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Electron Nanodiffraction

  • Jian-Min ZuoEmail author
Chapter
Part of the Springer Handbooks book series (SHB)

Abstract

This chapter introduces the practice and theory of electron nanodiffraction. After a brief introduction, the chapter provides a comprehensive description of electron diffraction techniques and their use for nanodiffraction. This is followed by discussions on electron probe properties, electron energy filtering and electron diffraction data analysis. Throughout the chapter, we emphasize different electron nanoprobes that can be formed inside an electron microscope, from a focused beam to parallel illumination, and how these probes can be used to extract structural information from different materials. For this purpose, we outline the electron diffraction theories based on both kinematic approximation and dynamic diffraction, which serve as the basis for the interpretation of electron nanodiffraction patterns. The principles and applications of scanning electron nanodiffraction and coherent diffraction imaging are covered in detail with applications for orientation mapping, imaging strain, 3-D nanostructure determination, and study of defects.

electron diffraction

nanostructure analysis convergent beam electron diffraction (CBED) diffractive imaging four-dimensional scanning transmission electron microscopy (4-D STEM) strain mapping atomic resolution tomography 

Here, electron nanodiffraction refers to a set of electron diffraction techniques that enable structural analysis at nanoscale. Specifically, the techniques we discuss here extend the study of electron diffraction to nanostructures, small crystals, defects, and materials' microstructure. We show how such studies can be carried out quantitatively at spatial resolutions ranging from sub-Å to nm and in two-dimensional () projections to three-dimensional () reconstruction.

Many of the instrumental requirements for electron nanodiffraction are similar to those for analytical electron microscopy. In fact, modern analytical transmission electron microscopes ( s) provide special convergent beam electron diffraction ( ) or nanobeam modes, or both, together with the modes for low and high magnification imaging, diffraction, and energy-dispersive x-ray ( ). This development resulted partly from the fact that the requirements for EDX (large tilt, small probe, low contamination) exactly match those for nanodiffraction. Analytical TEMs designed for scanning transmission electron microscopy ( ) also feature a bright field emission gun ( ), an improved vacuum system, and the instrument stability required for electron nanodiffraction. Most importantly, there is a clear scientific merit here since microanalysis complements electron diffraction in analytical transmission electron microscopy (TEM) by providing both chemical and crystallographic information.

The predecessor of electron nanodiffraction is electron microdiffraction using CBED. The first CBED pattern was recorded by Möllenstedt as early as 1939 using a two-magnetic lenses setup [18.1]. MacGillavry first attempted structure factor measurement by using the two-beam theory of Blackman for dynamic electron diffraction [18.2]. The multibeam theory of electron dynamic diffraction has its origin in the Bloch wave method originally formulated by Hans Bethe in his PhD thesis. Electron nanodiffraction started with the development of field emission gun (FEG) in the 1970s, which led to the development of dedicated STEM. Electron nanodiffraction took advantage of the small and highly coherent electron beam in these instruments (see Cowley's reviews [18.3, 18.4]). However, in these early STEMs, diffraction patterns were often recorded using TV cameras and on video tapes; the quality of the recorded diffraction patterns was poor, and the handling of video data was difficult. The development of array detectors, such as CCD cameras or imaging plates, enabled parallel recording of diffraction patterns and quantification of diffraction intensities over a large dynamic range that became widely available only in the 1990s [18.5, 18.6]. At about the same time, field emission instruments that combine STEM with TEM were developed. Electron energy filters, such as the in-column \(\Upomega\)-energy-filter , also became available, which allowed the inelastic background intensities from plasmon or higher electron energy losses, to be removed from recorded diffraction patterns with an energy resolution of a few eV [18.7]. These developments in electron diffraction hardware were accompanied by the development of efficient and accurate algorithms to simulate electron diffraction patterns [18.8] and modeling structures on a first-principle basis. By the late 1990s, many technical difficulties encountered in performing electron nanodiffraction using dedicated STEM [18.3] had been solved. Other more recent developments are time-resolved electron diffraction at the time resolution approaching femtoseconds [18.10, 18.11, 18.9], coherent nanoarea electron diffraction for the study of individual nanostructures [18.12] and electron diffraction using fast pixelated detectors or segmented detectors [18.13, 18.14]. Further developments of these techniques will significantly improve our ability to interrogate structures at high spatial and time resolution, which hitherto was not available before.

This chapter provides a comprehensive coverage of electron nanodiffraction. The intention is to treat the topic in a reference format that readers will find useful as a guide for materials characterization using electron nanodiffraction. The chapter is organized in five sections. In Sect. 18.1, various electron nanodiffraction techniques are described. This is followed by a discussion on electron probe formation in Sect. 18.2 and energy filtering in Sect. 18.3. Section 18.4 is on diffraction analysis. This part covers three major aspects:
  1. 1.

    Diffraction pattern indexing and mapping

     
  2. 2.

    Convergent beam electron diffraction (CBED)

     
  3. 3.

    Coherent electron nanodiffraction.

     
Examples included in Sect. 18.4 illustrate the applications of orientation mapping, strain analysis, 3-D nanostructure determination, crystal local symmetry determination, structure factor measurements, and diffractive imaging. Section 18.5 provides a brief conclusion.

Some background on transmission electron microscopy is needed and introductory materials and theoretical background that are not covered here can be found in several books [18.15, 18.16, 18.17, 18.18, 18.8].

18.1 Electron Diffraction Techniques

18.1.1 Selected Area Electron Diffraction (SAED)

Selected area electron diffraction ( ) is performed by illuminating the sample with a large defocused electron beam. The diffraction pattern is recorded from a selected sample area by placing an aperture at the image plane of the objective lens, as shown in Fig. 18.1. This plane is conjugate to the sample. Only electron beams passing through this aperture contribute to the diffraction pattern seen by the next intermediate lens. The electron beams come from the sample area defined by the virtual image of the selected area aperture at the specimen level. For an ideal lens, with the aperture centered on the optical axis, a small area at the center of the observed area is selected. This area is much smaller than the size of the aperture because of the objective lens magnification. A TEM equipped with an imaging (not probe) aberration corrector comes close to providing an ideal objective lens. Without the corrector, rays belonging to diffracted beams are at an angle to the optical axis, and they are displaced away from the center because of the spherical aberration of the objective lens (\(C_{\mathrm{s}}\)). The displacement is proportional to \(C_{\mathrm{s}}\alpha^{3}\), where \(\alpha\) is twice the Bragg angle (Fig. 18.1). The smallest area that can be selected in SAED is thus limited by objective lens aberrations. This limitation is largely removed when using an electron microscope equipped with a TEM aberration corrector placed behind the objective lens.

Fig. 18.1

Selected area electron diffraction in conventional TEM (Provided by Jun Yamasaki of Nagoya University, Japan)

Fig. 18.2a,b

Selected area diffraction in an aberration corrected TEM. (a) Diffraction pattern recorded from a silicon crystal along [110]. (b) TEM image of the selected area on the silicon crystal. The aperture hole is \({\mathrm{240}}\,{\mathrm{nm}}\) in diameter, fabricated in a copper thin plate using focused ion beams ( ). Reprinted from [18.19], with the permission of AIP Publishing

SAED is a popular diffraction technique in TEM. The technique can be applied to study both crystalline and noncrystalline materials. A large area illumination is useful for recording diffraction patterns from polycrystalline samples or for averaging over a large volume (for example, a large number of nanoparticles). SAED can also be used for low-dose electron diffraction, which is required for studying radiation sensitive materials, such as organic molecules. For small area analysis, the nanoarea electron diffraction technique described next is more appropriate. Alternatively, an aberration corrected TEM coupled with a small aperture can be used for electron nanodiffraction. For example, Morishita et al demonstrated that coherent diffraction can be achieved from areas as small as \(\approx{}{\mathrm{10}}\,{\mathrm{nm}}\) using this technique [18.19], also see Fig. 18.2a,b.

18.1.2 Nanoarea Electron Diffraction ( ) and Nanobeam Diffraction ( )

NAED uses a nanometer-sized parallel beam, with the condenser/objective setup shown in Fig. 18.3a-d [18.12], together with the use of a small condenser aperture. An auxiliary condenser lens, called condenser minilens or CM, placed immediately above the condenser-objective lens is also employed [18.20]. The CM lens takes the crossover formed by the last condenser lens (CL ) and images it onto the front focal plane of the objective prefield lens, which then forms a parallel beam on the specimen. Adjustment to the parallel beam can be made by changing the CL and CM lens excitations; the CL lens moves the beam crossover closer or further away from the CM lens, thus changing the beam divergence angle seen by the CM lens. For a condenser aperture of \({\mathrm{10}}\,{\mathrm{\upmu{}m}}\) in diameter, the probe diameter is \(\approx{}{\mathrm{50}}\,{\mathrm{nm}}\) with an overall magnification factor of \(1/200\) in the JEOL 2010 or 2100 electron microscopes (JEOL, USA). The beam size is much smaller than what can be achieved using a selected area aperture. Diffraction patterns recorded in this mode are similar to the SAED patterns. For crystals, the diffraction pattern consists of sharp diffraction spots. The major difference is that the diffraction volume is defined directly by the electron probe in NAED, since most electrons illuminating the sample are recorded in the diffraction pattern. NAED in an FEG microscope also provides higher beam intensity than SAED (the probe current intensity using a \({\mathrm{10}}\,{\mathrm{\upmu{}m}}\) condenser II aperture in a JEOL 2010F is \(\approx{}{\mathrm{10^{5}}}\,{\mathrm{e/(s{\,}nm^{2})}}\)) [18.12]. The small beam size allows the selection of an individual nanostructure and reduction of the background in the electron diffraction pattern from the surrounding materials. Figure 18.4a,b shows an example. The diffraction pattern was recorded from a single-wall carbon nanotube encapsulated with C60 molecules using a \({\mathrm{25}}\,{\mathrm{nm}}\) diameter electron probe (Fig. 18.4a,ba). The diffraction lines marked by the arrows are from C60 molecules. Together, \(\approx 25\) C60 molecules were selected and contributed to diffraction.

Fig. 18.3a-d

Comparison between (a) CBED, (b) NAED, (c) NBD, and (d) TEM illumination for SAED. The sample is located at the lower end of the diagrams

Fig. 18.4a,b

Nanoarea electron diffraction of a small diameter peapod (C60 molecules encapsulated inside a single-wall carbon nanotube). (a) High resolution TEM image of an isolated peapod. The inset shows the electron beam used for diffraction. (b) An experimental diffraction pattern recorded from the peapod shown in (a). The arrows indicate diffraction by C60 molecules. Reprinted from [18.21], with permission from Elsevier

A focused probe can be formed by weakening the CL lens and placing the crossover at the front focal plane of the CM lens. This results in a focused probe on the specimen, which is placed at the focal plane of the objective prefield lens. When using a small condenser aperture with a small convergence angle, the probe size becomes diffraction limited in an FEG TEM. The diffraction patterns recorded in this case consist of small disks (Fig. 18.13a-eb). This nanodiffraction technique was pioneered by Cowley [18.4], it is now called NBD.

18.1.3 Convergent-Beam Electron Diffraction ( )

CBED is recorded using a focused electron probe at the specimen. Compared to the diffraction techniques that we have discussed so far, CBED differs in terms of the beam convergence angle (\(\theta_{\mathrm{c}}\)) and the electron probe size. The convergence angle is several times larger than what is used in NBD, but it is still significantly smaller than the convergence angle used in an aberration corrected STEM. The convergence angle is largely determined by the size of the condenser aperture (CA ). The CA is considered to be conjugate to the diffraction pattern in CBED. Using an additional mini-lens placed above or in the objective prefield, it is also possible to vary the convergence angle by changing the strength of the mini-lens for CBED. In addition, in CBED performed using a thermionic electron source, the incident plane-wave components of the illumination are considered to be incoherently related.

The relatively large convergence angle used for CBED gives rise to transmitted and diffracted disks (Fig. 18.5); the size of the disk determines the range of excitation errors for each reflection (more in the Sect. 18.4.2 on the geometry of CBED). Thus, the convergence angle is a very important parameter in CBED. Its choice depends on the application. Along a zone axis, the ideal CBED disk size is twice the Bragg angle of the lowest order zero-order Laue zone ( ) reflection, in order to fill the diffraction space as nearly as possible with scattered rays. In an off-zone axis orientation, a large CBED disk can be used to extend the number of (high order Laue zone) lines recorded in the transmitted disk. As the desired convergence angle changes from one crystal to another or one application to another, a TEM designed for CBED provides a range of excitations of the CM lens so it can be used to vary the convergence angle as shown in Fig. 18.3a-d. The size of the CBED disk for a fixed CM lens excitation is determined by the condenser aperture size and the focal lengths of the probe-forming lenses. Experimentally, by having several condenser apertures from a few \(\mathrm{\upmu{}m}\) to several tens of \(\mathrm{\upmu{}m}\), it is possible to cover a range of convergence angles for many materials science applications.

Fig. 18.5

CBED pattern recorded from spinel (\(\mathrm{MgAl_{2}O_{4}}\)) at \({\mathrm{120}}\,{\mathrm{kV}}\), energy filtered, using LEO 912 TEM

If the CA is coherently illuminated using a field emission electron source, the electron probe diameter is less than the lattice spacing, when the coherent CBED disks are allowed to overlap, it then becomes possible to form a scanning transmission (STEM) lattice image. By observing this STEM lattice image, it thus becomes possible (in thin crystals) to stop the probe on the region at which a CBED pattern is required. Alternatively, electron diffraction can be recorded at each probe position for STEM (four-dimensional STEM, see Sect. 18.1.5). By these methods, it is quite possible to obtain CBED patterns from different regions within a single unit cell, and that these show different site symmetries, or alternatively, by averaging over one or several unit cells, to obtain their average symmetry. In order to obtain sufficient intensity from a probe of subnanometer dimensions, an instrument fitted with a field emission gun is needed for this type of work. For the analysis of large crystals, the benefit of a FEG is the improved plane-wave coherence at the specimen level. This also makes it sensitive to the contributions from defects in a real crystal. However, because of the small focused probe, the pattern has reduced contributions from thickness variations and bending under the probe.

For very thin crystals, the resulting patterns may be interpreted as electron holograms. Coherent CBED patterns formed with a very large illumination aperture have a special name, ronchigrams. The interpretation of ronchigrams is discussed in Chap.  2, on STEM, since these provide the simplest and most accurate method of aligning the instrument and measuring the optical constants of the probe-forming lens.

Fig. 18.6a-c

Experimental CBED patterns recorded from thin crystals of spinel (\(\mathrm{MgAl_{2}O_{4}}\)) along zone axes of (a) [100], (b) [110] and (c) [111]. Reprinted from [18.22], with the permission of AIP Publishing

CBED patterns from very thin crystals also show very few features within each diffracted disk (Fig. 18.6a-c). This leads to the possibility of using the average disk intensity to measure the magnitude of the structure factor based on the kinematic (single-scattering) theory. The angular width of the rocking curve feature in CBED is inversely proportional to the sample thickness, so that we might expect the intensity to be constant within each disk for a sufficiently thin crystal. Such a kinematic convergent beam ( ) (or blank disk) method has been investigated [18.22], and it was found to have the following advantages:
  1. 1.

    It allows use of the smallest electron beam diameter for solving true nanocrystal structures.

     
  2. 2.

    Since the beam energy is spread out throughout the disks, the (000) disk intensity may be measured without saturating the detector, so that absolute intensity measurements can be made, comparing the intensity of the zero-order beam with the Bragg intensities.

     
  3. 3.

    One has a test, which is independent of the (unknown) crystal structure, for the presence of unwanted multiple scattering, if the structure is known to be noncentrosymmetric. In that case, these CBED patterns will only be centrosymmetric (in accordance with Friedel's law) if the scattering is kinematic. (Friedel's law is violated in the presence of multiple scattering.)

     
Experimentally, one needs to obtain good quality CBED patterns from all the major zone axes of the crystal, which may be difficult for very small nanocrystals, depending on the degree of symmetry and radiation damage limitations. Full details of the method, as used to solve the structure of a spinel crystal with about \({\mathrm{0.03}}\,{\mathrm{nm}}\) resolution, are given in McKeown and Spence [18.22]. Here, a three-dimensional map of the crystal potential was obtained, including the positions of the oxygen atoms. To solve the phase problem , the remarkable charge-flipping algorithm was used [18.22]. (The charge-flipping algorithm is described in Sect. 18.4.4, Phase Retrieval Algorithms.)

18.1.4 Large-Angle Methods

Various instrumental techniques have been developed to obtain an angular view of a diffracted order that is greater than the Bragg angle. The earlier methods for doing this were reviewed by [18.23]. Such an angular expansion is required for space-group determination of crystals with a large unit cell, in which overlap of low orders may occur at such a small illumination angle \(\theta_{\mathrm{c}}\) that little or no rocking-curve structure can be seen within the orders. It has also been discovered that many narrow high-order reflections may be observed simultaneously using large-angle techniques.

Closely related to the large-angle methods are ronchigrams and shadow images, which are described in Chap.  2, however they differ according to the angular range over which the illumination is coherent. In this section, we deal only with incoherent conditions and the application of techniques used to prevent the overlap of orders.

The Tanaka or (large-angle convergent-beam electron diffraction) method [18.24] allows parallel detection of the entire wide-angle pattern and requires no instrumental modifications. The pattern is again, however, obtained from a rather large area of sample. A description of the method is given in [18.25]. Figure 18.7 shows the principle of the method, while Fig. 18.8 shows a pattern from (111) silicon taken at \({\mathrm{120}}\,{\mathrm{kV}}\) by this method. In Fig. 18.7, the CBED probe was focused on the object plane of the objective lens, while the sample was moved up by a distance \(\mathrm{d}S\), forming an image of the electron source in the plane of the selected area aperture. A source image is formed in every diffracted order, as shown. The aperture can then be used to isolate one source image and so prevent other diffracted beams from contributing to the image. Because the source images are small at the crossover, the illumination cone can be opened up to a semiangle that is larger than the Bragg angle. The tradeoff is the large area of sample illuminated by the out-of-focus probe at the sample. In addition, different regions of the sample contribute to different parts of the diffraction pattern. Patterns may be obtained with the probe focused either above or below the sample; the best results seem to be obtained with it below the sample (for TEM instruments).

Fig. 18.7

Tanaka's LACBED method. Here, \(\alpha\) and \(\theta\) denote the half-convergence and Bragg angles, respectively, and \(M\) is the objective lens magnification

Fig. 18.8

Large-angle CBED pattern recorded from Si [111] at \({\mathrm{120}}\,{\mathrm{kV}}\). Provided by John Steeds, Bristol University

An important finding is that the use of the smallest selected area aperture together with the largest permissible defocus minimizes the contribution of inelastic scattering to the pattern. This effect was studied in detail in [18.26]. A similar technique is used to image other diffracted orders. Here, the order of interest is brought onto the optical axis using the dark-field tilt controls.

If the geometric probe size and the effects of spherical aberration are both small, the diameter \(D\) of the region from which the pattern is obtained is given approximately by
$$D=2\alpha\,\mathrm{d}S\;,$$
(18.1)
where \(2\alpha\) is the beam convergence angle. The smallest \(\mathrm{d}S\) (Fig. 18.7) that allows separation of the diffraction orders should be used to minimize \(D\). A small geometric source image (consistent with sufficient intensity on the viewing screen) also facilitates the separation of orders; this is controlled by the demagnification settings of the condenser lenses. Applications of the above LACBED method can be found in [18.27]. A variation of this method has also been demonstrated, which makes it possible to record simultaneously on a single micrograph most of the CBED pattern, together with several diffracted orders at the Bragg condition [18.28].
Fig. 18.9

Experimental CBED pattern from silicon containing a dislocation as marked by arrows. The number of splits can be used to identify the dislocation's Burgers vector [18.29]

For defects, the LACBED or CBED technique can characterize individual dislocations, stacking faults and interfaces ([18.30, 18.31, 18.32], also Fig. 18.9). For applications to surfaces and interfaces, and structure without three-dimensional periodicity, parallel-beam illumination with a very small beam convergence is required.

The LACBED techniques that we have described so far are designed to avoid the overlap of diffraction disks by recording the intensities of a single reflection. Many applications require the intensities of multiple reflections, which can be obtained using the large-angle rocking-beam electron diffraction ( ) technique described by Koch [18.33] (Fig. 18.10a,b). The principle of LARBED is similar to the double-rocking beam technique developed earlier [18.34, 18.35]. It works by rocking the incident beam over a certain angular range, while ensuring that the same selected area of the sample contributes to the diffraction pattern and the same diffracted beam stay on the detector. For every incident-beam direction in this angular scan the intensity of the transmitted beam or a diffracted beam is displayed on a video monitor. The differences are: 1) LARBED uses a partial descan to produce a small diffraction circle from the precession of the incident beam, and 2) the diffraction rings are recorded on a charged-coupled device (CCD) camera instead of a point detector as in LACBED.

Fig. 18.10a,b

LARBED patterns collected with a parallel beam (a) and a convergent beam (b). In (a), for each tilt angle a separate diffraction pattern is recorded and the integrated background-subtracted intensity for each reflection is extracted for the corresponding beam tilt. In (b), for each beam tilt a CBED pattern is recorded. Provided by Christoph Koch, Humboldt University of Berlin

18.1.5 Scanning Electron Nanodiffraction, Scanning CBED, and 4D-STEM

Using the deflection coils, scanning electron nanodiffraction ( ) or scanning CBED ( ) patterns can be recorded from an area of the sample for every probe position, to provide spatially resolved structural information. This can be done either using a TEM or STEM. SCBED performed in a STEM with overlapping disks is also known as 4-D STEM. Diffraction patterns are recorded using a 2-D digital detector, for example, a CCD camera. Compared to position averaged CBED ( ) (Sect. 18.1.7), which records one diffraction pattern over many probe positions, SEND or SCBED collects the full 4-D data, in the form of two spatial coordinates, \((x,y)\) in the real space and \((k_{x},k_{y})\) in the reciprocal space. The only difference between SEND and SCBED is the beam convergence angle (which is larger for SCBED). For this reason, we will focus simply on SEND in the following discussion.

Fig. 18.11

Saddle yoke magnetic coils for electron beam deflection. A horizontal magnetic field is produced by a pair of coils of \(N\) turns with current \(I\) flowing in opposite directions. Electrons traveling vertically experience a force \(F\) as shown

When a pair of magnetic coils (Fig. 18.11) are arranged perpendicular to each other, they apply uniform forces on the beam electrons along the horizontal (\(x\) and \(y\)) directions. Four pairs can be used to shift or tilt the beam along any direction in the \(x\)\(y\) plane. Two pairs make a set of deflection coils covering the \(x\) and \(y\) directions. Double deflection coils are placed below the CL lens and above the CM lens. They are used to provide beam shift, bright-field beam tilt, and dark-field beam tilt. When driven by an external scan generator, they are used to scan the probe in a raster over the specimen and to form STEM images by coupling the scan together with a detector. In electron diffraction mode, they can be configured in a number of ways for beam rocking, conical scan, as used in precession [18.36], and scanning electron nanodiffraction [18.37].

Two deflectors working in opposite senses are used to shift or tilt the beam; the individual deflector excitations are different for these two operations. The beam shift is used for SEND, and the beam tilt is used in double-rocking LACBED or precession electron diffraction ( ). Figure 18.12 compares the beam shift with the beam tilt using the double-deflection coils in a TEM with a condenser-objective lens. For simplicity, the CM and the objective prefield lenses are shown as a single lens above the specimen. Consider a ray along the optical axis. To shift this ray at the specimen, it must be first deflected away from, and then toward, the optical axis by the first and second deflectors successively. Finally, the beam must intersect the optical axis at the front focal plane of the lens above the specimen, which then brings it to the specimen running parallel to the optical axis. To shift the beam, we actually tilt the beam. For other rays in the beam, because of the small convergence angle, the same tilt is achieved, so they all converge to the same point on the specimen. The amount of beam shift is proportional to the tilt angle. To tilt the beam, it is first deflected away from the axis, and then back towards the optical axis in such a way that all rays in the beam converge to the same point on the front focal plane as undeflected rays, but now shifted laterally.

Fig. 18.12

Beam deflection coils used for beam shift and beam tilt . The dark disk marks the pivot point and dashed lines mark the front and back focal planes of the prefield and objective lenses, respectively

Scanning electron diffraction can be carried out by first selecting an area of interest, dividing this area into a number of pixels, placing the electron probe at each of these pixels, and recording the diffraction patterns at each pixel [18.37]. Data acquisition is automated using either dedicated hardware to synchronize the scan and diffraction pattern (NanoMegas SPRL, Brussels, Belgium) or by using computer control of the TEM and the electron camera. An implementation of SEND using the second approach was reported by [18.38], which involves the automation of TEM deflection coils and diffraction pattern acquisition using a custom script written in the DigitalMicrograph® (DM, Gatan Inc., Pleasanton, CA) script language. The electron microscope is controlled using the script by communicating with the host processor built into the TEM. This technique does not require additional hardware other than the computer and the electron detector that are already installed on the TEM. The main drawback is that the speed of acquisition is limited by the camera readout speed or the speed of beam deflection inside the TEM, whichever is slower.

