High High Per erfor ormance mance Dummy ummy Fill Ins Fill - - PowerPoint PPT Presentation

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high high per erfor ormance mance dummy ummy fill ins

High High Per erfor ormance mance Dummy ummy Fill Ins Fill - - PowerPoint PPT Presentation

Jan 31, 2023 193 likes •479 views

High High Per erfor ormance mance Dummy ummy Fill Ins Fill nser ertion ion wit ith h Coupling oupling and and Unif Unifor ormit mity Cons onstraint aints Yibo Lin, Bei Yu, David Z. Pan Electrical and Computer Engineering

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SLIDE 1

Yibo Lin, Bei Yu, David Z. Pan Electrical and Computer Engineering University of Texas at Austin

High High Per erfor

rmance

mance Dummy ummy Fill Fill Ins nser ertion ion wit ith h Coupling

upling

and and Unif Unifor

rmit

mity Cons

nstraint

aints

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SLIDE 2

Outline

! Introduction ! Problem Formulation ! Algorithms ! Experimental Results ! Conclusion

2

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SLIDE 3

Chemical Mechanical Polishing (CMP)

3

Example of CMP [source: www.ntu.edu.sg]

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SLIDE 4

Uniformity

! Layout uniformity for CMP ! Coupling capacitance

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Metals Metals Dummy features

Uneven density distribution Even distribution

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SLIDE 5

Related Works

! Minimize density variation and number of fills

" Linear Programming (LP)

" [Kahng+,TCAD’99] " [Tian+, TCAD’01] " [Xiang+, TCAD’08]

" Monte Carlo and heuristic approaches

" [Chen+, ASPDAC’00] " [Chen+, DAC’00] " [Wong+, ISQED’05] ! Minimize density variation with coupling

capacitance constraints

" ILP

" [Chen+, DAC’03], [Xiang+, ISPD’07]

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SLIDE 6

Holistic Metrics for Uniformity

! Holistic metrics for layout uniformity from IBM

(ICCAD 2014 Contest)

" Variation (standard deviation) " Line hotspots " Outlier hotspots

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N Columns M Rows

µ µ + 3σ µ + 3σ µ − 3σ µ − 3σ

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SLIDE 7

Holistic Metrics for Uniformity

! Holistic metrics for layout uniformity from IBM

(ICCAD 2014 Contest)

" Variation (standard deviation) " Line hotspots " Outlier hotspots

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N Columns M Rows

µ µ + 3σ µ + 3σ µ − 3σ µ − 3σ

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SLIDE 8

Metrics for Coupling Capacitance

! Coupling capacitance

" Minimize overlay between layers

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Layer 1 Layer 2 Layer 3

Overlay Dummy fill

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SLIDE 9

Problem Formulation

Based on the ICCAD 2014 contest

! Input

" Layout with fill insertion regions " Signal wire density information across each window

! Quality score

" Overlay area (20%) " Variation/std. dev. (20%) " Line hotspot (20%) " Outlier hotspot (15%) " File size for dummy fill insertion (5%)

! Overall score

" Quality score (80%) " Runtime (15%) " Memory usage (5%)

! Output

" Dummy fill positions and dimensions with maximum quality score

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f(x) = max (0, 1 − x β )

Normalization function The higher score, the better

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SLIDE 10

Outline

! Introduction ! Problem Formulation ! Algorithms ! Experimental Results ! Conclusion

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SLIDE 11

Step 1: Density Planning

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Linear scan with a small step to find best target density

d1 d1 d2 d2 d3 d3 d4 d4 d5 d5 d6 d6 d7 d7 d8 d8 d9 d9

! Given density ranges of each window ! Find target density td for each window ! Maximize density scores

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SLIDE 12

Step 1: Density Planning

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Linear scan with a small step to find best target density

0.3 0.5 0.6 0.5 0.4 0.4 0.4 0.4 0.4

! Given density ranges of each window ! Find target density td for each window ! Maximize density scores

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SLIDE 13

Step 1: Density Planning

13

Linear scan with a small step to find best target density

0.3 0.5 0.6 0.5 0.4 0.4 0.4 0.4 0.4

! Given density ranges of each window ! Find target density td for each window ! Maximize density scores

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SLIDE 14

Step 1: Density Planning

14

Linear scan with a small step to find best target density

0.3 0.5 0.6 0.5 0.4 0.4 0.4 0.4 0.4

! Given density ranges of each window ! Find target density td for each window ! Maximize density scores

