Electromigration Reliability Analysis of Power Delivery ... · Electromigration Reliability...

Electromigration Reliability Analysis of Power Delivery

Networks in Integrated Circuits

by

Mohammad Fawaz

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Sciences

Graduate Department of Electrical & Computer EngineeringUniversity of Toronto

c© Copyright 2013 by Mohammad Fawaz

Abstract

Electromigration Reliability Analysis of Power Delivery Networks in Integrated Circuits

Mohammad Fawaz

Master of Applied Sciences

Graduate Department of Electrical & Computer Engineering

University of Toronto

2013

Electromigration in metal lines has re-emerged as a significant concern in modern VLSI

circuits. The higher levels of temperature and the large number of EM checking strate-

gies, have led to a situation where trying to guarantee EM reliability often leads to

conservative designs that may not meet the area or performance specs. Due to their

mostly-unidirectional currents, the problem is most significant in power grids. Thus, this

work is aimed at reducing the pessimism in EM prediction. There are two sources for

the pessimism: the use of the series model for EM checking, and the pessimistic assump-

tions about chip workload. Therefore, we propose an EM checking framework that allows

users to specify conditions-of-use type constraints to capture realistic chip workload, and

which includes the use of a novel mesh model for EM prediction in the grid, instead of

the traditional series model.

ii

Acknowledgements

It would not have been possible to write this thesis without the immense help and support

of the amazing people around me. I owe a very important debt to all of those who were

there when I needed them most.

First and above all, I would like to thank my supervisor Professor Farid N. Najm, who

supported me throughout my thesis with his great patience, regular encouragement, and

continuous advice. Professor Najm is easily the best advisor anyone could ever hope for;

his deep insight, professional leadership, and warm friendliness were key factors without

which the development of this work would not have been possible. Thank you professor

for your overwhelming efforts and for your mentorship on both professional and personal

levels.

I am also thankful for Professors Jason Anderson, Andreas Veneris, and Costas Sarris,

from the ECE department at the University of Toronto, for reviewing this work and

providing their valuable comments.

I would also like to thank Abhishek for his guidance and support throughout the first

half of my degree program. Abhishek was always there to answer my questions and to

discuss new research ideas. Special thanks go to my colleague and my friend Sandeep

Chatterjee who’s work was closely related to mine. A major part of this research was done

in collaboration with him, especially the content of Chapter 3. The many long discussions

we had helped a lot in understanding the problem and in shaping the proposed solutions.

Zahi “zehe” Moudallal, my colleague and one of my best friends, deserves a very

special mention. I thank him for all the great times we had inside and outside the lab.

The long working hours would not have been the same without his presence and his sense

of humor. I wish him best of luck in all his future endeavors.

I am also grateful to Noha “noni” Sinno for her friendship and her constant support

over the past two years. Thank you Noha for the fun times and for all the long discussions

we had about life in general; they helped me face the world with a better attitude. I

must also express my gratitude to Elias “ferzol” El-ferezli who helped me a lot when I

first arrived to Toronto. His friendship, advice, and assistance were key in surviving the

first few months away from home and in making me a better person overall.

Of my friends at the University of Toronto, I would like to thank Dr. Hayssam

Dahrouj for his motivation, Agop Koulakezian for all the help, as well as my office mates

in Pratt building, room 392, for making the lab a great and pleasant environment. I wish

them the best and the brightest futures.

