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

This paper presents an uncertainty-aware physics-informed surrogate model for one-dimensional $p$ - $n$ junction diodes that accurately predicts internal device states and key quantities of interest with minimal error, while providing a formalism to validate results against user-specified tolerances and extend to more complex junction configurations.

Original authors: Suchita Kulkarni

Published 2026-10-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Suchita Kulkarni

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Designing the tiny electronic components that power modern life often feels like navigating a vast, uncharted landscape. To build a semiconductor device, such as the diode that controls the flow of electricity in a circuit, engineers must solve a complex set of physical laws that describe how electric fields and charged particles interact within a specific shape. Traditionally, this is done using powerful computer simulations that act like a virtual laboratory. These simulations are incredibly accurate, but they are also slow. If a designer wants to test thousands of different variations—changing the length of the device, the concentration of impurities added to the material, or the voltage applied to it—they would have to run the simulation from scratch for every single new idea. This process is so time-consuming that it can bottleneck the entire design cycle. Scientists have long sought a faster way to explore this design space, one that could predict the behavior of a device without needing to run the full, heavy simulation every time.

In this search for speed, researchers have turned to a type of artificial intelligence known as a physics-informed surrogate. Think of this as a smart shortcut. Instead of just memorizing past answers, this computer model is taught the underlying laws of physics that govern the device. It learns to predict the internal state of the diode—the electric potential and the density of electrons and holes—by ensuring its predictions satisfy those physical laws. Once the model understands the internal state, it can calculate the important numbers engineers care about, such as the current flowing through the device or the width of the region where charge is stored. The challenge, however, has always been trust. If a model is faster, how do we know it isn't just guessing? Without a way to measure its own uncertainty, a fast answer is not necessarily a useful one.

A researcher at the University of Graz has developed a new method to solve this problem. They created a physics-informed surrogate specifically for p-n junction diodes, which are the fundamental building blocks of many electronic circuits. Their goal was not just to build a fast predictor, but to build one that knows when it is right and when it might be wrong. They trained their system on a vast collection of simulated diodes, varying the device length and the doping concentration, which is the measure of how many impurities are added to the semiconductor to make it conductive. The model learned to predict the internal state of the device based on these parameters and the applied voltage. From this internal state, the researcher derived five key quantities of interest: the terminal current, the depletion width, the peak electric field, and the stored charges for electrons and holes.

The researcher found that the way they calculated the current mattered immensely. One common method involves taking the mathematical derivative of the predicted internal state to find the flow of charge. However, they discovered that this approach fails dramatically for their model. Because the current depends on a slope that changes over many orders of magnitude within a very thin layer of the device, the model struggled to calculate it accurately, sometimes producing errors of a factor of ten or more. Instead, they used a different, derivative-free method based on an established relationship in semiconductor physics. This alternative approach, which relies on integrating the electric potential rather than differentiating it, proved to be far more reliable, accurately predicting the current across a wide range of conditions.

Beyond just predicting the numbers, the researcher introduced a system to quantify the uncertainty of every answer. They realized that the accuracy of the model depended heavily on how much of the device was "depleted," a state where the region is stripped of its mobile charge carriers. By analyzing how the error changed with the depleted fraction, they constructed a safety interval for every prediction. This interval acts like a confidence band, telling the user how wide the range of possible true values is. They tested this system on thousands of new devices that the model had never seen before. The results showed that the true value fell inside the predicted interval about ninety percent of the time. This means that when the model gives an answer, it can also tell the engineer, "I am ninety percent sure the true value is within this range."

The study also revealed the limits of this speed. The model works exceptionally well within the range of conditions it was trained on, but it begins to struggle when asked to predict behavior outside that range. Specifically, when the voltage applied to the device exceeds a certain threshold known as the built-in potential, the physical behavior of the device changes fundamentally. Because the model had never seen this regime during its training, it failed to predict the correct behavior, continuing to guess based on patterns that no longer applied. This highlights a crucial finding: while the surrogate is a powerful tool for exploring known design spaces, it cannot simply extrapolate to entirely new physical regimes without losing reliability.

Ultimately, this work demonstrates that it is possible to create a fast, physics-aware computer model that does more than just spit out numbers. By attaching a measure of reliability to every prediction, the researcher has turned a simple predictor into a trustworthy assistant. For a device physicist, this means they can explore a vast array of design options quickly, knowing exactly when the computer's answer is solid and when it is time to run the slower, more rigorous simulation to double-check. The approach successfully bridges the gap between the need for speed in design and the absolute necessity of accuracy in engineering, offering a path forward for designing the next generation of electronic components.

Technical Summary: A Physics-Informed Uncertainty-Aware Surrogate for p-n Junction Diodes

Problem Statement
Designing semiconductor devices typically requires solving coupled electrostatic and carrier-transport equations (drift-diffusion) for specific geometries, doping profiles, and boundary conditions using Technology Computer-Aided Design (TCAD) tools like Synopsys Sentaurus or Silvaco Atlas. While accurate, these solvers are computationally expensive, making them impractical for exploring large design spaces involving variations in device length, doping, and operating bias. Existing machine learning surrogates often fall into three categories: terminal-response models (fast but lacking internal state), physics-informed solvers (representing internal states but often lacking uncertainty quantification), and mesh-native operators (handling complex geometries but requiring specific grid structures). The challenge is to develop a surrogate that not only predicts the internal state of a device (potential and carrier densities) and derived quantities of interest (QoIs) across a family of devices but also provides a reliable, user-specified uncertainty estimate without access to ground truth.