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SLIDE 15

Step 1: Density Planning

15

Linear scan with a small step to find best target density

0.3 0.5 0.6 0.5 0.4 0.4 0.4 0.4 0.4

! Given density ranges of each window ! Find target density td for each window ! Maximize density scores

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SLIDE 16

Step 1: Density Planning

16

Linear scan with a small step to find best target density

0.3 0.5 0.6 0.5 0.4 0.4 0.4 0.4 0.4

! Given density ranges of each window ! Find target density td for each window ! Maximize density scores

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SLIDE 17

Step 2: Candidate Fill Generation

! Generate candidate fills with minimum overlay ! With the guidance of target density ! A final fill is a rectangle within a candidate fill

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Candidate fill Final fill

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SLIDE 18

Step 2: Candidate Fill Generation

! Generate candidate fills with minimum overlay ! With the guidance of target density ! A final fill is a rectangle within a candidate fill

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Layer 1 Fill Layer 2 Fill Layer 3 Fill

q = −fill overlay fill area + γ · fill area window area

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Step 3: Dummy Fill Insertion (1)

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Layer 1 Fill Layer 2 Fill Layer 1 Fill Layer 2 Fill

Shrink in horizontal direction Shrink in vertical direction

!

Given a set of candidate fills

!

Determine dimension of fills

!

Under DRC constraints

!

Minimize overlay area and density variation

!

Solve in an iterative manner

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SLIDE 20

B A C

x1

x1

x2

x2

x3

x3

x4

x4

x5

x5

x6

x6

min |(x2 − x1) · hA + (x4 − x3) · hB–td1 · Awin| + |(x6 − x5) · hC − td2 · Awin| + (x2 − x5) · hAC + (x6 − x3) · hBC s.t. x2 − x1 ≥ Wmin x4 − x3 ≥ Wmin x6 − x5 ≥ Wmin x3 − x2 ≥ Smin (x2 − x1) · hA ≥ Amin (x4 − x3) · hB ≥ Amin (x6 − x5) · hC ≥ Amin x2 − x5 ≥ 0 x6 − x3 ≥ 0 li ≤ xi ≤ ui, i = 1, 2, ..., 6

Step 3: Dummy Fill Insertion (2)

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Shrink in horizontal direction Density variation Overlay area DRC rules Overlay constraints

Awin: area of a window Wmin: minimum width Smin: minimum spacing Amin: minimum area

hA hC hB

hAC hBC

An example of the mathematical formulation in one iteration

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Step 3: Dummy Fill Insertion (3)

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min |(x2 − x1) · hA + (x4 − x3) · hB–td1 · Awin| + |(x6 − x5) · hC − td2 · Awin| + (x2 − x5) · hAC + (x6 − x3) · hBC

! Further relax to remove absolute operation ! Add tighter bound constraints to variables

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Step 3: Dual to Min-Cost Flow

! Convert bound constraints

to differential constraints

! Dual to min-cost flow

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min

xi N

X

i=1

cixi s.t. xi − xj ≥ bij, (i, j) ∈ E, li ≤ xi ≤ ui, i = 1, 2, ..., N, xi ∈ Z

c0

i =

⇢ ci i = 1, 2, ..., N − PN

i=1 ci

i = 0 b0

ij =

8 < : bij (i, j) ∈ E li i = 1, 2, ..., N, j = 0 −ui i = 0, j = 1, 2, ..., N

i j −b0

ij, ∞

c0

i

c0

j

Prime Dual

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SLIDE 23

Overall Flow

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Initial Fill Regions Density Planning Candidate Fill Generation Output Fills Density Planning Dummy Fill Insertion

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SLIDE 24

Experimental Environment

! Implemented in C++ ! 8-Core 3.4GHz Linux server ! 32GB RAM ! ICCAD 2014 contest benchmarks

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Experimental Results

! Compared with contest winners

" Quality Scores (13% better than the 1st place winner) " Overall Scores (10% better than the 1st place winner)

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0.2 0.4 0.6 0.8 1 s b m Comparison of T estcase Quality Score between Our Results and Contest T

p 3

1st 2nd 3rd

urs

0.2 0.4 0.6 0.8 1 s b m Comparison of T estcase Score between Our Results and Contest T

p 3

1st 2nd 3rd

urs

Quality Sores Overall Scores

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SLIDE 26

Experimental Results

! Detailed results

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Conclusion

! Methodology for fill optimization with holistic and

multiple objectives

! Validated on industry benchmarks

" ICCAD 2014 contest benchmark

! Future work

" Lithography related impacts

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Thank you!

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