Statistical Modeling and Simulation for VLSI Circuits and

Systems

Student: Fang Gong ([email protected]) Advisor: Lei He

EDA Lab (http://eda.ee.ucla.edu), Electrical Engineering Department, UCLA

Abstract

Introduction

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

65nm

90nm

Noisy response

45nm

: mean valuemean mean valuevalue

: mean valuestandard mean valuedeviation

Small Size

Large Variation

W

Results with Process Variation

Circuit Behavior Variation [ISPD11]

Static Timing Analysis: delay variation

Yield enhancement should consider process

variations

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

QuickYield

m(p)=worst

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

L

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

Incrementally

update

preconditioner

Yield boundary is the projection of intersection

boundary.

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]

Experiments:

Consider MOSFETs channel

width variations.

Period should be bounded

by [Tmin, Tmax].

Tmax

Augmenting DAE system with performance

constraint

Locating the yield boundary with global search

Calculate Cij with the charge

distribution.

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.

Tmin

3-stage ring oscillator.

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

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!

Device

variation

Parameter Domain

can be extended to multiple parameter cases with linear complexity.

Match with the time moment

of a LTI system

SPICE

Monte Carlo

Analysis

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

problems.

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

accuracy.

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]

1

h

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.

A

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

KF

C

CoxWL 2kT

Cn(0) 1k (tn )

4 (0) k

1

Cn 1 1 (tn 1 ) Cn(0) 2 1k (tn 2 )

2

3

3

Gn(0) 1k (tn )

h

3

m

2

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

method.

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

Ring-oscillator

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.