Position averaged convergent beam electron diffraction: Theory and applications

Convergent beam electron diffraction (CBED) at vertical grain boundaries (parallel to the electron beam) can be applied to determine the symmetry of bicrystals. It can also be used to investigate the structure of the boundary region itself when subnanometre probe sizes are employed. In this paper it is shown that (sub)nanometre-probe CBED patterns are largely influenced by the electron-beam geometry. In particular, simulations of coherent CBED patterns based on the multislice algorithm show that the CBED pattern of an edge-on interface depends on the defocus distance between the probe position and the specimen midplane, the probe size and the beam-convergence angle. The pattern symmetry may be lower than the theoretically predicted symmetry in case of large spherical aberration. This effect increases with smaller accelerating voltages. An increase in the beam-convergence angle also increases the possibility of a non-optimum symmetry due to spherical aberration of a coherent probe. Thus, for the determination of an interface structure using subnanometre (coherent) probes, the imaging conditions play an important role.

The convergent beam electron diffraction (CBED) patterns of twisted bilayer samples exhibit interference patterns in their CBED spots. Such interference patterns can be treated as off-axis holograms and the phase of the scattered waves, meaning the interlayer distance can be reconstructed. A detailed protocol of the reconstruction procedure is provided in this study. In addition, we derive an exact formula for reconstructing the interlayer distance from the recovered phase distribution, which takes into account the different chemical compositions of the individual monolayers. It is shown that one interference fringe in a CBED spot is sufficient to reconstruct the distance between the layers, which can be practical for imaging samples with a relatively small twist angle or when probing small sample regions. The quality of the reconstructed interlayer distance is studied as a function of the twist angle. At smaller twist angles, the reconstructed interlayer distance distribution is more precise and artefact free. At larger twist angles, artefacts due to the moiré structure appear in the reconstruction. A method for the reconstruction of the average interlayer distance is presented. As for resolution, the interlayer distance can be reconstructed by the holographic approach at an accuracy of ±0.5 Å, which is a few hundred times better than the intrinsic z-resolution of diffraction limited resolution, as expressed through the spread of the measured k-values. Moreover, we show that holographic CBED imaging can detect variations as small as 0.1 Å in the interlayer distance, though the quantitative reconstruction of such variations suffers from large errors.

Accurate values of foil thickness are required in many materials science applications, such as for measurement of defect concentrations and for x-ray microanalysis absorption corrections. Kelly et al. demonstrated that convergent beam electron diffraction (CBED) patterns can be analyzed using a simple graphical technique to give values for foil thickness with ±2% accuracy. More recently, Allen extended the treatment to make use of both maxima and minima in the CBED disks. The technique requires a knowledge of the d-spacing of the reflection, the electron wavelength, an evaluation of the deviation parameter, si, associated with the i-th fringe in the diffracted beam disk, and the assignment of a set of constants to "index" the fringes.

Convergent-beam and small-area-parallel-beam electron diffraction of icosahedral quasicrystals of a melt-quenched Al-Mn alloy

SnGe4N4O4 was synthesized at high pressure (16 and 20 GPa) and high temperature (1200 and 1500°C) in a large-volume press. Powder X-ray diffraction experiments using synchrotron radiation indicate that the derived samples are mixtures of known and unknown phases. However, the powder X-ray diffraction patterns are not sufficient for structural characterization. Transmission electron microscopy studies reveal crystals of several hundreds of nanometres in size with different chemical composition. Among them, crystals of a previously unknown phase with stoichiometry SnGe4N4O4 were detected and investigated using automated diffraction tomography (ADT), a three-dimensional electron diffraction method. Via ADT, the crystal structure could be determined from single nanocrystals in space group P63mc, exhibiting a nolanite-type structure. This was confirmed by density functional theory calculations and atomic resolution scanning transmission electron microscopy images. In one of the syntheses runs a rhombohedral 6R polytype of SnGe4N4O4 could be found together with the nolanite-type SnGe4N4O4. The structure of this polymorph was solved as well using ADT.

The fine structure of the first-order Laue zone (FOLZ) reflections in convergent-beam electron diffraction (CBED) patterns of Gaxln1–xAsyP1–y/InP heterostructures is investigated. The FOLZ reflection fine structure consists mainly of three bright lines; the position and intensity of these bright lines were used for a quantitative interpretation, using many-beam dynamical calculations. The main factor determining the position of the lines appears to be the composition. However, HOLZ-beam coupling, absorption and thermal vibrations of atoms have a considerable effect on the position and the intensity of the lines. Values of these parameters were estimated from the CBED patterns. A good agreement between calculation and experiment could only be obtained if small static displacements of the atoms from their ideal lattice sites were included. The calculations suggest that the static displacements in the quaternary compounds should be largest on the group-V sublattice. The quasi-periodic strain fields ...

