differences between COM and EH are well reproduced by the FE simulation if the PRE is included. Consequently, COM measurements in this case provide a more direct access to absolute electric fields due to the subsequent unbiased reference measurement, whereas EH is quantitatively understood by accompanying FE simulations, at least for the EH geometry used here.

[K. M.-C. acknowlegdes support under contract from the Helmholtz-Association and from moreSTEM VH-NG 1317]