1. GREMA: Graph Reduction Based Efficient Mask Assignment for Double Patterning Technology Yue Xu, Chris Chu Iowa State University Form ICCAD2009

2. Outline Introduction Problem formulation Candidate Stitch generation Mask assignment ▫Initial Mask assignment ▫Generate of Flipping Graph ▫Simplification of Flipping Graph ▫ILP Formulation for Max Cut Problem Experimental Results

5. Double Patterning 5 A B C D E Layout Decomposition min dp Mask 2 Mask 1 Stitch

7. DPT decomposition methodology 1. model-based decomposition approach is based on optical simulation => more accurate, but time-consuming 2. rule-based approach slices and assigns patterns based on geometric relationship between patterns => efficiency

8. Outline Introduction Problem formulation Candidate Stitch generation Mask assignment ▫Initial Mask assignment ▫Generate of Flipping Graph ▫Simplification of Flipping Graph ▫ILP Formulation for Max Cut Problem Experimental Results

9. Problem formulation Given : circuit layout,the minimum distance btw each patterns. Objective : minimize the patterns violating the minimum distance rule and the number of stitches

10. Overview( two stage approach )

11. Outline Introduction Problem formulation Candidate Stitch generation Mask assignment ▫Initial Mask assignment ▫Generate of Flipping Graph ▫Simplification of Flipping Graph ▫ILP Formulation for Max Cut Problem Experimental Results

12. Conflict Graph ACB L

13. Candidate Stitch generation The odd cycle in CG => infeasible We can slice one node in the odd cycle <= useful stitch Some slicing may be unused. (i.e A -> B -> L ) How to find a useful slicing node? => node projection idea introduced in [11]. GREMA uses breadth first search (BFS) to find odd cycles and choose all suitable candidate stitches.

15. Outline Introduction Problem formulation Candidate Stitch generation Mask assignment ▫Initial Mask assignment ▫Generate of Flipping Graph ▫Simplification of Flipping Graph ▫ILP Formulation for Max Cut Problem Experimental Results

16. Initial Mask assignment Cluster: conflict edge connected component. GREMA arbitrarily picks up a node in each cluster and denotes the node as cluster head. assign all of the cluster node to same partition.

17. Generate of Flipping Graph Abstract each cluster in DG into a single node. Flipping of Ci and Cj : let the cluster heads of Ci and Cj to be different partition (change which one?) Flipping gain ▫Before flipping: Ns After flipping: Nd ▫Flipping gain: Nd - Ns

18. Simplification of Flipping Graph Case1: tree structure

19. Simplification of Flipping Graph Case 2: Serial Edge

20. Simplification of Flipping Graph

21. Simplification procedure while( only exist two nodes or all of nodes have at least degree three ) while( exist degree one node ) detects all degree one node and remove it serial_edge_simplify() parellel_edge_simplify() while( exist degree one node ) detects all degree one node and remove it ILP_solve()

22. Example

23. ILP Formulation for Max Cut Problem

25. Outline Introduction Problem formulation Candidate Stitch generation Mask assignment ▫Initial Mask assignment ▫Generate of Flipping Graph ▫Simplification of Flipping Graph ▫ILP Formulation for Max Cut Problem Experimental Results

26. Experimental Result These benchmarks use cells from Artisan 90nm libraries with 140*0.4nm minimum spacing and 100*0.4nm minimum line width.