1. Practical Iterated Fill Synthesis for CMP Uniformity Supported by Cadence Design Systems, Inc. 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. CMP and Interlevel Dielectric Thickness Chemical -Mechanical Polishing (CMP) = wafer surface planarization Uneven features cause polishing pad to deform Dummy features ILD thickness Interlevel-dielectric (ILD) thickness is proportional to feature density Insert dummy features to decrease variation ILD thickness Features

4. Objectives of Density Control Objective for Manufacture = Min-Var minimize window 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  n region  window size = w  w  window density upper bound U Fill layout with Min-Var or Min-Fill objective such that no fill is added  within buffer distance B of any layout feature  into any overfilled window that has density  U

6. 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 with nr/w  nr/w fixed dissections Each w  w window is partitioned into r 2 tiles Overlapping windows w w/r n tile

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

8. Linear Programming Approach 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 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. 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 No Improvement

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

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 !