The Evolution of Lithographic Simulation
In the history of integrated circuit fabrication, the ability to predict how a pattern on a photomask will transfer onto a silicon wafer is fundamental to yield and performance. For decades, the industry relied on the Hopkins’ formulation of partially coherent imaging. This method uses Transmission Cross Coefficients (TCCs) to describe how different parts of the light source interact with the mask to form an image. However, as the wavelength of light shrunk to 13.5nm with the introduction of EUV lithography, the assumptions of the traditional TCC model began to fail.
The shift to High-NA EUV, which increases the numerical aperture from 0.33 to 0.55, introduces even more severe physical constraints. At these higher angles, the light no longer behaves as a simple scalar wave; instead, its vector nature—specifically its polarization—plays a dominant role in image formation. Furthermore, the physical structure of the mask itself, which consists of multilayer molybdenum-silicon (Mo/Si) stacks and an absorber layer, can no longer be treated as a thin, two-dimensional surface. The "shadowing" and "3D" effects caused by the thickness of these layers significantly distort the light, leading to placement errors and contrast loss. The Institute of Science Tokyo research directly solves these issues by extending the STCC formula to account for these multi-dimensional variables.
Technical Breakthrough: Integrating Polarization and M3D Effects
The core contribution of the paper is the extension of the STCC formula to include vector-based polarization. In standard EUV systems, polarization was often neglected because the angles of incidence were relatively small. In High-NA systems, the light hits the mask and the wafer at much steeper angles. This causes the transverse electric (TE) and transverse magnetic (TM) components of the light to behave differently, a phenomenon that can drastically alter the contrast of the printed features.
The researchers demonstrated that by making the TCC dependent on the source position, they could more accurately capture how the light’s electromagnetic field interacts with the three-dimensional topography of the mask. This "source-position-dependent" approach allows the simulation to account for the fact that light coming from different parts of the illuminator "sees" the mask 3D structures differently. By incorporating these effects into a unified STCC formula, the team has enabled a simulation environment that is fast enough for Optical Proximity Correction (OPC) cycles while maintaining the accuracy of rigorous Maxwell’s equation solvers.
Chronology of High-NA EUV Development
The publication of this paper occurs at a critical juncture in the semiconductor industry’s timeline. The journey toward this mathematical breakthrough can be traced through several key milestones:
- 2019: The first commercial high-volume manufacturing (HVM) using 0.33 NA EUV lithography begins, enabling the 7nm and 5nm nodes.
- 2021-2022: ASML and its partners identify "Mask 3D effects" and "polarization" as the two most significant hurdles for the upcoming High-NA transition.
- 2023: Early prototypes of the Twinscan EXE:5000 (ASML’s first High-NA scanner) are assembled. Initial testing confirms that traditional scalar imaging models result in significant discrepancies between simulated and printed wafers.
- 2024: The merger forming the Institute of Science Tokyo (combining Tokyo Institute of Technology and Tokyo Medical and Dental University) creates a concentrated hub for interdisciplinary semiconductor research.
- 2025: Leading chipmakers like Intel and TSMC begin receiving the first production-ready High-NA scanners. The need for rapid, accurate OPC models becomes urgent.
- May 2026: The research team led by Hiroyoshi Tanabe publishes the STCC formula in JM3, providing the mathematical foundation for the industry’s software tools to catch up with the hardware capabilities of 0.55 NA systems.
Supporting Data and Comparative Analysis
The paper provides rigorous data comparing the new STCC method against existing simulation models. One of the primary metrics used is the "Simulation Error vs. Computational Speed" trade-off. Traditional rigorous coupled-wave analysis (RCWA), which solves Maxwell’s equations for the mask, is highly accurate but too slow for full-chip simulations. On the other hand, traditional TCC models are fast but show an error rate exceeding 10% in edge placement when applied to High-NA conditions.
The researchers’ data indicates that the STCC formula reduces this error to less than 1% while maintaining a computational overhead that is only marginally higher than standard TCC models. Specifically, the paper highlights:
- Contrast Improvement: By accounting for polarization, the STCC formula predicted a 15% difference in image contrast between TE and TM polarization states at the 0.55 NA level, a factor that would be missed by scalar models.
- Shadowing Compensation: The formula accurately modeled the "mask shadowing" effect, where the absorber height causes a shift in the perceived position of features. The STCC model showed a high correlation with experimental data for feature sizes below 10nm.
- Throughput Gains: In a simulated OPC environment, the STCC-based algorithm processed complex logic patterns 50 times faster than a full vector-based M3D solver without losing critical accuracy.
Industry Implications and Official Context
The implications of this research extend across the entire semiconductor ecosystem. In the world of Electronic Design Automation (EDA), companies such as Synopsys, Cadence, and Siemens EDA are in a constant race to provide chip designers with tools that can predict manufacturing outcomes. The STCC formula is expected to be integrated into the next generation of OPC and Inverse Lithography Technology (ILT) software.
While official corporate statements from the major toolmakers were not part of the original paper, industry analysts suggest that this research provides the "missing link" for 2nm and 1.4nm design rules. Without the ability to simulate polarization-dependent imaging, chip designers would have to rely on excessive "guard-banding"—making designs less dense to avoid manufacturing errors—which would negate the benefits of moving to High-NA EUV in the first place.
"The purpose of this work is to extend the STCC formula to include the polarization effect and enable accurate and fast simulation of polarization-dependent imaging in high-NA EUV lithography," the authors noted in their summary. This objective aligns with the broader industry goal of "computational scaling," where software improvements are used to extract the maximum performance from multi-billion-dollar hardware investments.
Broader Impact on Global Technology
The successful implementation of the STCC formula has significant geopolitical and economic ramifications. As the global race for semiconductor supremacy intensifies, the ability to master High-NA EUV is seen as the ultimate differentiator between leading-edge foundries and those left behind. By providing a faster path to high-yield manufacturing, the Institute of Science Tokyo’s work supports the continued viability of Moore’s Law.
Furthermore, the research highlights the growing importance of the "Japan-Europe-US" research corridor. With ASML (Netherlands) providing the scanners, the Institute of Science Tokyo (Japan) providing the theoretical frameworks, and companies like Intel (US) and TSMC (Taiwan) implementing them, the paper serves as a testament to the collaborative nature of modern physics and engineering.
In the long term, the STCC formula may also find applications beyond EUV. The mathematical principles of source-position-dependent coefficients can be adapted for other forms of advanced microscopy and nanometrology, where high-angle light and three-dimensional structures interact. For now, however, its most critical role remains on the cleanroom floor, where it will help define the boundaries of the smallest features ever created by human engineering.
Future Research Directions
Following the publication of this paper, the research team suggests that the next frontier involves integrating these formulas with machine learning (ML) models. While the STCC formula provides a robust physical foundation, ML can be used to further accelerate the "training" of these models for specific mask materials and resist chemistries. As the industry looks toward "Hyper-NA" (NA values greater than 0.7), the foundations laid by Tanabe, Sugiyama, Shimoda, and Takahashi will likely serve as the starting point for even more advanced explorations into the physics of light and matter at the atomic scale.
The paper, titled "Source-position-dependent transmission cross coefficient formula including polarization and mask three-dimensional effects in high-numerical-aperture extreme ultraviolet lithography," is currently available in the SPIE Digital Library and stands as a cornerstone document for the 2026 lithography roadmap. It underscores the reality that as we reach the limits of physical scaling, the path forward is paved with increasingly sophisticated mathematical and computational innovations.
