Dynamical Theory Of Electron Diffraction Research Articles

Simulation of high energy photoelectron diffraction using many-beam dynamical Kikuchi-band theory

We use the many-beam dynamical theory of electron diffraction for the calculation of x-ray photoelectron diffraction $(\mathrm{XPD})$ patterns of the substrate emission. The reciprocity principle is used to apply a Bloch wave model for the diffraction of an incoming plane wave by a three-dimensional crystal. In this way, many-beam dynamical simulations of $\mathrm{XPD}$ in the context of Kikuchi-band theory can be carried out. This extends the results of the two-beam theory used so far and leads to quantitative descriptions of $\mathrm{XPD}$ patterns in the picture of photoelectrons reflected by lattice planes. The effects of forward scattering directions, substrate polarity, circular structures due to onedimensional diffraction, and emitter specific extinction of Kikuchi lines can be reproduced by Kikuchi-band theory. The results are compared with single scattering cluster calculations. In this way, the equivalence of the cluster approach and the Kikuchi-band picture can be demonstrated completely in both directions

Diffraction-contrast analysis of misfit strains around inclusions in a matrix: VN particles in α-Fe

A new method has been developed to determine the misfit of disc-shaped precipitates in a matrix using transmission electron microscopy (TEM). The method is applicable to specimens containing a high precipitate density, where the classical TEM method, based on the extent of the precipitate diffraction contrast cannot be applied. The new method is based on evaluation of the positions of extrema in the intensity distribution due to misfitting precipitates in both bright field (BF) and dark field (DF). A model system, consisting of a single disc-shaped misfitting precipitate placed centrally in a thin specimen, has been studied first. The dynamic theory of electron diffraction has been adopted for a four-beam case. The contrast lobes in BF and DF images have been calculated as a function of, in particular, the particle radius, the foil thickness and the particle thickness. Simultaneous fitting of calculated BF and DF diffraction contrast images to the experimental images leads to determination of the precipitate misfit, and the local thickness of the specimen foil. The method has been illustrated for a nitrided Fe-2 at. %V alloy with small disc-shaped VN precipitates and has led to a consistent interpretation in terms of particle size and misfit upon precipitation. The extent of elastic accommodation of misfit has been verified using high-resolution electron microscopy. The foil thickness values determined by diffraction contrast analysis agree well with independently obtained corresponding data. In addition X-ray diffraction (XRD) line profiles of the specimens have been recorded. The observed shifts and broadenings of the XRD profiles support the results obtained using TEM.

Analyse du contraste d'un sous-joint de torsion (0 0 1) dans le silicium en MET à deux ondes

A quantitative analysis of the image of a low angle (0 0 1) twist boundary in silicon is performed using the two-beam dynamical theory of electron diffraction. The contrast features are discussed as functions of the thickness of the foil and possible elastic relaxation effects of the low angle twist boundary in the thin foil. To cite this article: R. Bonnet et al., C. R. Physique 3 (2002) 657–663.

Review Electron backscatter Kikuchi diffraction in the scanning electron microscope for crystallographic analysis

The technique of electron backscatter Kikuchi diffraction patterns (BKDPs) in the scanning electron microscope is reviewed. The paper focuses mainly on the crystallographic applications of the technique, including discussions on point group and space group determination and strain analysis. Orientation microscopy is discussed but not reviewed. The geometrical configurations of BKDPs are reviewed in detail and the relationship between BKDPs and the technique of electron channelling patterns (ECPs) is explored briefly. Essential crystallography is discussed and methods of analysis of BKDPs to extract crystallographic information are analyzed in detail. Some important characteristics of diffraction contrast in BKDPs are analyzed with respect to the geometry of the technique, the dynamical theory of electron diffraction and crystallographic applications. Examples of the use of theoretical contrast in pattern interpretation are provided. Anomalous effects in BKDPs are analyzed in detail and ways of identifying anomalous contrast in practice are discussed. BKDPs included in the paper are zincblende (ZnS), silicon, germanium, GaAs, chalcopyrite (CuFeS2, TaTe4 and Er2Ge2O7.

