On the role of the second-order derivative term in the calculation of convergent beam diffraction patterns.

The simulation of (scanning) transmission electron microscopy images and diffraction patterns is most often performed using the forward-scattering approximation where the second-order derivative term in z is assumed to be small with respect to the first-order derivative term in the modified Schrödinger equation. This assumption is very good at high incident electron energies, but breaks down at low energies. In order to study the differences between first- and second-order methods, convergent beam electron diffraction patterns were simulated for silicon at the [111] zone-axis orientation at 20 keV and compared using electron intensity difference maps and integrated intensity profiles. The geometrical differences in the calculated diffraction patterns could be explained by an Ewald surface analysis. Furthermore, it was found that solutions based on the second-order derivative equation contained small amplitude oscillations that need to be resolved in order to ensure numerical integration stability. This required the use of very small integration steps resulting in significantly increased computation time compared to the first-order differential equation solution. Lastly, the efficiency of the numerical integration technique is discussed.

A study in the computation time required for the inclusion of strain field effects in Bloch-wave simulations of TEM diffraction contrast images.

As transmission electron microscopy (TEM) imaging techniques continue to become more quantitative, interpretation of the experimental images demands that accurate image simulations be computed incorporating all important aspects of the image including: compositional, crystallographic and microscope effects, as well as contrast due to strain fields arising from stresses created by lattice misfit or defects. Incorporation of the effects of strain fields in the simulation of diffraction-contrast TEM images in the Bloch-wave formalism requires the integration of a system of first-order differential equations in order to modify the excitation amplitudes and produce contrast in the image. This integration is computationally demanding with the time for integration scaling as the cube of the number of beams included in the calculation. In order to investigate the computational requirements of the integration, a variety of numerical integration packages were evaluated with respect to timing and accuracy in the simulation of quantum dot, spherical inclusion and screw dislocation images. It was determined that a class of Adams-multistep methods can provide a decrease in computation time ranging from 2 to 4 as compared to the standard Runge-Kutta 4(5) approach depending on the simulation conditions.