1. NTUEE 1 Coupling-Constrained Dummy Fill for Density Gradient Minimization Huang-Yu Chen 1, Szu-Jui Chou 2, and Yao-Wen Chang 1 1 National Taiwan University, Taiwan 2 Synopsys, Inc, Taiwan

2. NTUEE 2 Dummy Fill ․ Dummy fill is a general method to achieve layout uniformity before CMP (chemical-mechanical polishing) coupling capacitance density gradient ․ Two objectives for dummy fill: (1) minimize induced coupling capacitance of dummies (2) minimize density gradient of metal density Metals Dummies Metals smaller density gradientlarger density gradient

3. NTUEE 3 Previous Work: CDF Algorithm Slot partition by endpoints of segments A layout slots 123456 Coupling-free fill regions identification for each slot Filling max # of dummies into these regions ․ CDF Algorithm presented in [Xiang et al. TCAD’08] Do not consider density gradient! Use too many dummies!

4. NTUEE 4 Our Algorithm Flow CDF [TCAD’08] Density gradient-driven dummy fill Layout Coupling constraints Fill result ․ After identifying fill regions by the CDF algorithm, transform a slot-based layout into a tile-based one ․ Use density gradient-driven dummy fill slot-based tile-based Slot-to-tile conversion

5. NTUEE 5 Tile Density Bounds Computation ․ Compute bounds satisfying both coupling and foundry density rules ․ Tile density lower bound B l = max{F l, D s } ․ Tile density upper bound B u = min{F u, D c }  F l,F u : foundry density lower and upper bound rules  D s : segment density  D c : segment density + maximum dummy density in fill regions F l = 2/16 (12.5%), F u = 8/16 (50%) D s = 3/16, D c = 9/16  (B l, B u ) = (3/16, 8/16) segment dummy (B l, B u ) guarantees no coupling and density rule violations in the following stages

6. NTUEE 6 Multilevel Dummy Density Analysis cf. sliding window avoid the ordering problem reduce the discretization gap obtain a better global view cf. sliding window avoid the ordering problem reduce the discretization gap obtain a better global view Coarsening Uncoarsening G0G0 G1G1 G2G2 G2G2 G1G1 G0G0 Metal Density Low High (1) Gradient minimization by Gaussian smoothing (2) Density bounds update level by level (1) Density extraction (2) Density bounds update level by level

7. NTUEE 7 Multilevel Dummy Density Analysis 0.230.290.28 0.260.300.32 0.220.250.28 0.230.280.25 0.290.250.30 0.180.23 0.10.20.3 0.2 0.1 0.3 G0G0 0.260.300.31 0.280.300.32 0.270.300.31 0.2 0.3 0.2 0.3 0.2 0.3 0.270.28 0.29 0.30 0.260.270.28 0.300.31 0.32 0.33 0.290.300.31 0.28 0.38 0.28 0.38 0.28 0.38 Gradient minimization Average Density extraction G1G1 G2G2 G2G2 G1G1 G0G0 Coarsening Uncoarsening

8. NTUEE 8 Coarsening: Gradient Minimization ․ Gaussian smoothing at tile  D c (x,y): original density  g(x,y): weighting function = ․ Gaussian smoothing opens up a new direction for gradient minimization Gaussian smoothing (σ=1.0) 0.24 0.10.40.1 0.20.4 0.3 0.2 0.3 0.2 0.3 0.2 0.1 0.20.10.3 0.2 0.40.30.4 D c (x,y) density xyxy

9. NTUEE 9 Coarsening: Tile Density Bounds Update D c (x,y) after Gaussian smoothing 0.2 0.3 0.2 0.3 0.2 0.3 0.4 0.1 B u (x,y) B l (x,y) prune the value larger (smaller) than B u (B l ) G0G0 0.23 0.33 0.13 BuBu BlBl =0.23+min{B u (x,y) -D c (x,y)} =0.23+(0.4-0.3) =0.23-min{D c (x,y) -B l (x,y)} =0.23-(0.2-0.1) G1G1

10. NTUEE 10 Uncoarsening: Density Extraction 0.230.290.28 0.260.300.32 0.220.250.28 0.230.280.25 0.290.250.30 0.180.23 0.10.20.3 0.2 0.1 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.270.28 0.29 0.30 0.260.270.28 Gradient minimization Average Density extraction +0.03 +0.08 G0G0 G1G1 G2G2 0.28 0.38 0.28 0.38 0.28 0.38 G0G0 G1G1 0.300.31 0.32 0.33 0.290.300.31 0.260.300.31 0.280.300.32 0.270.300.31 G2G2

11. NTUEE 11 ILP-based Dummy Number Assignment ․ Optimally insert minimal # of dummies to satisfy the desirable tile density d d in a tile ․ For the tile with n fill regions R 1,…,R n, r i : # of dummies in R i d d : dummy density of tile a: tile area a i : area of one dummy in R i a max : max {a i } u i : max # of dummies in R i R1R1 R2R2 u 1 =5 a 1 =3 u 2 =3 a 2 =4 a max =4

