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

Performance Constraints

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, US2Nanyang Technological University, Singapore

3State Key Lab of ASIC, Fudan University, Shanghai, China

Presented by Fang Gong

OutlineOutline

Introduction

Algorithms

Experimental Results

Conclusions

Need of Fast Yield EstimationNeed 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.It is important to predict the Yield accurately.

* Data Source: Dr. Ralf Sommer, DATE 2006, COM BTS DAT DF AMF;

Monte Carlo SimulationMonte 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 Monte Carlo method is highly time-consuming!highly time-consuming!

Fast Yield Estimation Fast Yield Estimation becomes necessary!becomes necessary!

Yield Boundary in Parameter Space 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;

• γp is the process parameters subject to random variations.

• ƒworst is the worst-case performance that can be accepted.

Success region

Yield Boundary

ƒm(γp)=ƒworst

Success region

Fail region

Definition of Yield by BoundaryDefinition 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 = SYield = Ssuccesssuccess / S / Sentireentire

*Luc Devroye. Non-Uniform Random Variate Generation. New York: Springer-Verlag, 1986.

Framework of Existing Methods (1)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)Framework of Existing Methods (2)

p1

p2

p0

Multiple simulations for one point at boundary/surface.

Reference: P. Cox, P. Yang, and P. Chatterjee, IEDM’83; S. Srivastava and J. Roychowdhury, CICC’07; C. Gu and J. Roychowdhury, ASPDAC2008

1. Perform SPICE simulation with initial parameters;

2. Compare performance merit with performance constraints;

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

How to locate the yield boundary in the parameter space with global search?

OutlineOutline

Introduction

Algorithms Global-Search based Surface Building Global Search for Surface Point

Experimental Results

Conclusions

Surface Building: Init Step (1)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)Surface Building: Refinement (2)

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

1.1. XXP03P03 = X = XP04P04, Y, YP03P03=Y=YP05P05

2.2. Treat YTreat YP04P04 and X and XP05P05 as the as the

unknown parameters unknown parameters γp;;

3.3. Solve the augmented Solve the augmented system for unknowns.system for unknowns.

Y

XP03/XP04

XP05

YP04

YP03/YP05

X

One-time simulation for one boundary point!

Surface Building: Refinement (2)Surface Building: Refinement (2)

Refine the yield estimation by refining the linear approximation.

Key Idea of QuickYieldKey Idea of QuickYield

Existing Method QuickYield

Comparison to Existing WorkComparison 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.

OutlineOutline

Introduction

Algorithms Global-Search based Surface Building Global Search for Surface Point

Experimental Results

Conclusions

Representation of Global SearchRepresentation 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:

( ( ), ) ( ( ), ) 0

( ( ), ) ( ) 0

p p

p m p worst

dq x t f x t b

dt

H x t f f

The nonlinear system can be solved with Newton-Raphson Iterations.

( ( )) ( ( )) 0d

q x t f x t bdt

( ( ), )pH x t

Jacobian Matrix in Newton MethodJacobian Matrix in Newton Method

For DC analysis:

For Periodical Steady State (PSS) Analysis

( ( ), ) ( ( ), ) 0

( ( ), ) ( ) 0

p p

p m p worst

dq x t f x t b

dt

H x t f f

( ) [ ; ]( ; ) ( ; )

p Tp

p p p

f x fJ X where X x

H x x H x

1 1 11 1

11

1

1 1 1( )

1 1 1( )( )

nn

p p p p

n n n nn n n

p p p p

n p

qq f bC G C

h h h

q q f bJ XC C G

h h h

H H H

x x

;C q x

G f x

OutlineOutline

Introduction

Algorithms

Experimental Results

Conclusions

Schmitt TriggerSchmitt 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

ResultsResults

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

Method Yield (%) Time (s) Speedup

MC (6000) 70.185 197.1 1X

QuickYield 70.159 0.564 349X

QuickYield

3-stage Ring Oscillator3-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)

AccuracyAccuracy

Two performance constraints two yield boundaries

T> Tnorm * (1- 0.025)

T< Tnorm * (1+ 0.025)Tmin (QuickYield)

Tmax (QuickYield)

RuntimeRuntime

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

ScalabilityScalability

The scalability of QuickYield with the number of Surface Points:

Runtime increases linearly while the yield converges quickly.

High-dimensional CaseHigh-dimensional Case

The load capacitance C1 has been introduced random variation to increase the complexity.

QuickYield can achieve as low as 0.6% error.

RuntimeRuntime

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

OutlineOutline

Introduction

Algorithms

Experimental Results

Conclusions

Conclusions and Future WorkConclusions 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!