Lecture 8: Phase Shifting Mask

@luk036

2022-11-26

🗺️ Overview

• Background
• What is Phase Shifting Mask?
• Phase Conflict Graph
• Phase Assignment Problem
  • Greedy Approach
  • Planar Graph Approach

class: middle, center

Background

Background

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• In the past, chips have continued to get smaller and smaller, and therefore consume less and less power.

• However, we are rapidly approaching the end of the road and optical lithography cannot take us to the next place we need to go.

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Process of Lithography

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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

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Sub-wavelength Lithography

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• Feature size is much smaller than the lithography wavelength
  • 45nm vs. 193nm

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• What you see in the mask/layout is not what you get on the chip:
  • Features are distored
  • Yields are declined

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DFM Tool (Mentor Graphics)

OPC and PSM

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• Results of OPC on PSM:
  • A = original layout
  • B = uncorrected layout
  • C = after PSM and OPC

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Phase Shifting Mask

Phase Conflict Graph

• Edge between two features with separation of $\le b$ (dark field)
• Similar conflict graph for "bright field".
• Construction method: plane sweeping method + dynamic priority search tree

Phase Assignment Problem

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• Instance: Graph $G=(V,E)$
• Solution: A color assignment $c:V\to [\mathrm{1..}k]$ (here $k=2$)
• Goal: Minimize the weights of the monochromatic edges. (Question: How can we model the weights?)

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Phase Assignment Problem

• In general, the problem is NP-hard.
• It is solvable in polynomial time for planar graphs with $k=2$, since the problem is equivalent to the T-join problem in the dual graph [Hadlock75].
• For planar graphs with $k=2$, the problem can be solved approximately in the ratio of two using the primal-dual method.

Overview of Greedy Algorithm

• Create a maximum weighted spanning tree (MST) of $G$ (can be found in LEDA package)
• Assign colors to the nodes of the MST.
• Reinsert edges that do not conflict.
• Time complexity: $O(N\log N)$
• Can be applied to non-planar graphs.

Greedy Algorithm

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• Step 1: Construct a maximum spanning tree $T$ of $G$ (using e.g. Kruskal's algorithm, which is available in the LEDA package).

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Greedy Algorithm (Cont'd)

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• Step 2: Assign colors to the nodes of $T$.

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Greedy Algorithm (Cont'd)

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• Step 3: Reinsert edges that do not conflict.

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Other Approaches

• Reformulate the problem as a MAX-CUT problem. Note that the MAX-CUT problem is approximatable within a factor of 1.1383 using the "semi-definite programming" relaxation technique [Goemans and Williamson 93].

• Planar graph approach: Convert $G$ to a planar graph by removing the minimal edges, and then apply the methods to the resulting planar graph.

  👉 Note: the optimal "planar sub-graph" problem is NP-hard.

Overview of Planar Graph Approach (Hadlock's algorithm)

1. Approximate $G$ by a planar graph $G^{\prime }$
2. Decompose $G^{\prime }$ into its bi-connected components.
3. For each bi-connected component in $G^{\prime }$,
   1. construct a planar embedding
   2. construct a dual graph $G^{\ast }$
   3. construct a complete graph $C(V,E)$, where
      • $V$ is a set of odd-degree vertices in $G^{\ast }$
      • the weight of each edge is the shortest path of two vertices
   4. find the minimum perfect matching 💯👬🏻 solution. The matching edges are the conflict edges that have to be deleted.
4. Reinsert the non-conflicting edges from $G$.

Planar Graph Approach

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• Step 1: Approximate $G$ with a planar graph $G^{\prime }$
  • It is NP-hard.
  • The naive greedy algorithm takes $O(n^2)$ time.
  • Any good suggestion?

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Planar Graph Approach

• Step 2: Decompose $G^{\prime }$ into its bi-connected components in linear time (available in the LEDA package).

  

Planar Graph Approach

• Step 3: For each bi-connected component in $G^{\prime }$, construct a planar embedding in linear time (available in the LEDA package)

  

👉 Note: planar embedding may not be unique unless $G$ is tri-connected.

Planar Graph Approach

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• Step 4: For each bi-connected component, construct its dual graph $G^{\ast }$ in linear time.

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Planar Graph Approach

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• Step 5: Find the minimum weight perfect matching 💯👬🏻 of $G^{\ast }$.

  • Polynomial time solvable.
  • Can be formulated as a network flow problem.

  👉 Note: complete graph vs. Voronoi graph

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Planar Graph Approach

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• Step 6: reinsert the non-conflicting edges in $G$.

👉 Note: practically we keep track of conflicting edges.

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