Methodology
The author develops a physics-informed neural network (PINN) surrogate conditioned on device parameters (length $L$ , doping $n_D$ , and bias $V$ ).

Governing Equations & Parametrization: The system is modeled using the drift-diffusion equations with Shockley-Read-Hall (SRH) recombination. The author utilizes a quasi-Fermi-level (QFL) parametrization, solving for the electrostatic potential ( $\tilde{\phi}$ ) and electron/hole quasi-Fermi potentials ( $\tilde{v}_n, \tilde{v}_p$ ) rather than carrier densities directly. This avoids the numerical instability of balancing drift and diffusion terms that cancel to many orders of magnitude.
Network Architecture: Three separate Multi-Layer Perceptrons (MLPs) predict $\tilde{\phi}$ , $\tilde{v}_n$ , and $\tilde{v}_p$ . The networks use Feature-wise Linear Modulation (FiLM) to condition the hidden layers on device parameters ( $L, n_D, V$ , and geometry descriptors for 2D). The input includes spatial coordinates normalized by device length and the Debye length, allowing the network to resolve features across varying scales.
Loss Functions: The training minimizes a weighted sum of:
1. Data Loss: Mean squared error against reference data from the open-source TCAD solver DEVSIM.
2. Physics Loss: Residuals of the Poisson and continuity equations.
3. Boundary Condition Loss: Dirichlet conditions at contacts and Neumann conditions at insulating boundaries.
  Crucially, the author addresses the "density gating" problem where gradients vanish in low-density regions by including DEVSIM data, ensuring the network converges even where physics-only residuals are insensitive.
Derivative-Free Current Readout: A key methodological contribution is the method for calculating terminal current. Differentiating the network output to find current ( $J_{grad}$ ) fails in long, highly doped devices because the network cannot resolve the steep quasi-Fermi slopes required. Instead, the author uses a derivative-free readout ( $J_{RO}$ ) based on the Gummel relation, which integrates the potential and carrier profiles. This method recovers the current accurately even when the internal state gradients are imperfect.
Uncertainty Quantification: To estimate validity without ground truth, the author employs a localised conformal prediction framework. They define the error as a function of the depleted fraction of the device ( $\kappa = \log_{10}(W_{dep}/L)$ ). Using a calibration dataset, they construct a weighted 90% quantile error bar $b(\kappa)$ . At query time, if the calculated error bar is within a user-specified tolerance $\tau$ , the surrogate "vouches" for the answer.

Key Contributions

Derivative-Free Current Readout: The paper demonstrates that for PINNs, terminal current cannot be reliably obtained via automatic differentiation of the internal state in regimes with steep gradients. The proposed integral-based readout (Gummel relation) is shown to be robust, whereas the differentiated current can be wrong by an order of magnitude or have the wrong sign.
Uncertainty-Aware Surrogate: Unlike standard PINNs that provide point estimates, this work attaches a calibrated error interval to predictions. The interval is conditioned on a single physical variable (depleted fraction), allowing the surrogate to self-report when it is likely to be inaccurate (e.g., in fully depleted regimes).
Generalization Across Geometries: The formalism is validated on one-dimensional symmetric and asymmetric junctions, as well as a two-dimensional planar diode, showing that the architecture and readout methods are not tied to a specific geometry or doping symmetry.

Results

Accuracy: For a 1D symmetric junction, the surrogate achieves a median L1 error of 0.11% on the electrostatic potential, 1.5% on electron density, and 0.61% on hole density. The terminal current ( $J_{RO}$ ) shows a median relative error of 0.70% on test devices.
Uncertainty Coverage: The calibrated error bars achieve a marginal coverage of approximately 90% across test devices for all five quantities of interest (current, depletion width, peak field, stored electron charge, stored hole charge).
Extrapolation Limits: The surrogate performs well within the training range ( $V < 0.8 V_{bi}$ ). However, extrapolation beyond the built-in potential ( $V > V_{bi}$ ) fails, with median errors rising to 74%, as the device enters a resistance-limited regime absent from the training data.
Benchmarking: In 2D and asymmetric 1D benchmarks, the framework maintains low errors for internal states and QoIs, with the largest deviations occurring in stored charge predictions for specific regimes (e.g., 13% for $Q_n$ in 2D).

Significance and Claims
The paper claims to demonstrate a successful framework for a physics-informed surrogate that goes beyond simple prediction to provide reliability estimates. By utilizing conformal prediction methods conditioned on a physical state variable, the surrogate can inform a device physicist whether a specific prediction falls within a desired tolerance. The author emphasizes that while the method is currently limited to specific doping families and forward bias, it establishes a pathway for trustworthy surrogates in TCAD. The work explicitly notes that the uncertainty intervals are marginal (valid across the population of devices) rather than per-device guarantees, and they are only valid within the calibration support (specific ranges of depletion fraction). The study concludes that while the physics-informed approach is powerful, the integration of uncertainty quantification is essential for practical deployment in design exploration.

Technical Summary: A Physics-Informed Uncertainty-Aware Surrogate for p-n Junction Diodes

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