Statistical Modeling and Simulation for VLSI Circuits and

Statistical Modeling and Simulation for VLSI Circuits and

Statistical Modeling and Simulation for VLSI Circuits and
Student: Fang Gong ([email protected]) Advisor: Lei He
EDA Lab (, Electrical Engineering Department, UCLA


Process Variation
Variation Sources:
Optical Proximity Effects

As semiconductor industry enters into the
65nm and below, large process variations
and device noise become inevitable and
hence pose a serious threat to both design
and manufacturing of high-precision VLSI
systems and circuits. Therefore, stochastic
modeling and simulation has become the
frontier research topic in recent years in
combating such variation effects. To this
end, we propose accurate stochastic models
of those uncertainties and further develop
highly efficient algorithms using statistical
simulation techniques, for example, to
extract the variable capacitance in parallel,
estimate parametric yield, approximate the
arbitrary distribution of circuit behavior, and
perform efficient transient noise analysis.

IBM 90nm: Vt variation

Chemical Etching
Chemical-Mechanical Planarization Polishing

What we design is NOT what we got


Noisy response

: mean valuemean mean valuevalue
: mean valuestandard mean valuedeviation

Small Size
Large Variation


Results with Process Variation
Circuit Behavior Variation [ISPD11]
Static Timing Analysis: delay variation

Yield enhancement should consider process

Fast Yield Estimation considering Process Variations
Framework of Existing Methods

Capacitance Extraction Procedure [DAC09]
Discretize metal surface into small panels.
Form linear system P q v by collocation.
Results in dense potential coefficients.
Solve by iterative GMRES



Matrix-Vector-Product (MVP) with linear complexity.
Handle different variation sources incrementally with novel
precondition method.
Build Spectral preconditioner
Solve with GMRES

Geometric Moments

Potential Coefficient
Pij M (d 0 d1 d , w0 w1 w )

Existing Method

Table 1: Accuracy and Runtime (s) Comparison
between Monte Carlo and PiCAP.

Contribution in proposed PiCAP:
Develop one Parallel Fast Multi-pole Method (FMM) to evaluate

M (d 0 d1 d , w0 w1 w )

Significant impact on nanometer highprecision analogue/RF circuits.
CMOS PLL phase noise and jitter.
Noise-sensitive circuits: ADCs, PLLS, etc.
Thermal noise, flicker noise, shot noise, etc.
Traditional transient verification is difficult
nonlinear transient noise analysis
cannot be achieved
unknown how to analyze flicker noise

Serious yield loss issues [DAC10]
process variation will dominate yield loss


90nm NAND gate

Random Device Noise [DAC11]

Signal Integrity Analysis: parasitic RLC variation
Analog Mismatch Effect

Process W W W
Variations L L L

Parallel Variational Capacitance Extraction

Geometry Info ( d 0 , w0 )
Process Variation ( d , w )

Noise-free (nominal) response

Evaluate the MVP (Pxq)
with FMM in parallel

Table 2: Total Runtime (seconds) Comparison


Yield boundary is the projection of intersection
Many times of circuit simulations are required to
locate one point local search.
Local Search is inefficient, especially for
nonlinear circuits.

Contribution in QuickYield [DAC10]

Consider MOSFETs channel
width variations.
Period should be bounded
by [Tmin, Tmax].

Augmenting DAE system with performance
Locating the yield boundary with global search

Calculate Cij with the charge

With stochastic modeling, random process variation can be integrated into our parallel Fast Multi-pole
Method, and different variations can be considered by updating the nominal system incrementally.

Up to hundreds faster than Monte Carlo method
and up to 4.7X than state-of-the-art method.


3-stage ring oscillator.



Time (s)


MC (5000)




YENSS* (10 points)




QuickYield (10 points)




Stochastic Analog Circuit Behavior Modeling under Process Variations
Statistical Modeling of Performance Distribution [ISPD11]

Extract Behavioral Distribution pdf(f) Using RSM [ISPD11]
Synthesize analytical function
of performance using RSM

It is desired to extract the arbitrary distribution of performance merit
Such as oscillator period, voltage discharge, etc.
Monte Carlo method is usually used very time-consuming!