Identification of ε′-Fe 2N and ε′-Fe 3N superstructures studied with high-resolution electron microscopy

The identification of ε ′-Fe 2N and ε ′-Fe 3N nitrides, which have similar crystal structures, was studied according to the dynamical electron diffraction theory and image analysis. The nitrogen sub-lattice images in ε ′-Fe 2N, which are different from those in ε ′-Fe 3N, were observed by high-resolution electron microscopy.

Axial electron channeling statistical method of site occupancy determination

Multibeams dynamical theory of electron diffraction has been used to calculate the fast electron thickness-integrated probability density on Ti and Al sites in the λ-TiAl phase as a function of the incident electron beam orientation along [100], [110] and [011] zone axes, with the effect of absorption considered. Both of the calculation and experiments show that there are big differences in electron channeling effect for different zone axes or the same axis but with different orientations, so we should choose proper zone axis and suitable incident beam tilting angles when using the axial electron channeling statistical method to determine the site occupancies of impurities. It is suggested to calculate the channeling effect map before the experiments.

Measurement of charge of Cu and O in YBa 2Cu 3O y by energy-filtering convergent-beam electron diffraction

Large-angle convergent-beam electron diffraction (CBED) patterns were observed from small areas in nanometer scale in YBa 2Cu 3O y (Y123), using a new transmission electron microscope (TEM) JEM-2010FEF. An energy filter of omega type was used to subtract background intensities caused by inelastically scattered electron. Specimens were cooled with a liquid nitrogen stage to suppress thermal diffuse scattering that cannot be eliminated well by the energy filter. In order to measure specimen thickness, electron energy loss spectra were also measured from the same areas where CBED patterns were recorded. Charges of Cu and O on CuO chains and CuO 2 planes, and charge of O on Ba–O planes were determined by analyzing the intensities of CBED patterns with the dynamical theory of electron diffraction. The obtained values are Cu 2.0+ and O 1.65− on CuO chains, Cu 2.2+ and O 1.98− on CuO 2 planes and O 2.0− on BaO planes. The present results indicate that the energy-filtering CBED technique is useful for determination of charge distribution along the z-direction in Y123.

Image contrast of ultra HVEM and future ultra HREM

Review of the measurements and interpretation of energy dependence of extinction distance and transmissive power of electrons, which have appeared in the electron microscope images of crystals at the voltages ranging from 50 to 3000 kV, is presented. Applications of two beam dynamical theory of electron diffraction considering absorption effect of electrons made clear the importance of relativistic effect of the mass of electron to the variation of extinction distance and transmissive power at higher voltage and noted that they are proportional to (v/c) and (v/c)2 respectively, where v and c are the velocity of illuminating electrons and light. With elevating accelerating voltage of illuminating electrons, it was noted that many diffracted waves are excited and thus, for understanding of the image contrast and the improvement of transmissive power of electrons, many beam dynamical theory considering absorption of electron waves has to be applied. Detailed analysis of the excitation and absorption of component Bloch waves was carried out and flux (electron current) flow of Bloch waves and total waves was deduced. Improvements of image contrast and visibility of the imperfection in crystalline materials both in thin and thick regions of ultra-high voltages were realized by using the multi-beam imaging (MBI) method. In MBI of very thick crystals, it was noted that most of the inelastically scattered electrons which appeared in the background of the diffraction patterns contribute to the image contrast by the successive elastic scattering after inelastic scattering. It was noted that the improvement of maximum usable thickness at high voltage is highly dependent on the contribution of inelastically scattered electrons. The calculated flux flows of total electrons of various materials in various orientations were calculated for not only ultra-high voltages but low voltages such as 100 kV. It was suggested that the intensity distribution of such flux flow will be recorded by the ultra HREM in the near future and will become useful to identify types of atoms, such as the interface consisting of different kinds of atoms.

