3D
characterization using transmission electron diffraction, neural network optimization, and density functional theory
Authors: Pennington Robert S. , Koch Christoph T. 
Published: December 2016

Three‐dimensional characterization using the transmission electron microscope (TEM) can reveal complex nanoscale structural and chemical properties. Because the TEM generates two‐dimensional images and diffraction patterns, an inversion algorithm is necessary to retrieve the three‐dimensional specimen. An inversion algorithm we have developed includes multiple scattering, and thus can provide three‐dimensional nanoscale characterization of crystals from TEM diffraction data using artificial‐neural‐network optimization tools and GPU‐accelerated matrix exponentials ‐ we have previously reported retrieving strain and ferroelectric polarization on simulated data [1‐2]. Some algorithms reconstruct individual atomic positions [3]; our algorithm retrieves crystal properties and is suitable for larger specimens and structures. Mapping ferroelectric polarization domains and strain state variations in arbitrary geometries as a function of specimen depth with nanometer‐scale resolution can enable novel nanoscale analytical insights for a wide range of crystalline materials, including quantifying 3D structures and understanding surface‐induced artifacts.
However, our algorithm requires accurately modeling a layered crystalline specimen for Bloch‐wave‐type calculations. The conventional approach ‐ using isolated‐atom scattering factors (IASF) ‐ is fast, but neglects chemical bonding, while directly fitting the Ug structure‐factors might result in accurate pattern replication, but with drawbacks for this three‐dimensional application ‐ direct‐Ug‐fitting greatly increases the number of free parameters, and thus likely decreases the precision of each individual parameter while simultaneously complicating analysis [4]. The specifics of this problem enable a third method ‐ density functional theory (DFT) ‐ which provides self‐consistent ab‐initio structure factors with chemical bonding effects, and has been previously used to precompute key low‐order structure factors, comparing well with experiment [4‐5]. Here, we use DFT to generate all the structure factors, because the GPAW DFT code, which we have used for mean inner potential calculations, can provide the all‐electron density, which can then be processed to yield Ug structure factors [6‐7]. With modern hardware, thousands of small‐scale DFT simulations can be performed in a reasonable time, enabling DFT integration into our iterative inversion algorithm, which has been improved to be multi‐CPU+multi‐GPU parallelized.
Figures 1 and 2 show the results of combining DFT‐computed self‐consistent ab‐initio structure factors with our depth‐direction parameter retrieval algorithm on simulated data. Simulated SrTiO3 is our test material for simultaneous retrieval of ferroelectric atomic displacements (single‐atom property) and oxygen octahedral rotation (multi‐atom property); DFT is used both to generate the test data and during the retrieval routine. For perovskites, both of these specific parameters can be of interest for different systems; for other materials, the combination of single‐atom and multi‐atom parameters could be useful. For this noise‐free data, the results cease improving because the DFT simulations use a user‐selectable grid spacing; a finer grid can be used, at the cost of computational time.
In this work, we accurately retrieve ferroelectric atomic displacements and perovskite‐style octahedral oxygen rotation for SrTiO3 from simulated composite‐CBED‐type data using ab‐initio DFT structure factors [2]. Experimental applications of this technique to both 2D and 3D data will be discussed.