12. NTUEE 12 Experimental Setting ․ Programming language: C++ ․ Workstation: 2.0 GHz AMD-64 with 8GB memory ․ ILP solver: lp_solve ․ Parameters  Window size=3 × 3  Gaussian smoothing: σ=1.0  Foundry density lower and upper bounds: 20% and 80% ․ Test cases: MCNC and industrial Faraday benchmarks ․ Comparison with the CDF algorithm [TCAD’08] for all layers and layer 1  Metal density  Neighboring density difference (density gradient)

13. NTUEE 13 ․ Inserted dummy count is only 59% compared with CDF algorithm ․ Timing overhead is only 4% Runtime and Inserted Dummy Counts Circuit CDFOurs #DummyTime (s)#DummyTime (s) Mcc11262100160413735171 Mcc220117831724964144187292 Struct899622845684927572 Primary1708113532421090252 Primary22.5E+0742819140968489 S53782698762112044022 S92342301731410643416 S132076578407834165276 S1585072129899344549102 S3841721004673291033157336 S3858424600615171207341526 Dma28470067166574100 Dsp11371351290741346329 Dsp2941447189562385230 Risc142513682522463637312 Risc252912543963400666446 Comp.1.00 0.591.04

14. NTUEE 14 ․ The standard deviations are reduced by 55% and 49% among all layers and of layer 1, respectively Statistics of Metal Density (MCNC) Circuit CDF Analysis AlgorithmOurs Density among LayersDensity of Layer 1Density among LayersDensity of Layer 1 Avg.MaxStd.Avg.MaxStd.Avg.MaxStd.Avg.MaxStd. Mcc134.49%58.94%9.31%30.86%47.61%7.05%20.52%47.84%4.52%21.10%41.54%4.46% Mcc236.14%55.51%5.61%33.59%52.81%4.26%22.40%54.48%4.17%23.20%49.80%4.12% Struct22.63%33.70%5.08%16.55%27.79%2.26%18.68%5.19%1.98%16.32%19.99%1.58% Primary1 22.62%34.10%10.41%19.39%34.10%9.95%16.87%9.10%7.18%16.33%20.46%7.55% Primary2 22.35%33.43%3.65%18.50%23.74%1.21%19.39%10.09%1.12%18.42%19.99%1.06% S537827.37%40.15%3.84%27.10%36.95%3.28%19.93%30.29%0.67%19.81%25.92%0.45% S923428.30%48.00%4.97%27.90%40.80%3.62%20.06%30.80%1.02%19.79%21.60%0.31% S1320726.90%38.60%3.43%25.90%33.60%2.19%20.36%28.90%1.32%19.93%25.54%0.28% S1585026.80%38.60%3.05%26.30%33.30%2.19%20.23%30.00%1.06%19.91%21.57%0.09% S3841726.80%36.40%2.37%25.80%30.80%1.38%20.10%32.10%0.76%19.96%20.00%0.01% S3858426.36%32.91%1.94%25.90%32.82%1.53%20.04%28.42%0.49%19.96%19.99%0.01% Comp.1.00 0.730.760.450.770.730.51

15. NTUEE 15 ․ The standard deviations are reduced by 26% and 54% among all layers and of layer 1, respectively ․ Overall comparison (MCNC+Faraday) Statistics of Metal Density (Faraday) Circuit CDF Analysis AlgorithmOurs Density among LayersDensity of Layer 1Density among LayersDensity of Layer 1 Avg.MaxStd.Avg.MaxStd.Avg.MaxStd.Avg.MaxStd. Dma22.72%88.73%18.0%23.19%61.92%24.0%18.72%71.40%14.8%9.34%23.51%9.6% Dsp118.64%73.20%17.4%22.32%55.81%22.6%14.52%55.07%12.8%10.25%22.12%10.2% Dsp219.73%74.17%17.5%24.04%51.61%24.2%14.95%60.49%12.3%9.38%42.34%9.4% Risc1 18.53%66.18%17.3%21.67%54.15%21.5%13.98%58.11%12.2%10.35%42.61%10.2% Risc217.05%72.91%16.0%16.21%43.34%15.7%13.52%50.65%11.5%10.47%40.52%10.0% Comp.1.00 0.780.790.740.460.640.46 Circuit CDF Analysis AlgorithmOurs Density among LayersDensity of Layer 1Density among LayersDensity of Layer 1 Avg.MaxStd.Avg.MaxStd.Avg.MaxStd.Avg.MaxStd. Comp.1.00 0.740.770.630.69 0.47

16. NTUEE 16 Comparison of S5378 Layer 1 Filling Results CDF algorithm [TCAD’08] Ours

17. NTUEE 17 Conclusions and Future Work ․ Presented an effective and efficient dummy fill algorithm considering both gradient minimization and coupling constraints  Reduced 37% std. of metal density among all layers  Saved 41% dummy counts ․ Gaussian smoothing is effective for gradient- minimization dummy fill  Point out a new research direction on this topic ․ Future work: integration of gradient minimization and coupling constraints

18. NTUEE 18 Conclusions and Future Work ․ A dummy fill algorithm considering both gradient minimization and coupling constraints ․ Achieve more balanced metal density distribution with fewer dummy features and an acceptable timing overhead ․ Future work: integration of gradient minimization and coupling constraints  Simultaneously minimize the gradient and the coupling capacitance Thank You! Huang-Yu Chen yellowfish@eda.ee.ntu.edu.tw