1. Bibliographic Information

• Title: Curvilinear Masks: Motivations and Metrology
• Authors: Linyong (Leo) Pang¹, Dakota Seal², Tom Boettiger², Nagesh Shirali¹, Grace Dai¹, Aki Fujimura¹
  • Affiliations: ¹D2S, Inc. (San Jose, CA, USA), ²Micron Technology, Inc. (Boise, ID, USA).
  • Background: The authors represent a collaboration between a computational lithography software company (D2S) and a major semiconductor manufacturer (Micron). This blend of expertise provides both a theoretical/software perspective and a practical manufacturing perspective on the topic.
• Journal/Conference: The paper cites presentations at SPIE Advanced Lithography + Patterning 2024 [18] and Photomask Japan 2024 [19], and publication in the Journal of Micro/Nanopatterning, Materials, and Metrology (JM3) [20]. JM3 is a reputable, peer-reviewed journal published by SPIE, focusing on the science and technology of lithography and patterning. Publication in this venue signifies a high standard of technical rigor.
• Publication Year: 2024
• Abstract: The paper addresses the semiconductor industry's shift from traditional "Manhattan" (rectilinear) photomasks to curvilinear masks, a change enabled by multi-beam mask writers. It outlines the benefits of curvilinear masks, including improved process windows, simpler mask rules, lower Mask Error Enhancement Factor (MEEF), and reduced mask variability. The authors identify a critical challenge: the lack of a well-defined critical dimension (CD) specification for curvilinear patterns, which hinders communication between mask shops and wafer fabs. To solve this, the paper proposes a new CD specification based on Edge Placement Error (EPE) statistics. It concludes by presenting experimental evidence that curvilinear masks exhibit approximately 20% less mask variability than their Manhattan counterparts.
• Original Source Link: /files/papers/68ef218358c9cb7bcb2c7f7c/paper.pdf. The paper has been formally published in a peer-reviewed journal.

2. Executive Summary

• Background & Motivation (Why):

  • Core Problem: For decades, semiconductor manufacturing has relied on photomasks with rectilinear patterns (known as "Manhattan" geometry) because they were easier to write with single-beam systems. However, as chip features shrink to the nanometer scale, Manhattan approximations become a significant source of error, limiting manufacturing process windows. The ideal mask shapes to print complex wafer designs are curvilinear. While new multi-beam mask writers can create these shapes, the industry lacks a standardized metrology framework to measure and specify their quality, as the traditional concept of Critical Dimension (CD) is not applicable to complex curves.
  • Importance & Gaps: This lack of a metrology standard is a major barrier to the widespread adoption of curvilinear masks, which offer substantial benefits for yield and performance in advanced semiconductor nodes (e.g., DRAM, logic). Without a shared "language" for quality control, collaboration between design, mask manufacturing, and wafer fabrication is inefficient and error-prone.
  • Innovation: The paper introduces a novel metrology framework that generalizes the concepts of CD, Line Edge Roughness (LER), and CD Uniformity (CDU) for any arbitrary shape. It proposes using Edge Placement Error (EPE)—the deviation of the actual mask shape from its target—as the fundamental unit of measurement.

• Main Contributions / Findings (What):

  • Comprehensive Motivation: The paper provides a clear and consolidated argument for adopting curvilinear masks, summarizing benefits across process window, mask rules, MEEF, and mask variability.
  • Novel Metrology Framework: It proposes a new standard for curvilinear mask metrology based on EPE statistics (sEPE), defining equivalents for traditional metrics:
    • $ΔCD$ equivalent: 2X Average EPE
    • LER equivalent: Local EPE Variation (LEPV)
    • CDU equivalent: 2X sEPE Uniformity (sEPE-U)
  • Empirical Validation: The paper presents a direct experimental comparison between curvilinear ILT masks and traditional Manhattan OPC masks. The results demonstrate that curvilinear masks have ~20% lower mask variation across mean, standard deviation, and maximum deviation metrics, providing concrete evidence for their superior manufacturing consistency.

3. Prerequisite Knowledge & Related Work

• Foundational Concepts:

