Multislice Theory Research Articles

Observations of specimen morphology effects on near-zone-axis convergent-beam electron diffraction patterns.

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.

Optimal Angle Combination for Improving Electromagnetic Torque in Induction Motor With Double-Skewed Rotor

Double-skewed rotor is of great benefit to decrease harmonic magnetic fields in induction motors. But the fundamental magnetic field is damaged as well as the electromagnetic torque, like the case of normal skewed rotor. The effect of skew angle and stagger angle on the fundamental torque is studied in this paper. With the multi-slice theory, the double-skewing factor is proposed to determine the range of angle combination. Then, the range is shortened by the deduced torque coefficient with virtual displacement method. Compared with the general recommendation, the final angle combination is optimal and widely suitable for the double-skewed rotor. A higher pullout electromagnetic torque will be attained under this condition. With the finite elements-method, the validity of analysis model can be verified by the comparison of two kinds of motors.

A multislice theory of electron scattering in crystals including backscattering and inelastic effects

In the framework of the slice transition operator technique, a general multislice theory for electron scattering in crystals is developed. To achieve this generalization, we combine the approaches for inelastic scattering derived by Yoshioka [J. Phys. Soc. Jpn. 12, 6 (1957)] and backscattering based on the formalism of Chen and Van Dyck [Ultramicroscopy 70, 29–44 (1997)]. A computational realization of the obtained equations is suggested. The proposed computational scheme is tested on elastic backscattering of electrons, where we consider single backscattering in analogy to the computational scheme proposed by Chen and Van Dyck.

Back Cover: Transmission electron microscopy of single wall carbon nanotube/polymer nanocomposites: A first-principles study (Phys. Status Solidi RRL 8/2012)

Polymer nanocomposites filled with single wall carbon nanotubes (SWCNTs) are promising materials for aerospace applications. The control of homogeneous dispersion of the nanotubes in the polymeric matrix is important, e.g., for an optimum mechanical response of the nanocomposite. Furthermore, identification of the metallic or semiconducting nature of the nanotubes through chirality is needed. However, imaging SWCNTs in a polymer matrix is challenging both due to the small size and because the contrast is covered by scattering from the polymer matrix. Solá et al. (pp. 349–351) theoretically investigate the high-resolution transmission electron microscopy (HRTEM) images and electron diffraction (ED) of nanovolumes composed of polyethylene filled with SWCNTs using multislice theory and the optics of an FEG TEM at 80 kV. If the electron beam is oriented perpendicular to the nanotube walls, the results strongly depend on the nanocomposite thickness, HRTEM features of the nanofiller can even completely disappear, although chirality (and hence diameter) may be extracted by reflections of the ED pattern. The authors discuss the new limits set up for TEM investigation of dispersion and chirality.

Elastic scattering of high-energy electrons by dopant atoms within a crystal in transmission electron microscopy

A Bloch-wave model of dopant-atom scattering is developed using perturbation theory for parallel illumination in a transmission electron microscope. Dopant-atom scattering causes a change in the Bloch-wave excitations, with transitions from one Bloch state to another being governed by the amplitudes of the Bloch states at the dopant-atom position. The scattering mechanisms therefore depend on whether the dopant atom is substitutional or interstitial as well as the orientation of the crystal. The model is used to calculate the electron wavefunction for substitutional and interstitial Mo atoms in [111]- and [001]-oriented body-centred cubic Fe and the results overall are consistent with multislice simulations. However, subtle differences are also observed. For example, in the Bloch-wave model the phase change of the incident electrons (with respect to the perfect crystal) due to dopant-atom scattering varies with the crystallographic orientation while in multislice theory the phase change is constant. This is likely to be due to the assumptions made in the Bloch-wave model, such as neglect of elastic diffuse scattering. Apart from providing a fundamental understanding of dopant-atom scattering, the model can also potentially be extended to analyse dopant-atom imaging in a scanning transmission electron microscope.

