# Coherent Diffraction Imaging in Transmission Electron Microscopy for Atomic Resolution Quantitative Studies of the Matter

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## Abstract

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## 1. Introduction

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- In Section 2, the EDI/KEDI experimental set up is introduced. Experimentally, EDI/KEDI require the acquisition of one HRTEM image and one nanodiffraction pattern, by using the same illumination configuration on the same specimen area [7]. The need to fulfill the theoretical requirement necessary for a successful EDI/KEDI experiment poses some constraints on the experimental conditions and on the specimen illuminated area. The basis of the theory for phase reconstruction will be introduced, accounting for the Shannon theorem [10] and the generalized sampling theorem [11].
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- In Section 3, the data reduction strategies are discussed. The experimental images and nanodiffraction patterns need an appropriate data reduction prior to apply the phasing procedure. As will be shown, the data reduction can also involve methods to maximize the information that can be extracted from EDI/KEDI experiments resulting in an augmented EDI/KEDI quantitative imaging [12,13].
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- In Section 4, the phase retrieval processes are discussed. The phasing process requires algorithms capable of retrieving the phase lost in the nanodiffraction experiments. Different kinds of phase retrieval algorithms will be applied. In particular, we focus on recent developments capable of minimizing the reconstruction errors starting from low-resolution information of the support and from completely random phases [14].
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- In Section 5, we draw the conclusions and discuss the future perspectives of EDI/KEDI.

## 2. EDI/KEDI Experimental

**k**) of an electron diffraction pattern of a thin specimen illuminated by a wave function Ψ(

**r**) with wavelength λ is described in the kinematical approximation [5] by:

**r**) is the Coulomb potential of the sample medium, m and e are the mass and the charge of the electron respectively, and h is the Planck’s constant. For plane wave illumination, the electron diffraction intensity is related to the Fourier transform F(

**k**) of the potential U(

**r**) by [5]:

**Sf ≥**1. (

**Sf**)

**, with n the dimension of the image, is the oversampling ratio [8] and mathematically it must be at least 2 to tackle the phase retrieval process successfully [20]. This requirement and the coherent length of the electron sources nowadays available pose a constraint on the linear size of the object to be imaged by electron coherent diffraction imaging to about 10–20 nm [19]. The whole illuminated area from which the HRTEM image and the nanodiffraction patterns are acquired has a diameter of the order of about 50 nm, to satisfy the relevant oversampling requirements. An example of EDI experiment is reported in Figure 1. In this case the rod is about 18 nm in length and 5 nm wide, whereas the illuminated area is of 40 nm [7] (not all the illuminated area is shown in the picture).**

^{−n}_{2}, oriented along the [100] zone axis, at a resolution limited by the electron lens aberrations to 190 pm [7]. The HRTEM image is used: (i) to estimate the size and shape of the rod to be used as a priori information in the phase retrieval process; (ii) to estimate, by its Fast Fourier Transform (FFT), some of the low frequency diffracted intensities lost in the nanodiffraction pattern due to the beam stopper. Figure 1b reports the nanodiffraction pattern, complete with direct beam and some of the low frequency intensities, as derived from the FFT of the HRTEM after proper rotation and scaling [5,7]. Figure 1c shows the phase reconstructed image at a resolution of 70 pm [7], which makes it possible to distinguish the projection of the oxygen atomic columns on the TiO

_{2}anatase (100) crystallographic plane (see the anatase projected crystal cell in the inset). It is worthwhile to remark, that the oxygen atomic columns closer to the Ti atomic columns are invisible in the magnified HRTEM image in Figure 1d, due to the limited resolution caused by the electron lenses aberrations. The about threefold gain in resolution enables to detect the distortion of the TiO

_{2}anatase crystal cell, related to the photocatalytic properties of TiO

_{2}anatase at the nanoscale [7].