Fig. 18.13a-e

SEND of a small Au disk. (a) A BF image of a nanostructured Au disk and (b) a selected diffraction pattern acquired from SEND. The diffraction intensity is integrated for the areas of 1, 2, and 3 represented in (b). The corresponding intensity maps are shown in (ce), respectively. Reprinted from [18.38], with permission from Elsevier

In the method reported in [18.38], the electron beam scanning is performed in the TEM mode and carried out using the deflection coils to shift the beam under computer control. Two types of computer access to the TEM are used for the scanning process; the first retrieves the values of the illumination deflection coils and stores the values as real numbers \(x\) and \(y\), and the second shifts the electron beam by the amount \(x\) and \(y\). The \(x\) and \(y\) values, however, only refer to the setting of the deflection coils, which need to be calibrated into distances in nanometers. For this purpose, two scanning vectors are established along the vertical and horizontal directions. The calibration is carried out under a standard magnification in TEM mode. The reference value of \((x_{1},y_{1})\) is first obtained from the initial beam position. The electron beam is then horizontally shifted to position 2, and \((x_{2},y_{2})\) are obtained. Using the calibrated magnification, the distance (\(d\)) between 1 and 2 can be set to a fixed value. Then, the horizontal and vertical scanning vectors are calculated. Once calibrated, the electron beam can be shifted to a specific position by a combination of the two scanning vectors.

Once the 4-D dataset is collected using SEND, bright and dark-field STEM images can be obtained simultaneously from SEND in the simplest form of analysis by integrating the diffraction intensities of the direct beam and diffracted beams, respectively. This way, SEND works like STEM. A major distinction is that with the diffraction patterns recorded and stored, other information can be extracted offline to form images, beyond the simple integrated intensities. For example, diffraction patterns can be indexed and analyzed for orientation and phase mapping (Sect. 18.4.1, Orientation Mapping). This analysis can be done at nm resolution, which is unique to transmission electron diffraction. This last option is simply not available using the fixed, STEM, detectors. The tradeoff here, of course, is that one will be dealing with a far more complex, and larger, data set.

Figure 18.13a-e shows an example of SEND applied to a nanostructured Au disk. The SEND patterns were acquired over the area of \(210\times{\mathrm{210}}\,{\mathrm{nm^{2}}}\) in \(30\times 30\) pixels, corresponding to a step size of \({\mathrm{7}}\,{\mathrm{nm}}\). Figure 18.13a-eb shows one of 900 diffraction patterns acquired from SEND. The diffracted beams appear as small disks corresponding to \({\mathrm{4.2}}\,{\mathrm{mrad}}\) of full convergence angle. The electron probe was formed in a JEOL TEM with the \(\mathrm{LaB_{6}}\) gun at the low dose condition [18.38]. To demonstrate the imaging capability of SEND, the diffraction intensity between two circles as marked in Fig. 18.13a-eb was integrated from the diffraction patterns. The intensity sum for every single diffraction pattern was then mapped in the raster image, as shown in Fig. 18.13a-ec. For the mapping, three regions of the diffraction pattern were selected as marked in Fig. 18.13a-ec–e:
  1. 1.

    An annular area between the direct beam and the first ring (marked as 1)

     
  2. 2.

    The second ring (marked as 2)

     
  3. 3.

    The remaining area of the third ring, akin to the use of an annular dark field ( ) detector in STEM.

     
For the first region (Fig. 18.13a-ec), the amorphous region (C film) has high intensity, while the Au nanodisk shows low intensity. This is expected, since the amorphous scattering is strong where there are no Bragg spots from the Au nanodisk. Figure 18.13a-ed shows the variation in the integrated intensity over the grains of Au nanoparticles. This reflects the orientation change across the grains.

There are also major benefits in reducing radiation damage by using low-dose SEND to study radiation sensitive materials, including organic molecules. The work described in [18.39] showed that electron images recorded using illumination spots of \(100{-}200\,{\mathrm{nm}}\) from thin paraffin crystals and purple membrane improve the image contrast by a factor of \(3{-}5\) compared to electron images taken with a large illumination spot of \({\mathrm{3}}\,{\mathrm{\upmu{}m}}\). The improvement in image contrast was attributed to the reduced beam damage induced by specimen movement. In SEND, the beam damage is limited to only area of the specimen illuminated by the electron beam, and thus each diffraction pattern is recorded under nearly identical specimen conditions. The new direct electron detectors take advantage of this effect also, by summing many very brief exposures for which the effects of beam-induced motion are corrected during data merging.

18.1.6 Precession Electron Diffraction

Precession electron diffraction ( ) is a technique pioneered by Vincent and Midgley [18.36]. The principle of the method is illustrated in Fig. 18.14, in which, for simplicity, we omitted the CM and condenser-objective lenses above and below the specimen. In PED, the incident electron beam is made to rotate around the microscope's optical axis, maintaining a constant angle – the precession angle, by using beam deflectors [18.40]. To compensate for motion of the diffracted beams as the incident beam rotates, the outgoing beams are deflected back using the deflectors below the specimen. The technique is similar to the double-rocking technique we discussed for the recording of LACBED patterns [18.35], in which case the beam is made to scan over a rectangular area instead of precession around a circle. By recording electron diffraction patterns with the incident electron beam in precession, PED is able to provide the electron diffraction intensity integrated in angles across the Bragg condition for many reflections, provided that the recording time is much longer than the time it takes for one precession. Compared to CBED, which records the diffraction intensity for every incident beam direction, PED records one intensity integrated over the precession angle in a way similar to the rotational method in x-ray diffraction. It may be shown that this angular integration reduces the effects of multiple scattering, as first discussed by Blackman [18.42] and tested experimentally by Horstmann and Meyer [18.43]. Figure 18.15a-f shows an example. The sample is coesite, which is a high-pressure polymorph of silica with a monoclinic symmetry, space group \(C2/c\). The diffraction patterns, one without and one with precession, were taken on two sides of a twinned crystal. The patterns are only distinguishable using precession. Especially, the kinematically forbidden 001 reflection (\(00h\) with \(h\) odd) is not visible in the precessed pattern (arrowed in Fig. 18.15a-fc). The intensity of these reflections is quite strong in the conventional pattern; their absence in the precession pattern indicates a more kinematic-like behavior for the diffraction intensity [18.41]. This can be understood if we imagine that there is one extinction distance (in two-beam theory) associated with every point (every excitation error) within a CBED disk. By integrating over many such points, the precession signal averages over many extinction distances, and so smoothes out the oscillations with thickness due to the Pendellösung effect [18.44]. Because of this unique feature, PED has found major applications in electron crystallography for solving crystal structures.

Fig. 18.14

Precession electron diffraction setup and controls using the deflector coils above and below the specimen

Fig. 18.15a-f

Experimental electron diffraction patterns taken from coesite (a high-pressure polymorph of silica with a monoclinic symmetry, space group \(C2/c\)). Diffraction patterns marked as A and B are on each part of a twin for [110] and [101] orientations. Without precession: (a) and (b). With precession: (c) and (d). Kinematic simulated patterns: (e) and (f). From [18.41]

PED is implemented by driving the \(x\) and \(y\) deflection coils before and after the specimen synchronously using the oscillating sine wave obtained from a signal generator, which is phase shifted and amplitude adjusted for the \(x\) and \(y\) scan drivers. The same waveforms are used to drive the coils below the specimen. This is schematically illustrated in Fig. 18.14. The result after careful adjustments is that, at the lower part of the beam deflector coils, the incident beam scans sequentially around a circle, which is then brought back to the specimen ideally to a fixed point so the rotating incident beam form a cone of a constant angle. Thus, a focused beam should stay focused in PED and sharp diffraction spots should stay similarly sharp.

18.1.7 Selected Area Diffraction in STEM

The drawback of performing SAED in a conventional TEM, where the objective lens spherical aberration limits the selected area to about \({\mathrm{100}}\,{\mathrm{nm}}\) or more, can be largely avoided by performing electron nanodiffraction in a STEM. There are several ways to perform selected area electron diffraction in a STEM. Sharp diffraction spots can be obtained by using the objective prefield lens to form a small parallel probe on the specimen. The diameter of the region of the specimen with near-parallel illumination depends on the diameter of the condenser aperture. Using a small aperture (\({\mathrm{10}}\,{\mathrm{\upmu{}m}}\) or less), the illumination may be as small as a few tens of nanometers, and diffraction pattern spots are then as sharp as those obtained by a parallel beam in a TEM. For applications where sharp diffraction spots are not so critical, such as phase identification or orientation mapping, a focused probe can be used with correspondingly higher spatial resolution.

The recording of SAED patterns can be made in conjunction with STEM imaging using an annular dark-field detector with a low camera length setting and a large inner cutoff angle. Because STEM imaging is performed in diffraction mode, no additional optical adjustment is needed between imaging and diffraction. Once the image is obtained, nanodiffraction patterns can be recorded in several ways:
  1. 1.

    By positioning the electron probe at specific specimen positions, selected based on the STEM image.

     
  2. 2.

    By applying a small, fast scan of the beam during the recording of the pattern (\({\mathrm{0.1}}\,{\mathrm{s}}\) exposure time or longer). Then, the area giving rise to the diffraction pattern can be increased significantly beyond the diameter of the electron probe [18.4].

     
  3. 3.

    By recording scanning electron nanodiffraction patterns, which will be the subject of the next section.

     

Unlike SAED performed in TEM, the beam convergence angle is separately controlled from the selected area for electron diffraction in STEM. Because of this, some unique applications can be made. One is to acquire diffraction patterns over a small rectangular area defined by the STEM scan coils, which has special applications in atomic resolution STEM. The condenser aperture is coherently illuminated, so that large overlapping CBED disks interfere. The interference pattern changes sensitively as the electron probe moves from one atomic column to another, contributing to the image contrast observed in bright-field STEM [18.46]. Interpretation of coherent CBED (or coherent nanodiffraction) patterns, however, is complicated because we need to know the exact probe position as well as the phase of electron waves, including the phase from lens aberrations and electron multiple scattering. In this sense, the interpretation of these patterns is exactly as complicated as the interpretation of (high resolution electron microscopy) images. These patterns do, however, reveal the local point symmetry of the crystal as reckoned about the center of the beam, and this effect has been used to locate the STEM probe on particular atoms for collection of (electron energy loss spectroscopy) spectra [18.47]. This method has been used to determine the atomic structure, and to classify, the anti-phase domains that occur in alloys of CuAu [18.48], and has been reviewed by Cowley and Spence [18.49].

By averaging over a region of specimen, the PACBED removes all the interference between overlapping CBED disks [18.45, 18.50]. As the example in Fig. 18.16a-c shows, the patterns show a remarkable resemblance to CBED patterns recorded with an incoherent probe. In an aberration corrected STEM, the electron probe can be smaller than \({\mathrm{1}}\,{\mathrm{\AA{}}}\). The smallest specimen region that can be scanned in order to fully remove the coherence effect is a unit cell. The actual volume probed in a PACBED experiment depends on electron probe propagation. Since the electrons are no longer confined to a single atomic column as in a channeling situation, the actual volume is larger than the region scanned by the electron probe. Nonetheless, PACBED has the highest spatial resolution among all diffraction techniques for probing structure on the scale of the unit cell.

Fig. 18.16a-c

Position averaged CBED (PACBED ). (a) Experimental PACBED pattern recorded by scanning the electron probe across the boxed area in (b). (b) HAADF-STEM image of a \({\mathrm{5}}\,{\mathrm{nm}}\) thick \(\mathrm{LaNiO_{3}}\) film on \(\mathrm{(LaSr)AlTiO_{3}}\). (c) Same pattern as in (a), with pseudocubic Miller indices. Reprinted from [18.45], with the permission of AIP Publishing

A major application of PACBED is the determination of crystal thickness for quantitative analysis of STEM image contrast. This technique when combined with quantitative techniques described in later chapters could be used to study local symmetry, polarization, and crystal stoichiometry.

18.2 Electron Probes

18.2.1 Probe Formation

A variety of small electron probes are employed in electron nanodiffraction using coherent and incoherent illumination. Under incoherent conditions (i. e., the electron lateral coherence length is much smaller than the diameter of the condenser aperture) the total probe diameter \(d_{0}\) of a focused probe is given approximately at Gaussian focus by adding in quadrature the various contributions to \(d_{0}\). Thus
$$d_{0}^{2}=d_{\mathrm{s}}^{2}+d_{\mathrm{d}}^{2}+d_{\text{sa}}^{2}+d_{\mathrm{c}}^{2}+d_{\mathrm{f}}^{2}\;,$$
(18.2)
where \(d_{\mathrm{s}}\) is the geometrical source image diameter, \(d_{\mathrm{d}}\) is the diffraction broadening equal to \(0.6\lambda/\theta_{\mathrm{c}}\) with \(\theta_{\mathrm{c}}\) for the convergence angle, \(d_{\text{sa}}\) is the contribution from lens aberrations (in a TEM without a probe corrector, it is equal to \(0.5C_{\mathrm{s}}\theta_{\mathrm{c}}^{3}\) in the plane of least confusion, not the Gaussian image plane), and \(d_{\mathrm{c}}\) is the contribution from chromatic aberration, given by (\(\Updelta{}E_{0}/E_{0})C_{\mathrm{c}}\theta_{\mathrm{c}}\), with \(\Updelta E_{0}\) the energy spread in the electron beam. This last term can be neglected as a first approximation. In (18.2), \(d_{\mathrm{f}}\) is the contribution \(2\theta_{\mathrm{c}}\Updelta f\) from a small focusing error \(\Updelta f\). For a TEM instrument with \(C_{\mathrm{s}}={\mathrm{2}}\,{\mathrm{mm}}\) at \({\mathrm{100}}\,{\mathrm{kV}}\), the contributions of diffraction \(d_{\mathrm{d}}\) and spherical aberration \(d_{\text{sa}}\) are equal at an angle of about \({\mathrm{7}}\,{\mathrm{mrad}}\). Equation (18.2) cannot strictly be used for coherent conditions, with the widths of intensity distributions added in quadrature.

The smallest probe under the incoherent condition is obtained by minimizing all the quantities in (18.2). The \(d_{\mathrm{s}}\) can be made smaller than \(d_{\mathrm{d}}\) and \(d_{\text{sa}}\) by combining a small physical source with large demagnification. Then the probe formation becomes diffraction limited and the illumination necessarily coherent. Then detailed computations are required for the probe shape for particular values of the lens aberrations, the defocus \(\Updelta f\), \(\lambda\), and \(\theta_{\mathrm{c}}\).

Electron probe formation using a coherent illumination is affected by the aberrations of the objective prefield. For a convergent beam of electrons, the lens aberrations introduce an angle-dependent phase, \(\chi(k_{x},k_{y})\), with \(x\) and \(y\) standing for the coordinates perpendicular to the optical axis of the electron lens. The phase \(\chi(k_{x},k_{y})\) from the objective lens aberrations is described in Chaps.  13 and  2. For electron nanodiffraction, we must also consider the electron source wave function \(\phi_{\mathrm{S}}(x,y)\) formed by the last condenser lens and its contribution to the electron probe. The electron probe on the sample is an image of \(\phi_{\mathrm{S}}(x,y)\) magnified by the lens magnification \(M\). According to the image formation theory, the actual image is a convolution of \(\phi_{\mathrm{S}}(x,y)\) with the objective lens resolution function \(T(x,y)\)
$$\begin{aligned}\displaystyle\phi_{\mathrm{P}}(x,y)&\displaystyle=\phi_{\mathrm{S}}\left(\frac{-x}{M},\frac{-y}{M}\right)\otimes T(x,y)\\ \displaystyle&\displaystyle=\int_{-\infty}^{\infty}\phi_{\mathrm{S}}(-M\boldsymbol{k}_{\mathrm{t}})A(\boldsymbol{k}_{\mathrm{t}})\exp[\mathrm{i}\chi(\boldsymbol{k}_{\mathrm{t}})]\\ \displaystyle&\displaystyle\qquad\times\exp(2\uppi\mathrm{i}\boldsymbol{k}_{\mathrm{t}}\cdot\boldsymbol{r})\mathrm{d}\boldsymbol{k}_{\mathrm{t}}\\ \displaystyle&\displaystyle=\mathfrak{F}\left\{\phi_{\mathrm{S}}(-M\boldsymbol{k}_{\mathrm{t}})A(\boldsymbol{k}_{\mathrm{t}})\exp[\mathrm{i}\chi(\boldsymbol{k}_{\mathrm{t}})]\right\}.\end{aligned}$$
(18.3)
Here, we have used \(\boldsymbol{k}_{\mathrm{t}}=k_{\mathrm{x}}\boldsymbol{x}+k_{\mathrm{y}}\boldsymbol{y}\) and \(\mathfrak{F}\) as a shorthand for the Fourier transform; \(A(\boldsymbol{k}_{\mathrm{t}})\) is the aperture function with a value of 1 for \(|\boldsymbol{k}_{\mathrm{t}}|<\theta_{\mathrm{c}}/\lambda\) and 0 beyond. The electron beam energy spread and the chromatic aberration are neglected in (18.3). The equation also assumes that the illuminating electron wave is perfectly coherent across the condenser aperture.
The intensity distribution of the probe at the sample is given by
$$I_{\mathrm{p}}=|\phi_{\mathrm{p}}(x,y)|^{2}=\phi_{\mathrm{p}}(x,y)\phi_{\mathrm{p}}^{*}(x,y)\;.$$
(18.4)
For a conventional TEM without the probe \(C_{\mathrm{s}}\) corrector, the wave-front aberration function \(\chi\) is given by
$$\chi(\boldsymbol{k}_{\mathrm{t}})=\uppi\left(\Updelta f\lambda k_{\mathrm{t}}^{2}+\frac{1}{2}C_{\mathrm{s}}\lambda^{3}k_{\mathrm{t}}^{4}+2\boldsymbol{k}_{\mathrm{t}}\cdot\boldsymbol{r}_{\mathrm{p}}\right).$$
(18.5)
Here \(\boldsymbol{r}_{\mathrm{p}}\) is the probe coordinate.

A focused electron probe on the sample is formed by placing the electron beam crossover far away from the front focal plane of the objective lens. This gives a demagnified, sharp, electron source image on the sample with magnification \(M\ll 1\). The size of the electron probe, in this case, is largely determined by the objective lens resolution function \(T(x,y)\). In reciprocal space, the demagnified electron source has a broad, spherical wave-like spectrum of wave vectors.

For a conventional TEM, the calculations for optimized probe sizes have been published [18.51]. The required focus setting \(\Updelta f\) can be defined as that which minimizes the radius of the probe area that contains, say, \({\mathrm{70}}\%\) of the beam intensity [18.51]. Calculations based on (18.4) then show that the following values must be used to obtain this most compact probe
$$\theta_{\mathrm{c}} ={1.27}C_{\mathrm{s}}^{-\frac{1}{4}}\lambda^{\frac{1}{4}}\;,$$
(18.6)
$$\Updelta f =-{0.75}C_{\mathrm{s}}^{\frac{1}{2}}\lambda^{\frac{1}{2}}\;.$$
(18.7)
This gives the minimum probe diameter (containing \({\mathrm{70}}\%\) of the intensity) as
$$d({\mathrm{70}}\%)={0.66}\lambda^{\frac{3}{4}}C_{\mathrm{s}}^{\frac{1}{4}}\;.$$
(18.8)
For experimental reasons, it may be easier to measure the probe displacement for which the intensity at a sharp edge in a STEM image falls from 80 to \({\mathrm{20}}\%\). As judged by this criterion, the smallest probe is obtained with the constant \(\mathrm{0.66}\) above replaced by \(\mathrm{0.4}\). Experimental measurements of coherent probe widths, in rough agreement with the above theoretical estimates, can be found in [18.52].
To form a small parallel beam, the electron beam crossover is placed close to, or at, the front focal plane. The electron source in this case is magnified (\(M\gg 1\)), which is used to reduce the electron beam convergence angle for this parallel-beam diffraction. Meanwhile, the sample remains at the back focal plane of the objective prefield lens, far away from the electron source image. Thus, the electron beam seen at the specimen level is a defocused image of the source. This large underfocus must be included as a part of the lens aberration function in (18.5) [18.12]. To demonstrate this, we assume a Gaussian distribution for the magnified electron source after the objective prefield lens (with a magnification of \(M\))
$$\phi_{\mathrm{S}}\left(\frac{r}{M}\right)=A\exp\left(-\frac{a^{2}r^{2}}{M^{2}}\right),$$
(18.9)
where \(a\) gives the crossover half-width at the amplitude of \(A/e\). The Fourier transform of this Gaussian probe is
$$\phi_{\mathrm{S}}(k_{\mathrm{t}})=\frac{A\sqrt{\uppi}}{a}\exp\left[-\dfrac{{k_{\mathrm{t}}^{2}}}{\left(\frac{a}{M\uppi}\right)^{2}}\right].$$
(18.10)
Substituting (18.10) into (18.3) gives
$$\begin{aligned}\displaystyle&\displaystyle\phi_{\mathrm{P}}(x,y)\\ \displaystyle&\displaystyle=\mathfrak{F}\left\{\frac{A\sqrt{\uppi}}{a}\exp\left[\dfrac{-k_{\mathrm{t}}^{2}}{\left(\frac{a}{M\uppi}\right)^{2}}\right]A(\boldsymbol{k}_{\mathrm{t}})\exp[\mathrm{i}\chi(\boldsymbol{k}_{\mathrm{t}})]\right\}.\end{aligned}$$
(18.11)
Thus, the width of the beam in the reciprocal space is reduced by a factor of \(1/M\). The Gaussian half-width of the defocused electron beam is \(\approx{\mathrm{0.05}}\,{\mathrm{mrad}}\) in the JEOL 2010F TEM formed using a \({\mathrm{10}}\,{\mathrm{\upmu{}m}}\) condenser aperture [18.53]. The real-space probe observed at the specimen level is a convolution of the magnified source with
$$T(x,y)=\mathfrak{F}\{A(\boldsymbol{k}_{\mathrm{t}})\exp[i\chi(\boldsymbol{k}_{\mathrm{t}})]\}\;.$$
(18.12)
The dominant probe features come from \(T(x,y)\), as shown in Fig. 18.17 for a comparison between an experimental probe and simulation based on \(T(x,y)\) alone [18.12].
Fig. 18.17

Experimental and simulated electron nanobeam used in nanoarea electron diffraction (NAED). The simulation used \(C_{\mathrm{s}}={\mathrm{1}}\,{\mathrm{mm}}\) and \(\Updelta f=-{\mathrm{360}}\,{\mathrm{nm}}\)

In electron nanodiffraction using a focused probe and coherent illumination, a small condenser aperture is used so that the diffraction pattern recorded consists of small diffraction disks. The small disks are helpful for the determination of diffraction peak positions, for example, in local strain measurements [18.4]. For a small convergent angle at \(\approx{\mathrm{1}}\,{\mathrm{mrad}}\), the size of the focused probe is diffraction limited, with its intensity distribution given by
$$I(r)\propto\left[\frac{J_{1}\left(\frac{2\uppi r\sin\theta}{\lambda}\right)}{\frac{\uppi r\sin\theta}{\lambda}}\right]^{2},$$
(18.13)
where \(\theta\) is the beam's half-convergence angle, and \(J_{1}\) is the first-order Bessel function. The first zero of \(J_{1}(x)\) occurs at \(x=3.832\), which gives the so-called Rayleigh criterion for resolution
$$r_{0}=0.61\frac{\lambda}{\theta}\;.$$
(18.14)
The intensity distribution in (18.13) can be fitted approximately by a Gaussian function with a full width at half maximum ( ) of
$$d_{\text{FWHM}}=0.52\frac{\lambda}{\theta}\;.$$
(18.15)
The diffraction-limited probe size increases as the convergence angle decreases. Since the probe size defines the spatial resolution for a beam with a small convergence angle, improvements in angular resolution in the diffraction pattern are thus obtained at the expense of spatial resolution.
Fig. 18.18

(a) Electron probe formed inside an FEI Titan TEM. The image was recorded as the probe passing through a (110)Si crystal with the lattice fringes clearly seen within the probe. (b) Intensity profile along the line in (a). (c) Electron nanodiffraction pattern recorded with the probe in (a). Reprinted from [18.54], with the permission of AIP Publishing

Figure 18.18 shows an example of a small focused probe formed inside a probe \(C_{\mathrm{s}}\)-corrected FEI Titan microscope operated at \({\mathrm{300}}\,{\mathrm{kV}}\) using a condenser aperture of \({\mathrm{50}}\,{\mathrm{\upmu{}m}}\) in diameter. The microscope was operated in the so-called \(\upmu{}\)-probe TEM scan mode, in which the probe corrector was operated like an additional condenser lens. The probe recorded has a FWHM of \({\mathrm{2.7}}\,{\mathrm{nm}}\). Diffraction patterns recorded using this probe consist of small diffraction disks with a convergence semi-angle of \({\mathrm{0.37}}\,{\mathrm{mrad}}\), according to (18.15).

