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Electron image contrast analysis of mosaicity in rutile nanocrystals using direct electron detection

aMaterials Science and Engineering, University of Illinois at Urbana-Champaign, 1304 W. Green Street, Illinois 61801, USA, bMaterials Research Laboratory, University of Illinois at Urbana-Champaign, 106 S. Goodwin Avenue, Urbana, Illinois 61801, USA, and cHitachi High-Technologies America Inc., USA
*Correspondence e-mail: jianzuo@illinois.edu

Edited by L. D. Marks, Northwestern University, USA (Received 14 April 2020; accepted 11 August 2020; online 23 October 2020)

Direct electron detection provides high detective quantum efficiency, significantly improved point spread function and fast read-out which have revolutionized the field of cryogenic electron microscopy. However, these benefits for high-resolution electron microscopy (HREM) are much less exploited, especially for in situ study where major impacts on crystallographic structural studies could be made. By using direct detection in electron counting mode, rutile nanocrystals have been imaged at high temperature inside an environmental transmission electron microscope. The improvements in image contrast are quantified by comparison with a charge-coupled device (CCD) camera and by image matching with simulations using an automated approach based on template matching. Together, these approaches enable a direct measurement of 3D shape and mosaicity (∼1°) of a vacuum-reduced TiO2 nanocrystal about 50 nm in size. Thus, this work demonstrates the possibility of quantitative HREM image analysis based on direct electron detection.

1. Introduction

Three-dimensional (3D) nanocrystals are building blocks for functional materials, such as quantum dots, catalysts and energy-storage materials. Critical to their properties is nanocrystal structure, which differs from the bulk crystals by the small size and the size-related effects. However, quantitative determination of nanocrystal structure is still a major challenge in crystallography and materials research in general, despite considerable interest in this subject and the progress that has been made in nanostructure characterization using X-ray diffraction and electron microscopy. At the atomic and nanometre scale, much of the progress being made is in the form of scanning transmission electron microscopy (STEM) imaging using a high-angle annular dark-field (HAADF) detector. For example, electron tomography based on HAADF-STEM reconstructs 3D structure from multiple projections [for a review, see Levin et al. (2016[Levin, B. D. A., Padgett, E., Chen, C.-C., Scott, M. C., Xu, R., Theis, W., Jiang, Y., Yang, Y., Ophus, C., Zhang, H., Ha, D.-H., Wang, D., Yu, Y., Abruña, H. D., Robinson, R. D., Ercius, P., Kourkoutis, L. F., Miao, J., Muller, D. A. & Hovden, R. (2016). Sci. Data, 3, 160041.])]. Three-dimensional structural information can also be obtained by taking a focal series of the HAADF-STEM images (Borisevich et al., 2006[Borisevich, A. Y., Lupini, A. R. & Pennycook, S. J. (2006). Proc. Natl Acad. Sci. USA, 103, 3044-3048.]; Nellist et al., 2008[Nellist, P. D., Cosgriff, E. C., Behan, G. & Kirkland, A. I. (2008). Microsc. Microanal. 14, 82-88.]; Xin & Muller, 2009[Xin, H. L. & Muller, D. A. (2009). J. Electron Microsc. 58, 157-165.]; Gao et al., 2015[Gao, W. P., Sivaramakrishnan, S., Wen, J. & Zuo, J. M. (2015). Nano Lett. 15, 2548-2554.]). The main advantage of HAADF-STEM has been the simplified image contrast, in the form of Z-dependent atomically resolved columns (Pennycook & Nellist, 2011[Pennycook, S. & Nellist, P. (2011). Editors. Scanning Transmission Electron Microscopy, Imaging and Analysis. New York: Springer.]), while the main drawback is the experimental difficulty associated with acquiring multiple electron images at a larger electron dose compared with transmission electron microscopy (TEM) and in the presence of scan noises and sample instabilities. In the case of atomic resolution electron tomography, very thin samples are also required to achieve atomic resolution (Lu et al., 2015[Lu, X. W., Gao, W. P., Zuo, J. M. & Yuan, J. B. (2015). Ultramicroscopy, 149, 64-73.]).

HREM (high-resolution electron microscopy) is traditionally used for imaging crystals and their defects. HREM images are formed by interference of transmitted and diffracted electron beams as brought together by the objective lens (Spence, 2013[Spence, J. C. H. (2013). High-resolution Electron Microscopy. Oxford University Press.]). The image contrast is sensitive to the phase difference of the transmitted and diffracted beams, which can be quantified as demonstrated by the remarkable success of single-particle imaging using cryo-electron microscopy (see eg. Vulović et al., 2014[Vulović, M., Voortman, L. M., van Vliet, L. J. & Rieger, B. (2014). Ultramicroscopy, 136, 61-66.]; Cheng, 2015[Cheng, Y. (2015). Cell, 161, 450-457.]). In an inorganic crystal, the phase of diffracted beams is strongly influenced by electron scattering from the periodic crystal potential. Consequently, the phase shift depends sensitively on the crystal thickness, crystallinity, orientation and electron acceleration voltage, giving rise to complex interference patterns. Furthermore, the contrast of a HREM image also depends sensitively on the objective lens' properties, such as the lens aberrations and defocus (Spence, 2013[Spence, J. C. H. (2013). High-resolution Electron Microscopy. Oxford University Press.]; Zuo & Spence, 2017[Zuo, J. M. & Spence, J. C. H. (2017). Advanced Transmission Electron Microscopy, Imaging and Diffraction in Nanoscience. New York: Springer.]). Because of these factors, extracting information from the HREM images requires a high-quality electron image to start with. For the above reasons, success in 3D structure determination using HREM has so far been limited to very thin crystals (see eg. Jia et al., 2014[Jia, C. L., Mi, S. B., Barthel, J., Wang, R. E., Dunin-Borkowski, K. W., Urban, K. W. & Thust, A. (2014). Nat. Mater. 13, 1044-1049. ]).

