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Hybrid approach to reconstruct nanoscale grating dimensions using scattering and fluorescence with soft X-rays

This paper proposes a hybrid approach combining soft X-ray scatterometry and fluorescence analysis to resolve ambiguous inverse solutions and achieve sub-nanometer accuracy in reconstructing complex nanoscale grating dimensions.

Original authors: Leonhard M. Lohr, Richard Ciesielski, Vinh-Binh Truong, Victor Soltwisch

Published 2026-02-19

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CC BY 4.0

Original authors: Leonhard M. Lohr, Richard Ciesielski, Vinh-Binh Truong, Victor Soltwisch

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

Imagine you are trying to figure out the exact shape of a tiny, intricate castle built from Lego bricks, but the castle is so small (nanoscale) that you can't see it with your eyes, and you can't touch it without breaking it. This is the daily challenge for engineers making modern computer chips. They need to know the precise width, height, and angles of these microscopic structures to ensure the chips work.

This paper describes a clever new way to "see" these invisible structures by combining two different types of "flashlights" and using a bit of detective work.

The Problem: The "Shape-Shifting" Puzzle

Traditionally, scientists use a technique called scatterometry. Imagine shining a laser pointer at a grating (like a comb with tiny teeth) and looking at the pattern of light that bounces off. By analyzing that pattern, you can mathematically guess the shape of the comb.

However, there's a catch. Because these structures are so small and the math is complex, the light pattern can be ambiguous. It's like looking at a shadow on a wall: a round ball and a flat disk can cast the exact same shadow from a certain angle. In the world of nanotechnology, this means the computer might give you three or four different possible shapes for the "castle," and it doesn't know which one is the real one. This is called the multimodality problem (having multiple valid answers).

The Solution: A Hybrid "Flashlight" Strategy

To solve this, the researchers at the Physikalisch-Technische Bundesanstalt (PTB) in Germany decided to use two flashlights at once instead of just one.

Flashlight A (Scattering): This is the traditional method. They shine soft X-rays (a type of light with a very short wavelength) at the sample. Some of this light bounces off, creating a diffraction pattern. This tells them about the shape and structure of the object.
Flashlight B (Fluorescence): This is the new trick. When the soft X-rays hit the material, they don't just bounce; they also make the atoms inside the material glow (emit their own light, called fluorescence). This is like the Lego bricks themselves lighting up. This glow tells them exactly where the different materials (like silicon and nitrogen) are located and how much of each there is.

The Analogy: The Blindfolded Sculptor

Think of the researchers as blindfolded sculptors trying to recreate a statue based on clues.

Scattering is like feeling the outline of the statue with your hands. You know it's tall and has a curve, but you might confuse a smooth curve with a sharp edge.
Fluorescence is like someone telling you, "Hey, the statue is made of clay on the left and marble on the right." This gives you a map of the materials.

If you only use your hands (scattering), you might get the shape wrong. If you only use the material map (fluorescence), you might not know the exact height. But if you combine both clues, you can build a perfect, unique model of the statue.

The "Mixing" Secret Sauce

The paper's biggest breakthrough isn't just using two methods; it's figuring out how much to trust each method.

The researchers created a mathematical "recipe" (called a weighted function) to mix the data from the scattering and the fluorescence. They tested different ratios:

Too much scattering? The answer gets stuck in the "shadow" ambiguity.
Too much fluorescence? You lose the fine details of the shape.
Just the right mix? The "fog" clears.

They found a "sweet spot" (a specific weighting parameter) where the two methods cancel out each other's weaknesses. When they hit this sweet spot, the computer stopped giving them three confusing answers and finally said, "Aha! This is the only shape that fits both the shadow and the glowing material map."

The Result

By using this Hybrid Approach, they successfully reconstructed the dimensions of a silicon nitride grating (a tiny, periodic structure) with high confidence. They proved that by listening to both the "echo" (scattering) and the "glow" (fluorescence), they could solve the puzzle of ambiguous shapes that stumped them before.

In short: They stopped guessing which shadow was real by adding a glowing map to the mix, allowing them to see the true shape of the microscopic world with crystal-clear certainty.

1. Problem Statement

As semiconductor manufacturing transitions to the nanoscale regime (driven by EUV lithography), the demand for precise, non-destructive, and in-line metrology has increased. Traditional scatterometry, which measures periodic structures by analyzing diffraction patterns, faces significant challenges:

Ambiguity (Multimodality): Solving the inverse problem (reconstructing geometry from diffraction data) often yields multiple valid geometric solutions (multimodality) that fit the data equally well, making it difficult to identify the true physical structure.
Sensitivity Limits: While using short-wavelength radiation (soft X-rays) improves resolution, the loss of phase information in far-field diffraction measurements exacerbates the ambiguity problem.
Material Complexity: Modern structures often consist of low-Z materials (e.g., Silicon, Nitrogen, Oxygen) where distinguishing between different material distributions and shapes is difficult using scattering alone.

The paper addresses the need for a hybrid metrology approach that combines Soft X-ray Scatterometry and Soft X-ray Fluorescence (XRF) to resolve these ambiguities and achieve sub-nanometer accuracy.

2. Methodology

The authors developed a hybrid measurement and reconstruction framework using a specific experimental setup and a combined regression model.