Last but not least, I would like thank my parents Bassam Fawaz and Jamila Fawaz,

to whom I dedicate this work, for always encouraging me and investing their time and

iii

money in my future. Thank you for your constant support and advice, and for always

believing in me and making me who I am today. I would also like to thank my two

younger brothers Hassan and Hussein and wish them the best of luck in achieving their

future goals.

iv

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Background 4

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Electromigration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Flux Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.2 Blech Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.3 Failure Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Reliability Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.2 Reliability Measures . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.3 Time-to-Failure Distributions . . . . . . . . . . . . . . . . . . . . 10

2.4 Traditional Electromigration Checking . . . . . . . . . . . . . . . . . . . 13

2.4.1 Current Density Limits . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.2 Statistical Electromigration Budgeting (SEB) . . . . . . . . . . . 13

2.5 Electromigration in the Power Grid . . . . . . . . . . . . . . . . . . . . . 14

2.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5.2 Power Grid Model . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6 Sampling and Statistical Estimation . . . . . . . . . . . . . . . . . . . . . 20

2.6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6.2 Sampling from the Standard Normal . . . . . . . . . . . . . . . . 20

2.6.3 Sampling from the Lognormal . . . . . . . . . . . . . . . . . . . . 21

2.6.4 Mean Estimation by Random Sampling . . . . . . . . . . . . . . . 21

2.6.5 Probability Estimation by Random Sampling . . . . . . . . . . . 24

2.7 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

v

3 Vector-Based Power Grid Electromigration Checking 26

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 The ‘Mesh’ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Estimation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.1 MTF and Survival Probability Estimation . . . . . . . . . . . . . 28

3.3.2 Resistance Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.3 Generating Time-to-Failure Samples . . . . . . . . . . . . . . . . 29

3.4 Computing Voltage Drops . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.1 Sherman-Morrison-Woodbury Formula . . . . . . . . . . . . . . . 32

3.4.2 The Banachiewicz-Schur Form . . . . . . . . . . . . . . . . . . . . 33

3.4.3 Case of Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Vectorless Power Grid Electromigration Checking 43

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.1 Modal Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2.2 Current Feasible Space . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3.1 Local Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3.2 Exact Global Optimization . . . . . . . . . . . . . . . . . . . . . . 55


4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5 Simulated Annealing Based Electromigration Checking 60

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2 Simulated Annealing for Continuous Problems . . . . . . . . . . . . . . . 60

5.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2.2 Main Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2.3 The Acceptance Function . . . . . . . . . . . . . . . . . . . . . . 62

5.2.4 Cooling Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.2.5 Next Candidate Distribution . . . . . . . . . . . . . . . . . . . . . 63

5.2.6 Stopping Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.3 Simulated Annealing with Local Optimization . . . . . . . . . . . . . . . 67

5.4 Optimization with Changing Currents . . . . . . . . . . . . . . . . . . . 67

vi

5.4.1 Estimating EM Statistics for Step Currents . . . . . . . . . . . . 67

5.4.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.5 Optimization with Selective Updates . . . . . . . . . . . . . . . . . . . . 70


5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6 Conclusion and Future Work 76

Bibliography 78

vii

List of Tables

3.1 Comparison of power grid MTF estimated using the series model and the

mesh model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Survival probability estimation . . . . . . . . . . . . . . . . . . . . . . . 40

4.1 Exact average minimum TTF computation . . . . . . . . . . . . . . . . . 58

5.1 Speed and accuracy comparison between the first Simulated Annealing

based method and the exact solution of chapter 4 . . . . . . . . . . . . . 70

5.2 Comparison of power grid average minimum TTF and CPU time for the

three Simulated Annealing based methods . . . . . . . . . . . . . . . . . 71

viii

List of Figures

2.1 A triple point in a wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Standard normal and lognormal distributions . . . . . . . . . . . . . . . . 12

2.3 High level model of the power grid . . . . . . . . . . . . . . . . . . . . . 16

2.4 A resistive model of a power grid . . . . . . . . . . . . . . . . . . . . . . 17

2.5 A small resistive grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1 Resistance evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Mesh model MTF estimation . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 CPU time of MTF estimation using the mesh model . . . . . . . . . . . . 41

3.4 Estimated statistics for grid DC3 (200K nodes) . . . . . . . . . . . . . . 42

4.1 Choosing the next starting point I(2) . . . . . . . . . . . . . . . . . . . . 55

4.2 CPU time of the exact approach versus the number of grid nodes . . . . 59

5.1 Generating lambda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 One way of reflecting yk+1 back into X to obtain yk+1 = yk+1 . . . . . . . 66

