SRAF Insertion via Supervised Dictionary Learning Hao Geng 1 , Haoyu - - PowerPoint PPT Presentation

SRAF Insertion via Supervised Dictionary Learning

Hao Geng1, Haoyu Yang1, Yuzhe Ma1, Joydeep Mitra2, Bei Yu1

1The Chinese University of Hong Kong 2Cadence Inc.

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Moore’s Law to Extreme Scaling

1940 1950 1960 1970 1980 1990 2000 2010 2020 10,000,000,000 1 10 100 1,000 10,000 100,000 1,000,000 10,000,000 100,000,000 1,000,000,000 Intel Microprocessors

Invention of the Transistor

10 1 0.1 0.01

45nm

Year Number of Transistors per Integrated Circuit

Moore’s Law

Process Technology (µm

)

4004 8086 286 386 486 Pentium Pentium II Pentium 4 Core 2 Duo Core i7

Doubles every 2.1 yrs 2 / 19

Nanometer Era of Manufacturing: An Inverter Example

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Optical Proximity Correction (OPC)

Design target

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Optical Proximity Correction (OPC)

Design target Mask Wafer

without OPC

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Optical Proximity Correction (OPC)

Design target Mask Wafer

without OPC with OPC

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What is SRAF?

◮ Patterns deliver light to target features without printing themselves ◮ Make isolated features more dense ◮ Improve the robustness of the target patterns ◮ Rule-based [Jun+,SPIE’15], Model-based [Shang+,Mentor’05], Machine learning

model-based [Xu+,ISPD’16]

(a) (b)

Target OPC SRAF PV band

(a) Printing with OPC only (2688 nm2 PV band area); (b) Printing with both OPC and SRAF (2318

nm2 PV band area).

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Outline

Supervised Feature Revision SRAF Insertion Experimental Results

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Outline

Supervised Feature Revision SRAF Insertion Experimental Results

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Concentric Circle Area Sampling

◮ Initial feature extraction method in SRAF generation

Label: 1 Label: 0

(a)

0 1 2 N%1 sub%sampling0point

(b)

(a) SRAF label; (b) CCAS feature extraction method in machine learning model-based SRAF generation.

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Introduction to Dictionary Learning

Overview

Originally, the dictionary learning model is composed of two parts. One is sparse coding and the other is dictionary constructing. The joint objective function with respect to D and

x is below.

min

x,D

1 N

N

• t=1

{1 2 yt − Dxt2

2 + λ xtp},

(1)

◮ yt ∈ R(n): the t-th input data vector ◮ D = {dj}s

j=1 , dj ∈ R(n): the dictionary where every column is called an atom.

◮ xt ∈ R(s): the sparse code ◮ λ: hyper-parameter ◮ p: the norm type of penalty term, e.g. l1 norm

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The Illustration for Dictionary Learning

yt>

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The Illustration for Dictionary Learning

yt> D

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The Illustration for Dictionary Learning

yt> D

xt>

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The Illustration for Dictionary Learning

yt> D

xt>

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Online Learning Framework

Sparse Coding

The subproblem with D fixed is convex. The objective function for sparse coding of i-th training data vector in memory is

xt

∆

= arg min

x

1 2 yt − Dx2

2 + λxp.

(2)

Solver Details ◮ p = 0: l0 norm and NP-hard [Mallat+,TIP’93], [Pati+,ACSSC’93] ◮ p = 1: LASSO problem [Friedman+,JSS’10], [Beck+,SIIMS’09]

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Online Learning Framework

Dictionary Constructing

The subproblem with x fixed is also convex. The objective function for dictionary constructing is

D

∆

= arg min

D

1 N

N

• t=1

1 2 yt − Dxt2

2 + λxtp.

(3) Solver Details

◮ Block coordinate descent method with

warm start

◮ Introducing two auxiliary variables B and C to speed up convergence rate ◮ Sequentially updating atoms in a

dictionary D

• Bt ← t − 1

t

• Bt−1 + 1

t yt x⊤

t ,

(4)

• Ct ← t − 1

t

• Ct−1 + 1

t xt x⊤

t .