  • Photomask (or Mask): A high-purity quartz plate coated with an opaque film (typically chrome-based). Patterns are etched into this film, creating a stencil used in photolithography to transfer a circuit design onto a silicon wafer.
  • Manhattan vs. Curvilinear Masks:
    • Manhattan Geometry: Refers to patterns composed exclusively of horizontal and vertical lines, resembling a city grid like Manhattan's. This was the standard due to limitations of older mask writing technology.
    • Curvilinear Geometry: Refers to patterns with smooth curves and arbitrary angles. These shapes are optically superior for printing complex designs on a wafer.
  • Inverse Lithography Technology (ILT): A computational technique that starts with the desired pattern on the wafer and works backward to calculate the optimal, often non-intuitive, mask pattern needed to produce it. ILT naturally generates curvilinear mask shapes (as seen in Image 1) because the physics of light diffraction smooths out sharp corners.
  • Optical Proximity Correction (OPC): A less aggressive computational technique than ILT. OPC modifies mask patterns by adding small serifs or jogs to line edges to compensate for optical distortions during lithography. Standard OPC is typically rule-based or model-based and produces Manhattan-like shapes.
  • Process Window: The range of process parameters (e.g., focus and exposure dose) within which a feature can be manufactured on a wafer while meeting all specifications (like size and shape). A wider process window means a more robust and higher-yielding manufacturing process.
  • Mask Error Enhancement Factor (MEEF): A measure of how much errors on the photomask (e.g., variations in line width) are amplified when transferred to the wafer. A lower MEEF is desirable, as it means the wafer pattern is less sensitive to imperfections in the mask.
  • Critical Dimension (CD): The width of a critical feature (like a transistor gate or a contact hole) on a mask or wafer. It's a key metric for process control. CD Uniformity (CDU) measures the consistency of CD across the mask or wafer.
  • Line Edge Roughness (LER): The random, high-frequency variation along the edge of a printed line. High LER can degrade device performance.

• Technological Evolution:

  • The paper situates its work at a pivotal moment in lithography. For decades, single-beam mask writers were the standard. These systems write patterns using rectangular "shots," making them efficient for Manhattan geometry but impractical for complex curvilinear shapes.
  • The advent of multi-beam mask writers (e.g., from IMS Nanofabrication, NuFlare) changed the landscape. These systems use thousands of tiny parallel electron beams to "paint" the mask pattern pixel by pixel, making it just as easy to write a curve as a straight line.
  • Simultaneously, computational power, particularly GPU acceleration, has made full-chip ILT feasible. Previously, ILT was too computationally expensive and had to be run on small, partitioned sections of a chip, leading to "stitching" errors where the sections met. Modern hardware allows for partition-free, full-chip ILT, unlocking its full potential.
  • This convergence of multi-beam writing and GPU-accelerated ILT has made curvilinear masks a practical reality, driving the need for the metrology solutions proposed in this paper.

• Differentiation:

  • This work differentiates itself from previous research on ILT (which focused on demonstrating wafer-level benefits) by tackling the crucial, practical problem of mask-level metrology. While others proved why curvilinear is better, this paper proposes how to measure and control it in a high-volume manufacturing environment, bridging a critical gap between theory and implementation.

4. Methodology (Core Technology & Implementation)

The paper's core contribution is a new metrology framework for curvilinear masks that overcomes the limitations of traditional CD-based measurements.

• Principles:

  • The fundamental problem is that "width" (CD) is a one-dimensional concept that is ill-defined for complex two-dimensional curvilinear shapes.
  • The proposed solution is to shift the measurement paradigm from a 1D CD to a 2D contour-based metric: Edge Placement Error (EPE).
  • EPE is defined as the positional difference between the actual fabricated mask contour and the intended target mask contour, measured perpendicularly to the target edge. This concept is universally applicable to any shape, whether Manhattan or curvilinear.

• Steps & Procedures: The proposed workflow is as follows:

  1. Obtain a high-resolution image of the mask pattern using a Scanning Electron Microscope (SEM).

  2. Extract the contour of the actual fabricated pattern from the SEM image.

  3. Align the extracted contour with the target design contour.

  4. Measure the EPE at discrete points along the entire target contour (e.g., every 4 nm, as illustrated in Image 10).

  5. Compute statistics from the collection of EPE measurements to characterize the mask quality.

     该图像是一个示意图，展示了设计目标、掩模目标和实际掩模之间的位置关系，以及每隔4nm测量一次EPE（边缘偏差误差）的过程，反映了曲线型掩模的度量方法。

• Proposed Curvilinear Metrics (Generalizations of Traditional Metrics):

  • Equivalent $ΔCD$: In a simple Manhattan line, the change in CD ($ΔCD$) is twice the EPE on one edge (assuming symmetric error). The paper generalizes this concept.
    • Proposed Metric: 2X Average EPE. The average of all EPE measurements along the contour is calculated. Multiplying this by two provides a single value analogous to the average $ΔCD$ for the entire feature.
  • Equivalent CD Standard: Instead of a single CD value, the authors propose using a statistical description of the EPE distribution.
    • Proposed Metric: Statistics-based EPE (sEPE). This includes the mean EPE, standard deviation (STD) of EPE, and a histogram of the EPE distribution. This provides a much richer description of the mask's dimensional fidelity.
  • Equivalent Line Edge Roughness (LER): LER is the high-frequency, local variation of an edge.
    • Proposed Metric: Local EPE Variation (LEPV). This is calculated as the standard deviation of EPE values within a small, local window around each measurement point. It directly captures the "roughness" of the contour deviation.
  • Equivalent CD Uniformity (CDU): CDU describes the variation of CD across a larger area.
    • Proposed Metric: 2X sEPE Uniformity (sEPE-U). This refers to the variation (e.g., standard deviation or range) of the 2X Average EPE metric when measured on multiple instances of the same feature across the mask.
  • Mask Variation Band: To visualize and quantify manufacturing variability, multiple SEM images of the same intended feature (from different locations on the mask or different masks) are captured and their contours are overlaid. The resulting "band" represents the total process variation. Statistics like the mean width, standard deviation of the width, and maximum width (range) of this band can be calculated to provide a comprehensive view of mask consistency.