5.3 CPU time of the Simulated Annealing based methods versus the number

of grid nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.4 Average minimum TTF estimated for the three Simulated Annealing based

methods versus the number of grid nodes . . . . . . . . . . . . . . . . . . 74

5.5 Simulated Annealing progress for a particular TTF sample using all the

three proposed methods (33K grid) . . . . . . . . . . . . . . . . . . . . . 75

ix

Chapter 1

Introduction

1.1 Motivation

The on-die power grid in integrated circuits (IC) is the electric network that provides

power from the power supply pins on the package to the on-die transistors. The power

grid must supply a source of power that is fairly free from fluctuations over time. A large

drop in supply voltage may lead to timing violations or logic failure. With technology

scaling, power grid verification, which involves checking that the voltage levels provided

to the underlying logic are within an acceptable range, has become a critical step in any

IC design. Unfortunately, it is not enough to check the performance of the grid at the

fabrication time; a well designed power grid must continue to deliver the required voltage

levels to all circuit nodes for a certain number of years before failing.

Electromigration (EM), a long term failure mechanism that affects metal lines, is a

key problem in VLSI especially in the power grid. The gradual transport of metal atoms

caused by electromigration leads to the creation of a void which significantly increases

the resistance of the line in consideration and can lead to an open circuit. This affects the

power distribution to the underlying logic and may cause harmful voltage fluctuations.

Checking for electromigration in a power grid involves computing its mean time-to-failure,

which gives the designer an idea about the robustness of the grid and whether it needs

to be redesigned or not.

What is most worrying is that existing electromigration checking tools provide pes-

simistic results, and hence the safety margins between the predicted EM stress and the

EM design rules are becoming smaller. Historically, electromigration checking tools relied

on worst-case current density limits for individual grid lines. Later on, Statistical Electro-

migration Budgeting (SEB) was introduced in [1] in which the series model is employed

with other simplifying assumptions leading to a simple expression of the failure rate as

1

Chapter 1. Introduction 2

the sum of failure rates of individual components, and became a standard technique in

many industrial CAD tools. SEB is appealing because it relates the reliability of circuit

components to the reliability of the whole system. In addition, SEB is simple to use and

allows some components to have high failure rates as long as the sum of all the failure

rates is acceptable.

Nonetheless, modern power grids are meshes rather than the traditional “comb” struc-

ture. The mesh structure allows multiple paths between any two nodes, and accordingly,

modern grids have some level of redundancy that must be considered to get a better

prediction of the lifetime of the grid. Moreover, the rate of EM degradation in power

grid lines depends on the current density, and hence on the patterns of current drawn by

the underlying circuitry. It is impractical to assume that the exact current waveforms are

available for all the chip workload scenarios. Also, one might need to verify the grid early

in the design flow where a limited amount of workload information is available. There-

fore, a vectorless approach is needed to deal with the uncertainties about the underlying

logic behavior. A vectorless technique is a technique that does not require the exact

current waveforms nor specific chip input vectors; it can verify the chip using limited

information about the circuit operation.

1.2 Contributions

The goal of this research is to develop an efficient, less pessimistic, and vectorless

electromigration checking tool for mesh power grids. We first propose a vector-based tech-

nique which computes the mean time-to-failure of the power grid using a more accurate

model than SEB and which assumes that the current waveforms are available exactly. A

vector-based technique is a technique that requires the exact currents drawn by the chip

based on a specific chip input vector. This step is necessary to explain our model and

will be a basis for our other contributions. The engine developed can also be used to

compute the survival probability of the grid for a certain number of years as well as to

derive its reliability function.

To overcome user uncertainty about the chip workload, we also propose a vector-

less framework which extends the vector-based engine to the case where partial current

specifications are available in the form of constraints on the currents and on the usage

frequencies of different power modes. In this domain, our first contribution is an exact

but expensive approach which relies on solving a set of linear and mixed integer opti-

mization problems. The exact approach is interesting and only useful when the grid is

small or when only certain parts of the grid need to be verified.

