Analysis of Variation
Sources in Ring Oscillator
Layouts
Thomas Chandler and Shion Hung
Non-Members, IEEE
14 May 2003
6.780 Term Project
1
Agenda
Acknowledgment of Original work
Methods
Areas for Exploration
„
„
„
Polysilicon Density Effects
Single chips vs. ‘superchip’
Two way effects
Results
Conclusions
14 May 2003
6.780 Term Project
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Acknowledgement of Original Work
Based on Karen M. Gonzalez-Valentin’s
Master thesis: “Extraction of Variation
Sources due to Layout Practices”
Multiple ring oscillator structures on 35
chips
Conclusions:
„
„
„
Poly Density changes frequency by 2.5%
Finger spacing has significant effect
Vertical ring oscillators also significant
14 May 2003
6.780 Term Project
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Methods
Compiled data for aggregate ‘superchip’
Used JMP-IN
„
„
Fit model for polysilicon density
Two way interactions – least squares
regression
Compared whether individual results
within ‘superchip’ confidence interval
14 May 2003
6.780 Term Project
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Areas for Exploration –
Poly Density Effects
Determine a model for the effect of
polysilicon density on frequency
Poly
Poly
Poly
Poly
Ring Oscillator
Poly
Poly
Poly
Poly
Based on measurements for poly density =
0%, 12.5%, 25%, 50%
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Fit Model for Poly Density
frequency = 4.477 − 0.213 * density
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Actual by Predicted Poly Density
4900000
Column 2 Actual
4800000
4700000
4600000
4500000
4400000
4300000
4200000
4200000
4400000 4600000 4800000
Column 2 Predicted P<.0001 RSq=0.21
RMSE=79410
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6.780 Term Project
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Single chips vs. ‘superchip’
Any differences between individual chip
behavior and the aggregate ‘superchip’?
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6.780 Term Project
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Results – Poly Density
Slope Coefficient Confidence Intervals
0
-50,000
-100,000
slope estimate
-150,000
-200,000
upper
-250,000
lower
-300,000
mean
-350,000
-400,000
-450,000
su
pe
r
ch
ip
ch
ip
ch 5
ip
2
ch 6
ip
20
ch
ip
ch 8
ip
28
ch
ip
ch 4
ip
1
ch 8
ip
2
ch 7
ip
2
ch 4
ip
35
ch
ip
ch 7
ip
1
ch 1
ip
2
ch 9
ip
3
ch 0
ip
2
ch 3
ip
16
ch
ip
9
-500,000
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6.780 Term Project
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Intercept Confidence Intervals
Intercept Confidence Intervals
4,600,000
mean estimate
4,550,000
4,500,000
upper
lower
4,450,000
`
mean
4,400,000
su
pe
r
st
a
ch r
ip
ch 5
ip
2
ch 6
ip
20
ch
ip
ch 8
ip
28
ch
ip
ch 4
ip
1
ch 8
ip
2
ch 7
ip
2
ch 4
ip
35
ch
ip
ch 7
ip
1
ch 1
ip
2
ch 9
ip
3
ch 0
ip
2
ch 3
ip
16
ch
ip
9
4,350,000
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6.780 Term Project
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Areas for Exploration Two way effects with Vertical RO’s
Gonzalez-Valentin’s demonstrated that
Vertical RO’s may be different (lower µ,
larger σ)
„
May be due to mask making bias or ion
implantation effect
Next question:
„
„
Are these vertical effects truly significant?
Are there any effect interactions in the data
set?
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Vertical RO’s versus 3x spacing
Poly
Poly
Poly
Does line spacing have a significant effect
on frequency?
Significant interaction with Vertical RO’s?
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Vertical RO’s vs Single Finger
Canonical RO has multiple fingers
„
Does a single finger with same effective gate
length have a different effect?
Significant interaction with Vertical RO’s?
14 May 2003
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Two way interactions
Vertical vs. 3x spacing
Both main effects are significant (<0.001)
Cross term not significant
Vertical vs. single finger
Both main effects are significant (<0.001)
Cross term is significant (<0.001), but
magnitude is minimal
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Summary of Fit
0.442632
RSquare
0.442533
RSquare Adj
121798.8
Root Mean Square Error
4411380
Mean of Response
17010
Observations (or Sum Wgts)
Avg. RSquare for
individual chips:
0.887206
Parameter Estimates
Term
Intercept
vertical
single finger
vertical*single finger
14 May 2003
Estimate Std Error
4363792.7
-22979.12
-110366.5
7898.9525
t Ratio Prob>|t|
1153.876 3781.9
1153.876 -19.91
1153.876 -95.65
1153.876
6.85
6.780 Term Project
0.0000
<.0001
0.0000
<.0001
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Fit of ‘superchip’ poor
‘Superchip’ R2 values were much worse
than the individual chips
Fit affected by some significant outliers:
Y Actual
13142857
9857142.9
6571428.6
4000000 7000000 10000000 13000000
Y Predicted P0.0000 RSq=0.44
RMSE=121799
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Conclusions
All main effects inspected in paper were
significant
Model fit for polysilicon density effect
„
„
50% ∆ in density = 3% ∆ in frequency
Density has a consistent effect over multiple
individual chips
Significant but small interaction between
vertical RO’s and single finger
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6.780 Term Project
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Questions?
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6.780 Term Project
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