Uncovering polar vortex structures by inversion of multiple scattering with a stacked Bloch wave model

TL;DR

This paper introduces a stacked Bloch wave model to analyze electron diffraction in thick samples, enabling the extraction of local structural properties and vortex topologies in complex crystalline materials.

Contribution

The paper develops a fast, reduced-parameter stacked Bloch wave method for modeling multiple scattering in thick samples with varying structure along the beam.

Findings

Successfully disentangled tilt from polarization in multilayer samples.

Revealed complex vortex structures in PbTiO3 layers.

Applied to large fields of view with high accuracy.

Abstract

Nanobeam electron diffraction can probe local structural properties of complex crystalline materials including phase, orientation, tilt, strain, and polarization. Ideally, each diffraction pattern from a projected area of a few unit cells would produce clear a Bragg diffraction pattern, where the reciprocal lattice vectors can be measured from the spacing of the diffracted spots, and the spot intensities are equal to the square of the structure factor amplitudes. However, many samples are too thick for this simple interpretation of their diffraction patterns, as multiple scattering of the electron beam can produce a highly nonlinear relationship between the spot intensities and the underlying structure. Here, we develop a stacked Bloch wave method to model the diffracted intensities from thick samples with structure that varies along the electron beam. Our method reduces the large parameter space of electron scattering to just a few structural variables per probe position, making it fast enough to apply to very large fields of view. We apply our method to SrTiO$_3$/PbTiO$_3$/SrTiO$_3$ multilayer samples, and successfully disentangle specimen tilt from the mean polarization of the PbTiO$_3$ layers. We elucidate the structure of complex vortex topologies in the PbTiO$_3$ layers, demonstrating the promise of our method to extract material properties from thick samples.