Supported by Cadence Design Systems, Inc.,
NSF, the Packard Foundation, and
State of Georgia’s Yamacraw Initiative
Monte-Carlo Methods for
Chemical-Mechanical Planarization on
Multiple-Layer and Dual-Material Models
Y. Chen, A. B. Kahng, G. Robins, A. Zelikovsky
(UCLA, UCSD, UVA and GSU)
http://vlsicad.ucsd.edu
Outline
 Layout Density Control for CMP
 Our Contributions
 STI Dual-Material Dummy Fill
 Multiple-layer Oxide CMP Dummy Fill
 Summary and Future Research
CMP and Interlevel Dielectric Thickness
 Chemical-Mechanical Planarization (CMP)
= wafer surface planarization
 Uneven features cause polishing pad to deform
Features
ILD thickness
 Interlevel-dielectric (ILD) thickness  feature density
 Insert dummy features to decrease variation
Dummy
features
ILD thickness
Layout Density Model
 Effective Density Model
window density  weighted sum of tiles' feature area

weights decrease from center tile to neighboring tiles
Filling Problem
 Given
 rule-correct
 upper
layout in n  n region
bound U on tile density
 Fill layout subject to the given constraints
 Min-Var objective
 minimize
density variation subject to upper bound
 Min-Fill objective
total amount of filling subject to fixed
density variation
 minimize
LP and Monte-Carlo Methods
 Single-layer fill problem  linear programming
problem
 impractical
 essential
runtime for large layouts
rounding error for small tiles
 Monte-Carlo method (accurate and efficient)
 calculate
priority of each tile according to its effective
density
 higher priority of a tile  higher probability to be filled
 pick the tile for next filling randomly
 if the tile is overfilled, lock all neighboring tiles
 update priorities of all neighboring tiles
Outline
 Layout Density Control for CMP
 Our Contributions

new Monte-Carlo methods for STI Min-Var and Min-Fill
objectives

LP formulations for a new multiple-layer fill objective

new Monte-Carlo methods for multiple-layer fill problem
 STI Dual-Material Dummy Fill
 Multiple-layer Oxide CMP Dummy Fill
 Summary and Future Research
Our Contributions
 Fill problem in STI dual-material CMP
 new
Monte-Carlo methods for STI Min-Var objective
 new
Monte-Carlo/Greedy methods with removal phase
for STI Min-Fill objective
 Fill problem in Multiple-layer oxide CMP
a
LP formulation for a new multiple-layer fill objective
 new
Monte-Carlo methods
Outline
 Layout Density Control for CMP
 STI Dual-Material Dummy Fill
 new
Monte-Carlo methods for Min-Varr and Min-Fill
objectives
 Multiple-layer Oxide CMP Dummy Fill
 Summary and Future Research
Shallow Trench Isolation Process
Nitride
Silicon
nitride deposition on silicon
height difference H
etch shallow trenches
through nitride silicon
Oxide
remove excess oxide
and partially nitride by CMP
oxide deposition
 Uniformity requirement on
CMP in STI
nitride stripping

under polish

over polish
STI CMP Model
 STI post-CMP variation can be controlled by
changing the feature density distribution
using dummy features insertion
 Compressible pad model
 polishing
occurs on both up and down areas after
some step height
 Dual-Material polish model
 two
different materials are for top and bottom
surfaces
STI Fill Problem
 Non-linear programming problem
 Min-Var objective: minimize max height variation
 Previous
method (Motorola)
 Drawbacks
of previous work
 dummy feature is added at the location having the
cansmallest
not guarantee
find a global minimum since it
effectiveto
density
is deterministic
 terminates when there is no feasible fill position left
 for Min-Fill, simple termination when the bound is
first met
is not sufficient
tototal
yieldnumber
optimal/sub Min-Fill
objective:
minimize
of inserted
optimal solutions.

fill, while keeping the given lower bound
 Previous


method (Motorola)
adds dummy features greedily
concludes once the given bound for ΔΗ is satisfied
Monte-Carlo Methods for STI Min-Var
 Monte-Carlo method
 calculate
 pick

priority of tile(i,j) as H - H (i, j, i’, j’)
the tile for next filling randomly
if the tile is overfilled, lock all neighboring tiles
 update
tile priority
 Iterated Monte-Carlo method
 repeat
forever
 run
Min-Var Monte-Carlo with max height difference H
 exit
if no change in minimum height difference
 delete
as much as possible pre-inserted dummy
features while keeping min height difference M
MC/Greedy methods for STI Min-Fill
 Find a solution with Min-Var objective to satisfy the
given lower bound
 Modify the solution with respect to Min-Fill objective
 Algorithm
 Run
Min-Var Monte-Carlo / Greedy algorithm
 Compute
 WHILE
removal priority of each tile
there exist an unlocked tile DO