Stacking-Fault Fringes

Abstract Crystalline interfaces associated with lattice translations such as stacking-faults (SFs) and twin boundaries may be characterized in the transmission electron microscope (TEM) by oscillatory fringe contrast known as SF, or rigid-body-translation, fringes. Such fringe contrast is usually obtained using weak-beam bright-field (BF) or dark-field (DF) conditions with ideally only one reflection, g, excited. The nature of the SF contrast has been explained by dynamical theory of electron diffraction in terms of the Howie-Whelan equations [1]. The fringe contrast arises from an interference caused by a phase shift the diffracted electron beams, undergo upon propagating across the interface. The phase shift, α, is proportional to the projection of the translation vector, R, onto the diffraction vector, g (i.e. α # g.R). The direction of the translation state can be determined explicitly from the SF contrast in a manner analogous to the analysis of dislocation Burgers vector. An analysis of SF fringes with respect to diffraction theory permits the magnitude of the lattice translation to be evaluated in experimental images.

3D morphology of II–VI semiconductor nanocrystals grown in inverted micelles

Colloidal CdS nanocrystals have been elaborated by using a technique of the chemistry of colloids. Their mean size range between 3 and 10 nm. The 3D morphology of the nanocrystals has been studied by using high-resolution transmission electron microscopy (HRTEM). The method is based on the measurement of I g, hr ( r) , the intensity of a family of g lattice fringes as a function of the position r(x, y) in the high-resolution electron micrograph, assuming that I g, hr increases monotonously as a function of the thickness t of the crystal along the electron beam. The range of thickness for the validity of the method has been calculated in the frame of the dynamical theory of electron diffraction. CdS colloids are mainly in the cubic structure of blende-type (B) and a few of them are in the hexagonal structure of wurtzite-type (W). In the cubic structure of B-type, the morphologies are found to be the octahedron and tetrahedron. The hexagonal structure of W-type, have a platelet-like morphology with well-developed basal faces.

Kikuchi-band formation in medium-energy electron-diffraction patterns

of close-packed ~110! planes. The formation of the Kikuchi bands is shown to be governed by two types of electron scattering in a crystal. The dominant contribution to enhanced electron-scattering intensity within a band comes from the forward-focusing effect as the electrons move along the numerous interatomic directions in the ~110! planes. The other mechanism responsible for the sharp edge regions in the Kikuchi bands involves electron scattering from the nearest planes. It is proposed to use the specific profile of the Kikuchi bands in estimating the shape and size of light-element crystallites forming during initial stages of island-film growth. © 1999 American Institute of Physics.@S1063-7834~99!00903-X# Studies of surfaces and interfaces have been making increasing use of methods of atomic structure analysis based on medium-energy electron diffraction ~hundreds of eV to ;2 keV!. This relates primarily to the diffraction of x-ray generated photo- and Auger electrons, 1‐3 as well as of incoherently scattered primary electrons. 3‐6 The spatial distributions of all these three groups of electrons have a similar structure for close energies. As a rule, it may be considered as a superposition of maxima oriented along the closepacked directions in a crystal. This specific pattern of the diffractograms allows their fairly straightforward interpretation and provides information regarding the object under study in real space. Besides, one usually observes bands of a higher intensity oriented along the projections of the closestpacked atomic planes. For crystals made up of light elements, these bands exhibit a high contrast and have sharp boundaries. The maxima lying along interatomic directions are usually associated with the forward electron focusing effect in crystals. 1‐3 The mechanism responsible for formation of the higher-intensity bands, which in the electron backscattering patterns are traditionally called Kikuchi bands, and in photoand Auger-electron diffraction, Kikuchi-like or Bragg features, is less clear. In the early stages of these studies, when the potential of using these patterns for structural analysis of surfaces was not yet recognized, the Kikuchi bands were considered within the dynamical theory of electron diffraction. 7 The corresponding calculations were performed, as a rule, in the simplest two-wave approximation, which permitted interpretation of the observed Kikuchi-band profile and their width equal to twice the Bragg angle. Later studies, 8‐12 where the emphasis shifted to the electron focusing effect and numerical simulation of the patterns in terms of the single-scattering cluster approximation, put forward various suggestions concerning the formation mechanism of these features. The most widely used viewpoint associates the sharp drops in intensity at the Kikuchiband edges with electron Bragg diffraction from a system of planes, which is taken into account by introducing large clusters. For instance, the cluster used to analyze photoelectron diffraction from CaF2~111! at 1139 eV was 303303 30 Ai n

Precise and accurate refinements of the 220 structure factor for silicon by the systematic-row CBED method

The 220 structure factor for silicon has been refined by matching rocking curves extracted from energy filtered convergent beam electron diffraction patterns to those calculated from a Bloch-wave formulation of the dynamical theory of electron diffraction. There is an excellent agreement between the refined results that are obtained from rocking curves recorded from different crystal thicknesses and published results. Purpose-written computer codes have been used to perform the fitting. Three efficient criteria for automatic beam selection have been incorporated in these computer programs to guarantee the convergence of a calculation. The importance of minimization criteria and treatment of anomalous absorption on the convergence of the calculations have been tested. It is demonstrated that the refined structure factors are independent of minimization criteria and that the computationally less intensive perturbation treatment of absorption can be used to refine a structure factor with high precision (less than 0.1%) as well as high relative accuracy.

Defocussed TEM contrast from small helium bubbles in copper

The theoretical defocussed contrast of the TEM images of bubbles is computed using simulations based on the two-beam dynamical theory of electron diffraction. The bubble parameters used are chosen to be typical of the small (∼ 2 nm in diameter) bubbles found in gas-bubble superlattices. Simulated images are calculated for single equilibrium bubbles, for isolated columns of equilibrium bubbles and for single overpressured bubbles in copper. The calculated images are found to depend strongly on many parameters: the depth, pressure and radius of bubbles; foil thickness; diffracting vector; excitation error; defocus level, small deviations from the Bragg angle and number of bubbles included in the column. Representative examples are presented and comparisons made with experiment. It is concluded that simulations provide a very useful framework for guiding the broad choice of imaging parameters and provide a basis for the interpretation of the complicated imaging behavior often encountered in practice. Various methods for measuring bubble radii from TEM micrographs are assessed.

Effects of Debye-Waller factors and compositional uncertainties on the 200 structure factor refinement eta in-TiAl

The effects of uncertainties in Debye-Waller factors and chemical composition on 200 structure factor refinement in eta-TiAl have been investigated. Energy-filtered convergent-beam electron diffraction has been used to record 200 rocking curves from two different thicknesses of a thin foil taken from a Ti-51.3 1Al alloy (all compositions in atomic per cent). The structure factor has been refined by matching rocking curves determined by experiment to those theoretically calculated using the Bloch wave formalism of the dynamical theory of electron diffraction. The influence of uncertainties in the Debye-Waller factors and sample composition have been assessed: for samples deviating from stoichiometry, the need to use accurate site-specific Debye-Waller factors has been demonstrated. Also, to extract accurate structure factors, it is necessary to make use of homogeneous samples, the compositions of which must be known to better than 0.2%, a rather restrictive limitation.

Dynamical calculation of high-resolution reflection electron microscopy lattice fringes from the Si(111)7 × 7 surface

High-resolution reflection electron microscopy (HR-REM) lattice fringes for the flat Si(111)7 × 7 surface are simulated using our earlier reported theory of REM image formation. This simulation procedure employs a 2D Bloch wave formulation of dynamical (multiple scattering) elastic reflection high-energy electron diffraction (RHEED) theory combined with conventional electron microscopy (Abbé) imaging theory. One of the purposes of this study is to provide the first critical comparison of HR-REM images simulated using this procedure with experimental images for the Si(111)7 × 7 surface. It is found that it is possible to reproduce in the simulations all of the fringe periodicities observed in real HR-REM images from this surface. The incorporation into the simulation procedure of the effects of primary electron beam divergence and chromatic aberration is described in detail. The primary beam divergence is essential to reproduce in simulated images the correct defocusing behaviour. When these details are incorporated, the simulation procedure gives good agreement with experiment. This work shows that quantitative HR-REM image simulation constitutes a valuable interpretative tool to be used in support of experimental HR-REM. Published by Elsevier Science B.V.

Direct retrieval of crystal and surface structures using high energy electrons

A direct scheme has been developed for crystal and surface structure determination. This scheme is based on a partition of the crystal potential V( r) into a known part V 0( r) and a perturbation δV( r), i.e. V( r)= V 0( r)+ δV( r). The full dynamical electron diffraction theory is used for treating the dynamical electron diffraction by V 0( r) and perturbation theory is used for dealing with the residual scattering by δV( r). It is shown that this partition is applicable for a wide range of situations of practical importance, including scattering by the re-distributed valence electrons and by a reconstructed surface layer. In some favourable cases, when the crystal thickness is not exceedingly large and atomic displacements from the known structure giving rise to V 0( r) are small (as in the case of structure refinement), crystal structure factors and crystal and surface structure can be directly inverted from dynamical diffraction data. Both the perturbation method and the direct scheme have been extensively tested, and in particular the direct scheme has been used successfully to invert crystal structure factors directly from experimental CBED data obtained from a single crystal of MgO.