Multislice theory of fast electron scattering incorporating atomic inner-shell ionization

It is demonstrated how atomic inner-shell ionization can be incorporated into a multislice theory of fast electron scattering. The resulting theory therefore accounts for both inelastic scattering due to inner-shell ionization and dynamical elastic scattering. The theory uses a description of the ionization process based on the angular momentum representation for both the initial and final states of the atomic electron. For energy losses near threshold, only a small number of independent states of the ejected atomic electron need to be considered, reducing demands on computing time, and eliminating the need for tabulated inelastic scattering factors. The theory is used to investigate the influence of the collection aperture size on the spatial origin of the silicon K-shell EELS signal generated by a STEM probe. The validity of a so-called local approximation is also considered.

Thermal diffuse scattering in sub-angstrom quantitative electron microscopy—phenomenon, effects and approaches

This paper reviews the recent progress in the following areas. (1) In quantitative high-resolution transmission electron microscopy, the theoretically calculated images usually give better contrast than the experimentally observed ones although all of the factors have been accounted for. This discrepancy is suggested due to thermal diffusely scattered (TDS) electrons, which were not included in the image calculation. The contribution from TDS electrons is especially important if the image resolution is approaching 0.1 nm and beyond with the introduction of Cs corrected microscopes. A more rigorous multislice theory has been developed to account for this effect. (2) We proved that the off-axis holography is an ideal energy filter that even filters away the contribution made by TDS electrons in the electron wave function, but conventional high-resolution microscopy do contain the contribution made by phonon scattered electrons. (3) In electron scattering, most of the existing dynamical theories have been developed under the first order diffuse scattering approximation, thus, they are restricted to cases where the lattice distortion is small. A formal dynamical theory is presented for calculating diffuse scattering with the inclusion of multiple diffuse scattering. By inclusion of a complex potential in dynamical calculation, a rigorous proof is given to show that the high order diffuse scattering are fully recovered in the calculations using the equation derived under the distorted wave Born approximation, and more importantly, the statistical time and structure averages over the distorted crystal lattices are evaluated analytically prior numerical calculation. This conclusion establishes the basis for expanding the applications of the existing theories. (4) The ‘frozen lattice’ model is a semi-classical approach for calculating electron diffuse scattering in crystals arisen from thermal vibration of crystal atoms. Based on a rigorous quantum mechanical phonon excitation theory, we have proved that the frozen lattice mode is an excellent approximation and no detectable error would be possible under normal experimental conditions.

Electron Microscopy Structural Characterisation of Nano-Materials: Image Simulation and Image Processing

High resolution electron microscopy is now often used to determine the structure of nano-materials: very small clusters, multilayers, precipitates etc... The images are often poor, because of the small size of the object, and their complete interpretations is very difficult. Image simulation has to be done in order to obtain a correct interpretation of these experimental images. This simulation is usually done using the multislice theory: however, a cluster or an atomic layer cannot be considered as a 3D-periodical object, though it is sort of crystallised matter but nano-samples are different objects as they have only a small and deformed crystalline component. Image processing can then help to interpret the images if one is conscious of all the bugs which can be introduced in the process!. In this paper, we want to forewarn newcomers in the field of HREM imaging for nano-clusters and explain some of the problems linked with simulating and processing nano-samples, insisting on all the wrong results which can be obtained by applying all the possible logicals which can be bought.

An optical potential approach to incoherent multiple thermal diffuse scattering in quantitative HRTEM

