QuickYield: An Efficient Global-Search Based Parametric Yield Estimation with Performance Constraints

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QuickYield An Efficient Global-Search Based
Parametric Yield Estimation with Performance
Constraints

• Fang Gong1, Hao Yu2, Yiyu Shi1, Daesoo Kim1,
  Junyan Ren3, and Lei He1

1University of California, Los Angeles, Los
Angeles, US 2Nanyang Technological University,
Singapore 3State Key Lab of ASIC, Fudan
University, Shanghai, China
Presented by Fang Gong
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Outline

• Introduction
• Algorithms
• Experimental Results
• Conclusions

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Need of Fast Yield Estimation

• Process Variation is an major source of yield
  loss
• lithography, CMP and etc.
• Threshold voltage, timing delay, and etc.
• Larger Variation with scaling leads to lower
  Yield Rate.

Data Source Dr. Ralf Sommer, DATE 2006, COM
BTS DAT DF AMF
It is important to predict the Yield accurately.
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Monte Carlo Simulation

• Generate huge number of parameter samplings
  according to probability distribution
• Perform simulation to find performance merit of
  interest
• Compare with performance constraint to identify
  success points in the parameter space.

Monte Carlo method is highly time-consuming! Fast
Yield Estimation becomes necessary!
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Yield Boundary in Parameter Space

• Parameter space is the space bounded by the min
  and max of all process parameters around nominal
  values (blue square)
• Yield Boundary separates success region from fail
  region
• success region ? where parameters lead to
  acceptable performance.

• ƒm is the performance merit of interest
• ?p is the process parameters subject to random
  variations.
• ƒworst is the worst-case performance that can be
  accepted.

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Definition of Yield by Boundary

• Ssuccess the region where parameters lead to
  successful performance.
• Sentire the entire space that variable
  parameters can be reached around their nominal
  values (blue square).
• In the parameter domain, all parameters follow
  uniform distributions. Other distributions can be
  transformed into uniform ones.

Yield Ssuccess / Sentire
Luc Devroye. Non-Uniform Random Variate
Generation. New York Springer-Verlag, 1986.
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Framework of Existing Methods (1)

• Any circuit can be described by differential
  algebraic equation (DAE) system ? performance
  surface
• Performance constraints ? Constraint Plane
• Yield Boundary is the projection of intersection
  boundary of these two surfaces in the parameter
  domain.

Local searches in existing work
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Framework of Existing Methods (2)

1. Perform SPICE simulation with initial parameters

1. Compare performance merit with performance
   constraints

1. If not satisfied, select new parameters and
   repeat.

p2
p1
p0
Multiple simulations for one point at
boundary/surface.
How to locate the yield boundary in the parameter
space with global search?
Reference P. Cox, P. Yang, and P. Chatterjee,
IEDM83 S. Srivastava and J. Roychowdhury,
CICC07 C. Gu and J. Roychowdhury, ASPDAC2008
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Outline

• Introduction
• Algorithms
• Global-Search based Surface Building
• Global Search for Surface Point
• Experimental Results
• Conclusions

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Surface Building Init Step (1)

• Calculate Intersection points (P01 and P02) at
  each parameter axis
• By connecting two points (P01 and P02) , the
  boundary can be estimated with piecewise linear
  approximation.

Linear Approximation to the Boundary/Surface
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Surface Building Refinement (2)

• Start from the middle point (P03) of line
  (P01P02) to find additional surface points P04
  and P05

1. XP03 XP04, YP03YP05
2. Treat YP04 and XP05 as the unknown parameters ?p
3. Solve the augmented system for unknowns.

One-time simulation for one boundary point!
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Surface Building Refinement (2)

• Refine the yield estimation by refining the
  linear approximation.

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Key Idea of QuickYield
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Comparison to Existing Work

• QuickYield performs one simulation for each
  surface point.
• Yield Estimation via Nonlinear Surface Sampling
  (YENSS)
• locally searches along the tangent direction of
  performance surface to locate one surface point.
• Several simulations are performed to locate one
  single surface point.
• Search requires expensive sensitivity analysis.

C. Gu and J. Roychowdhury, An efficient, fully
nonlinear, variability-aware non-monte-carlo
yield estimation procedure with applications to
sram cells and ring oscillators, in ASP-DAC 08,
2008.
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Outline

• Introduction
• Algorithms
• Global-Search based Surface Building
• Global Search for Surface Point
• Experimental Results
• Conclusions

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Representation of Global Search

• Parameter finding is initially used for device
  optimization, and we will discuss its application
  in yield estimation.
• Original circuit in differential algebra equation
  (DAE) representation
• Integrating the performance constraint
  can be integrated into DAE system as

The nonlinear system can be solved with
Newton-Raphson Iterations.
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Jacobian Matrix in Newton Method

• For DC analysis
• For Periodical Steady State (PSS) Analysis

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Outline

• Introduction
• Algorithms
• Experimental Results
• Conclusions

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Schmitt Trigger

• Consider channel widths of Mn1 and Mp2 as
  variational parameters.
• 30 variations from nominal values
• Other process parameters can be handled in the
  same way.
• Performance Constraint
• Lower switching threshold VTL as performance
  merit
• When the input VTL is 0.4V and the output is
  pre-charged to Vdd, the output VOUH should be
  greater than 1.7V

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Results

• QuickYield can achieve only 0.4 error compared
  with Monte Carlo method, and gain 349X speedup.

QuickYield
Method Yield () Time (s) Speedup
MC (6000) 70.185 197.1 1X
QuickYield 70.159 0.564 349X
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3-stage Ring Oscillator

• Performance Constraint oscillator period
• Determined by the delay of inverters
• Nominal Tnorm is 7.2028ns
• Performance Constraint period variation ?T
  should be within 2.5 of nominal Tnorm.
• MOSFETs in the first stage have channel width
  variations with 3s40 perturbation range
  (Gaussian distribution)

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Accuracy

• Two performance constraints ? two yield
  boundaries
• Tgt Tnorm (1- 0.025)
• Tlt Tnorm (1 0.025)

Tmin (QuickYield)
Tmax (QuickYield)
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Runtime

• YENSS results are normalized with respect to
  Monte Carlo method from published paper.
• QuickYield can obtain 519X speedup over Monte
  Carlo at a similar accuracy.

Method Yield Time (s) Speedup
MC (5000) 0.62658 44073.8 1X
YENSS (10 points) 0.6482 317 139X
QuickYield (10 points) 0.6463 84.9 519X
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Scalability

• The scalability of QuickYield with the number of
  Surface Points
• Runtime increases linearly while the yield
  converges quickly.

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High-dimensional Case

• The load capacitance C1 has been introduced
  random variation to increase the complexity.
• QuickYield can achieve as low as 0.6 error.

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Runtime

• QuickYield can be up to 267X faster than MC and
  4.6X faster than YENSS.

Method Yield Time (s) Speedup
MC (5000) 0.617 63128 1X
YENSS (20 points) 0.623 1107.5 57X
QuickYield (20 points) 0.621 236.9 267X
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Outline

• Introduction
• Algorithms
• Experimental Results
• Conclusions

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Conclusions and Future Work

• A fast algorithm, QuickYield, was proposed
• Augmenting DAE system with performance
  constraint.
• Locating the yield boundary with global search by
  solving the augmented system.
• Up to 519X faster than MC and up to 4.7X than
  YENSS, while keeping the same accuracy.
• Future work
• handle more variables and multiple performance
  constraints simultaneously.

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Thanks!