The ordering scheme in N b aluminides with the B2 crystal structure

This research may be summarized as follows. 1. 1. The OTLs for the three alloys Nb-15Al-xTi, where x = 10,125 or 40, have been determined. The trends in site occupancy in these compounds has been identified, and the apparent sublattice compositions have been determined. 2. 2. From these OTLs, it has been shown that while both compounds containing 10Ti and 40Ti exhibit the B2 crystal structure, their ordering schemes are rather different. 3. 3. Simulation of the ALCHEMI results using the dynamical theory of electron diffraction has provided estimates of the end-points of the OTLs. The estimated long range order parameters for the alloys with Ti contents of 10,25 and 40 have been estimated to be 0.88,0.25 and 0.5, respectivley. 4. 4. In all three compounds studied, the most ordered state would correspond to the Al atoms being confined to one sublattice, and this appears to be the most significant factor influencing the stability of the B2 crystal structure adopted by these alloys.

Application of the small-angle cleavage technique to thickness measurement of TEM samples.

In modelling transmission electron microscopy (TEM) images with the dynamical theory of electron diffraction, the sample thickness at the region of interest must be accurately known. A technique of sample preparation for cross-sectional single-crystal samples has been developed to provide this information. This technique, based on the small-angle cleavage technique, can allow a direct measurement of the sample thickness at the region where cross-sectional TEM analysis has been undertaken. Several cross-sectional samples of GaAs-based multiple quantum wells were prepared with parallel cleaved faces and thicknesses in the range of 50 to 200 nm. The samples were initially mounted so that cross-sectional images were obtained, then on suitable samples the mount was adjusted to obtain a plan view image of the same area. With suitable care, this technique provided an accurate thickness measurement of a cross-sectional region of a sample, which allowed subsequent modelling and analysis.

Emission channeling study of annealing of radiation damage in heavy-ion implanted diamond

The annealing of lattice damage in diamond produced by the implantation of heavy ions was studied by the emission channeling technique. Radioactive 111In and 73As ions were implanted at 300 K into IIa diamonds with 〈110〉 surface orientations, at energies of 120 and 60 keV and doses of 5 × 10 12 and 1 × 10 13 ions cm −2, respectively. Axial channeling measurements were performed on the emitted conversion electrons after annealing the diamonds at temperatures up to 1673 K. Two distinct stages for annealing of implantation damages are observed, the first between 300 and 600 K, and the second setting in at 1200 K. The lattice sites of the implanted ions were determined from comparisons of the measured yields with channeling effects calculated using the dynamical theory of electron diffraction. For In ions the fraction at substitutional lattice sites was about 40%, while for As ions the fraction at such sites exceeded 50%. Implantation of In ions in diamond at high temperature (1373 K) gave the same maximum channeling yield as that obtained with implantation at 300 K followed by annealing above 1200 K.

Calculation of many-beam dynamic electron diffraction without high-energy approximation

High energy approximation in dynamic theory of electron diffraction involves some intrinsic problems. First, the loss of theoretical strictness makes it difficult to comprehend the phenomena of electron diffraction. Secondly, it is difficult to believe that the approximation is reasonable especially in the following cases: 1) when accelerating voltage is not sufficiently high, 2) when the specimen is thick, 3) when the angle between the surface normal of the specimen and zone axis is large, and 4) when diffracted beam with large diffraction angle is included in the calculation. However, until now the method to calculate the many beam dynamic electron diffraction without the high energy approximation has not been proposed. For this reason, the authors propose a method to eliminate the high energy approximation in the calculation of many beam dynamic electron diffraction. In this method, a perfect crystal with flat surface was assumed. The method was applied to the calculation of [111] zone axis CBED patterns of Si.