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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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Presentation Outline
What is Yield?
Difficulties in Yield Prediction
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.
Difficulties in Yield Prediction
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Presentation Outline
What is Yield?
Difficulties in Yield Prediction
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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Presentation Outline
What is Yield?
Difficulties in Yield Prediction
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
Max Thickness Cd Cd ≤θ NextActionCritical Non-Critical
S1 93 85, 78, 81 8 Yes Quadrisect
S2 97 83, 79, 86 11 No Retain
S3 95 76, 73, 80 15 No Retain
S4 94 88, 84, 89 5 Yes Quadrisect
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
Max Thickness Cd Cd ≤θ NextActionCritical Non-Critical
S11 85 72, 74, 79 6 Yes Quadrisect
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
Max Thickness Cd Cd ≤θ NextActionCritical Non-Critical
S41 94 82, 78, 87 7 Yes Quadrisect
S42 88 75, 73, 67 13 No Retain
S43 84 71, 66, 69 11 No Retain
S44 89 86, 81, 78 3 Yes Quadrisect
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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Presentation Outline
What is Yield?
Difficulties in Yield Prediction
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%
Case 1 - α = 2 Case 1 - α = 3 Case 1 - α = 4
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%
Case 1 - α = 2 Case 1 - α = 3 Case 1 - α = 4
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
Case 1 - α = 2 Case 1 - α = 3 Case 1 - α = 4
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%
Case 1 - α = 2 Case 1 - α = 3 Case 1 - α = 4
Correlation Cases
Yiel
d
After OPCA - 300 Variables After OPCA - 200 Variables
OPCA Simulations - CPU Run Times
481 482
476
470 469
463
450455460465470475480485
Case 1 - α = 2 Case 1 - α = 3 Case 1 - α = 4
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%
Case 1 - α = 2 Case 1 - α = 3 Case 1 - α = 4
Correlation Cases
Yie
ld
θ = 0.09 µm θ = 0.075 µm
HAQ Simulations - CPU Run Times
372
239
361312
221
316
050
100150200250300350400
Case 1 - α = 2 Case 1 - α = 3 Case 1 - α = 4
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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Presentation Outline
What is Yield?
Difficulties in Yield Prediction
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