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Double-Patterning Aware DSA TemplateGuided Cut Redistribution for Advanced1-D Gridded Designs
Zhi-Wen Lin and Yao-Wen Chang
National Taiwan University
ISPD 2016
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Outline
Introduction
Conclusions
Contributions
Algorithm
Experimental Results
Problem Formulation
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Directed Self-Assembly Block Copolymer
․The directed self-assembly (DSA) technology hasemerged as a promising candidate for sub-10 nmdesigns
[Xiao et al., SPIE’13]
(a)
(b)
(c)
(d)
+
=
Self-assemblyblock copolymer
Topographicaltemplates
Smaller anddenser patterns
Template
Feature
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1-D Style Layout Design with DSA
․DSA has applications in
Contact hole patterning [Wong et al., SPIE’12]
Cut printing of 1-D gridded designs [Ma et al., SPIE’14]
․1-D layouts are popular for their regularity [Lam et al.,SPIE’11]
․DSA can be used to pattern dense cuts in 1-D layouts
Dummy wire (gap)
DSA templates
The resulting 1-D layout
Real wire
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Challenge of Cut Printing with DSA
․The size of the feasible-template library is limited
․Conflict cost: two close templates might interfere witheach other [Ma et al., SPIE’14]
# cut pairs with spacing conflicts should be minimized
[Ma et al., SPIE’14]
Complex template
Poor printability
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Cut Redistribution
․Redistribute cuts to reduce conflict costs
․Wire cost: the total length of extended real wires shouldbe minimized considering the performance impact
․Given a layout, minimize its conflict cost and wire cost
DSA template guided cut redistribution problem (DSA-TCR)
vs.
Extended wire
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Double Patterning
․Double patterning can be adopted to further improvethe image fidelity of templates in cut printing with DSA[Ma et al., SPIE’14]
․Integrate DSA cut redistribution with double patterning
Conflict costs and wire costs can be further reduced
(a) Dense templates
(b) Double patterning
decomposition
Mask 1
Mask 2
(c) Patterned cuts
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Outline
Introduction
Conclusion
Contributions
Algorithm
Experimental Results
Problem Formulation
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Main Contributions: Algorithms
․A linear-time optimal algorithm for the special case withlimited contiguous rows is identified
Simultaneously minimize conflict costs and wire costs
․A linear-time algorithm for decomposing a generalproblem into subproblems conforming to the special case
With double patterning, a criterion for non-interferingsubproblems is proposed
Large distance
No conflict
Different mask
No conflict
Subproblems with contiguous rows
Subproblem
of special case
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Main Contributions: Performance
․The total wire cost of a derived template distribution islinear to the number of cuts
․Experimental results show that our algorithm can resolveall conflicts, with smaller running times, compared withthe related previous works
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Outline
Introduction
Conclusion
Contributions
Algorithm
Experimental Results
Problem Formulation
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Problem Formulation
․Given
A library of templates, design rules, a layout, and originaldistribution of cuts
․Output
A template distribution patterning required cuts
A mask assignment of all derived templates
․Objective
An ordered optimization on the conflict cost first and then thewire cost
Our algorithm can readily apply to the linear combination ofconflict cost and wire cost
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Outline
Introduction
Conclusion
Contributions
Algorithm
Experimental Results
Problem Formulation
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Overall Flow
․Propose an algorithm (DPDSA-Core), incorporatingDSA with double patterning
Divide a general problem into subproblems
Subproblem candidate construction
Solving every subproblem candidate
Subproblem candidate selection
Input
Output
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Subproblem Candidate Construction
Subproblem candidate construction
Solving every subproblem candidate
Subproblem candidate selection
Input
Output
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Special Case with Contiguous Rows
No conflict
Different mask
No conflict
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Solutions of Partitions
․Subproblems with fewer rows can have less internalwire cost and conflict cost
․Also consider the subproblems with fewer rows when asubproblem partition with fewer inter-costs exists
․A linear-time partition algorithm is proposed
Subproblem withfewer rows
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Solving Subproblem Candidates
Subproblem candidate construction
Solving every subproblem candidate
Subproblem candidate selection
Input
Output
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Overview
․Given a fixed rM, solve a subproblem optimally in lineartime to the number of gaps in the subproblem
․Find a path of gap solutions with minimized conflictcosts and wire costs
Templatespecifier
Distributionspecifier
g2
g1
h
th,j
dh,j
…
…
…
…
…
…
h
h+1
h+2
…
…
…
…
vh,j
vh+2,j
vh+1,j
vh,j+2
vh+2,j+2
vh+1,j+2
vh,j+1
vh+2,j+1
vh+1,j+1
Divide a sub-probleminto multiple columns
For every column,generate feasiblegap solutions
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Data Structure of Column Solutions
․Template specifier specifies the topologies of cuts inthe current column and restrict the templates next to it
․To consider cut conflict completely, store thedistribution of previous cuts in a column solution
․For every column solution, find the best compatibleprevious column solution
Horizontalcolumnindex
Previouscutdistribution
