HRTEM (Fultz and Howe, 2008) is a parallel imaging technique for atomic-resolution analysis of crystallographic structural information. The contrast of HRTEM images arises from the interference of electron waves as they are phase-shifted by their interaction with the interatomic potential of the specimen, generating ‘phase contrast’ (Williams and Carter, 2009c). Under the proper conditions, cation columns in a metal-oxide can be directly imaged and peak-fitting routines (see Real-Space Structure Analysis Software Tools) can be used extract cation positions with picometer precision (Lawrence et al., 2021).

The primary challenge of HRTEM imaging is that it is only representative of the projected crystal structure, or directly interpretable, under specific conditions so image simulations are typically required for accurate image contrast interpretation (MacLaren and Ramasse, 2014). For instance, in an uncorrected TEM, where the spherical aberration coefficient, C_s, of the objective or image-forming lens is always positive, images taken at the special ‘Scherzer defocus’ condition (Scherzer, 1949) will show atomic columns as dark regions (dark atom contrast). However, if the focus, specimen mass-thickness, or crystallographic tilt changes, the resulting image may display contrast reversals where atomic columns contrast may vary across the image, appearing either bright or dark depending on local sample conditions. Two common methods to bypass this issue are exit-wave reconstruction (EWR) and crystallographic image processing (CIP). To determine aperiodic atomic structure, such as in the disordered or defective regions of disordered crystalline systems, exit-wave reconstruction is preferred since CIP relies on structural periodicity (Thust et al., 1996).

EWR can be used to reconstruct a representative image of the projected crystal structure through the acquisition of a ‘focal-series,’ or a stack of HRTEM images taken over a range of focus values. EWR has been used to discern the structure, orientation relationship, and matrix coherency of Y₂Ti₂O₇ pyrochlore nano-oxides in a nanostructured ferritic alloy (NFA) (Figure 3A). Figure 3B demonstrates the complexity of determining structure from HRTEM images, showing the difference in apparent structure in underfocused, focused, and overfocused (Figures 3Bi–iii) conditions. After acquisition of a focal series, the reconstructed phase image matches with a Y₂Ti₂O₇ pyrochlore structure simulated under compressive strain (Figure 3C) (Wu et al., 2016).

An additional challenge of HRTEM arises due to the difference in scattering power between light and heavy elements. In the case of pyrochlores (and metal oxides in general), the phase-contrast generated from cations is typically strong due to their high nuclear charge, while oxygen generates poor phase contrast and has a relatively low scattering power (Jia et al., 2003) which can make imaging of the oxygen sublattice challenging. With the development of aberration-correctors, it is possible to precisely tune the optical parameters of the (S)TEM (MacLaren and Ramasse, 2014). When the spherical aberration coefficient (Cs) is set to a small negative value, ‘negative Cs imaging,’ (NCSI) (Jia et al., 2003; Jia et al., 2004) can be performed. This method delivers directly interpretable images where both heavy and light elements show strong bright-atom contrast simultaneously (Dunin-Borkowski et al., 2016). Figures 3D–F shows a HRTEM image of a Bi-containing pyrochlore (Bi_1.5ZnNb_1.5O₇, BZN) acquired under NCSI conditions, where cation columns are bright and well-defined and oxygen columns are bright, yet diffuse. 2D Gaussian peak-fitting is performed to determine precise atomic positions and displacements from ideal pyrochlore lattice positions, providing direct evidence of atomic-scale displacive disordering on the A and O′ sites (Levin et al., 2002; Jia et al., 2018). In this case, EWR and NCSI imaging are combined to optimize the bright atom contrast condition (Wu et al., 2016).

Although HRTEM imaging is being progressively replaced with scanning TEM (STEM) techniques due to the relative ease of image interpretation, there remains a clear advantage to using a static, broad electron beam for dynamic experiments over a scanned electron probe. Rapid specimen evolution during in situ experiments necessitates high temporal resolution image acquisition. In TEM mode, the acquisition rate is mainly limited by the desired signal-to-noise ratio and the frame rate of the signal detection system. However, the STEM acquisition rate is additionally limited by scan rate of the probe. For modern systems, this is on the order of a few frames per second (fps) (Ishikawa et al., 2020), where faster scan speeds can lead to significant image distortions (Sang et al., 2016). Although newly developed probe-scanning systems have demonstrated acquisitions of 25 fps (512 × 512 pixels) in STEM, approaching the necessary frame rate for many in situ experiments (Levin, 2021), TEM mode is still presently the choice for in situ experiments due the high frame rates of modern CMOS cameras (e.g., 1,600 fps at 512 × 512, K3 Gatan Inc.). In situ TEM experiments are routinely performed at atomic resolution, for example to track picoscale atomic surface rearrangements of CeO₂ catalytic nanoparticles imaged under low-dose NCSI imaging conditions with millisecond temporal resolution (Levin et al., 2020a).

