Automated Layout and Phase Assignment for Dark Field PSM

1. Automated Layout and Phase Assignment for Dark Field PSM Andrew B. Kahng, Huijuan Wang, Alex Zelikovsky UCLA Computer Science Department http://vlsicad.cs.ucla.edu 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. Features Conflict areas (<B) < B > B 0 180 0 Phase Assignment Problem Assign phases 0, 180 to all features s.t. pairs with separation < B have opposite phases  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. 3 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

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