A physics-informed uncertainty-aware surrogate for $p$-$n$ junction diodes

The drift-diffusion equations are the fundamental building block for the modeling, design, and optimization of modern semiconductor devices. Solutions of the drift-diffusion equations rarely give way to analytical forms, and exhibit strong nonlinearity, with the solutions ranging over several orders of magnitude. These attributes have led to research into numerical solvers, and while many commercial Technology Computer Aided Design (TCAD) packages now exist, they may require long simulation times, exhibit sensitivity to initial guesses, and rely on mesh generation. Further, these solvers are not well suited for inverse problems where the design parameters of a device are retrieved from its desired functional characteristics. To address all these challenges, this work demonstrates the usage of Physics Informed Neural Networks (PINNs) for the solution of the drift-diffusion equations with errors comparable to traditional simulation software using advanced training techniques to solve problems beyond the well-behaved equilibrium regime. For this study, we specifically choose an abrupt PN-junction in both one and two dimensions that features extreme gradients of the carrier concentrations at the Ohmic contacts and an abrupt discontinuity in the doping profile. Using the proposed PINN architecture, we are able to retrieve accurate results for all observables under all bias conditions. Additionally, we apply the PINN solver towards the inverse problem of retrieving doping profiles directly from desired electric field distributions or electrostatic potentials under non-zero applied biases. Our work establishes a powerful PINNs-based approach for semiconductor device modeling with the ability to solve inverse problems under non-equilibrium conditions for microelectronics research and discovery.

Physics-informed neural networks (PINNs) offer a promising framework for modeling semiconductor devices, yet standard architectures struggle with severe numerical stiffness and multiscale spatial discrepancies inherent to oxide heterostructures. Here, we demonstrate a cascaded PINN architecture coupled with a custom second-order Chebyshev second kind polynomial spectral optimizer (DSO V2 Hybrid) to model ion-electronic drift-diffusion transport in Pt/SrTiO$_3$/Si memristive heterostructures across a 20 nm STO film on a 380 $μ$m Si substrate. By isolating potential, carrier density, and vacancy transport into four sequentially trained sub-neural-networks, our model circumvents condition numbers exceeding $10^{16}$ without operator splitting. The trained surrogate reproduces experimental conductive-AFM current-voltage hysteresis ($R^2 > 0.96$) while ensuring strict Poisson consistency across continuous space. Compared to conventional finite-element solvers (e.g., COMSOL), the PINN surrogate enables differentiable inverse parameter estimation and linear time inference.

High-fidelity TCAD simulation of drift-diffusion transport remains the workhorse of emerging FinFET device design, but it is computationally expensive, especially for 3D structures where runtime escalates steeply with mesh complexity. This sharply limits multi-objective design space exploration. Existing machine-learning surrogates map a fixed set of design parameters to a few scalar device metrics, discarding the underlying physics and losing transferability across device geometries and families. A physics-informed graph attention network (GAT) surrogate is proposed. It operates directly on the tetrahedral TCAD mesh and predicts, at every mesh node, the electrostatic potential together with the electron and hole quasi-Fermi levels, the fundamental unknowns of the drift-diffusion system. Training combines a data loss with finite-volume current-continuity residuals, embedding carrier-transport physics into the objective. Operating on the mesh as a graph, the surrogate inherits size generalization: a model trained on few-fin meshes applies unchanged to substantially larger arrays, bounded at inference only by GPU memory. Per-node uncertainty from a deep ensemble drives an active-learning loop that screens large candidate pools in seconds and forwards only the most informative designs for full simulation. Benchmarked against Sentaurus Device on multi-fin tri-gate FinFETs, the surrogate reproduces the three drift-diffusion fields with sub-volt per-field RMSE and reaches a per-design throughput orders of magnitude higher than the full simulator. The advantage grows with device size: on large multi-fin arrays that are prohibitively slow to simulate directly, inference still completes in under a second per device, enabling Pareto-front exploration across device scales infeasible for direct TCAD sweeps.