Physics-informed neural network for rapid ion trajectory prediction in ion-etch reactors

Accurate prediction of plasma sheath behavior near material and geometric discontinuities is essential in plasma-based microelectronic manufacturing, where small variations in ion energy and angular distributions can significantly affect semiconductor yield. This work presents a sheath plasma physics-informed neural network (SPPINN) framework for rapid modeling of plasma sheaths in reactive ion etch reactors. The two-dimensional, mesh-free model embeds Poisson's, ion continuity, and ion momentum equations directly into the loss function, ensuring physically constrained solutions. Reactor geometry, including focus ring height, is incorporated as an input parameter, enabling high-dimensional design exploration within a single model. The framework solves ion dynamics and electrostatic potential in a weakly self-consistent manner, where electrons are modeled using a trainable Boltzmann relation based on a prescribed ion source. The model achieves residuals ≤ 10−2 for Poisson's equation and ≤10−3 for continuity and momentum. Predictions are compared against an analogous two-dimensional reference study, showing good agreement in overall sheath structure, with relative L2 errors of order 10−2, and limited agreement of ion tilt. Localized discrepancies near sharp geometric transitions arise from differentiable geometry smoothing required for stable training. Ion tilt results were corroborated as physical solutions by a Pearson correlation coefficient ≥0.93, implying the geometry smoothing approximations preserve the dominant physical trends in sheath dynamics while introducing only localized, nonsystematic bias near discontinuities. Once trained, SPPINN generates millisecond-scale predictions across the parametric domain, providing 4 orders-of-magnitude speedup compared to conventional solvers. These results demonstrate SPPINN's viability for rapid, geometry-aware plasma sheath prediction and reactor design exploration.