(5)

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Further Exploration: Supervised Dictionary Learning

Exploring Latent Label Information min

x,D,A

1 N

N

• t=1

{1 2

• y⊤

t , √αq⊤ t

⊤ − D √αA

• xt
• 2

2

+ λxtp}.

(6)

Exploiting both Latent and Direct Label Information min

x,D,A,W

1 N

N

• t=1

{1 2

• y⊤

t , √αq⊤ t ,

• βht

⊤ −   D √αA √βW   xt

• 2

2

+ λxtp}.

(7)

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The Illustration for Supervised Online Dictionary Learning

xi>for i ≤ t

yi>, √ αq>

t ,

p βht

  D √αA √βW  

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Outline

Supervised Feature Revision SRAF Insertion Experimental Results

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SRAF Insertion

Preliminary Work ◮ SRAF probability learning for each grid: Logistic regression ◮ SRAF grid model construction: Merging c(x, y) =

• (i,j)∈(x,y) p(i, j),

if ∃ p(i, j) ≥ threshold,

−1,

if all p(i, j) < threshold. (8)

◮ p(i, j): the probability of a grid with index

(i,j)

◮ c(x, y): the summed probability value of

merged grid with index (x,y)

(x, y) (i, j)

10nm

SRAF grid model construction.

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SRAF Insertion via ILP

max

a(x,y)

• x,y

c(x, y) · a(x, y)

(9a)

s.t. a(x, y) + a(x − 1, y − 1) ≤ 1, ∀(x, y),

(9b)

a(x, y) + a(x − 1, y + 1) ≤ 1, ∀(x, y),

(9c)

a(x, y) + a(x + 1, y − 1) ≤ 1, ∀(x, y),

(9d)

a(x, y) + a(x + 1, y + 1) ≤ 1, ∀(x, y),

(9e)

a(x, y) + a(x, y + 1) + x(x, y + 2) + a(x, y + 3) ≤ 3, ∀(x, y),

(9f)

a(x, y) + a(x + 1, y) + x(x + 2, y) + a(x + 3, y) ≤ 3, ∀(x, y),

(9g)

a(x, y) ∈ {0, 1}, ∀(x, y).

(9h)

Wmin Wmax 40nm

X X X X

SRAF insertion design rule under the grid model.

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Outline

Supervised Feature Revision SRAF Insertion Experimental Results

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The Overall Flow

CCAS Feature Extraction Layout Pattern Supervised Feature Revision SRAF Probability Learning SRAF Generation via ILP SRAFed Pattern Feature Extraction SRAF Insertion

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Experimental Bed

Benchmark Set ◮ The same benchmark set as applied in [Xu+,ISPD’16] ◮ 8 dense layouts and 10 sparse layouts with contacts sized 70nm ◮ 70nm spacing for dense and ≥ 70nm spacing for sparse layouts

(a) (b)

(a) Dense layout with golden SRAFs; (b) Sparse layout with golden SRAFs.

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Results

Dense-Average Sparse-Average Total-Average 85 90 95 F1 score (%) ISPD’16 SODL

(a)

Dense-Average Sparse-Average Total-Average 2.4 2.6 2.8 PV band area (0.001µm2) ISPD’16 SODL+Greedy SODL+ILP

(b)

Dense-Average Sparse-Average Total-Average 0.6 0.8 1 EPE (nm) ISPD’16 SODL+Greedy SODL+ILP

(c)

Dense-Average Sparse-Average Total-Average 20 40 Runtime (s) ISPD’16 SODL+Greedy SODL+ILP

(d)

Lithographic performance comparisons with a state-of-the-art machine learning based SRAF insertion tool.

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Conclusion

Summary:

◮ First introduced the concept of dictionary learning into the layout feature extraction

stage and further proposed a supervised online dictionary learning algorithm.

◮ ILP for SRAF generation in a global view. ◮ Boost F1 score and enhance lithographic performance with less time overhead.

Future Work:

◮ Speed up SRAF insertion process ◮ Consider more SRAF design rules into ILP ◮ ...

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