In high-angle annular dark-field STEM (HAADF-STEM), elastic, incoherent, high-angle Rutherford scatter is collected (β _inner > α) (MacLaren and Ramasse, 2014). Thus, the image intensity is proportional to the mean value of Zⁿ; where Z is the effective atomic number, and n is a constant between 1.5 and 2 (Kirkland, 1998; Nellist et al., 2007). This technique is the most widely used because the atomic positions are clear and there are no contrast reversals (as in HRTEM). However, it is not ideal for imaging light elements since the Rutherford scattering process is inefficient for light atoms with low nuclear charge at high angles. In medium- (or low-) angle annular dark-field STEM (MAADF-/LAADF-STEM), lower angle coherent scatter is collected (β _inner ≥ α) where β _inner is just slightly larger than α (e.g., for α = 15–30 mrad, β _inner = 30–60 mrad). These images retain some of the characteristics of HAADF images (e.g., no contrast inversions, direct interpretability of atomic columns over a large focus range) but also reveal strongly diffracting features (strained areas such as dislocations or other crystalline defects, nanosized coherent or semi-coherent precipitates etc.) (MacLaren and Ramasse, 2014).

In annular bright-field STEM (ABF-STEM) the outer section of the bright-field (BF) disc is collected (β _inner < α < β _outer ) (Hammel and Rose, 1995; Findlay et al., 2010a). In contrast to images formed by the center of the BF disc which are dominated by phase contrast and are subject to contrast reversals as in HRTEM, the outer part of the BF disc produces images that are more incoherent with dark atomic contrast, including light atoms such as oxygen (Findlay et al., 2010a) or even hydrogen (Findlay et al., 2010b). Bright-field STEM (BF-STEM) is analogous to bright-field TEM images attained in TEM, especially at very low collection angles. Here the low angle scatter is small (β ≤ α), and the signal is coherent and dominated by phase contrast. Just as in HRTEM, the contrast is critically dependent on sample thickness and microscope defocus, and can show contrast inversions. (MacLaren and Ramasse, 2014).

As described previously, strain-fields in pyrochlores can dramatically affect their properties, thus, analysis of strain-fields around structural defects formed in pyrochlores formed during synthesis are critical to understand (Shafieizadeh et al., 2018). Geometric phase analysis (GPA) (Hÿtch et al., 1998) and peak analysis (Bayle et al., 1994; Anthony and Granick, 2009) are two commonly used digital processing approaches that facilitate extraction of lattice parameter at high resolution, based in reciprocal- and real-space, respectively (Zhu et al., 2013). In GPA, atomic displacements are measured by calculating the ‘local’ Fourier components of the lattice fringes of a real-space image (Hytch, 1997), whereas peak-finding methods find atomic positions from the contrast maxima of images directly, often through center of mass calculations or by 2D Gaussian fitting which can be more precise (Anthony and Granick, 2009). In either case, contrast maxima in the image are identified as a periodic lattice so deviations of the real, local lattice with respect to an unstrained reference lattice can be calculated and mapped (Zhu et al., 2013). Although these methods are generally robust, precautions must be taken to minimize microscope and specimen effects that may produce inaccurate strains, especially from HRTEM images (Hÿtch and Plamann, 2001) where contrast interpretation is often less straightforward than in HAADF-STEM.

GPA has been used to map strain fields (Figure 7) around superdislocations and other extended crystalline defects in Yb₂Ti₂O₇ formed during synthesis (Shafieizadeh et al., 2018). HAADF-STEM reveals extended defects such as anti-phase boundaries and dissociated superdislocations with unique core structures and anomalously large Burgers vectors (Figure 7A) compared with other oxides (Jia et al., 2005). GPA has also been used to map strain at La₂Zr₂O₇(LZO)/YSZ (111) interfaces, highlighting the presence of misfit dislocations that may impact the accumulation of radiation damage at such features (Kaspar et al., 2017). This approach has also been used to demonstrate the effect of strain at Σ 3 ( 11 1 ¯ ) < 1 1 ¯ 0 > twin boundaries in Gd₂Ti₂O₇, showing a local reduction of the oxygen migration barrier compared with the bulk (Gupta et al., 2020).