^{−2}). This set up makes it possible to acquire the diffraction pattern without using the beam stopper [9]. This is an immediate advantage with respect to the experimental set up used in EDI where the use of the beam stopper makes the diffraction pattern incomplete. In KEDI experiments, to achieve a resolution ρ, it is necessary to measure in the reciprocal space at least a frequency of (1/ρ) pm

^{−1}, and the pixel size of the detector should be at least two times smaller than ρ to produce an image not pixelated. For example, to achieve an image resolution of 70 pm, using a 1024 × 1024 array detector, with physical pixel size of about 19 microns, the detector has a total field of view (FoV) of about 30 nm. To achieve the Nyquist’s oversampling conditions the area of the specimen illuminated by the electron beam has to be $\le {2}^{-\frac{1}{2}}\xb7\left(\mathrm{FoV}\right)\cong 21\mathrm{nm}$. In Figure 2, an example of KEDI experiment, applied to a bulk specimen of Si oriented along the [112] direction, is shown. A silicon specimen in [112] orientation represents a benchmark for the spatial resolution as in this configuration the Si atoms in the dumbbell are 78 pm apart. The HRTEM image in Figure 2a shows the self-confined illuminated area, which represents the support in KEDI experiments. The interference fringes for Si (112) are visible within the illuminated area, but the resolution, aberration-limited at 190 pm at optimum defocus [1], prevents to resolve finer details of the interference pattern. The size of the illuminated area is of about 9 nm and hence the conditions for resolving the Si dumbbell at 78 pm in KEDI are largely satisfied. In Figure 2b the nanodiffraction pattern is reported; the arrows point the highest spatial frequency reflections measured, corresponding to a spacing of 72 pm; hence the highest spatial frequency information is enough to resolve the Si dumbbell. In Figure 2c, the image after phase recovery is reported; the red square marks the region shown magnified in Figure 2d. For reader convenience, in the top left part of Figure 2d, the Si crystal cell (in blue) and the Si atomic column positions (in red), projected on the (112) crystal plane, are reported. In the inset below the red line, the simulation of the Si (112) projected potential is reported. The experimental intensity profile along the red line in Figure 2d is shown in Figure 2e (solid line) together with the relevant simulation of the Si (112) (red dots) projected potential intensity profile. As expected, the resolution of the phase recovered image enables to safely achieve the resolution of 78 pm, necessary to distinguish the Si atoms in the dumbbells. Some small artifacts at low intensities, in between the projected atomic columns, are visible in Figure 2d. These artifacts are similar to those shown in Figure 6 of Ref. [9] and could be related to small dynamical diffraction contributions to the diffracted intensities.

## 3. Data Reduction

_{3}specimen in [001] orientation.

_{3}atoms, in [001] projection, by dots of different colors (see caption of Figure 3). The simulation points that the specimen area is about 25 nm thick, in agreement with the measurement made by convergent beam electron diffraction, which is definitely not thin enough to neglect the dynamical diffraction effects [22]. The other element pointed by the simulation is that the maxima in the experimental image are centered on the Sr and Ti+O be atomic columns, whereas the O columns are invisible in the HRTEM image, even if the image has been acquired at Scherzer’s defocus [22].

_{3}potentials projected on the (001) crystallographic plane. The resolution of the image is related to the finer spatial frequency in the pattern in Figure 3, which is of 65 pm. Figure 4c,d are the simulated and experimental SrTiO

_{3}projected potentials, respectively, and the agreement between them, is not only qualitative, but also quantitative in the intensity of each projected potential atomic columns, within an accuracy of 5%. This is a good agreement, if we consider that the crystal potential calculations performed by linear combination of atomic potentials, as those in Figure 4c, are affected by an error of about 10% [24].