Templatespecifier
Distributionspecifier
th,j
dh,j
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Column-wise Searching
․Construct column solutions from left to right
․For every pair of a column solution and its predecessor,evaluate the current conflict costs and wire costs
․Then trace back according to predecessors for a path
g2
g1
Add 1 conflict
Add 1 grid wire cost
Right cut of g2 hasn’taddressed yet
Predecessor
Current solution
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Reduced Column-Space
․Prune redundant columns to speed up the process
․For ever gap, calculate the remaining columns within it
First, calculate the possible range for cuts
Scan from left to right, and use the information of possiblerange to get the remaining columns
․Restrict the complexity of the number of remainingcolumns and the total wire cost to be linear
Thus the running time grows linearly
Possible range
Remainingcolumns
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Subproblem Candidate Selection
Subproblem candidate construction
Solving every subproblem candidate
Subproblem candidate selection
Input
Output
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Overview of Partition Algorithm
․Divide the whole layout into regions
Subproblem candidate selection
Region construction
Region solution
graph construction
Optimal region solution
path determination
Template distributionconstruction
R1
R0
R2
R1
R0
R2
v1
v2
v3
v4
v5
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Region Solutions
․Every region solution is a set of solved subproblems
Subproblem candidate selection
Region construction
Region solution
graph construction
Optimal region solution
path determination
Template distributionconstruction
R1
R2
2
1
2
1
R1
R2
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Region Solution Graph
․An edge exists between two region solutions whenthey share a subproblem
Subproblem candidate selection
Region construction
Region solution
graph construction
Optimal region solution
path determination
Template distributionconstruction
R2
R1
R2
R1
․Inter-subproblem conflicts can occur only betweenadjacent regions for the large enough height of regions
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Partition from Region Solution
․Get subproblems by selected region solutions
․Construct the whole solution by selected subproblems
3
2
1
5
4
3
2
1
5
3
2
1
5
4
4
Subproblem candidate selection
Region solution
graph construction
Optimal region solution
path determination
Template distributionconstruction
Region construction
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Extension
․A framework to relax the same-mask constraint in a row
Input
Output
Terminal condition
Double-patterning for templates
DPDSA-Core for everyconflicting component
Not satisfied
Satisfied
Conflicting componentselection and merging
Connected componentconstruction
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Outline
Introduction
Conclusion
Contributions
Algorithm
Experimental Results
Problem Formulation
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Experimental Setting
․Platform: 2.13 GHz Linux workstation with 16 GB memory
․Programming language: C++
․Resource of benchmarks: the authors of [Xiao et al.,SPIE’13]
․Compared with
[Xiao et al., SPIE’13] [Ou et al., GLSVLSI’15]
Extend these works by a general high-level framework [Badr et al.,SPIE’15] for DSA-multiple patterning hybrid lithography
The framework consists of the two stages GP-MP (GM) and MP-GP (MG)
GP: group cuts into templates
MP: Assign masks
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Conflict Costs
![Our algorithm can resolve all conflicts
[Xiao et al., SPIE’13]: S13
[Ou et al., SPIE’15]: S15 (ILP-based algorithm), S15sp (ILP-based speed-up heuristic), SC (set-cover-based algorithm)
“-”: running time ≥ an hour](data/images/img37.png)
no conflict!!
Numberof cuts
Initialconflict
Conflict cost
GP-MP
MP-GP
S13
S15sp
S15
SC
S13
S15sp
S15
SC
ours
50
12
0
0
0
0
0
0
0
1
0
100
46
0
0
0
0
0
1
0
0
0
200
83
0
0
0
0
3
0
0
4
0
500
223
0
0
-
0
4
0
0
5
0
1000
421
2
0
-
1
7
3
0
8
0
2000
855
0
0
-
2
17
2
0
11
0
Average
0.33
0
-
0.50
5.17
1.00
0
4.83
0
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Wire Cost
․Our wire cost is 6X smaller than S15sp with GP-MP
․Except pure-ILP works, our wire costs are consistentlysmaller than others with the same numbers of conflicts
Number ofcuts
Wire cost
GP-MP
MP-GP
S13
S15sp
S15
SC
S13
S15sp
S15
SC
ours
50
48
8
6
3
13
1
1
0
1
100
109
29
24
30
48
5
2
6
6
200
272
58
48
63
150
27
11
23
10
500
665
153
-
166
235
33
14
29
27
1000
1466
307
-
289
542
48
32
45
49
2000
2647
593
-
576
1055
89
65
76
96
Average
867.83
191.33
-
187.83
340.50
33.83
20.83
29.83
31.50
Normalized
average
27.55
6.07
0.00
5.96
10.81
1.07
0.66
0.95
1.00
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Running Time
․Our algorithm has better scalability than others
Betterscalability
Numberof cuts
Running time (seconds)
GP-MP
MP-GP
S13
S15sp
S15
SC
S13
S15sp
S15
SC
ours
50
0.01
0.09
0.22
0.01
0.01
0.08
0.04
0.01
0.01
100
0.02
0.45
9.78
0.03
0.02
0.16
0.07
0.03
0.02
200
0.04
0.53
609
0.09
0.04
0.33
0.45
0.07
0.03
500
0.12
1.95
-
0.34
0.12
0.84
1.06
0.2
0.08
1000
0.31
3.33
-
1.22
0.32
1.84
6.12
0.64
0.15
2000
0.91
7.99
-
4.38
0.92
4.49
55.1
2.45
0.3
Average
0.24
2.39
-
1.01
0.24
1.29
10.47
0.57
0.10
Normalized
average
2.39
24.31
-
10.29
2.42
13.12
106.51
5.76
1.00
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Outline
Introduction
Conclusion
Contributions
Algorithm
Experimental Results
Problem Formulation
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Conclusions
․A linear-time optimal algorithm for the special case ofDSA-TCR
․A linear-time algorithm for decomposing general doublepatterning DSA-TCR problems and reduce inter-subproblem conflicts cost
․Better experimental results compared with the previouswork
․Future Work
Incorporate DSA with multiple patterning technology
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Thank You!