Chapter 1. Introduction 3

To deal with larger grids, we propose three other approximate approaches that are

based on the use of Simulated Annealing [2]. The proposed approaches provide fairly

accurate results as well as significant speed up over the exact solution.

1.3 Organization

This thesis is organized as follows: Chapter 2 develops all the necessary background ma-

terial on electromigration, reliability mathematics, and the power grid model. Chapter 3

describes the new vector-based checking model which takes the redundancy of the grid

into account. Chapter 4 presents our exact approach for vectorless power grid EM check-

ing, and Chapter 5 shows Simulated Annealing-based approaches. We conclude with

future research directions in Chapter 6.

Chapter 2

Background

2.1 Introduction

In this chapter, we present a review of all the background material needed for this work.

Section 2.2 discusses the the physics of electromigration as well as the basic mathematical

models associated with it. In Section 2.3, we cover the mathematical functions describing

the reliability of a physical system. In Section 2.4 we present the existing electromigra-

tion checking techniques including current density limits checks and most importantly

Statistical Electromigration Budgeting. In Section 2.5 we turn our focus on the power

grid and its model. We also discuss the reasons why checking for electromigration in the

power grid is critical for the safety of the chip. Section 2.6 presents a summary of the

basic sampling techniques as well some of the existing mean and probability estimation

methods. The last Section introduces few notations that will be useful throughout the

thesis.

2.2 Electromigration

Electromigration in metal lines is the gradual transport of metal caused by the momentum

exchange between the conducting electrons and the diffusing metal atoms. Over time,

metal diffusion causes a depletion of enough material so as to create an open circuit. A

pile-up of metal (called hillock) can also occur and can cause a short circuit between

neighboring wires, but this phenomena is usually suppressed and ignored in modern IC

due to the layers of other material around the wires. In this work, we will only consider

the effect of voids on the lifetime of the power grid while ignoring the effect of shorts

that could occur between neighboring lines due to hillocks.

4

Chapter 2. Background 5

Triple Point

Figure 2.1: A triple point in a wire

Electromigration has been known for over 100 years but became of interest after

the commercialization of integrated circuits. Since then, many EM failure models have

been generated and analyzed for individual interconnects, as well as several full-chip

estimation and checking techniques. In the following, we review some key points about

Electromigration relevant to our work. Some of our main references include [3] by J. R.

Black, [4] by J. W. McPherson, [5] by I. A. Blech, and [6] by A. Christou.

2.2.1 Flux Divergence

A non-vanishing divergence of atomic flux is required for electromigration to occur. This

basically means that the flow of metal atoms into a region should not be equal to the

flow of atoms out of the region. Otherwise, the inward flow would compensate for the

outward flow and hence no deformation would occur. A depletion occurs when the

number of atoms flowing out is greater than the number of atoms flowing in.

Flux divergence usually occurs close to vias because vias are generally made of a hard

metal that would not move as easily as the metal surrounding the via. A flux divergence

can also occur away from vias especially at triple points. A triple point is a location in

the metal line where three grain-boundaries meet as shown in figure 2.1. The geometrical

structure of a triple point affects its vulnerability and its failure time [7]. Generally, long

metal lines are more likely to fail early because they are more likely to contain triple

points.

2.2.2 Blech Effect

For sufficiently short lines, the back-stress developed due to accumulation of atoms at

the ends of a line can overcome the build-up of the critical stress required for creation

of a void in the line. In other words, a reversed migration process can occur due to the

accumulation of atoms, and this reduces or even compensates the effective material flow

towards the anode [8]. For that reason, short lines generally have very long lifetimes and


in many cases, can be considered immortal; this is called the Blech Effect [5].

The Blech effect is quantified is terms of a critical value of the product of current

density (J) and length of a line (L), denoted βc. For modern IC, βc ranges between

2000A/cm and 10, 000A/cm.

