handling partial correlations in yield prediction...`objective – transform correlated random...

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Handling Partial Correlations in Yield Prediction

Sridhar VaradanDept of ECE

Texas A&M University

Jiang HuDept of ECE

Texas A&M University

Janet WangDept of ECE

University of Arizona

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Presentation Outline

What is Yield?

Difficulties in Yield Prediction

Previous Research

Proposed Research

Simulation Results

Conclusion

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Yield - Probability of any Manufacturing or Parametric spec satisfying its limits.

Manufacturing Yield – for manufacturing specs.

Parametric Yield – performance measures (timing, power etc.)

Process variations affect yield prediction.

Intra-die process variations no longer negligible.

What is Yield?

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Process Variations

Chip manufacturing involves complex chemical and physical processes.

Tighter pitches and bounds make process variations unavoidable.

Types of process variations –1. Systematic process variations – layout dependent2. Random process variations -

a. Inter-die Random variations – depend on circuit designb. Intra-die Random variations – dominant components

(1) Independent random variations(2) Partially correlated random variations

3. Overall intra-die variations at n locations –

where µ(n) – systematic intra-die variationsε(n) – random intra-die variations

)()()( nnnp εμ +=

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Chemical Mechanical Planarization (CMP) – used in patterning Cu interconnects.

CMP model – Yield is probability of thicknesses at all locations lying within the Upper and Lower thickness limits.

For simplicity, a chip is meshed into a no. of tiles.

Each tile is a location monitored for interconnect thickness.

Meshing a chip into small tiles –Dimension – 100 µm x 90 µm.Size of each tile – 10 µm x 10 µmTotal no. of tiles – 90No. of locations monitored - 90

CMP Yield

100 um

90 um

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Process variations in interconnect thicknesses at n locations –

CMP Yield –Probability for thickness at n locations to lie in the shaded region.

Illustrating a CMP Model

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Factors making Yield Prediction important -1. Presence of Process Variations2. Shrinking feature sizes

Dishing – Excessive polishing of Cu.

Erosion – Loss in field oxide betweeninterconnects.

Potential open and short faults in interconnects.

Predict Yield in circuit design stages to get Yield friendly design.

Need for Predicting CMP Yield

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∑

−∑−=

−

n

T ppp)2(

)()()(1

πμμφ

∫ ∫ ∫=U

L

U

L

U

LndpdpdppY .......21).(...... φ

Where ∑ - covariance matrix for the n variables – {p1, p2,…,pn}

Yield is obtained via numerical integration of a joint PDF -

…..(1)

…..(2)

U, L & µ - upper and lower thickness limits, & mean thickness value.

Equations for Yield Prediction

Yield equation (1) can be decomposed as –

Where YU (High Yield) - probability for thickness at all locations to stay below upper thickness limit.

YL (or Low Yield) - probability for thickness at all locations to stay above lower thickness limit.

1−+= LU YYYield ….(3)

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What is Yield?


Previous Research

Proposed Research

Simulation Results

Conclusion

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Issues Affecting Yield Prediction –

1. Large number of locations to monitor (104-106).

2. Independent & partial correlations between locations.

3. Large memory requirements.

4. Complexity of numerical integration due to problem size.


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What is Yield?


Previous Research

Proposed Research

Simulation Results

Conclusion

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Perfect Correlation Circles (PCC) approach – to reduce no. of tiles.

Luo, et al., DAC 2006

Previous Research

1. Find tile with maximum thickness MAX1.

2. Form PCC -CIRCLE1 (centre at MAX1, pre-fixed radius).

3. Find tile with maximum thickness MAX2 outside CIRCLE1.

4. Form PCC CIRCLE2 (centre at MAX2).

5. Form similar PCCs until no tiles are left uncovered by PCCs.

6. Centers of PCCs (MAX1, ….., MAXm) form reduced set of variables.

7. Use Genz algorithm to compute yield.

Algorithm for PCC Approach

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Let the setup look like this after reduction

Reduction from 90 tilesto 14 variables (the centresof PCCs - MAX1, ….., MAX14.)

PCCs are formed in a sequence –MAX1 – CIRCLE1, MAX2 – CIRCLE2,…………………..,MAX13 – CIRCLE13,MAX14 – CIRCLE14.

Compute Low Yield using similar procedure.

Example Showing Reduction

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Advantages -1. Reduction in problem complexity. 2. Reduced run-time.

Disadvantages –1. Yield Accuracy is affected.

a. Large PCC radius Heavy reduction in variables.(over-estimation in yield)

b. Small PCC radius Lesser reduction in Variables.more accurate yield estimate(but larger run-time)

Pros and Cons of the PCC Approach

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What is Yield?


Previous Research

Proposed Research

Simulation Results

Conclusion

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Develop reduction methods to –1. Reduce problem complexity.2. Reduce effect on yield accuracy.