The theory for the absorption potential (or optical potential) in electron scattering was first proposed by Yoshioka in 1957 based on an approximation that the Green's function is replaced by its form in free-space. This approximation has dramatically simplified the calculation, but the function of the optical potential has been partially lost. In this paper, a rigorous theoretical proof is given based on the quantum inelastic excitation theory to show that the inclusion of the optical potential in the dynamic calculation automatically recovers the contributions made by the high-order diffuse scattering although the calculation is done using the equation derived for single diffuse scattering. This conclusion generalizes the existing first-order diffuse scattering theories to cases in which the incoherent multiple diffuse scattering are important. It is suggested that multiple thermal diffuse scattering (or phonon excitations) is likely to be the dominant source for affecting image contrast in quantitative electron microscopy because of the channeling effect and the localized scattering nature of phonon excitation. Surprisingly, phonon scattering is anticipated to contribute fine atomic-scale contrast in the image, which is believed to be more pronounced than the “white noise” effect of conventional understanding. The optical potential is no longer a simple potential function, rather it is a non-local function strongly dependent on the dynamic diffraction in the crystal because of the involvement of Green's function. These characteristics can be properly taken into the computation using a proposed Bloch wave–multislice approach, in which the non-local effect is resolved in the Bloch wave matrix diagonalization and the dynamical effect is taken care of using Green's function calculated by the multislice theory under the small angle (or high-energy) scattering approximation. In the ground-state approximation, by which we mean that the crystal is in its ground state before each and every inelastic excitation, an introduction of the mixed dynamic form factor and the density matrix automatically produces the incoherence between different order and different phonon excitation processes, resolving a big problem encountered by many other theories. In the conventional dynamical calculation for elastic wave, the inclusion of the Debye–Waller factor and the optical potential accounts only for the effects of the phonon excitation on the elastic wave rather than the contribution made by the diffusely scattered electrons to the image/diffraction pattern.

Characterization of nanostructured materials using SEM and HREM techniques.

A microstructural characterization based on analytical scanning electron microscopy (SEM) and high resolution transmission electron microscopy (HREM) was carried out on nanostructured M50-type steel and also on nanometer-sized gold particles. Both nanostructured materials were prepared with different chemical methods recently reported in the literature. The obtained nanostructured steel powders were subsequently consolidated into bulk samples. The SEM studies of the M50 compound probes show the presence of pores of different sizes. The composition of the specimens indicates small differences with the M50-type steel alloy, with strong variations of the vanadium amount in the cavities of the compound. The HREM images show the presence of small precipitates in the range of a few nanometers in size. The structural characteristics of the grain boundaries between the nanometric crystallites were also explored. The geometrical relationships between adjacent alpha-Fe grains were obtained for some particular boundary arrangements. The nanometric gold particles show diameters which vary from 4 to 11 nm. Some of these particles display twin boundary arrangements. The nature of these twin arrangements was also explored. Theoretical simulations based on the multislice theory of the electron diffraction dynamical theory were carried out mainly to explore the nature of the twin boundaries obtained in the gold particles. Comparisons between the simulated images and the experimental results are presented.

A physical optics approach to kinematical diffraction.

In this work the physical optics approach (Fourier Optics) that has been so successful in formulating the (multislice) high energy theory of electron diffraction is used to formulate the kinematical theory of electron diffraction. The resulting expressions already have the required phase factors and Ewald's sphere built in. A simple link to the weak phase object approximation and to the multislice theory is provided.

Accurate multislice theory for elastic electron scattering in transmission electron microscopy

Combining the concept of the slice-transmission-operator matrix with the transfer matrix technique, the complete Schrödinger equation for elastic electron scattering in solids is rigorously solved within the framework of the multislice scheme. This new approach extends the range in which the multislice theory is accurate. It is shown that the multislice theory can in principle be as accurate as the Bloch wave theory and the Green function theory in all cases of elastic electron scattering. It is pointed out that in some cases such as the calculation of high-order Laue zone (HOLZ) reflections, more accurate corrections should be made for the conventional high-energy approximation and therefore the conventional multislice method (CMS). A new multislice formula is developed for calculating the transmitted wave. This formula takes back-scattering effects into account and may therefore be accurate for large beam tilt. It is also shown that the new multislice approach keeps all the advantages of the CMS with respect to the Bloch wave method.

Towards quantitative simulations of inelastic electron diffraction patterns and images

A new dynamical theory has been developed based on Yoshioka's coupled equations for describing inelastic electron scattering in thin crystals. Compared to existing theories, the primary advantage of this theory is that the incoherent summation of the diffracted intensities contributed by electrons after exciting vast numbers of different excited states has been evaluated before any numerical calculation. An additional advantage is that the phase correlations of atomic vibrations are considered, so that full lattice dynamics can be combined in the phonon scattering calculation. The new theory has been proven to be equivalent to the inelastic multislice theory, and has been applied to calculate energy-filtered diffraction patterns and images formed by phonon, single electron and valence scattered electrons.A calculated diffraction pattern of elastic and phonon scattered electrons for a parallel incident beam case is in agreement with the one observed (Fig. 1), showing thermal diffuse scattering (TDS) streaks and Kikuchi pattern.

