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=10 5, W=20um/10nm=2000 R=W/4=500 Total windows ≈ 9.6x10 9 Sliding windows ≈ 3.88x10 4 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. H1H1 H2H2 H3H3 H 1 +H 2 =H 3 s s

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 {d 1,d 2,d 3,d 4 } ? d1d1 d2d2 d3d3 d4d4

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. TestcaseLayout Area (um 2 )#rectangles Test1576x576 191,967 Test2576x576 360,799 Test3512x512 449,828 Test41248x1216 762,412 Test5512x5121,375,605 Test6992x9923,106,559 Test7992x9924,632,445 Test8992x9925,033,242 Test91216x12165,287,136 Test10992x9925,583,589

12. Experimental Results (Cont) TestAlg3MDA (err ≤ 2%)Our Algorithm Max DensCUP (s)Max DensCPU (s)Max DensCPU (s) Test157.54%2202758.41%30057.54%2 Test242.83%8325443.26%22442.83%4 Test328.99%51h30m29.32%17028.99%42 Test484.48%4623185.52%82184.48%3 Test5-> 24h19.61%19719.35%110 Test6-> 24h56.33%13655.57%39 Test7-> 24h47.95%68747.34%195 Test8-> 24h26.93%13826.64%73 Test9-> 24h86.88%13585.90%15 Test10-> 24h39.30%34638.96%74 Test Results with a window size 32um

13. Experimental Results (Cont) TestAlg3MDA (err ≤ 2%)Our Algorithm Max DensCUP (s)Max DensCPU (s)Max DensCPU (s) Test167.23%1058768.06%43667.23%1 Test247.40%4228948.02%81747.40%3 Test329.82%9324230.20%20129.82%32 Test484.42%2287685.57%142184.42%5 Test5-> 24h21.05%16620.94%88 Test6-> 24h58.62%27057.56%28 Test7-> 24h50.82%77950.04%96 Test8-> 24h28.49%12828.08%64 Test9-> 24h88.24%10486.84%15 Test10-> 24h43.51%21342.92%46 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.