This threshold value is very useful in circuit design. It determines whether a line is

immortal or not as follows: given a line ℓ of length Lℓ, subject to a current density Jℓ,

then ℓ is considered EM -immune (i.e. immortal) if JℓLℓ < βc and EM -susceptible if

JℓLℓ ≥ βc.

2.2.3 Failure Models

Since the degradation rate depends on the microstructure of a line which varies from

chip to chip, electromigration is considered to be a statistical phenomena. This means

that the time-to-failure of a mortal line under the effect of electromigration is a random

variable. It has been established for a while that EM failure times have a good fit to

a lognormal (LN) distribution, i.e. its logarithm has a normal (Gaussian) distribution.

Other, possibly more accurate models have been proposed such as the multilognormal

distribution [9] and the shifted lognormal distribution [7], however, the lognormal remains

the simplest and the most practical distribution to use.

The most commonly used expression for the mean time-to-failure (MTF) of a mortal

line is Black’s equation [6]:

MTF =a

AJ−η exp

(

Ea

kT

)

(2.1)

where A is an experimental constant that depends on the physical properties of the metal

line (volume resistivity, etc.), a is the cross sectional area of the line, J is the effective

current density, η is the current exponent that depends on the material of the wire and

the failure stage (η > 0), k is the Boltzmann’s constant, T is the temperature in Kelvin,

and Ea is the activation energy for EM.

Most of the references report a value between 1 and 2 for η. A value close 1 usually

indicates that the lifetime is dominated by the time taken by the void to grow, while

a value close 2 indicates that void nucleation (the accumulation of vacancies at sites of

flux divergence) is the dominant phase of the lifetime. Other references such as [4] report

different values for different metal systems: typical values are ≈ 2 for aluminum alloys

and ≈ 1 for copper.


2.3 Reliability Mathematics

2.3.1 Overview

In this section, we cover the reliability measures of a physical system based on a system

theoretic approach. This means that only the input-output properties of the system are

of interest, and not how it is built internally [10]. We start by a definition of reliability

and then we introduce the mathematical functions that describe it.

Definition 1. Reliability is the probability of performing a certain function without

failure for specific period of time.

This definition has the following four main elements:

1. Probability: The exact time-to-failure of a system is usually unpredictable be-

cause it depends on several stochastic physical phenomena. Accordingly, reliability

is a probability, i.e. a number between zero and one.

2. Function: The system under consideration must be evaluated based on a specific

functionality.

3. Failure: What constitutes a failure in a physical system must be well defined before

one can estimate its reliability. A system is said to fail when it becomes unable to

perform as intended.

4. Time: The system must perform for a period of time and hence reliability almost

always depends on time.

There are many mathematical metrics that describe the reliability of a system. In the

following we present the ones relevant to our work.

2.3.2 Reliability Measures

Let T be the time-to-failure of a system. We assume that T is the time to first failure

and that the system remains failed for all future time (i.e. the system is non-repairable).

Also, we assume that the system is working properly at time t = 0. This allows defin-

ing T as a continuous random variable (RV) with the following cumulative distribution

function (CDF):

F (t) = Pr{T ≤ t}, t > 0 (2.2)


F (t) is sometimes called the unreliability of the system. It represents the probability

that the system fails in the interval [0, t]. The probability of failure in the interval (t1, t2]

is simply F (t2)− F (t1).The reliability function R(t) is defined as follows:

R(t) = 1− F (t) = Pr{T > t} (2.3)

It represents the probability that the first failure occurs after time t. Being a cumula-

tive distribution function, F (t) is non-negative and non-decreasing with F (0) = 0 and

F (∞) = 1. Accordingly, R(t) is non-negative and non-increasing with R(0) = 1 and

R(∞) = 0.