18.2.2 Probe Current

The brightness \(\beta_{\mathrm{s}}\) of a source is conserved and constant in a TEM, even if aberrations and aperture stops are permitted. In an electron illumination system, the source is demagnified using the C1 lens. The net effect is to increase current density \(J\) and increase the source angle \(\alpha\) (proportional to \(1/M\)) for the effective source, leaving brightness constant. The condenser aperture, which is placed at or after the second condenser lens (CL), only lets through a part of the illuminating cone. Its diaphragm size determines the size of the electron cone and the numerical aperture produced by the CL lens. Thus, the beam current after the condenser aperture is given by
$$I_{\mathrm{b}}=\beta_{\mathrm{s}}\uppi^{2}\alpha_{\mathrm{c}}^{2}\left(\frac{d_{\mathrm{s}}}{2}\right)^{2},$$
(18.16)
where \(\alpha_{\mathrm{c}}\) is the condenser aperture half angle. The beam current can be increased by enlarging the illumination aperture \(\alpha_{\mathrm{c}}\) (provided that it remains filled with electrons), or by increasing the effective source diameter \(d_{\mathrm{s}}\) using a smaller source demagnification, or by increasing the intrinsic source brightness.
At the specimen level, and a fixed beam current, the beam size is determined by
$$d=\sqrt{\frac{4I_{\mathrm{b}}}{\beta_{\mathrm{s}}\uppi^{2}}}\frac{1}{\alpha_{\mathrm{c}}}\;.$$
(18.17)
Figure 18.19 shows the calculated probe radius plotted against the probe convergence angle for a source brightness of \({\mathrm{10^{12}}}\,{\mathrm{A/(m^{2}{\,}sr)}}\) and a range of beam currents from \({\mathrm{1}}\,{\mathrm{pA}}\) to \({\mathrm{100}}\,{\mathrm{nA}}\). The typical current employed in STEM is between 10 and hundreds of pA, which gives a convergence angle on the order of tens of mrad for an ångström-sized probe.
Fig. 18.19

The probe radius plotted against the probe convergence angle \(\alpha_{\mathrm{c}}\) for brightness \({\mathrm{10^{12}}}\,{\mathrm{A/(m^{2}{\,}sr)}}\)

18.2.3 Probe Coherence and Coherent Current

The smallest probes can only be obtained using a fully coherent illumination. For the STEM or CBED modes using a focused probe, the electron beam coherence is defined by the coherence width \(L\) at the aperture which illuminates the sample. According to the Zernike–Van Cittert theorem , the degree of coherence between the electron wave functions at two different points in an aperture plane far away from the electron source is given by the Fourier transform of the source intensity distribution. The source seen by the condenser aperture is the source image formed before the CL lens. If we assume the source has a uniform and ideally incoherent intensity distribution within a circular disk, the coherence function at the aperture is then given by
$$\gamma_{12}(r,0)=\lambda J_{1}\dfrac{\left(\frac{2\uppi\alpha r}{\lambda}\right)}{2\alpha r}\;,$$
(18.18)
with \(J_{1}\) a first-order Bessel function, \(r\) the radial distance in the aperture plane, and \(\alpha\) the angle subtended by the electron source (Fig. 18.20). The lateral coherence length \(L\), which is used in the literature, is defined by the distance \(r\) to the first zero of \(J_{1}\), which has the value of
$$L=0.6\frac{\lambda}{\alpha}\;.$$
(18.19)
This provides an estimate of the distance between points in the plane of CA at which the wave field is capable of producing strong interference fringes in a Young's slit experiment with the slit spacing \(L\). If the diameter of the CA is \(2R_{\mathrm{a}}\), we then have following possibilities:
  1. 1.

    If \(2L\ll 2R_{\mathrm{a}}\), the illumination aperture can be considered to be incoherently filled and so treated as an ideally incoherent effective source. This is the situation for conventional TEM systems using a tungsten or \(\mathrm{LaB_{6}}\) source under most operating conditions. Then CA can be treated as an ideally incoherent source, within which each point acts as a statistically independent emitter of electrons. (A useful exercise is to calculate \(L\) for a \(\mathrm{LaB_{6}}\) source operating at the smallest probe size.) The probe formed further downstream by this incoherent source, filling the illumination aperture, will then be partially coherent.

     
  2. 2.

    If \(2L> 2R_{\mathrm{a}}\), the illumination aperture is coherently filled, and the radiation can be considered to originate from a point source. Then the entire optical system beyond CA is filled with perfectly coherent radiation and the probe may be treated as perfectly coherent. This is often a good approximation for field emission gun (FEG) instruments. With a focused probe, the sample is then illuminated by an aberrated, converging spherical wave.

     
We see that coherence is increased by decreasing \(d_{\mathrm{s}}\), that is, by increased demagnification of the source, by using lower accelerating voltage, and by decreasing the size of CA. For a Schottky emission source, the emission diameter is between 20 and \({\mathrm{30}}\,{\mathrm{nm}}\) according to [18.55]. At a distance of \({\mathrm{10}}\,{\mathrm{cm}}\) away from the electron source image, a factor of 10 source demagnification provides a coherence length from 100 to \({\mathrm{150}}\,{\mathrm{\upmu{}m}}\). However, the actual coherence length is smaller because of the gun lens aberrations.
Fig. 18.20

Relationship between a finite source and the lateral coherence length \(L\) at the condenser aperture. The source size is defined by the angle (\(\alpha\)) shown subtended at the aperture

Note that in forming a focused probe, the source is imaged onto the sample by the probe-forming lens, while the illumination aperture CA (called the objective aperture on STEM instruments) is imaged onto the detector in STEM by the objective lens. CA subtends a semiangle \(\theta_{\mathrm{c}}\), at the sample, while the geometrical electron source image (of diameter \(d_{\mathrm{s}}\)) subtends a semiconvergence angle \(\alpha\) at CA. Thus, a measurement of the coherence can be made by examining interference in overlapping CBED disks obtained with a crystal (Chaps.  2 and  13). In this chapter, we shall refer loosely to coherent CBED as the case \(2L\gg 2\textit{R}_{\mathrm{a}}\), and incoherent CBED as the case \(2L\ll 2\textit{R}_{\mathrm{a}}\), noting that these labels refer to the coherence conditions in the illumination aperture, and that in the second case the probe itself is partially coherent.

A figure of merit for imaging or diffraction using coherent electrons is the coherent current, defined by the electron current available at a certain degree of coherence [18.56]. For a coherently illuminated CA, the coherent current \(I_{\text{coh}}\) is given by [18.18]
$$I_{\text{coh}}(\theta)=-\ln[\gamma(\theta,0)]\beta_{\text{s}}\lambda^{2}\;,$$
(18.20)
which shows that the available coherent current is proportional to the log of the degree of coherence (\(\gamma(\theta,0)\)) and the so-called reduced brightness \(\beta_{\mathrm{s}}\uplambda^{2}\) of the electron source. The electron source brightness increases linearly with accelerating voltage \(\Phi\), while the electron wavelength is inversely proportional to the square root of the accelerating voltage. Thus, it can be concluded that both the reduced brightness and the coherent current are a constant property of the emitter, independent of the electron wavelength or accelerating voltage.

18.3 Energy Filtering

Removal of the inelastic background is an option when using an electron microscope equipped with an electron energy filter. The purpose of the filter is to remove from diffraction patterns all those electrons which, on traversing the sample, lose more than a few electron volts in energy and contribute to the background intensity. The important inelastic processes are phonon or plasmon scattering and single-electron excitation. (For a review of energy-loss processes in electron microscopy, see [18.55, 18.57].) Phonon scattering involves relatively large inelastic scattering angles, but very small energy losses (perhaps \({\mathrm{30}}\,{\mathrm{meV}}\)). These are not excluded by elastic filtering. Plasmon losses involve larger energies (about \({\mathrm{15}}\,{\mathrm{eV}}\)) and small scattering angles. Plasmon excitations or higher energy losses can be filtered out by dispersing the electrons according their energies using the magnetic or electrostatic fields inside an electron energy filter and using a slit equivalent to a few eV in width around the elastic (zero-loss) electron beam. The advantages of zero-loss energy filtering are outlined in more detail below in the discussion of imaging filters, however a glance at Fig. 18.22 will indicate the improvement in the quality of data to be expected. This improvement affects all the techniques discussed here, especially for quantitative CBED. Duval et al [18.58] was the first to demonstrate that, by placing the diffraction pattern at the object plane of the imaging filter, most of the background intensities disappeared, especially at small scattering angles. The other major application of energy-filtering is electron spectroscopic imaging ( ) used for composition mapping [18.7].

The thickness of the sample is important in determining the need for filtering. In very thin samples there is little inelastic scattering, and hence no requirement for elastic filtering. However, such ideal samples only exist in 2-D materials like graphene. For other materials, very thin samples are known to be strongly influenced by thin-film relaxation effects [18.59]. Thus, for the study of thicker material, which is more representative of the bulk, elastic energy filtering combined with a detector system with large dynamic range is required for quantitative intensity analysis. For example, in order to obtain accuracy in structure-factor measurement or atomic position determination using dynamical scattering comparable to that achieved in x-ray crystallography, an energy filter, tuned to the elastic peak, is essential. An image of a wedge formed from elastically filtered scattering will be seen to reach a maximum intensity at some thickness, beyond which it becomes dark, since at larger thickness virtually all electrons have been inelastically scattered. Other benefits of zero-loss energy filtering are:
  1. 1.

    The dramatic reduction in background significantly improves the contrast of fine high-order Laue zone ( ) lines and other features and thus enables the measurement of strain and determination of symmetry at greater accuracy.

     
  2. 2.

    It enables the study of diffuse scattering whether it is thermal or comes from defects or from modulations of the crystal structure. In relatively thick samples, the diffuse scattering is buried by the inelastic background if elastic energy filtering is not used.

     
  3. 3.

    Filtering allows much thicker crystals to be examined without incurring the penalty of radiation damage, which would result if higher accelerating voltages were used.

     
  4. 4.

    The use of greater thickness (without background) for the study of defects. This is new information, which was not previously extractable due to the presence of the background.

     

There are several types of in-column energy filters that are named according to the shape of the electron path, such as the \(\Upomega\)-, \(\upalpha\)-, or \(\upgamma\)-energy filters. The other is the postcolumn Gatan imaging filter (GIF ). The in-column filter is placed between the intermediate and the projector lenses of the TEM and can be used in combination with all forms of electron detectors. The GIF is integrated with an electron camera and placed below the camera chamber. Its use for electron diffraction typically requires switching the TEM to a special low camera-length setting.

The optics of an imaging energy filter is illustrated in Fig. 18.21. While the details can differ, all energy filters have the optical elements of entrance image plane, achromatic image plane (where a focused image is formed without the separation of color or energy) and an energy dispersion plane, where the energy selection slit is placed. The intermediate and projector lenses here refer to the lenses immediately above and below the energy filter, which are part of the intermediate and projection lens systems in a TEM for the in-column energy filters. In GIF, they are replaced by focusing coils and additional multipoles. The function of the intermediate lens is to transfer the image or diffraction pattern to the image entrance plane, while the projection lens looks at the achromatic image plane or the energy dispersion plane in the imaging and diffraction mode or spectroscopic mode, respectively. A diffraction pattern is formed at the energy dispersion plane when we have an image at the achromatic image plane, or, in reverse, an image is formed at the energy dispersion plane when we have a diffraction pattern at the achromatic image plane.

Fig. 18.21

The optics of the energy filter for electron imaging, diffraction, and energy-loss spectroscopy

Figure 18.22 shows the striking improvement that results in the quality of CBED patterns from the use of a \(\Upomega\)-energy filter, even when film recording is used. These patterns were recorded on the Zeiss Omega model 912 TEM-STEM. The exposure time used in Fig. 18.22 was \({\mathrm{1}}\,{\mathrm{s}}\) for the unfiltered recording and \({\mathrm{3}}\,{\mathrm{s}}\) for the filtered pattern. These times were arranged to produce an approximately equal optical density on the film, so that a valid comparison could be made. Before the imaging filters became widely available, energy filtering was performed by scanning the electron image or diffraction pattern over an EELS system with a point or array detector, by measuring the zero-loss peak intensity. The image recording time for a \({\mathrm{10^{3}}}\times{\mathrm{10^{3}}}\) pixel image using the Omega system is at least \(\mathrm{1000}\) times less than that required by the scanned readout system, for the same dose. (This assumes a parallel detector for the EELS capable of one-dimensional ( ) imaging.) The advantages of the Omega filter increase rapidly with the number of pixels. Since the experimental observation in Fig. 18.22 is that the background between the Bragg reflections is almost entirely removed by elastic filtering, we must conclude that this background is due to multiple, coupled phonon, and plasmon scattering. The phonon scattering events provide the large angular change, and the associated plasmon losses then allow these electrons to be removed by elastic filtering. This interpretation is consistent with the relatively large thickness used (\(t={\mathrm{270}}\,{\mathrm{nm}}\)).

Fig. 18.22

Comparison of filtered and unfiltered CBED patterns, recorded on film. Provided by Joachim Mayer, MPI

For CBED, an optimum sample thickness exists with energy-filtered data. For very thin crystals, there is little inelastic scattering and so no requirement for filtering, however the CBED disks show no useful contrast variation. At very large thickness, all scattering is inelastic, and no elastic signal can be recorded. In the simplest model [18.60], the thickness dependence of, for example, the plasmon-loss electrons is given by the product of the multiply-scattered Bragg-beam intensity with an appropriate term of the Poisson distribution.

18.4 Diffraction Analysis

An electron diffraction pattern records the intensity of scattered electrons as a function of the scattering angle. Electron diffraction patterns in general appear as:
  1. 1.

    Broad halos for amorphous materials or liquids

     
  2. 2.

    Multiple concentric rings for powder samples

     
  3. 3.

    Sharp diffraction spots (spot diffraction pattern)

     
  4. 4.

    Disks for CBED.

     
In thick crystals, a diffuse background may be observed including sharp lines. The lines, so-called Kikuchi lines (Fig 18.3a-d), are formed by Bragg diffraction of the diffuse background.
Structural information obtainable from electron diffraction patterns include the following:
  1. 1.

    The diffraction peak position can be used to measure the \(d\)-spacing of individual reflections. The combination of diffraction peak positions and their indexing can be used to determine the crystal lattice, its repeating unit cell, cell parameters, and cell orientation.

     
  2. 2.

    Diffraction pattern symmetry recorded in CBED and the dynamic extinction in the form of Gjonnes–Moodie lines can be used to determine the crystal symmetry or the lack of symmetry.

     
  3. 3.

    Diffraction pattern indexing, when it is done together with SEND, can be used to determine nanograins, their crystallographic orientation, and distribution.

     
  4. 4.

    The change in diffraction peak intensity and the diffuse scattering around the Bragg peaks can be used to identify structural defects.

     
  5. 5.

    For very thin samples, the Fourier transform of the diffraction pattern gives the projected interatomic distances. For nanocrystalline or amorphous structures, the Fourier transform of the radial diffraction intensity gives the pair distances and their distribution. The diffraction patterns can also be used to obtain information about medium-range ordering.

     
  6. 6.

    Electron diffraction intensity can be used to determine the atomic positions. In cases where multiple scattering effects in the measured diffraction intensities are strong, multiple scattering must be included in order to determine the atomic structure.

     
  7. 7.

    Accurate structure factor measurement from diffraction intensities can be used to determine the atomic thermal vibrations (the Debye–Waller factors), and crystal potential, or charge density.

     
Procedures used to retrieve the above structural information range from relatively straightforward to highly sophisticated, involving quantitative analysis of diffraction intensities. In general, the complexity of diffraction analysis increases with the order of the above list. Here, we introduce diffraction analysis for spot diffraction patterns, CBED, and coherent electron nanodiffraction.

18.4.1 Crystal Diffraction Pattern Indexing and Orientation Mapping

The Geometry and Kinematical Intensity of Spot Diffraction Patterns

Here, we first provide a general description of spot diffraction pattern geometry and intensity and the relationship between the crystallographic and experimental coordinates.

Fig. 18.23

Transmission electron diffraction geometry

A spot diffraction pattern is formed when a collimated electron beam traverses a thin sample, giving rise to the diffraction spots recorded on a distant detector. The incident wave is represented by its wave vector inside the crystal \(\boldsymbol{k}_{0}\). Figure 18.23 illustrates the geometry of electron diffraction, where the diffracted wave \(\boldsymbol{k}\) associated with the reflection \(\boldsymbol{g}\) yields the diffraction spot at the diffraction plane \((g_{x},g_{y})\). According to the Laue diffraction condition, we have
$$\boldsymbol{k}-\boldsymbol{k}_{0}=\boldsymbol{g}+\boldsymbol{S}_{g}\;.$$
(18.21)
Where \(\boldsymbol{g}\) is a reciprocal lattice vector
$$\boldsymbol{g}=h\boldsymbol{a}^{*}+k\boldsymbol{b}^{*}+l\boldsymbol{c}^{*}\;,$$
(18.22)
and \(\boldsymbol{S}_{g}\) is the so-called excitation error. It describes the deviation from the Bragg condition . At the Bragg condition, \(\boldsymbol{S}_{g}=0\). The \(S_{g}\) is positive when the length of \(|\boldsymbol{k}_{0}+\boldsymbol{g}|\) is shorter than \(k_{0}\). The direction of \(\boldsymbol{S}_{g}\) is taken along the sample surface normal direction.
The incident wave number \(k_{0}\approx 1/\lambda\) and \(\lambda\) is the electron wavelength determined by the electron accelerating voltage, \(\Phi\) in volts,
$$\lambda=\frac{12.2643}{\sqrt{\Phi(1+0.97845\times{\mathrm{10^{-6}}}\Phi)}}\;,$$
(18.23)
in ångströms (Å).
By the requirement of elastic scattering, the diffracted wave vector must fall on the sphere of radius \(1/\lambda\) (the Ewald sphere), and thus
$$|\boldsymbol{k}_{0}+\boldsymbol{g}+\boldsymbol{S}_{g}|^{2}=|\boldsymbol{k}_{0}|^{2}=\frac{1}{\lambda^{2}}\;.$$
(18.24)
Using (18.24), it can be shown that
$$S_{g}\approx\frac{\left(k_{0}^{2}-|\boldsymbol{k}_{0}+\boldsymbol{g}|^{2}\right)}{2k_{0}}\;.$$
(18.25)
The diffracted wave intersects the diffraction plane at the position \((g_{x},g_{y})\), which is given by
$$\begin{aligned}\displaystyle g_{x}&\displaystyle=(\boldsymbol{k}_{0}+\boldsymbol{g}+\boldsymbol{S}_{g})\cdot\boldsymbol{x}\approx(\boldsymbol{k}_{0}+\boldsymbol{g})\cdot\boldsymbol{x}\;,\\ \displaystyle g_{y}&\displaystyle=(\boldsymbol{k}_{0}+\boldsymbol{g}+\boldsymbol{S}_{g})\cdot\boldsymbol{y}\approx(\boldsymbol{k}_{0}+\boldsymbol{g})\cdot\boldsymbol{y}\;.\end{aligned}$$
Electrons are diffracted by the Coulomb potential of the positive nuclei and that of all the electrons, including the core electrons surrounding each nucleus. The relationship between this potential and the electron density is given by Poisson's equation
$$\nabla^{2}V(\boldsymbol{r})=-\frac{e[Z(\boldsymbol{r})-\rho(\boldsymbol{r})]}{\varepsilon_{0}}\;,$$
(18.26)
where \(Z(\boldsymbol{r})\) is the nuclear charge density. For a \({\mathrm{100}}\,{\mathrm{kV}}\) electron traveling at a speed of \(\approx 1/3\) the speed of light, the time it spends inside a TEM specimen is \(\approx{\mathrm{10^{-15}}}\,{\mathrm{s}}\), whereas the typical phonon frequency is \(\approx{}{\mathrm{10^{12}}}\,{\mathrm{Hz}}\). Thus, the fast electrons see the instantaneous frozen configuration of a vibrating lattice, a snapshot. To a good approximation, the average experimental diffraction intensity can then be modeled by electron interaction with an average potential
$$\langle V(\boldsymbol{r})\rangle=\frac{|e|}{4\uppi\varepsilon_{0}}\int\mathrm{d}^{3}\boldsymbol{r}^{\prime}\frac{\langle Z(\boldsymbol{r}^{\prime})-\rho(\boldsymbol{r}^{\prime})\rangle}{|\boldsymbol{r}-\boldsymbol{r}^{\prime}|}\;.$$
(18.27)
The electron structure factor is obtained from
$$V_{g}=\frac{|e|}{4\uppi\varepsilon_{0}g^{2}}(Z_{g}-F_{g})\;,$$
(18.28)
where
$$Z_{g}=\sum_{i}Z_{i}T(\boldsymbol{g})\exp(2\uppi\mathrm{i}\boldsymbol{g}\cdot\boldsymbol{r})$$
(18.29)
and \(F_{g}\) is the average x-ray structure factor and \(T(\boldsymbol{g})\) is the so-called temperature factor .
The diffracted beam intensity in the kinematical approximation for a crystal of thickness \(t\) is given by
$$I_{g}(t,S_{g})=\uppi^{2}\lambda^{2}\frac{\sin^{2}(\uppi S_{g}t)}{(\uppi S_{g}t)^{2}}|U_{g}|^{2}t^{2}\;.$$
(18.30)
Where \(U_{g}\) is the electron structure factor in the unit of \(\AA{}^{-2}\) with
$$U_{g}=\frac{2m|e|}{h^{2}}V_{g}\;.$$
(18.31)
In the approximation that the crystal potential is approximated by a superposition of atomic potential, the \(V_{g}\) is then given by
$$\begin{aligned}\displaystyle V_{g}&\displaystyle=\frac{1.145887}{V_{\mathrm{c}}}\sum_{i=1}^{n}\frac{\left[Z_{i}-f_{i}^{x}(s)\right]}{s^{2}}\\ \displaystyle&\displaystyle\quad\,\times\exp\left(-B_{i}s^{2}\right)\exp\left[-2\uppi\mathrm{i}(hx_{i}+ky_{i}+lz_{i})\right].\end{aligned}$$
(18.32)
Here, the sum is over the atoms within the unit cell, \(f_{i}^{x}(s)\), \((x_{i},y_{i},z_{i})\), and \(B_{i}\) are the x-ray atomic scattering factor, atomic position, and the Debye–Waller factor of the \(i\)th atom, respectively.
For a reasonable thick crystal, where the electron multiple scattering effect is significant, dynamical theory is required for describing electron diffraction intensity. In the so-called two-beam approximation with only one strongly diffracted beam and no absorption, the two-beam dynamical theory gives
$$I_{g}(t,S_{g})=\frac{1}{1+\omega^{2}}\sin^{2}\left(\frac{\uppi t}{\xi_{g}}\sqrt{1+\omega^{2}}\right),$$
(18.33)
here, the dimensionless parameter \(\omega=S_{g}\xi_{g}\) is used, with \(\xi_{g}=1/\lambda|U_{g}|\). The variation of the intensity with thickness is known as Pendellösung or thickness fringe oscillations. The kinematical result is equal to the two-beam theory in the limit of \(|\omega|=|S_{g}|\xi_{g}\gg 1\) or \(|S_{g}|\gg 1/\xi_{g}\) or at very small thickness with \(|S_{g}|\approx 0\) and
$$I_{g}(z)\approx\frac{\uppi^{2}t^{2}}{\xi_{g}^{2}}\;,$$
where \(t\ll 1/(\uppi\xi_{g})\) (which together with \(|S_{g}|\gg 1/\xi_{g}\) define the limits of the kinematic approximation).
As we will see later, the task of indexing electron diffraction patterns is helped greatly by using simulated spot diffraction patterns as templates (Sect. 18.4.1, Automated Indexing of Electron Diffraction Patterns). For this purpose, only an estimate of the diffraction peak intensity is needed. Often, an ad hoc formula is used to assign the spot peak intensity. The formula below is an example
$$P(g)=A\log\left[\dfrac{|U_{g}|^{2}}{1+\left(\frac{S_{g,\min}}{\lambda|U_{g}|}\right)^{2}}\right]-B(g)\;,$$
(18.34)
where \(U_{g}\) is the electron structure factor in the unit of \(\AA{}^{-2}\), \(S_{g,\min}\) is smallest excitation error of the reflection \(g\) for an incident convergent beam, \(A\) is a scaling constant, and \(B(g)\) is a length-dependent correction function for the intensity of high-order reflections.
To define the crystal orientation in an electron diffraction experiment, we need to consider the incident beam, which is vertical in a TEM, in the crystallographic axes, where a direction \(\boldsymbol{n}\) is taken as
$$\boldsymbol{n}=u\boldsymbol{a}+v\boldsymbol{b}+w\boldsymbol{c}\;.$$
(18.35)
An incident beam along this direction can be written as
$$\boldsymbol{k}_{0}=-\frac{\boldsymbol{n}}{\lambda|\boldsymbol{n}|}\;.$$
(18.36)
A transformation matrix is then required to relate the crystallographic axes and the microscope coordinates of (\(x,y,z\)), where
$$(\boldsymbol{a},\boldsymbol{b},\boldsymbol{c})=(\boldsymbol{x},\boldsymbol{y},\boldsymbol{z})\begin{pmatrix}a_{x}&b_{x}&c_{x}\\ a_{y}&b_{y}&c_{y}\\ a_{z}&b_{z}&c_{z}\end{pmatrix}=(\boldsymbol{x},\boldsymbol{y},\boldsymbol{z})\mathbf{T}\;.$$
(18.37)
A determination of crystal orientation involves the determination of the \(\mathbf{T}\) matrix using the indexed diffraction patterns; results of diffraction pattern indexing give the \(z\)-axis in the crystallographic coordinates and the position of at least one indexed spot, which can used to determine the orientation of the \(x\) and \(y\)-axes.
Once the incident beam direction has been specified, a systematic method is needed to determine which diffracted beams should be included in the simulated diffraction pattern. The selection can be established based on three criteria [18.61]:
  1. 1.

