Helium focused ion beam damage in silicon: Physics-informed neural network modeling of helium bubble nucleation and early growth

Currently, the time and cost required to obtain large datasets limit the application of data-driven machine learning in nanoscale manufacturing. Here, we focus on predicting the nanoscale damage induced by helium focused ion beams (He-FIBs) on silicon substrates. We briefly review the most relevant atomistic defects and the partial differential equations (PDEs), or rate equations, that describe the mutual creation and annihilation of the defects, eventually leading to the amorphization of the substrate and, the nucleation and early growth of helium bubbles. The novelty comes from the use of a physics-informed neural network (PINN) to simulate quantitatively the evolution of the bubbles, thus bypassing the dataset availability problem. As usual, the proposed PINN learns the underlying physics through the incorporation of the residuals of the PDEs and corresponding Initial Conditions (ICs) and Boundary Conditions (BCs) in the network’s loss function. Meanwhile, the system of PDEs poses some challenges to the PINN modeling strategy. We find that (i) hard constraints need to be imposed on the network output in order to satisfy both BCs and ICs, (ii) all the inputs and outputs of the PINN need to be cautiously normalized to ensure convergence during training, and (iii) customized weights need to be carefully applied to all the PDE loss terms in order to balance their contributions, thus improving the accuracy of the PINN predictions. Once trained, the network achieves good prediction accuracy over the entire space-time domain for various ion beam energies and doses. Comparisons are provided against previous experiments and traditional numerical simulations, which are also implemented in this study using the Finite Difference Method (FDM). While the L2 relative errors for all collocated points remain below 10%, the accuracy of the PINN decreases at lower beam energies and larger ion doses, due to the presence of higher numerical gradients.