Is Your Layout Density Verification Exact ?

1. Is Your Layout Density Verification Exact ? Hua Xiang*, Kai-Yuan Chao‡, Ruchir Puri* and D.F. Wong+ *IBM T.J. Watson Research Center +Univ. of Illinois at Urbana-Champaign ‡Intel Corporation

2. Density Calculation • Density Calculation is a fundamental operation in deep-submicron chip designs. • Density Control • Find the max/min density window in a given layout • Several manufacturing processes (CMP, etch, CD, lithography etc.) are sensitive to pattern density. • Density check • Foundries have density range requirements. • Density rules are associated with many process layers • Dummy fills / slotting are based on density calculation. • Existing Methods: • Exact density calculation  Running time is very long (days) • Approximate algorithm  No exact solution • Fix-dissection approach

3. Fix-Dissection Approach • Total windows: (M-W+1)x(N-W+1) • Sliding windows: [(M-W)/R+1]x[(N-W)/R+1] e.g. M=N=1mm/10nm=105, W=20um/10nm=2000 R=W/4=500 Total windows ≈ 9.6x109 Sliding windows ≈ 3.88x104 Lemma: If R is larger than the minimum feature size, fix-dissection approach cannot guarantee to solve the density problem exactly.

4. Density Bound Theorems • Theorem 1: Any window Win can be fully covered by four sliding windows, and its density d satisfies where D is the max/min density of the four sliding windows.

5. Density Bound Theorems • Theorem 2 For any given region, there exists a maximum density window whose two adjacent edges overlap with two rectangle edges, and the overlapped window edges and rectangle edges are in the same direction. H1 H3 H2 s s H1+H2=H3

6. Density Bound Theorems • Theorem 3 For any given region, there exists a minimum density window whose two adjacent edges overlap with two rectangle edges, and the overlapped window edges and rectangle edges are in the different direction.

7. Theorem Extension • A layout with rectangular and overlap shapes can be converted to a layout only with rectangles. • All theorems can be applied on the converted layout.

8. Density Calculation Algorithm • Main Ideas • Start from fix-dissection. Let d be the max density of this iteration. • Prune regions based on Theorem 1 • For selected regions, call detail_density with finer grids • When the region size is small enough, call exact_density which is based on Theorems 2 d  R/W – (R/2W)2 + max {d1,d2,d3,d4} ? d4 d1 d2 d3

9. Detail_Density Region Properties • Region Size L = W + B • All windows share the center (W-B)x(W-B) area • The left bottom corner of any window falls in the pink region • The number of sliding windows is (k+1)2, where k=B/R

10. Exact_Density • The grid is set up based on rectangle edges. • Only the left rectangles within the left column are considered. Similarly for other directions. • The density of the center (W-B)x(W-B) area is obtained from previous iterations.

11. Experimental Results • Implemented in C on a linux workstation (2.3GHz) • Test cases are derived from industry designs • Compared with two algorithms • ALG3 is an exact algorithm. • Jobs were killed when the running time was longer than 24 hour • Our algorithm reduces the running time from hours/days to secs/mins • MDA is an approximate algorithm. • Our algorithm can report exact max/min density numbers; while the running time is equivalent or even shorter.

12. Experimental Results (Cont) Test Results with a window size 32um

13. Experimental Results (Cont) Test Results with a window size 24um

14. Conclusion • Density calculation is a fundamental operation in many manufacturing processes. • A fast and exact density algorithm is proposed to identify the maximum/minimum density window for a given layout. • The algorithm fully utilize the density calculation results from previous iterations so that the running time can be greatly reduced. • Compared with the existing exact algorithms, the running time is reduced from hours/days to seconds/minutes. • The running time is equivalent to the existing approximate algorithms in literature.