    Proximity to the Ewald sphere with \(|S_{g}|<S_{\max}\)

     
  2. 2.

    Length of \(g\) with \(g<g_{\max}\)

     
  3. 3.

    The dimensionless parameter \(\omega=S_{g}\xi_{g}\) with \(\omega<\omega_{\max}\).

     
The three parameters, \(S_{\max}\), \(g_{\max}\), and \(\omega_{\max}\) are adjusted for the best match with the experimental pattern.
The electron diffraction pattern recorded on the detector is a magnified image of the diffraction plane. At the detector, the measured peak position can be specified by \((D_{x},D_{y})\) in the length unit of mm and
$$D=\sqrt{D_{x}^{2}+D_{y}^{2}}\;.$$
The lattice \(d\)-spacing is given by \(d_{hkl}=1/|\boldsymbol{g}|=1/|h\boldsymbol{a}^{*}+k\boldsymbol{b}^{*}+l\boldsymbol{c}^{*}|\). The \(d\)-spacing can be obtained by measuring the length of \(\boldsymbol{g}\) in the experimental diffraction pattern (\(D\)) using
$$d\approx\frac{L\lambda}{D}\;.$$
(18.38)
Where \(L\) is the experimental camera length. Experimentally, only the camera constant of the product of \(L\lambda\) is needed, which can be calibrated using a known \(d\)-spacing.

Kikuchi Lines and HOLZ Lines

Kikuchi lines are produced by Bragg diffraction of electron diffuse scattering produced by inelastic scattering and appear as background intensities in the diffraction pattern. The average electron energy loss is small, and the wavelength of the inelastic background is approximately the same as the incident electron beam. High-order Laue zone (HOLZ) lines are sharp lines observed in the CBED disks. They are produced by Bragg diffraction of lattice planes of high-order reflections. The rapid increase in the excitation error of a high-order reflection away from the Bragg condition results in a rapid decrease in the diffraction intensity. The maximum diffraction intensity occurs at the Bragg condition under the kinematic approximation, which appears as a straight line within the CBED disk. Since Kikuchi lines also mark the Bragg conditions, Kikuchi lines superimpose on HOLZ lines inside the disk and continue outside the disk. A major difference between the two is the contrast. Kikuchi lines are mostly observed for low and medium-order reflections, while HOLZ lines can be observed for high-order reflections.

The position of HOLZ lines is very sensitive to small changes in lattice parameters and the local strain. The sensitivity comes from the large scattering angle. This can be seen in the case of a cubic crystal for which
$$\theta\approx\frac{g\lambda}{2}=\sqrt{h^{2}+k^{2}+l^{2}}\frac{\lambda}{2a}\;.$$
A small change in \(a\) gives
$$\delta\theta\approx 0.5\frac{g\lambda\delta a}{a}\;.$$
The amount of change in the Bragg angle is proportional to the length of \(g\). The positions of these lines move relative to each other when the lattice parameters change. This effect can be used for accurate measurement of lattice parameters.
The direction of a HOLZ line is normal to the reciprocal lattice vector, and its position is decided by the Bragg condition. In diffraction analysis, it is useful to express HOLZ lines using line equations in an orthogonal coordinate system (\(x,y,z\)), with \(z\) parallel to the zone axis direction. The \(x\) direction can be taken along the horizontal direction of the experimental pattern, and \(y\) is normal to \(x\). The Bragg diffraction expressed in this coordinate is given by
$$k_{y}=-\frac{g_{x}}{g_{y}}k_{x}+\frac{2g_{z}-g^{2}}{2g_{y}}|k_{z}|\;.$$
(18.39)
Here,
$$|k_{z}|=\sqrt{k_{0}^{2}-k_{x}^{2}-k_{y}^{2}}\approx k_{0}=\frac{1}{\lambda}\;.$$
(18.40)
The approximation holds for high-energy electrons of small wavelengths and the typical acceptance angles in electron diffraction. Within this approximation, beams that satisfy the Bragg condition form straight lines.

Automated Indexing of Electron Diffraction Patterns

With the development of scanning-based electron diffraction techniques, a large number of diffraction patterns ( s) can be acquired over a short time. For example, \(\mathrm{1000}\) diffraction patterns can be acquired within minutes using the SEND technique described in Sect. 18.1.5, Scanning Electron Nanodiffraction and Scanning CBED. Indexing of such large diffraction data sets requires a fast and robust approach to DP indexing.

Various schemes of automatic indexing based on the comparison of experimental and calculated diffraction patterns have been proposed and developed for x-ray and electron diffraction [18.62, 18.63, 18.64, 18.65, 18.66]. For electron diffraction, the most successful approach so far is based on the template matching algorithm to compare the acquired DPs to precalculated ones [18.67]. In template matching, the crystal orientation (and phase) is determined from the best fit, which is identified among a large number of precalculated diffraction pattern templates. The template-based approach has also demonstrated some success of indexing superimposed diffraction patterns that were recorded from grain, or twin, boundaries [18.67].

In what follows, we will describe the normalized cross-correlation ( ) algorithm based on the comparison of an experiment DP with a set of simulated DPs and by searching for the best match [18.66, 18.68]. The NCC algorithm was initially developed by Rauch et al [18.66, 18.69] and improved by Wu and Zaefferer [18.68] and Meng and Zuo [18.70].

The basic steps of the NCC algorithm as described by Meng and Zuo [18.70] are:
  1. 1.

    Determine the beam center position and camera length (magnification) of the experimental DP.

     
  2. 2.

    Calculate the circular projection of diffraction intensities in the experimental DP. Intensities of all pixels located at one polar angle are integrated to generate a 1-D circular profile.

     
  3. 3.

    Calculate the radial projection of diffraction intensities in the experimental DP. Intensities of all pixels on one radius are integrated to generate a 1-D radial profile.

     
  4. 4.

    Calculate the circular and radial projections for all simulated DPs based on the experimental camera length.

     
  5. 5.

    Compare the circular and radial profiles of the experimental DP with those of the simulated ones using the 1-D NCC. Record the pattern rotation angle for each simulated DP.

     
  6. 6.

    Compare the experimental DP with the simulated DPs using the radial and circular profiles. Select an amount of simulated DPs that are most similar to the experimental ones.

     
  7. 7.

    Compare the experimental DP with the selected simulated DPs using the direct 2-D NCC [18.71].

     
The correlation coefficient \(\gamma\) is defined in the following equation
$$\gamma=\dfrac{{\displaystyle\sum_{x,y}}\left(\left[I_{\mathrm{A}}(x,y)-\overline{I}_{\mathrm{A}}\right]\left[I_{\mathrm{B}}(x,y)-\overline{I}_{\mathrm{B}}\right]\right)}{\sqrt{\left({\displaystyle\sum_{x,y}}\left[I_{\mathrm{A}}(x,y)-\overline{I}_{\mathrm{A}}\right]^{2}\right)\left({\displaystyle\sum_{x,y}}\left[I_{\mathrm{B}}(x,y)-\overline{I}_{\mathrm{B}}\right]^{2}\right)}}\;,$$
(18.41)
where \(I_{\mathrm{A}}(x,y)\) and \(I_{\mathrm{B}}(x,y)\) are intensities of the pixel \((x,y)\) in images \(A\) and \(B\), respectively, and \(\overline{I}_{\mathrm{A}}\) and \(\overline{I}_{\mathrm{B}}\) are mean intensities of images \(A\) and \(B\).

Circular and radial profiles calculated in steps (4) and (5) can be used achieve a faster indexing result. Since a 1-D profile contains a much smaller amount of pixels than the whole DP, the calculation of the correlation coefficient of circular and radial profiles is significantly faster than comparing the whole DPs. Thus, a comparison using the circular and radial profiles can be used to select a small number of simulated patterns. A final NCC comparison is executed with the selected patterns. Figure 18.24 shows the indexing result of an experimental DP.

Fig. 18.24

(a) An experimental spot DP. (b) The corresponding simulated DP found using the NCC algorithm. (c) The correlation factor map

The DP center plays a crucial role in NCC-based algorithms. The most accurate way to determine the DP center is to use the 2-D lattice observed in an experimental pattern selected for calibration. The 2-D lattice determination involves fitting two lattice vectors and the origin of the lattice to match the calibration pattern. The fitting provides an accurate determination of the DP center position. If no well-defined lattice is observed, the disk center of the direct beam can be simply taken as the DP center. The DP magnification (camera length) is utilized to scale the simulated patterns to match the experimental patterns. The DP magnification can be calibrated using a standard sample such as Si.

The significant challenges for achieving reliable diffraction pattern indexing results are the quality of experimentally acquired DPs, which is poor when the crystals are thick, with strong diffuse inelastic scattering background, and overlapping DPs from different grains.

Orientation Mapping

Fine-grained and nanostructured materials are very much at the core of materials research. Significant properties arise in nanostructures from surface/interface effects: A significant fraction (or the majority) of atoms are located on or near the surfaces or interfaces in a nanostructure. Since the chemical bonds of surface or interface atoms can differ significantly from interior (or bulk) atoms, interfacial atoms give rise to distinct chemical, mechanic, thermodynamic, electronic, magnetic, and optical properties. The surface/interface effects are expected to increase as the ratio of surface to bulk atoms increases.

However, nanostructure characterization is challenging because of the small sizes and the difficulty of characterizing small interfaces. Electron backscatter diffraction ( ) is commonly used to characterize the microstructure of granular materials, such as metals and alloys, down to the submicron scale [18.72]. However, the spatial resolution of the EBSD technique is limited by the relatively large electron probe size in a SEM and the interaction volume, as well as the need to tilt the sample to a high angle, which is typically \(70^{\circ}\) from horizontal, for the collection of the EBSD patterns. The achievable spatial resolution is above \(\approx 20{-}30\,{\mathrm{nm}}\) using a field emission SEM and the accuracy of orientation determination is in a range of \(0.5^{\circ}\). To improve upon the resolution of EBSD, transmission EBSD ( ) was developed [18.73]. An electron transparent sample is used in t-EBSD and the sample is mounted in such way that the diffraction pattern projected from the lower surface of the sample can be recorded. Studies have shown that a spatial resolution of \(\approx{\mathrm{10}}\,{\mathrm{nm}}\) can be achieved by t-EBSD, but the resolution is sample thickness and acceleration voltage dependent, and the success of t-EBSD also depends strongly on these factors.

For very fine grains, electron nanodiffraction techniques, such as SEND performed in a TEM, offers a spatial resolution of \(\approx 1{-}5\,{\mathrm{nm}}\). The major limitation with the use of electron nanodiffraction for orientation mapping has been diffraction pattern indexing, which was not as fast as the EBSD technique. This limitation is largely improved now with the development of automated indexing methods (see the previous section for the discussion). CBED, which gives much better contrast for the recorded Kikuchi lines as well as HOLZ lines, provides the best accuracy for orientation determination, and it can be combined with electron nanodiffraction, in principle to improve the accuracy of orientation determination.

Fig. 18.25a-c

Orientation mapping of nanocrystalline Au thin film using SEND. (a) A bright-field TEM image of a region where diffraction patterns were recorded, (b) reconstructed bright-field image from the SEND data, and (c) a color-coded orientation map of (b). Provided by Piyush Vivek Deshpande, UIUC

In what follows, we will illustrate the principle of TEM-based orientation mapping using the Au thin film of Fig. 18.25a-c as an example. The film has a thickness of \({\mathrm{50}}\,{\mathrm{nm}}\) deposited on a silicon substrate. A plane view Au thin film sample was prepared using tripod polishing followed by ion milling, and a region of interest was selected for electron diffraction study (Fig. 18.25a-cb). The SEND dataset was acquired on a \(31\times 31\) mesh using an electron probe of \({\mathrm{2.8}}\,{\mathrm{nm}}\) in FWHM, at an exposure time of \({\mathrm{0.1}}\,{\mathrm{s}}\) for each diffraction pattern and a step size of \({\mathrm{8}}\,{\mathrm{nm}}\). The corresponding scanned area of the sample is \(240\times 240\,\mathrm{nm^{2}}\). The diffraction patterns were recorded on a Gatan UltraScan® 1000XP (Model 894) CCD camera with Peltier-cooled CCD and \(2048\times 2048\) pixels. The recorded diffraction patterns were binned eight times.

In order to generate an orientation map from the Au film sample, a correlation image is first generated from the SEND patterns. The aim of the correlation analysis method is to identify similar diffraction patterns, average these similar patterns, and obtain a correlation map where the distribution of similar diffraction patterns is identified. This correlation image is further processed to form the orientation map. The similarity of DPs is quantified by the value of the NCC coefficient calculated for two individual diffraction patterns. The value of an NCC coefficient between two DPs will range from \(-1\) to 1, with NCC \(=0\) indicating complete dissimilarity, and NCC \(=1\) indicating complete similarity. We use (18.41) to compute the NCC value between two diffraction patterns, say 1 and 2, with 1 for A and 2 for B. Next, the DPs are grouped using the values of NCC between the pairs of diffraction patterns. A group is defined as one with all DPs belonging to the group having values of the NCC values amongst themselves to be equal to or greater than a fixed, predefined, threshold value. This fixed threshold of NCC is called correlation threshold (CT ). The formation and number of groups will be directly affected by a change in this CT value, which is schematically shown in Fig. 18.26 using the analysis of seven DPs as an example. When CT is set at \(\mathrm{0.3}\), all seven DPs fall into one single group, as the NCC values amongst these seven DPs are equal to or higher than \(\mathrm{0.3}\). Also, if a CT value of 0.8 is selected, each DP is considered as a different group, i. e., grouping will not occur for these DPs, as the NCC values are \(<0.8\).

Fig. 18.26

Grouping of diffraction patterns using correlation analysis. Provided by Piyush Vivek Deshpande, UIUC

Fig. 18.27a,b

Correlation image and diffraction pattern templates for various sample regions. Provided by Piyush Vivek Deshpande, UIUC

A correlation image is obtained from the diffraction grouping analysis, each group is assigned a unique index number and color (Fig. 18.27a,b), and diffraction patterns belonging to the same group are summed together and averaged to obtain a diffraction template for that group. Thus, the correlation image displays areas belonging to different diffraction templates.

For the example of Fig. 18.25a-c, the correlation analysis yielded 54 diffraction templates, which are analyzed for orientation determination using the automatic indexing technique described in Sect. 18.4.1, Automated Indexing of Electron Diffraction Patterns. Figure 18.25a-cc shows the results of the diffraction pattern indexing result. The orientation of the grain is color coded, with red for [001], green for [011], and blue for [111].

The results of orientation mapping can be validated using the constraints of virtual dark-field imaging. A virtual dark-field image is obtained from the SEND data by integrating diffraction intensities within a  (region of interest) of the diffraction patterns; each DP gives the intensity of a pixel in the virtual dark-field or bright-field image, where the DP is recorded.

Fig. 18.28a-c

Overlay of dark field ( ) reconstruction on correlation image for validation of algorithm: (a) Selected spot for DF reconstruction, (b) reconstructed DF image, (c) overlay result. Provided by Piyush Vivek Deshpande, UIUC

Figure 18.28a-cc shows the result of overlay of one of the reconstructed DF images onto the orientation map. The diffraction spot selected for virtual DF image construction as indicated in Fig. 18.28a-ca yields the virtual DF image of Fig. 18.28a-cb. The grain corresponding to the selected spot is associated with bright intensities as compared to other regions of the nanostructure. It is important to note that this is the same grain that corresponds to the template 7 in Fig. 18.27a,b. This grain is blue-colored in the orientation map, which is close to [111]. This procedure can be performed for other grains identified in the orientation map.

The orientation map of Fig. 18.25a-c reveals that most of the grains are closer to the [111] zone axis. This is expected for the Au system, as Au has an fcc crystal structure and, consequently, it would crystallize in a way such that the normal direction would be predominantly [111]. However, interestingly, we also see grains very close to [001], which is colored in red. Thus, the orientation information of the fine nanostructure of the Au thin film sample is successfully determined by the method described above.

Fig. 18.29a-c

Intergranular fracture mode of TiN hard coating revealed by orientation mapping and in-situ TEM. (a) Orientation map of a FIB fabricated beam with a notch, the map is obtained from the boxed region using SEND. The diffraction pattern for each grain is shown in (b). (c) The crack growth path. Provided by Yang Hu, UIUC

Applications of TEM-based orientation mapping have been demonstrated for the characterization of copper interconnect lines fabricated by the damascene process in microelectronic devices [18.74], highly-deformed metals [18.75], nanocrystalline Cu [18.76], and crack growth in nanocrystalline TiN thin films [18.77]. Figure 18.29a-c shows intergranular fracture in nanocrystalline TiN observed in a FIB fabricated beam. In order to better distinguish the grain boundary and grain shape, SEND was performed before the bending test. The scanning region was \({\mathrm{200}}\,{\mathrm{nm}}\times{\mathrm{150}}\,{\mathrm{nm}}\) with an electron probe size \(\approx{\mathrm{2}}\,{\mathrm{nm}}\) and step size \({\mathrm{8}}\,{\mathrm{nm}}\). The orientation of each grain was determined by automatic electron diffraction pattern indexing. The orientation map and the corresponding diffraction pattern for each grain are shown in Fig. 18.29a-ca,b. The majority of diffraction patterns are from single crystal, indicating that very few grain-overlaps exist along the electron beam direction. From the correlation map color-coded by the crystal orientation of each grain, most of the grains are separated by high angle grain boundaries. The crack propagation path is shown in Fig. 18.29a-cc with a post-mortem bright-field TEM image and a schematic figure illustrating the configuration of grains and crack. The crack front proceeds along the grain boundary between G5 and G6, and then diverges into two crack propagation paths (Fig. 18.29a-cc). The crack on the right keeps propagating along the grain boundary of G3 until it meets G2, where the crack is arrested due to large resistance exerted by the grain boundary perpendicular to the crack propagation direction. The crack on the left continues to propagate along the grain boundary of G2, G4, and G5 and keeps being deflected by the grain boundaries.

Strain Mapping Using SEND

Using the TEM deflection (or STEM scan) coils, SEND patterns can be recorded from an area of the specimen to provide spatially resolved strain information for strain mapping. The diffraction peak positions are determined from the recorded diffraction patterns and used to map the local strain in real space. Additionally, from the recorded diffraction patterns, bright and dark-field STEM images can be obtained from SEND by integrating the diffraction intensities of the direct beam or the diffracted beams, respectively, which can be used to register the strain map to these images. SEND can be performed in either STEM or TEM mode. When NBD is performed in STEM, the STEM ADF detector can be used to visualize the ROI by collecting the scattered electrons to large angles during scanning. This way, diffraction can be correlated with the probe location during data collection.

The basis of strain mapping is that a series of diffraction patterns are acquired and compared to a reference diffraction pattern from an unstrained region. The strain is then calculated as the relative lattice mismatch by determination of the exact position of the diffraction spots positions. Using the FEI-Titan microscope operated in the \(\upmu{}\)-Probe STEM mode with a \({\mathrm{2.7}}\,{\mathrm{nm}}\) probe size and \({\mathrm{0.5}}\,{\mathrm{mrad}}\) convergence angle, a precision of \(6\times{\mathrm{10^{-4}}}\) was reportedly achieved [18.54]. The accuracy depends on the sample preparation and related strain relaxation, and is estimated to be about \(\mathrm{10^{-3}}\).

The projected 2-D strain \(\varepsilon\) can be obtained using the 2-D deformation matrix (\(\mathbf{D}\)) obtained from two measure reciprocal vectors, \(\boldsymbol{g}_{1}\) and \(\boldsymbol{g}_{2}\). They can be taken as the basis vectors for the zone axis diffraction pattern or any two nonparallel vectors recorded in the diffraction pattern. Each vector is defined by its components along the \(x\) and \(y\) directions perpendicular to the zone axis. They give the following \(\mathbf{G}\)-matrix
$$\mathbf{G}=\begin{pmatrix}g_{1x}&g_{2x}\\ g_{1y}&g_{2y}\end{pmatrix}$$
(18.42)
and \(\mathbf{D}\) is simply given by
$$\mathbf{D}=(\mathbf{G}^{\mathrm{T}})^{-1}\mathbf{G}_{0}^{\mathrm{T}}-\mathbf{I}\;,$$
(18.43)
where \(\mathbf{T}\) represents the transverse, and \(\mathbf{I}\) is a unit diagonal matrix and \(\mathbf{G}_{0}^{\mathrm{T}}\) is the transverse of the \(\mathbf{G}\) matrix of the reference crystal. The strain and crystal rotation are obtained from \(\mathbf{D}\) using
$$\varepsilon=\frac{1}{2}(\mathbf{D}+\mathbf{D}^{\mathrm{T}})\text{ and }\omega=\frac{1}{2}(\mathbf{D}-\mathbf{D}^{\mathrm{T}})\;.$$
(18.44)
In SEND, dynamical effects can lead to rapid changes in diffraction spot intensities, with thickness and changes in the crystal orientation. By recording electron diffraction patterns with the incident electron beam in precession, PED is able to provide the integrated electron diffraction intensity across the Bragg condition for many reflections. The same principle of reducing dynamic effects by precession can also be used to improve measurements of strain in nanostructures [18.78]. PED has been shown to greatly improve the quality and robustness of electron diffraction strain analysis. It can work with a larger convergence angles and, thus, a small probe size, which offers increased flexibility in the experimental conditions.
Fig. 18.30

(a) HAADF-STEM image of the observed sample composed of four SiGe layers deposited on a (001) silicon substrate. (b\(\varepsilon_{xx}\) strain profiles obtained from N-PED, NBD, and simulations. Provided by Jean-Luc Rouviere, CEA, Grenoble, France

Rouviere et al reported on an implementation of scanning procession electron diffraction on a FEI-TITAN TEM [18.78]. The scan coils of the STEM unit were used to precess the incident beam and to perform a descan of the diffracted beams at a precession speed of \({\mathrm{0.1}}\,{\mathrm{s}}\). Diffraction patterns were recorded on a \({\mathrm{2}}\,{\mathrm{k}}\times{\mathrm{2}}\,{\mathrm{k}}\) Gatan ultrascan CCD camera with acquisition times of \(\approx{\mathrm{1}}\,{\mathrm{s}}\). The incident beam used for strain measurements had a convergence angle of \({\mathrm{1.8}}\,{\mathrm{mrad}}\) and a size of \({\mathrm{2.4}}\,{\mathrm{nm}}\). Thus, the diffraction patterns consisted of small disks. By using PED, the intensities within a given diffraction disk are made more uniform than the CBED-like patterns recorded without precession. Diffracted beam positions were measured by detecting the edges of the diffraction disks instead of their peak intensity. Because of the improvements in the intensity distribution within the diffraction disk, a better measurement accuracy could be obtained with the help of precession. Precession also helps by spreading the intensity across to high-index diffraction spots and making them more amenable for detection. Additionally, precession improves the robustness of measurements by reducing the crystal misorientation effects by averaging over the precessed incident beam directions at the cost of slightly larger beam diameter, and increased crystal volume from the tilted incident beam and its precession.

Figure 18.30b shows the measured strain profiles from a Si/SiGe multilayer grown by reduced pressure chemical vapor deposition ( ) using NBD with and without precession. The sample contains four SiGe layers, each \({\mathrm{11}}\,{\mathrm{nm}}\) thick and of different Ge compositions: \(\mathrm{20}\), \(\mathrm{31}\), \(\mathrm{38}\), and \({\mathrm{45}}\%\). The composition was determined using secondary ion mass spectroscopy ( ). The SiGe layers are biaxially strained by the Si substrate, and the lattice parameter is larger than the reference substrate only in the direction perpendicular to the layers. Figure 18.30 also plots the strain profile obtained by finite element simulations of the structure to take into account the strain relaxation in the thin TEM lamella. The profile was plotted by averaging the strain along the \(\langle 011\rangle\) beam direction and convoluting the obtained profiles with the measured electron beam size, i. e., \({\mathrm{2.5}}\,{\mathrm{nm}}\). This reduced the strain in the layers slightly; this effect is greater in the layer with the higher Ge concentration, where the strain is reduced from 2.76 down to \({\mathrm{2.6}}\%\). As can be seen in Fig. 18.30, in the SiGe layers, the difference between the measurements of NBD with and without precession is small. At the center of the SiGe layers, the difference between the calculated and the experimental strain obtained with precession is about \({\mathrm{0.1}}\%\) for the three layers that have the lowest Ge concentration. Large differences are observed inside the silicon; the profile obtained with precession is slightly negative inside the silicon, which fits very well with the simulation result, while the NBD measurement without precession gives far larger negative strain than the simulation result indicates, especially near the Si-SiGe layer interface. Away from the SiGe layers, the strain profile obtained with precession is very smooth, with a root mean square of fluctuations of \({\mathrm{1.5\times 10^{-4}}}\).

