layout: true class: typo, typo-selection --- count: false class: nord-dark, middle, center # 🔬 Phase Shifting Mask & Optical Proximity Correction ### PSM + OPC for Sub-wavelength Lithography 🖨️ @luk036 👨‍💻 · 2026 📅 --- ### 📋 Agenda .pull-left[ **1️⃣ Background** 🏛️ - Why DFM? Chip Scaling vs Optics - Sub-wavelength Lithography **2️⃣ OPC** 🎯 - Diffraction Problem - OPC + SRAF **3️⃣ Phase Shifting Mask** 🎨 - PSM Principle - Phase Conflict Graph ] .pull-right[ **4️⃣ Phase Assignment Problem** 🧩 - Greedy Approach 🤑 - Planar Graph Approach 🗺️ **5️⃣ Hadlock's Algorithm** 🔍 - 6-Step Planar Method - Approach Comparison **6️⃣ Summary** 🎯 - Key Takeaways - Q&A ] --- class: nord-light, middle, center ## 🏛️ Part 1: Background --- ### 🌍 Why DFM Matters .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 🚧 - **Design for Manufacturing (DFM)** bridges layout and fabrication 🛠️ ] .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[ .mermaid[ <pre> graph TD A["🎨 Photo-resist\ncoating"] --> B["💡 Illumination"] B --> C["📸 Exposure"] C --> D["⚒️ Etching"] D --> E["🧪 Doping"] E --> F["🔗 Metal\nconnection"] style A fill:#e3f2fd,stroke:#1565c0,color:#2e3440,stroke-width:3px style B fill:#fff9c4,stroke:#f57f17,color:#2e3440,stroke-width:3px style C fill:#ffe0b2,stroke:#e65100,color:#2e3440,stroke-width:3px style D fill:#c8e6c9,stroke:#2e7d32,color:#2e3440,stroke-width:3px style E fill:#f3e5f5,stroke:#7b1fa2,color:#2e3440,stroke-width:3px style F fill:#e8eaf6,stroke:#283593,color:#2e3440,stroke-width:3px </pre> ] **The wafer's image quality is set at the *exposure* step 📸** ] --- ### 🌊 Sub-wavelength Lithography .pull-left[ - Feature size (**45nm**) much smaller than wavelength (**193nm**) 📏 - Resolution limit (Rayleigh criterion): **$R = k_1 \dfrac{\lambda}{\mathrm{NA}}$** - $k_1$: process constant (PSM lowers it! ✨) - $\lambda$: exposure wavelength - $\mathrm{NA}$: numerical aperture  ] .pull-right[ - Mask layout ≠ final chip 😱 - Features distort 🌀 - Yields decline 📉 - Diffraction blurs edges beyond resolution  ] --- ### ⚙️ 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 📊 .pull-left[  ] .pull-right[  ] --- class: nord-light, middle, center ## 🎯 Part 2: Optical Proximity Correction --- ### 🔍 The Diffraction Problem .pull-left[ **Light doesn't play fair** 🌊 - Edges of the mask pattern **diffract** light - Corners round off, lines blur - Adjacent features bleed into each other - Result: **wafer ≠ mask** 😱 ] .pull-right[ .mermaid[ <pre> graph LR M["🎭 Ideal mask\nlayout"] --> D["🌊 Diffraction\n+ image errors"] D --> W["😖 Distorted wafer\nlow yield"] MC["🎭 Mask + OPC\n(pre-compensated)"] --> DC["🌊 Diffraction\n(cancels errors)"] DC --> WC["😍 Faithful wafer\nhigh yield"] style M fill:#e3f2fd,stroke:#1565c0,color:#2e3440,stroke-width:3px style D fill:#ffcdd2,stroke:#c62828,color:#2e3440,stroke-width:3px style W fill:#ffe0b2,stroke:#e65100,color:#2e3440,stroke-width:3px style MC fill:#fff9c4,stroke:#f57f17,color:#2e3440,stroke-width:3px style DC fill:#e8eaf6,stroke:#283593,color:#2e3440,stroke-width:3px style WC fill:#c8e6c9,stroke:#2e7d32,color:#2e3440,stroke-width:3px </pre> ] ] --- ### 🔍 What is OPC? .pull-left[ **Optical Proximity Correction** — a photolithography enhancement technique: - Compensates for **diffraction/image errors** 🌀 - Modifies the **mask layout** for better wafer fidelity 🖨️ - Adds **biasing**, **serifs**, and **assist features** ✂️ ] .pull-right[  ] --- ### 🛠️ OPC + SRAF .pull-left[ **Sub-Resolution Assist Features (SRAF)**: - Tiny bars too small to print ⚙️ - Sculpt the light field to sharpen main features 🎯 - Combined with OPC for maximum fidelity ] .pull-right[  ] --- ### 🔄 OPC & PSM Synergy .pull-left[ **Results comparison:** - **A** = Original layout 🅰️ - **B** = Uncorrected 🅱️ - **C** = PSM + OPC ©️ *PSM fixes phase, OPC fixes amplitude — together they hit the target* 🎯 ] .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:#e3f2fd,stroke:#1565c0,color:#2e3440,stroke-width:3px style B fill:#f3e5f5,stroke:#7b1fa2,color:#2e3440,stroke-width:3px style C fill:#fff3e0,stroke:#e65100,color:#2e3440,stroke-width:3px style D fill:#c8e6c9,stroke:#2e7d32,color:#2e3440,stroke-width:3px style E fill:#ffcdd2,stroke:#c62828,color:#2e3440,stroke-width:3px </pre> ] **Today's focus:** the **Phase Assignment** step 🎨 — a graph coloring problem! --- class: nord-light, middle, center ## 🎨 Part 3: Phase Shifting Mask --- ### 🌓 PSM Principle .pull-left[ - Adjacent apertures get **opposite phases** (0° / 180°) 🔄 - At the boundary, light **destructively interferes** 🌑 - Produces a sharp **intensity null** where the dark line belongs 🎯 - Resolution doubles without shrinking $\lambda$ ✨ **$I = |E_1 + E_2\,e^{i\Delta\varphi}|^2$** with **$\Delta\varphi = \pi$** ⇒ **$I \to 0$** ] .pull-right[  ] --- ### 💡 PSM: Destructive Interference .mermaid[ <pre> graph TD subgraph "Conventional Mask" A1["Aperture\n0°"] --> W1["+ Bright edge\nblurry boundary 🌫️"] A2["Aperture\n0°"] --> W1 end subgraph "Phase Shifting Mask" B1["Aperture\n0°"] --> W2["|E1 − E2|² ≈ 0\nsharp dark line ✅"] B2["Aperture\n180°"] --> W2 end style A1 fill:#ffe0b2,stroke:#e65100,color:#2e3440,stroke-width:3px style A2 fill:#ffe0b2,stroke:#e65100,color:#2e3440,stroke-width:3px style W1 fill:#ffcdd2,stroke:#c62828,color:#2e3440,stroke-width:3px style B1 fill:#e3f2fd,stroke:#1565c0,color:#2e3440,stroke-width:3px style B2 fill:#f3e5f5,stroke:#7b1fa2,color:#2e3440,stroke-width:3px style W2 fill:#c8e6c9,stroke:#2e7d32,color:#2e3440,stroke-width:3px </pre> ] **The catch:** adjacent features that *must* be same-phase create **conflicts** ⚠️ --- ### ⚖️ Phase Conflict Graph .pull-left[ - **Nodes** = features on the mask 🧩 - **Edges** connect features with separation $\leq b$ (dark field) ⚫ - Similar logic for bright field ⚪ - Construction: **Plane sweeping** + **dynamic priority search tree** 🌳 ] .pull-right[  ] --- ### 🕸️ Conflict Graph Example .mermaid[ <pre> graph LR F1["🧩 Feature A"] -- "⚠️ conflict\n≤ b" --> F2["🧩 Feature B"] F2 -- "⚠️ conflict\n≤ b" --> F3["🧩 Feature C"] F3 -- "⚠️ conflict\n≤ b" --> F4["🧩 Feature D"] F1 -. "✅ ok\n> b" .-> F4 F2 -. "✅ ok\n> b" .