Choose unlock tile Tij randomly according to priority

Delete a dummy feature from Tij

Update the tile’s priority
STI Fill Results
Greedy
testcase
L1/32/4
L1/32/8
L2/28/4
L2/28/8
L3/28/4
Orig H
695.2
999.6
801.8
1124.6
1095.2
H
305.3
426.1
487.9
569.8
577.8
CPU
3.2
3.8
5.1
5.7
8.5
MC
H
335
325.8
374
536.1
563.2
Igreedy
CPU
3.2
3.4
5.2
5.2
8.3
H
304.5
407.4
487.9
546.3
577.8
CPU
3.5
7.1
5.7
10.1
8.9
IMC
H
290.2
307.2
348
482.7
563.2
CPU
3.4
4.2
5.6
6.6
9.1
Methods (Greedy, MC, IGreedy ad IMC) for STI Fill under Min-Var objective
testcase
L1/32/4
L1/32/8
L2/28/4
L2/28/8
L3/28/4
Orig H
695.2
999.6
801.8
1124.6
1095.2
GreedyI
final H Area CPU
395.1 10336
3.1
462.7 22091
3.9
526.2 7491
4.8
639.8 16808
5.7
563.2 24274
8.2
MCI
Area CPU
12003
3.1
20679
3.4
15164
4.9
26114
5.5
27114
8
GreedyII
Area CPU
8962
3.2
15615
3.6
7593
5.1
8367
5.9
16628
8.6
MCII
Area CPU
9141
3.2
14754
3.4
6543
5.1
7142
5.5
16142
8.5
Methods(GreedyI, MCI, GreedyII and MCII) for STI Fill under Min-Fill objective
Outline
 Layout Density Control for CMP
 Our Contributions
 STI Dual-Material Dummy Fill
 Multiple-layer Oxide CMP Dummy Fill

LP formulations for a new multiple-layer fill objective

new Monte-Carlo methods
 Summary and Future Research
Multiple-Layer Oxide CMP
 Each layer except the bottom one can’t assume a
perfect flat starting surface
Layer 1
Layer 0
 Multiple-layer density model
^ : fast Fourier transform operator
:effective local density
: step height
: local density for layer k
Multiple-Layer Oxide Fill Objectives
 (Min-Var objective) minimize
 sum


of density variations on all layers
can not guarantee the Min-Var objective on each layer
A bad polishing result on intermediate layer may cause problems on
upper layers
 maximum
density variation across all layers
 LP formulation
 Min
M
 Subject to:
Multiple-Layer Monte-Carlo Approach
 Tile stack

column of tiles having the same positions on all layers
 Effective density of tile stack

sum of effective densities of all tiles in tile stack
tile stack
tiles on each layer
layer 3
layer 2
layer 1
Multiple-Layer Monte-Carlo Approach
 Compute slack area and cumulative effective
density for each tile stack
 Calculate priority of each tile stack according to its
cumulative effective density
 WHILE ( sum of priorities > 0 ) DO

randomly select a tile stack according to its priority

from its bottom layer to top layer, check whether it is
feasible to insert a dummy feature in

update slack area and priority of the tile stack

if no slack area left, lock the tile stack
Multiple-Layer Fill Results
LP0
testcase SumVar
CPU
Greedy
SumVar
CPU
MC
SumVar
CPU
IGreedy
SumVar
CPU
IMC
SumVar
CPU
L4/16/8
L4/16/5
L4/8/8
L4/8/5
L5/8/8
L5/8/5
0.642
0.5541
0.7794
0.7882
2.0526
1.345
0.6285
0.5535
0.7766
0.7804
2.0913
1.3943
0.6285
0.5535
0.7762
0.7804
2.0526
1.3252
0.6285
0.5535
0.7762
0.7804
2.0716
1.3476
0.6626
0.5435
0.9031
0.8351
2.2118
1.3494
33.1
30.7
140.1
33.4
8093
8879
36.3
32.2
48.1
35.4
102.8
65
36.6
33
36.2
32.7
65.4
54.2
37.7
31.1
74.7
39.1
111.7
79.6
33.9
30.5
34.5
30.7
67.6
59.3
LP1
testcase MaxDen CPU
Greedy
MaxDen CPU
MC
MaxDen CPU
IGreedy
MaxDen CPU
IMC
MaxDen CPU
L4/16/8
L4/16/5
L4/8/8
L4/8/5
L5/8/8
L5/8/5
0.4459
0.3638
0.5437
0.5576
1.1081
0.6857
0.4454
0.3635
0.541
0.5497
1.1089
0.705
0.4454
0.3635
0.5406
0.5497
1.1081
0.6698
0.4454
0.3635
0.5406
0.5497
1.1081
0.6746
0.4696
0.3638
0.6255
0.5897
1.2174
0.6886
34.8
36.5
120.8
33.2
761.3
524
36.3
30.2
48.1
35.4
102.8
65
36.6
33
36.2
32.7
65.4
54.2
37.7
31.1
74.7
39.1
111.7
79.6
33.9
32.5
34.5
30.7
67.6
59.3
Performance of LP0, LP1, Greedy, MC, IGreedy and IMC for Min-Var-Sum
Outline
 Layout Density Control for CMP
 Multiple-layer Oxide CMP Dummy Fill
 STI Dual-Material Dummy Fill
 Summary and Future Research
Summary and Future Research
 STI fill problem
 Monte-Carlo
methods for STI Min-Var
 Monte-Carlo / Greedy methods for STI Min-Fill
 Multiple-layer fill problem
 LP
formulation for a new Min-Var objective
 efficient multiple-layer Monte-Carlo approaches
 Ongoing research
 further
study of multiple-layer fill objectives
 more powerful Monte-Carlo methods for multiplelayer fill problem
 CMP simulation tool
Thank you!