In addition to experimental factors, such as high-quality TEM specimen preparation, and proper optical setup and image acquisition parameters, software tools can be used to correct scan distortions of STEM images and automated procedures can perform highly accurate assessment of atomic positions. There are many software programs available to do this (e.g., Ranger (Jones and Nellist, 2013), qHAADF (Galindo et al., 2007), iMtools, StatSTEM (De Backer et al., 2016), and oxygen octahedra picker [110], as well as Atomap [111], among others [112] (Jones et al., 2015)). This is still a highly active area of research, where machine learning tools are being developed to improve image quality and atom detection (Lin et al., 2021). Image simulation is useful to aid the interpretation and validation of STEM data, as conveyed through multiple examples above. The two most commonly used image simulation methods are the Bloch wave and multislice methods, see (Kirkland, 1998) for detailed descriptions. Some popular software options for STEM image simulation are Dr. Probe (Barthel, 2018) Prismatic (Ophus, 2017; Rangel DaCosta et al., 2021), and pyQSTEM (Koch, 2002), among many others.

Often, the presence of additional scattering effects (satellite and diffuse scatter) in fluorite and pyrochlore systems are a result of local, strain-driven compositional and displacive structural modulations (Withers et al., 1991; Tabira et al., 1999; Tabira et al., 2001; Withers et al., 2004; Liu et al., 2006; Withers and Hawkes, 2008; Welberry and Weber, 2016; Trump et al., 2018) and sometimes can be related to the formation of locally-ordered domains (García-Martín et al., 2005; Lau et al., 2008; Reid et al., 2012). For instance, the presence of local, C-type ordering in lanthanide oxides on short to intermediate length scales can give rise to ED fine structure; selective imaging of signal originating from single satellite peaks (satellite dark-field imaging) has been used to reveal the distribution of C-type ordered nanodomains on the order of ∼50–100 Å in real-space (Reid et al., 2012).

In most (S)TEM techniques, data interpretation is complicated by multiple scattering which commonly occurs due to the strong interaction of electrons with the crystal potential, even in thin specimens (<100 nm). This ‘dynamical’ diffraction perturbs the relative intensities of Bragg reflections and diffuse scatter in ED, and leads to inaccurate measurement of CN in ePDF analysis (Ishimaru, 2006). The first line of defense to minimize multiple scattering is optimization of specimen thickness and/or electron energy (Gorelik et al., 2019), however, many other methods have been devised (Eggeman et al., 2012) specifically to avoid its effect on the relative intensities of Bragg peaks in ED patterns. PED (Ishimaru et al., 2002) is one of these methods, whereby the incident electron beam (often set to ΝΒED conditions), is tilted and precessed to form a conical electron probe at the specimen. PED can be used to identify crystal structure, determine the local crystal orientation (Bowman et al., 2020), or to investigate crystal texture, grain rotation, and strain. Compared with a steady (unprecessed) beam, precession provides many more reflections by sampling additional layers of reciprocal-space (both the zero-order and higher-order Laue Zones, ZOLZ and HOLZ, respectively) allowing for greater sensitivity to local orientation changes as well as changes in lattice parameter. It also integrates the intensities through the Bragg condition over the precession cycle, averaging out relative intensity fluctuations due to dynamical scattering (Moeck and Rouvimov, 2010; Midgley and Eggeman, 2015).

Until this point, the diffraction techniques discussed have utilized parallel (α ≈ 0), or quasi-parallel (small α) illumination that produce localized diffracted signals originating from a broad specimen area. CBED requires convergent illumination, where the electron beam is focused into a probe that impinges the specimen over a range of angles (Williams and Carter, 2009b) and delocalizes the diffracted signal into discs that may contain complex, and information-rich intensity distributions (Zuo et al., 2017b). Generally, the convergence angle used for CBED is several times larger than what is used in μ/N/ΑBED, but still significantly smaller than that used in an aberration-corrected STEM. The convergent illumination condition generates a spherical electron-wavefront capable of sampling a greater portion of reciprocal-space, thus, CBED is better suited than parallel-beam ED approaches for involved crystallographic studies because it enables unique determination of the point group from one or few zone axes, and with analysis of the systematic absences it is possible to uniquely determine most space groups. The complex intensity distributions within the Bragg discs, and 3D information in the higher-order Laue zone (HOLZ) lines (Zou et al., 2011) provide additional signatures of the underlying crystal symmetry. Positions of the HOLZ lines are sensitive to the unit cell parameters, so deviation from a standard can be used to probe any local unit cell distortions. See (Zuo et al., 2019) for an in-depth discussion of S-BED techniques and their utility.

