Physics Informed Neural Networks for Solving the Drift-Diffusion Equations
Authors: Luke Ragan, Matteo G. C. Alasio, Enrico Bellotti, Luca Dal Negro
Organizations: Department of Electrical & Computer Engineering, Boston University, Boston, MA, USA · Dipartimento di Elettronica e Telecomunicazioni, Politecnico di Torino, Torino, TO, Italy · Department of Electrical & Computer Engineering, Division of Materials Science & Engineering, Boston University, Boston, MA, USA · Department of Electrical & Computer Engineering, Division of Materials Science & Engineering, Physics Department, and Photonics Center Boston University, Boston, MA, USA
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.
Figures & tables
Quantity
Definition
Thermal voltage
VT=kBT/q
Reference density
Nref=max(NA,ND)
Intrinsic density
n^i=ni/Nref
Electron density
n^=n^ieψ−ϕn
Hole density
p^=n^ieϕp−ψ
Electron current density
Jn=−n^∇ϕn
Table 1: Normalized quantities in the drift-diffusion equations.
Figure 1: Structure of the PINN solver. Three fully connected networks predict the electrostatic potential ψ and the two carrier variables (the quasi-Fermi levels ϕn , ϕp or the normalized Slotboom variables u~ , v~ ) from the position; the Ohmic contact values are built into the outputs exactly (Sec. 2.3.1 ), so the loss contains no contact terms. Automatic differentiation of the outputs gives the residuals of the Poisson and continuity equations and, in two dimensions only, of the insulating (Neumann) boundary conditions, which form the loss terms Lp , Lc,n , Lc,p and Lbc . In Stage 1 all networks minimize the total loss jointly; in Stage 2 (Gummel split) each network is updated only with the loss of its own equation and the boundary loss (Sec. 2.4.1 ).
Figure 2: One-dimensional pn-junction ( L=2μ m, NA=ND=1016cm−3 ) at a forward bias of 0.8 V, solved with the normalized Slotboom variables and the two-stage training with the Gummel split (Sec. 2.4.1 ). (a) Band diagram with the conduction and valence band edges and the electron and hole quasi-Fermi levels, (b) electron and hole densities, (c) electric field and (d) electron, hole and total current densities. Solid lines: PINN; dashed lines: FEM. The relative L2 errors, given in the boxes, are 0.26% for ψ , 0.02% and 0.01% for log10n and log10p , 0.03% and 0.28% for the quasi-Fermi levels and 0.45% for the total current density. The steep minority-carrier boundary layers at both contacts (b) and the bend of the quasi-Fermi levels at the contacts (a) are resolved. The electric-field error (5.1%) is concentrated at the junction peak, where the smooth doping step of the PINN differs from the abrupt profile of the reference.
Figure 3: (a) Band diagram, with the conduction and valence band edges and the electron and hole quasi-Fermi levels, and (b) electron and hole densities of the one-dimensional pn-junction at a reverse bias of −0.6 V, solved with the quasi-Fermi level variables and the SOAP training (Sec. 2.4.2 ). Solid lines: PINN; dashed lines: FEM; the relative L2 errors are given in the boxes. (c) Reverse-bias I-V characteristic between −1.0 and −0.1 V on a linear scale and (d) forward-bias I-V characteristic between 0.1 and 1.0 V on a logarithmic scale, with the same data on a linear scale in the inset; PINN terminal currents (circles) and FEM (squares and line), with the relative L2 error of the terminal current over each curve given in the boxes. The reverse-bias currents are obtained with the SOAP training, where the runs at −0.8 to −1.0 V are initialized from the solution at the neighbouring bias, and the forward-bias currents with the two-stage training (Sec. 2.4.1 ). The FEM currents at −0.9 and −1.0 V are the median over the device, since the FEM solutions at these biases contain numerical current spikes.