Experimentally, the recording of HREM images has been a challenge as the performance of conventional detectors, such as films and charge-coupled device (CCD) cameras, is limited as measured by the detective quantum efficiency (DQE) and detector resolution (Zuo, 2000[Zuo, J. M. (2000). Microsc. Res. Tech. 49, 245-268.]; McMullan et al., 2009[McMullan, G., Chen, S., Henderson, R. & Faruqi, A. R. (2009). Ultramicroscopy, 109, 1126-1143.]). Previous attempts to match experimental HREM image contrast with theory have revealed major differences by a factor of two to four, which is known as the Stobbs factor (Hÿtch & Stobbs, 1994[Hÿtch, M. J. & Stobbs, W. M. (1994). Ultramicroscopy, 53, 191-203.]; Boothroyd, 1998[Boothroyd, C. B. (1998). J. Microsc. 190, 99-108.]). The cause(s) of the Stobbs factor are still being debated (Howie, 2004[Howie, A. (2004). Ultramicroscopy, 98, 73-79.]; Krause et al., 2013[Krause, F. F., Müller, K., Zillmann, D., Jansen, J., Schowalter, M. & Rosenauer, A. (2013). Ultramicroscopy, 134, 94-101.]), while the works by Thust and Van Dyck attributed the mismatch, respectively, to the camera's modulation transfer function (MTF) (Thust, 2009[Thust, A. (2009). Phys. Rev. Lett. 102, 220801.]) and thermal diffuse scattering (Van Dyck, 2011[Van Dyck, D. (2011). Ultramicroscopy, 111, 894-900.]). As direct electron detection and electron counting provide a significantly improved signal-to-noise level, reduced camera point spread function (PSF) and capability for low-dose imaging (McMullan et al., 2014[McMullan, G., Faruqi, A. R., Clare, D. & Henderson, R. (2014). Ultramicroscopy, 147, 156-163.]), it is thus worthwhile to examine the performance of direct electron detectors for quantitative HREM imaging.

For quantitative HREM, extracting structural information from the recorded HREM images is a critical step. Several methods have been proposed before, and these methods fall into the following categories. The first is the comparison between the experimental and simulated images using the cross-correlation coefficients (Thust & Urban, 1992[Thust, A. & Urban, K. (1992). Ultramicroscopy, 45, 23-42.]). Many simulated images are calculated using a structure model for different defocus–thickness planes and the match with the experimental image allows the sample thickness and the defocus value of HREM micrographs to be determined. The second method is to use iterative digital image matching where HREM image analysis is treated as an optimization problem. The parameters to be optimized include both structural and imaging parameters (Möbus & Rühle, 1994[Möbus, G. & Rühle, M. (1994). Ultramicroscopy, 56, 54-70.]). The third method is based on exit-wave reconstruction using a series of HREM images to reconstruct the complex electron wavefunction at the exit plane of the specimen (Zandbergen & Van Dyck, 2000[Zandbergen, H. W. & Van Dyck, D. (2000). Microsc. Res. Tech. 49, 301-323.]; Kirkland & Meyer, 2004[Kirkland, A. I. & Meyer, R. R. (2004). Microsc. Microanal. 10, 401-413.]; Ophus & Ewalds, 2012[Ophus, C. & Ewalds, T. (2012). Ultramicroscopy, 113, 88-95.]). The multiple experimental images recorded at different defocuses can be used to reconstruct the exit wavefunction, and a large improvement in the interpretable information may be achieved over a single image. These approaches tend to focus on the problem of structure determination of a relatively uniform and thin crystal. For nanocrystals, HREM analysis must consider their shapes or geometric factors.

Here, we report on the electron imaging and image analysis of rutile (TiO2) nanocrystals using a Gatan K2 direct electron detector in the summit mode inside an environmental transmission electron microscope. TiO2 is an excellent photocatalyst for water splitting, dye-sensitized solar cells or degradation of the organic pollutants that has attracted significant interest (Grätzel, 2001[Grätzel, M. (2001). Nature, 414, 338-344.]; Diebold, 2003[Diebold, U. (2003). Surf. Sci. Rep. 48, 53-229.]; Maeda & Domen, 2010[Maeda, K. & Domen, K. (2010). J. Phys. Chem. Lett. 1, 2655-2661.]). The light absorption and surface redox processes are strongly enhanced in nanocrystals because of the close coupling of surface and defect states. Thus, there is strong interest in understanding the structure of TiO2 nanocrystals and their surfaces. Among three polymorphs that TiO2 is known to adopt, rutile is the most stable phase, as well as the most studied phase. Because of this, there is also a long history in electron imaging of rutile crystals and their defects (Bursill & Hyde, 1972[Bursill, L. A. & Hyde, B. G. (1972). Prog. Solid State Chem. 7, 177-253. ]; Wood & Bursill, 1981[Wood, G. J. & Bursill, L. A. (1981). Proc. R. Soc. London Ser. A, 375, 105-125.]; Smith et al., 1987[Smith, D. J., McCartney, M. R. & Bursill, L. A. (1987). Ultramicroscopy, 23, 299-303.]; Kubo et al., 2007[Kubo, T. H., Orita, H. & Nozoye, H. (2007). J. Am. Chem. Soc. 129, 10474-10478.]; Gao et al., 2015[Gao, W. P., Sivaramakrishnan, S., Wen, J. & Zuo, J. M. (2015). Nano Lett. 15, 2548-2554.]). To examine what can be learnt about the structure of TiO2 nanocrystals from the contrast of HREM images using the improvements of direct electron detection and an environmental transmission electron microscope, we developed an analysis technique of auto-HREM to automate image matching between experimental and simulated images, and used the matching to determine four major parameters: crystal orientation (tilt angle in x and y directions), sample thickness and defocus. The technique was then applied to identify the shape of the rutile nanocrystals and crystal mosaicity at the high temperature of 550°C.

