Practical Iterated Fill Synthesis for CMP Uniformity

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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
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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

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CMP and Interlevel Dielectric Thickness

• Chemical -Mechanical Polishing (CMP)
• wafer surface planarization
• Uneven features cause polishing pad to deform

• Interlevel-dielectric (ILD) thickness is
  proportional to feature density
• Insert dummy features to decrease variation

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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

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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

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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 r2 tiles

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Layout Density Models

• Spatial Density Model
• window density ? sum of tiles feature area
  

• Effective Density Model (more accurate)
• window density ? weighted sum of of tiless
  feature area
• elliptical weights decrease from window center to
  boundaries

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Linear Programming Approach

• Min-Var Objective
• (Kahng et al.)
• Maximize M
• Subject to
• for any tile
• 0 ? pT ? slackT
• for any window
• ? T?W (pTareaT) ? U
• M ? ? T?W (pT areaT)
• pT fill area of tile
• spatial density model

• Min-Fill Objective
• (Wong et al.)
• Minimize fill amount
• Subject to
• for any tile
• 0 ? pT ? slackT
• LowerB ? ?0(T) ? UpperB
• UpperB - LowerB ? ?
• ?0(T) the effective density of tile T
• effective density model

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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

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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

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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 lt L) lock the
  tile
• randomly select unlocked tile according to its
  priority
• delete a filling geometry from tile
• update priorities of tiles

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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

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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

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Computational Experience

• Iterated Monte-Carlo approach is more accurate
  than standard MC approach and faster than LP
  approach

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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

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Thank you !