⚡ electrical engineering

Uncertainty-aware calibration from SEM profiles to electrical response using coupled process and device TCAD

This paper introduces an uncertainty-aware, closed-loop computational framework that maps SEM profiles to electrical responses by propagating boundary distributions through coupled process and device TCAD simulations, thereby improving electrical prediction accuracy and annotation efficiency compared to conventional hard-boundary calibration.

Original authors: qinao hu

Published 2026-08-28

📖 6 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 invisible world of modern electronics, the smallest transistors are built layer by layer, like microscopic skyscrapers. To ensure these tiny structures work correctly, engineers must inspect them with powerful microscopes that can see features at the nanometer scale. These images, known as scanning electron microscope profiles, show the exact shape of a transistor's cross-section. However, these images are never perfectly sharp; the edges appear fuzzy due to SEM blur and inter-annotator variation, and different experts might trace the outline of a shape in slightly different ways. This uncertainty matters because the precise geometry of a transistor dictates how electricity flows through it. If the shape is even a few nanometers off, the device might fail to switch on or off properly, or it might consume too much power. For decades, engineers have tried to fix this by taking a single, best-guess outline from a blurry image and feeding it into a computer simulation to predict how the device will behave. But this approach throws away the valuable information contained in the fuzziness itself, treating a range of possibilities as if only one outcome were real.

A new study by Qinao Hu at East China Normal University proposes a different way to handle this problem. Instead of forcing a single, rigid line onto a blurry image, the researchers built a system that acknowledges the uncertainty from the very beginning. They treated the edge of the transistor not as a fixed line, but as a cloud of possibilities. This cloud represents all the plausible ways an expert might draw the boundary given the image's blur. The researchers then fed this entire cloud of possibilities into their computer models, rather than just one guess. By doing this, they allowed the computer to calculate a range of likely electrical behaviors, capturing the full picture of what the device might do. The goal was to see if keeping this uncertainty alive through the entire process would lead to more accurate predictions of how the transistor actually performs.

The researchers focused on a specific type of advanced transistor called a nanosheet field-effect transistor, which is used in the most powerful computer chips today. They examined the spacer, a tiny insulating wall next to the transistor channel, which is critical for controlling the flow of electricity. In their experiments, they used a dataset of 180 real images taken from five different batches of manufacturing. Some of these images were traced by two different experts, allowing the team to measure exactly how much human judgment varied when defining the edges. The team trained a computer program to look at these images and output a probability map, showing where the edge was likely to be and how confident it was in that location. This program was taught to minimize errors in the overall shape, the distance between lines, and specific critical measurements like the width of the channel and the angle of the walls.

Once the computer generated a set of possible shapes for each image, the researchers passed these shapes through a sophisticated simulation that mimics the manufacturing process. This step, known as process calibration, adjusted the internal settings of the simulation to match the observed shapes. Because the input was a cloud of shapes rather than a single line, the output was also a cloud of possible manufacturing settings. This uncertainty was then carried forward into a second simulation that predicted the electrical performance of the device. This final step calculated key metrics like the voltage needed to turn the transistor on and the amount of current it could carry. The researchers compared these predictions against the actual, high-fidelity results from the device simulations to see how close they came.

The results showed that embracing uncertainty led to significantly better predictions. When the researchers used their new method, the error in predicting the voltage required to turn the transistor on dropped to 11.9 millivolts, a substantial improvement over the 21.8 millivolts error seen with the traditional method that uses a single hard boundary. Similarly, the error in predicting the current-voltage curve, which describes how the device behaves under different conditions, fell from 5.8 percent to 3.1 percent. The new approach also proved more reliable in its confidence intervals; it correctly captured the true electrical behavior within its predicted range 91 percent of the time, compared to only 63 percent for the older method. This suggests that by acknowledging the fuzziness of the microscope images, the computer models could navigate the manufacturing variations more effectively.

The study also explored how to use this information to improve the data itself. Since the system could identify which images had the most uncertainty that actually mattered for the final electrical performance, it could prioritize those specific images for re-examination by human experts. In a test where the team added just 30 new labels to their dataset, this smart selection strategy reduced the prediction error to 1.54 percent. In contrast, simply picking images at random to re-label, or picking them based only on how blurry they looked without considering their electrical impact, resulted in much higher errors of 2.36 percent and 2.1 percent respectively. This finding highlights that not all uncertainty is equal; some blurry edges matter far more for the final product than others, and the new system learned to spot the difference.

Ultimately, this work demonstrates that in the high-stakes world of semiconductor manufacturing, ignoring the limits of our measurements can lead to worse predictions. By treating the edge of a transistor as a distribution of possibilities rather than a single fact, the researchers created a more robust digital twin of the manufacturing process. This approach does not just improve the accuracy of the numbers; it provides a clearer picture of the risks and variations inherent in building the world's smallest machines. The method successfully linked the visual imperfections of a microscope image to the electrical reality of a functioning chip, proving that a careful accounting of what we do not know can lead to a better understanding of what we do.

Technical Summary: Uncertainty-aware Calibration from SEM Profiles to Electrical Response

Problem Statement
In semiconductor process development, calibrating Scanning Electron Microscopy (SEM) cross-sectional profiles to process parameters and subsequent device electrical responses is a critical inverse problem. Conventional workflows typically extract a single deterministic boundary (contour) from an SEM image to drive process and device Technology Computer-Aided Design (TCAD) simulations. This approach discards two vital sources of uncertainty:

Measurement Uncertainty: Arising from finite SEM contrast, blur, and charging artifacts.
Annotation Uncertainty: Arising from inter-annotator variation when defining boundaries in low-contrast regions.