The most straightforward analysis that can be performed on a 4D-STEM dataset is to integrate the diffracted signal within a (usually circular or annular) mask generated in reciprocal-space to construct a 2D-STEM image (Levin et al., 2020b). The ability to flexibly select signals in reciprocal-space makes it possible to reconstruct images equivalent to any of the traditional STEM detectors (see STEM Imaging Modes), even with atomic resolution (Figures 13C–E) (Hachtel et al., 2018; Levin et al., 2020b). The left column of images in Figure 12ei (Savitzky et al., 2021) shows an example of virtual imaging with circular and annular masks at low magnification, used to generate virtual-BF (green) and virtual-DF (blue/purple) images of the irradiated GTO sample, ED patterns in the <110> projection. The second column of Figure 12Ei shows the unique power of virtual imaging, where specific diffraction spots, sets of spots, or other diffraction fine-structure can be selected to form images that may be otherwise impossible to generate through DF imaging techniques. Here, the virtual-DF images are constructed from sets of cation ordering reflections (Lian et al., 2003) that correspond to the pyrochlore structure (red), and from a set of reflections shared by both pyrochlore and fluorite (blue), demonstrating how local regions of cation ordering can be imaged by filtering electrons from the diffraction pattern based on their scattering vectors. Another approach to mapping cation-ordering achieved by template matching is described in the following Structure Classification, Orientation, and Strain Mapping. In theory, imaging of the anion-ordered regions could be achieved in a similar manner by selecting spots with indices h + k + l = 4n (i.e., 022 and 422) (Lian et al., 2003) to form virtual-DF images. However, the scattering from oxygen is likely much weaker and may require methods to improve the signal-to-noise ratio.

By the same logic, virtual imaging could also be used to image locally-ordered domains (O’Quinn et al., 2020) in heterogeneously disordered pyrochlores (e.g., to reveal the formation of Weberite-type (O’Quinn et al., 2020; Shamblin et al., 2016), or bixbyite (C-type) structural units (Reid et al., 2012). It could also be used to investigate the spatial the origins of diffuse electron scatter (see Diffraction Fine-Structure) from the specimen, however, this presents several challenges. A more sophisticated approach is achieved by Cepstral STEM (Shao et al., 2021), which is a new 4D-STEM technique for imaging severe lattice distortions within disordered crystals using fluctuations in electron diffuse scatter that arise due to crystal disorder. Cepstral analysis is a sensitive signal processing technique for detecting weak harmonic signals by performing a Cepstral transform of the coherent NBED pattern. The Cepstrum transform has been applied previously to ED patterns for lattice strain analysis based on Bragg scatter (Padgett et al., 2020). Recently, the difference Cepstrum (dC_p) was developed, which represents the difference between the Cepstral transforms of local NBED patterns and the region-averaged NBED pattern within a 4D dataset (Shao et al., 2021). The dC_p separates electron diffuse scattering from Bragg diffraction so the diffuse signal can be analyzed to determine the distortive part of electron scattering potential. This part can then be imaged via construction of a virtual-DF image (Figure 13) as in the previous example, only by masking the dC_p rather than the NBED pattern directly. The sharp harmonic signals detected by Cepstral STEM make numerical quantification of lattice distortions and their mapping possible. Figure 13 shows the dC_p of a high-entropy alloy, where the composite virtual-image from select signals in the dC_p demonstrate the complex structural distortions of and HEA. See (Ophus, 2019) for more applications of virtual imaging to various systems.

For structure classification, phase, orientation, and strain mapping applications, precise determination of Bragg disc positions is critical, thus, the discs must be well-separated and well-defined so automated analysis can be carried out to determine their locations. The overlap of discs can be prevented by reducing the convergence angle such that the distance between the Bragg spots is at least twice the Bragg angle (θ_B ≈ few mrad) which can be achieved by ensuring α < θ_B. Phase and orientation mapping rely on a template matching approach, where stacks of simulated diffraction data are matched to the experimental data to determine the local phase, orientation, or strain at each probe position (Rauch et al., 2010). Figure 12Eii shows classification of the GTO sample into three distinct structural phases, the pyrochlore (parent) structure, the recrystallized and amorphous regions, which can then be mapped to observe their spatial distributions. 4D-STEM has been used to quantify and map the degree of cation disorder in pyrochlore materials at the nanoscale by quantifying the intensity ratio of pyrochlore superlattice reflections to the main fluorite reflections for each pattern in the scanning NBED (4D-STEM) dataset can be generated (Janish et al., 2019). Again, a template matching approach is implemented to match each experimental pattern with a dictionary of simulated diffraction patterns (Allen et al., 2015) generated for calculated structures with varying degrees of disorder (Janish et al., 2019; Janish et al., 2020).

