New Graph Bipartizations for Double-Exposure, Bright Field Alternating Phase-Shift Mask Layout

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New Graph Bipartizations for Double-Exposure,
Bright Field Alternating Phase-Shift Mask Layout

• Andrew B. Kahng (UCSD) abk_at_ucsd.edu
• Shailesh Vaya (UCLA)
• Alex Zelikovsky (GSU)

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Outline

• Subwavelength lithography
• Alternating PSM
• Phase assignment problem
• Minimum perturbation problem
• Bipartizing feature graph
• Fast algorithm for edge-deletion
• Approximation algorithm node-deletion
• Experimental results
• Conclusions

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Subwavelength Optical Lithography
Subwavelength Gap since .35 ?m
Numerical Technologies, Inc.
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Alternating PSM for Subwavelength Technology
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Double-Exposure Bright-Field PSM
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The Phase Assignment Problem

• Assign 0, 180 phase regions such that critical
  features with width lt B are induced by adjacent
  phase regions with opposite phases

0
180
ltB
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Phase Assignment for Bright-Field PSM

• PROPER Phase Assignment
• Opposite phases for opposite shifters
• Same phase for overlapping shifters

Overlapping shifters
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Key Global 2-Colorability

• Odd cycle of phase implications layout
  cannot be manufactured
• layout verification becomes a global, not local,
  issue

?
180
0
180
0
180
180
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Critical features F1,F2,F3,F4
F2
F4
F1
F3
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F2
F4
F1
F3
Opposite-Phase Shifters (0,180)
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F2
S3
S4
F4
S7
S8
S1
F1
S2
F3
S5
S6
Shifters S1-S8

• PROPER Phase
  Assignment
• Opposite phases for opposite shifters
• Same phase for overlapping shifters

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Phase Conflict
F2
S3
S4
F4
S7
S8
S1
F1
S2
F3
S5
S6
Phase Conflict
Proper Phase Assignment is IMPOSSIBLE
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Conflict Resolution Shifting
F2
S3
S4
F4
S7
S8
S1
F1
S2
F3
S5
S6
Phase Conflict
feature shifting to remove overlap
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Conflict Resolution Widening
F2
S3
S4
F4
S7
S8
S1
F1
S2
F3
Phase Conflict
feature widening to turn conflict into
non-conflict
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Minimum Perturbation Problem

• Layout modifications
• feature shifting
• feature widening
• ? area increase, slowing down
• ? manual fixing, design cost increase

• Minimum Perturbation Problem
• Find min of layout modifications leading to
  proper phase assignment

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Feature Graph
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Odd Cycles in Feature Graph
Feature graph has ODD CYCLE
Proper Phase Assignment IMPOSSIBLE
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Shifting in Feature Graph I
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Shifting in Feature Graph II
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Widening in Feature Graph
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Graph Bipartization

• Proper phase assignment ? Feature graph bipartite
• Minimum Perturbation Problem
• ? Graph Bipartization Problem

• Layout modifications ? Graph modifications
• feature shifting ? edge deletion (blue-pink
  edge) or
  ? node deletion (blue node)
• feature widening ? node deletion (black node)

• both types with weights ? node-weighted deletion

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Edge-Deletion Graph Bipartization

• In general graphs
• NP-hard
• Constant-factor approximation

• In planar graphs ? T-join problem (exact
  solution)
• ? reduction to min-weight matching O(n3)
  (Hadlock)
• ? LP-based solution (Barahona) O(n3/2logn)
• no known implementation
• ? fast reduction to matching via gadgets O(n3/2
  log n)

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Edge Deletion Turns Odd Faces ? Even Faces
broken edges in original G
original graph G
dual graph D
T-join of odd-degree nodes in D
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The T-join Problem

• How to delete minimum-cost set of edges from
  conflict graph G to eliminate odd cycles?
• Construct geometric dual graph Ddual(G)
• Find odd-degree vertices T in D
• Solve the T-join problem in D
• find min-weight edge set J in D such that
• all T-vertices has odd degree
• all other vertices have even degree
• Solution J corresponds to desired min-cost edge
  set in conflict graph G

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T-join Problem Reduction to Matching

• Desirable properties of reduction to matching
• exact (i.e., optimal)
• not much memory (say 2-3Xmore)
• results in a very fast solution
• Solution gadgets
• replace each edge/vertex with gadgets s.t.
• matching all vertices in gadgeted graph
• Û T-join in original graph

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T-join Problem Reduction to Matching

