A hydrodynamic approach to electron beam imaging using a Bloch wave representation
Introduction
The wave-particle duality of electrons causes a segregation between the types of simulation techniques employed in the field of electron microscopy. Simulations are categorized as either image simulations, where the probability distribution of the electron wave function is used to obtain contour maps relating to the exit plane or diffraction pattern, or particle scattering simulations, using classical methods to obtain intensities associated mostly to X-ray emission events and particle penetration [1]. Image simulations use techniques such as multislice [2], [3], [4] or Bloch wave [5], [6], [7], [8] which simulate the probability density in real or reciprocal space of the electron upon exit of the material. Conversely, scattered particle trajectories are typically simulated through Monte Carlo techniques where electrons are assumed to be classical spheres that undergo a forward scattering random walk process, where the scattering and energy loss parameters are calculated by physical models [9], [10]. As of yet, there is no technique which simultaneously simulates both the wave and particle characteristics of electrons within the confines of an electron microscope.
Here, we couple a Bloch wave representation of the electron wave function with the propagation of quantum trajectories to simulate electron-matter interactions inside a crystalline material under various probing conditions. The quantum trajectory method arises from the hydrodynamic formulation of a quantum process [11], [12]. Given an initial position, the particle will follow a specific path dictated by the wave function. The uncertainty then comes in the choice of the initial position [13]. Nonlocality is preserved because each point along the trajectory is determined by the entire wave function in configuration space. By virtue of the nonlocality of the standard quantum theory, wherein the state of the system at one point in time affects the state at each subsequent point, trajectories computed using the same quantum mechanical structure demonstrate the same property of nonlocality [13], [14]. Such simulations have typically been performed for particle diffraction experiments and small scale quantum processes [15], [16], [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 a low voltage scanning transmission electron microscope (STEM) [18]. 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 [18]. The algorithm developed in this study constitutes a completely new and different approach. The trajectories are no longer calculated using a spectral decomposition and the split-operator method is substituted for the Bloch wave method. Other work has been done using a multislice approach to simulate a STEM probe located at different positions along a unit cell [19], [20]. However, there was little analysis done of the beam interaction with the material, specifically at different probing conditions [19]. The use of the multislice method also limits usability of the calculation because the trajectories may only be computed within the confines of the chosen grid. An important advantage of the Bloch wave method is the possibility of computing the wave function at any point in space without being restricted to a structured grid. Furthermore, computations may be performed at lower accelerating voltages, making the method applicable for wave function simulations at energies typically used in a low voltage STEM [6], [21]. While Bloch wave calculations coupled with quantum trajectories have been previously investigated by Cheng et al. [22], criteria such as the number of beams and the initial wave function used in the computational method were not indicated, making it difficult to reproduce their findings. There was also no explanation of the calculation process used in the electron backscattered diffraction (EBSD) simulations and consequently other, more explicit and reproducible studies, are necessary.
In this study, the electron wave function along the particle path was computed using the Bloch wave expression to simulate electrons traveling through a single crystal of Cu. Simulations were performed at 200 and 30 keV to distinguish the difference in electron transport in conventional TEM versus low voltage STEM-like conditions. The initial positions of the trajectories were chosen uniformly to map out the wave function over the entire unit cell. Two probing conditions were investigated, the (100) zone axis orientation and the two beam Bragg condition. The quantum potential and quantum force are also computed at the exit plane. These values represent the quantum effects which cannot be explained through classical particle propagation [11]. As a result, these parameters show how two beam diffraction theory can be explained in real-space via quantum trajectories. It is also shown through the quantum force that, at normal incidence, electrons are drawn to and from the atom columns, resulting in their channeling through the material. The quantum force about the atom columns then acts as an attractor. The method allows the evaluation of electron-matter interactions while retaining the quantum nature of the process.
Section snippets
Method
A Bloch wave expression was used to compute the electron wave function along the trajectory paths. The wave function, Ψ(r), of an electron in a periodic potential can be expressed as a sum of Bloch waves weighted by excitation coefficients for each Bloch wave j [23],The coefficients and contributions γ(j) of each Bloch wave are obtained by solving the eigenvalue equation,where k0 is the incident
Normal incidence
Simulations were first performed at normal incidence in the (100) zone axis orientation for copper. For an incident energy of 200 keV, and which resulted in a computation of 45 strong beams and 8 weak beams. For an entire 3D analysis of the wave function propagation, trajectories were first positioned equidistant from each other on a 10 × 10 grid spanning the cross-sectional area of a single atom column. Then, to generate a 2D projection of the system, 50 trajectories along a line
Conclusions
The quantum trajectory method was coupled with a Bloch wave calculation of the transmitted wave function of an electron beam through a thin copper foil. Simulations were performed in the zone axis case and the two beam condition. It was shown that quantum trajectories can provide useful insight into electron-matter interactions by displaying where the particles may pass as they are transmitted through an imaged material. In the zone axis orientation, it is shown that the electrons are channeled
Declaration of Competing Interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
Acknowledgments
The authors would like to thank Marc DeGraef and Scott Findlay for their insightful and helpful discussions. Financial support was provided by the Alexander Graham Bell Graduate Scholarship from the Natural Sciences and Engineering Research Council (NSERC) of Canada.
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