layout: true class: typo, typo-selection --- count: false class: nord-dark, middle, center # π¬ Phase Shifting Mask & Optical Proximity Correction @luk036 π¨βπ» Β· 2026 π --- ### π Overview .pull-left[ - **Background** ποΈ - **What is Phase Shifting Mask?** β - **Phase Conflict Graph** βοΈ - **Phase Assignment Problem** 𧩠- Greedy Approach π€ - Planar Graph Approach π ] .pull-right[  ] --- ### ποΈ Background .pull-left[ - Chip sizes continue to shrink, leading to lower power consumption β‘π - We're approaching optical lithography limits π - Current methods can't achieve next-generation requirements π§ ] .pull-right[  ] --- ### π¨οΈ Lithography Process .pull-left[ 1. **Photo-resist coating** (ε ι»ζΆε±) π¨ 2. **Illumination** (ε η §) π‘ 3. **Exposure** (ζε ) πΈ 4. **Etching** (θε») βοΈ 5. **Impurities doping** (ζ质ζΊζ) π§ͺ 6. **Metal connection** π ] .pull-right[  ] --- ### π Sub-wavelength Lithography .pull-left[ - Feature size (45nm) much smaller than wavelength (193nm) π  ] .pull-right[ - Mask layout β final chip: π± - Features distort π - Yields decline π  ] ---  --- ### βοΈ Lithography's DFM Impact Lithography critically defines IC physical features. DFM techniques like phase shifting masks address manufacturing issues. π οΈ **Main challenges:** - Variations in feature dimensions/shapes π - Light intensity/wavelength/angle deviations π - Process-induced variations affecting performance/yield --- ### π Lithography DFM Impact (Visual) .pull-left[  ] .pull-right[  ] --- ### π» DFM Tool (Mentor Graphics)  --- ### π Optical Proximity Correction (OPC) .pull-left[ Photolithography enhancement technique: - Compensates for diffraction/image errors - Modifies mask layout for better wafer fidelity π¨οΈ ] .pull-right[  ] --- ### π οΈ OPC + SRAF  --- ### π OPC & PSM Synergy .pull-left[ **Results comparison:** - A = Original layout π °οΈ - B = Uncorrected π ±οΈ - C = PSM + OPC Β©οΈ ] .pull-right[  ] --- ### π PSM & OPC Flow .mermaid[ <pre> graph LR A[Layout] --> B[Phase Assignment] B --> C[OPC] C --> D[Mask] D --> E[Lithography] style A fill:#e3f2fd,stroke:#1565c0,stroke-width:3px style B fill:#f3e5f5,stroke:#7b1fa2,stroke-width:3px style C fill:#fff3e0,stroke:#e65100,stroke-width:3px style D fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px style E fill:#ffcdd2,stroke:#c62828,stroke-width:3px </pre> ] --- class: nord-light, middle, center ## PSM & Phase Assignment π¨ --- ### π Phase Shifting Mask  --- ### βοΈ Phase Conflict Graph - Edges connect features with separation β€ b (dark field) β« - Similar logic for bright field βͺ - Construction: Plane sweeping + dynamic priority search tree π³  --- ### 𧩠Phase Assignment Problem .pull-left[ - **Input**: Graph G = ({\color{salmon}V}, {\color{lime}E}) π - **Solution**: Color assignment ${\color{coral}c}: {\color{salmon}V} \to [1..{\color{orange}k}] \ (k=2)$ π¨ - **Goal**: Minimize monochromatic edge weights βοΈ ] .pull-right[  ] --- ### 𧩠Problem Complexity - Generally NP-hard π€― - Polynomial solution for planar graphs (k=2) via T-join problem in dual graph [Hadlock75] β±οΈ - 2-approximation possible using primal-dual method 2οΈβ£ --- class: nord-light, middle, center ## Greedy Approach π€ --- ### π€ Greedy Algorithm Overview 1. Build max weighted spanning tree (MST) π³ 2. Color MST nodes π¨ 3. Reinsert non-conflicting edges π - **Time**: O(N log N) β±οΈ - Works for non-planar graphs π --- ### π Greedy Algorithm Steps .pull-left[ **Step 1**: Construct MST (Kruskal's algorithm) π¦ ] .pull-right[  ] --- ### π¨ Greedy Algorithm (Cont'd) .pull-left[ **Step 2**: Color MST nodes π¨ ] .pull-right[  ] --- ### β Greedy Algorithm (Cont'd) .pull-left[ **Step 3**: Reinsert compatible edges β ] .pull-right[  ] ---  --- class: nord-light, middle, center ## Planar Graph Approach πΊοΈ --- ### π Alternative Approaches - **MAX-CUT formulation**: 1.1383-approximation via semi-definite programming [Goemans-Williamson 93] π - **Planar graph method**: Remove minimal edges β apply planar techniques π *(Note: Optimal planar subgraph is NP-hard) π€―* --- ### πΊοΈ Planar Graph Approach (Hadlock) 1. Decompose G into biconnected components π¨ 2. For each component G': a. Approximate as planar graph P π b. Build planar embedding ποΈ c. Construct dual graph P* π d. Create complete graph of odd-degree vertices 𧩠e. Solve min perfect matching (conflict edges) π 3. Reinsert non-conflicting edges π --- ### π Hadlock's Algorithm **Key Steps**: - **Decomposition**: Linear-time biconnected components (LEDA) πͺ - **Planar Approximation**: Greedy O(nΒ²) ποΈ - **Component Processing**: Embedding β dual graph β odd-vertex matching π - **Edge Management**: Delete conflicts, keep others π€ --- ### ποΈ Planar Approach Visualization - **Step 1**: Biconnected decomposition  --- ### ποΈ Planar Approach (Cont'd) .pull-left[ **Step 2**: Planar approximation - NP-hard core - O(nΒ²) greedy baseline - Open for better solutions ] .pull-right[  ] --- ### ποΈ Planar Approach (Cont'd) - **Step 3**: Linear-time planar embedding  *Note: Unique only for tri-connected graphs* --- ### π Planar Approach (Cont'd) .pull-left[ **Step 4**: Dual graph construction (linear time) ] .pull-right[  ] --- ### 𧩠Planar Approach (Cont'd) .pull-left[ **Step 5**: Min perfect matching - Solvable in polynomial time - Network flow formulation *Complete vs Voronoi graphs* ] .pull-right[  ] --- ### π Planar Approach (Cont'd) .pull-left[ **Step 6**: Non-conflicting edge reinsertion *Track conflicts during process* ] .pull-right[  ] --- count: false class: nord-dark, middle, center .pull-left[ ## Q&A π€ ] .pull-right[  ]