The main advantages of the approach developed here, as compared with HAADF-STEM, are (i) HREM images can be recorded at a much reduced electron dose level using direct electron detection, (ii) the time resolution is seconds or less dependent on the electron dose rate, and (iii) a large field of view, including several nanocrystals for example. These advantages are particularly useful for in situ TEM, as demonstrated here. The image sampling required is similar to that of quantitative HAADF-STEM. Significant gains in time resolution and field of view are obtained by parallel recording and by the use of a large-format detector. The dose efficiency comes from bright-field (BF) imaging.

2. Experimental methods

The experiment was performed using a Hitachi H9500 transmission electron microscope with a LaB6 electron source, which was operated at 300 kV. The microscope was designed with differential pumping and a custom-built gas delivery system for use as an environmental transmission electron microscope (Gao et al., 2017[Gao, W. J., Wu, A., Yoon, P., Lu, L., Qi, J., Wen, J., Miller, J. C., Mabon, W. L., Wilson, H., Yang, H. & Zuo, J.-M. (2017). Sci. Rep. 7, 17243.]). For image recording, the microscope is equipped with a dual electron detection system of a CCD camera (Orius SC200, Gatan) and a direct electron detection camera (K2-IS, Gatan). For the study here, electron images were recorded using the Summit counting mode of the K2-IS camera at a dose rate of less than 110 (e Å−2) s−1 and the exposure time of a few seconds.

The rutile nanocrystals studied here were purchased from SigmaAldrich (CAS No. 13463-67-7). The nanocrystals are ∼50 nm for the average size (see Fig. 1[link] for an example). Most of the as-received nanocrystals were found to be free of obvious defects. For the experimental results reported here, the specimen was heated to 550°C inside the microscope vacuum, which kept the surface free of contamination. The sample heating was done using a MEMS-based sample-heating stage (Hitachi Blaze heating holder, Norcada MEMS chips) (Thompson et al., 2017[Thompson, M. S., Zega, T. J. & Howe, J. Y. (2017). Meteorit. Planet. Sci. 52, 413-427. ]).

[Figure 1]
Figure 1
Image contrast comparison between two cameras, the CCD camera in (a), the direct electron detection camera in (b). (c) and (d) are the intensity profiles projected over a width of 5 nm in (a) and (b). The nearest local maxima and local minima in (c) and (d) are used to compute the contrast plotted in (e).

The camera's impact on the experimental image contrast is evaluated in Fig. 1[link] by comparing two images recorded from the same nanocrystal using the Orius SC200 [Fig. 1[link](a)] and the K2-IS cameras [Fig. 1[link](b)], at the same imaging conditions. At first sight, Fig. 1[link](a) features a bright background with non-uniform noise, while Fig. 1[link](b) has a more uniform background intensity. For quantitative comparison, we define the image contrast as the difference between the maximum and minimum intensity, calculated locally for the lattice images with multiple local maxima and local minima:

[{\rm Contrast} = {{I_{\max}-I_{\min}}\over {I_{\max}+I_{\min}}}.\eqno(1)]

Figs. 1[link](c) and 1[link](d) show the intensity profiles taken along the lines and the arrow directions, which are also projected along the width direction. From each profile, the contrast is calculated and plotted in Fig. 1[link](e). The results show that there is a large difference in the image contrast between the two images. Taking the average of the contrast from the inner region of the nanocrystal at 10–40 nm away from the edge of the nanocrystal, the mean contrast of the Orius image is 0.01, while the mean contrast of the K2 image is 0.05 [Fig. 1[link](e)]. Thus, the K2 camera offers an image contrast that is about five times higher than that of the Orius camera for the inner part of the nanocrystal.

The significant contrast improvement showed in the image recorded by K2 can be directly attributed to the improved MTF with electron counting (McMullan et al., 2014[McMullan, G., Faruqi, A. R., Clare, D. & Henderson, R. (2014). Ultramicroscopy, 147, 156-163.]). Because of the direct conversion, the direct electron detection camera also produces a DQE as high as 80% at zero frequency and 40% at half Nyquist frequency, which is the reason for the perceived smoothness of the background intensity. For the Orius CCD camera, the electrons are detected by the phosphor/fibre-optic/CCD sensor, which lowers the DQE to 7–10% at half Nyquist frequency, and the reduced image contrast can be attributed to the PSF of the CCD camera.