• replace each vertex with a chain of triangles
• one more edge for T-vertices
• in graph D m edges, n vertices, t T
• in gadgeted graph 4m-2n-t vertices, 7m-5n-t
  edges
• cost of red edges original dual edge costs
  cost of (black) edges in
  triangles 0

vertex in T
vertex ? T
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Example of Gadgeted Graph
Gadgeted graph
Dual Graph
black red edges min-cost perfect matching
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Node-Deletion Graph Bipartization

• Difficult for general graphs
• MAX SNP-hard ? no very good approximation
• Complexity unknown for planar graphs
• For planar graphs
• Primal-dual algorithm (GW98)
• takes in account weights
  
  (to distinguish two modification types)
• simple for implementation
• provably good 9/4 approximation
• quadratic runtime
• Greedy Vertex Cover Algorithm

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Primal-Dual Approximation Algorithm (GW)
Input Planar graph (with node weights)
Output Bipartite subgraph
For each face F age(F)? 0 While there are odd
faces do for each odd face F age(F) ?
age(F)1 delete v with max weight (v) sum of
ages of faces with v for new face F age(F)?
0 In reverse order of node deletions do bring
node v back if an odd face appears, then
delete v permanently
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Example of GW Algorithm
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Example of GW Algorithm
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Example of GW Algorithm
1
1
3
2
3
1
2
2
2
0
1
0
1
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Example of GW Algorithm
2
2
5
4
5
2
4
4
4
1
2
0
2
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Example of GW Algorithm
2
2
5
4
5
2
4
4
4
1
2
0
2
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Example of GW Algorithm
2
2
5
4
2
4
4
4
1
2
0
2
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Example of GW Algorithm
2
2
5
5
2
6
6
5
1
3
0
3
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Example of GW Algorithm
2
2
5
5
2
6
6
5
1
3
0
3
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Example of GW Algorithm
2
2
5
5
2
6
5
1
3
0
3
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Greedy Vertex Cover Algorithm (GVC)
Input Planar graph (with node weights)
Output Bipartite subgraph
Color all nodes into 2 colors using BFS node
traversal Find the set T of all violating edges
(endpoints of the same color) Greedily cover
with vertices violating edges Wile there
are violating edges do Delete node incident to
maximum of violating edges
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Experiment Setting

• Compact layouts aggressively
• design rule between features should be single
  shifter
• Determine shifter overlaps
• Find minimum of modifications
  to resolve all phase conflicts

• Two industrial benchmarks Metal Layers
• wires 8622 (L1) and 4539 (L2)
• overlaps 7805 (L1) and 5439 (L2)

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Experimental Results

Benchmark Algorithm Cost Ratio L1
L2
Edge-deletion 314
234

• GW algorithm nearly 2x better than Greedy Vertex
  Cover algorithm

• Exact edge-deletion algorithm is better than GW
  for cost ratio gt 2

• Runtime
• GVC is linear and very fast
• Exact edge-deletion is 2x faster than GW for
  benchmarks

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Conclusions/Future Work

• Contributions
• first formulation of the minimum perturbation
  problem for bright-field Alternating PSM
  technology
• unified approach for feature widening and
  shifting
• optimal solution for feature shifting and
  approximate solution when feature when both
  modifications are allowed

• Ongoing work
• develop a model for PSM in hierarchical designs
• standard cell overlapping
• composability of standard cells
• multiple PSM-aware versions of master cells
• synergy with OPC and filling
• cost- and function-driven PSM

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• THANK YOU !
• (extra slides)

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Standard-Cell PSM

• Hierarchical layout vs flat layout
• Free composability of standard cells
• Cells may overlap unique master cell causes
  area loss
• Multiple PSM-aware versions of master cell
• Version-composability matrix

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Taxonomy of Composability

• (Same) Same row composability any cell can be
  placed immediately adjacent to any other
• (Adj) Adjacent row composability any two
  cells from adjacent rows are freely combined
• Four cases of cell libraries
  Gguaranteed composability, NGnot
  guaranteed
• Adj-G/Same-G free composability
• Adj-G/Same-NG
• Adj-NG/Same-G
• Adj-NG/Same-NG

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Taxonomy of Composability
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Adj-G/Same-NG

• GIVEN
• order of cells in a row
• version compatibility matrix
• FIND version assignment
• such that versions of adjacent cells are
  compatible
• (BFS) traversal of DAG
• nodes versions
• arcs compatibility

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Adj-G/Same-NG

• GIVEN
• order of cells in a row (or optimal placement)
• version compatibility weighted matrix (weight
  extra sites)
• FIND version assignment minimizing
• either total of extra sites
• or total/max displacement from optimal
  placement
• Dynamic Programming O(kV), kmax displacement
• Restricted DP