edfas.org ELECTRONIC DEVICE FAILURE ANALYSIS | VOLUME 26 NO. 4 8 of this approximation can be visualized in Fig. 1, Panel IIc. In the ray diagram, the probe is focused at the entrance surface of the sample and, because the sample is sufficiently thick, the probe becomes increasingly divergent as it approaches the exit surface. As a result, a single probe function cannot accurately represent a range of diverse states. Relationships describing the limit of the multiplicative approximation have been proposed[9,37,38] and these models predict a value on the order of a few nanometers for commonly used experimental conditions. However, this simplistic picture does not account for dynamical scattering and electron-sample interactions, which modify the phase and modulus of the electron probe, meaning the multiplicative approximation can potentially fail in even thinner samples.[19] To address this problem, an extension to the single slice or 2D ptychography algorithms was introduced.[12,39] It is coined the “multislice” approach, deriving its name from the multislice image simulation technique, which treats the sample in a similar fashion[40] and models the specimen as a number of slices or layers, such that the multiplicative approximation remains valid for each individual slice. The algorithm then solves for a series of transmission functions rather than one. Accordingly, the exit wave of the first layer is calculated as the product of the probe and the transmission function, as we introduced for a 2D ptychography algorithm. It is then propagated to the next object slice and is used as the probe function for the next slice and so on until the final slice, as illustrated in Fig. 1, Panel IIc. Finally, the exit wave is propagated using a Fourier transform to calculate the intensity at the detector plane. The back calculation proceeds in a similar fashion. This approach was first demonstrated by examining two overlapping carbon nanotubes (CNTs).[41] A traditional STEM image would be a projection of the entire sample showing the two CNTs simultaneously; where the depth sampling of the ptychographic reconstruction was sufficiently fine that the upper and lower CNTs could be resolved separately and with sufficient spatial resolution to discern the structure of each (Fig. 3, Panel I). Importantly, this technique has been demonstrated to work on samples that are “thick” and would fail to reconstruct accurately in a single slice or 2D ptychography algorithm,[19] enabling ptychography to be used on a much broader set of samples. Thus far, the multislice approach has been used to characterize the depth dependent lattice variations due to strain and polarization around a dislocation core in SrTiO3 (Fig. 3, Panel II) [42] as well as the distribution of oxygen vacancies in a zeolite.[23] (a) (c) (b) Fig. 3 Panel I: Optical sectioning of two carbon nanotubes overlapping in projection. The phase images reconstructed at different depths can resolve the CNTs individually while reconstructing the lattice. Scale bars are equal to 10 nm. Reproduced under the terms of a Creative Commons Attribution 4.0 International License.[41] Panel II: Optical sectioning of a kinked edge dislocation in SrTiO3. The sum of all phase images (a), along with individual slices at 2.4nm, 6.4nm, and 12.0nm in depth (c). Schematics of the kink configuration (b). Scale bars are equal to 0.5 nm. Reproduced under the terms of a Creative Commons Attribution 4.0 International License.[42]
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