1. Path Finding for 3D Power Distribution Networks A. B. Kahng and C. K. Cheng UC San Diego Feb 18, 2011

2. Power Grid Optimization Based on Rent’s Rule 2 Higher current density in the inner grid Lowest current density Highest current density We consider one quarter of the power grid Vdd

3. Power Grid Topology Quarter of Die: 200um X 200um Four Metal Layers: M1, M3, M6, AP Wire Direction: M1- horizontal, M3-vertical, M6-Horizontal, AP- vertical

4. Power Grid Parameters Pitch Initial Width Range Local Density Constraint Min-max Constraint M12.5um0.17umN/A M38.0um0.25umN/A M620um4.2um 2um- 8um 15%-80% 2um- 12um AP40um10um 3um- 16um 15%-80% 2um- 35um “Local Density “ is defined as (2*width)/pitch. “Width Range” is determined by intersection of “Local Density Constraint” and “Min-max Constraint”. Total metal area for M6 and AP layers are fixed.

5. Current Sources Based on Rent’s Rule Current source density function: I(d) =c*d^α ; S={(x, y)| (x, y) is the position of a node in M1} ; We put a input source I (x,y) for every (x,y) in S such that ; The total power in an area of d*d is c*d^β where β=(α+2)/2; 5

6. Problem Formulation Inputs from the user: – Topology of power grid; – Default resistances of branches; – Possible current distributions satisfying Rent’s rule; Optimization for static voltage drop: Minimize (Maximum IR drop for all possible current distributions) Subject to – Total wire areas for M6 and AP are fixed; – Lower bound and upper bound for resistances of branches; 6

7. Previous Work P. Gupta and A.B. Kahng, "Efficient Design and Analysis of Robust Power Distribution Meshes", Proc. International Conference on VLSI Design, Jan. 2006, pp. 337-342. W. Zhang, L. Zhang, etc, “On-chip power network optimization with decoupling capacitors and controlled-ESRs”, ASP-DAC, 2010, pp. 119-124. A. Ghosh, S. Boyd and A. Saberi, “Minimizing effective resistance of a graph”, SIAM Review, problems and techniques section, Feb. 2008, 50(1): pp. 37-66. L. Vandenberghe, S. Boyd and A. El Gamal, “Optimal Wire and Transistor Sizing for Circuits with Non-Tree Topology”, IEEE/ACM International Conference on Computer-Aided Design, Nov 1997, pp. 252-259. S. Boyd, “Convex Optimization of Graph Laplacian Eigenvalues”, Proceedings International Congress of Mathematicians, 2006, 3: pp. 1311-1319.

8. Design of Experiments Two optimization methods – Nonlinear programming – Heuristic search Fourteen current peak positions (red dots in the left figure) and four β values 0.25,0.5,0.75,1.0 for testing. The coordinates of the fourteen peak positions are (0,0), (50,0),(50,50), (100,0),(100,50),(100,100), (150,0),(150,50),(150,100),(150,150), (200,0),(200,50),(200,100),(200,150). VD = worst voltage drop of the power grid over all locations and all current distributions satisfying power law.

9. Method 1: nonlinear programming (NLP) The whole flow of NLP for wire sizing optimization with fixed current distribution. The current peak locates at origin.

10. Sizing Results of NLP Segment, β=0.75, VD=0.2941Wire, β=0.75, VD=0.2936 Segment, β=1.0, VD=0.2945Wire, β=1.0, VD=0.2957 VD for uniform sizing = 0.3054

11. Sizing Results of NLP Segment, β=0.25, VD=0.2921Wire, β=0.25, VD=0.2934 Segment, β=0.5, VD=0.2932Wire, β=0.5, VD=0.2945 VD for uniform sizing = 0.3054

12. Observations When β is large (i.e. current sources distribute uniformly), the results suggest putting most of wire resources near the voltage source. When β is small (i.e. most of current sources gather at origin), we should give some wire resources to segments near the origin. “Segment” optimization results are more stable than “Wire” optimization results relative to change of β.

13. Method 2: Heuristic search The candidates include all combinations of w l,w h,p l,p h. The curve part is fitted by a polynomial function satisfying area constraints. The best wire sizing result is chosen to minimize the worst voltage drop over all locations and all possible current distributions with different peaks and β value.

14. Sizing Results of Heuristic Search Each wire is assumed to have the same width. VD for uniform sizing = 0.3054. VD for optimized sizing = 0.2902.

15. Width Range Adjustment for M6 M6 : 3um-7um AP : 3um-16um VD = 0.2918 M6 : 4um-6um AP : 3um-16um VD = 0.2932 Original Setup M6 : 2um-8um AP : 3um-16um VD = 0.2902

16. Width Range Adjustment for AP M6 : 2um-8um AP : 5um-14um VD = 0.2961 M6 : 2um-8um AP : 7um-12um VD = 0.2975 Original Setup M6 : 2um-8um AP : 3um-16um VD = 0.2902

17. Width Range Adjustment for Both M6 and AP M6 : 4um-6um AP : 5um-14um VD = 0.2965 M6 : 3um-7um AP : 5um-14um VD = 0.2932 Original Setup M6 : 2um-8um AP : 3um-16um VD = 0.2902 M6 : 3um-7um AP : 7um-12um VD = 0.2953 M6 : 4um-6um AP : 7um-12um VD = 0.2983

18. Observations The heuristic search method performs better than NLP methods on the objective of minimizing maximum voltage drop over all locations and current distributions. Adjustment of width range of AP has more effect on performance of optimized sizing results than adjustment of width range of M6.

19. Area Budget Adjustment between M6 and AP The sizing results of both methods achieve smaller voltage drop when more area resources are allocated from AP to M6. M6 Initial Width AP Initial Width 4.2um-90% Satisfying Area Constraints 4.2um-75% 4.2um-60% … 4.2um+75% 4.2um+90%