Two new methods for predicting yield –

1. Orthogonal Principal Component Analysis (OPCA)

2. Hierarchical Adaptive Quadrisection (HAQ)

Proposed Research

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Let vector be metal thicknesses at n locations -

This vector an be decomposed as follows –and

where - nominal value- systematic variation- random variation

pT

npppp ),.....,,( 21=

iiip δμ += ii Δ+= μμ

μiΔiδ

Yield Model used in this Work

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Objective – Transform correlated random variables to a reduced & uncorrelated set through an orthogonal base

Procedure –1. Form initial thickness vector, correlation & covariance matrices. 2. Perform Eigenvalue Decomposition.3. Transform into to set of uncorrelated variables through a mapping matrix.4. Discern unwanted eigenvalues to get reduced set of uncorrelated variables.

Initial Setup for OPCA –Let the initial thickness variations at n locations be –

Let and be the corresponding correlation and covariance matrices.Let be the variance.

Orthogonal Principal Component Analysis

Tn},.....,,{ 21 δδδδ = ….(1)

nxnΣnxnΓ2

iσjinxn σσδδ ⋅⋅Γ=∑ )()(nxnij)()( Γ=Γ δ and ….(2)

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Re-express covariance matrix using Eigenvalue Decomposition –

where - eigenvalue (diagonal) matrix- corresponding eigenvector matrix

The diagonal matrix will look like -

such that

Eigenvalue decomposition gives dominant directions in covariancerelationship between a correlated set of variables.

TQQ ⋅Λ⋅=∑ )()( δδ

Using Eigenvalue Decomposition

)(δΛQ

nn ×Λ )(δ

nλλλ ≥≥≥ ........21

….(3)

….(4)

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Let be the new set of uncorrelated variables such that –

Without loss of generality, assume B follows a Gaussian Distribution –

The matrices and are related as follows –

where

Transforming through an Orthogonal Base – Let be the mapping matrix -

Mapping into a New Set of Variables

εδ ⋅= B

0)( =εμ I=Λ )(ε1×nδ

TJJ ⋅Λ⋅=Λ )()( εδ

1×nε

&

1×nε

B

JQB ⋅=

….(5)

….(6)

….(7)

….(8)

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Correspondingly, we have –

This transforms the initial set of correlated random variables to an uncorrelated set through an orthogonal base.

Reducing the no. of uncorrelated variables –1. After reduction, if we have k variables, then matrices and are –

2. The corresponding sizes of matrices and become , thus giving reduction.

Transforming through an Orthogonal Base …. Contd..

εεδ ⋅⋅=⋅= JQB

TT JQJQQQ )()()()( ⋅⋅Λ⋅⋅=⋅Λ⋅=∑ εδδ

)(δΛ kkJ ×

&

B Q kn×

….(9)

….(10)

….(11)

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Conquer and divide based clustering approach.

Clustering done using sub-regions (similar to PCCs).

Clustering in sub-regions is based on thickness variations.

Sizes of clusters are not homogeneous.

Hierarchical Adaptive Quadrisection

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Consider entire chip as one basic sub-region S.

Sub-region S consists of tiles used in evaluating yield.

Threshold thickness value θ decides possibility of clustering.

Threshold θ tells on variations in thickness of tiles in a sub-region.

Computing High Yield using HAQ

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Stage 1:- Sub-region S covers the entire chip. Let θ be 10.

Sub-regionMonitored

Max Thickness Cd Cd ≤θ NextActionCritical Non-Critical

S 97 93, 95, 94 2 Yes Quadrisect

Working Model for Computing High Yield

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Stage 2 – After forming sub-regions S1, S2, S3 and S4.

Sub-regionMonitored


S1 93 85, 78, 81 8 Yes Quadrisect

S2 97 83, 79, 86 11 No Retain

S3 95 76, 73, 80 15 No Retain


Working model for High Yield ……. Stage 2

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Stage 3 – Inside sub-regions {S11, S12, S13 , S14} & {S41, S42, S43 , S44}.

Sub-regionMonitored



S12 78 63, 65, 60 13 No Retain

S13 81 70, 68, 66 11 No Retain

S14 93 79, 77, 75 16 No Retain

Sub-regionMonitored



S42 88 75, 73, 67 13 No Retain

S43 84 71, 66, 69 11 No Retain


Working Model for High Yield …. Stage 3

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• After Stage 3 in the HAQ algorithm, the setup will look like -

• Stage 3, the chip is covered by 19 basic sub-regions.

• Further clustering based on thickness variations in new sub-regions.

Working Model for High Yield …. After Stage 3

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Clustering based on minimum thickness variations in sub-regions.