## 4. Phasing Process

_{3}(001), showing the atomic columns of oxygen not visible in the experimental HRTEM. From the results in Figure 6 the area shown in Figure 4d has been extracted. The latter is compared with the simulated SrTiO

_{3}(001) crystal projected potential shown in Figure 4c. The experimental projected potentials intensities are quantitatively in agreement, within 5%, with what expected by simulating the projected potentials of Sr, Ti+O and O columns projected on the (001) crystallographic plane and hence represent a structural and chemical map of the SrTiO

_{3}specimen. The widths of the projected potential peaks are also in agreement with the spatial resolution of 65 pm of the retrieved image, as evidenced by the comparison with the simulated projected potentials in Figure 4c.

## 5. Conclusions and Future Perspectives

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) HRTEM image of a TiO

_{2}nanorod in [100] zone axis; (

**b**) nanodiffraction pattern complemented with the Fast Fourier Transform (FFT) of the HRTEM after rotation and scaling; (

**c**) image after phase reconstruction; (

**d**) magnified view of a portion of the HRTEM image in (

**a**). (Reprinted with permission from De Caro et al. [7]).

**Figure 2.**Keyhole electron diffraction imaging (KEDI) experiment on Si [112]: (

**a**) HRTEM imaged of the self-confined illuminated area (support); (

**b**) nanodiffraction pattern from the area shown in the HRTEM experiment in (

**a**), the arrows point the diffracted spots at the highest frequency corresponding to a lattice spacing of 72 pm; (

**c**) image reconstruction in the illuminated area after phase retrieval process; (

**d**) magnified view of the image in (

**c**) together with the atomic columns position within the Si crystal cell in [112] projection. The inset shows the simulation of the atomic projected potential in [112] projection; (

**e**) simulated (dots) and measured (solid line) intensity profile of the Si atomic columns in [112] projection showing the Si dumbbell spacing at 78 pm well resolved.

**Figure 3.**Left- HRTEM image of a nano-region of a SrTiO

_{3}extended sample in [001] zone axis, with a zoom in the left inset, and the relevant simulation in the right inset (objective lens underfocus of 41.3 nm and specimen thickness of 25.0 nm); the dots in the simulation point to the structural positions of the SrTiO

_{3}atomic species in [001] projection: Sr = Blue, Ti+O = green, O = red. (

**a**) KEDI raw experimental nano-diffraction pattern (logarithmic scale); (

**b**) sup{I(s)} rescaled pattern; (

**c**) difference between patterns shown in (

**a**,

**b**); (

**d**) comparison in a logarithmic scale between the line profile along the dashed blue line in (

**a**) (blue curve) and along the dashed red line in (

**b**) (red curve) after rescaling. Black curve is the corresponding profile of sup{I(s)} × I

_{max}constraint. (Reprinted with permission from De Caro et al. [12]).

**Figure 5.**Description of the Memetic Phase Retrieval (MPR) approach. The standard approach can be interpreted as MPR deprived of genetic operations of Selection, Crossover and Mutation.

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**MDPI and ACS Style**

Carlino, E.; Scattarella, F.; Caro, L.D.; Giannini, C.; Siliqi, D.; Colombo, A.; Galli, D.E.
Coherent Diffraction Imaging in Transmission Electron Microscopy for Atomic Resolution Quantitative Studies of the Matter. *Materials* **2018**, *11*, 2323.
https://doi.org/10.3390/ma11112323

**AMA Style**

Carlino E, Scattarella F, Caro LD, Giannini C, Siliqi D, Colombo A, Galli DE.
Coherent Diffraction Imaging in Transmission Electron Microscopy for Atomic Resolution Quantitative Studies of the Matter. *Materials*. 2018; 11(11):2323.
https://doi.org/10.3390/ma11112323

**Chicago/Turabian Style**

Carlino, Elvio, Francesco Scattarella, Liberato De Caro, Cinzia Giannini, Dritan Siliqi, Alessandro Colombo, and Davide Emilio Galli.
2018. "Coherent Diffraction Imaging in Transmission Electron Microscopy for Atomic Resolution Quantitative Studies of the Matter" *Materials* 11, no. 11: 2323.
https://doi.org/10.3390/ma11112323