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Layout Decomposition for Quadruple Patterning Lithography and Beyond Bei Yu , David Z. Pan Department of Electrical & Computer Engineering University of Texas at Austin, TX USA 06/03/2014 Supported by IBM scholarship, NSF, NSFC, SRC 1 /

SLIDE 1

Layout Decomposition for Quadruple Patterning Lithography and Beyond

Bei Yu, David Z. Pan

Department of Electrical & Computer Engineering University of Texas at Austin, TX USA

06/03/2014

Supported by IBM scholarship, NSF, NSFC, SRC

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SLIDE 2

Quadruple Patterning Lithography (QPL)

◮ Natural extension of triple patterning lithography (TPL) ◮ But with one more mask

Why QPL?

◮ Delay of EUVL ◮ CAD tools: need to be prepared ◮ Resolve native conflict from triple patterning mask 1 mask 2 mask 3 mask 4

a b c d a b c d

(a) (b)

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SLIDE 3

Problem Formulation

Input:

◮ Input layout patterns ◮ Minimum coloring distance mins

b c d a e

Stitch Candidate

b c d a1 e1 b e2 a2 d c b c d a1 e1 b e2 a2 d c

Output:

◮ Decomposed layout ◮ Minimize the conflict number & the stitch number

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SLIDE 4

Layout Decomposition – From Double To Quadruple

Double Patterning

ILP Odd-Cycle Partition Matching

Triple Patterning

ILP Graph Method Heuristic SDP

Quadruple Patterning ...

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SLIDE 5

SDP Formulation with NEW Color Representations

x z y

(0, 0, 1) (0, 2

√ 2 3 , − 1 3)

(

√ 6 3 , − √ 2 3 , − 1 3)

(−

√ 6 3 , − √ 2 3 , − 1 3)

◮ Four unit vectors ◮ same color:

vi · vj = 1

◮ different color:

vi · vj = −1/3

Semidefinite Programming (SDP) Formulation

min

eij∈CE
vi ·

vj − α

eij∈SE
vi ·

vj s.t. vi · vi = 1, ∀i ∈ V

vi ·

vj ≥ −1 3, ∀eij ∈ CE

◮ Greedy Mapping v.s. Backtrack Mapping

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SLIDE 6

New Linear Color Assignment

◮ Linear runtime complexity: resolved one by one ◮ But, Any coloring order results in Local Optimality ◮ Example: order a-b-c-d-e

b d c a e a c d e b

(a)

b d c a e a c d e b

(b)

Color-Friendly Rules:

b d c a e

Half Pitch

(c)

b d c a e a c d e b

(d)

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SLIDE 7

New Linear Color Assignment

Peer Selection:

◮ Three orders would be processed simultaneously ◮ Best solution would be selected ◮ Still Linear runtime complexity

3Round-Coloring

Degree-Coloring

Sequence-Coloring

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SLIDE 8

New Linear Color Assignment

Peer Selection: Better results than any single order

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SLIDE 9

New: 3-Cut Removal

◮ Reduce the problem size ◮ Example:

component 2 component 1

a b c d e f (a)

component 2 component 1

a b c d e f

component 2 component 1

a b c d e f (b) (c)

rotated by 1

color 0 color 1 color 2 color 3

GH-Tree to find 3-Cuts

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SLIDE 10

Experimental Results – Quadruple Patterning

Runtime Comparison:

C 3 , 5 4 C 5 , 3 1 5 C 6 , 2 8 8 C 7 , 5 5 2 S 1 , 4 8 8 S 3 8 , 4 1 7 S 3 5 , 9 3 2 S 3 8 , 5 8 4 S 1 5 , 8 5 Scaled Runtime

SDP+Backtrack SDP+Greedy Linear

20 40 60 80 100 C 4 3 2 C 4 9 9 C 8 8 C 1 , 3 5 5 C 1 , 9 8 C 2 , 6 7

Linear algorithm achieves

◮ 500× cf. SDP+Backtrack ◮ 60× cf. SDP+Greedy

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SLIDE 11

Experimental Results – Quadruple Patterning

Conflict # Comparison:

S 3 5 , 9 3 2 S 3 8 , 5 8 4 S 1 5 , 8 5 Conflict #

SDP+Backtrack SDP+Greedy Linear

1 10 100 1,000 C 4 3 2 C 4 9 9 C 8 8 C 1 , 3 5 5 C 1 , 9 8 C 2 , 6 7 C 3 , 5 4 C 5 , 3 1 5 C 6 , 2 8 8 C 7 , 5 5 2 S 1 , 4 8 8 S 3 8 , 4 1 7

Linear algorithm achieves:

◮ Similar conflict # cf. SDP+Backtrack ◮ 67% Conflict # reduction cf. SDP+Greedy

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SLIDE 12

Extend Algorithms to General K-Patterning

◮ Semidefinite programming (SDP) ◮ Linear color assignment ◮ GH-Tree based graph division

Results for Pentuple Patterning (K = 5)

Circuit SDP+Backtrack SDP+Greedy Linear cn# st# CPU(s) cn# st# CPU(s) cn# st# CPU(s) C6288 19 2 2.4 19 2 0.49 19 5 0.005 C7552 1 1 0.3 1 1 0.05 1 4 0.001 S38417 4 1.45 4 0.21 4 0.001 S35932 5 20 8.11 5 20 0.62 5 25 0.009 S38584 3 4 1.66 7 3 0.3 3 6 0.008 S15850 6 5 2.7 7 5 0.4 5 15 0.007 avg. 5.7 6.0 2.77 6.5 5.83 0.35 5.5 9.8 0.005 ratio 1.0 1.0 1.0 1.15 0.97 0.12 0.97 1.64 0.002 ◮ Linear color assignment: best conflict #, 500× speed-up

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SLIDE 13

Conclusions

Double Patterning

ILP Odd-Cycle Partition Matching

Triple Patterning

ILP Graph Method Heuristic SDP Linear 3-Cut Removal Graph Method SDP

Quadruple Patterning ...

◮ First attempt for Quadrule Patterning and Beyond ◮ Generic & Robust to General Patterning ◮ Facilitaing the advancement of Multiple Patterning

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SLIDE 14

Thank You !

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