Probability Density Function

The local behavior of a system at a time t is captured by the probability density function

(PDF) defined as follows:

f(t) =dF (t)

dt=d(1−R(t))

dt= −dR(t)

dt(2.4)

with,

f(t) ≥ 0 and

∫ ∞

0

f(x)dx = 1 (2.5)

As a result,

F (t) =

∫ t

0

f(x)dx = 1−∫ ∞

t

f(x)dx = 1−R(t) (2.6)

Failure Rate

The failure rate λ(t) describes the conditional probability of failure around a time t. It

can be expressed as follows:

λ(t) = lim∆t→0

Pr{t < T < t+∆t|T > t}∆t

= lim∆t→0

1

∆t

Pr{t < T < t+∆t and T > t}Pr{T > t}

= lim∆t→0

1

∆t

F (t+∆t)− F (t)Pr{T > t}

=1

R(t)lim∆t→0

F (t+∆t)− F (t)∆t

=f(t)

R(t)


Basically, for small ∆t, the product λ(t)∆t represents the probability of failure in the

interval [t, t+∆t] under the condition that the system has survived until time t.

It is sometimes useful to express R(t) as a function of λ(t). To do that, we use (2.4)

to write:

λ(t) =f(t)

R(t)= − 1

R(t)

dR(t)

dt= − d

dt(lnR(t)) (2.7)

Integrating both sides between 0 to t:

∫ t

0

λ(x)dx = − (lnR(t)− lnR(0)) (2.8)

Because R(0) = 1, we obtain:

R(t) = exp

(

−∫ t

0

λ(x)dx

)

(2.9)

The Mean time-to-failure (MTF)

The mean time-to-failure is the expected value of the random variable T:

MTF = E[T] =

∫ ∞

0

tf(t)dt (2.10)

Knowing the fact that f(t) = −dR(t)dt

, we can write:

MTF = −∫ ∞

0

tdR(t) (2.11)

Integrating by parts:

MTF = −[

tR(t)

∣

∣

∣

∣

∞

0

−∫ ∞

0

R(t)dt

]

(2.12)

Clearly, tR(t)

∣

∣

∣

∣

0

= 0. Also, for most statistical distributions encountered in the study of

circuit reliability, R(t) falls faster than 1/t, meaning:

limt→∞

tR(t) = 0 (2.13)

Thus,

MTF =

∫ ∞

0

R(t)dt (2.14)

Therefore, the MTF is equal to the area under the reliability curve.


The α-Percentile

For a given α ∈ [0, 1], the α-percentile is the time instant tα for which F (tα) = α. Because

F is continuous and increasing, the inverse function F−1 exists and thus, tα = F−1(α)

is unique. Basically, tα is the time by which a fraction α of the population is expected

to fail. Computing tα is generally done using statistical tables or using existing software

routines (such as erf() function which is used when the distribution in hand is the

standard normal).

2.3.3 Time-to-Failure Distributions

A variety of statistical distributions are found to be useful to describe the reliability

of a system subject to a certain failure mechanism. Because it is very hard to derive

them from the basic physics of the failure, these distributions are generally determined

empirically where the distribution that best fits the observed data is the one used to

describe the phenomena under consideration. Below, we cover two of the mostly widely

used distributions in the study of reliability: the Normal distribution and the Lognormal

distribution.

The Normal distribution

The Normal (Gaussian) distribution has been found to describe many natural phenomena

and it very useful in many statistical techniques such as random sampling. The PDF of

the normal distribution is bell shaped and is given by:

f(t) =1

σ√2π

exp

[

−1

2

(

t− µσ

)2]

, −∞ < t < +∞ (2.15)

where µ is the mean and σ2 is the variance. The bell curve is symmetric around µ and it

can be shown that∫∞−∞ f(t)dt = 1. For a normal distribution, F (t), R(t), and λ(t) can

be expressed as integrals but they don’t have closed forms.

The standard normal distribution, whose PDF is shown in figure 2.2a, is a special

form of the normal distribution, where µ = 0 and σ = 1. Its PDF function is given by:

φ(z) =1√2π

exp

(

−1

2z2)

(2.16)

The CDF of the standard normal is usually denoted Φ(·), and is shown in figure 2.2b.

