Experimental study

Momentum-resolved 4D-STEM data of a PbZr_0.2Ti_0.8O₃ specimen have been collected by scanning a focused aberration-corrected electron probe across the sample and recording a full diffraction pattern at each scan position with an EMPAD detector³¹. The direction of the incident beam was along the [010] zone axis of the crystal. An FEI Titan G² scanning transmission electron microscope (STEM) equipped with an aberration corrector for the illumination was operated at 200 keV electron energy. The STEM pixel size was set to 35 pm, and a dwell time of 1 ms was chosen for each STEM pixel, synchronized with the frame recording of the EMPAD detector. The probe semi-convergence angle has been measured to 24.6 mrad, and the pixel size in the recorded diffraction patterns was 0.88 mrad.

Initial tilt (θ_x = 0.92 mrad, θ_y = − 0.36 mrad) and thickness (20 nm)²¹, and the L₁ loss was minimized (see Supplementary Fig. ⁴⁾ as in the theoretical study. The whole recorded diffraction pattern was subject to loss calculation up to a detector-imposed maximum scattering angle of 57 mrad. Note that the actual multislice inversion extended much further up to a scattering angle of 110 mrad. Spatial coherence parameters were initialized by sampling a Gaussian with FWHM of one Ångstrom. The defocus offset of the temporal coherence was initialized with zero. Defocus and all other aberrations were initialized to zero, too. Using the recovered phase grating from a single epoch, an atomic model was constructed, again periodic in beam direction. To refine the initial thickness estimate, the loss was calculated for different thicknesses covering a range of ± 2.5 nm, and the minimum was taken (see Supplementary Fig. ⁵⁾. The model already captured the ferroelectric polarization, however only qualitatively, as the measurements did not agree with the literature values as summarized in Supplementary Table ¹ for comparison. Because the oxygen atom near the (ZrTi) column was invisible, it was placed on the column. Then, the atomic positions were optimized, first for a model without frozen phonons, but with potentials taking the absorptive form factors¹⁷ as well as a Debye-Waller damping into account. This comprises a parametrized single state model for the specimen, which sometimes leads to inaccurate oxygen and (ZrTi) column positions, respectively, as shown in Supplementary Fig. ⁶. Other parameters, namely probe positions, aberrations, coherence parameters and crystallographic tilt were optimized alongside the atomic positions. These parameters were kept when switching to a full frozen phonon reconstruction, while the atomic positions were reverted to the starting positions. Using Adam and an initial learning rate of 0.05, we optimized for 80 more epochs. All parameters except for the atomic positions were fixed for the first 20 epochs. By this model, which incorporates the exact inversion of the contemporary scattering theory in the presence of TDS and finite coherence, all atomic columns have been reconstructed correctly, as seen in Fig. 4. Also, the loss dropped by approximately 13% as compared to the Debye-Waller based reconstruction scheme with the specimen in a single state.