Physics-informed Bayesian Optimization for Quantitative High-Resolution Transmission Electron Microscopy

Abstract

Quantitative high-resolution transmission electron microscopy (HRTEM) provides an indispensable means to understand the structure-property relationships of a material in atomic dimensions. Successful quantification requires reliable retrieval of essential atomic structural information despite artifacts arising from unwanted but practically unavoidable imaging imperfections. Experimental observation carried out in tandem with model-based iterative image simulation shows vast applications in quantitative structural and chemical determination of objects spanning zero to three dimensions [Prog. Mater. Sci. 133, 101037, 2023]. However, the large number of parameters involved in the simulations make the current multi-step, user-guided iterative approach highly time consuming, thereby restricting its application primarily to small sample areas and to experienced users. In this work, we implement and apply a physics-informed Bayesian optimization (BO) framework to advance HRTEM quantification towards full automation and large-field-of-view analysis. Unlike conventional optimization approaches, our method adopts a stepwise strategy that fully leverages the strength of BO in handling high-dimensional parameters, while its probabilistic engine rigorously and efficiently refines the parameter space to enable rapid quantification. Using a BaTiO3 single crystal that contains heavy, medium and light elements as a model system, we demonstrate that the three-dimensional crystal structure can be determined from a single HRTEM image with a three to four order-of-magnitude improvement in time efficiency. This approach thus opens new avenues for fast and automated image quantification over larger sample volumes and, potentially, in the time domain.

Figures (5)

• Figure 1: Conventional and proposed Bayesian optimization boosted workflow for quantitative HRTEM. (a) Conventional workflow based on user-supervised, multi-step iteration. The experimental image is set as the input. The unknown parameters are initialized in the first iteration and iteratively updated until the best fit between simulation and experiment is achieved. (b) Implemented BO workflow replacing the user-supervised procedure in (a). In BO, a batch of input parameter vectors $\mathbf{X} \in \mathbb{R}^n$ is initialized for subsequent optimization. Respective difference images between experiment and simulation are used to train a surrogate model and to dynamically adjust the TuR in $\mathbb{R}^n$, within which a new search is conducted using a batch of parameter vectors proposed by the acquisition function. Continuous relaxation is introduced to handle both continuous and discrete parameters and physical constraints are also enforced before generating new candidate $\mathbf{X}$.
• Figure 2: Determination of imaging parameters based on averaged experimental image. (a) Normalized experimental image of BTO and (b) averaged image from ROI2, ROI6 and ROI10 in (a) used as input (i.e., Exp) for global imaging parameter fitting. Inset to (a) shows a cubic BTO model projected along the [110] direction. The model size outlined by the black frame corresponds to a single ROI. (c) The best-fit simulated image (i.e., Sim) and (d) difference image between Exp and Sim after image parameter fitting. A unity value is added to the mean intensity of (d) to allow all images to be displayed on the same absolute intensity scale (see greyscale bar alongside (c)). Scale bars: 0.2 nm in (a) and 0.1 nm in (b)-(d). (e) Minimum $\mathcal{L}_\mathrm{MSE}$ value at each iteration plotted as a function of simulation time, showing a stable convergence after 75 s for imaging parameter fitting. The inset shows that a sample thickness of 2.835 nm (i.e., 20 slices) is applied. (f) Minimum $\mathcal{L}_\mathrm{MSE}$ value plotted as a function of time for subsequent structure parameter fitting. Arrows in the inset denote atomic column shifts after structure parameter optimization. Scale bar: 5 pm. Red dashed lines mark $\mathcal{L}_\mathrm{MSE}$ measured from vacuum. The time preceding the first data point corresponds to initialization and scales linearly with the number of optimized parameters.
• Figure 3: Parallel matching for 3D structural determination of ROIs. Introduced parameters $N$ (i.e., the number of atoms for each type in each atomic column) to optimize the 3D atomic model of each ROI. Individual vacancies inside a column (small circle) are not treated explicitly, and isolated surface atoms (large circle) are considered energetically unfavorable. (b) Plots of $L_{\rm MSE}$ as a function of iteration time for all ROIs, evidencing a steady convergence after about 300 s. (c)-(e) Experimental images, best-matching simulations, and corresponding difference images for all ROIs, respectively, displayed at the same greyscale level. (f) Determined 3D atomic models for all ROIs, showing the distinct increase of sample thickness from top to bottom (wedge shape), resulting from TEM sample preparation.
• Figure 4: Determination of the global $C1$ and relative position relationship between ROIs. (a) and (b) Simulated images for two BTO supercell models under identical simulation parameters. The model in (b) consists of the same crystalline part of (a) plus a vacuum with thickness of 0.567 nm, equivalent to 4 slices of BTO. (c) Difference image between (a) and (b) showing a negligible contribution of added vacuum at the electron entrance plane. (d) Simulated image for a BTO model, now with a vacuum added below the sample exit plane, while other parameters are maintained the same. A change in the image contrast (std) is detected. (e) Simulated image for the same BTO model as in (a), but with an addition to $C1$ of 0.567 nm, compensating the vacuum volume added below the sample in (d). (f) Difference image between (d) and (e) showing their equivalence. (g) and (h) illustrate how structure models of separately evaluated patches can be combined into one model by interpreting local defocus differences as relative z-position differences with respect to a common reference plane. Scale bar: 0.1 nm.
• Figure 5: Simulation using the integrated 3D atomic structure. (a) Integrated supercell model constructed from all the individual models. Colored circles represent the number of atoms in atomic columns. Large: Ba; Medium: Ti; Small: O. (b) Reproduction of the experimentally recorded image shown in Fig. \ref{['fig: imaging parameters']}(a) and (c) the simulated image using the integrated 3D atomic model for the entire area. The $NCC_{\rm P}$ between (b) and (c) is 96.3%. (d) and (e) Original and mean-intensity-adjusted difference images between (b) and (c), showing the residual image contrast. Scale bars: 0.2 nm. All images are displayed at the same grey-scale level.