A. Experimental Setup (XFS)

Sample: A 1D nanoscale grating consisting of Silicon Nitride ( $Si_3N_4$ ) lines on a Silicon Dioxide ( $SiO_2$ ) layer, deposited on a Silicon ($Si$) substrate. The pitch is 100 nm, with nominal line width and height of 50 nm and 100 nm, respectively.
Instrumentation: A dedicated chamber at the BESSY II synchrotron facility equipped with:
- A monochromatized soft X-ray beam (incident energy $E_i = 680.0$ eV, chosen to excite N and O K-edges).
- A Charge-Coupled Device (CCD) to capture elastic scattering (diffraction efficiencies).
- A Silicon Drift Detector (SDD) positioned $30^\circ$ off the plane of incidence to capture fluorescence spectra ( $N-K\alpha$ and $O-K\alpha$ ).
Measurement Strategy: An angular scan of the grazing incidence angle ( $\alpha_i$ ) from $2^\circ$ to $6^\circ$ . This range captures the transition from total external reflection (dominated by scattering) to deeper penetration (dominated by absorption and fluorescence).

B. Theoretical Modeling

Forward Model: The standing wave field inside the grating is calculated using the Finite Element Method (FEM) via JCMsuite software. This allows for precise modeling of complex geometries (rounding, sidewall angles) and near-field effects.
Scattering: Diffraction efficiency is modeled as the squared Fourier transform of the standing wave field, including a Debye-Waller damping factor to account for line edge roughness.
Fluorescence: Fluorescence intensity is modeled using a modified Sherman equation, integrating the standing wave field intensity over the material volume, accounting for self-absorption and the specific detection geometry (off-plane incidence).

C. Inverse Problem & Hybrid Regression

Parameter Space: The reconstruction seeks to determine geometric parameters: height ( $h$ ), width ( $w$ ), sidewall angle ( $\beta$ ), corner radii ( $r_{top}, r_{bottom}$ ), layer thicknesses ( $d_{groove}, d_{line}$ ), and roughness ( $\xi$ ).
Weighted $\chi^2$ Function: To combine the two data sets, a weighted objective function ( $\chi^2_\gamma$ ) is minimized:
$\chi^2_\gamma = \gamma \frac{\chi^2_{scat}}{\hat{\chi}^2_{scat}} + (1-\gamma) \frac{\chi^2_{fluo}}{\hat{\chi}^2_{fluo}}$
Where $\gamma$ is a weighting parameter ( $0 \le \gamma \le 1$ ). The normalization by individual best-fit values ( $\hat{\chi}^2$ ) compensates for differences in data scale and uncertainty between scattering and fluorescence.
Optimization: Differential evolution is used to minimize the function. Principal Component Analysis (PCA) and K-means clustering are applied to analyze the solution space and identify clusters of solutions.

3. Key Contributions

Hybrid XFS Metrology: Demonstrated a novel experimental setup that simultaneously collects soft X-ray scattering and fluorescence data from the same sample volume, leveraging the complementary nature of the two signals.
Resolution of Multimodality: Showed that while scattering-only or fluorescence-only reconstructions yield ambiguous solutions (multiple clusters), the hybrid approach converges to a unique solution cluster that is robust against the weighting parameter $\gamma$ .
Weighting Parameter Analysis: Introduced a systematic method to determine the optimal balance between data sets. The study revealed that the "true" solution lies in a region where the reconstruction is independent of the weighting parameter $\gamma$ , whereas ambiguous solutions drift significantly as $\gamma$ changes.
Off-Plane Fluorescence Sensitivity: Highlighted the advantage of measuring fluorescence $30^\circ$ off the plane of incidence, which increases sensitivity to the grating profile due to shadowing effects, particularly for Oxygen K $\alpha$ emission.

4. Results

Cluster Identification: The optimization over 161 values of $\gamma$ $γ$ resulted in three distinct solution clusters (Orange, Blue, Purple).
- Orange & Blue Clusters: These solutions were highly dependent on $\gamma$ , drifting significantly as the weighting shifted between scattering and fluorescence. They represent ambiguous solutions dominated by one data type.
- Purple Cluster: This cluster remained stable (low relative range) regardless of $\gamma$ . It represents the unique, physically correct solution.
Optimal Solution: The Purple cluster yielded the following reconstructed parameters (mean values):
- Line Height ( $h$ ): $\approx 102.2$ nm
- Line Width ( $w$ ): $\approx 52.3$ nm
- Sidewall Angle ( $\beta$ ): $\approx 4.4^\circ$
- These values were consistent with the expected geometry and showed minimal sensitivity to the weighting parameter.
Data Complementarity: The results confirmed that scattering data is highly sensitive to the overall shape and periodicity, while fluorescence data provides specific information about material distribution and depth. The combination eliminates the "blind spots" of each individual technique.
Optimal Weighting: The study identified a range of $\gamma \approx 0.16$ to $0.67$ where the probability of finding the unique solution is highest. The optimal point was found at $\gamma \approx 0.56$ , slightly favoring fluorescence, likely due to the higher sensitivity of scattering to angular misalignment in this specific setup.

5. Significance

Reference Metrology: This work establishes soft X-ray hybrid scatterometry as a viable candidate for reference metrology in semiconductor manufacturing, capable of providing sub-nanometer accuracy without destructive cross-sectioning.
Solving the Inverse Problem: It provides a robust statistical framework for handling the inverse problem in nanometrology, demonstrating that combining orthogonal physical phenomena (scattering vs. fluorescence) is an effective strategy to resolve multimodality.
Process Control: The ability to uniquely determine complex 3D profiles of low-Z material structures is critical for the control of advanced nodes in EUV lithography, where traditional optical methods lack sufficient resolution and sensitivity.
Methodological Framework: The approach of using a weighting parameter to normalize data sets and identify solution stability offers a generalizable method for future hybrid metrology applications involving other combinations of techniques (e.g., XRR, GISAXS, AFM).