-> F4 style F1 fill:#e3f2fd,stroke:#1565c0,color:#2e3440,stroke-width:3px style F2 fill:#e3f2fd,stroke:#1565c0,color:#2e3440,stroke-width:3px style F3 fill:#e3f2fd,stroke:#1565c0,color:#2e3440,stroke-width:3px style F4 fill:#e3f2fd,stroke:#1565c0,color:#2e3440,stroke-width:3px </pre> ] **Conflict edges** force different phases (0°/180°) — coloring must be **proper** 🎨 --- ### 🧩 Phase Assignment Problem .pull-left[ - **Input**: Graph $G = ({\color{salmon}V}, {\color{lime}E})$ 📊 - **Solution**: Color assignment ${\color{royalblue}c}: {\color{salmon}V} \to [1..{\color{orange}k}] \ (k=2)$ 🎨 - **Goal**: Minimize monochromatic edge weights ⚖️ $$\min \sum_{e \in {\color{lime}E}} {\color{orange}w_e} \cdot \mathbf{1}\!\big[\,{\color{royalblue}c}(u) = {\color{royalblue}c}(v)\,\big]$$ - Each vertex must be colored **0° or 180°** 🔄 - A monochromatic edge = **phase conflict** = dark line lost ⚠️ ] .pull-right[  ] --- ### 🤯 Problem Complexity - **Generally NP-hard** — no polynomial exact algorithm (unless P = NP) 😱 - **Planar graphs** ($k=2$): polynomial via **T-join problem in the dual graph** [Hadlock 75] ⏱️ - **2-approximation** possible using the **primal-dual method** 2️⃣ **Plan of attack:** 1. 🤑 Greedy approach — fast, works on any graph 2. 🗺️ Planar graph approach — optimal, restricted to planar instances --- class: nord-light, middle, center ## 🤑 Part 4: Greedy Approach --- ### 🌳 Greedy Algorithm Overview .pull-left[ 1. **Build max weighted spanning tree** (MST) 🌳 2. **Color MST nodes** — 2-coloring is trivial on a tree 🎨 3. **Reinsert non-conflicting edges** — those whose endpoints already differ 🔄 **Time:** O(N log N) ⏱️ **Works for non-planar graphs** 🌐 ] .pull-right[  ] --- ### 🎨 Greedy Step 2: Color the MST .pull-left[ - Root the tree, alternate **0° / 180°** by depth 🎨 - Tree edges are automatically **properly colored** ✅ - Heavy edges (high weight) get priority via MST 🌳 ] .pull-right[  ] --- ### ➕ Greedy Step 3: Reinsert Edges .pull-left[ - Scan all edges in decreasing weight 📊 - If endpoints differ in color → **keep** ✅ - If endpoints are same-colored → **conflict** ⚠️ (monochromatic edge weight adds up) ] .pull-right[  ] --- ### 🔄 Greedy Flow .mermaid[ <pre> graph LR G["📊 Graph G\nweighted"] --> MST["🌳 Max-weight\nspanning tree"] MST --> COL["🎨 2-color\nthe tree"] COL --> R{"🔍 Scan edge:"} R -->|"endpoints differ"| K["✅ Keep"] R -->|"same color"| X["⚠️ Conflict\n+ weight"] K --> D["🏁 Assignment\nO(N log N)"] X --> D style G fill:#e3f2fd,stroke:#1565c0,color:#2e3440,stroke-width:3px style MST fill:#c8e6c9,stroke:#2e7d32,color:#2e3440,stroke-width:3px style COL fill:#fff9c4,stroke:#f57f17,color:#2e3440,stroke-width:3px style R fill:#ffe0b2,stroke:#e65100,color:#2e3440,stroke-width:3px style K fill:#c8e6c9,stroke:#2e7d32,color:#2e3440,stroke-width:3px style X fill:#ffcdd2,stroke:#c62828,color:#2e3440,stroke-width:3px style D fill:#f3e5f5,stroke:#7b1fa2,color:#2e3440,stroke-width:3px </pre> ] **Greedy = fast + general.** But can we do *better* on planar layouts? 