Practical Iterated Fill Synthesis for CMP Uniformity

1. Supported by Cadence Design Systems, Inc. Practical Iterated Fill Synthesis for CMP Uniformity Y. Chen, A. B. Kahng, G. Robins, A. Zelikovsky (UCLA, UVA and GSU) http://vlsicad.cs.ucla.edu

2. Outline • Chemical-Mechanical Polishing (CMP) • Filling Problem in fixed-dissection regime • LP and Monte-Carlo (MC) approaches • Our contributions: • MC approach with Min-Fill objective • Iterated MC method • Computational experience • Summary and research directions

3. Chemical -Mechanical Polishing (CMP) = wafer surface planarization Uneven features cause polishing pad to deform Features ILD thickness Dummy features ILD thickness CMP and Interlevel Dielectric Thickness • Interlevel-dielectric (ILD) thickness is proportional to feature density • Insert dummy features to decrease variation

4. Objectives of Density Control • Objective for Manufacture = Min-Var minimizewindow density variation subject to upper bound on window density • Objective for Design = Min-Fill minimize total amount of filling subject to fixed density variation

5. Filling Problem • Given • rule-correct layout in n  nregion • window size = w  w • window density upper bound U • Fill layout with Min-Var or Min-Fill objective such that nofill is added • within buffer distance B of any layout feature • into any overfilled window that has density  U

6. w w/r tile Overlapping windows n Fixed-Dissection Regime • Monitor only fixed set of w  w windows • “offset” = w/r (example shown: w = 4, r = 4) • Partition n x n layout withnr/w nr/w fixed dissections • Each w w window is partitioned into r2tiles

7. tile Slack Area Feature Area Layout Density Models • Spatial Density Model window density  sum of tiles’ feature area • Effective Density Model (more accurate) window density weighted sum of of tiles’s feature area • elliptical weights decrease from window center to boundaries

8. Min-Var Objective (Kahng et al.) Maximize: M Subject to: for any tile 0  p[T]  slack[T] for any window  TW (p[T]+area[T])  U M   TW (p[T] + area[T]) p[T] = fill area of tile spatial density model Linear Programming Approach • Min-Fill Objective (Wong et al.) • Minimize: fill amount • Subject to: for any tile 0  p[T]  slack[T] LowerB  0(T)  UpperB UpperB - LowerB   0(T)= the effective density of tile T • effective density model

9. Monte-Carlo Approach • Fill layout randomly • pick the tile for next filling geometry randomly • higher priority of a tile  higher probability to be filled • lock tile if any containing window is overfilled • Tile priorities • slack • min density of any windows containing the tile • max density of any windows containing the tile • Heuristics for updating priorities • update priorities of all affected tiles • update priorities only of tiles which belong to newly locked window

10. LP vs. Monte-Carlo • LP • impractical runtime for large layouts • r-dissection solution may be suboptimal for 2r dissections • essential rounding error for small tiles • Monte-Carlo • very efficient: O((nr/w)log(nr/w)) time • scalability: handle large values of r • accuracy: reasonably high comparing with LP • drawback: excessive amount of fill features for Min-Var

11. Monte-Carlo with Min-Fill Objective • Delete excessive fill ! • Fill-Deletion problem • delete as much fill as possible while maintaining min window density  L. • Min-Fill Monte-Carlo Algorithm • if (min covering-window density < L) lock the tile • randomly select unlocked tile according to its priority • delete a filling geometry from tile • update priorities of tiles

12. Min-Var No Improvement Min-Fill Iterated Monte-Carlo Approach • Repeat forever • run Min-Var Monte-Carlo with maximum window density U • exit if no change in minimum window density • run Min-Fill Monte-Carlo Algorithm with minimum window density M

13. Testcases Metal layers from industry standard-cell layouts Computational Experience • Implementation features • grid slack computation • doughnut area computation • wraparound density analysis and synthesis • different pattern types • Testbed • GDSII input • hierarchical polygon database • C++ under Solaris • open-source code

14. Computational Experience Iterated Monte-Carlo approach is more accurate than standard MC approach and faster than LP approach

15. Summary and research directions • Monte-Carlo approach with Min-Fill objective • Iterated Monte-Carlo approach • more accurate and practical • several practical features • simultaneously address different filling objectives for different density models • Ongoing research • hierarchical filling • grounded fill generation • multi-layer density control

16. Thank you !