Closing the Smoothness and Uniformity Gap in Area Fill Synthesis Y. - - PowerPoint PPT Presentation

Closing the Smoothness and Uniformity Gap in Area Fill Synthesis

Supported by Cadence Design Systems, Inc., NSF, the Packard Foundation, and State of Georgia’s Yamacraw Initiative

• Y. Chen
• Y. Chen,

, A. B. Kahng, G. Robins, A. Zelikovsky

• A. B. Kahng, G. Robins, A. Zelikovsky

(UCLA, UCSD, UVA and GSU) (UCLA, UCSD, UVA and GSU)

http:// http://vlsicad.ucsd.edu vlsicad.ucsd.edu

Outline

Layout Density Control for CMP Our Contributions Layout Density Analysis Local Density Variation Summary and Future Research

CMP and Interlevel Dielectric Thickness

Chemical-Mechanical Planarization (CMP)

= wafer surface planarization

Uneven features cause polishing pad to deform

Dummy features ILD thickness

Interlevel-dielectric (ILD) thickness ≈ feature density Insert dummy features to decrease variation

ILD thickness Features

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

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

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 into nr/w × × × × nr/w fixed dissections Each w × × × × w window is partitioned into r2 tiles

Overlapping windows

w w/r n tile

Previous Works

Kahng et al.

first formulation for fill problem layout density analysis algorithms first LP based approach for Min-Var objective Monte-Carlo/Greedy iterated Monte-Carlo/Greedy hierarchical fill problem

Wong et al.

Min-Fill objective dual-material fill problem

Outline

Layout Density Control for CMP Our Contributions Layout Density Analysis Local Density Variation Summary and Future Research

Our Contributions

Smoothness gap in existing fill methods

large difference between fixed-dissection and floating

window density analysis

fill result will not satisfy the given upper bounds

New smoothness criteria: local uniformity

three new relevant Lipschitz-like definitions of local density

variation are proposed

Outline

Layout Density Control for CMP Our Contributions Layout Density Analysis Local Density Variation Summary and Future Research

Oxide CMP Pattern Dependent Model

) , ( y x K dt dz ρ − =

   − < − > =

1 1

1 ) , ( ) , , ( z z z z z z y x z y x ρ ρ

z = final oxide thickness over metal features Ki = blanket oxide removal rate t = polish time ρ0 = local pattern density

Removal rate inversely proportional to density Density assumed constant (equal to pattern) until local step has been removed: Final Oxide thickness related to local pattern density      > + − − <         − =

i i i i

K z t z y x t K z z K z t y x t K z z / ) ( ) , ( / ) ( ) , (

1 1 1 1

ρ ρ ρ ρ

(Stine et al. 1997)

pattern density is crucial element of the model.

) , ( y x ρ

Layout Density Models

Spatial Density Model window density ≈ ≈ ≈ ≈ sum of tiles feature area Effective Density Model (more accurate) window density ≈ ≈ ≈ ≈ weighted sum of tiles' feature area

weights decrease from window center to boundaries

Feature Area

tile tile

The Smoothness Gap

Fill result will not satisfy the given bounds Despite this gap observed in 1998, all published filling methods fail to consider this smoothness gap floating window with maximum density Fixed-dissection analysis floating window analysis fixed dissection window with maximum density

Gap!

Accurate Layout Density Analysis

fixed dissection window arbitrary window W shrunk fixed dissection window bloated fixed dissection window tile Optimal extremal-density analysis with complexity inefficient Multi-level density analysis algorithm

An arbitrary floating window contains a shrunk window and is

covered by a bloated window of fixed r-dissection

) (

2

K O

Multi-Level Density Analysis

Make a list ActiveTiles of all tiles Accuracy = ∞ ∞ ∞ ∞, r = 1 WHILE Accuracy > 1 + 2ε ε ε ε DO

find all rectangles in tiles from ActiveTiles add windows consisting of ActiveTiles to WINDOWS Max = maximum area of window with tiles from ActiveTiles BloatMax = maximum area of bloated window with tiles from

ActiveTiles

FOR each tile T from ActiveTiles which do not belong to any

bloated window of area > Max DO

remove T from ActiveTiles

replace in ActiveTiles each tile with four of its subtiles Accuracy = BloatMax/Max, r = 2r

Output max window density = (Max + BloatMax)/(2*w2)

