Simulation study of the performance of neural network-enhanced PACBED for characterizing atomic-scale deformations in 2D van der Waals materials

Two dimensional (2D) van der Waals (vdW) materials have attractive mechanical, electronic, optical, and catalytic properties that are highly tunable especially when they are thin. However, they are rarely perfect and flat, and their properties are strongly influenced by local crystal lattice deformations that include the 2D strain tensor, in-plane rotation and corrugation, where the latter is manifested as local sample tilt. Therefore, to gain more control over their properties, a detailed understanding of these deformations is needed. Position averaged convergent beam electron diffraction (PACBED) is a powerful technique for providing information about local atomic structure. In this work, we perform a comprehensive simulation study of the performance of PACBED in combination with convolutional neural networks (CNNs) for prediction of deformations of 2D materials. We generate around 100,000 simulated PACBED patterns from 2H MoS${}_2$ for thicknesses from 1 to 20 atomic layers where strain, rotation, and tilt parameters are varied. Five convergence angles are explored which vary from conventional nano beam electron diffraction (6.35 mrad) to atomic resolution conditions (32.94 mrad). From this simulated PACBED library, we train regression CNNs to simultaneously predict the 2D strain tensor, in-plane rotation, and tilt of the sample. For different convergence angles and thicknesses, we study the prediction performance for each of the deformation parameters. We find that there is a trade-off between better prediction performance (small convergence angles) and probe size (large convergence angles). For smaller convergence angles like those used for conventional NBED conditions, the strain prediction error can be as low as 0.0003 %, while for larger convergence angles like those used for atomic resolution probes, the strain error increases to 0.001 - 0.003 %. The impressive prediction performance even for large convergence angles suggests that PACBED combined with CNNs is a feasible method for predicting deformation parameters using atomic resolution electron probes. Further, we conclude that the prediction can be difficult for monolayers, and suggest two remedies: excluding tilt from the predictions and performing nonlinear intensity rescaling of the training data. This work contributes to the optimal design of PACBED experiments for characterization of local crystal deformations and, therefore, to an improved understanding of how 2D vdW materials respond to imperfections.