3. Image matching using auto-HREM

3.1. Automated image matching

To analyse the HREM image from a nanocrystal, we developed the automated HREM (auto-HREM) method based on: (i) HREM image acquisition by using a direct electron detection camera (as described in Section 2[link]), (ii) determination of sample orientation using template-based image matching, and (iii) determination of sample thickness and defocus by analysing the nanocrystal Pendellösung fringes. The image matching is performed in real space rather than in Fourier space as in the work of Thust & Urban (1992[Thust, A. & Urban, K. (1992). Ultramicroscopy, 45, 23-42.]). Also, by determining the crystal orientation together with crystal thickness, the method is thus capable of determining the 3D shape and orientation of the nanocrystal. The implementation of the auto-HREM algorithm combines several steps of image simulations, template generation and pattern matching. Fig. 2[link] illustrates the workflow of auto-HREM.

[Figure 2]
Figure 2
The workflow of auto-HREM for image matching.

The first step in auto-HREM is to extract image templates from the recorded experimental image. Two approaches are used here: the first approach is to subdivide the image into smaller pieces like a chessboard with each piece labelled by its row (n) and column (m) numbers. This approach generates a total of n × m number of templates. The second approach is to generate the spatially averaged templates by using the template matching algorithm (TeMA) method (Zuo et al., 2014[Zuo, J. M., Shah, A. B., Kim, H., Meng, Y., Gao, W. & Rouviére, J. L. (2014). Ultramicroscopy, 136, 50-60.]). A region of the original image is selected and compared with other regions of the original image to find the similar regions. Then, templates from the similar region are spatially averaged to provide the enhanced contrast. By spatial averaging, the second approach also reduces the computational time, as the number of templates is significantly reduced. This method may not work for sample regions with localized features. In the first approach, the individual templates are noisy, but the method is useful for the determination of local structure.

The second step in auto-HREM is to build a library of simulated images. To do this, we built a structural model of the TiO2 nanocrystal using the reference data (ICDD #00-021-1276). We simulated the HREM images systematically by varying the simulation parameters using the multislice program described by Spence & Zuo (1992[Spence, J. C. H. & Zuo, J. M. (1992). Electron Microdiffraction. New York: Plenum.]). The multislice simulation parameters include the crystal orientation, which is simulated using beam tilt x (Tx), tilt y (Ty) and crystal thickness (Th). We used the TiO2 structure projected along the [001] direction with the crystal thicknesses ranging from 30 to 89 nm, the beam tilt from +30 to −30 pixel (−4.5 to 4.5 mrad). For image simulation, we used Cs = 0.7 mm, Cc = 1.4 mm, ΔE = 1.5 eV, defocus from −80 to 40 nm and a beam convergence angle of 2 mrad (full width at half-maximum). The convergence angle is used as a catch-all parameter for the damping in the contrast transfer function, which is important to reduce the excessive details predicted by coherent imaging theory. All simulations were automated by a Python-based script. The simulated images were stitched together to construct `the thickness–defocus map' or `the tilt map', and these maps were used for comparison with the experimental templates. Fig. 3[link] shows examples of the thickness–defocus map and the tilt map.

[Figure 3]
Figure 3
Examples of the thickness–defocus map (a) and the tilt map (b) constructed from the simulated images. The parameters for (a) are Tx = 0, Ty = 10 px (1 px = 1.5 mrad) and for (b) Th = 26 nm and Δf = 0. The actual map used for the cross-correlation analysis is larger than the examples shown here.

For image matching, as a third step in auto-HREM, we used the normalized cross-correlation (XC) as defined by equation (2)[link], which is often used to determine the similarity of two images. The XC coefficient r(u,v) of the template t and the image f is defined by

[\eqalignno{&r(u,v) = &\cr &{{\sum _{x,y}\left[f(x-u,y-v)- { \overline{f}}_{u,v}\right]\left[t(x,y)- \overline{t}\right]}\over{{\left\{\sum _{x,y}{\left[f(x-u,y-v)- { \overline{f}}_{u,v}\right]}^{2}\sum _{x,y}{\left[t(x,y)- \overline{t}\right]}^{2}\right\}}^{1/2}}}&(2)}]

where f is the image, [\bar t] is the mean of the template and [{\bar f_{x,y}}] is the mean of f(x,y ) in the region under the template. In auto-HREM, a template image t is one of the experimental templates obtained from the first step, while the f image is one of the constructed maps of the simulated images that we built during the second step. After running the XC analysis, the matching simulated image templates are extracted and sorted based on the XC values, starting with the highest. The sorting is used as the highest XC coefficient is often not enough to uniquely determine the parameters of the experimental image at the initial stage of image matching. Further analysis on the sorted list is used to select the best match.

The above template matching process runs for each template extracted from the experimental image, independent of each other. However, the best match ideally should provide similar parameters to those from the neighbouring templates. For example, two nearest experimental templates should have similar thickness and crystal orientation in the absence of large defects such as phase boundaries. Further, thickness usually increases from the edge to the inside of the nanocrystal. In order to consider these similarities, we define the continuity constraint (CC): neighbouring templates should share similar parameters within a small range, so that there is the shortest `Euclidean distance' between the determined parameters. The idea of the CC is illustrated in Fig. 4[link]. In auto-HREM, an additional step is provided where the candidate parameters are sorted by the continuity as the process is repeated until consistent results emerge.

