1. Automated Layout and Phase Assignment for Dark Field PSM
Andrew B. Kahng, Huijuan Wang, Alex Zelikovsky
UCLA Computer Science Department
Supported by a grant from Cadence Design Systems, Inc.

2. Outline Phase assignment for dark field Alt PSM
Removing odd cycles from conflict graph
previous work
proposed methods
Algorithms for odd cycle elimination
Implementation experience
Conclusions

3. Outline Phase assignment for dark field Alt PSM
Removing odd cycles from conflict graph
previous work
proposed methods
Algorithms for odd cycle elimination
Implementation experience
Conclusions

4. Alternating PSM conventional mask phase shifting mask glass Chrome
Phase shifter
0 E at mask 0
0 E at wafer 0
0 I at wafer 0

5. Phase Assignment Problem
Assign phases 0, 180 to all features s.t. pairs
with separation < B have opposite phases
Features
Conflict areas (<B)
180
< B
> B
 b
b  minimum separation
B  minimum separation between same-phase features

6. Conflict Graph Vertices: features Edges: conflicts
(feature pairs with separation < B )
< B

7. Odd Cycles in Conflict Graph
No valid phase assignment exists, because of odd cycle (triangle) in conflict graph
Valid assignment  2-colorable  bipartite  no odd cycles

8. Breaking an Odd Cycle
 B

9. Outline Phase assignment for dark field Alt PSM
Removing odd cycles from conflict graph
previous work
proposed methods
Algorithms for odd cycle elimination
Implementation experience
Conclusions

10. Previous Work Interactive methods (Ooi et al., Moniwa et al.)
detect odd cycles
manually widen spacing for chosen pairs
Compaction method (Ooi et al.)
symbolic layout from mask layout
phase assignment in symbolic layout
PSM design rules
compaction of symbolic layout

11. Proposed Methods Iterative coloring and compaction
One-shot phase assignment
Conflict edge weight
Splitting of features
Vertical/horizontal spacing
Layer assignment

12. Iterative Phase Assignment and Compaction
Iterate until conflict graph becomes bipartite:
Compact the layout and find conflict graph
Find minimum set of edges to be deleted from conflict graph for 2-colorability
Add new separation constraints: one per deleted edge

13. Iterative Phase Assignment and Compaction
conflict graph
find minimum #
edges to be deleted
for 2-colorobility
already
2-colorable
yes
phase assignment no
PSM constraints
compaction

14. One-Shot Phase Assignment
Find conflict graph
Find minimum set of edges to be deleted from conflict graph for 2-colorability
Assign phases such that only chosen conflict edges connect features of the same phase
Compact layout with PSM design rules:
B-separation if features have the same phase
b-separation if features have different phase

15. One-Shot Phase Assignment
conflict graph
find minimum #
edges to be deleted
for 2-colorobility
phase assignment
compaction

16. Conflict Edge Weight Compaction moves all features left
Constraint graph contains arcs between edges
Critical path between leftmost, rightmost features
Conflict edges not on critical path: break for free
critical path

17. Feature Splitting Splitting features may eliminate odd cycle
Green areas: phase shift between 0, 180 degrees

18. Vertical / Horizontal Spacing
Introducing a vertical or horizontal gap eliminates all conflict edges that cross gap
Optimal algorithm to find min # gaps

19. Layer Assignment

20. Outline Phase assignment for dark field Alt PSM
Removing odd cycles from conflict graph
previous work
proposed methods
Algorithms for odd cycle elimination
Implementation experience
Conclusions

21. Optimal Odd Cycle Elimination
Construct conflict graph G
Construct dual graph D
Find odd-degree vertices ODD in D
Find minimum weighted perfect matching of ODD (weights = the length of path)
Delete all edges of G which correspond to paths of the minimum matching of ODD

22. Optimal Odd Cycle Elimination
blue features/red conflicts
conflict graph
matching of odd degree nodes
dual graph

23. Optimal Odd Cycle Elimination
blue features/red conflicts
delete green conflicts
matching of odd degree nodes
conflict graph

24. Fast Algorithm For each odd degree vertex V in dual graph
Voronoi region  even degree vertices which are closer to V than to any other odd degree vertex
Connect two vertices if there is an edge between their Voronoi regions
edge weight  path cost in dual graph
Find matching between odd degree nodes in Voronoi graph 3

25. Outline Phase assignment for dark field alt PSM
Removing odd cycles from conflict graph
previous work
proposed methods
Algorithms algorithm for odd cycle elimination
Implementation experience
Conclusions

26. Compaction Shape constraints Connectivity constraints
Spacing constraints (PSM design rules)
Bellman-Ford solution for constraint graph for one-dimensional constraint graph in x-direction
Flip design and solve in y-direction

27. Data Flow GDSII  CIF CIF  internal layout representation
New layer with phase shift  CIF

28. Results

29. Outline Phase assignment for dark field alt PSM
Removing odd cycles from conflict graph
previous work
proposed methods
Algorithms algorithm for odd cycle elimination
Implementation experience
Conclusions

30. Conclusions
New fast, optimal algorithms for minimum-cost conflict removal
Integration with GDSII reader, polygon database, layout compactor
More direct integrations with layout under investigation
Preliminary results (speed, capacity) promising