edfas.org ELECTRONIC DEVICE FAILURE ANALYSIS | VOLUME 26 NO. 4 4 EDFAAO (2024) 4:4-11 1537-0755/$19.00 ©ASM International® FOUR-DIMENSIONAL SCANNING TRANSMISSION ELECTRON MICROSCOPY: PART III, PTYCHOGRAPHY Aaron C. Johnston-Peck and Andrew A. Herzing National Institute of Standards and Technology, Gaithersburg, Maryland aaron.johnston-peck@nist.gov INTRODUCTION The final part of this series on four-dimensional scanning transmission electron microscopy (4D-STEM) covers the topic of ptychography. Ptychography is a form of computational imaging that recovers the phase information imparted to an electron beam as it interacts with a specimen, and which is subsequently lost during the detection process. STEM detectors are only sensitive to the amplitude of the electron exit wave. This amplitude information is encoded as intensity within a diffraction pattern. The phase information of the electron exit wave is lost in this process, the so-called “phase problem.” Ptychography can algorithmically recover this phase information to produce an image of the sample in the form of the transmission function. The computed transmission function represents both the modulus and the accumulated phase difference, relative to free space, of an electron wave transmitted through the sample. Its successful retrieval offers a route to several materials characterization methods since it contains information about the structure and properties of the specimen. In ptychography, the first step is to acquire electron scattering data from multiple points on the specimen with a known spacing and redundancy in the sampling. As in other 4D-STEM techniques, a 4D dataset is assembled by acquiring 2D diffraction patterns at each position of a 2D sampling grid. A key aspect of ptychography is that the spatial coverage of the 2D sampling grid is deliberately oversampled such that the illuminated area of the specimen at each position overlaps, as depicted in the schematic of Fig. 1, Panel I. This overlap in sampling eliminates ambiguities in the phase solution and aids convergence of the computation. It was Walter Hoppe who postulated this principal of using multiple diffraction patterns to eliminate ambiguities when solving the phase problem and who also coined the term ptychography,[1,2] although the current implementations of ptychography have evolved considerably from what Hoppe first described. More information on sampling requirements,[3,4] as well as additional information on the history, development, and fundamentals of ptychography can be found elsewhere.[5,6] Several algorithms have been used for ptychographic treatment of electron microscopy datasets.[7-13] This re- view limits the discussion to STEM data collected in the far-field where there are two primary experimental implementations defined by whether the electron probe is focused or defocused at the sample plane (Fig. 1, Panel II). There are considerations and benefits to each approach. For example, using a focused probe, as one would for conventional imaging, affords the possibility to acquire complementary datasets with high spatial resolution. This complementary signal could be a conventional imaging signal or a spectroscopic signal such as electron energy loss spectroscopy (EELS) or energy dispersive x-ray spectroscopy (EDX). By contrast, using a defocused probe illuminates a larger area of the sample, which can increase throughput and lower the applied dose rate to reduce radiation damage. A practical example of how ptychography functions is the commonly used extended ptychographical iterative engine (ePIE) algorithm. A schematic of the workflow is shown in Fig. 1, Panel III.[8] Essentially ePIE solves an optimization problem, where a model describing the interaction between the electron probe and the sample and the subsequent formation of the diffraction pattern is iteratively updated and compared to experimental data until a convergence criteria is satisfied. The exit wave, or the electron wave emanating from the sample, ψ(r, R), is modeled as the product of two complex functions describing the electron probe, P(r), and the object, O(r), as the probe is shifted relative to the sample by a distance, R. The propagation of the exit wave to the detector is then represented by a Fourier transform and accordingly the intensity at the detector is equal to I(u) = |F[O(r)P(r - R)]|2. Therefore, by identifying the correct phases for each diffraction pattern, each exit wave and the functions O(r) and P(r) can be determined. Guesses of the probe and object functions initiate the algorithm, from which an exit wave
RkJQdWJsaXNoZXIy MTYyMzk3NQ==