Abstract text (incl. figure legends and references)

Quantitative measurements of nanoscale electromagnetic fields are of great interest for fundamental research, as well as for the development of next-generation electronic and spintronic devices. Off-axis electron holography (EH) is a well-established transmission electron microscopy (TEM) technique for measuring potentials and fields within and outside materials with nm spatial resolution.[1,2] 4-dimensional scanning TEM (4D STEM) is a rapidly developing technique for measuring potentials and fields, which is usually based on the recording of diffracted intensity on a pixelated detector.[3,4]

Here, we present a quantitative comparison of long-range electrostatic field and potential measurements recorded from the same sample using EH and 4D STEM based on scanning nanobeam electron diffraction. The sample comprised two Au needles, which were loaded into a Nanofactory STM-TEM specimen holder and separated by a distance of 200 nm. A bias of +50 V was applied to one of the needles, as shown in Fig. 1. Surprisingly, measurements recorded using the two techniques revealed that the projected potential and the magnitude of the y component of the projected in-plane electric field (Ey) were much smaller when measured using EH than 4D STEM, whereas the magnitude of the x component of the projected in-plane electric field (Ex) was almost the same. The difference between the two measurements is thought to originate from the effect of a perturbed reference wave (PRW) when using EH, as the long-range electric field from the needles affects the vacuum reference and hence primarily Ey, thereby reducing the potential and field measured using EH. In order to confirm this hypothesis, electrostatic potentials were calculated in various ways, as shown in Fig. 2. A measurement of the potential based on an analytical equation that does not consider the PRW (Fig. 2d) is similar to that measured using 4D STEM (Fig. 2b). The inclusion of the PRW effect into the analytical expression (Fig. 2e) closely reproduces the apparent potential measured using EH. This comparison is consistent with the explanation that the PRW can significantly affect measurements of long-range potentials and fields using EH.

[1] A. Tonomura, Rev. Mod. Phys., 59, 639 (1987)

[2] H. Lichte et al., Ann. Rev. Mat., 37, 539 (2008)

[3] J.N. Chapman et al., Ultramicroscopy, 3, 203 (1978)

[4] N. Shibata et al., Sci. Rep. 5, 10040 (2015)

Fig. 1. Comparison between measurements recorded from two biased Au needles using (a-d) EH and (e-h) 4D STEM. The figures show (a, e) projected electrostatic potential; (b, f) magnitude of the projected in-plane electric field E; (c, g) x component of the projected in-plane electric field Ex (d, h) y component of the projected in-plane electric field (Ey) . The arrows in (b, f) represent the direction of the projected in-plane electric field. The same scales and scale bars are used for the same physical quantities. The units are V∙nm and V for the projected electrostatic potential and projected electric field, respectively.

Fig. 2. Comparison between measurements of projected electrostatic potential around the biased Au needles recorded using (a) EH; (b) 4D STEM. (c) Application of model-based iterative algorithm to the phase image recorded using EH; (d) Application of analytical expression based on line charges; (e) Application of analytical expression based on line charges but including PRW effect. The black dashed lines in the figures represent the outlines of the Au needles. The units are V∙nm for the projected electrostatic potential. The contours have a spacing of 10⁴ V∙nm.