The differential phase contrast uncertainty relation: Connection between electron dose and field resolution

Differential phase contrast (DPC) microscopy is a STEM imaging technique, which is used to measure magnetic and electric fields of mesoscopic and nanoscopic dimensions, i.e. interatomic distances (Chapman et al. 1978; Chapman et al. 1981; Chapman, 1984; Chapman et al. 1985; Chapman et al. 1997; Lohr et al. 2012; Shibata et al. 2015; Bauer et al. 2014; Carvalho et al. 2016; Lohr et al. 2016; Mueller-Caspary et al. 2019a,2019b; Mueller-Caspary et al. 2018; Mueller-Caspary et al. 2017; Mueller-Caspary et al. 2014; Winkler et al. 2020; Toyama et al. 2020). In this paper we will demonstrate that the electron dose per pixel deposited on the specimen is decisive to the precision and resolution of measurements of a field's local strength. Relations are given which connect a given electron dose per pixel to the fundamentally achievable precision to which the specimen's interaction with the electrons may be determined, taking into account quantum mechanical considerations. Vice versa, given a certain required precision, the required dose per pixel can be easily predicted for reliable measurements of a desired property. First, these relations are given for the case of a continuous, i.e. non-pixelated, detector followed by simulations which show that the same relations hold for pixelated detectors. Then, the achievable precision for detectors with different pixel counts in combination with different camera lengths is discussed and the maximum measurable field amplitude per set-up is determined. Finally, the effect of inhomogeneities within the diffraction disk is discussed and possible deviations from the derived relations are considered. We also demonstrate that Heisenberg's uncertainty relation determines the possible field resolution in differential phase contrast microscopy, and that the achievable local field resolution is a function of the applied electron dose per pixel.