Contribution on High Order Moments Calculation
approximate high order moments with a weighted sum of sampling
values of f(x) without analytical function efficiently and accurately.

f p0 1 1 N N

Calculate time moments

Try to estimate unknown behavioral distribution in performance domain under known
stochastic variations in parameter domain Find link between them!

Parameter Domain

can be extended to multiple parameter cases with linear complexity.
Match with the time moment
of a LTI system

Monte Carlo

Experiment Results

Performance Domain
h(t) can be used to estimate pdf(f)

Existing method:
Assume performance merits follow Gaussian distribution. not realistic!
Response Surface Model (RSM): approximate circuit performance as an
analytical polynomial function of all process variations
f p0 1 1 N N

Limitations of RSM based Method:
synthesis of analytical function becomes highly difficult for large scale
calculation of high order moments is too complicated or prohibitive

consider 6-T SRAM Cell and discharge
behavior during reading.
all threshold voltages of MOSFETs are
independent variable.
Proposed method (PEM) can provide
high accuracy as Monte Carlo and
existing method called APEX.
On average, PEM can achieve up to
181X speedup over MC and up to 15X
speedup over APEX with similar

Fast Non-Monte-Carlo Transient Noise Analysis
Noise Models [DAC11]

Synthesize Flicker Noise in Time Domain

Thermal Noise: noise-free element and a Gaussian white noise current
source in parallel.

NMC Transient Noise Analysis [DAC11]


2 Rm Cm

Model with Summation of Lorentzian spectra:

Expand all random variables with SoPs;
Take inner-product with SoPs due to orthogonal property;
Obtain the SoP expansion of noise at each time-step.

g (t ) g m

Flicker Noise: modeled by a noise current in parallel with the MOSFET.
Power spectrum density of flicker noise in MOSFET

W: channel width
L: channel length
Cox: gate oxide capacitance
per unit area
KF: flicker noise coefficient,
process-dependent constant

References & Collaborators


CoxWL 2kT

Cn(0) 1k (tn )

4 (0) k
Cn 1 1 (tn 1 ) Cn(0) 2 1k (tn 2 )
Gn(0) 1k (tn )
1 m
Tk 1k (tn ) g r ( xn 1 ) g r ( xn 2 ) 0
3 k
3 r 1
r 1

Numeric Experiment

achieve 488X
speedup over MC
with 0.5% error;

Stochastic Orthogonal Polynomials (SoPs)
Any stochastic random variable can be represented by stochastic orthogonal polynomials.

can be 6.8X
faster than existing

Gaussian distribution can be described with Hermite Polynomials:

Orthogonal Property:

Fang Gong, Hao Yu, Lei He, PiCAP: A Parallel and Incremental Capacitance Extraction Considering Stochastic Process
Variation, ACM/IEEE 46th Annual Design Automation Conference (DAC09), 2009
Fang Gong, Hao Yu, Yiyu Shi, Daesoo Kim, Junyan Ren, Lei He, QuickYield: An Efficient Global-Search Based Parametric
Yield Estimation With Performance Constraints, ACM/IEEE 47th Annual Design Automation Conference (DAC10), 2010

Flicker Noise Modeling


provide high
accuracy in the
entire range.

Fang Gong, Hao Yu, Lei He, Stochastic Analog Circuit Behavior Modeling by Point Estimation Method, International Symposium on Physical Design (ISPD'11), 2011.
Fang Gong, Hao Yu, Lei He, "Fast Non-Monte-Carlo Transient Noise Analysis for High-Precision Analog/RF Circuits by Stochastic Orthogonal Polynomials",
ACM/IEEE 48th Annual Design Automation Conference (DAC11), 2011
Collaborators: Dr. Hao Yu, Dr. Yiyu Shi, Dr. Junyan Ren, Mr. Daesoo Kim.

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