Using Bethe Potentials in the Scattering Matrix for Defect Image Simulations

Bethe potentials were introduced by Bethe in 1928 to reduce the size of the dynamical matrix in electron scattering problems using first order perturbation theory [1]. The standard approach is to start from the dynamical equations in the Bloch wave representation, and split the set of diffracted beams into two subsets, namely strong and weak beams, according to some appropriate criterion. While there is some early work on the application of the perturbation approach to the formulation of the dynamical scattering problem via the scattering matrix formalism [2], Bethe potentials have not been used extensively for simulation approaches other than the Bloch wave approach. For defect image contrast simulations, the Bloch wave approach is somewhat tedious, and, traditionally, one has resorted to solving the dynamical equations in differential form, while replacing the standard excitation error by a position dependent, effective excitation error. This works well for defects with continuous displacement fields, such as dislocations and semi-coherent inclusions, but for planar defects, with a discontinuous displacement field, one must include their effect as a separate phase shift of the potential Fourier coefficients. A more coherent approach would be to describe all defect displacements in terms of phase shifted Fourier coefficients. In this contribution, we show that by using Bethe potentials in combination with the scattering matrix, one can reduce the size of the dynamical matrix and hence the computation time required for defect image simulations.