In a previous section, we saw the utility of the ePDF/eRDF for analyzing the structure of amorphous materials from diffraction patterns. This approach becomes much more powerful when coupled with a scanning beam in order to map out real-space distributions of SRO (Lu and Gauntt, 2013). Most implementations of so-called “STEM-PDF” (Mu et al., 2021) or “STEM-RDF” (Mu et al., 2016a) require some type of multivariate statistical analysis (MVSA) used to mine large 4D datasets to identify relevant co-varying variables in order to map statistically significant structural components (Lu and Gauntt, 2013; Mu et al., 2019). For example, to disentangle the local atomic bonding and packing information mixed in NBED data to characterize nanoscale heterogeneous amorphous materials such as nanoglass (Mu et al., 2021). Or to reveal interfacial layers from an amorphous ZrO₂/Zr_0.2Fe_0.8 mutilayer system with unique atomic arrangements that could not be seen with EDX or EELS (Figures 14A,B) (Mu et al., 2016a). As in other 4D-STEM approaches, the size the of the beam on the sample will limit the spatial resolution of structural information maps. A small beam (down to 2 nm), can be used, but speckled regions of the pattern that appear as the beam size is reduced (recall Figure 11C) must be removed with masking in analysis (McBride and Cockayne, 2003). Overall, the nm-scale spatial resolution of STEM-PDF is a dramatic improvement to the 10 μm spatial resolution of x-ray synchrotron PDF mapping (Billinge, 2019).

Like ePDF, fluctuation electron microscopy (FEM) is used to characterize the structure of amorphous materials in the (S)TEM. However, in contrast to ePDF which is ideal for measuring pair-correlations out to distances of the first few shells of neighboring atoms, it is better suited for measuring MRO which extends beyond the first few shells. It has been shown that the two-body correlations are not sufficient to distinguish between competing structural models of some amorphous materials (Treacy and Borisenko, 2012), therefore access to MRO is necessary for full structural analysis. It can be performed in both TEM and STEM mode, although STEM mode facilitates easier convergence-angle adjustment to investigate different sizes of atomic clusters (Ophus, 2019; Savitzky et al., 2021) as well as to determine correlation lengths of locally-ordered regions (Voyles and Muller, 2002; Voyles et al., 2011). When these clusters deviate from a fully random distribution, they lead to “speckles” in the amorphous halo as described previously, FEM provides structural information by quantification of the degree of variability as a function of scattering angle and probe size (Treacy et al., 2005; Hilke et al., 2019). Like ePDF and ABED analyses, FEM has primarily been applied to metallic glasses however, there are some examples of its use in oxide systems. For example, to study crystal nucleation in TiO₂-SiO₂ glasses upon heat treatment (Rezikyan and Moore, 2020), and to analyze structural disordering in ion-irradiated amorphous zircons (Zhao et al., 2010). For a review of FEM theory and development see (Voyles et al., 2011).

Figure 12Eiii demonstrates crystalline strain mapping from determination of Bragg disc position. Like crystalline materials, a change of the average atomic spacing in an amorphous material will result in a change in the radius of the diffuse halo. When this halo is fit to an elliptical function local strain can be measured (Ebner et al., 2016) and mapped (Savitzky et al., 2021; Gammer et al., 2018) if acquired with a scanning beam. The py4dSTEM implementation of this analysis (Savitzky et al., 2021) on the amorphized region of the GTO sample is shown in Figures 14C,D, where the 1 / 2 ( ε x x + ε y y ) map shows the local dilation, which is amplified in the vicinity of amorphous-polycrystal interfacial regions. This method has also been combined with in situ tensile testing to reveal the accumulation of strain under mechanical deformation in a bulk metallic glass (Gammer et al., 2018).

Many variations of 4D-STEM exist, for example, to map electromagnetic fields (Hachtel et al., 2018). Other applications aim to increase the sensitivity to light elements (Hachtel et al., 2018; Ahmed et al., 2020) and to retrieve lost electron-wave phase information to achieve super-resolutions (Jiang et al., 2018; Chen et al., 2021). A detailed account of the various 4D-STEM methods can be found in (Ophus, 2019).

4D-STEM approaches are becoming increasingly accessible with the development of more suitable hardware, larger data storage capacity, more powerful computation, and the continued development of software packages for simulation and analysis that facilitates easier analysis workflows. The analysis of such large, multidimensional datasets requires powerful computers and specialized software, many of which are home-grown in various labs around the world. Some available packages are Hyperspy, py4DSTEM, LiberTEM, Pycroscopy, pixSTEM, and the Cornell Spectrum Imager plugin for ImageJ. 4D-STEM simulation software (Pelz et al., 2021) is also critical component of analysis of 4D datasets.