Modeling dynamical scattering using Bethe potentials
Our research focuses on determing the optimal criterion for using the so-called Bethe potentials to reduce the matrix size in electron dynamical scattering calculations and thus the computational time.
Electron micrographs display a rich variety of contrast features such as thickness change, bending and defects in the materials. It is important to understand these features both qualitatively and quantitatively. Therefore, we need to be able to model the electron dynamical scattering process, which involves in solving the Schrodinger equation. The electron wave function can be predicted by the Schrodinger equation as a function of the electrostatic lattice potential, sample thickness and the incident electron beam (direction and energy). There are two main approaches to solve the wave function: Bloch wave approach and scattering matrix approach. The former approach is a matrix factorization approach whereas the latter one is a matrix multiplication process. Our study focuses mainly on using the scattering matrix approach since it is easier to implement for defect imaging simulations and takes less computational time.
In general, many electron beams are involved in the dynamical scattering process. Thus, we need to take into account many diffracted beams in the simulations. This results in large sizes of matrices in calculation which may take a long computational time (size of matrix is the same as the number of beams in calculation). Bethe, in 1928, introduced the so-called Bethe potentials to reduce the size of the dynamical matrix in electron scattering problems. In the top figure on the right, diffracted beams in a crystalline diffraction pattern is shown schematically separated into three regions, namely strong, weak and very weak beams, according to their contributions to imaging. The contribution of each beam can be determined by its excitation error (distance from the diffraction spot to the Ewald sphere) and the potential interaction with other beams. Bethe incorporated the contributions from the weak beams into the strong beams using first order perturbation theory and we can simply ignored the very weak beams. Thus, the dynamical matrix is the size of the number of strong beams. An example of Cu (111) systemmatic row case at 200keV is presented. In the middle figure, intensity profile of the central beam as a function of thickness is shown. And in the bottom figure, intensity profile of the (111) diffracted beam is shown. The black curve is computed from all 17 beams given the same weight whereas the red-dot curve is computed from taking only the 8 strongest beams into account incorporating with 4 weak beams. It is clear that the red-dot curve matches the black curve and the computational time is much lesser.
We are applying this approach to various crystalline structures and a wide range of microscope accelerating voltage to determine the optimal criterion for separating the strong and weak beams. This approach can benefit image simulations from different techniques, such as electron back scatter diffraction, precession electron diffraction and so on. Also, we will apply this approach to several types of defect imaging.