Electron diffraction contrast of small coherent particles. I. Ellipsoidal inclusion in an isotropic matrix

Abstract

The computer simulation technique is applied t o the electron diffraction contrast of inclusions. A new formulation of t h e Bloch wave theory of diffraction contrast in the twobeam case and a cubic interpolation scheme are used t o diminish the calculation time. Bright and dark field images of coherent ellipsoid-shaped particles in an elastically isotropic crystal have been calculated and printed using exact elastic distortion fields. Taking into account only the dipole part of the elastic fields has not great influence on the images. Contrast details are discussed on the basis of perturbation expansions for the beam intensities.Die Methode der Computersimulation des Beugungskontrasts wird auf Einschliisse angewendet. Um Rechenzeit zu sparen, werden eine neue Formulierung der Blochwellengleichungen fur den Beugungskontrast im Zweistrahlfall und ein kubisches Interpolationsverfahren eingefiihrt. Theoretische Hell-und Dunkelfeldbilder von kohkrenten ellipsoidformigen Einschliissen in einem elastisch isotropen Kristall werden angegeben, die mit Hilfe des exakt berechneten elastischen Distortionsfeldes in der Umgebung der Teilchen gewonnen wurden. Es zeigt sich, daB der Beugungskontrast im wesentlichen durch den Dipolanteil der elastischen Felder bestimmt wird. Kontrasteinzelheiten werden unter Verwendung storungstheoretischer Ausdriicke fur die Strahlintensitaten diskutiert.

Characterization of Structure and Composition of Quantum Dots by Transmission Electron Microscopy

“…For R o >~/2 (curve iii, upper bound [72]) a black-white contrast without oscillations, also called black lobe or coffee-bean contrast is obtained (see also Fig. 3.2).…”

“…Here, better results may be revealed using the diffraction contrast analysis of non-spherical inclusions [77], cubes or ellipsoids in anisotropic media [72], or applying finite element methods (cf., e.g., [78,40,44] but restricted to conical islands and neglecting the modified BF) to evaluate the strain fields. The calculations in [25] reveal the contrast inversion with increasing thickness and the sensitivity to imaging conditions as discussed above.…”

“…Dots are invisible for very small particle diameters 2R o or for small strains E. Using a 10% contrast rule with respect to the background and exact two-beam orientation, the visibility limit is approximately gER6/e = 0.1 (curve i). A more complicated rule exists for large particles (curve iv), which are invisible due to decreasing strains [72,73]. However, very large particles may be considered as separate phases, and the contrast is determined by the effects neglected above: interface, Moire or structure factor contrast.…”

“…The structure factor contrast is in a first order approximation equivalent to an effective thickness change of Ro(l/~o -l/~m), which is a negligible effect for typical QD radii of about Ro = 15 nm (the extinction distances of matrix~m and dot~o are about 16 nm-30 nm for GaAs in the twobeam approximation for (111) and (200) reflections). Following the discussion of Chick et al [71], Lepski [72], and others (see, e.g., [73] and references therein), the strain contrast in a two-beam excitation may be approximated analytically in first order perturbation and by using the Ashby-Brown [74] displacement field of a spherical inclusion in isotropic media. The different investigations may be summarized to a diagram as given in Fig.…”

On the computer simulation of the electron microscopic diffraction contrast of dislocations in jog and dipole configurations

“…For saving time and to simplify the basic relations of the two-beam case suitable transformation relations are used. For the Bloch wave formulation such a transformation was given by Lepski [9], but also in the case of the plane wave formulation of Howie-Whelan suitable transformation relations can be used to reduce the number of necessary differential equations [ 101.…”