[Figure 4]
Figure 4
Schematic diagram showing how the fit parameters are determined using the continuity constraint. Among the candidates of a match, the new candidates are selected from the overlapping candidates between two templates.

3.2. Pendellösung fringes and sample thickness

To determine sample thickness, we use the Pendellösung fringes as recorded in the intensity modulations of an electron image. The basic idea is that by examining the intensity modulation in the nanocrystal, it is possible to determine the projected nanocrystal thickness more rigorously than the thickness determined solely based on template matching. The Pendellösung fringes are obtained from an experimental image by Fourier filtering using the selected Bragg spots. The oscillating intensity in the filtered images is used to determine the extinction condition for the selected reflections. By comparing the measured extinction with simulations, we can determine the thickness.

3.3. Computational requirements

The auto-HREM described here is implemented on a personal computer using a combination of Python and Matlab scripting together with the computer programs for image processing and multislice simulations as described in the above sections. The most time-consuming part is the generation of the image templates library. Once this is done, the matching process can be performed relatively quickly. The prerequisite is the crystal structure, which must be known for image simulations. For image analysis, the part of auto-HREM for image matching is fully automated, while the 3D crystal shape determination part still requires user experience in HREM and crystallography.

4. Results and discussion

4.1. Experimental image templates

Fig. 5[link] shows the HREM image of a TiO2 nanocrystal taken with the K2 camera during vacuum reduction at 550°C under the TEM column vacuum of 2 × 10−5 Pa. Figs. 5[link](a) and 5[link](b) were taken at 105 (e Å−2) s−1 with a 10 s exposure time. At a glance, the contrast of HREM in a single image is not uniform. The local contrast in the nanocrystal is also expected to change due to the change in local thicknesses. The contrast difference is also expected by the change in defocus for a 3D-shaped nanocrystal. Experimentally, the microscope defocus was adjusted to keep the image focused at the edge of the nanocrystal.

[Figure 5]
Figure 5
The HREM image of a TiO2 nanocrystal and selection of experimental templates. (a) shows the overall shape of the nanocrystal, which was imaged with a dose of 105 (e Å−2) s−1 and 10 s exposure time under the environmental conditions of 2 × 10−5 Pa and 550°C. (b) shows a magnified view of the boxed region in (a). (c) displays the templates extracted from (b). Images in the left column are cropped directly from (b). Images in the middle column are spatially averaged over the masked area of (c). (c) is the masked image of Fig. 5(b). The coloured region indicates the area where the XC coefficient is above the threshold.

Fig. 5[link](c) shows the experimental image templates extracted from Fig. 5[link](b). Selected examples are shown. Two types are highlighted, one directly cropped from the original image, and one averaged using TeMA (Zuo et al., 2014[Zuo, J. M., Shah, A. B., Kim, H., Meng, Y., Gao, W. & Rouviére, J. L. (2014). Ultramicroscopy, 136, 50-60.]). The areas where the averaging took place are marked and shown in Fig. 5[link](c).

The spatially averaged templates have the enhanced contrast from the reduction of image noise. For example, a higher XC correlation value (>0.8) can be obtained using the averaged template compared with the templates directly obtained from the raw image, which usually yields 0.4–0.5 for the correlation coefficient at most.

4.2. Nanocrystal orientation determination

To determine the crystal orientation, we start with the assumption that the nanocrystal is a single crystal so that the tilt angle is constant in a single image. This assumption helps to decrease the initial computational cost and enables us to estimate the values of the crystal thickness and defocus for the next image matching step. In the later image analysis, the single-crystal approximation is relaxed to allow the consideration of crystal mosaicity.

The initial matches were made over a broad range of beam tilt angles (Tx, Ty) and a narrow range of thickness (Th) and defocus (Δf). Since all four parameters, sample thickness, sample orientation (equivalent to beam tilt angles) and defocus, have an impact on the image contrast, the pattern matching was performed in the four dimensions. Fig. 6[link](a) plots the histogram of template matches made for each beam tilt angle. These matches were selected from the simulated templates according to the method described in Section 3.1[link]. In total, 13 experimentally averaged templates were compared with the simulated templates, which were calculated with parameters varying in the range of Tx = [−19, 19] px, Ty = [−19, 19] px, Th = [44, 88] nm and Δf = [−80, 0] nm (here, px stands for pixels with 1 px = 1.5 mrad). For each experimental template, we kept 100 candidate matches with top XC values. Fig. 6[link](a) plots their distribution for all 13 experimental templates. In Fig. 6[link](b), the candidate templates were further selected using the CC of Section 3.1[link]. Among the 100 candidate templates for the match found by the XC search for each experimental template, the candidates shared by two neighbouring experimental templates (iso members) were included. The consideration of CC narrows down the distribution of peaks from two separated peaks in Fig. 6[link](a) to neighbouring peaks near (2, −12) in Fig. 6[link](b). A second iteration in a reduced search region was then carried out to identify the best fit. In Figs. 6[link](c) and 6[link](d), we modified the range of comparison to Tx = [0, 16] px, Ty = [−16, 0] px, Th = [16, 88] nm and Δf = [−80, 40] nm. The search region was selected by visual inspection in this case, the selected region encompassing all nonzero pixels near (2, −12). We found that the increase of the search region by a few pixels in this case does not change the search results.