🤔 --- class: nord-light, middle, center ## 🗺️ Part 5: Planar Graph Approach --- ### 📐 Alternative Approaches .pull-left[ **MAX-CUT formulation** 📐 - 1.1383-approximation via **semi-definite programming** [Goemans–Williamson 93] 🧮 - Powerful but heavy (SDP solver) ] .pull-right[ **Planar graph method** 🗺️ - Remove **minimal edges** → apply planar techniques 📊 - *Note: Optimal planar subgraph is NP-hard* 🤯 - **Hadlock's algorithm**: optimal *phase conflicts* on planar graphs 🎯 ] --- ### 🗺️ 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 🤝 .pull-left[  ] .pull-right[  ] --- ### 🔍 Hadlock: Planar Tricks .pull-left[ **Step 3**: Linear-time planar embedding 🏗️ *Unique only for tri-connected graphs*  ] .pull-right[ **Step 4**: Dual graph construction (linear time) 🔄  ] --- ### 🧩 Hadlock: Matching & Reinsertion .pull-left[ **Step 5**: Min perfect matching 🧩 - Odd-degree vertices of the dual - Solvable in polynomial time - Network flow formulation - *Complete vs Voronoi graphs*  ] .pull-right[ **Step 6**: Non-conflicting edge reinsertion 🔙 *Track conflicts during process*  ] --- ### 📊 Hadlock Flow .mermaid[ <pre> graph TD G["📊 Graph G"] --> D1["🪓 Biconnected\ndecomposition"] D1 --> D2["🏞️ Planar\napproximation\nO(n²)"] D2 --> D3["🏗️ Planar\nembedding"] D3 --> D4["🔄 Dual graph\nP*"] D4 --> D5["🧩 Min perfect\nmatching (odd V)"] D5 --> D6["🔙 Reinsert\nnon-conflict edges"] D6 --> O["🏁 Optimal phase\nassignment ✅"] style G fill:#e3f2fd,stroke:#1565c0,color:#2e3440,stroke-width:3px style D1 fill:#ffe0b2,stroke:#e65100,color:#2e3440,stroke-width:3px style D2 fill:#fff9c4,stroke:#f57f17,color:#2e3440,stroke-width:3px style D3 fill:#e8eaf6,stroke:#283593,color:#2e3440,stroke-width:3px style D4 fill:#f3e5f5,stroke:#7b1fa2,color:#2e3440,stroke-width:3px style D5 fill:#ffcdd2,stroke:#c62828,color:#2e3440,stroke-width:3px style D6 fill:#e3f2fd,stroke:#1565c0,color:#2e3440,stroke-width:3px style O fill:#c8e6c9,stroke:#2e7d32,color:#2e3440,stroke-width:3px </pre> ] **Duality insight:** *min phase conflicts in G ⇔ min T-join in dual P\*** 🧠 --- ### 📊 Approach Comparison | Method | Quality | Time | Graph Type | |--------|---------|------|------------| | 🤑 Greedy (MST) | Approx ⭐⭐ | O(N log N) ⏱️ | Any 🌐 | | 🧮 SDP MAX-CUT | 1.1383-approx ⭐⭐⭐⭐ | Heavy 🐌 | Any 🌐 | | 🗺️ Hadlock (planar) | Optimal ⭐⭐⭐⭐⭐ | Polynomial ⏱️ | Planar 📊 | | 2️⃣ Primal-dual | 2-approx ⭐⭐⭐ | Fast ⚡ | Any 🌐 | **Rule of thumb:** planar layout → **Hadlock**. Otherwise → greedy or SDP 🎯 --- ### 🔑 Key Takeaways **1️⃣ Sub-wavelength lithography breaks the mask→wafer promise** 🌊 — DFM must compensate **2️⃣ OPC pre-distorts the mask** so diffraction *cancels itself* ✂️🎯 **3️⃣ PSM uses π phase shifts** for sharp dark lines — resolution without shorter $\lambda$ ✨ **4️⃣ Phase assignment = weighted 2-coloring** — NP-hard in general 🤯 **5️⃣ Planar graphs are tractable** — Hadlock via dual T-join is optimal 🗺️ **6️⃣ Greedy stays relevant** — O(N log N), works on any graph 🤑 --- count: false class: nord-dark, middle, center .pull-left[ ## Q&A 🎤 ### Questions? Discussion? ] .pull-right[  ] --- count: false class: nord-dark, middle, center # 👏 Thank You ### Phase Shifting Mask & Optical Proximity Correction @luk036 👨‍💻 · 2026 📅 **Slides built with Remark.js 🎉 | KaTeX | Mermaid | Nord Theme** Key formula: **min conflicts = min T-join in dual graph** 🧠 **Links:** - Slides: `luk036.github.io/algo4dfm/psm-opc-remark.html` - Code: `github.com/luk036/luk036.github.io`