Fig. 18.31

(a) Bright-field image of a transistor with a \({\mathrm{22}}\,{\mathrm{nm}}\) channel (C), recessed SiGe source (S), and drain (D). Some dislocations are indicated by arrows. (be) The different strain and rotation maps obtained from the analysis of N-PED patterns. Provided by J.-C. Rouviere, CEA, Grenoble, France

Application of SEND with precession is demonstrated in Fig. 18.31 for the analysis of a transistor with recessed \(\mathrm{Si_{0.65}Ge_{0.35}}\) source and drain. Figure 18.31a is a bright-field image of the device. Strain and rotation maps have been obtained by using (18.44). The DPs used in the analysis were obtained using a beam semiconvergence angle of \({\mathrm{1.8}}\,{\mathrm{mrad}}\), a precession angle of \(0.5^{\circ}\), and a beam diameter of \({\mathrm{2.5}}\,{\mathrm{nm}}\). The root mean square of the strain in the Si substrate, far from the layers, can be used to measure the strain measurement precision, i. e., the reproducibility of the technique. For NBD, the strain precision can be as low as \({\mathrm{6\times 10^{-4}}}\). The best precision obtained experimentally on NBD with precession was \({\mathrm{9\times 10^{-5}}}\) with a probe size of \({\mathrm{2.5}}\,{\mathrm{nm}}\). In the above examples, only small maps, \(20\times 20\) pixels, were obtained. With improved precession speed, larger memory, and a shorter camera acquisition time, larger maps can be realized. Overall, initial results demonstrate that NBD with precession is a very efficient technique to measure strain in nanostructures.

Determination of Three-Dimensional Nanostructures

The structure of nanocrystalline materials is determined by the constitutive phases, composition, 3-D grain morphology, orientation, and distribution, which can only be obtained from a 3-D structure determination. Previously, 3-D x-ray diffraction microscopy ( ) [18.79, 18.80] was developed for the study of polycrystalline materials at mesoscale. Recently, two new x-ray diffraction ( ) techniques, differential-aperture x-ray microscopy ( ) [18.81] and diffraction contrast tomography ( ) [18.82], achieved sub-\(\mathrm{\upmu{}m}\) spatial resolution in 3-D. Using a combination of a SEM and focused ion beam, 3-D electron backscattered diffraction ( ) [18.83] is the technique for obtaining 3-D orientation maps in bulk polycrystalline samples. However, the destructive nature of 3-D-EBSD makes it unfavorable for multitechnique or in-situ analysis. There are fewer attempts in 3-D microstructure determination using TEM. Liu et al reported a 3-D orientation mapping technique called 3-D-OMiTEM, which was developed based on the conical scanning dark field imaging technique [18.84]. More recently, Midgley's group at Cambridge [18.85] demonstrated a determination of the three-dimensional precipitate morphology in an Ni-based superalloy using scanning precession electron diffraction [18.36] and a principal component-based separation algorithm to separate the matrix and precipitate diffraction patterns (Fig. 18.32).

Fig. 18.32

Scanning precession electron tomography of an Ni-based superalloy. 3-D reconstruction of the superalloy volume from \(\mathrm{14400}\) precession electron diffraction data collected over \(5^{\circ}\) intervals from the tilt angles of \(\mathrm{-60}\) to \(+70^{\circ}\). The faceted blue particle is metal carbide, while the elongated shape is the \(\upeta\)-phase (green, approximate composition \(\mathrm{Ni_{6}Nb}\)(Al,Ti)) and the surrounding matrix phase is colored orange. From [18.85]

In what follows, we show how SEND can be used for 3-D nanostructure determination . Specifically, by taking advantage of diffraction information, we show that the principles of electron tomography can be extended for nanostructure characterization beyond the capabilities of the traditional mass-thickness contrast approach.

Electron tomography works by recording a series of electron images at various sample rotations (or tilts) and reconstructing a 3-D image from the projected images. The contrast of recorded images is assumed to be proportional to the sample thickness and mass. The 3-D image construction is carried using algorithms developed through cryoelectron microscopy, based on well-established methods such as the Radon transform. The range of sample rotations plays a crucial role in the accurate reconstruction of targeted objects. The tilt angle of a specimen can be limited by the sample thickness, shadowing effects from the sample holder or the supporting grid [18.86].

For electron diffraction, a custom tomography holder that allows \(\pm 87^{\circ}\) rotation of the specimen was developed by Meng and Zuo [18.87]. Their design employs a needle-shaped specimen mounted on a regular JEOL single-tilt holder (Fig. 18.33a,b). The sample is placed on the top of the tungsten substrate using the FIB lift-out technique. The sample is annularly milled to the desired diameter (usually between \(\mathrm{100}\) and \({\mathrm{300}}\,{\mathrm{nm}}\)). The small diameter of the mounting tube (part C) allows a free sample rotation in a small polepiece gap. The needle-like sample is also parallel to the rotation axis of the holder, which provides a rotation with the minimum precession movement.

Fig. 18.33a,b

Tomographic holder for tip-shaped samples. (a) Design of the customized tomographic holder. (b) SEM image of the polished W wire. Provided by Yifei Meng, UIUC

Three-dimensional diffraction data can be obtained by tilting the sample and performing SEND at each sample tilt. In the experiment reported by Meng and Zuo, an exposure time of \({\mathrm{0.1}}\,{\mathrm{s}}\) and a diffraction pattern size of \(256\times 256\) pixels were used in the diffraction pattern recording. A small camera length was also used to include as many diffraction spots as possible without too much degradation in the resolution of the diffraction pattern. The step size of sample tilting was selected based on a balance between the time cost of the data acquisition, the data size, and the accuracy of the reconstructed grain morphology. A smaller step size gives a more reliable 3-D morphology of a grain at the cost of an increase in time and data size. It has been suggested to use a tilting step size smaller than \(10^{\circ}\).

The first step of 3-D reconstruction is to examine the projected (2-D) morphology of the grains. The 2-D morphology of a grain is identified by constructing dark-field images from the recorded DPs through the following sorting process. At a specific sample rotation angle, the position and intensity of all diffraction spots are recorded using the template matching method [18.88]. Using this information, a dark-field image is then constructed for each diffraction spot. Two dark-field images will be similar to each other if their diffraction spots belong to the same DP of the single crystal grain. Thus, dark-field images with similar contrasts are grouped using normalized cross-correlation. A correlation threshold is used for grouping based on the trial and error method. The 2-D morphology is extracted from the averaged dark-field image, after applying an intensity threshold. Meanwhile, diffraction spots belonging to a single grain are grouped into a single crystal diffraction pattern. Unlike an experimental DP, this DP only contains a subset of measured diffraction spots. Figure 18.34 shows the sorting results of the TiN sample at \(-5^{\circ}\). This step is repeated for every sample rotation angle.

Fig. 18.34

Identification of grains and grain DPs by sorting. Eight grains are identified here for the TiN sample at \(-5^{\circ}\) rotation. The image-filtered DPs are shown along with the averaged dark-field image. Provided by Yifei Meng, UIUC

Next, the filtered DPs are indexed for the determination of crystal orientation. If the number of diffraction spots in an image-filtered DP is not sufficient for a reliable indexing, we index the averaged experimental DPs within the identified 2-D grain. In this case, the averaged experimental DP may contain diffraction spots from other overlapping grains. Therefore, multiple local maxima may appear in the indexing correlation factor map [18.89]. Electron DP indexing is done using automatic indexing (Sect. 18.4.1, Automated Indexing of Electron Diffraction Patterns). Most diffraction patterns can be indexed this way with the exception of a few DPs far away from zone axes. However, a small number of failed index attempts does not affect the overall analysis.

To reconstruct a 3-D grain from the projected dark-field images, the algebraic reconstruction technique ( ) is used [18.90]. The commonly used backprojection method is not applicable here, since the dark-field images are not monotonic to any physical properties of the grain because of electron multiple scattering. Additionally, the use of ART is justified by the following reasons. First, one grain may only be identified from a part of the rotation. It is also possible that the data at a particular rotation angle is not usable because of weak diffraction spots or strong multiple scattering. In either case, we found that the number of available projections is often limited in the 3-D diffraction dataset. ART is designed for incomplete projection data. Secondly, ART allows inputs of prior information about the object. The outline of the needle-shaped sample introduces a strong constraint that can be included in the reconstruction (it is used for setting up the ray–voxel interaction matrix; details are described later). This step improves the accuracy of reconstruction results. Prior to 3-D reconstruction using ART, the 2-D dark-field images are identified as belonging to the same grain. This is done by confirming two projected grain images belonging to the same grain from neighbouring rotations using two criteria:
  1. 1.

    The difference between the two beam directions is equal to the sample rotation step size

     
  2. 2.

    The 2-D grain images overlap with each other. Figure 18.35 shows the 2-D images of one grain from \(\mathrm{-75}\) to \(-5^{\circ}\).

     
Fig. 18.35

Dark-field images identified as belonging to the same grain from \(-75^{\circ}\) to \(-5^{\circ}\). Provided by Yifei Meng, UIUC

ART requires discretization of the sample and the projection. The projection is already discretized by stepwise beam scanning. We assume that the beam scanning is performed in an \(m\times n\) area, and the number of sample tilting angles is \(r\). Then, the sample is discretized into an \(m\times n\times n\) space. Each voxel is a cube with an edge length of the scanning step size. The problem of the 3-D reconstruction can be reduced to a linear algebraic equation
$$Ax=p\;,$$
(18.45)
where \(A\) is a ray–voxel interaction matrix ( ), \(x\) is a column vector representing the object distribution, and \(p\) is a column vector representing the projection data. The height of \(A\) is equal to the number of rays (electron beams) applied in the experiment. The width of \(A\) is equal to the number of voxels in the sample space. Under the previous assumption, the size of \(A\) is \(mnr\)-by-\(\mathrm{mn}^{2}\). The value of an element \(a_{ij}\) in \(A\) is same as the length of the segment of the \(i\)th ray inside the \(j\)th voxel; \(a_{ij}\) represents the contribution of the \(j\)th voxel to the projection result of the \(i\)th ray; \(A\) is pre-calculated using the fast ray-tracing algorithm proposed by Amanatides and Woo [18.91]. An element \(x_{j}\) in \(x\) represents the distribution of the object in the \(j\)th voxel; \(x\) is the unknown variable. An element \(p_{i}\) in \(p\) represents the measured projection under the \(i\)th ray; \(p\) is determined based on the acquired dark-field images. The value of \(p_{i}\) is set to 1 if the projection of the \(i\)th ray is within the outline of the 2-D grain morphology. Otherwise, it is set to 0. Various algorithms were developed for solving (18.45). Here, we use the algebraic iterative algorithm first proposed by Kaczmarz [18.92]; \(x\) is additively modified in each cycle to approximate the ideal solution. We stop the iteration when \(x\) is stable.

The output of the ART is a 3-D map of voxel contribution to the target grain. By creating an isosurface of the map, the morphology of the grain can be plotted in the 3-D space. The isosurface value is adjusted so that the isosurface is continuous.

The orientation of a grain is determined from the indexing results of all available projections. In stereo projection, the indexing results are expected to form a line. The orientation of a grain is defined by the transformation matrix that transforms a vector in the crystal coordinate onto one in the holder coordinate. We define the transformation relation as \(\boldsymbol{h}=\mathbf{T}\boldsymbol{c}\); \(\boldsymbol{c}\) is a 3-by-1 vector, which represents a direction in the crystal coordinate; \(\boldsymbol{h}\) is a 3-by-1 vector, which represents the same direction in the holder coordinate; \(\mathbf{T}\) is a 3-by-3 matrix, which transforms the direction from the crystal coordinate into the holder coordinate.

Ideally, the matrix \(\mathbf{T}\) can be determined with two arbitrary observations. In practice, the indexing results contain noise, and a more accurate \(\mathbf{T}\) is obtained by minimizing the following error function
$$E(M)=\sum_{k=1}^{n}\parallel\boldsymbol{h}_{k}-\mathbf{T}\boldsymbol{c}_{k}\parallel^{2}\;,$$
(18.46)
where \(n\) is the number of useful projections. The solution for \(\mathbf{T}\) can be found using a single-value decomposition method [18.93]. First we calculate a 3-by-3 matrix \(\mathbf{B}\) as
$$\mathbf{B}=\sum_{k=1}^{n}\boldsymbol{h}_{k}\boldsymbol{c}_{k}^{\mathrm{T}}\;,$$
where \(\boldsymbol{c}_{k}^{\mathrm{T}}\) is the transpose of \(\boldsymbol{c}_{k}\). Next, we compute the single-value decomposition of \(\mathbf{B}\) as follows
$$\mathbf{B}=\mathbf{USV}^{\mathrm{T}}\;.$$
The transformation matrix is
$$\mathbf{T}=\mathbf{UMV}^{\mathrm{T}}\;,$$
(18.47)
where
$$\mathbf{M}=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&\text{det }(\mathbf{U})\text{ det }(\mathbf{V})\end{bmatrix}.$$
Fig. 18.36a-d

Reconstructed grains and their orientations. Side (a), front (b), and top view (c) of the 3-D morphologies of reconstructed grains. (d) Orientations of the seven grains. Each cube is labeled by the color used to represent the grain. Provided by Yifei Meng, UIUC

Figure 18.36a-d shows a reconstructed 3-D image of TiN nanograins using 3-D-SEND. The experiment was performed on a TiN thin-film nanocrystalline sample. TiN is a material that is widely applied in the electronics industry, as well as in protective and decorative coating [18.94]. Compared with polycrystalline TiN, nanocrystalline TiN exhibits improved mechanical properties such as hardness, and wear and corrosion resistance [18.95, 18.96, 18.97]. This is achieved by controlling intrinsic properties such as grain size, morphology, and texture [18.98]. Nanocrystalline TiN thin film can be grown by chemical vapor deposition ( ) and physical vapor deposition ( ) [18.99]. Here, the TiN sample was grown on a p-type Si(100) substrate by an unbalanced magnetron sputtering ( ) system [18.96].

Seven major grains in the TiN sample were reconstructed, as well as their crystallographic orientations. Previous experimental studies found that the grain is elongated along the growth direction, and multiple grains are stacked along the elongation direction [18.94]. However, the morphology of the grains is not known. The 3-D-SEND reconstruction results show that grains are elongated and the grain boundaries are not ideally round or flat.

The spatial resolution of the reconstructed 3-D image equals the step size used during the beam scanning under certain conditions. How it works is that the reconstructed data is a set of scattered points assigned with values. These points are then interpolated to obtain the grain morphology using a cubic kernel, which usually gives a rendering resolution lower than the data resolution. The spatial resolution of 3-D-SEND is ultimately limited by the electron probe size \(d_{0}\) and the column diameter under the column approximation. The diameter of the cone \(d_{\text{AB}}\) is defined approximately as
$$d_{\text{AB}}=2\alpha t\;,$$
(18.48)
where \(\alpha\) is the convergence semi-angle and \(t\) is the sample thickness. The radius of the first Fresnel zone \(\rho_{1}\) used to represent the diffraction column is calculated using
$$\rho_{1}=\sqrt{\lambda t}\;,$$
(18.49)
where \(\lambda\) is the electron beam wavelength. At \({\mathrm{200}}\,{\mathrm{kV}}\), \(\lambda={\mathrm{2.5}}\,{\mathrm{pm}}\). For a JEOL 2100 TEM, we can form a probe with a FWHM of \({\mathrm{2.3}}\,{\mathrm{nm}}\) using a \({\mathrm{10}}\,{\mathrm{\upmu{}m}}\) condenser aperture in the CBD mode with a full convergence angle of \({\mathrm{4.2}}\,{\mathrm{mrad}}\). If the sample thickness is \({\mathrm{200}}\,{\mathrm{nm}}\), \(d_{\text{AB}}\) is \({\mathrm{0.8}}\,{\mathrm{nm}}\) and \(\rho_{1}\) is \({\mathrm{0.7}}\,{\mathrm{nm}}\). This means that the best spatial resolution of 3-D-SEND is probe limited at around \({\mathrm{2}}\,{\mathrm{nm}}\) in a JEOL 2100 TEM.

18.4.2 Convergent Beam Electron Diffraction of Complex Crystals

Compared to electron nanodiffraction, which uses a small condenser aperture to achieve a small convergence angle that yields a spot-like diffraction pattern, CBED records diffraction patterns using a larger convergence angle that gives rise to diffraction disks, instead of diffraction spots. The extra diffraction intensities recorded in CBED make it ideal for the study of crystals, especially complex ones.

The Geometry of CBED

At medium convergence angles, the interpretation of electron nanodiffraction patterns recorded from crystals uses the same theory for CBED. The starting point for understanding CBED is the Ewald sphere construction. Figure 18.37 shows one example. By the requirement of elastic scattering, all transmitted and diffracted beams are on the Ewald sphere. Let us take the incident beam \(P\), which satisfies the Bragg condition for \(g\). For an incident beam \(P^{\prime}\), to the left of \(P\), the diffracted beam also moves to the left. The difference between the incident wave and the diffracted wave is the vector \(\boldsymbol{g}\). The deviation of the diffracted beam away from the Bragg condition is defined by the so-called excitation error given in (18.25).

Fig. 18.37

Ray diagram of CBED. Variation of excitation errors at different positions of the CBED disk. The beam marked by the red line (\(P\)) is at the Bragg condition, while beam marked by the blue line is associated with a positive excitation error (\(S_{\mathrm{g}}\))

The change in the excitation error across the CBED disk is important for understanding the rich diffraction intensity patterns often observed in CBED. To see how the excitation error changes within a CBED disk for a particular reflection, let us take the component of the wave vector \(\boldsymbol{k}_{0}\) along a reciprocal lattice vector \(\boldsymbol{g}\) as
$$k_{g}=-\frac{g}{2+\Delta}\;.$$
The positive \(\Delta\) corresponds to a beam tilt towards \(\boldsymbol{g}\). Substituting \(k_{g}\) into (18.25), we have
$$\begin{aligned}\displaystyle S_{g}&\displaystyle=\dfrac{\lambda\left[\left(-\frac{g}{2}+\Delta\right)^{2}-\left(\frac{g}{2}+\Delta\right)^{2}\right]}{2}\\ \displaystyle&\displaystyle=\frac{-g\Delta}{\left(\frac{1}{\lambda}\right)}\approx-g\partial\theta\;.\end{aligned}$$
Here, \(\partial\theta\) is the deviation angle from the Bragg condition. The excitation error given above is approximately the distance from the end of the diffracted wave vector to the Ewald sphere. The excitation error also has a sign. The \(S_{g}\) is negative for a positive \(\Delta\) and positive for a negative \(\Delta\). Thus, an incident beam moving to the left gives a positive excitation error. Correspondingly, a beam, moving to the right of \(P\), gives a negative excitation error. Generally, to a good approximation, the excitation error changes linearly across the CBED disk and along the direction of \(\boldsymbol{g}\) for each diffracted beam. The slope of the change is the length of \(\boldsymbol{g}\). The range of excitation errors within each disk is proportional to the length of \(\boldsymbol{g}\) and the convergence angle. Consequently, the excitation error changes much faster for a high-order Laue zone (HOLZ) reflection than a reflection in ZOLZ close to the direct beam. The excitation error and the crystal thickness are the two parameters controlling the diffraction intensity beside the crystal potential and the underlying atomic structure. This dependence can be directly seen in the expression for the diffraction intensity in the case of the two-beam approximation (18.33), which assumes only two strong beams in the diffraction pattern with the direct beam and the diffracted beam of \(g\). In CBED, the crystal thickness is approximately constant under the electron beam, the intensity variations observed in CBED disks are mostly caused by change in the excitation errors of the diffracted beams.

CBED Intensities

Interpretation of diffraction intensities is required for CBED analysis. There are several approaches to treat electron multiple scattering in electron diffraction [18.100, 18.18]. For perfect crystals, the Bloch wave approach is the most useful, which is based on an expansion of the electron wave function in plane waves, which was formulated first by Hans Bethe. In Bloch wave theory, when diffraction disks do not overlap, the diffraction intensity for a point inside the diffraction disk belonging to the reflection \(\mathbf{g}\) is given by
$$\begin{aligned}\displaystyle I_{g}(x,y)&\displaystyle=|\phi_{g}(x,y)|^{2}\\ \displaystyle&\displaystyle=\left|\sum_{i}c_{i}(x,y)C_{g}^{i}(x,y)\exp\left[2\uppi\mathrm{i}\gamma^{i}(x,y)t\right]\right|^{2}.\end{aligned}$$
(18.50)
Here the eigenvalue \(\gamma\) and eigenvector \(C_{g}\) are obtained from diagonalizing the equation
$$2KS_{g}C_{g}+\sum_{h}U_{gh}C_{h}=2K\gamma C_{g}\;,$$
(18.51)
where
$$\begin{aligned}\displaystyle U_{g}&\displaystyle=\frac{2me}{h^{2}}V_{g}=\frac{2me}{h^{2}}\sum_{i}\left(f_{i}+\mathrm{i}f_{i}^{\prime}\right)T_{i}(\boldsymbol{g})\\ \displaystyle&\displaystyle\quad\,\times\exp(-2\uppi\mathrm{i}\boldsymbol{g}\cdot\boldsymbol{r}_{i})\;,\end{aligned}$$
(18.52)
with \(U_{g}\) for the electron interaction structure factor including absorption. Details on the evaluation of the absorption potential (\(f^{\prime}\) in (18.52)) can be found in [18.101, 18.102, 18.103]; \(S_{g}\) is the excitation error as defined in (18.25). The coefficients, \(c_{i}\), are obtained from the first column of the inverse eigenvector matrix as determined by the incident beam boundary condition. The solution of (18.16) generally converges as the number of beams included in the calculation increases. In numerical calculations, the strong beams are included in the diagonalization, while weak beams can be treated by perturbation. In practice, an initial list of beams is selected using the criteria of maximum \(g\) length, maximum excitation error, and their perturbational strength. Additional criteria are used for selecting strong beams [18.61].
For electron diffraction from the crystals with defects that can be characterized by a lattice-dependent displacement field, \(\boldsymbol{u}(\boldsymbol{R}_{nml})\), dynamic scattering of the defects can be approximately calculated by using the scattering matrix method [18.104, 18.60]. In this method, the crystal is divided into parallel slices. Each slice contains a different atomic displacement. The number of slices, \(n\), is selected to give a good approximation of the displacement field along the beam direction. The electron wave function in the reciprocal space at thickness \(t\), \(\Psi=(\phi_{0}(t),\phi_{g}(t),\dots)^{\mathrm{T}}\), is related to the incident wave \(\Psi_{0}=(1,0,\dots)^{\mathrm{T}}\) through
$$\Psi=\mathbf{P}_{n}\mathbf{P}_{n-1}\dots\mathbf{P}_{1}\Psi_{0}\;,$$
(18.53)
where \(\mathbf{P}_{n}\) stands for the scattering matrix of the \(n\)th slice of the imperfect crystal and \(\mathbf{P}\) can be calculated using the above Bloch wave method [18.8] using
$$\mathbf{P}=\mathbf{Q}C\boldsymbol{\Upupsilon}C^{-1}\mathbf{Q}^{-1}\;.$$
Here, \(\mathbf{C}\) is the Bloch wave eigenvector matrix obtained from (18.16) and both \(\mathbf{Q}\) and \(\boldsymbol{\Upupsilon}\) are diagonal matrices, and
$$\begin{aligned}\displaystyle\mathbf{Q}&\displaystyle=\exp[2\uppi\mathrm{i}\boldsymbol{g}\cdot\boldsymbol{u}(z)]\quad\text{and}\\ \displaystyle\boldsymbol{\Upupsilon}&\displaystyle=\exp(2\uppi\mathrm{i}\gamma^{i}\Updelta z)\;,\end{aligned}$$
with \(\Updelta z\) the slice thickness. The above scattering matrix method works, as long as there is a common set of reflections perpendicular to the beam direction. The number of these reflections should not be too large, so the scattering matrix can be calculated and stored in a computer at reasonable computational cost. Small changes in composition can also be included in the scattering matrix, as long as the change in the lattice perpendicular to the electron beam is small. This approach is particularly useful for studying coherent interfaces, including epitaxial multilayer structures in plane view geometry. For example, the scattering matrix method has been successfully used in simulations of CBED patterns from a buried quantum well [18.105] and multilayers [18.106].
Another commonly used method for dynamic electron diffraction simulation is the multislice method developed by Cowley and Moodie, and others [18.107, 18.108]. As a numerical method, multislice has the advantage that it can treat both crystals and nonperiodic structures, including amorphous structures. Because of this, the multislice method is particularly suitable for electron nanodiffraction simulation. The multislice method models the forward propagation of the electron waves through successive thin slices of potentials (Fig. 18.38). The basic equation is the relationship between the incident wave \(\phi_{n}(x,y)\) and the exit wave \(\phi_{n+1}(x,y)\) of the \(n\)th slice
$$\begin{aligned}\displaystyle\phi_{n+1}(x,y)&\displaystyle=\left(\phi_{n}(x,y)\exp\left[i\sigma\overline{V}_{n}(x,y)\right]\right)\\ \displaystyle&\displaystyle\quad\,\otimes P(x,y,\Updelta z_{n})\;.\end{aligned}$$
(18.54)
The term inside the brackets [18.99] describes a modification to the phase of the electron wave by the slice's projected potential, which is an integration of the potential over the slice thickness
$$\overline{V}_{n}(x,y)=\int_{\Updelta z_{n}}V(x,y,z)\mathrm{d}z\;.$$
(18.55)
The assumption in (18.54) is that the change in the electron wave amplitude by the slice potential is small enough and can be neglected when the slice is selected as thin enough. This approximation is known as phase grating approximation ( ). It should be noted that the kinematical approximation as in (18.1) uses the first-order term of the PGA. The convolution in (18.54) describes the wave propagation over the distance of the slice thickness \(\Updelta z_{n}\). The propagation between two waves over a short distance is described by the Fresnel propagator
$$P(x,y,\Updelta z_{n})=\frac{1}{\Updelta z_{n}}\frac{1}{\lambda\mathrm{i}}\exp\left(\uppi\mathrm{i}\frac{x^{2}+y^{2}}{\lambda\Updelta z_{n}}\right).$$
(18.56)
For electron nanodiffraction, the incident electron wave is set to the electron probe function as described in Sect. 18.2.1, e. g.,
$$\phi_{1}(x,y)=\phi_{\mathrm{P}}(x,y)\;.$$
The electron exit wave, \(\phi_{\text{exit}}(x,y)\), can be obtained by applying (18.54) sequentially from the first slice to the last. A model for the potential can be constructed based on the approximation of the superposition of individual atomic potentials. The electron diffraction pattern recorded at the far field away from the sample is the intensity of the Fourier transform of the exit wave function
$$\begin{aligned}\displaystyle&\displaystyle I(S_{x},S_{y})\\ \displaystyle&\displaystyle=\left|\int\int\phi_{\text{exit}}(x,y)\times\exp[-2\uppi\mathrm{i}(S_{x}x+S_{y}y)]\mathrm{d}S_{x}\mathrm{d}S_{y}\right|^{2}.\end{aligned}$$
(18.57)
The convolution used for the wave propagation in (18.54) can be evaluated numerically using the fast Fourier transform. In this method, the electron wave function is first multiplied by the PGA. The product is then Fourier transformed and multiplied with the Fourier transform of the Fresnel propagator. The result is then inversely transformed back to obtain the next electron wave function.
Fig. 18.38

Atomic potential sampling in the multislice method. The potential is divided into slices of thickness \(\Updelta z\) and averaged along \(z\) for each slice. Typical slice thickness is about \({\mathrm{2}}\,{\mathrm{\AA{}}}\). The choice of slice thickness affects the numerical convergence of the calculation and accuracy of high-order Laue zone reflections. Along the \(x\) and \(y\) directions, the potential is sampled in discrete points with a fixed interval or pixels

The main limitation of the multislice method is the number of atoms that can be included realistically in a simulation. The limitation comes from the atomic potential sampling considerations, as illustrated in Fig. 18.38. The 3-D sample potential in a multislice calculation is represented in a 2-D numerical array for each slice along the beam direction. The representation of the atomic potentials requires a minimum number of sampling points. For example, a minimum of five points are required to represent the center, the size, and the gap of the atomic potential. For a \({\mathrm{1}}\,{\mathrm{\AA{}}}\) sized atom, the spacing between these points is \({\mathrm{0.2}}\,{\mathrm{\AA{}}}\), defining a minimum pixel size in the real space. In this case, \({\mathrm{1}}\,{\mathrm{k}}\times{\mathrm{1}}\,{\mathrm{k}}\) represents a sample area of \(20\times{\mathrm{20}}\,{\mathrm{nm^{2}}}\).

