Eigenstructure Analysis of Bloch Wave and Multislice Formulations for Dynamical Scattering in Transmission Electron Microscopy

TL;DR

This work establishes a matrix-based bridge between Bloch-wave and multislice dynamical scattering models in TEM by introducing a transmission matrix $\mathbf{A}$ that decouples the input wave from the crystal. It proves that the eigenstructure of the multislice transmission matrix $\hat{\mathbf{S}}$ and the Bloch-wave scattering matrix $\mathbf{S}$ are equivalent under a 2D Fourier relation of eigenvectors and a modulo $2\pi$ alignment of eigenvalue phases, and it demonstrates these ideas on GaAs, SrTiO$_3$, and Au, with validation against Bragg patterns and projected potentials. Furthermore, the authors show that the mean inner potential (MIP) can be estimated from diffraction data either via the determinant of $\hat{\mathbf{S}}$ or by phase-unwrapping the slice potentials, achieving good agreement with DFT and experimental values, including an MoS$_2$ 4D-STEM case. The framework yields insights into the dynamical scattering process, offers a computationally efficient route relative to the full scattering-matrix approach, and provides practical tools for quantitative structure retrieval from TEM diffraction data.

Abstract

We investigate the eigenstructure of matrix formulations used for modeling scattering processes within materials in transmission electron microscopy. Dynamical scattering is crucial for describing the interaction between an electron wave and the material under investigation. Unlike the Bloch wave formulation, which defines the transmission function via the scattering matrix, the traditional multislice method is lacking a pure transmission function due to the entanglement of electron waves with the propagation function. To address this, we reformulate the multislice method into a matrix framework, which we refer to as the transmission matrix. This allows a direct comparison to the scattering matrix derived from Bloch waves in terms of their eigenstructures. Through theory, we demonstrate their equivalence with eigenvectors related by a two-dimensional Fourier matrix, given that the eigenvalue angles differ by modulo $2πn$ (integer $n$). We numerically verify our findings as well as demonstrate the application of the eigenstructure for the estimation of the mean inner potential.

Figures (11)

• Figure 1: Comparison of the Bloch wave and the multislice approach. a, visualization of a SrTiO$_3$ crystal. b, projected potential representation of the SrTiO$_3$ crystal in a used in the multislice method. c, the construction of the respective scattering matrix $\mathbf{S}$ and, d, transmission matrix $\mathbf{\hat{S}}$. e, f, diffraction patterns generated from matrices $\mathbf{S}$ and $\mathbf{\hat{S}}$ using plane wave illumination are similar, yet do not explicitly characterize the differences in eigenstructure of matrices in c and d.
• Figure 2: Frobenius norm as a function of crystal thickness, calculated using the sorted index of eigenvalues and eigenvectors. a, Frobenius norm between eigenvalues $\Sigma$ (transmission matrix) and $\Lambda$ (scattering matrix). b, Frobenius norm between eigenvectors $\mathbf W$ (transmission matrix) and $\mathbf C$ (scattering matrix). c, Frobenius norm between transmission matrix $\mathbf{\hat{S}}$ and scattering matrix $\mathbf S$. Results for GaAs, SrTiO$_3$ and Au are plotted.
• Figure 3: The alignment of the eigenvalue angles. Since the total of eigenvalues are the same for both transmission and scattering matrices, both axis represent the value of eigenvalue angles in radian. The x- and y-coordinates are the eigenvalue angles of the scattering and transmission matrix, respectively. a, GaAs with a thickness of $169.6$ Å and a standard deviation (SD) of $0.037$ rad. b, SrTiO$_3$ with a thickness of $78.1$ Å and a SD $=0.034$ rad. c, Au with a thickness of $102$ Å and a SD $=0.035$ rad.
• Figure 4: Plane wave diffraction patterns along the [001] zone axis calculated using the transmission matrix (top row) and the scattering matrix (bottom row). a, d, GaAs with a simulated thickness of $175.2523$ Å. b, e, SrTiO$_3$ with a simulated thickness of $126.48$ Å. c, f, Au with a simulated thickness of $121.055$ Å. The amplitude of the diffraction spots is given in arbitrary units. Crystalline materials are simulated with $2\times 2$ unit cells with the parameters in Table \ref{['tab:params_multislice']}.
• Figure 5: Bragg beam amplitude as a function of specimen thickness, calculated using the transmission matrix (top row) and the scattering matrix (bottom row). a, d, GaAs. b, e, SrTiO$_3$. c, f Au.