⚡ electrical engineering

Uncertainty propagation from SEM profiles to TCAD-referenced electrical predictions using coupled process and device models

This paper presents a framework that propagates uncertainty from cross-sectional SEM profiles through coupled process and device TCAD models to generate spatially coherent contours and calibrated electrical predictions, achieving high geometric accuracy and reliable threshold-voltage interval coverage for semiconductor device characterization.

Original authors: Qinao Hu

Published 2026-10-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Qinao Hu

Original paper licensed under CC BY 4.0 (https://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

In the world of modern electronics, the smallest components are built layer by layer, like a microscopic skyscraper made of silicon and metal. To ensure these tiny structures work correctly, engineers rely on powerful computer simulations that predict how electricity will flow through them. These simulations need a precise blueprint of the device's shape. However, when scientists look at these real-world devices under an electron microscope, the images are often fuzzy or ambiguous at the edges. It is difficult to tell exactly where one material ends and another begins. This uncertainty in the picture creates a problem: if the computer simulation starts with a blurry or guessed shape, its prediction of the electrical performance will also be uncertain. The question researchers face is whether they can use the imperfect microscope image to narrow down the possible shapes enough to make a reliable electrical prediction, even without knowing the exact manufacturing settings that created the device.

A researcher at East China Normal University tackled this challenge by creating a new method that treats the blurry edges of a microscope image not as a single mistake to be fixed, but as a range of possibilities to be explored. Instead of forcing the computer to pick one single line to represent the edge of a material, their system generates hundreds of slightly different, plausible outlines that all fit the image. They then feed each of these possible shapes into a detailed physics simulator to see how the electricity would behave in each version. By combining the results of all these simulations, they can produce a prediction that includes a measure of confidence, telling engineers not just what the electrical performance likely is, but also how much that performance might vary due to the uncertainty in the image.

The researcher tested this approach on a dataset of 180 real microscope images taken from a specific type of advanced transistor. They compared their method against several standard techniques used in the industry. The standard methods usually try to find just one best line to represent the edge, or they use simpler statistical guesses. The new method, which couples the image analysis with the physics simulation, proved to be more accurate. When predicting a key electrical property called the threshold voltage—which determines when the transistor turns on—the new method was off by an average of only 11.9 millivolts. In contrast, the best standard method using a single fixed line was off by a larger margin, and a more complex statistical baseline was slightly less accurate. Furthermore, the new method provided a range of likely values that was both precise and reliable. In 91 percent of the test cases, the actual electrical behavior fell within the predicted range, and the width of this range was tight enough to be useful for engineering decisions.

A crucial part of the study was ensuring that the computer's predictions were based on the image data and not just on assumptions about how the device was made. The researcher used a "gold standard" reference created by an expert who independently traced the device shape without seeing the computer's guesses. This allowed them to measure the true electrical consequence of the shape extraction. They found that their method successfully translated the visual uncertainty from the microscope into a realistic electrical uncertainty. For example, they could predict that the current flowing through the device would vary by a small, calculable amount, rather than giving a single, potentially misleading number. This ability to quantify uncertainty is vital because it helps engineers understand the risks in their designs without needing to build and test every single physical variation.

The study also explored how to make the process even better by asking human experts to double-check specific images. The researcher developed a smart system that decides which images would benefit most from a second look. Instead of picking images at random or just looking for the ones that look the most confusing, the system calculated which images, if re-examined, would most improve the final electrical prediction. When they added just 30 extra expert annotations using this smart selection method, the error in the electrical prediction dropped to 2.12 percent. This was better than picking images randomly or simply choosing the ones with the most visual ambiguity. This finding suggests that in the future, engineers could save time and money by focusing their limited human attention on the specific data points that matter most for the final product's performance.

However, the researcher was careful to note the limits of their success. When they tried to apply their trained system to a completely new batch of devices made under slightly different conditions, the accuracy dropped, and the confidence intervals became less reliable. This indicates that while the method works very well within a specific family of devices, it is not yet a universal solution that works perfectly across all manufacturing variations without some adjustment. They found that adding just a few new examples from the new batch allowed the system to recalibrate and recover most of its accuracy. This highlights that while the method is powerful, it still relies on the manufacturing process staying relatively consistent.

Ultimately, this work demonstrates a practical way to bridge the gap between the imperfect images we can take of tiny devices and the precise electrical predictions needed to design them. By acknowledging that the edge of a material is not a single line but a zone of possibility, and by mathematically tracking how that zone affects the flow of electricity, the researcher has provided a tool that is more accurate and more honest about its own limitations than previous methods. For the industry, this means a step toward more reliable computer-aided design, where engineers can trust that their simulations reflect the true variability of the physical world, leading to better chips and more efficient electronics.

Technical Summary: Uncertainty Propagation from SEM Profiles to TCAD-Referenced Electrical Predictions

Problem Statement
In semiconductor manufacturing, Scanning Electron Microscopy (SEM) provides cross-sectional images of fabricated devices, but the exact process parameters (e.g., etch bias, depth, taper) that generated these profiles are often unknown. Traditional forward simulators predict profiles from inputs but cannot uniquely determine the inputs from an observed structure. Furthermore, standard segmentation methods often produce a single "hard" boundary, ignoring the inherent ambiguity in low-contrast regions. This uncertainty in the geometric boundary propagates to device simulations, potentially leading to inaccurate electrical predictions. The core challenge is whether SEM evidence can sufficiently narrow the distribution of compatible process structures to improve electrical predictions compared to nominal forward calculations, while rigorously quantifying the associated uncertainty.