Crystal Symmetry and Symmetry Mapping

A major application of CBED is to determine the crystal symmetry. Since the electron probe size is very small, the symmetry determination can be carried out locally at high spatial resolution, for example inside nanodomains in ferroelectrics. The crystal symmetry is reflected in the diffraction patterns. For example, if the crystal has a rotation axis, two diffraction patterns related by rotation should be the same. The same is true for mirror symmetry. Additional symmetries are produced in electron diffraction because of 1) the principle of reciprocity and 2) the projection along the zone axis for ZOLZ [18.100]. The principle of reciprocity states that the intensity of the diffracted beam (B) with a source (A) is the same as the intensity detected at A with the source at B by the same scatter. The projection of crystal structure along the zone axis orientation used in observation produces a mirror symmetry at the middle of the sample, which may or may not exist in the crystal. The combination of reciprocity and projection with the crystal point groups produces 31 diffraction groups , whose relationships with the 32 point groups were tabulated by Buxton et al [18.109]. The correspondence is often not unique. The determination of crystal point groups comes down to elimination of multiple choices using the symmetry of diffraction patterns recorded along several major symmetric orientations and/or using information about the lattice determined from the diffraction pattern geometry. The diffraction pattern symmetries used in the determination are those of the whole pattern, the transmitted beam (bright), the diffracted beams (dark-field), and the symmetry between \(+g\) and \(-g\) beams. It should be emphasized that the Friedel symmetry (\(I_{g}=I_{-g}\)) is absent in electron diffraction because of dynamic scattering. The point groups can be uniquely determined by electron diffraction.

The screw and glide axes present in the crystals can be determined by observing dynamic extinction in kinematically forbidden reflections (zero structure factor due to the glide or screw axes). These reflections generally show some intensities due to electron multiple scattering. The dynamic extinction is observed when the incident beam is in the glide plane in the case of a glide; this was first reported by Gjonnes and Moodie using CBED (the extinction appears as dark lines, subsequently named G–M lines). The dynamic extinction of a screw axis is more complicated and is described in detail in [18.18].

The combination of point group determination and identification of translation symmetry allows the unique identification of space groups [18.110, 18.111]. Both CBED and LACBED techniques can be used for this purpose. Applications of symmetry determination by CBED include phase identification and as part of the determination of unknown structures. Methods for quantifying and autodetection of the CBED symmetry can be found in [18.112, 18.113].

Real crystals often have local symmetry dependent on sample position. An obvious case is the breakdown of symmetry to surface and interfacial stress and strain or the presence of defects. Another case is ferroelectric crystals, electric polarization removes the inversion symmetry, and its direction coincides with the principal symmetry axis in tetragonal and rhombohedral crystals. Thus, measurement of local symmetry can be used to determine the polarization direction. There are many other examples, where local symmetry can help with phase identification and microstructure determination.

In using CBED for local symmetry determination, it is helpful to quantify the amount of symmetry recorded in CBED patterns [18.112, 18.114, 18.115]. The basic idea is to measure, from the diffraction intensities, the similarity between points inside the CBED disks that are related by symmetry. The similarity can be measured by the standard normalized cross-correlation coefficient (\(\gamma\) as defined in (18.41)) [18.71]. In order to measure the symmetry, regions in CBED patterns must be selected and are aligned. Figure 18.39a-c shows an experimental CBED pattern from the Si[110] zone axis. We use this pattern to demonstrate the image processing procedures employed for the dark-field symmetry quantification. The discussion below is specific to mirror symmetry, but the principle also applies to rotational symmetry. First, two diffraction discs are selected on two sides of the mirror plane (marked by the yellow line) as shown in Fig. 18.39a-ca. For the discussion, the selected CBED discs are named template \(\mathrm{A}\) and template \(\mathrm{A}^{\prime}\) (Fig. 18.39a-cb,c), respectively. Each template is then rotated by an angle \(\theta\), so that the mirror is aligned. The template A is used as the reference motif so that the symmetry element is calculated by comparison with template \(\mathrm{A}^{\prime}\). For the mirror operation, the template \(\mathrm{A}^{\prime}\) is flipped to obtain a mirror image. The mirror-applied image will be referred to as \(\mathrm{A}^{\prime}_{m}\). For the rotational operation, the template \(\mathrm{A}^{\prime}\) is rotated by \(180^{\circ}\), \(120^{\circ}\), \(90^{\circ}\), and \(60^{\circ}\) for the two, three, four, and sixfold rotation, respectively. The rotated template \(\mathrm{A}^{\prime}\) will be referred to as \(\mathrm{A}^{\prime}_{n}\) (\(n=2,3,4,6\)). The circular mask shown in Fig. 18.39a-cb,c is used to remove areas affected by the CBED disk edge. Thus, the final templates are obtained by multiplying the mask image to the templates A and \(\mathrm{A}^{\prime}_{m}\).

Fig. 18.39a-c

Image processing procedures used for symmetry quantification. The example here is for mirror symmetry. Two diffraction discs related by the mirror are selected as indicated by the dotted circles A and \(\mathrm{A}^{\prime}\) in (a). The two discs are then processed to give two templates (A (b) and B (c)) as shown above. Provided by Kyouhyun Kim, UIUC

The normalized cross-correlation coefficient (\(\gamma\)) is used to quantify the similarity between A and \(\mathrm{B}=\mathrm{A}^{\prime}_{m\text{ or }n}\). For a pattern with perfect symmetry, \(\gamma=1\). For the experimental pattern in Fig. 18.39a-c, which was recorded using a JEOL 2100 \(\mathrm{LaB_{6}}\) TEM at \({\mathrm{200}}\,{\mathrm{kV}}\), the \(\gamma\) values range from 0.981 to 0.991 for the mirror. A test of the robustness of the symmetry quantification procedure over 20 experimental Si[110] CBED patterns gave the \(\gamma\) values ranging from \(\mathrm{0.981}\) to \(\mathrm{0.991}\) for all quantification results.

Fig. 18.40a-e

Determination of nanodomains in \((1-x)\mathrm{Pb(Zn_{1/3}Nb_{2/3})O_{3}}\)-\(x\mathrm{PbTiO_{3}}\) (\(x={\mathrm{0.08}}\)) using SCBED. (a) The mirror symmetry as measured by \(\gamma\) varies across two types of domains. The red dashed line indicates the domain boundary. The red arrows indicate the projected polarization directions for each type of domain. The polarization directions were determined with help of simulations. (bd) show experimental and simulated CBED patterns. The mirror plane in the (b) type-1 and (d) type-2 domains is rotated by \(45^{\circ}\). (c,e) Simulated patterns using the Bloch wave method assuming \(Pm\) phase symmetry. Indexing is based on simulated diffraction patterns. Provided by Y.T. Shao, UIUC

The symmetry quantification method we have described can be combined with the scanning electron diffraction technique described in Sect. 18.1.5 for symmetry mapping. In scanning CBED, a series of CBED patterns are recorded and stored in a 4-D dataset. The 4-D dataset consists of \(m\) by \(n\) patterns; the \(m\) and \(n\) correspond to the number of sampling points along the two edges of the rectangular grid. Figure 18.40a-e shows an application of the symmetry mapping technique to the determination of the ferroelectric domain boundary in \((1-x)\mathrm{Pb(Zn_{1/3}Nb_{2/3})O_{3}}\)-\(x\mathrm{PbTiO_{3}}\) (\(x=0.08\)) (PZN-PT). The principle of domain identification is based on the above described CBED measurement of crystal symmetry. The ferroelectric polarization direction lies in the mirror plane, which can be determined by measuring the mirror symmetry in the recorded CBED patterns. Thus, ferroelectric domains can be identified by the change of CBED pattern symmetry (Fig. 18.40a-e). The change in CBED pattern symmetry is quantified using the normalized cross-correlation (\(\gamma\)) value of a pair of diffraction discs related by mirror symmetry. The SCBED experiments of Fig. 18.40a-e were carried out using a JEOL 2010F FEG TEM operated at \({\mathrm{200}}\,{\mathrm{kV}}\) with a convergent beam of \({\mathrm{2.6}}\,{\mathrm{nm}}\) in FWHM. Energy-filtering ( ), which improves the contrast of CBED patterns, was performed using a Gatan imaging filter (GIF). EF-SCBED was performed by scanning the focused electron probe over a selected area on a \(15\times 15\) grid, a step size of \({\mathrm{2}}\,{\mathrm{nm}}\), and through a postcolumn GIF energy window of \({\mathrm{10}}\,{\mathrm{eV}}\). The shift and tilt of diffraction patterns during beam scanning were minimized and calibrated using a silicon single crystal. The symmetry of PZN-8%PT was determined as monoclinic \(Pm\), which agrees with the x-ray diffraction result [18.116]. Nanodomains are observed using EF-SCBED as demonstrated by the symmetry variations across these domains in an EF-SCBED dataset from a \(30\times{\mathrm{30}}\,{\mathrm{nm^{2}}}\) sample area are shown in Fig. 18.40a-ea. The scan consists of 15 by 15 points, with a step size of \({\mathrm{2}}\,{\mathrm{nm}}\). The \(\gamma\) values are shown in grayscale. Two domains are identified (type-1 and type-2) with different mirror symmetry. The type-1 and type-2 domains are associated with two distinguishable CBED patterns that were observed along the \([100]_{\mathrm{C}}\) incident direction (Fig. 18.40a-eb,d). The highest \(\gamma\) values of type-1 and type-2 patterns are detected along two different directions (A and B) as shown in Fig. 18.40a-eb,d. The A and B directions are rotated by \(45^{\circ}\) along the \([100]_{\mathrm{C}}\) zone axis. The corresponding simulated patterns for type-1 and type-2 domains are along the monoclinic \(Pm\) zone axes \([100]_{Pm}\) and \([010]_{Pm}\), as shown in Fig. 18.40a-ec,e, respectively. In the \(Pm\) structure model, the polarization direction is \(\boldsymbol{P}_{\mathrm{S}}=[u,0,v]_{Pm}=[3,0,4]_{Pm}\), which lies in the mirror plane of \(Pm\) symmetry [18.116]. Along the \([100]_{Pm}\) incident direction, the mirror plane is superimposed on \((001)/(00\overline{1})\) reflections, which is parallel to the A direction in Fig. 18.40a-eb. This mirror is not observed along the \([010]_{Pm}\) incident direction. The projection of the polarization lies approximately on the \((101)/(\overline{1}0\overline{1})\) reflections, which is parallel to the B direction in Fig. 18.40a-ed. The highest mirror symmetry in this case is detected along direction B in the simulated pattern (Fig. 18.40a-ee) with \(\gamma={\mathrm{60}}\%\).

Fig. 18.41a,b

Symmetry mapping in single-crystal silicon. (a) Mirror symmetry map obtained from a \({\mathrm{25}}\,{\mathrm{nm}}\times{\mathrm{25}}\,{\mathrm{nm}}\) sample region. (b) Averaged CBED pattern from pixels with the highest symmetry. Provided by Y.T. Shao, UIUC

Another application of symmetry mapping is to determine the highest symmetry in a single crystal and obtain highly symmetrical experimental CBED patterns. The symmetry of recorded CBED patterns is very sensitive to the quality of sample surface and strain in prepared single crystal samples, including modifications to the surface by ion milling or focused ion beam irradiation. Because of this, the measured symmetry in experimental patterns often vary from area to area. For symmetry determination, it is critical to determine the highest symmetry. This can be achieved using SCBED and symmetry mapping. Figure 18.41a,b shows an example for Si[110] CBED patterns. The sample was prepared by precision ion milling. To quantify the mirror symmetry in the CBED patterns, the symmetry related pairs of diffraction discs from \((\mathrm{A},\mathrm{A}^{\prime})\) to \((\mathrm{G},\mathrm{G}^{\prime})\) are selected (orange circles) about the mirror plane (blue circles). The intensity weighted cross-correlation value is used to map the symmetry, the highest symmetry measured is at \(\gamma={\mathrm{98.1}}\%\) [18.117].

Fig. 18.42a-c

CBED pattern symmetry observed at three different temperatures in single-crystal \(\mathrm{BaTiO_{3}}\) for (a) tetragonal, (b) orthorhombic and (c) rhombohedral symmetry. Provided by Y.T. Shao, UIUC

Figure 18.42a-c shows an application of highest symmetry determination using SCBED for barium titanate (\(\mathrm{BaTiO_{3}}\)). Ferroelectricity of \(\mathrm{BaTiO_{3}}\) was discovered in the 1940s. Since then, it is regarded as a model of perovskite-type ferroelectrics. \(\mathrm{BaTiO_{3}}\) undergoes successive phase transformations from the high-temperature cubic (C) paraelectric phase to three low-temperature ferroelectric phases with tetragonal (T), orthorhombic (O), and rhombohedral (R). In Fig. 18.42a-c, the highest symmetry was identified from a total of 625 CBED patterns over an area of \({\mathrm{25}}\,{\mathrm{nm}}\times{\mathrm{25}}\,{\mathrm{nm}}\) and the most symmetrical CBED patterns taken along the pseudocubic incidence \([100]_{\text{pc}}\) at different temperatures are shown. In the T (\(P4mm\)), O (\(Amm2\)), and R (\(R3m\)) phases, \(\boldsymbol{P}_{\mathrm{S}}\) lies in one of the \(\langle 100\rangle_{\mathrm{T}}\), \(\langle 110\rangle_{0}\), and \(\langle 111\rangle_{\mathrm{R}}\) directions, respectively. Thus, for CBED patterns taken along the \([100]_{\text{pc}}\) incidence, we should expect mirror planes normal to \([010]_{\mathrm{T}}\), \([001]_{\mathrm{T}}\), \([011]_{0}\), \([0\overline{1}1]_{0}\), \([011]_{\mathrm{R}}\), and \([0\overline{1}1]_{\mathrm{R}}\) directions. Figure 18.42a-ca shows the CBED pattern of T phase with mirror plane \(m\parallel[001]_{\mathrm{T}}\), and \(\upgamma={\mathrm{98.5}}\%\) is consistent with the space group \(P4mm\). Figure 18.42a-cb,c shows the CBED patterns taken approximately at \({\mathrm{263}}\,{\mathrm{K}}\) (O phase) and \({\mathrm{95}}\,{\mathrm{K}}\) (R phase) with mirror plane \(m\parallel[011]_{0}\) and \(\upgamma={\mathrm{95.8}}\%\), \(m\parallel[011]_{\mathrm{R}}\) and \(\upgamma={\mathrm{94.6}}\%\), respectively.

18.4.3 Measurement of Crystal Structure Factors

Electron structure factors are obtained from experimental CBED patterns by using the refinement method [18.118, 18.119, 18.120, 18.121]). The refinement method works like the Rietveld method in powder x-ray or neutron diffraction, where the structure factors are treated as structural parameters, which together with other parameters, are obtained by comparing experimental and theoretical intensities and optimizing for the best fit. Multiple scattering effects are taken into consideration by using dynamical theory to calculate diffraction intensities during the refinement. In this way, the failures of the kinematical approximation in electron diffraction are avoided. Further, electron interference due to coherent multiple scattering actually enhances the sensitivity of the diffracted intensities to the crystal potential and crystal thickness, and thus improves the electron diffraction measurement accuracy.

The refinement is automated by defining a goodness of fit ( ) parameter and using a numerical optimization routine to do the search in a computer. One of the most useful GOFs for a direct comparison between experimental and theoretical intensities is the value of \(\chi^{2}\) as defined by
$$\begin{aligned}\displaystyle\chi^{2}&\displaystyle=\frac{1}{n-p-1}\\ \displaystyle&\displaystyle\quad\,\times\sum_{i,j}\frac{1}{\sigma_{i,j}^{2}}\left[{I_{i,j}}^{\exp}-c{I_{i,j}}^{\text{Model}}(a_{1},a_{2},\dots,a_{p})\right]^{2}.\end{aligned}$$
(18.58)
Here, \(I_{i,j}^{\exp}\) is the experimental intensity (in units of counts) measured from an energy-filtered CBED pattern, \(i\) and \(j\) are the pixel coordinates on the detector, \(n\) is the total number of data points, and \(a\) and \(p\) are the adjusted parameter and the number of parameters; \(I_{i,j}^{\text{Model}}\) is the model intensity calculated with parameters \(a_{1}\) to \(a_{p}\), and \(c\) is the normalization coefficient. The other commonly used GOF is the \(R\)-factor
$$R=\frac{\sum_{i,j}\left|I_{i,j}^{\exp}-cI_{i,j}^{\text{Model}}(a_{1},a_{2},\dots,a_{p})\right|}{\sum_{i,j}\left|I_{i,j}^{\exp}\right|}\;.$$
(18.59)
The optimum \(\chi^{2}\) has a value close to unity, which is obtained when the differences between theory and experiment are normally distributed, and the variance \(\sigma\) is correctly estimated. The value of \(\chi^{2}\) smaller than 1 indicates an overestimation of the variance in the experimental data. The \(R\)-factor simply measures the residual difference in percentage and does not need the estimate of \(\sigma\). Both criteria have been used in electron diffraction refinement techniques.
Experimental electron diffraction data are collected by recording diffraction patterns for each reflection at or near its Bragg condition. The diffraction pattern must be energy filtered to remove the inelastic background (a thickness difference technique was developed by Nakashima and Muddle that avoids this requirement [18.121]). Experimental issues involved in the diffraction pattern recording include the geometric distortions of the diffraction patterns, the detector resolution and noise, the optimum sample thickness, and diffraction geometry. These are discussed in details in the literature [18.122, 18.123, 18.124, 18.125]. The noise in the experimental data can be estimated using the measured detector quantum efficiency ( )
$$\text{var}(I)=\frac{mgI}{DQE(I)}\;.$$
(18.60)
Here, \(I\) is the estimated experimental intensity, var denotes the variance, \(m\) is the area under the modulated transfer function ( ), and \(g\) is the gain of the detector [18.6]. Details of energy filters and electron detectors are given in [18.18].
Fig. 18.43

(a) A systematic row CBED pattern of rutile (\(\mathrm{TiO_{2}}\)), recorded at \({\mathrm{120}}\,{\mathrm{kV}}\) with inelastic background removed by energy filtering, after removing the detector PSF response by deconvolution. (b) The best fit, obtained from dynamical electron diffraction simulations and refinement is also shown. From [18.126]. Reproduced with permission of the International Union of Crystallography. https://journals.iucr.org/

Three types of diffraction conditions have been used for electron structure factor measurements. One is the systematic row diffraction condition (Fig. 18.43), in which a set of parallel lattice planes is close to the Bragg diffraction condition. The CBED pattern appears one-dimensional with relatively uniform intensity normal to the systematic direction, because diffraction is dominated by reflections belonging to the same set of lattice planes (a systematic row). The largest effect of a small change in the structure factor of a reflection on its diffraction intensity is near the Bragg condition, which can be accomplished for two reflections in a systematic row CBED pattern. Other choices of diffraction conditions include the symmetric zone-axis orientation and slightly off-zone axis orientations [18.125, 18.127]. The advantage of these orientations is that a larger number of reflections can be refined and measured simultaneously. This is done at the cost of increased complexity in comparing two-dimensional patterns and computing time, because a large number of reflections contribute to diffraction in the zone axis orientation.

Fig. 18.44

\(\upchi^{2}\) map plotted as function of structure factor amplitudes for the (220) and (110) reflections in Fig. 18.43

To model diffraction intensities, both the detector response and the background intensity from thermal diffuse scattering must be included. A general expression for the model intensity including both factors is
$$I_{i,j}^{\text{Model}}=I^{\text{Theory}}(i,j)\otimes H^{\prime}(i,j)+B(i,j)\;.$$
(18.61)
Here, the theoretical intensity \(I^{\text{Theory}}\)is convoluted with the detector response function \(H^{\prime}\) plus the background \(B\). The theoretical intensity is integrated over the area of a pixel. For a pixelated detector with a fixed size, the electron microscope camera length determines the resolution of the recorded diffraction patterns. At sufficiently large camera length, we can approximate \(H^{\prime}\) by a delta function for a deconvoluted diffraction pattern. The background intensity \(B\), in general, is slowly varying, which can be subtracted or approximated by a constant for each reflection.

To calculate the theoretical intensities, an approximate model of the potential is needed. In the case of electron density measurements, the crystal structure (the atomic species and their coordinates, the cell constants) is first determined very accurately. This is usually done by x-ray or neutron diffraction. The unknowns for electron refinement are the low-order structure factors (which are the most sensitive to bonding effects, as we have seen), the absorption coefficients, and the experimental parameters related to diffraction geometry and specimen thickness. The structure factors calculated from a spherical atom or ionic model can be used as a starting point. Absorption coefficients are estimated using the Einstein model with known Debye–Waller factors either from direct measurement of x-ray or neutron diffraction [18.128] or theory.

The actual refinement is divided into two steps. In the first step, the theoretical diffraction pattern is calculated based on the starting parameters. The pattern can be the whole, or part of, the experimental diffraction pattern. In some cases, such as a systematic row, a few line scans across the experimental diffraction pattern contain enough data points for the refinement purpose. In the second step, the calculated pattern is placed on top of the experimental pattern. The two patterns are matched by shifting, scaling, and rotating the theoretical pattern. Both steps are automated by optimization. The first step optimizes structural parameters, and the second step is for the experimental parameters. For the experimental parameters, we have:
  1. 1.

    The zone axis center (in practice, the tangential wave vector \(\boldsymbol{K}_{\mathrm{t}}\) for a specific pixel)

     
  2. 2.

    The length and angle of the \(x\)-axis in the zone axis coordinate used for simulations

     
  3. 3.

    The specimen thickness, an intensity normalization coefficient, and the background intensity model.

     
The structural parameters include:
  1. 1.

    Structure factor amplitude and phase (in the case of an acentric crystal) of the selected reflection (hkl at or near the Bragg condition)

     
  2. 2.

    Fourier amplitude (and phase in some cases) of the absorption potential of the selected reflection for the same hkl as above.

     
Given a set of calculated theoretical intensities, their corresponding values in the experimental pattern can be found by adjusting some experimental parameters (e. g., orientation) without the need for dynamical calculations.

Figure 18.44 shows the \(\chi^{2}\) map as a function of the structure factors of the (110) and (220) reflections, for a rutile (110) systematic refinement. It clearly shows that near the global minima there is no other local minimum. This property ensures that the refinement program can find the true global minima. It is interesting to note that, for the (110) reflection, the minimum point is almost independent of the (220) reflection.