[Figure 6]
Figure 6
The histogram of template matches for each beam tilt angle. The intensity colours indicate the number of theoretical candidates for a template match. The CC significantly reduces the number of theoretical candidates. (a) and (b) were obtained after the first iteration with the starting simulation parameters indicated at the top-right corners. Before applying the CC (a), the maxima are observed at tilt (6, −14) and (−6, −14). After applying the CC (b), the maxima frequencies are shifted and clustered near (2, −12). (c) and (d) are the result from the second iteration. The maximum frequency peaks at (0, −16) before the CC is applied (c), and at (8, −2) after the CC is applied. Here the tilt is specified in px with 1 px equal to 1.5 mrad.

The above match yielded the clustered occurrence near (Tx, Ty) = (2, −12) px as the most probable sample orientation after the first iteration [Fig. 6[link](b)], and (8, −2) px as the most probable sample orientation after the second iteration [Fig. 6[link](d)]. The above process was repeated until the results converged with the parameters stabilized over subsequent iterations. Normally, this takes a few iterations, depending on the search region.

4.3. Thickness determination and nanocrystal shape reconstruction

In the next step, we used the Pendellösung fringes to determine nanocrystal thicknesses and shape. Fig. 7[link](b) shows an example of the Pendellösung fringes obtained from Fourier filtering of the as-recorded image in Fig. 7[link](a). The first dark fringe associated with ±(010) appears at 4 nm from the top surface. The positions of the intensity minima for other diffracted beams were similarly determined. Theoretically, to compare with the experiment directly, the extinction distance can be calculated by performing Fourier transform on the simulated images. At (Tx, Ty) = (8, −2) px, the calculated intensity minima appear at the crystal thicknesses of 30 and 84 nm for g = (010), 30 and 77 nm for g = (100), 30 and 75 nm for g = (110), and 31 and 81 nm for g = (1[\bar 1]0). Thus, the crystal thickness at 4–5 nm away from the surface is either 31 ± 0.5 nm or 79 ± 4 nm.

[Figure 7]
Figure 7
Nanocrystal shape determination. (a) shows the projected image of the nanocrystal along the [001] direction [same as Fig. 5[link](a)]. (b) shows the Fourier-filtered image of (a) with the filtering mask at g± = (010) as shown at the bottom-right corner. (c)–(e) The model shape constructed based on the experimental observation and the theoretical facet energy ratio. (f)–(g) The thickness z profiles along the y and x directions as marked in (c). The thicknesses are estimated from the Pendellösung fringes.

The observed rutile nanocrystal is faceted according to Fig. 7[link](a). To determine the crystal shape, we consider crystal faceting from the knowledge of equilibrium shape. The equilibrium shape model based on Wulff construction is readily available using the knowledge of the surface energy. The known surface energies of TiO2 are listed in Table 1[link]. The equilibrium shape of rutile crystals is a rod shape that is elongated along the [001] direction. The equilibrium crystal shape consists of 16 facets: four {110}, four {100} and eight {101}. The {110} planes are the dominant facet as they have the lowest surface energy. The rod shape of TiO2 has been reported before with 12 or 16 facets (Barnard & Zapol, 2004[Barnard, A. S. & Zapol, P. (2004). Phys. Rev. B, 70, 235403.]; Esch et al., 2014[Esch, T. R., Gadaczek, I. & Bredow, T. (2014). Appl. Surf. Sci. 288, 275-287.]). In our experiment, other than the stable {110} and {100} facets, {100}, {001} and {111} facets were also frequently observed. Additionally, the vicinal surfaces at the rounded corners give rise to variations in the nanocrystal shapes.

Table 1
The computed values for the surface energies of rutile, taken from Labat et al. (2008[Labat, F. P., Baranek, P. & Adamo, C. (2008). J. Chem. Theory Comput. 4, 341-352.]) and Esch et al. (2014[Esch, T. R., Gadaczek, I. & Bredow, T. (2014). Appl. Surf. Sci. 288, 275-287.])

Surface Energy (J m−2)
(110) 0.62
(100) 0.85
(101) 1.20
(001) 1.47

The projected shape of the rutile nanocrystal [Fig. 7[link](a)] can be combined with information about the crystal facets obtained from the Wulff theorem and experimentally observed Pendellösung fringes to construct the 3D shape of the nanocrystal. The distances to (110)/(100) surfaces are directly obtained by measuring the distance from the centre to the facet baseline. The thickness at the point where the extinction occurs is determined using information from the calculated Pendellösung fringes. For the nanocrystal imaged in Fig. 7[link](a), the crystal thickness at the first extinction can be 30 nm or 75–85 nm, according to the Pendellösung fringe calculation. Since the nanocrystal is likely to be elongated along the [001] direction according to Wulff construction, we chose the thickness value of 75–85 nm. We then reconstructed the 3D shape of the nanocrystal as shown in Figs. 7[link](c)–7[link](e) using trigonometry. The minimum thickness calculated at the edge of the specimen is 54 nm, while the maximum at the nanocrystal centre is 91 nm. The error in the determined thickness using this method is estimated at 5 nm. This estimation is further supported by the electron nano­diffraction result. Figs. 7[link](c)–7[link](e) show the model shape for the nanocrystal imaged in Fig. 7[link](a). As described above, the model was constructed based on the experimental observations of the projected shape and the measured thicknesses. We also assumed that the nanocrystal is faceted and, locally, the facet distance ratios follow the theoretical surface energy ratios as closely as possible. With these assumptions, we obtained the thickness z profiles of Figs. 7[link](f) and 7[link](g) along the marked lines in Fig. 7[link](c).

