Quantitative measurements of nanoscale electrostatic and mean inner potentials in crystals by electron beam refraction using
CBED
and
DPC
Authors: Wu Mingjian , Spiecker Erdmann 
Published: December 2016

Probing nanometer scale electrostatic and mean inner potential (MIP) and establishing structure–properties relationship at this length scale in advanced functional materials are not only of fundamental interest but also of technological relevance and importance, especially for materials with application in the ever miniaturizing electronics. As pure phase object, these potentials can only be “seen” by phase of the wave of probing radiation/particle; and electron microscopy (EM) based phase contrast methods are the most suitable, if not the only, tool for this purpose. Here, MIP is, by definition, the local volume (or unit cell) average of the Coulomb potential of the sample, which can be theoretically calculated or accurately measured (cf. Ref [1]). While the measurable, the local scattering potential being probed by the high‐energy electrons, can be considered (assuming homogeneous potential in projection) superposition of MIP and electrostatic potential, which is the deviation of local Coulomb potential from the MIP of bulk from the perspective of measurement.

Differential phase contrast (DPC) using electrons based on scanning transmission EM (STEM) has seen a renaissance of interest mainly due to the recent demonstration of probing electrostatic potential in real space at atomic resolution
[3]
, beside its demonstrated robustness in studying magnetic properties in the past decades, e.g. Ref. [2]. The DPC‐STEM signal composes the difference of intensity from opposite quadrants from a segmented annual bright field detector. In principle, a DPC‐STEM experiment effectively records, under kinematic approximation, a two dimensional vector of electron beam deflection (i.e., phase gradient) and an absorption/amplitude contrast signal (i.e., sum intensity of all quadrants) simultaneously at each probing position/pixel which raster over the specimen at desired field of view (FOV). Therefore, DPC‐STEM is expected to show advantages of 1) direct interpretability, 2) sharp features in focus, 3) flexible FOV and 4) simultaneous phase and amplitude contrast, compared to interference‐ (e.g. electron holography) and propagation‐ (e.g. Fresnel contrast) based phase contrast methods. Despite these simple descriptions, there are, however, very limited applications of DPC‐STEM in the quantitative and systematic study of electrostatic potential and MIP in crystals. Moreover, as STEM based method using convergent illumination, convergent beam electron diffraction (CBED) patterns under identical condition are indispensable to evaluate the DPC‐STEM results quantitatively.

In this contribution, we focus on experimental studies of quantitative measurements of MIP from crystal wedges by compiling the results from DPC‐STEM and CBED raster arrays under identical diffraction conditions. The experiments were performed on a Titan Themis
3
TEM equipped with Cs correctors, working at 200 kV in μ‐probe STEM mode. The camera length and probe convergence angle are carefully calibrated and chosen to balance resolution (to about 1 nm) and detection sensitivity. Figure 1 shows the representative results of measuring MIP from a cleaved 90° Si wedge. Under quasi‐kinematic condition (cf. Fig. 1a), remarkable beam refraction is observed when the probe is moved from vacuum to inside sample and the refraction angle is constant to a considerablely large sample thickness. Meanwhile the total intensity decreases homogeneously within the beam disk and exponentially as a function of the local thickness, as expected. The refraction angle measured from a CBED raster array is quantified to sub‐pixel accuracy, which corresponds to a MIP of Si to be 12.52±0.21 V. The calibrated DPC‐STEM signals deliever very close mean value of the magnitude of refraction, but with much greater variance, due to the orders‐of‐magnitude shorter dwell time and thus noisier signal (Fig. 1g,h). The same measurements have been carried out with a 90° GaAs wedge, from which we derived the MIP of GaAs to be 14.10±0.33 V from CBED measurement and a very close mean value from DPC‐STEM data. The MIP values agrees very well with previous measurements based on electron holography and theoretical calculations
[1]
.

Further examples on the application of the method for mapping electrostatic potentials in semiconductor nanostructures, as well as attempts to map piezo‐electric potentials will be presented at the conference.