Estimates of errors in refined parameters can also be obtained by repeating the measurement. For CBED, this can be done by using different regions of the pattern or patterns recorded at different diffraction conditions and sample thicknesses. A test of electron diffraction accuracy has been reported by [18.120, 18.127, 18.129]. They measured the low-order structure factors of silicon, which are known from x-ray Pendellösung measurements. The experimental data of [18.129] has since then been re-refined using 279 beams, which were selected using more stringent beam selection criteria. This lowered \(\chi^{2}\) to \(\mathrm{1.58}\). The electron structure factors obtained are \(U(\overline{1}11)={\mathrm{0.04738}}(5)\ \AA{}^{-2}\) and \(U(\overline{2}22)={\mathrm{0.00095}}(5)\ \AA{}^{-2}\). The x-ray Pendellösung measurements, converted to electron structure factors, give \(U(\overline{1}11)={\mathrm{0.04736}}(4)\ \AA{}^{-2}\) and \(U(\overline{2}22)={\mathrm{0.000943}}(5)\ \AA{}^{-2}\). In this case, the different electron diffraction measurements and the best x-ray data agree within the experimental measurement error.

A major application of the accurate measurement of crystal structure factors is the determination of the crystal charge density. Since the crystal charge density is a ground state property of the electronic structure of the crystal, quantitative CBED has the potential to probe local electronic structures. This is based on the conversion of the accurately measured electron structure factors to x-ray crystal structure factors (Fourier transform of charge density) and from them to map electron distributions in crystals. In particular, the significant improvement in the accuracy of experimental structure factors, achieved by the development of quantitative CBED, has resulted in a number of accurate studies of electron density of several inorganic crystals of different bonding types (for a review, see [18.130]). Critical to the success was to correct extinction in x-ray diffraction, from which the intensity of a few strong low-order reflections deviates significantly from the ideal imperfect crystal (mosaic blocks) model and limits the accuracy of experimental structure factors extracted from diffraction intensities using the kinematical approximation.

18.4.4 Inversion of Diffraction Patterns and Nanostructure Determination

Inversion of electron diffraction patterns provides a direct solution to the analysis of electron diffraction data. In the limit of coherent diffraction, the diffraction pattern records the intensity of the Fourier transform of the electron exit wave function. Inverse Fourier transform of the diffraction pattern thus requires the phase missing in the diffraction pattern. This is known as the phase problem in diffraction. Critical to the inversion of diffraction patterns is to find the phases of the diffracted waves.

In crystallography, the phase problem is solved based on a priori information about the crystal structure. The a priori information includes the sharply peaked atomic charge density and the periodicity of the crystal. By solving the inversion problem of crystal diffraction, crystallographers routinely image atoms in 3-D molecules as long as they can be crystallized. There is a long history of attempts at inverting the recorded electron diffraction patterns. One was to use interference between diffraction disks in electron ptychography [18.131]. The concept of ptychography was first proposed by Hoppe [18.132] and then further developed by Rodenburg [18.131]. In the original ptychography, electron diffraction patterns are recorded over an area of a crystal using a coherent probe with a diameter less than the size of the crystal unit cell, and the diffraction intensity at the middle of the overlapping disks is processed as a function of the probe position to form atomic resolution images [18.131]. For electron nanodiffraction, we must consider objects that are not perfect, infinite, crystals.

The ability to invert diffraction patterns to form images has attracted considerable interest recently in the x-ray diffraction community, where the lack of a high-resolution imaging lens has been a major obstacle toward x-ray imaging (Chap.  20). In electron diffraction, the additional phase introduced by the lens aberrations does not affect the diffraction intensity, and diffractive imaging by solving the phase problem provides atomic resolution imaging at diffraction-limited resolution. The inversion of electron diffraction patterns of nanometer-sized objects is helped by the fact that the small object leads to broadened diffraction peaks and, in the case of coherent electron diffraction, the broadening gives additional diffraction information and under not so restricted conditions can lead to inversion of diffraction patterns [18.133].

Here, we introduce coherent electron diffraction techniques for nanometer sized objects. We start with a description of the noncrystallographic phase problem, which is followed by a discussion on the different iterative transformation algorithms for solving the noncrystallographic phase problem and their requirements. We then use quantum dots as an example to demonstrate the phasing of experimental electron diffraction patterns and introduce techniques for achieving this.

The Noncrystallographic Phase Problem

The scattered waves at the back focal plane of the objective lens satisfy the so-called Fraunhofer diffraction condition [18.18] and can be described as the Fourier transform of the exit wave function \(\psi_{\text{exit}}(\boldsymbol{r})\)
$$\begin{aligned}\displaystyle\Psi(\boldsymbol{k})&\displaystyle=\int\psi_{\text{exit}}(\boldsymbol{r})\exp(-\mathrm{i}2\uppi\boldsymbol{k}\cdot\boldsymbol{r})\mathrm{d}^{3}\boldsymbol{r}\\ \displaystyle&\displaystyle=|\Psi(\boldsymbol{k})|\exp[-\mathrm{i}\varphi(\boldsymbol{k})]\;,\end{aligned}$$
(18.62)
where \(\boldsymbol{k}\) is the wave vector of the scattered wave and \(\varphi(\boldsymbol{k})\) is the phase of the complex amplitude of \(\Psi(\boldsymbol{k})\). For small nanostructures, such as carbon nanotubes, electron diffraction is well described by the kinematical approximation. At a given scattering angle, the scattered electron wave is a sum of the scattered waves over the volume of the structure
$$\begin{aligned}\displaystyle\Psi(\boldsymbol{k})&\displaystyle\approx\int[1+\mathrm{i}\uppi\lambda U(\boldsymbol{r})]\mathrm{e}^{-2\uppi\mathrm{i}\boldsymbol{k}\cdot\boldsymbol{r}}\varphi_{0}(\boldsymbol{r})\mathrm{d}^{3}\boldsymbol{r}\\ \displaystyle&\displaystyle=\varphi_{0}(\boldsymbol{k})+\mathrm{i}\uppi\lambda\int U(\boldsymbol{r})\mathrm{e}^{-2\uppi\mathrm{i}\boldsymbol{k}\cdot\boldsymbol{r}}\varphi_{0}(\boldsymbol{r})\mathrm{d}^{3}\boldsymbol{r}\;.\end{aligned}$$
(18.63)
The illuminating electron wave function \(\varphi_{0}(\boldsymbol{r})\) is formed by the electron lens, as described in Sect. 18.2.1.

The exit wave function \(\psi_{\text{exit}}(\boldsymbol{r})\) can be reconstructed by inverse Fourier transform of \(\Psi(\boldsymbol{k})\). Experimentally, however, one can only measure the length of the complex vector (\(|\Psi(\boldsymbol{k})|\)), while the phase angle \(\varphi(\boldsymbol{k})\) cannot be measured directly from the diffraction pattern. This is known as the phase problem. The missing phase that is commonly referred to in crystallography is the phase of structure factors. The phase \(\varphi(\boldsymbol{k})\) here is more general, including the effects of electron multiple scattering. The phase problem thus prevents one from direct inversion of diffraction using Fourier transformation.

The exit wave phase, in principle, can be measured by holography. A reference wave is used in holography to interfere with the object exit wave, which gives a set of interference fringes. The maxima of the fringes are locations where the phase of the scattered wave matches that of the reference wave, therefore the phase of the scattered wave can be measured from the intensities recorded in the interference pattern. When the hologram is illuminated with the same reference wave, one can reconstruct the object exit wave by a backward propagation. The re-illumination stage is equivalent to the inverse Fourier transformation of the hologram. Lichte et al showed experimentally that in off-axis electron holography, the complete information about amplitude and phase of the electron exit wave can be reconstructed numerically from a single hologram [18.134, 18.135]. The reference wave is created by splitting the illumination using an electron briprism in imaging. This is performed in the imaging mode, and because of this, the measured wave function also contains the phases due to the lens aberrations, which is the a major limiting factor to the ultimate information that can be obtained by electron holography [18.56].

The same holographic experiment in diffraction requires diffractive waves from an aperture. This has been demonstrated in the case of soft x-ray diffraction [18.136].

There are a number of established crystallographic methods to solve the crystallographic phase problem. For inorganic crystals and organic molecules with a small number of atoms, direct methods  [18.137, 18.138] are widely used. Direct methods are a group of ab initio phase determination techniques based on mathematical procedures that compare structure factor amplitudes derived from a single crystal. For example, by using the statistically correct phase relation proposed by Karle et al [18.137, 18.138], \(\varphi(\boldsymbol{h})\approx\varphi(\boldsymbol{k})+\varphi(\boldsymbol{h}-\boldsymbol{k})\), one can obtain the phase of \(\boldsymbol{h}\) from the phases of \(\boldsymbol{k}\) and \(\boldsymbol{h}-\boldsymbol{k}\). For macromolecules such as proteins, the large number of atoms (of the order of \(E2{-}E5\)) makes deriving structures using direct methods computationally prohibitive. Alternative methods have been developed based on atomic replacement using chemically modified molecules, which are more efficient for macromolecular phasing.

Solving crystal structure by direct methods or other crystallographic methods requires the preparation of a crystalline specimen. Many biologically important macromolecules, such as viruses and cells, cannot be crystallized. In materials science, diffraction of noncrystalline materials gives broad peaks and continuous background. The difficulty of crystallizing nonperiodic structures prevents structure determination at the atomic resolution through conventional crystallography methods. Overcoming this difficulty requires the solution of the phase problem for nonperiodic structures or the so-called noncrystallographic phase problem.

Coherent Diffractive Imagining of Finite Objects

A number of experimental methods have been proposed to image nonperiodic structures from diffraction patterns. Gabor first proposed a two-stage imaging process [18.139]. In the first stage, a diffraction pattern of the specimen is recorded on a photographic plate using a divergent electron beam emerging from a point source. The diffraction pattern recorded this way is essentially a hologram (since scattered and unscattered beams overlap and interfere) and thus carries both the phase and amplitude of the electron wavefront. In the second stage, the plate is illuminated using visible light, and the electron wavefront emerged from the specimen is reconstructed using optical lenses. Since the spherical aberrations of the optical lenses are easier to correct than electron lenses, the image resolution could be improved in the second stage. However, atomic resolution was never achieved using Gabor's idea due to a number of technical difficulties. Firstly, it is very difficult to realize an ideal point source. Secondly, the electron beam that Gabor used had limited coherence, and, therefore, not enough inference was formed to carry the phases. More recently, Spence and his co-workers experimented with a field emission point source for point projection microscopy [18.140, 18.141].

Recent breakthroughs in coherent diffractive imaging come from the convergence of several ideas that has led to a working solution of the noncrystallographic phase problem.

Oversampling
The idea of oversampling is based on the information theory for sampling a continuous, but finite object, which was first formulated by Harry Nyquist in 1928 [18.142] and further developed by Claude E. Shannon [18.143] in 1949. The Nyquist–Shannon theorem states that [18.143]

if a function \(f(x)\) vanishes outside the points \(x=\pm a/2\), then its Fourier transform \(F(k)\) is completely specified by the values which it assumes at the points \(k=0,\pm 1/a,\pm 2/a,\dots\)

The minimum sampling frequency of \(1/a\) is called the Nyquist frequency .
Discrete sampling is used in digital representation of the object or diffraction patterns, where a continuous object of dimension \(a\) is approximated by discrete points denoted by \(x=0,\dots,N-1\), and the Fourier transform is carried out via summation for a set of discrete frequencies denoted by \(k\)
$$F(k) =\sum_{x=0}^{N-1}f(x)\exp\left(\frac{2\uppi\mathrm{i}k\,x}{N}\right),$$
(18.64)
$$f(x) =\frac{1}{N^{2}}\sum_{k=0}^{N-1}F(k)\exp\left(\frac{2\uppi\mathrm{i}k\,x}{N}\right),$$
(18.65)
where equations (18.64) and (18.65) denote the forward and inverse Fourier transform, respectively. The smallest frequency in Fourier transform is \(1/a\).
In a diffraction experiment, what is measured is \(|F(k)|^{2}\) instead of \(F(k)\). Inverse Fourier transforming \(|F(k)|^{2}\) gives the autocorrelation function of \(f(x)\)
$$f(x)\otimes f(x)=\mathfrak{F}^{-1}\{|F(k)|^{2}\}\;,$$
where \(\mathfrak{F}^{-1}\) denotes the inverse Fourier transform and \(\otimes\) denotes convolution. The autocorrelation function \(f(x)\otimes f(x)\) has exactly twice the dimension of the original function \(f(x)\), e. g., if \(f(x)\) has a dimension of \(a\), its autocorrelation function then has a dimension of \(2a\). To obtain the complete autocorrelation function, one must sample the diffraction pattern \(|F(k)|^{2}\) at an interval of \(1/2a\). According to the nature of discrete Fourier transformation, the corresponding object \(f(x)\) should have a physical dimension of \(2a\), meaning that the total field of view should be twice the object size. Having a field of view larger than the actual dimension of the object is called oversampling the object.
The following argument advanced by Miao et al [18.144] suggests that oversampling diffraction experiment gives the extra information which can be used to solve the phase problem. The inverse problem can be phrased as the following equation
$$|F(k)|=\left|\sum_{x=0}^{N-1}f(x)\exp\left(\frac{2\uppi\mathrm{i}k\,x}{N}\right)\right|,$$
(18.66)
which is a set of equations, and the inverse problem is to solve these equations for \(f(x)\) at each pixel. In the simplest case of a 1-D real-valued object, the total number of equations in (18.66) is \(N/2\) (the factor of 2 comes from the symmetry of FT), and the total number of unknown variables is \(N\) (all the pixels in the real space). Such a set of equations is mathematically under-determined. However, if one oversamples the object by padding zero pixels around it and enlarging the field of view by a factor of 2, the number of unknown is then reduced to \(N/2\) since the other half of pixels are known to be zero. Therefore, by oversampling, the inverse problem becomes mathematically over-determined. The above argument also links the degree of oversampling with the over-determination. Therefore we can define an oversampling ratio \(\sigma\)
$$\sigma=\frac{N_{\text{total}}}{N_{\text{object}}}1\;,$$
where \(N_{\text{total}}\) and \(N_{\text{object}}\) denote the number of pixels in the total field of view and in the object respectively. The larger the \(\sigma\), the more over-determined is the phase problem.
Sampling Experimental Diffraction Patterns and the Field of View
Experimentally, the smallest measured frequency in the Fourier space is determined by the detection geometry. The first detection parameter is the camera length \(L\), which is the equivalent distance between the object and the detector plane when diffraction patterns are recorded using TEM. The other parameters are the detector pixel size and the number of pixels. When the diffraction pattern of an object with a lattice spacing of \(d\) is recorded at a distance of \(L\) away from the object, from Bragg's law, we have
$$\frac{1}{d}=\frac{np}{L\lambda}\;.$$
This means that \(n\) pixels in the detector plane record a spatial frequency of \(1/d\), and each pixel records a spatial frequency of \(1/(nd)\), or
$$\text{spatial frequency of each pixel }=\frac{p}{L\lambda}\;.$$
The total field of view is the reciprocal of the spatial frequency of the pixel
$$\text{field of view }=\frac{L\lambda}{p}\;.$$
For a \({\mathrm{200}}\,{\mathrm{kV}}\) electron with a wavelength of \({\mathrm{0.0251}}\,{\mathrm{\AA{}}}\) and a camera length of \({\mathrm{80}}\,{\mathrm{cm}}\), the pixel size is \({\mathrm{50}}\,{\mathrm{\upmu{}m}}\), and the total field of view according to the above equation is \({\mathrm{40}}\,{\mathrm{nm}}\). The oversampling ratio can be calculated by dividing the total field of view by the size of the object for a perfectly coherent beam. A nanoparticle of \({\mathrm{10}}\,{\mathrm{nm}}\) in size, for example, has an oversampling ratio in one dimension of 4 under the above experimental conditions.

The minimum oversampling ratio required to solve the phase problem according to Miao et al [18.144] is \(> 2\) for a 1-D object, \(> 2^{1/2}\) in each dimension for a 2-D square object, and \(> 2^{1/3}\) in each dimension for a 3-D cubic object.

Requirements on Beam Coherence
Experimentally, the maximum oversampling is determined by the coherence length; the diffracting object must be fully contained in the coherence volume of the illuminating beam, which is defined by the coherence lengths in the lateral and temporal directions. The coherence length perpendicular to the electron beam, or the lateral coherence length, is related to the beam divergence angle from a finite source [18.145], according to the Van-Cittert–Zernike theorem
$$X_{\mathrm{c}}=\frac{\lambda}{\theta_{\mathrm{c}}}\;,$$
where \(X_{\mathrm{c}}\) denotes the lateral coherence length and \(\theta_{\mathrm{c}}\) the beam divergent angle.

Partial coherence leads to a reduction of the oversampling ratio. In the ideal case, this has no effect as long as the oversampling ratio meets the minimum requirement. In practice, the combination of a reduced oversampling ratio and the noise in recorded experimental diffraction patterns leads to a loss of information in the reconstructed object function. This effect has been demonstrated by Huang et al in a simulation study [18.146].

Phase Retrieval Algorithms

Given the phase problem is overdetermined by oversampling, the question becomes how to retrieve the phase from the recorded diffraction patterns and solve the inverse problem numerically. Several iterative transformation algorithms ( ) [18.147, 18.148, 18.149, 18.150, 18.151] have been proposed to retrieve the missing phases from diffraction data. Common to all of these algorithms is iteration between two domains, typically the real space and the reciprocal space, and the iterant is forced to satisfy what is known in each domain, called constraints in each domain. This iterative algorithm was first proposed by Gerchberg and Saxton. The iteration continues until some error metric reaches a certain level.

Fig. 18.45

Flow chart of a generic iterative phase retrieval algorithm

Figure 18.45 illustrates a generic ITA for phase retrieval. An estimate of the object function \(g(x)\) is Fourier transformed into the Fourier domain, which yields \(G(k)\)
$$G(k)=\mathfrak{F}\{g(x)\}\;.$$
Generally, \(G(k)\) are complex. The amplitudes of \(G(k)\) are then made to satisfy the amplitude constraint by replacing the amplitudes with the experimentally measured ones, \(|F(k)|\), which leads to \(G^{\prime}(k)\)
$$G^{\prime}(k)=G(k)\,\frac{|F(k)|}{|G(k)|}\;.$$
(18.67)
Then \(G^{\prime}(k)\) is inverse Fourier transformed back to the object domain giving \(g^{\prime}(x)\)
$$g^{\prime}(x)=\mathfrak{F}^{-1}\{G^{\prime}(k)\}\;.$$
(18.68)
A set of constraints in the object domain is applied to modify \(g^{\prime}(x)\) into \(g_{\text{new}}(x)\), which will then be fed into the next iteration cycle. The specific constraints, or modifications, change depending on the particular algorithm; below is a summary of the most popular algorithms (a further discussion on these and other algorithms can be found in Chap.  20):
  1. 1.

    Gerchberg–Saxton Algorithm ()  [18.150]

    In the GS algorithm, what is aimed to reconstruct is a complex wave field, and the algorithm assumes that the amplitudes are known in both the object and the Fourier domains. Therefore, the last step in the loop is
    $$g_{\text{new}}=g^{\prime}(x)\cdot\frac{|A(x)|}{|g^{\prime}(x)|}\;,$$
    (18.69)
    where \(|A(x)|\) is the amplitude measured in the object domain.
     
  2. 2.

    Error Reduction Algorithm (ER)  [18.148]

    Fienup [18.148] modified the GS algorithm to extend its application to situations where only the amplitudes in the reciprocal space are measured, such as in x-ray diffraction. Instead of using the amplitude constraint in the object domain as the GS algorithm requires, he suggested using a support constraint
    $$g_{\text{new}}(x)=\begin{cases}g^{\prime}(x)\;,&x\in S\quad\text{ and}\\ 0\;,&x\notin S\;,\end{cases}$$
    (18.70)
    where \(S\) denotes the support of the object, which is the region where the object has nonzero density. Equation (18.70) essentially applies the information gained by oversampling, which is the zero-valued region surrounding the support. For a real-valued object under kinematical diffraction, for example, two more constraints can be applied on top of the support constraint. They are the real constraint and the positivity constraint
    $$g_{\text{new}}(x) =\mathfrak{Re}\{g^{\prime}(x)\}\;,$$
    (18.71)
    $$g_{\text{new}}(x) =0\;,\quad\text{ if }\mathrm{g}^{\prime}(x)<0\;.$$
    (18.72)
    For kinematical x-ray diffraction without the absorption and refraction effects, the electron density is nonnegative everywhere. Therefore, the conditions of (18.71) and (18.72) generally hold. For kinematical electron diffraction, the atomic potential of ionic materials can have both signs simultaneously. Fieunp [18.149] showed that the ER algorithm can work with a complex-valued object by removing the positivity constraint. The object function becomes complex as a result of dynamical scattering, or in the case of kinematical diffraction, the projection of three-dimensional diffraction, or absorption or refraction.
    Fig. 18.46a,b

    The HIO algorithm. (a) The input and output of the grouped operation. (b) The input with a small change and the corresponding linear response in the output

     
  3. 3.

    Hybrid-Input-Output Algorithm (HIO)  [18.148]

    The ER algorithm suffers from slow convergence and a tendency to stagnate at local minima in the solution space [18.148]. To solve this problem, Fienup introduced a feedback mechanism into the ER as illustrated in Fig. 18.46a,b. The input and output of the Fourier domain modification are \(g(x)\) and \(g^{\prime}(x)\), respectively. This operation produces the function, \(g^{\prime}(x)\), which satisfies the amplitude constraint in the Fourier domain. The algorithm ultimately seeks a solution that satisfies the constraints in both the Fourier and the object (support) domains. To take account of this, Fienup treated the \(g(x)\) as a driving function, rather than a solution, to drive the \(g^{\prime}(x)\) toward a solution that satisfies the object domain constraints. Although, in general, the overall procedure is nonlinear, for a small change in \(g(x)\), say \(\Updelta g(x)\), it produces approximately a linear response (Fig. 18.46a,b), \(g^{\prime}(x)+\alpha\,\Updelta g(x)\). To satisfy the support constraint, the property of the small change should have
    $$\alpha\,\Updelta g(x)=\begin{cases}0\;,&x\in S\text{ and }\\ -g^{\prime}(x)\;,&x\notin S\;.\end{cases}$$
    (18.73)
    That is, the operation drives the output toward zero outside the support. Therefore, the desired input for the next iteration should take the following form
    $$g_{\text{new}}(x)=\begin{cases}0\;,&x\in S\text{ and }\\ g(x)-\beta g^{\prime}(x)\;,&x\notin S\;,\end{cases}$$
    (18.74)
    where \(\beta\) is \(\alpha^{-1}\). Since the new input mixes the previous input and output of the operation in a linear combination, this algorithm was named the hybrid-input-output (HIO) algorithm.
     
  4. 4.

    Charge Flipping Algorithm ()  [18.151]:

    Oszlanyi and Suto [18.151] proposed a very simple phase retrieval scheme for x-ray crystallography, based on the fact that electron density is positive everywhere
    $$g_{\text{new}}(x)=-g^{\prime}(x)\,\text{ if }\mathrm{g}^{\prime}(x)<\delta\;,$$
    (18.75)
    where \(\delta\) is a positive threshold for flipping. Note that in (18.75), no support constraint is needed. Therefore, the CF algorithm can also be applied without oversampling. Wu and Spence [18.152] extended the CF algorithm to phase a complex-valued object using a support constraint: for pixels outside the support, the algorithm flips the signs of their real parts.
     
Use of Image Information

In a TEM, an image of the diffraction object can be recorded directly up to microscope resolution. At the minimum, the electron image provides accurate support information. At the maximum, the electron images can be used to obtain both the amplitude and phase of the exit wave function. Imaging, both the amplitude and the phase are affected by the microscope contrast transfer function (CTF ). In general, the phase is more reliably recorded up to the frequency where the CTF changes sign.

In electron diffraction, the recorded diffraction patterns are also far from ideal and contain only limited information. For example, weak intensities between diffraction spots are often lost because of detector noise. The central peak is often missing or saturated in experimentally recorded diffraction patterns because of its strong intensities. Even if the central peak is recorded, its intensity is mixed with other small angle scatterings, such as inelastic scattering from apertures, which makes small angle scattering intensities less reliable for diffractive imaging. The effect of noise is twofold: it limits the amount information that can be recorded about the shape factors of the particles in the diffraction pattern, and it also limits the amount of information about the weak interference originating from local defects.

The information obtained in electron images complements those recorded in diffraction patterns. The shape information of the nanometer-sized objects, which is partially lost due to noise, can be compensated by having an accurately determined boundary, or support, from the direct image. For certain types of defects, the change in structure introduces contrast variation within the objects. The direct image provides this contrast albeit at a lower resolution. Therefore, the loss of weak diffuse scattering can be compensated in the real space by using a direct image in two aspects:
  1. 1.

    The image provides an accurate determination of the object boundary as support.

     
  2. 2.

    At a reduced resolution, the contrast of the reconstructed object should agree with that of the direct image.

     
Additionally, by knowing which parts of the diffraction pattern are noise and which are data using the image information, one can further reduce the amount of diffraction noise that does not play any meaningful role in the object function reconstruction.