4.4. Image contrast

Here, we evaluate how well the simulated image matches the experimental image by comparing the intensity patterns. Fig. 8[link] shows one of the matching results taken at a thin region of the nanocrystal after we run the auto-HREM analysis as described in Section 4.2[link].

[Figure 8]
Figure 8
Image contrast comparison between the experiment (a) and the simulation (b) for a thin region of the TiO2 nanocrystal. The image dimensions are 10 nm in width and 13 nm in height. (c) and (d) are plotted from the horizontal lines of i and iii in (a) and (b), respectively. Likewise, (e) and (f) are plotted from the vertical lines ii and iv in (a) and (b), respectively. The maximum intensity per px (in vacuum) for (c) and (e) is 1, and 6.1 × 10−5 for (d) and (f).

Fig. 8[link](b) is the best match obtained for the experimental template of Fig. 8[link](a). The simulation parameters corresponding to Fig. 8[link](b) are Tx = 12, Ty = 0, Th = 41.4 nm and Δf = 5 nm. The simulation reproduces the brick-and-mortar-like pattern seen in the experimental template, with the XC coefficient of 0.91. The thickness also agrees well with the nanocrystal shape analysis. The estimated thickness of the experimental template in Fig. 8[link](a) is 54 nm, while Th = 41.4 nm for Fig. 8[link](b).

The image intensities are compared in Figs. 8[link](c)–8[link](e). The shape of the intensity profiles of the simulated pattern resembles the experimental one. The contrast of the experimental profile is 0.25 [Fig. 8[link](c)] and 0.27 [Fig. 8[link](e)] according to equation (1)[link], while the contrast of the simulation profile is 0.9 [Fig. 8[link](d)] and 0.52 [Fig. 8[link](f)]. The experimental contrast is thus two to three times lower than the theoretical contrast, indicating the presence of the Stobbs factor despite the improved PSF of direct electron detection. The main difference between the experimental and simulated images is the minimum (or background) intensity observed in the experimental template. The background intensity of the experimental image is about two times higher than the background in the simulated image. This difference in the minimum intensity explains most of the discrepancy in the experimental and simulated image contrast.

The higher contrast level of the simulated images than the experiment can be attributed to three major factors. The first is the assumption of coherent imaging used in our simulation, which treats the effect of beam convergence by the use of a damping envelope. This approximation breaks down when the electron multiple scattering effect becomes significant for a relatively thick crystal. Then, the appropriate approach is to integrate image intensity over the range of incident-beam angles. Our calculations show that such integration can reduce the image contrast by ∼30%. The second factor is the background intensity from inelastic scattering, mainly thermal diffuse scattering (TDS) at the elevated temperature and the increased phonon density of states for nanocrystals. TDS is distributed broadly in the reciprocal space. The energy loss of TDS is relatively small, less than 0.1 eV. It has been argued that the high-angle, incoherently scattered TDS electrons are subjected to strong phase shifts from the rapidly varying portion of the lens transfer function and will therefore largely contribute a constant background to the image (Boothroyd, 1998[Boothroyd, C. B. (1998). J. Microsc. 190, 99-108.]). In BF-STEM, a small BF detector can be used to effectively remove TDS and its contribution to the HREM image contrast (LeBeau et al., 2009[LeBeau, J. M., D'Alfonso, A. J., Findlay, S. D., Stemmer, S. & Allen, L. J. (2009). Phys. Rev. B, 80, 174106.]). In a conventional transmission electron microscope, a hard aperture can be used in the back focal plane of the objective length to reduce the contribution of high-angle scattering to HREM image contrast (but not as effectively as in BF-STEM). In an in situ experiment, however, the use of such a hard aperture is difficult for a number of experimental reasons, chiefly the compatibility with the sample holder. The images reported here were thus recorded without the objective aperture. A consequence of this is that high-angle TDS, which is detected in HAADF-STEM for example, contributes to the background of our experimental images. The TDS also affects the coherent contrast through the absorption effect, which we have considered through the use of the Debye–Waller factor and the calculation of the anomalous absorption effect in the electron image simulations.

The third factor is the mechanical instability of the sample, including sample motion, the stability of sample support and sample holder drift. The contribution of this factor can be minimized through the use of dose fractionation by counting electrons at a high frame rate and capturing the electron image in many subframes. Computational alignment of these subframes before averaging them can correct for motion-induced image blurring as demonstrated in cryo-electron microscopy [for a review, see Cheng (2015[Cheng, Y. (2015). Cell, 161, 450-457.])].

4.5. Image match and measurement of nanocrystal mosaicity

With the nanocrystal shape model established and the HREM image contrast validated, we then proceeded to match the experimental image with simulations. For this purpose, we selected a section of the experimental image [Fig. 9[link](a)]. To match the selected experimental image, we used two best estimates of the nanocrystal orientation, as identified in Section 4.2[link]. The crystal thickness was determined by the shape of the nanocrystal, and so was the defocus change. The only free parameter is the defocus value at the edge of the nanocrystal. Figs. 9[link](b) and 9[link](c) show the best matching simulated images using the above parameters under the single-crystal assumption. The thickness and defocus profiles used to simulate these images are shown in Figs. 9[link](d) and 9[link](e). Close inspection of images in Figs. 9[link](a) to 9[link](c) shows that a better match with the experiment is made with (Tx, Ty) = (8, −2) px for the thinner part of the nanocrystal (left side of the image) and with (Tx, Ty) = (0, −8) px for the thicker part of the nanocrystal (right side of the image). Thus, the image matching suggests that the nanocrystal consisted of mosaic blocks with a range of crystal orientations.