Given any normally distributed random variable T, with mean µ and variance σ2, the


random variable T−µσ

has a standard normal distribution, therefore the PDF of T is

f(t) = φ(

t−µσ

)

and its CDF is F (t) = Φ(

t−µσ

)

The Lognormal Distribution

A random variable T is said to have a lognormal distribution if the logarithm of T has a

normal distribution. The PDF of T can be shown to be:

f(t) =1

tσ√2π

exp

[

−1

2

(

ln t− µln

σln

)2]

, 0 < t < +∞ (2.17)

where µln is the mean of lnT, and σ2ln is its standard deviation. It can be shown that the

mean and variance of T can be expressed as follows:

µ = E[T] = exp

(

µln +1

2σ2ln

)

(2.18)

σ2 = Var(T) =(

exp(

σ2ln

)

− 1)

exp(

2µln + σ2ln

)

=(

exp(

σ2ln

)

− 1)

µ2 (2.19)

Also, it is easy to see that the CDF of the lognormal is the following:

F (t) = Pr{T ≤ t} = Pr{lnT ≤ ln t} = Φ

(

ln t− µln

σln

)

(2.20)

From this, we can write:

f(t) =d

dtF (t) =

d

dtΦ

(

ln t− µln

σln

)

=1

σlntΦ

(

ln t− µln

σln

)

(2.21)

Therefore,

λ(t) =f(t)

R(t)=

f(t)

1− F (t) =

1σlnt

Φ(

ln t−µln

σln

)

1− Φ(

ln t−µln

σln

) (2.22)

Again, the standard lognormal distribution is a special form of the lognormal for which

µln = 0 and σln = 1. The PDF and the CDF of the standard lognormal are shown in

figures 2.2c and 2.2d.


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

T

f(t)

(a) PDF of the standard normaldistribution (µ = 0 and σ = 1)

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T

F(t

)(b) CDF of the standard normaldistribution (µ = 0 and σ = 1)

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

T

f(t)

(c) PDF of the standard lognormaldistribution (µln = 0 and σln = 1)

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T

F(t

)

(d) CDF of the standard lognormaldistribution (µ = 0 and σ = 1)

Figure 2.2: Standard normal and lognormal distributions


2.4 Traditional Electromigration Checking

2.4.1 Current Density Limits

Historically, electromigration checking tools compared interconnect average current per

unit width Ieff (computed by averaging the current waveform over time and dividing the

result by the width of the line), to a conservative fixed limit to determine whether a line

is reliable or not. For every line, the following ratio is computed:

S =Actual Ieff

Design Limit Ieff(2.23)

Appropriate modifications are made to S when the line under consideration is a contact

or is holding a bipolar current. When S ≤ 1, the line is deemed reliable; otherwise, it

has to be redesigned. The designer has to guarantee that S ≤ 1 for all the lines in the

chip.

2.4.2 Statistical Electromigration Budgeting (SEB)

Because of the statistical nature of electromigration, identical lines subject to identical

current stress may show very different failure times, and hence, the procedure explained

in the previous section is not sufficient to guarantee a reliable interconnect. Moreover,

when chip-level reliability is in question, the current density limits above become math-

ematically arbitrary. This means that the chip is not necessarily reliable if S ≤ 1 for all

the lines. Similarly, the chip is not necessarily unreliable if S > 1 for some lines. To verify

a chip design, one must check that the whole metal structure is reliable, not so much the

individual lines. In [11] the authors proposed the treatment of the whole on-die metal

structure as a series system making use of a Weibull approximation to perform the series

scaling.

Definition 2. A system is said to be a series system if it is deemed to have failed if any

one of its components fails.

The time-to-failure of a series system composed of k components is the RV:

T = min(T1,T2, . . . ,Tk) (2.24)

where T1,T2, . . . ,Tk are the RVs representing the time to failure of the k components.


