Yue%20Hu

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Fast Estimation of Rare Circuit Events and Its
Application to SRAM Design

• Yue Hu
• Zhong Zheng

1 A. Singhee, et al., Statistical Blockade
Very Fast Statistical Simulation and Modeling of
Rare Circuit Events and Its Application to Memory
Design, IEEE Transactions on Computer-Aided
Design of Integrated Circuits and Systems, Vol.
28, No. 8, Aug 2009.
2 S. Srivastava, et al., Rapid Estimation of
the Probability of SRAM Failure due to MOS
Threshold Variations, IEEE CICC, 2007.
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Table of Contents

• Background
• GPD with Statistical Blockade Method
• Boundary Curve Method

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Table of Contents

• Background
• GPD with Statistical Blockade Method
• Boundary Curve Method

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Circuit Reliability of HRCs

• In nanoscale regime
• Worst case corners no longer suffice
• For high-replication circuits (HRCs) like SRAM,
  high yield rate of a system -gt exceedingly rare
  failures of cells

Fig. 1. A top-level SRAM structure
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Circuit Reliability of HRCs

• 1-Mb SRAM Example
• If chip yield rate 99
• Cell yield rate 99.999999
• i.e., 5.6 s on std normal distribution
• Thus, for MC approach
• 100 million simulations -gt just ONE sample point
• lack of statistical confidence

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Circuit Reliability of HRCs

• Fast estimation method needed for robust memories
  design
• For SRAM, variation mostly comes from
• Doping process
• Poly-Si crystal orientation

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Failure Rate and Cell Failure Rate
Y is some performance metric
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Failure Rate and Cell Failure Rate

• Redundant columns for fault tolerance improvement
• Poisson model

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Table of Contents

• Background
• GPD with Statistical Blockade Method
• Boundary Curve Method

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GPD with Statistical Blockade Method

• Problem of modeling rare event statistics
• MC method only fits body accurately, but not
  tail

Fig. 2. Possible skewed distribution for some
SRAM metric
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GPD model

• Generalized Pareto Distribution (GPD)
• Good approach of fitting tails

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GPD model
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Fitting the GPD to Data

• Probability-weighted moment (PWM) matching

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Statistical Blockade

• For purpose of simulation speed, we need a
  classifier, support vector machine (SVM), to
  block body points
• A classifier needs training, while the training
  set typically have more body points than expected
• To minimize misclassification, we penalize the
  misclassification of tail points by ?t, more than
  that of body points, ?b

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Statistical Blockade Algorithm

• Generate MC sample points
• Run standard simulation
• Define threshold pt and safety margin pc
• Build a classifier to block body points

• Generate tail sample points only
• Run simulation for tail points
• Fit data to GPD function
• Calculate probability of rare event

Fig. 3. Statistical Blockade
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Examples

• A 6T SRAM cell (Vt, t_ox) ? Id ? write/read time

Fig. 4. A typical SRAM cell
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Examples

• A 6T SRAM cell (contd)

Table 1.
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Examples

• A 6T SRAM cell (contd)

Fig. 5. Comparison of GPD tail model from
blockade and the empirical tail CDF
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Examples

• A 64-b SRAM column
• Large dimension, 400
• Reduce dimensionality by Spearmans rank
  correlation coefficient ?s

Fig. 6. A 64-bit SRAM column
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Examples

• Given ?sgt 0.1 as threshold
• Reducing dimensionality to only 11

Fig. 7. Sorted Parameter Index
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Examples

• A 64-b SRAM column (contd)

Table 2.
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Examples

• A 64-b SRAM column (contd)

Fig. 8. Comparison of GPD tail model from
blockade and the empirical tail CDF
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Table of Contents

• Background
• GPD with Statistical Blockade Method
• Boundary Curve Method

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Boundary Curve Method

• Does not rely on MC techniques
• Finds the boundary in Vt between success and
  failure regions
• Via an Euler-Newton curve tracing technique

Fig. 9. Access-time failure and successful regions
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Example Read 1
Fig. 10. Read-1 Operation
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Example Read 1

• Fig. 11.
• Convergence of a point via Newton-Raphson on the
  solution curve
• Curve tracing using the Euler-Newton method

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Example Read 1

• Fig. 12.
• Vt curve obtained by Euler-Newton method. This
  curve partitions the region of Vts of M1 and M2
  into the failure and successful regions.
• Bit-differential (dBL) surface generated using MC
  simulations, which produces the same Vt curve

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Summary

• Method 1
• Generating samples in the tails of distributions
• Deriving sound statistical models of these tails
• Significantly higher accuracy than std MC
• Speedups of one to two orders
• Method 2
• Finding boundary curves of Vt
• Calculating areas enclosed by curves
• Significantly higher accuracy than std MC
• Speedups of 11.2x

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• Thank you

Q A?