In scanning transmission electron microscopy (STEM), one can obtain a variety of STEM images such as bright‐field (BF) and annular dark‐field (ADF) STEM images by changing the shape of the scintillator. However, the intensity distribution of convergent beam electron diffraction (CBED) patterns at the detector plane is yet to be fully utilized. Meanwhile, direct electron detectors with fast frame rate have recently been commercialized and used in electron microscopy. Such detectors, when used for recording CBED pattern images for each STEM probe position, are called pixelated STEM detectors. With the obtained 4‐dimensional (4D) dataset, any shape of STEM detector can be synthesized in a post processing by a free selection of the integration area. Therefore, we can synthesize variety of STEM images such as differential phase contrast (DPC) and annular bright field (ABF) images, if we once record the image signal with a pixelated detector. The 4D dataset can also be used for the advanced image processing techniques such as ptychography, which has been shown to provide high efficiency for reconstructing the phase image of an object [1,2]. Using ptychography, not only the phase contrast can be enhanced but also the effect of lens aberrations to the image such as defocus can be corrected by the post processing using the information collected by a pixelated detector. Experiments were performed using an aberration corrected microscope (JEOL JEM‐ARM200F) equipped with a pixelated detector (pnDetector pnCCD), the fast direct electron detector, which can record images at a speed of 1,000 fps in a full frame mode (264 x 264 pixels). Binning or windowing can increase the speed. The camera was placed below the ADF detector to enable simultaneous recording. Figure 1 shows STEM images of a monolayer graphene obtained at 80 kV. Fig. 1a shows an ADF image obtained with the probe current of approximately 0.2 pA and the dwell time for a pixel of 0.5 ms. Because of a combination of low dose and residual uncorrected aberrations, the lattice contrast of graphene is almost buried in the noise originated from 50 Hz commercial frequency. Fig. 1b shows a reconstructed phase image using ptychography. The image contrast is significantly improved compared to the simultaneous ADF image, but the contrast transfer of the image (see the Fourier transform displayed in the inset) is anisotropic, resulting in uncertain positions of carbon atoms. This is because there remains large two‐fold astigmatism and defocus in the image shown in Fig. 1(b), because we could not adjust those by observation of the ADF image due to the weak image signal. Fig. 1c is a phase image in which the aberrations are corrected through post processing using the same 4D dataset by applying correction functions in the spatial frequency domain. The image is no longer anisotropic and the carbon atomic positions can be unambiguously determined. Although the same amount of electron dose is used to form the ADF and the corrected phase images, the result clearly shows the benefit of ptychographic phase reconstruction in improving image signal to noise and being able to correct aberrations through post processing. Figure 2 shows the ptychographic phase maps of the 4D dataset at a certain spatial frequency. With an aberration‐free electron probe, they would be flat phase on the two sidebands, and the phase difference between the bands be π. In Fig. 2a, the map of the original dataset, there is a phase gradient inside each sideband because the aberrations were present in the electron probe. Fig. 2b is the correction function that compensates for the aberration seen in Fig. 2a. Fig. 2c is the corrected phase map and corresponding to the image in Fig. 1c. Here, only defocus and two‐fold astigmatism are corrected but corrections of other higher order aberrations are in principle possible. Figure 3 shows through focus images created in the same way as above. As we know the full information on the electron wave at the condenser aperture plane, we can induce any aberrations such as defocus.

Convergent beam electron diffraction (CBED) is widely used as a microanalysis tool. By the relative position of HOLZ-lines (Higher Order Laue Zone) in CBED-patterns one can determine the unit cell parameters with a high accuracy up to 0.1%. For this purpose, maps of HOLZ-lines are simulated with the help of a computer so that the best matching of maps with experimental CBED-pattern should be reached. In maps, HOLZ-lines are approximated, as a rule, by straight lines. The actual HOLZ-lines, however, are different from the straights. If we decrease accelerating voltage, the difference is increased and, thus, the accuracy of the unit cell parameters determination by the method becomes lower.To improve the accuracy of measurements it is necessary to give up the HOLZ-lines substitution by the straights. According to the kinematical theory a HOLZ-line is merely a fragment of ellipse arc described by the parametric equationwith arc corresponding to change of β parameter from -90° to +90°, wherevector, h - the distance between Laue zones, g - the value of the reciprocal lattice vector, g‖ - the value of the reciprocal lattice vector projection on zero Laue zone.