By reducing the profile to a single "hard" contour, standard methods fail to propagate the uncertainty inherent in the image measurement through to the final electrical predictions (e.g., threshold voltage, subthreshold swing, drive current). Consequently, a high degree of mask overlap (segmentation accuracy) does not necessarily guarantee improved electrical prediction accuracy, as different boundary locations within the uncertainty band may have vastly different impacts on device physics.

Methodology
The paper proposes a closed-loop computational framework that treats the SEM boundary as a probability distribution rather than a fixed line, propagating this uncertainty through Bayesian process calibration to device-level electrical prediction. The workflow consists of three integrated stages:

Probabilistic SEM Boundary Modeling:
Instead of predicting a single binary mask, a segmentation network outputs a foreground probability field and a signed-distance estimate. The model is trained using a composite loss function that combines:
- Binary Cross-Entropy (pixel-wise).
- Signed Distance Field (SDF) loss.
- Chamfer distance loss (boundary displacement).
- Geometry-aware loss (critical dimensions, sidewall angles, corner radii).
  This approach captures the distribution of possible boundary locations, particularly for dual-annotated profiles where inter-annotator variation is explicitly modeled.
Shape-Aware Bayesian Process Calibration:
Boundary samples extracted from the probabilistic model are propagated through a Bayesian calibration of the process TCAD model. The process posterior $p(\theta | \mathcal{X}_{\partial S})$ is estimated by comparing sampled SEM boundaries against simulated process profiles using shape-aware dissimilarity measures (signed-distance and Chamfer terms). This yields a posterior distribution over process parameters (lateral etch bias, depth scale, sidewall-angle modifier, corner-radius modifier) rather than a single point estimate, acknowledging that multiple parameter combinations can produce similar profiles.
Uncertainty Propagation and Sensitivity-Driven Relabeling:
The posterior process-profile samples are mapped through a device solver to generate a posterior-predictive distribution of electrical responses ( $V_T$ , subthreshold swing, $I_{on}$ , and full I–V curves).
- Evaluation: The system validates the coverage of the predictive intervals against held-out reference TCAD data.
- Active Learning: A sensitivity score is computed for unlabeled profiles, defined as the product of boundary uncertainty and the local electrical sensitivity ( $\|\nabla_S f_d(S)\|$ ). This score prioritizes profiles for additional annotation where boundary uncertainty most significantly impacts downstream electrical performance, optimizing the labeling budget.

Key Contributions

Probabilistic Formulation: The author formulates SEM boundary extraction as a probabilistic contour problem, utilizing pixel, distance-field, Chamfer, and critical-dimension losses to retain boundary distribution information.
End-to-End Uncertainty Propagation: The framework connects the boundary distribution to shape-aware Bayesian process calibration and propagates the resulting process-profile posterior into a posterior-predictive electrical distribution, preserving uncertainty from image to electrical metric.
Comprehensive Evaluation: The study evaluates geometry error, interval coverage, and electrical prediction accuracy under matched data, annotation, and TCAD budgets. It demonstrates that retaining boundary uncertainty reduces electrical prediction error compared to hard-boundary calibration.
Sensitivity-Driven Relabeling: The paper introduces a strategy for targeted relabeling that ranks SEM profiles by downstream electrical sensitivity, showing that this approach outperforms random or pure-uncertainty selection in reducing I–V curve errors.

Results
Evaluated on a dataset of 180 cross-sectional nanosheet-FET spacer profiles:

Segmentation Accuracy: The proposed method achieved a Dice score of 0.934, a 95th-percentile Hausdorff distance (HD95) of 6.8 nm, and a critical-dimension mean absolute error (CD MAE) of 1.9 nm, outperforming baselines including U-Net, DeepLabV3+, and Canny with active contours.
Electrical Prediction: Compared to a hard-boundary least-squares baseline, the proposed Bayesian pipeline reduced the threshold voltage ( $V_T$ ) mean absolute error from 21.8 mV to 11.9 mV and the I–V normalized root-mean-square error (NRMSE) from 5.8% to 3.1%.
Calibration Coverage: The method achieved an empirical 90% interval coverage of 0.91, closely matching the nominal level, whereas the hard-boundary baseline achieved only 0.63.
Relabeling Efficiency: Under a fixed labeling budget, the sensitivity-driven relabeling strategy reduced the I–V NRMSE to 1.54% after 30 additional labels, significantly outperforming random selection (2.36%).

Significance and Claims
The paper claims that its primary contribution is demonstrating that propagating SEM boundary uncertainty through process and device TCAD improves held-out electrical prediction accuracy under matched computational and annotation budgets.

The author emphasizes that their framework separates three distinct effects often conflated in prior work:

Improvements in boundary displacement and critical-dimension error (not just mask overlap).
The production of well-calibrated posterior intervals via Bayesian inference.
The persistence of electrical improvements after propagation through the device solver.

The study concludes that while image uncertainty alone is not the most efficient criterion for annotation, incorporating downstream electrical sensitivity into the selection process allows for more efficient use of labeling resources. The work is presented as a step toward more robust semiconductor digital twins where uncertainty is traceable from image measurement to electrical prediction, though the author notes that transfer to other device architectures or process windows would require independent recalibration.

Technical Summary: Uncertainty-aware Calibration from SEM Profiles to Electrical Response

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