Figure 4: Two-dimensional pn-junction at equilibrium: a 10×5μ m p-type substrate ( NA=1017cm−3 ) with an embedded 4×1μ m n-type region ( ND=1017cm−3 , dashed outline) at the top surface, contacted by the Anode ( 0≤x≤1μ m) and the Cathode ( 6.5≤x≤8.5μ m) with all other boundaries insulating. The PINN uses the quasi-Fermi level variables, the distance-function construction for the Ohmic contacts (Sec. 2.3.1 ), the Neumann conditions as penalty terms, and collocation points taken from the FEM mesh. Columns: electrostatic potential ψ (left), log10n (center) and log10p (right); rows: PINN (top) and FEM reference (bottom), on a common color scale per column. The relative L2 errors, given in the boxes, are 3.9% for ψ , 2.6% for log10n and 1.9% for log10p ; the depletion shell around the n-region and its corners are reproduced, and the remaining error is concentrated at the right edge of the n-region near the surface.
Figure 5: Doping profiles ND−NA recovered by the inverse PINN (solid) compared with the true abrupt profile used in the FEM reference (dashed), (a) The electric field training data at a bias voltage of 0.5 V. (b) The electrostatic potential training data at a bias voltage of 0.4 V. (c) doping profile inversely retrieved from the electric field at 0.5 V and (d) doping profile inversely retrieved from the electrostatic potential at 0.4 V. The doping profiles have a relative L2 errors of 3.89% and 8.02% for the electric field and electrostatic potential training respectively. For training on the electric field, Adam training refined with L-BFGS was used. Both plateaus are recovered to within 0.4% and the junction to within 0.2 nm. With the electrostatic potential, SOAP training followed by a continuation at a lower learning rate was used, and the junction was recovered to within 0.4 nm, and the remaining error comes from an offset of the plateaus of about 4%.
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/SrTiO3/Si memristive heterostructures across a 20 nm STO film on a 380 μm Si substrate. By isolating potential, carrier density, and vacancy transport into four sequentially trained sub-neural-networks, our model circumvents condition numbers exceeding 1016 without operator splitting. The trained surrogate reproduces experimental conductive-AFM current-voltage hysteresis (R2>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.
Rodion Podorozhny, Nikoleta Theodoropoulou, Jelena Tešić
Dept. of Computer Science, Texas State University, San Marcos, TX, USA · Dept. of Physics, Texas State University, San Marcos, TX, USA
Many physical and engineering systems require solving direct problems to predict behavior and inverse problems to determine unknown parameters from measurement. In this work, we study both aspects for systems governed by differential equations, contrasting well-established numerical methods with new AI-based techniques, specifically Physics-Informed Neural Networks (PINNs). We first analyze the logistic differential equation, using its closed-form solution to verify numerical schemes and validate PINN performance. We then address the Porous Medium Equation (PME), a nonlinear partial differential equation with no general closed-form solution, building strong solvers of the direct problem and testing techniques for parameter estimation in the inverse problem. Our results suggest that PINNs can closely estimate solutions at competitive computational cost, and thus propose an effective tool for solving both direct and inverse problems for complex systems.
Noura Al Helwani, Sophie Moufawad, Georges Sakr
Department of Physics, American University of Beirut, Lebanon · Department of Mathematics, American University of Beirut, Lebanon · The authors would like to acknowledge
Physics-Informed Neural Networks (PINNs) recast PDE solving as an optimisation problem in function space by minimising a residual-based objective, yet many applications require additional derivative-based relations that are just as fundamental as the governing equations. In this paper, we present Derivative-Constrained PINNs (DC-PINNs), a general framework that treats constrained PDE solving as an optimisation guided by a minimum objective function criterion where the physics resides in the minimum principle. DC-PINNs embed general nonlinear constraints on states and derivatives, e.g., bounds, monotonicity, convexity, incompressibility, computed efficiently via automatic differentiation, and they employ self-adaptive loss balancing to tune the influence of each objective, reducing reliance on manual hyperparameters and problem-specific architectures. DC-PINNs consistently reduce constraint violations and improve physical fidelity versus baseline PINN variants, representative hard-constraint formulations on benchmarks, including heat diffusion with bounds, financial volatilities with arbitrage-free, and fluid flow with vortices shed. Explicitly encoding derivative constraints stabilises training and steers optimisation toward physically admissible minima even when the PDE residual alone is small, providing reliable solutions of constrained PDEs grounded in energy minimum principles.
Kentaro Hoshisashi, Carolyn E Phelan, Paolo Barucca
Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK