A novel quantum dynamical approach in electron microscopy combining wave-packet propagation with Bohmian trajectories

Abstract

The numerical analysis of the diffraction features rendered by transmission electron microscopy (TEM) typically relies either on classical approximations (Monte Carlo simulations) or quantum paraxial tomography (the multislice method and any of its variants). Although numerically advantageous (relatively simple implementations and low computational costs), they involve important approximations and thus their range of applicability is limited. To overcome such limitations, an alternative, more general approach is proposed, based on an optimal combination of wave-packet propagation with the on-the-fly computation of associated Bohmian trajectories. For the sake of clarity, but without loss of generality, the approach is used to analyze the diffraction of an electron beam by a thin aluminum slab as a function of three different incidence (work) conditions which are of interest in electron microscopy: the probe width, the tilting angle, and the beam energy. Specifically, it is shown that, because there is a dependence on particular thresholds of the beam energy, this approach provides a clear description of the diffraction process at any energy, revealing at the same time any diversion of the beam inside the material towards directions that cannot be accounted for by other conventional methods, which is of much interest when dealing with relatively low energies and/or relatively large tilting angles.

A hydrodynamic approach to electron beam imaging using a Bloch wave representation

“…This is in contrast to the parametrization of the real space potential utilised in [17]. Here, only the Fourier coefficients of the potential are required.…”

“…Once the real space wave function is computed, it can be used to calculate trajectories of the associated quantum particle. The quantum trajectory method, described in previous work [17], provides a visual representation of electron-matter interactions as the beam is propagated through a material. In this formulation, the polar form of the wave function is used to solve the Schrödinger equation resulting in a continuity equation and a quantum form of the Hamilton-Jacobi equation (QHJ) [11].…”

“…Such simulations have mostly been performed for particle diffraction experiments and small scale quantum processes [14][15][16]. In previous work, the method was applied in 2D to simulate the time-dependent propagation of a Gaussian wave-packet under conditions similar to those of an SEM [17]. It was found however that there were a number of limiting numerical factors, such as the energy bandwidth, grid size, and film thickness, which restricted the applicability of a time-dependent propagation scheme [17].…”

“…In previous work, the method was applied in 2D to simulate the time-dependent propagation of a Gaussian wave-packet under conditions similar to those of an SEM [17]. It was found however that there were a number of limiting numerical factors, such as the energy bandwidth, grid size, and film thickness, which restricted the applicability of a time-dependent propagation scheme [17]. The algorithm developed in this study constitutes a completely new and different approach.…”

Wave-packet numerical investigation of thermal diffuse scattering: A time-dependent quantum approach to electron diffraction simulations

“…In our case, the potential simulates a thin Al(100) film in two dimensions (2D). Once the potential model is set, the numerical simulation of the electron transmission has been carried out according to the method described in earlier work for fully coherent scattering [19]. The vibration amplitudes investigated and the corresponding material temperatures are displayed in Table I.…”