layout: true class: typo, typo-selection --- class: nord-dark, middle, center count: false # Phase Shifting Mask & Optical Proximity Correction @luk036 👨‍💻 · 2026 📅 --- ### Overview 📋✨ .pull-left[ - **Background** 🏛️ - **What is Phase Shifting Mask?** ❓ - **Phase Conflict Graph** ⚖️ - **Phase Assignment Problem** 🧩 - Greedy Approach 🤑 - Planar Graph Approach 📊 ] .pull-right[  ] --- ### Background 🏛️ .pull-left[ - Chip sizes continue to shrink, leading to lower power consumption ⚡📉 - We're approaching optical lithography limits 🛑 - Current methods can't achieve next-generation requirements � ] .pull-right[  ] --- ### Lithography Process 🖨️ .pull-left[ 1. **Photo-resist coating** (光阻涂层) 🎨 2. **Illumination** (光照) 💡 3. **Exposure** (曝光) 📸 4. **Etching** (蚀刻) ⚒️ 5. **Impurities doping** (杂质掺杂) 🧪 6. **Metal connection** 🔗 ] .pull-right[  ] --- ### Sub-wavelength Lithography 🌊📏 .pull-left[ - Feature size (45nm) much smaller than wavelength (193nm) 📏  ] .pull-right[ - Mask layout ≠ final chip: 😱 - Features distort 🌀 - Yields decline 📉  ] ---  --- ### Lithography's DFM Impact ❓⚙️ Lithography critically defines IC physical features. DFM techniques like phase shifting masks address manufacturing issues. 🛠️ **Main challenges:** - Variations in feature dimensions/shapes 📉 - Light intensity/wavelength/angle deviations 🔍 - Process-induced variations affecting performance/yield --- ### Lithography DFM Impact (Visual) ⚙️📊 .pull-left[  ] .pull-right[  ] --- ### DFM Tool (Mentor Graphics) 🛠️💻  --- ### Optical Proximity Correction (OPC) 🔍✨ .pull-left[ Photolithography enhancement technique: - Compensates for diffraction/image errors - Modifies mask layout for better wafer fidelity 🖨️ ] .pull-right[  ] --- ### OPC + SRAF 🛠️➕  --- ### OPC & PSM Synergy 🔄⚙️ .pull-left[ **Results comparison:** - A = Original layout 🅰️ - B = Uncorrected 🅱️ - C = PSM + OPC ©️ ] .pull-right[  ] --- ### PSM & OPC Flow 🔄 .mermaid[ <pre> graph LR A[Layout] --> B[Phase Assignment] B --> C[OPC] C --> D[Mask] D --> E[Lithography] style A fill:#2196f3 style B fill:#9c27b0 style C fill:#ff9800 style D fill:#4caf50 style E fill:#f44336 </pre> ] --- ### Phase Shifting Mask ⚙️🌓  --- ### Phase Conflict Graph ⚖️📊 - Edges connect features with separation ≤ b (dark field) ⚫ - Similar logic for bright field ⚪ - Construction: Plane sweeping + dynamic priority search tree 🌳  --- ### Phase Assignment Problem 🧩🎨 .pull-left[ - **Input**: Graph G = ({\color{salmon}V}, {\color{lime}E}) 📊 - **Solution**: Color assignment c: V → [1..k] (k=2) 🎨 - **Goal**: Minimize monochromatic edge weights ⚖️ ] .pull-right[  ] --- ### Problem Complexity 🧩 - Generally NP-hard 🤯 - Polynomial solution for planar graphs (k=2) via T-join problem in dual graph [Hadlock75] ⏱️ - 2-approximation possible using primal-dual method 2️⃣ --- ### Greedy Algorithm Overview 🤑📈 1. Build max weighted spanning tree (MST) 🌳 2. Color MST nodes 🎨 3. Reinsert non-conflicting edges 🔄 - **Time**: O(N log N) ⏱️ - Works for non-planar graphs 🌐 --- ### Greedy Algorithm Steps .pull-left[ **Step 1**: Construct MST (Kruskal's algorithm) 📦 ] .pull-right[  ] --- ### Greedy Algorithm (Cont'd) .pull-left[ **Step 2**: Color MST nodes 🎨 ] .pull-right[  ] --- ### Greedy Algorithm (Cont'd) .pull-left[ **Step 3**: Reinsert compatible edges ➕ ] .pull-right[  ] ---  --- ### Alternative Approaches 🔄🧠 - **MAX-CUT formulation**: 1.1383-approximation via semi-definite programming [Goemans-Williamson 93] 📐 - **Planar graph method**: Remove minimal edges → apply planar techniques 📊 *(Note: Optimal planar subgraph is NP-hard) 🤯* --- ### Planar Graph Approach (Hadlock) 1. Decompose G into biconnected components 🔨 2. For each component G': a. Approximate as planar graph P 🔄 b. Build planar embedding 🏗️ c. Construct dual graph P* 🔄 d. Create complete graph of odd-degree vertices 🧩 e. Solve min perfect matching (conflict edges) 🔍 3. Reinsert non-conflicting edges 🔙 --- ### Hadlock's Algorithm 🔍 **Key Steps**: - **Decomposition**: Linear-time biconnected components (LEDA) 🪓 - **Planar Approximation**: Greedy O(n²) 🏞️ - **Component Processing**: Embedding → dual graph → odd-vertex matching 🔄 - **Edge Management**: Delete conflicts, keep others 🤝 --- ### Planar Approach Visualization - **Step 1**: Biconnected decomposition  --- ### Planar Approach (Cont'd) .pull-left[ **Step 2**: Planar approximation - NP-hard core - O(n²) greedy baseline - Open for better solutions ] .pull-right[  ] --- ### Planar Approach (Cont'd) - **Step 3**: Linear-time planar embedding  *Note: Unique only for tri-connected graphs* --- ### Planar Approach (Cont'd) .pull-left[ **Step 4**: Dual graph construction (linear time) ] .pull-right[  ] --- ### Planar Approach (Cont'd) .pull-left[ **Step 5**: Min perfect matching - Solvable in polynomial time - Network flow formulation *Complete vs Voronoi graphs* ] .pull-right[  ] --- ### Planar Approach (Cont'd) .pull-left[ **Step 6**: Non-conflicting edge reinsertion *Track conflicts during process* ] .pull-right[  ] --- class: nord-dark, middle, center count: false .pull-left[ # Q&A 🎤 ] .pull-right[  ]