Theoretical and experimental studies of electron resonance effects in reflection high-energy electron diffraction

The electron resonance effect for 100 keV electrons incident on the surface of GaAs (110) in RHEED is studied by theoretical simulations using multislice theory and experimental observations. The exact resonance conditions, effective resonance region, effective penetration depth of electrons at or near the resonance condition and scattering processes involved for the resonance effect are investigated. It is found that the intensity of the 440 specularly reflected beam is mainly due to direct reflection from the surface atomic layer and beam enhancement due to surface channeling effects under the resonance conditions seems to be insignificant. For the 880 specularly reflected beam most of the electron intensity penetrates the surface and is diffracted by the crystal. The resonance condition for the 880 specular beam is satisfied when the transmitted beam excites a strong surface wave which propagates in the direction parallel or nearly parallel to the surface and is localized in the surface region by the surface potential barrier; double diffraction from the surface beam to the specular beam then enhances the total intensity in the specular beam. The exact resonance condition for the 880 beam is found to be at the glancing angle of 35.7 mrad and the azimuth angle of 29.7 mrad. A strong wave field is localized in the surface region at the resonance condition with an effective electron penetration depth of ~ 5 A, which increases to ~ 35 A on going to the nonresonance conditions. The effective resonance region for the 880 spot is ~ 2 mrad about the azimuth angle and ~1 mrad about the glancing angle.

Contribution of Atomic-Level TEM to Resolution of Structure

Atomic Resolution Electron Microscopes are now producing useful results in many fields of science and technology. This success was obtained not only by the improvement of resolution of TEM but also through the developments of theories and experiments of diffraction crystallography, image formation and recording technique over the past 44 years. Boersch (1946-47) discussed the image of atoms and crystals by the phase object approximation; the image contrast is due to the phase shift of electron waves passing through them. Scherzer (1949) discussed the effect of the phase shift by the electron lens and proposed the phase contrast transfer function. He pointed out that the key to observe images of single atoms is by contrast enhancement, which might be possible by dark field images if resolution is improved. This proposal was attempted by improving the resolution using tilted-beam dark fields (Th atoms, Hg & Pt atoms 1971). Crystal lattice fringes were observed by Menter in 1956 who discussed the contrast by kinematical theory. The interpretation of lattice images by dynamical theories were carried out by Hashimoto et al. (Bethe theory) and Cowley (Multi-slice theory) in 1958-60 and noted the correct position of atoms is in neither black fringe regions nor white ones when the images are taken in the Bragg reflecting position. However Miyake et al in 1964, showed that black or white contrast peaks appear at atom positions if the crystal is in symmetry position. Two dimensional lattice structure images were first photographed in 1970-71, which stimulated strongly the application of electron microscopy to materials science, crystallography and engineering at many laboratories such as the National Center at Berkeley. For the identification of correct position of atoms, two types of image contrast calculation have been proposed (e.g. multi-slice 1972, Bethe method 1975). The partially coherent theory originally developed in optics was introduced into the contrast calculation in 1979-80. Around this time, many observations of defect structures have been carried out, some of which are shown in Table 1. In situ observation of moving atoms and atom columns in molecules and crystals have been carried out in 1978 by using a TV system, which enable us to see the transition phenomena with a speed of 1/30 or 1/60 seconds. More recently atomic imaging at 1.8 A-1.6 A played an important role in the structure research of new materials and phenomena such as superconducting materials (metallic A15, 1989, ceramic high Tc, 1988), metal ceramic interface (Nb/Al2O3 1984), superstructure (GaAs/AlAs/GaAs 1985) quasi crystals (1987) etc. Many subsequent observations are presented in this congress. In spite of these developments, there are still some problems to be solved, e.g. imaging of atoms in the correct positions and the identification of different kinds of atoms in the materials especially with unknown crystal structure.

