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
Outline
 Introduction
 Algorithms
 Experimental Results
 Conclusions
Need of Fast Yield Estimation
* Data Source: Dr. Ralf Sommer, DATE 2006, COM BTS DAT DF AMF;
 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.
It is important to predict the Yield accurately.
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!
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.
Success region
Yield Boundary
ƒm(γp)=ƒworst
• ƒm is the performance merit of interest;
• γp is the process parameters subject to random variations.
• ƒworst is the worst-case performance that can be accepted.
Definition of Yield by Boundary

Yield = Ssuccess / Sentire
 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*.
Success region
Fail region
*Luc Devroye. Non-Uniform Random Variate Generation. New York: Springer-Verlag, 1986.
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
ƒm(γp)=ƒworst
Framework of Existing Methods (2)
1. Perform SPICE simulation with initial parameters;
2. Compare performance merit with performance constraints;
3. If not satisfied, select new parameters and repeat.
p1
p2
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, IEDM’83; S. Srivastava and J. Roychowdhury, CICC’07; C. Gu and J.
Roychowdhury, ASPDAC2008
Outline
 Introduction
 Algorithms
 Global-Search based Surface Building
 Global Search for Surface Point
 Experimental Results
 Conclusions
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
Surface Building: Refinement (2)
 Start from the middle point (P03) of line (P01P02) to find additional
surface points P04 and P05
Y
YP0
1. XP03 = XP04, YP03=YP05
4
2. Treat YP04 and XP05 as the
unknown parameters γp;
YP03/YP05
3. Solve the augmented
system for unknowns.
XP05
X
XP03/XP04
One-time simulation for one boundary point!
Surface Building: Refinement (2)
 Refine the yield estimation by refining the linear approximation.
Key Idea of QuickYield
Existing Method
QuickYield
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.
Outline
 Introduction
 Algorithms
 Global-Search based Surface Building
 Global Search for Surface Point
 Experimental Results
 Conclusions
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:
d
q ( x(t ))  f ( x(t ))  b  0
dt
 Integrating the performance constraint H ( x(t ),  p ) can be
integrated into DAE system as:
d
 q( x(t ),  p )  f ( x(t ),  p )  b  0
 dt
 H ( x(t ),  p )  f m ( p )  f worst  0

The nonlinear system can be solved with Newton-Raphson Iterations.
Jacobian Matrix in Newton Method
 For DC analysis:
f  p
 f x

T
J (X )  
where
X

[
x
; p ]

H ( x;  p ) x H ( x;  p )  p 
 For Periodical Steady State (PSS) Analysis
d
 q( x(t ),  p )  f ( x(t ),  p )  b  0
 dt
 H ( x(t ),  p )  f m ( p )  f worst  0

1
 h C1  G1



1
J(X )  
 Cn 1

h

 H
 x
1

1
 Cn
h
1
Cn  Gn
h
H
xn
f
b 
1 q1 qn
(

) 1  1 
h  p  p  p  p



f n bn 
1 qn qn 1
(

)

h  p  p
 p  p 

H


 p

C  q x ;
G  f x
Outline
 Introduction
 Algorithms
 Experimental Results
 Conclusions
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
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
3-stage Ring Oscillator
 Performance Constraint: oscillator period
 Determined by the delay of inverters;
 Nominal Tnorm is 7.2028ns;
 Performance Constraint: period variation ΔT should be within
± 2.5% of nominal Tnorm.
 MOSFETs in the first stage have channel width variations
with 3σ=40% perturbation range (Gaussian distribution)
Accuracy
 Two performance constraints  two yield boundaries
 T> Tnorm * (1- 0.025)
 T< Tnorm * (1+ 0.025)
Tmax (QuickYield)
Tmin (QuickYield)
Runtime
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


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.
Scalability
 The scalability of QuickYield with the number of Surface
Points:
 Runtime increases linearly while the yield converges
quickly.
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.
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
Outline
 Introduction
 Algorithms
 Experimental Results
 Conclusions
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.
Thanks!