The approach is to take limited information in an electron image and diffraction pattern and reconstruct the object function at resolution of, or near, the diffraction limit based on the additional information obtained by oversampling. The authors of this chapter have developed techniques for achieving this. The information used for resolution improvement mainly comes from the diffraction intensity. The electron image recorded at the resolution available from the instrument is used 1) to provide an initial set of phases for low frequency diffraction intensities and 2) to estimate the object boundary for real space constraints or support. To use image and diffraction information effectively, the electron image and the diffraction pattern are recorded from the same area of the sample. As the recorded electron diffraction pattern and the image are then aligned and scaled to match each other using image processing techniques and used for image reconstruction with iterative phase retrieval techniques. Details about these procedures can be found in [18.153] and below.

Phasing CdS Quantum Dots

Cadmium selenide quantum dots were prepared by solution based chemical methods by Dr. K.W. Kwon of Professor Moonsub Shim's group at the University of Illinois. A solution containing \({\mathrm{300}}\,{\mathrm{mg}}\) of trioctylphosphine oxide, \({\mathrm{315}}\,{\mathrm{mg}}\) 1,2-hexadecanediol, and \({\mathrm{10}}\,{\mathrm{mL}}\) of octyl ether was vacuum degassed at \({\mathrm{100}}\,{\mathrm{{}^{\circ}\mathrm{C}}}\) for \({\mathrm{30}}\,{\mathrm{min}}\). Sulfur powder (\({\mathrm{15}}\,{\mathrm{mg}}\)) was added at \({\mathrm{100}}\,{\mathrm{{}^{\circ}\mathrm{C}}}\) under \(\mathrm{N_{2}}\) and stirred for \({\mathrm{5}}\,{\mathrm{min}}\). After cooling to \({\mathrm{80}}\,{\mathrm{{}^{\circ}\mathrm{C}}}\), cadmium acetylacetonate (\({\mathrm{150}}\,{\mathrm{mg}}\)) was added and stirred for \({\mathrm{10}}\,{\mathrm{min}}\). The reaction mixture was heated to \({\mathrm{280}}\,{\mathrm{{}^{\circ}\mathrm{C}}}\) and annealed for \({\mathrm{30}}\,{\mathrm{min}}\). The final CdS nanocrystals were precipitated with ethanol, centrifuged to remove excess capping molecules, and redissolved in chloroform. As-synthesized CdS nanocrystals were found by x-ray powder diffraction to be a mixture of wurzite and zinc blend structures.

The CdS quantum dots were supported on ultrathin graphene sheets or carbon nanotubes bundles. Lacey carbon films attached to a copper TEM grid were used to support the graphene and nanotubes. The TEM grid was first immersed in chloroform for \({\mathrm{10}}\,{\mathrm{s}}\) to dissolve the Formvar backing layer and then heated in argon ambient to remove the residue Formvar. Formvar was found to cause contamination problems under the nanoarea electron beam. Commercial double-walled carbon nanotubes were then dispersed onto the grid. Most of the nanotubes form bundles with each other. Graphene sheets were also dispersed onto the grid. Diluted solutions of CdS quantum dots were finally dispersed onto the grid. The quantum dot density was controlled to a very low level, about 1 quantum dot in every \(900{-}1600\,{\mathrm{nm^{2}}}\).

Fig. 18.47a,b

As recorded HREM image (a) of a CdS quantum dot supported on graphene and its Fourier spectrum (b). Provided by Weijie Huang, UIUC

Figure 18.47a,b shows a single crystalline CdS quantum dot of \({\mathrm{9}}\,{\mathrm{nm}}\) in diameter. The diffraction pattern from this quantum dot is shown in Fig. 18.48a,b. By indexing the diffraction pattern, the quantum dot was identified to have a wurzite structure, and the zone axis along which the pattern was recorded was near its \(c\)-axis, or [0001]. The structural model suggests that along this orientation the image should display a honeycomb-like structure, while it now appears to be a close-packed structure instead. This is because the resolution in the direct image is not enough to resolve the pair of atoms separated by \({\mathrm{2.5}}\,{\mathrm{\AA{}}}\).

Fig. 18.48a,b

Electron diffraction of a CdS quantum dot and diffraction background subtraction. (a) As-recorded electron diffraction pattern from a CdS quantum dot supported on graphene and (b) diffraction pattern after background subtraction. Provided by Weijie Huang, UIUC

To phase the diffraction pattern, we found that suppressing the noise in between the Bragg peaks is very useful in improving the quality of the reconstructed image. To do that, we separate DP into three different regions of \(G\) above or \(B\) below the background noise level and \(M\) where diffraction information is not available. We apply the diffraction intensity constraint in \(G\). In region \(B\), we place an upper limit of three times the background noise on the intensity. While in region \(M\), the intensity is allowed to float. In the reciprocal space, we replace the amplitudes of the Fourier transform according to
$$F(u,v)=\begin{cases}F(u,v)\,\frac{\sqrt{I^{\mathrm{D}}(u,v)}}{|F(u,v)|}\\ \qquad\text{if }(u,v)\in G\;,\\ F(u,v)\,\alpha\,\frac{\min\left(\sqrt{I^{D}(u,v)},\sqrt{3\sigma}\right)}{|F(u,v)|}\\ \qquad\text{else if }(u,v)\in B\;,\\ F(u,v)\\ \qquad\text{if }(u,v)\in M\;,\end{cases}$$
(18.76)
where \(G\), \(B\), and \(M\) mark different regions in the reciprocal space mask as described before. The \(I^{\mathrm{D}}\) is the experimental diffraction intensity, \(\alpha\) is a fractional number that was used to reduce the effect of background intensity on reconstructed image, and \(\sigma\) is the standard deviation of the background intensity. The maximum of \(3\sigma\) is imposed on the background intensity to remove artefacts from the background subtraction. The \(\alpha\) has a value between 0 and 1. It is used to reduce the amplitudes outside the diffraction support, where the background noise prevails. It is found that when \(\alpha\) is between \(\mathrm{0.1}\) and \(\mathrm{0.4}\), satisfactory reconstructions can be achieved. When \(\alpha\) is larger than \(\mathrm{0.4}\) this resulted in a noisy and blurred reconstruction, while \(\gamma<{\mathrm{0.1}}\) generated artificial unphysical structures in the reconstruction.
Fig. 18.49a,b

Reconstructed CdS quantum image. (a) Reconstructed object function and its power spectrum (b). The diffraction pattern and image used for reconstruction are shown in Figs. 18.47a,b and 18.48a,b. Provided by Weijie Huang, UIUC

The object function reconstructed from the diffraction pattern (shown in Fig. 18.48a,b from the quantum dot imaged in Fig. 18.47a,b) is shown in Fig. 18.49a,b. The reconstruction process is as follows:
  1. 1.

    An initial estimate of the object function is obtained from the start image. A small background noise about \(1/5\) of the maximum in the starting image is generated with a random seed number and added to the estimated object function.

     
  2. 2.

    An iterative phase retrieval is performed starting with the estimated object function and using the HIO algorithm. A real object constraint is applied during this step.

     
  3. 3.

    Step 2 is followed by iterations using the error reduction algorithm with the real object constraint.

     
  4. 4.

    Step 3 is followed by iterations using the HIO algorithm without the real object constraint to reconstruct the complex exit wave function.

     
  5. 5.

    Steps 1 to 4 are repeated using a different random seed number.

     
  6. 6.

    The object functions obtained from the above steps are averaged.

     
The experimental diffraction pattern always contains a certain amount of noise. The noise in the diffraction pattern is expected to transmit to the reconstructed object function. The background noise can be considered during the reconstruction process by defining a tolerance factor \(\varepsilon\) [18.154]. The function of \(g^{\prime}(x)\), whose Fourier transform satisfies the diffraction amplitude constraint (18.20), is considered to meet the real space support constraint if it is below the tolerance factor. In HIO, a new estimate of the object function is obtained by driving the current estimate of the object function toward the support constraint, if \(g^{\prime}(x)\) is above the tolerance factor (18.26). In all iterations, we use a factor of \(\varepsilon={\mathrm{1\times 10^{-4}}}\).

Compared to the as-recorded HREM image, the reconstructed image clearly shows a honeycomb structure, due to the \({\mathrm{0.72}}\,{\mathrm{\AA{}}}\) information transfer in the recorded diffraction pattern. The resolution improvement can also be seen in the comparison between the power spectra of the as-recorded image and the reconstructed image (Fig. 18.49a,bb).

Fig. 18.50a-d

Sub-Å resolution diffractive imaging of a CdS quantum dot \({\mathrm{7}}\,{\mathrm{nm}}\) in diameter along the cubic CdS crystal [18.112] orientation. The reconstructed image in (b) uses information from the starting image in (a). The insets show a magnification of the outlined region and part of the power spectrum. An intensity profile taken along the line shown in the inset demonstrates that the \({\mathrm{0.84}}\,{\mathrm{\AA{}}}\) separation between the Cd and S atomic columns is well resolved, and the lower peak can be attributed to S. Provided by Weijie Huang, UIUC

The resolution improvement of the reconstructed image to sub-Å using information from diffraction pattern over the as-recorded TEM image is further evidenced in Fig. 18.50a-d. The information in the reconstructed image extends to the cubic (\(62\overline{4}\)) reflection of \({\mathrm{0.78}}\,{\mathrm{\AA{}}}\) \(d\)-spacing. In comparison, only the \(\pm(111)\) cubic reflections of \({\mathrm{3.3}}\,{\mathrm{\AA{}}}\) \(d\)-spacing are present in the power spectrum of the starting image. Thus, the resolution improvement from the diffraction pattern is about a factor of 4. The reconstructed image from the diffraction pattern shows clearly resolved atomic columns in areas near the highlighted region. At the orientation where the diffraction pattern was recorded, the smallest separation between the Cd and S atomic columns is \({\mathrm{0.84}}\,{\mathrm{\AA{}}}\). This is clearly resolved in Fig. 18.50a-d. Asymmetric peaks are also seen in the reconstructed image from the intensity profile, taken across a pair of atoms, as shown in Fig. 18.50a-d. In the weak-phase-object approximation, the real part of the complex object is proportional to the object potential and, therefore, a larger peak is expected for Cd than S. Using this information, the crystal polarity can be directly determined from the reconstructed image. Interestingly, both the low-resolution starting image and the reconstructed image indicate an interface present within the particle. The lattices on the two sides of the boundary have a very small misorientation, as can be seen from the change in the contrast of the atomic columns. The lattice is also shifted across the boundary. The shift, measured directly from the reconstructed image, is \({\mathrm{1.6}}\,{\mathrm{\AA{}}}\). The small-angle boundary is also consistent with split Bragg peaks for some reflections, most notably \(\pm(111)\) and \(\pm(222)\), but not others such as (\(2\overline{2}0\)) in the diffraction pattern, as is shown in Fig. 18.51, because the lattice shift occurs only for the \(\pm(111)\) lattice. The reconstructed image thus provides a direct interpretation of the complex diffraction pattern here.

Fig. 18.51

Coherent electron diffraction pattern from a CdS nanocrystal \({\mathrm{7}}\,{\mathrm{nm}}\) in diameter in the [112] cubic orientation. The intensity distributions of selected diffraction spots are shown at the bottom. Provided by Weijie Huang, UIUC

A major difficulty of object reconstruction without image information is that the results of ITA are quite often trapped in local minima, or iterations starting with different set of random noises produce different reconstructions. Often, these constructions are similar to those measured by merit metrics designed to monitor the progress of ITA, such as examining the difference between the calculated intensities and the experimental ones. Thus, none are accurate enough to be selected as the right reconstruction [18.155]. To overcome this problem, the guided HIO method was developed based on the idea of a set of images close to the sought-after solution can be used efficiently to guide ITA towards to the solution. In the proposal by Chen et al, the images used to guide the search are selected from a set of images reconstructed using different random started phases based on the level of fit to experimental diffraction intensities. The guided HIO method has been used successfully for the reconstruction of crystalline particles and cells without starting image information [18.156, 18.157, 18.158]. The role of the starting image is thus similar in guiding the search of ITA closer to the final solution. However, unlike guided HIO, where the initial selection of starting images can be subjective, there is no ambiguity in the starting image obtained experimentally with its phase and amplitude information, albeit limited in resolution. Further, random noises can be added to the object function estimated from the starting image. The effect of adding noise perturbs the starting phases for the spatial frequencies that are recorded in the image. For high spatial frequencies beyond the image information limit, the random noises introduce new starting phases. Results obtained from different starting random noises can be averaged to provide a robust reconstruction.

Atomic Resolution Tomography

Three-dimensional reconstruction at atomic resolution is a major challenge in the characterization of the structure of materials. Recent advances in electron optics have enabled a direct determination of the atomic structure at sub-ångström resolution in 2-D projection [18.159, 18.160, 18.161, 18.162]. However, imaging atoms inside, and at the surface of, a nanoparticle or determining the structure of a 3-D defect requires information of the 3-D atomic structure. Tomographic reconstruction at atomic resolution provides a way forward to 3-D structure determination for any objects.

At atomic resolution, several groups have reported 3-D reconstruction using Z-contrast images obtained in a scanning transmission electron microscope (STEM) equipped with an annular dark field ( ) detector. Van Aert et al reported a successful reconstruction of the 3-D atomic structure of Ag precipitates in the Al matrix using discrete tomography [18.163]. This method only requires electron images recorded in a few zone-axis projections, but a prior knowledge of the structure is necessary for the reconstruction. The other approach is to detect atomic position in 3-D using the STEM depth sectioning method [18.164, 18.165, 18.166, 18.77] or using a combination of quantitative STEM and multislice simulation [18.167]. The resolution of these methods, however, is limited along the beam direction by the probe elongation effect resulting from the small electron beam convergence angle and by electron multiple scattering along atomic columns [18.164, 18.168]. Recently, Jianwei Miao's group at UCLA demonstrated the 3-D reconstruction of an Au nanoparticle at \({\mathrm{2.4}}\,{\mathrm{\AA{}}}\) resolution by using tomographic reconstruction based on the so-called equal-sloped fast Fourier transform [18.169]. The reconstruction is based on the \(Z\)-contrast image data recorded in a tilt series from \(\mathrm{-72.6}\) to \(72.6^{\circ}\) in equal-slope increment. This method has been further applied to image dislocations in an Au nanoparticle in 3-D [18.170].

In what follows, we describe a 3-D tomographic reconstruction algorithm based on the hybrid input-output (HIO) algorithm developed by Fienup [18.148] and polar Fourier fast transform ( ). Only coherent diffraction data obtained in a tilt series are needed for reconstruction. Because information recorded in a diffraction pattern is limited only by scattering, the method described here has the potential to achieve the highest resolution under the kinematical diffraction conditions.

Iteration in HIO requires forward and backward fast Fourier transformation (FFT). However, conventional FFT uses equispaced rectilinear sampling, which cannot be directly extended to tomography reconstruction of diffraction data recorded in a tilt series. To overcome this issue, polar FFT as implemented in the general category of non-equispaced FFT ( ) methods is used. This avoids resampling in the diffraction space and its related issues [18.172]. The basic concept of NFFT computation is to use conventional FFT in a Cartesian grid, while the Fourier frequencies are oversampled, and a window function is used to interpolate between equally and nonequally sampled space. The inverse FT is obtained through a least squares minimization [18.173]. In the 3-D HIO algorithm using polar FFT, the object function is sampled in a Cartesian grid, while the Fourier space is sampled in the cylindrical grid, where in the \(xy\)-plane the sampling is in the polar grid with points equally spaced on concentric circles (Fig. 18.52a,b).

Fig. 18.52a,b

Sampling in real and reciprocal space. (a) Sampling in 3-D Fourier space by a single-axis tilting series around the first tilt axis, and (b) the 3-D object is sampled in the Cartesian coordinate. Reprinted from [18.171], with permission from Elsevier

For 3-D reconstruction, the computational cost increases dramatically in order to achieve high resolution. Furthermore, because of oversampling, polar FFT requires a significant increase in the data size, which makes it challenging to implement the HIO algorithm not only in Fourier space but also in object space, while maintaining computational efficiency.

Reconstruction of the 3-D object using the HIO algorithm is performed in both Fourier and object spaces. A set of 2-D diffraction patterns are recorded in a tilt series. The diffraction patterns are combined in 3-D by taking the rotation axis as the \(z\)-axis in the cylindrical coordinate, as illustrated in Fig. 18.52a,ba. For each \(xy\)-plane of constant \(z\), the data is sampled in the polar coordinate of \(\boldsymbol{k}=(k,\theta)\) with \(k_{n}=n\Updelta k\) and \(\theta_{m}=m\Updelta\theta\). Along \(\boldsymbol{z}\), the sampling is performed in equal space with \(z_{l}=l\Updelta z\). Together, the sampling points of \((n,m,l)\) constitute a cylindrical grid. In the object space, a 3-D object is sampled in the equispaced Cartesian coordinate, as shown in Fig. 18.52a,bb.

The 3-D forward FT is achieved by first performing 2-D polar FFT in the \(xy\)-plane, followed by 1-D FFT along \(z\). We call this approach forward cylindrical FT ( ) in what follows. The 2-D polar FFT is applied in each slice of 3-D object data (\(x\) and \(y\)-planes) as step I. Step II uses regular 1-D FFT in the \(z\)-dimension of the 3-D object data. The 3-D inverse FT is carried out by performing 2-D inverse polar FT in the \(xy\)-plane first, and then 1-D inverse FFT along \(z\). We call this inverse CFT (CFT\({}^{-1}\)) in what follows.

For 2-D polar FFT in the \(xy\)-plane, let \(M\) denote the total number of nonequispaced sampling points in the Fourier space. The points are arranged in a polar grid with number of radial sampling points of \(R\) and a number of angular sampling of \(T\), \(R,T\in\mathrm{N}^{*}\), and total sampling points \(M=R\times T\). Taking
$$\{\boldsymbol{k}_{0},\boldsymbol{k}_{1},\dots,\boldsymbol{k}_{M-2},\boldsymbol{k}_{M-1}\}$$
as the sampling points on the polar grid in the Fourier space, \(\mathbb{R}^{2}\) is represented by the unit square \([-1/2,1/2]^{2}\), \(\boldsymbol{k}_{j}\in\mathbb{R}^{2}\) and \(j=0,\dots,M-1\). Let \(f_{j}\) denote the complex Fourier coefficient at the Fourier spatial frequency \(\boldsymbol{k}_{j}\). Let
$$I_{N}=\{\boldsymbol{r}_{0},\boldsymbol{r}_{1},\dots,\boldsymbol{r}_{\mathrm{N}-2},\boldsymbol{r}_{N-1}\}$$
denote the set of equispaced Cartesian grid points in the \(xy\)-plane. For a finite number of complex Fourier coefficients \(f_{j}\), the 2-D forward and inverse polar FT can be calculated as
$$f_{j} =\sum_{\boldsymbol{r}_{n}\in I_{N}}\rho(\boldsymbol{r}_{n})\mathrm{e}^{2\uppi\mathrm{i}\boldsymbol{k}_{j}\cdot\boldsymbol{r}_{n}}\;,$$
(18.77)
$$\rho(\boldsymbol{r}_{n}) =\sum_{\boldsymbol{k}_{j}\in\mathbb{R}^{2}}f_{j}\mathrm{e}^{-2\uppi\mathrm{i}\boldsymbol{k}_{j}\cdot\boldsymbol{r}_{n}}\;.$$
(18.78)
Details of the computation of (18.77) as implemented in NFFT can be found in [18.174, 18.175]. The computation involves three basic steps:
  1. 1.

    Selection and precomputation of the FT of a window function

     
  2. 2.

    FFT of \(\rho(\boldsymbol{r})\) on an oversampled Cartesian grid

     
  3. 3.

    Calculation of the Fourier frequency on the polar grid by interpolation using convolution between the selected window function and the oversampled Fourier frequencies.

     
Step 3 is calculated by using regular FFT. The accuracy of NFFT is adjusted to the requirements by choosing an oversampling factor and a cut-off parameter for the window function, called the convolution kernel width \(m\) in NFFT [18.174].
Inverse 2-D polar FT of (18.78) is obtained using a least-squares approach by solving the unconstrained minimization problem of
$$\sum_{j=1}^{M}w_{j}|f_{j}-y_{j}(\boldsymbol{r}_{n})|^{2}\overset{\tilde{\rho}}{\rightarrow}\min\;,$$
where
$$y_{j}=\sum_{\boldsymbol{r}_{n}\in I_{N}}\tilde{\rho}(\boldsymbol{r}_{n})\mathrm{e}^{2\uppi\mathrm{i}\boldsymbol{k}_{j}\cdot\boldsymbol{r}_{n}}$$
and \(\tilde{\rho}(\boldsymbol{r}_{n})\) is the estimated object function. We use the modified conjugated-gradient method as implemented in NFFT for solving the least-squares inverse problem. The number of sampling points on the polar grid is forced to be larger than the number of points in the Cartesian grid (\(M\geq I_{N}\)), so the inverse problem is overdetermined. The weight \(w_{j}> 0\) in the least square is introduced to account for the sampling density variations in the polar grid. Each step in the iterative optimization process involves a computation of 2-D polar FT. Thus, the inverse CFT is computationally intensive [18.174].
Fig. 18.53a-i

The intensity profiles of the projected 2-D images. (a,d) and (g) are 2-D images obtained by integrating the intensity from the 3-D reconstructed images along the same projection direction for the ideal structural model (a), the reconstruction using \({\mathrm{1}}\,{\mathrm{\AA{}}}\) probe (d), and \({\mathrm{1}}\,{\mathrm{\AA{}}}\) probe with 30 missing wedges (g). In each image, two intensity profiles along the horizontal and vertical directions shown by the orange and blue lines are presented on the right-hand side. (b,e) and (h) are the horizontal profiles and (c,f) and (i) are the vertical profiles for (a,d) and (g), respectively. Reprinted from [18.171], with permission from Elsevier

Lu et al describe a 3-D reconstruction algorithm based on the cylindrical Fourier transform computed by the nonequispaced fast Fourier transform [18.171]. The algorithm is accelerated and implemented using computer unified device architecture (CUDA ) on a graphics processing unit (GPU ). The algorithm was used to test 3-D atomic resolution reconstruction for a 309-atom Au nanoparticle using calculated diffraction patterns based on kinematical approximation. Results using simulated test data show that the algorithm is capable of reconstructing the 3-D structure at atomic resolution. The reconstructed 3-D object is highly dependent on the quality of the input 2-D diffraction patterns as described by resolution, noise, and missing wedges in the simulated image tilt series (see Fig. 18.53a-i for an example). Without the missing wedge, the algorithm reconstructs the full atomic structure information up to the resolution limit in the simulated diffraction patterns. The missing wedge introduces additional degradation in resolution in the direction normal to the missing wedge. Overall, for the nanoparticle with 309 atoms, the algorithm was capable of reconstructing its 3-D structure by resolving individual atoms using simulated data, provided that the experimental condition is controlled within the tolerance of a \({\mathrm{25}}\%\) noise level, \({\mathrm{1}}\,{\mathrm{\AA{}}}\) information transfer, and a sample tilting range of \(150^{\circ}\) with a \(2^{\circ}\) increment.

When applied to real experimental results, in principle, the utilization of diffraction patterns should be able to promote the image quality, because of the higher signal/noise ratio, the absence of both objective lens aberration and scanning noise. However, significant challenges remain for experimental realization of a truly universal atomic resolution tomography. The unanswered questions include: could the recent success of atomic resolution tomography based on STEM be extended to nonperiodic structures, such as amorphous glass? Is there a limit to the electron multiple scattering effect in a tilt series 3-D reconstruction? How do we interpret reconstructed data in the presence of varying electron multiple scattering effects? Most importantly, 3-D diffraction data collection from a single nanoparticle remains an unsolved experimental challenge, especially for radiation sensitive materials.

18.5 Conclusions

The advantages of electron nanodiffraction are the small probes, the strong elastic scattering cross-sections of the high-energy electrons, and the high information content. These advantages allow the recording of diffraction patterns from very small nanostructures, for example, a single-walled carbon nanotube. The challenge in electron nanodiffraction is to relate diffraction information to the atomic structure. Since electron diffraction is not affected by lens aberrations (except geometric distortions at large diffraction angles from the projector lens of the electron microscope), the relationship between the electron diffraction pattern and the structure is simpler than electron imaging. In this chapter, we have demonstrated how this simpler relationship can be made to be a greater benefit for quantitative and high-resolution structure analysis. Specifically, we have outlined an electron diffraction theory based on both kinematic approximation and dynamic diffraction, which can serve as the basis for the interpretation of electron nanodiffraction patterns. We also emphasized the different electron nanoprobes that can be formed inside an electron microscope, which range from a focused beam to parallel illuminations. The flexibility of the electron illumination system for forming different probes is another advantage of electron nanodiffraction. In particular, the use of parallel beams for diffraction imaging is very promising for achieving diffraction-limited resolution. We also demonstrated the principles and applications of scanning electron nanodiffraction and diffraction imaging for orientation mapping, imaging strain, 3-D nanostructure determination, and the study of defects.

Notes

Acknowledgements

The writing of this chapter was made possible with the support by US Department of Energy, Grant DEFG02-01ER45923 and NSF DMR 1410596. The work described here would not have been possible without the outstanding efforts of students and postdoc students, especially work by Weijie Huang, Yifei Meng, Yu-Tsun Shao, Kyouhyun Kim, Xiangwen Lu, Wenpei Gao, and Piyush Vivek Deshpande has contributed directly to the writing of this chapter.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dept. of Materials Science & Engineering and Materials Research LaboratoryUniversity of Illinois at Urbana-ChampaignUrbana, ILUSA

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