Atomic Thermal Vibration Phase Correlation in Electron Diffraction and Imaging

Consideration of phase correlation between localized inelastic events occurring at different atomic sites remains one of the major obstacles in the dynamic calculations of either the Bloch wave theory or the generalized multislice theory. The former is restricted by the 3-D periodic assumption of the transition matrix and the latter is limited by the coherent treatment of the inelastic excitation within the same slice. To approach this problem properly, one starts from the coupled Schrodinger equations,Ψ0 is the elastically scattered wave of energy E0, Ψn describes the inelastically scattered wave of the nth excited state of energy En = E0 - εn, and H'nm are the transition matrix elements. Taking Ψn = ϕn(z)Ψn0(r), where Ψn0(r) satisfies the elastic scattering Schrodinger equation of different wave vectors,and under the small angle approximation, Eq. (1) becomesEquation (2) can be solved with established elastic scattering theory, such as the multislice method.

Intensity distribution of electron diffraction pattern from Ti-14Wt%Mo alloy containing ωphase plate crystal

The study of ω-phase in various alloys such as Ti-Mo, Ti-Cr, Zr-Nb etc. have been carried out so far by electron diffraction and electron microscopy. However, the shape of ω-phase particles has not been fully clarified yet. This is due to the facts that ω-phase is formed in ω-phase crystal, and rather difficult to obtain isolated crystal particles and also that w-phase is produced only by the displacement of atoms in characteristic directions which makes no contrast in the electron microscope images projected along this directions and hence the crosssectional shape can not be studied. Since the intensity of electron diffraction patterns is very sensitive to the amount of ω-phase existing in β-phase, its thickness dependence has been studied in the present paper by using multi-slice dynamical theory of electron diffraction for the various thickness combination of the ω- and β-phases so that the thickness and location of ω-phase can be seen.

Dynamical inelastic scattering in high-energy electron diffraction and imaging: A new theoretical approach.

A generalized multislice theory is proposed from quantum mechanics to approach the multiple elastic and multiple inelastic scattering of high-energy electrons in a solid. The nonperiodic structure of crystals can be introduced in the calculations for the scattering geometries of transmission electron microscopy and reflection electron microscopy. Detailed applications of this generalized theory will be given for calculating (a) the energy-filtered-plasmon energy-loss diffraction patterns and images, (b) the energy-filtered diffraction patterns from atomic inner-shell losses, and (c) the contribution of thermal diffuse scattering to the high-angle annular-dark-field (ADF) scanning-transmission-electron-microscopy (STEM) lattice images. An ``incoherent'' imaging theory is presented for simulating the ADF STEM images and the detailed calculations are addressed for Ge/Si interfaces.

Electron virtual inelastic scattering in the multislice scheme

A multislice theory has been developed for including the virtual inelastic scattering in dynamical calculations of high-energy electron diffraction. The effects on elastic waves of all inelastic processes, such as single-electron excitation, plasmon excitation and phonon scattering, can be characterized by a complex correction potential. Its real part describes the virtual inelastic process and its imaginary part represents the inelastic absorption effect. This potential is directly related to the generalized dielectric response function of the crystal.

Simulating high-angle annular dark-field stem images including inelastic thermal diffuse scattering

A generalized multislice theory is employed to simulate the annular dark-field (ADF) images, in scanning transmission electron microscopy (STEM), taking into account the inelastic thermal diffuse scattering (TDS). It is shown that the ADF image has the following characteristics if the first-order Laue zone (FOLZ) (or high-order Laue zones (HOLZ)) is excluded: (1) there is no contrast reversal with change of focus; (2) there is no contrast reversal with change of specimen thickness and (3) the ADF image shows highly localized scattering from each atomic column and can provide some chemical information for a specimen having two or more atoms with very different atomic numbers. It is shown that the TDS may take a dominant role in determining the ADF imaging contrast, depending strongly on the atomic number, the sizes (or shapes) of the atoms and their relative vibration amplitudes. TDS may increase the visibility of light elements. A full dynamical calculation including TDS is required to interpret the ADF imaging contrast. It is therefore not proper to consider the ADF imaging contrast as purely Z dependent. The HOLZ can introduce strong diffraction contrast in the ADF image.