GREMA: Graph Reduction Based Efficient Mask Assignment for Double Patterning Technology

1. GREMA: Graph Reduction Based Efficient MaskAssignment 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. C D E A mindp B Layout Decomposition Mask 2 Mask 1 Double Patterning 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 A B C 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.