5. Experimental Setup

• Datasets:
  • The experiment used specially designed test masks containing both curvilinear and Manhattan patterns placed side-by-side for direct comparison.
  • Curvilinear patterns: Generated using D2S's full-chip ILT software (TrueMask® ILT).
  • Manhattan patterns: Generated using a conventional OPC flow at Micron.
  • The masks were manufactured using a NuFlare MBM-2000 PLUS multi-beam mask writer, a state-of-the-art tool capable of producing curvilinear shapes.
• Evaluation Metrics:
  • The primary metrics were derived from the Mask Variation Band, as described in the methodology. The analysis focused on comparing the statistics of this band for the curvilinear patterns versus the Manhattan patterns.
  • The key reported statistics are the mean, standard deviation (STD), and maximum width of the variation band, measured across four different sites on the mask.
• Baselines:
  • The baseline for the comparison is the traditional Manhattan OPC mask.
  • The proposed approach is the curvilinear ILT mask.
  • The experiment is designed to isolate the effect of the mask shape (curvilinear vs. Manhattan) on manufacturing variability.

6. Results & Analysis

The paper presents compelling evidence that curvilinear masks are not only theoretically superior but also practically more consistent to manufacture than Manhattan masks.

• Benefit 1: Improved Process Window

  • Multiple studies are cited to show that ILT-generated curvilinear masks yield significantly wider process windows.

  • A 2020 study from Micron (Image 2) showed an 85% improvement in depth of focus (DoF) for DRAM contacts when using curvilinear ILT compared to standard OPC.

    该图像是一个图表，比较了DRAM接触孔中曲线反演光刻（Curvilinear ILT）与标准光刻修正（Standard OPC）的工艺窗口差异。图中显示曲线ILT相比标准OPC，焦深（DoF）约提升了85%，并通过PV带和强度阈值曲线直观展示了两者的性能区别。

  • Another study with D2S and Micron (Image 3) demonstrated a >100% increase in the process window for random contact layers. The tables in Image 3 show the number of points within a 10% CD variation spec across different dose and defocus conditions. The curvilinear TrueMask® ILT maintains good CD control over a much wider range of conditions.

    该图像是比较OPC工艺窗口和TrueMask® ILT工艺窗口的对比图，展示不同焦距和曝光剂量下的CD变化及对应SEM图像。图中绿色表示小于10%的CD变化，反映TrueMask® ILT工艺具有更好的工艺宽容度和CD一致性。

  • Transcription of Table Data from Image 3 (OPC Process Window):

    Dose (mJ)	-60	-40	-20	0	20	40	60	Dose(%)
    21	55.4	59.6	62.5	61.2	60.5	51.6	41.8	93.3%
    21.5	56.3	57.5	63.7	61.0	58.1	52.9	40.3	95.6%
    22	57.3	60.8	61.0	60.8	57.1	48.3	41.6	97.8%
    22.5	54.0	52.8	60.1	55.8	54.7	52.1	39.3	100.0%
    23	49.4	54.4	55.3	60.3	54.5	49.2	36.1	102.2%
    23.5	49.2	50.9	54.5	55.0	51.6	46.1	33.9	104.4%
    24	54.0	56.2	54.8	49.9	50.9	44.7	28.6	106.7%

  • Transcription of Table Data from Image 3 (TrueMask® ILT Process Window):

    Dose (mJ)	-60	-40	-20	0	20	40	60	Dose(%)
    21	57.7	60.5	59.6	64.1	58.8	62.1	58.4	93.3%
    21.5	54.0	56.9	58.1	60.9	59.0	62.8	59.4	95.6%
    22	60.6	55.4	59.3	60.3	57.9	59.5	58.8	97.8%
    22.5	54.5	57.7	57.6	60.6	56.2	59.4	57.9	100.0%
    23	52.0	56.5	57.0	56.2	57.3	60.9	58.3	102.2%
    23.5	52.2	56.7	55.1	54.6	54.9	55.6	54.7	104.4%
    24	48.6	52.7	51.6	50.8	56.1	52.3	56.1	106.7%

    (Note: Green cells indicate <10% CD Variation)

• Benefit 2: Denser Layouts and Simpler Rules

  • Curvilinear shapes allow for denser packing. As shown in Image 4, by replacing squares with circles, the minimum corner-to-corner distance constraint is relaxed, potentially enabling a 14% pitch reduction.