Methodology
The author proposes a coupled workflow that propagates boundary uncertainty through process and device Technology Computer-Aided Design (TCAD) models. The approach consists of three main stages:

Spatially Coherent Boundary Modeling:
Instead of generating a single contour, the method employs a U-Net encoder-decoder to predict a mean signed-distance field (SDF) and a set of smooth basis fields. A stochastic signed-distance model generates spatially coherent contours by sampling from a Gaussian distribution defined by these fields. This approach retains a boundary distribution rather than a single point estimate. The model is trained using a composite loss function including binary cross-entropy, SDF supervision, a differentiable contour term (soft Chamfer distance), geometry penalties (width, depth, angle), and a negative log-likelihood (NLL) term to calibrate the distribution against paired annotations.
Process-Profile Calibration via Bayesian Inference:
The learned boundary distribution is coupled with a forward process model (a ViennaPS level-set etch/deposition simulator) to infer a bounded posterior distribution of process parameters. The parameters include lateral bias, depth scale, taper modifier, and corner-radius modifier.
- Likelihood: A geometric compatibility likelihood is defined based on the overlap between the learned boundary distribution and the forward geometric noise model. This avoids treating stochastic draws as independent measurements.
- Posterior: Markov Chain Monte Carlo (MCMC) methods (Metropolis-Hastings) are used to sample the posterior distribution of the four effective parameters, constrained by a bounded prior and geometric validity checks (e.g., no self-intersections).
Device TCAD Propagation:
Each sample from the process posterior is mapped to a specific device geometry using a fixed structure-construction operator. This operator generates the material regions, contacts, and mesh for a Sentaurus Device simulation. The device solver computes the electrical response (I-V curves, threshold voltage $V_T$ , subthreshold swing $SS$ , on-current $I_{on}$ ). The final output is a probabilistic distribution of electrical metrics derived from the ensemble of simulated devices.

Key Contributions

Uncertainty Propagation Framework: The study establishes a pipeline that propagates boundary uncertainty from SEM images through process calibration to electrical predictions, explicitly distinguishing between segmentation uncertainty and process-parameter uncertainty.
Geometric Compatibility Likelihood: The author introduces a likelihood function that operates on geometric observations (signed distances at fixed anchors) rather than raw pixels, effectively coupling the segmentation model with the physical process model.
Electrical-Sensitivity Annotation Strategy: A method is proposed to select additional annotation candidates based on their electrical sensitivity (using a central difference score on geometry descriptors), demonstrating that targeted labeling reduces prediction error more effectively than random or boundary-uncertainty-based selection.
Rigorous Evaluation Protocol: The work utilizes an independent, adjudicated reference contour (or hidden synthetic geometry) to generate TCAD electrical targets. This ensures that the evaluation measures computational consistency and uncertainty propagation without circularity (i.e., the target is not derived from the method's own inferred contour).

Results
The method was evaluated on a dataset of 180 cross-sectional SEM profiles from five batches (108 training, 36 validation, 36 test).

Segmentation Accuracy: The proposed boundary model achieved a mean Dice score of 0.934 and a 95th-percentile Hausdorff distance (HD95) of 6.8 nm, outperforming deterministic baselines (e.g., DeepLabV3+) and probabilistic baselines (e.g., Deep Ensemble).
Electrical Prediction: The proposed pipeline achieved a threshold-voltage ( $V_T$ ) mean absolute error (MAE) of 11.9 mV and an I-V normalized root-mean-square error (NRMSE) of 3.1%. This represents an improvement over hard-boundary calibration (5.8% NRMSE) and the deep-ensemble Bayesian baseline (3.4% NRMSE).
Uncertainty Calibration: The nominal 90% $V_T$ intervals covered 164 of 180 repeated predictions (91.1%) with a mean width of 54 mV. This coverage was achieved without significantly widening the intervals compared to the ensemble baseline, indicating efficient uncertainty quantification.
Ablation Studies: Removing the joint shape-supervision group increased the I-V NRMSE to 3.8%, while replacing posterior calibration with point calibration reduced coverage to 110/180. This confirms that shape supervision improves geometry, while posterior calibration is critical for downstream prediction and coverage.
Batch Transfer: When applied to held-out batches without retraining (frozen transfer), the error increased to 4.24% and coverage dropped to 85.0%. However, adapting the image model with just six additional labels per batch restored coverage to 89.3% and reduced error to 3.6%.
Annotation Efficiency: With 30 additional annotations, electrical-sensitivity selection achieved a curve error of 2.12%, outperforming boundary-uncertainty selection (2.24%) and random selection (2.42%).

Significance and Claims
The paper claims that propagating SEM boundary distributions through coupled process and device models supports accurate, uncertainty-aware electrical prediction within a fixed device family. The results demonstrate that:

SEM-conditioned uncertainty propagation yields lower observed electrical errors than nominal forward predictions or hard-boundary calibrations.
The proposed method provides well-calibrated uncertainty intervals (close to the nominal 90% level) with reasonable interval widths.
The distinction between upstream geometry improvements (shape supervision) and downstream inference effects (posterior calibration) is critical for optimizing the pipeline.
While the method requires significant computational setup (surrogate training, MCMC), it offers a viable alternative for repeated inverse inference within a stable device/process window, provided that batch shifts are managed through minimal adaptation.

The author explicitly states that these conclusions apply to the studied nanosheet device family and the specific TCAD references used. The work does not claim agreement with measured device I-V characteristics, as the evaluation relies on TCAD-generated targets from adjudicated contours, not direct experimental electrical measurements.

Technical Summary: Uncertainty Propagation from SEM Profiles to TCAD-Referenced Electrical Predictions

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