1. Sambuddha Bhattacharya Subramanian Rajagopalan Shabbir H. Batterywala Fixing Double Patterning Violations With Look-Ahead ASD-DAC’14

2. Outline Introduction DPT Fixing As Legalization Co-Resolution of DPT and DRC Constraint Feasibility Check Preventing New DPT Violations Experimental Results Conclusion

3. Introduction Lithography at 20nm and below necessitate double patterning (DP) or splitting of a mask into two sub-mask patterns A graph is free of DPT conflicts if it is bi- colorable

4. The main idea of this paper is that model the non-DPT edges in the input layout, address the issue of new DPT conflicts Legalization based approach to fix DPT violations

5. Key contributions  A look-ahead method that predicts new DPT edges  A tightly coupled interaction between a constraint feasibility checker and DPT conflict verifier  Fix existing violations without introducing new ones

6. DPT Fixing As Legalization A legalization engine reads in an input layout and fractures its mask layer geometry into rectangles All applicable design rules are modeled as linear difference constraints

7. Generally, the constraint set obtained from a realistic layout is often infeasible due to conflicts between complex rules and design intents An infeasibility in the corresponding linear program can be detected as positive cycle in the constraint graph(G c )

8. Modeling DPT Constraints DPT conflicts express themselves as odd cycles of spacing between shapes

9. Note that the DPT constraints are naturally disjunctive If constraint set is infeasible, then replace the representative constraint of a disjunctive set with an alternative and repeat the feasibility check

10. Methodology Overview

12. Co-Resolution of DPT and DRC There is a strong relationship between G D and G C G C should be made feasible and G D should be freed of odd cycles

14. Detecting DPT Conflicts in BCC The total number of odd cycles in a graph can be exponential We identify the odd-cycles in a cycle basis from G D

15. Constraint Feasibility Check Each edge in the constraint graph G C models corresponding constraints of the form x i − x j ≥ w ij Generally, a pair of rectangles can have multiple constraints between them

16. Inconsistencies in the constraints express themselves as positively weighted cycles in G C Positive cycles are detected by invoking a single source longest path algorithm

17. Assume that the DPT constraint were absent If w DRC > then the edge weight gets reduced to Resolving positive cycles in this way have two properties  If a constraint is not satisfiable, the input layout value is retained  The input layout positions represent a feasible solution

18. Once the infeasibilities are resolved in GC the graph is augmented with the DPT constraints If an edge was already relaxed to current layout value, the corresponding DPT difference constraint is deactivated and swapped with the next constraint

19. If none of the disjunctions for a DPT conflict are satisfiable, then the corresponding DPT conflict can not be resolved

20. Preventing New DPT Violations As DPT conflict cycles are broken by increasing spacing, this can result in reduction of spacing in the neighborhood Simulate the behavior of the LP solver to predict new DPT violations

22. Layout-aware Bellman-Ford Algorithm (LABF) Forward LABF generates the upper bound Reverse LABF generates the lower bound The ranges generated by LABF includes all points in the solution space where minimum layout perturbation

24. Experimental Results Using designs in 40nm and 20nm technology with appropriate representative DPT rules and DRC rules 2.8GHz, 4 core linux machine with 8 GB RAM

26. Conclusion Presented a DPT fixing approach based on wire spreading The framework of legalization ensures that DRC violations are modeled and fixed simultaneously Feasible constraint selection, new DPT edge prediction and DPT conflict cycle verification