The standard technique for sub-pixel estimation of atom positions from atomic resolution scanning transmission electron microscopy images relies on fitting intensity maxima or minima with a two-dimensional Gaussian function. While this is a widespread method of measurement, it can be error prone in images with non-zero aberrations, strong intensity differences between adjacent atoms or in situations where the neighboring atom positions approach the resolution limit of the microscope. Here we demonstrate mpfit, an atom finding algorithm that iteratively calculates a series of overlapping two-dimensional Gaussian functions to fit the experimental dataset and then subsequently uses a subset of the calculated Gaussian functions to perform sub-pixel refinement of atom positions. Based on both simulated and experimental datasets presented in this work, this approach gives lower errors when compared to the commonly used single Gaussian peak fitting approach and demonstrates increased robustness over a wider range of experimental conditions.

In this chapter, the basic principles of atomic resolution scanning transmission electron microscopy (STEM) will be described. Particular attention will be paid to the benefits of the incoherent Z-contrast imaging technique for structural determination and the benefits of aberration correction for improved spatial resolution and sensitivity in the acquired images. In addition, the effect that the increased beam current in aberration corrected systems has on electron beam-induced structural modifications of inorganic systems will be discussed. Procedures for controlling the electron dose will be described along with image processing methods that enable quantified information to be extracted from STEM images. Several examples of the use of aberration-corrected STEM for the study of nanoscale systems will be presented; a quantification of vacancies in clathrate systems, a quantification of N doping in GaAs, a quantification of the size distribution in nanoparticle catalysts, and an observation of variability in dislocation core composition along a low-angle grain boundary in SrTiO3. The potential for future standardized methods to reproducibly quantify structures determined by STEM and/or high-resolution TEM will also be discussed.

This work presents observations of symmetry breakages in the intensity distributions of near-zone-axis convergent-beam electron diffraction (CBED) patterns that can only be explained by the symmetry of the specimen and not the symmetry of the unit cell describing the atomic structure of the material. The specimen is an aluminium-copper-tin alloy containing voids many tens of nanometres in size within continuous single crystals of the aluminium host matrix. Several CBED patterns where the incident beam enters and exits parallel void facets without the incident beam being perpendicular to these facets are examined. The symmetries in their intensity distributions are explained by the specimen morphology alone using a geometric argument based on the multislice theory. This work shows that it is possible to deduce nanoscale morphological information about the specimen in the direction of the electron beam - the elusive third dimension in transmission electron microscopy - from the inspection of CBED patterns.

A systematic method to identify the space group from PED and CBED patterns part I - theory

To accurately and easily determine the polarity of AlN using transmission electron microscopy, we compare the convergent-beam electron diffraction (CBED) patterns along the widely used <112̄0 > and <11̄00 > zone-axes. For the <112̄0 > zone-axis, the diffraction disk of g= 0002 differs from that of g= 0002̄, while for <11̄00 >, the diffraction disks of g= 0002 and 0002̄ are similar. The preferential clarity of these two disks is explained using Bloch-wave dynamical theory. To further support the explanation, we compare the results of GaN case. On the basis of our analysis, we conclude that the CBED patterns of the <112̄0 > zone-axis are more useful for accurately determining AlN polarity compared to the CBED patterns along the <11̄00 > zone-axis.

Under almost all circumstances, electron diffraction patterns contain information about the phases of structure factors, a consequence of the short wavelength of an electron and its strong Coulombic interaction with matter. However, extracting this information remains a challenge and no generic method exists. In this work, a set of simple analytical expressions is derived for the intensity distribution in convergent-beam electron diffraction (CBED) patterns recorded under three-beam conditions. It is shown that these expressions can be used to identify features in three-beam CBED patterns from which three-phase invariants can be extracted directly, without any iterative refinement processes. The octant, in which the three-phase invariant lies, can be determined simply by inspection of the indexed CBED patterns (i.e. the uncertainty of the phase measurement is ±22.5°). This approach is demonstrated with the experimental measurement of three-phase invariants in two simple test cases: centrosymmetric Si and non-centrosymmetric GaAs. This method may complement existing structure determination methods by providing direct measurements of three-phase invariants to replace 'guessed' invariants in ab initio phasing methods and hence provide more stringent constraints to the structure solution.

Convergent beam electron diffraction (CBED) on cross-sectional transmission electron microscopy (TEM) specimens can map stress in the Si substrate of microelectronics devices with high spatial resolution. However, at shallow depths below the interface the diffraction lines within a CBED pattern are split, rendering pattern interpretation impossible in the classical way. This effect is believed to derive from a bending of the diffracting planes through the sample thickness. In this work, a variety of specimens, ranging from polycrystalline to epitaxial and amorphous materials, have been systematically analyzed. A similar qualitative distribution of the splitting effects is present in each case, but quantitative differences are observed which can be related to the stress induced by the different materials. Capping layers can partially avoid the splitting but seem to modify the strain distribution. The surface relaxation in the thin TEM specimen foil is modelled by finite elements simulations as well as with an epitaxial-layer model. The results predict well the experimental magnitude of the splitting but fail to simulate the intensity of the secondary lines. Possible explanations for such discrepancies are discussed and critically assessed.