Abstract text (incl. figure legends and references)

Ferroelectrics remain an important research field due to a wide range of possible nanoelectronic devices, e.g., high electron mobility transistors or ferroelectric tunnel junctions. Thus, (S)TEM methods that allow for the quantitative mapping of polarisation-induced electric fields down to the unit-cell scale are desirable to characterize and fundamentally understand such devices.

Nowadays, ultrafast cameras allow for high momentum and spatial resolution in aberration-corrected STEM. Especially, momentum-resolved STEM (MR-STEM) enabled first moment-based imaging of subatomic electric fields in thin specimens [1,2]. Recently, polarisation-induced fields in semiconductors were mapped by unit-cell averaged (uc.a.) MR-STEM data [3]. This method assumes that long-range components of the electric field are dominant over atomic ones in the uc.a. first moments. The validity of this assumption depends on crystal symmetry and experimental parameters such as specimen thickness or tilt, leading to a systematic error δ whose magnitude needs to be addressed in simulations to allow for a quantitative interpretation of mesoscopic electric field measurements.

Here, MR-STEM of non-ferroelectric SrTiO₃ (STO) and ferroelectric PbZr_0.2Ti_0.8O₃ (PZT) is considered in comprehensive experimental and simulation studies. A unit cell segmentation is performed to yield the uc.a momentum transfers P. Ideally, this corresponds to a potentially present local polarisation-induced electric field, however, a detailed assessment of δ reveals systematic errors of at least the same magnitude. First, we determine the mistilt from [100] zone axis and the in-plane rotation from Kikuchi bands. This exhibits strong domain contrast in PZT, too, which requires a careful interpretation of the apparent electric field contrast among domains determined from uc.a P, because also δ varies strongly in dependence of tilt. A striking example is STO, where the uc.a P takes partly large values although no internal electric fields are present.

The experimental uc.a P are compared to thickness- and tilt-dependent multislice simulations for each PZT domain with the experimental conditions as input. Using an isolated atom model with periodic boundary conditions in STEMsim [4], we find that all non-zero uc.a. first moments arise from dynamical scattering and symmetry breaking (mistilt and atom displacements in PZT). We finally address sources of ambiguities in measuring the momentum space origin arising from perturbed reference data, similar to perturbed reference waves in electron holography.

For non-ferroelectric STO, Fig. 1 shows that the uc.a. P varies with tilt and thickness of the sample. Here, only the systematic error δ can be the cause. For ferroelectric PZT, Fig. 2 shows that the uc.a. P gives domain contrast, that could erroneously be interpreted as different polarisation-induced electric fields. However, the contrast is mainly caused by crystallographic tilt and rotation.

To conclude, the contrast in unit-cell averaged first moments is mainly caused by structural effects and dynamical scattering and not by a polarisation-induced electric field.

Fig. 1: a) P, b) uc.a. P and c) tilt of an STO data set.

Fig. 2: a) P, b) uc.a. P and c) tilt of a PZT data set.

[1] Nat. Commun. 5 p. 5653 (2014)

[2] Ultramicroscopy 178 p. 62 (2017)

[3] Phys. Rev. Lett. 122 p. 106102 (2019)

[4] Mat. 120 p. 1316 (2007)

[5] Funding: Initiative and Network Fund (Helmholtz), contract VH-NG-1317