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

1
New Graph Bipartizations for Double-Exposure,
Bright Field Alternating Phase-Shift Mask Layout

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

2
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

3
Subwavelength Optical Lithography
Subwavelength Gap since .35 ?m
Numerical Technologies, Inc.
4
Alternating PSM for Subwavelength Technology
5
Double-Exposure Bright-Field PSM
6
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
7
Phase Assignment for Bright-Field PSM

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

Overlapping shifters
8
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
9
Critical features F1,F2,F3,F4
F2
F4
F1
F3
10
F2
F4
F1
F3
Opposite-Phase Shifters (0,180)
11
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

12
Phase Conflict
F2
S3
S4
F4
S7
S8
S1
F1
S2
F3
S5
S6
Phase Conflict
Proper Phase Assignment is IMPOSSIBLE
13
Conflict Resolution Shifting
F2
S3
S4
F4
S7
S8
S1
F1
S2
F3
S5
S6
Phase Conflict
feature shifting to remove overlap
14
Conflict Resolution Widening
F2
S3
S4
F4
S7
S8
S1
F1
S2
F3
Phase Conflict
feature widening to turn conflict into
non-conflict
15
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

16
Feature Graph
17
Odd Cycles in Feature Graph
Feature graph has ODD CYCLE
Proper Phase Assignment IMPOSSIBLE
18
Shifting in Feature Graph I
19
Shifting in Feature Graph II
20
Widening in Feature Graph
21
Graph Bipartization

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

• Layout modifications ? Graph modifications
• feature shifting ? edge deletion
• feature widening ? node deletion

• both types with weights ? node-weighted deletion

22
Edge-Deletion Graph Bipartization

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

• In planar graphs ? T-join problem
  can be solved efficiently
• 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)

23
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

24
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

25
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
26
Example of Gadgeted Graph
Gadgeted graph
Dual Graph
black red edges min-cost perfect matching
27
Node-Deletion Graph Bipartization

• Difficult for general graphs
• MAX SNP-hard ? no very good approximation
• 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

28
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
29
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) Wile there are
violating edges do Delete node incident to
maximum of violating edges
30
Experimental Results
cost of feature widening cost of feature shifting
Cost Ratio
31
Experimental Graph Bipartization

• GW algorithm is
• 2-2.5 times better than Greedy Vertex Cover
  Algorithm
• quite good for small cost ratio
• may be very bad (gt50 worse) for large cost
  ratio
• Exact edge deletion algorithm is
• better than GW for cost ratio gt 2
• faster than GW

32
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

• Future work develop a model for PSM in
  hierarchical designs

33

• EXTRA SLIDES

34
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

35
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

36
Taxonomy of Composability
37
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

38
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

39
T-join Problem in Sparse Graphs

• Reduction to matching
• construct a complete graph T(G)
• vertices T-vertices
• edge costs shortest-path cost
• find minimum-cost perfect matching
• Typical example sparse (not always planar)
  graph
• note that conflict graphs are sparse
• vertices 1,000,000
• edges ? 5 ? vertices
• T-vertices ? 10 of vertices 100,000
• Drawback finding all shortest paths too slow and
  memory-consuming
• vertices 100,000 edges 5,000,000,000