Computing Low Yield using HAQ

Comparing HAQ and PCC approachesHAQ Approach PCC Approach

Heterogeneous cluster sizes

Clustering based on variations and sensitivity inside sub-regions

No. of Clusters in working model –Stage-1 4Stage-2 10Stage-3 19

Homogeneous cluster sizes

No importance for sensitivity in variations for clustering

No. of Clusters in each stage of the working model

Stage-1 4Stage-2 16Stage-3 64

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What is Yield?


Previous Research

Proposed Research

Simulation Results

Conclusion

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Experiments simulated –1. Monte Carlo (MC) Simulations2. PCC method3. OPCA method4. HAQ method

Yield evaluated for three cases of correlation –where = {2, 3, 4} and - distance between centres of different tiles.

Simulation Inputs –1. Input thickness –

Mean thickness value – 0.3580 µmUpper thickness limit – 0.4580 µmLower thickness limit – 0.2580 µmStandard deviation – 0.02 µm

9958.0)10( 5 +×− − xα

xα

Simulation Results

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Correlation Equation:Initial seed = 5

9958.0103 5 +×− − x

Monte Carlo Simulations

Monte Carlo Run Times

6165 6191 6158

6516 65186472

59006000610062006300640065006600

Case 1 - α = 2 Case 1 - α = 3 Case 1 - α = 4

Correlation CasesC

PU R

un T

ime

(sec

)

MC without OPCA MC with OPCA

Monte Carlo Yield Values

60% 60% 60%

76% 74% 71%

0%10%20%30%40%50%60%70%80%


COrrelation Cases

Yiel

d

MC without OPCA MC with OPCA

PCC Simulations - Yield Values

89%88%

86%

90%

88%87%

84%85%86%87%88%89%90%91%


Correlation Cases

Yiel

d

PCC Size - 150 µm PCC Size - 250 µm

PCC SImulations - Run Times

2242 2238 2214

1636 1619 1649

0

500

1000

1500

2000

2500


Correlation Cases

CPU

Run

Tim

e (s

ec)

PCC Size - 150 µm PCC Size - 250 µm

Correlation Equation PCC Size No. of Variables150 µm250 µm

431/435305/310

150 µm250 µm

432/427305/310

150 µm250 µm

429/425307/308

9958.0102 5 +×− − x

9958.0104 5 +×− − x

9958.0103 5 +×− − x

OPCA Simulations - Yield Values

77%76%

73%

78%77%

74%

70%71%72%73%74%75%76%77%78%79%


Correlation Cases

Yiel

d

After OPCA - 300 Variables After OPCA - 200 Variables

OPCA Simulations - CPU Run Times

481 482

476

470 469

463

450455460465470475480485


Correlation CasesC

PU R

un T

ime

(sec

)

After OPCA - 300 Variables After OPCA - 200 Variables

HAQ Simulations - Yield Values

80%

77%75%

82%

79%

76%

70%72%74%76%78%80%82%84%


Correlation Cases

Yie

ld

θ = 0.09 µm θ = 0.075 µm

HAQ Simulations - CPU Run Times

372

239

361312

221

316

050

100150200250300350400


Correlation Cases

CPU

Run

Tim

e (s

ec)

θ = 0.09 µm θ = 0.075 µm

Correlation Equation θ No. of Variables0.09 µm

0.075 µm175/178153/155

0.09 µm0.075 µm

80/7961/61

0.09 µm0.075 µm

172/170148/143

9958.0102 5 +×− − x

9958.0104 5 +×− − x

9958.0103 5 +×− − x

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Monte Carlo without OPCA –Neglecting correlation under-estimates yield.

OPCA –Less variable reduction better accuracy, yield is closer to Monte Carlo.

PCC –Larger PCC sizes more reduction over-estimated yield valueSmaller PCC sizes improves accuracy in yield longer run time

HAQ –Higher threshold values less reduction (fine-grained grid)

improved accuracy Smaller threshold values over-estimated yield

Observations in Results

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Comparing yield accuracy and algorithm run time -

Correlation Equation Method Yield Error

Speedup

PCCOPCAHAQ

18.9%2.7%4.1%

1x4.6x9.4x

PCCOPCAHAQ

21.1%2.8%5.6%

1x4.7x6.2x

PCCOPCAHAQ

17.1%1.3%5.3%

1x4.7x6x9958.0102 5 +×− − x

9958.0104 5 +×− − x

9958.0103 5 +×− − x

Comparisons in Results

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What is Yield?


Previous Research

Proposed Research

Simulation Results

Conclusion

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Yield prediction is complex -1. Large number of locations monitored2. Partial & independent correlations between locations

New methods used in yield prediction –1. Orthogonal Principal Component Analysis 2. Hierarchical Adaptive Quadrisection

Both reduce complexity & have less impact on Yield Accuracy.

Conclusion

Scope for Future WorkExtend same methods to predict timing yield in sequential circuits.

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Thank You

handling partial correlations in yield prediction...`objective – transform correlated random...

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