Dynamical Electron Diffraction Calculations Research Articles

Analysis of Kikuchi band contrast reversal in electron backscatter diffraction patterns of silicon

We analyze the contrast reversal of Kikuchi bands that can be seen in electron backscatter diffraction (EBSD) patterns under specific experimental conditions. The observed effect can be reproduced using dynamical electron diffraction calculations. Two crucial contributions are identified to be at work: First, the incident beam creates a depth distribution of incoherently backscattered electrons which depends on the incidence angle of the beam. Second, the localized inelastic scattering in the outgoing path leads to pronounced anomalous absorption effects for electrons at grazing emission angles, as these electrons have to go through the largest amount of material. We use simple model depth distributions to account for the incident beam effect, and we assume an exit angle dependent effective crystal thickness in the dynamical electron diffraction calculations. Very good agreement is obtained with experimental observations for silicon at 20 keV primary beam energy.

Microstructure of ZrO 2–CeO 2 hetero-multi-layer films grown on YSZ substrate

Multi-layer films of pure ZrO 2 and CeO 2 were grown using oxygen plasma assisted molecular beam epitaxy on yttria stabilized zirconia substrates. The microstructure of the film was analyzed using X-ray diffraction, conventional and high-resolution transmission electron microscopy, electron energy-loss spectroscopy, energy dispersive X-ray elemental mapping, selected area electron diffraction, and dynamical electron diffraction calculations. The deposited pure CeO 2 layers exist in the cubic fluorite structure, and the ZrO 2 layers show a good epitaxial orientation with respect to the CeO 2 layers. However, distinctive forbidden diffraction spots of (odd, odd, even) type were observed on the selected area electron diffraction patterns of the film. Dark-field imaging clearly reveals that these forbidden diffraction spots were contributed solely by the ZrO 2 layers. Dynamical electron diffraction calculation based on the tetragonal phase of unity tetragonality (space group P4 2/ nmc) with oxygen displaced along the c-axis does not match with the experimental pattern. Instead, a diffraction pattern calculated based on a cubic structure (space group P 4 ¯ 3 m ) for which the oxygen sub-lattice was displaced along the 〈1 1 1〉 matches with the experimental data. It is further suggested that the displacement of the oxygen from the ideal (1/4, 1/4, 1/4) position was introduced during the film growth process.

Polarity Determination of Wurtzite and Zincblende Structures by TEM

The applicability of transmission electron microscopy (TEM) to determine polarity was examined for GaN, ZnO, AlN and GaAs. The adopted methods are the TEM-EDS (energy dispersive X-ray spectroscopy) and the conventional convergent beam electron diffraction (CBED) methods. TEM observations and dynamical electron diffraction calculations were carried out. The results show that the CBED method is useful for ZnO and GaN, while the TEM-EDS method must be used for AlN and GaAs and can also be used for GaN and ZnO. It is clearly indicated that the TEM-EDS method is of use for the case where the difference in atomic scattering factor between the two constituent elements is small, while the CBED method is useful for the case where the difference is large.

On the phase-object function, used in dynamical electron diffraction calculations

A systematic study is presented of methods employed to calculate a phase-object function which can be used in HRTEM image calculations (multislice and real space). The influence of errors introduced by the discrete sampling of the crystal potential and the electron beam wavefunction is examined. Several possible ways of suppressing these errors are quantitatively tested. Using an anti-aliased phase-object function in a dynamical calculation improves the accuracy considerably for a given number of calculation points. The calculation of an anti-aliased phase-object function is very time-consuming, which will become a problem when there is a structural variation in the direction of the electron beam, and the calculation has to be done for each slice. Two new methods are presented for calculating anti-aliased phase-object functions. One, which applies the anti-aliasing a priori to the individual atoms, is quite generally applicable but some accuracy is lost. The other, which applies anti-aliasing a priori to rectangular building blocks, is not so general in application but is very fast and accurate.

The real space method for dynamical electron diffraction calculations in high resolution electron microscopy: III. A computational algorithm for the electron propagation with its practical applications

The real space description of the high energy electron diffraction as used for the simulation of high resolution electron microscope images is analysed with respect to its numerical performance. Recent arguments reported in the literature, which question the practical utility of the real space method, are refuted. It is shown that these arguments do not concern the idea of real space calculations in se but are caused by an inappropriate approximation for the real space electron propagator. In this work a new and proper algorithm for the propagator is introduced by which the method becomes numerically equivalent and competitive in calculation speed with the multislice method. Furthermore, a real space “patching” method is devised, by which the dynamical electron diffraction in extended areas such as for crystal defects can be calculated in patches, hereby avoiding artificial periodic continuation and removing the limitations imposed by the computer memory.

The real space method for dynamical electron diffraction calculations in high resolution electron microscopy: I. Principles of the method

The real space method for dynamical electron diffraction calculations has been established on a more rigorous basis than in the original paper of 1980. First, the validity of the high energy approximation for deriving the high energy differential equation is critically examined. Then, the second-order slice expansion, which is the basis of the real space method, is compared with other existing slice methods, and it is shown why the originally proposed algorithm can lead to computational divergencies and how they can be avoided. Then the real space method is applied to calculate the wavefunction in Au, to show that, in the limit of small slice thickness and sampling distance, even for heavily scattering material, the results are identical with those obtained from the multislice and the iterative methods. Finally, it is shown that in the case of light scattering materials such as SiF 4 and foil thicknesses of the order of 10 to 20 nm, the accuracy obtained with the real space method can be several times better than that of the first-order multislice method, using the same slice thickness.

Observations on the real space computation of dynamical electron diffraction intensities

SUMMARYThe computational procedures and theoretical approximations used in the real space approach to dynamical electron diffraction calculations are discussed. Although the method is theoretically sound it is found to be unsuitable for rapid and accurate computation of electron wavefunctions.