Multi-level Density Analysis on Effective Density Model

Assume that the effective density is calculated with the value of r-dissection used in filling process The window phase-shift will be smaller Each cell on the left side has the same dimension as the one on right side

cell window tile

Accurate Analysis of Existing Methods

.3097 .5711 .2533 .2916 .5691 .2302 .3161 .58 .2617 .2946 .5845 .2417 .2987 .5753 .2417 .5577 L2/28/16 .0926 .324 .0397 .1013 .3186 .048 .0974 .334 .0481 .1097 .3385 .063 .106 .3419 .0291 .1008 .2977 L2/28/4 .2086 .4818 .1811 .2167 .4818 .185 .2215 .5169 .1811 .2787 .5091 .2488 .2283 .4818 .2156 .4816 L1/16/16 .0724 .4251 .0499 .0693 .4245 .0481 .0713 .4286 .052 .0904 .4251 .0788 .0703 .4244 .0512 .1073 .4161 L1/16/4 Effective Density Model .0655 .1919 .0559 .0646 .1921 .0544 .0658 .1932 .0613 .0721 .1941 .0672 .0643 .1911 .0577 .0497 .1887 L2/28/16 .0898 .2181 .0328 .0908 .2202 .0326 .0973 .2236 .0482 .0986 .2244 .0529 .1012 .2288 .0326 .05 .1887 L2/28/4 .0753 .2653 .0705 .0755 .2653 .0705 .0758 .2676 .0705 .0773 .2696 .0705 .0915 .2653 .0896 .0417 .2643 L1/16/16 .0727 .2653 .0621 .084 .2653 .0621 .0756 .2679 .0621 .0783 .2706 .0621 .0855 .2653 .0639 .0516 .2572 L1/16/4

Spatial Density Model

Denv MaxD DenV Denv MaxD DenV Denv MaxD DenV Denv MaxD DenV Denv MaxD DenV MinD MaxD T/W/r Multi-Level FD Multi-Level FD Multi-Level FD Multi-Level FD Multi-Level FD OrgDen Testcase

IMC IGreedy MC Greedy LP

Multi-level density analysis on results from existing fixed-dissection filling methods

• The window density variation and violation of the maximum window density in

fixed-dissection filling are underestimated

Outline

Layout Density Control for CMP Our Contributions Layout Density Analysis Local Density Variation Summary and Future Research

Local Density Variation

Global density variation does not take into account that CMP polishing pad can adjust the pressure and rotation speed according to pattern distribution The influence of density variation between far-apart regions can be reduced by pressure adjustment Only a significant density variation between neighboring windows will complicate polishing pad control and cause either dishing or underpolishing Density variations between neighboring neighboring windows

Lipschitz-like Definitions

Local density variation definitions

Type I:

max density variation of every r neighboring windows in each

row of the fixed-dissection

The polishing pad move along window rows and only

• verlapping windows in the same row are neighbored

Type II:

max density variation of every cluster of windows which cover

• ne tile

The polishing pad touch all overlapping windows

simultaneously

Type III:

max density variation of every cluster of windows which cover

tiles

The polishing pad is moving slowly and touching overlapping

windows simultaneously

2 2 r r ×

Behaviors of Existing Methods on Smoothness Objectives

1.189 1.204 1.112 1.128 1.136 .936 1.23 1.235 1.115 1.147 1.147 1.061 1.159 1.159 1 L2/28/16 4.981 5.64 2.532 5.678 6.317 2.702 5.092 5.579 1.498 6.565 6.587 2.694 4.855 5.782 2.882 L2/28/4 .77 .77 .763 .847 .847 .839 .847 .847 .814 1.051 1.051 .978 .835 .843 .843 L1/16/16 3.315 4.481 4.245 3.132 4.254 3.994 3.448 4.166 3.631 5.19 5.619 5.332 3.864 4.333 4.048 L1/16/4

Effective Density Model

.0606 .0631 .0265 .0604 .0619 .0272 .0658 .0658 .0248 .0707 .0713 .0388 .0632 .0642 .033 L2/28/16 .0766 .0873 .0286 .0755 .0883 .0333 .0852 .0947 .0289 .0893 .096 .0412 .0841 .0989 .0414 L2/28/4 .061 .073 .0707 .0617 .0725 .0724 .0643 .0742 .0708 .0644 .0742 .073 .0711 .0868 .0854 L1/16/16 .0597 .0698 .0673 .063 .0824 .0818 .06 .0709 .0678 .0627 .0738 .0712 .0713 .0837 .0832 L1/16/4

Spatila Density Model

LipIII LipII LipI LipIII LipII LipI LipIII LipII LipI LipIII LipII LipI LipIII LipII LipI T/W/r