[Figure 9]
Figure 9
Image matches made with the single-crystal assumption for two selected nanocrystal orientations. (a) A section of raw experimental image taken from Fig. 7[link](a) along the vertical direction. (b), (c) The best matching simulated images using the nanocrystal shape model of Fig. 7[link] for the beam directions of (Tx, Ty) = (0, −8) px and (Tx, Ty) = (8, −2) px, which are identified from the orientation determination. (d) and (e) show the thickness and defocus profiles used for simulating images in (b) and (c).

The concept of mosaic crystals comes from X-ray diffraction. Most crystals contain defects and the quality of a crystal for X-ray diffraction is described by the parameter of mosaicity (also known as mosaic spread or rocking angle), which involves the degree of perfection of the lattice translations throughout the crystal. For electron diffraction, since electron diffraction patterns are often recorded with a microprobe or nanoprobe away from defects, the model of a perfect crystal often works remarkably well, as demonstrated by quantitative convergent-beam electron diffraction (CBED) (Zuo, 2004[Zuo, J. M. (2004). Rep. Prog. Phys. 67, 2053-2103. ]). However, there are few data about the mosaicity of nano­crystals. In X-ray diffraction, the way of describing mosaicity is to approximate the crystal as being composed of mosaic blocks, each block a perfect crystal. The diffraction peak is broadened when the mosaic blocks are disordered and distributed within a range of angles. Thus, mosaicity is given in degrees.

To measure the nanocrystal mosaicity, we performed image matching for individual experimental templates as shown in Fig. 10[link](a). The experimental templates were extracted directly from the experimental image without spatial averaging as shown in Figs. 10[link](b) and 10[link](c). For image matching, the beam direction is allowed to vary, as is the thickness (for the reason given below). The selection of matching images was made based on the highest XC value within a defined range of thicknesses. The matching results are displayed together with the original image and the spatially averaged templates. Excluding the two matching templates within the red box in Fig. 10[link](a), the determined beam direction shifts systematically from (12, 0) to (0, −6) in Fig. 10[link](a). The shift as measured in degrees amounts to 20 mrad or 1.15°. This shift occurs from the edge to the middle of the nanocrystal, and represents its mosaicity.

[Figure 10]
Figure 10
Final matching result after the second iteration result for Fig. 5[link](b): (a) the simulated image of the best fit; (b), (c) raw experimental image; (d) spatially averaged templates.

The crystal thickness is allowed to change for the measurement of mosaicity for the reason that electron diffraction along the thickness direction is also affected by the mosaic spread, which tends to reduce the extent of the dynamical effect, known in X-ray diffraction as primary extinction. This is reflected by the fact that the determined thicknesses from image matching in Fig. 10[link] tend to be smaller than the thicknesses determined by the nanocrystal shape, except the two templates inside the red box. In these two cases, the simulated images clearly deviate from the contrast observed in the spatially averaged experimental templates [Fig. 10[link](d)], which indicates the failure of image matching using the perfect crystal model at these local positions.

5. Conclusions

We have shown that by using direct electron detection and developing a quantitative image matching algorithm, auto-HREM, quantitative HREM analysis of TiO2 nanocrystals is possible. It is demonstrated here for the 3D shape analysis and the measurement of nanocrystal mosaicity. The main results include:

(i) Experimentally, in addition to the improvement of the signal/noise ratio, direct electron detection improves the experimental image contrast by a factor of two to five over the CCD camera.

(ii) The experimental HREM image contrast is two to three times lower than the theoretical contrast at the crystal thickness of ∼40 nm, indicating the presence of the Stobbs factor despite the improved PSF of direct electron detection.

(iii) The main difference between the experiment and simulation is the background intensity. The experimental background is about two times higher than the background in the simulated images.

(iv) The image match can be improved using a proper model to take account of the electron multiple scattering effect of the beam convergence angle.

(v) By subtracting the average local image intensity in the cross-correlation analysis, image matching between the experiment and simulations can be carried out quantitatively.

(vi) The determination of crystal orientation and thickness is critical for quantitative image analysis. This is helped by the 3D shape of nanocrystals.

(vii) Analysis of the experimental image of a rutile nanocrystal shows that local crystal orientation changes from the near-surface region to inside the nanocrystal. The amount of orientation change amounts to 1.25° for the observed nanocrystal (∼50 nm in size).

(viii) The determined thicknesses from image matching tend to be smaller than the thicknesses determined by the nanocrystal shape, which indicates the failure of the perfect crystal model along the electron beam direction.

(ix) Image matching fails in some region of the nanocrystal, where the best matching simulated images clearly deviate from the contrast observed in the spatially averaged experimental templates. To improve this, the mosaicity of the crystal structure along the electron beam direction should be considered in future electron image simulations.

Acknowledgements

We thank Drs James Mabon of the University of Illinois and Cory Czarnik of Gatan Inc. for technical help. We also thank Hitachi High-Technologies America Inc. for the loan of the MEMS sample heating holder.

Funding information

This work was supported by NSF grants DMR-1410596, DOE BES DEFG02-01ER45923 and NSF MRI-1229454.

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