  • Mask design rules are also simplified. Manhattan masks require complex rules for corners, jogs, and line-ends. Curvilinear masks can be defined by simpler rules: minimum curvature, minimum CD, and minimum spacing (Image 5).

    该图像是示意图，展示了从方形（矩形）图形到圆形（曲线）图形时，最小角对角距离的变化。图中通过红色箭头标注了两个图形间的最小间距，表明采用曲线图形可缩小图案间距。

    该图像是示意图，展示了不同颜色球体对应的曲线边界上的接触点，反映了曲线掩膜设计中边界与关键点的关系。

• Benefit 3: Lower MEEF and Better Dose Margin

  • Manhattan shapes have sharp 90-degree corners that are difficult to write consistently, leading to poor dose margin (Image 7). Curvilinear shapes have no sharp corners, resulting in uniform and better dose margins.

  • A 2022 study showed that smoother curvilinear lines exhibit an ~28% improvement (reduction) in MEEF compared to Manhattanized diagonal lines (Image 8). This is because MEEF is proportional to the perimeter-to-area ratio ($MEEF \propto \frac{Perimeter}{Area}$), and the jagged "staircase" edges of Manhattan approximations (Image 6) have a much larger perimeter than a smooth curve for the same area.

    该图像是对比传统曼哈顿掩模（左图红色方形边缘）与曲线掩模（右图绿色圆滑边缘）的示意图，展示了设计尺寸为掩模144纳米、晶圆36纳米的同一结构轮廓的差异。

    该图像是图表，展示了不同形状（方形边缘、方形角和圆形）剂量裕度随相对CD变化的曲线及曼哈顿阵列与曲线阵列的剂量裕度对比，图中用红绿颜色区分剂量裕度的好坏。

    该图像是一个示意图，展示了曼哈顿（Manhattan）和曲线（Smooth）掩模板在不同测试图案（Mid-Edge和Line-End）上的掩模误差增强因子（MEEF）对比，包含公式 $$MEEF \propto \frac{Perimeter}{Area}$$，显示曲线掩模板MEEF改善约28%。

  • Transcription of Table Data from Image 8 (MEEF Comparison):

    Test Pattern	Manhattan	Smooth
    Line-End	7.8	5.6
    Mid-Edge	3.3	2.4

• Core Result: Smaller Mask Variation

  • The experimental comparison of mask variation bands for curvilinear and Manhattan patterns yielded the paper's key finding.
  • The histograms of the variation band width (Figure 21 in the paper) show a clear difference: the distribution for the curvilinear mask is tighter and shifted closer to zero, indicating smaller and more consistent manufacturing errors.
  • The quantitative analysis (Figures 22 and 23 in the paper) confirms this. Across all four measured sites, the curvilinear masks showed an approximate 20% improvement (reduction) in the mean, STD, and maximum values of the mask variation band compared to the Manhattan masks. This is a significant improvement in manufacturing consistency.

7. Conclusion & Reflections

• Conclusion Summary: The paper successfully argues that the move to curvilinear masks is a necessary and beneficial evolution in semiconductor manufacturing. It highlights the significant advantages in process window, layout density, MEEF, and manufacturing consistency. The central contribution is a practical and extensible metrology framework based on EPE statistics, which provides the missing "standard" needed for high-volume production. The experimental results, showing a ~20% reduction in mask variability, provide strong validation for the superiority of curvilinear shapes.

• Limitations & Future Work:

  • The authors note that the analysis was performed using their own proprietary software (D2S). While effective, broader adoption of the proposed metrology would require standardization and implementation across different commercial SEM and software platforms.
  • The paper focuses on mask-level variation. Future work could extend this analysis to correlate the proposed EPE-based mask metrics directly with wafer-level performance and yield, closing the loop between mask quality and final device outcome.

• Personal Insights & Critique:

  • This paper is highly impactful because it addresses a critical, practical barrier to technological advancement. The transition from 1D CD to 2D EPE is a fundamental paradigm shift in metrology, akin to moving from simple rulers to advanced 3D scanners. It's a necessary step to manage the complexity of modern chip designs.
  • The collaboration between a software vendor (D2S) and a leading manufacturer (Micron) lends significant credibility to the findings. This is not just a theoretical proposal but a solution tested and validated in a real-world manufacturing context.
  • The proposed sEPE framework is elegant in its simplicity and power. By capturing the full distribution of edge placement errors, it offers a far more comprehensive picture of mask quality than a few sparse CD measurements ever could. This rich data can be invaluable for process control and diagnosing manufacturing issues. The paper effectively makes the case that for 2D patterns, we need 2D metrology.