IMC IGreedy MC Greedy LP

Testcase

Comparison among the behaviors of existing methods w.r.t Lipschitz objectives

• The solution with the best Min-Var objective value does not always have the

best value in terms of “smoothness” objectives

Linear Programming Formulations

Lipschitz Type I

r j i r j i m l L j i Den j i Den w nr k j i j i Den W j i Den

lm

+ − = ≤ − − = ≤ ≤ ) ( , , ) ( ) ( ) , ( min ) , ( max 1 ,..., , , ) , ( max ) , ( min K

1 / ,..., , , 1 / ,..., , ) ( 1 / ,..., , ) ( 1 / ,..., ,

2 1 1

− = ≤ − − = −

• ≤

− = ≤ − = ≥

∑ ∑

− + = − + =

w nr k j i L W W w nr j i area w U p w nr j i T slack p w nr j i p

ik ij ij ij r i i s r j j t st ij ij ij

α

Lipschitz Type II

∑ ∑ ∑ ∑

− + = − + = − + = − + =

+ =

1 1 1 1

) (

r i i s r j j t r i j s r j j t st st ij

p T area W

here,

Linear Programming Formulations

II I

L C L C M C Minimize ∗ + ∗ + ∗

2 1

:

Lipschitz Type III

2 ) ( , , 2 ) ( ) ( ) , ( min ) , ( max 1 ,..., , , ) , ( max ) , ( min r j i r j i m l L j i Den j i Den w nr k j i j i Den W j i Den

lm

+ − = ≤ − − = ≤ ≤ K

Combined Objectives

linear summation of Min-Var, Lip-I and Lip-II objectives

with specific coefficients

add Lip-I and Lip-II constraints as well as

1 / , , , − = ≤ w nr j i W M

ij

K

Computational Experience

.0024 .0024 .0007 .1527 .0022 .0023 .0015 .1483 L2/28/8 .0061 .0061 .0013 .1051 .0049 .0058 .0029 .106 L2/28/4 .0052 .006 .002 .3939 .0023 .0025 .0025 .1709 L1/16/8 .01 .0154 .004 .2662 .0039 .0043 .0045 .0703 L1/16/4 Effective Density Model .0744 .0825 .0264 .0871 .0654 .0658 .034 .0666 L2/28/8 .0693 .072 .0251 .0724 .0841 .0989 .0414 .1012 L2/28/4 .1428 .1932 .0938 .1972 .067 .0777 .0734 .0814 L1/16/8 .1268 .167 .0553 .1725 .0713 .0837 .0832 .0855 L1/16/4 Spatial Density Model Lip3 Lip2 Lip1 Den V Lip3 Lip2 Lip1 Den V T/W/r LipI LP Min-Var LP Testcase

Comparison among LP methods on Min-Var and Lipschitz condition objectives (1)

Computational Experience

.0021 .0021 .0007 .1382 .0022 .0032 .0018 .2063 .0022 .0023 .0015 .1559 L2/28/8 .0049 .0057 .0015 .0953 .0052 .0064 .0026 .1039 .0054 .0064 .0029 .1022 L2/28/4 .0019 .0022 .0021 .268 .0018 .0029 .0028 .2906 .0018 .0025 .0025 .2902 L1/16/8 .0034 .0045 .004 .1753 .003 .0051 .0043 .1792 .0033 .0047 .0039 .1594 L1/16/4 Effective Density Model .0656 .0708 .0255 .0747 .0714 .1033 .0594 .1188 .0661 .0697 .0331 .07 L2/28/8 .0758 .0809 .0242 .0825 .0895 .0928 .0462 .0943 .0836 .0871 .0467 .0888 L2/28/4 .0766 .1005 .0937 .1707 .0664 .1224 .1158 .1835 .0756 .1027 .1016 .1702 L1/16/8 .0409 .0619 .0574 .1143 .0433 .0734 .0733 .1273 .0434 .0663 .0649 .1265 L1/16/4 Spatial Density Model Lip3 Lip2 Lip1 DenV Lip3 Lip2 Lip1 DenV Lip3 Lip2 Lip1 DenV T/W/r Comb LP LipIII LP LipII LP Testcase

Comparison among LP methods on Min-Var and Lipschitz condition objectives (2)

• LP with combined objective achieves the best comprehensive solutions

Outline

Layout Density Control for CMP Our Contributions Layout Density Analysis Local Density Variation Summary and Future Research

Summary and Future Research

Ongoing research

extension of multi-level density analysis to measuring

local uniformity w.r.t. other CMP models

improved methods for optimizing fill synthesis w.r.t. new

local uniformity objectives

Smoothness gap in existing fill methods

for the first time, we show the viability of gridless window

analysis for both spatial density model and effective density model

New smoothness criteria: local uniformity

three new relevant Lipschitz-like definitions of local

density variation are proposed