If the components are independent, then the reliability of the system is:

R(t) = Pr{T > t} =k∏

i=1

Pr{Ti > t} =k∏

i=1

Ri(t) (2.25)

Using (2.9), (2.25) can be written as:

exp

(

−∫ t

0

λ(x)dx

)

=k∏

i=1

exp

(

−∫ t

0

λi(x)dx

)

= exp

(

−k∑

i=1

∫ t

0

λi(x)dx

)

Taking the natural logarithm on both sides, and then differentiating with respect to t,

we get:

λ(t) =k∑

i=1

λi(t) (2.26)

This result leads to what is called the part count method, which was found applicable,

approximately, to electromigration in IC chips [11]. The key advantage of using the

series system model is that some lines, where it is hard to meet the design rules, may be

allowed to have high failure rates as long as the overall failure rate is acceptable. This

observation led, later on, to Statistical Electromigration Budgeting (SEB), introduced

in [1] and applied to the Alpha 21164 microprocessor. SEB also assumes a series system

model of the chip, where the failure rates of the chip components are budgeted over the

various interconnect classes. Again, the benefit is that designers are allowed to exceed the

design limits in some critical paths to push performance without compromising overall

chip reliability. Overall, SEB became the standard technique for EM checking in modern

IC design and verification.

2.5 Electromigration in the Power Grid

2.5.1 Overview

The power distribution network, commonly referred to as the “power grid”, is a multiple-

layer metallic mesh that connects the external power supply pins to the chip circuitry

thus providing the supply voltage connections to the underlying circuit components.

Ideally, every node in the power grid should have a voltage level equal to the supply

voltage level (vdd). However, due to the RLC behavior of grid transmission lines, and


due to circuit activity and coupling effects, the voltage levels at the nodes drop below

vdd. Similarly, the voltage levels at the ground grid nodes (which are supposed to be zero

Volts) may rise above zero.

With today’s deep sub-micron (DSM) technologies running at GHz clock speeds and

exhibiting small feature sizes, the voltage drops in the power grid are approaching serious

levels while affecting the performance, reliability, and correctness of the underlying logic.

Soft errors (glitches (errors) in signal lines which are not catastrophic and normally do

not destroy the device) as well as unwanted circuit delays have been observed in the cases

where the voltage drops are sufficiently high [12]. As a result, the performance of a power

grid is generally evaluated based on how well the supply voltage vdd is being delivered

to grid nodes. Every node in the power grid should be able to provide a certain voltage

level to the underlying components. This condition is generally quantified using a certain

threshold on the voltage drop at any given node. If the voltage drop at a node turns

out to be larger than its corresponding threshold, the node is considered unsafe, and

accordingly, the whole grid is deemed to be obsolete or failing. The process of checking

the validity of every node in the grid is called Power Grid Verification, which is a major

step in the design of any chip.

To make things worse, power grid rails suffer from all kinds of wear-out mechanisms

such as contact and via migration, corrosion, and most importantly electromigration [10].

These problems generally have an effect on the long-term reliability, and often can cause

sharp rises in the resistance of grid interconnects resulting in a poor grid performance.

Consequently, checking that the power grid performs as intended at the fabrication time

is not enough. A well designed grid should continue to deliver the required voltage levels

to all circuit nodes for a certain number of years before failing.

With technology scaling, electromigration seems to be the most serious of all wear-out

mechanisms. It is forecast that the metal line reliability due to EM will get dramatically

worse as we move towards the 14nm node [13]. Even today, design groups are reporting

that foundries are requiring very strict EM rules, creating tight bottlenecks for designers.

Although electromigration affects signal and clock lines, there are good reasons to be

more concerned about EM in the rails of the power grid:

1. First, signal and clock lines usually carry bidirectional currents, and hence they

tend to have longer lifetimes under EM due to healing. Healing occurs when the

damage done due to EM is reversed by an atomic flow in the direction opposite

to the electron wind force that caused the damage in first place. Power grid lines

carry mostly unidirectional current with no benefit from healing, and thus they fail

early.


Power GridMetal Lines

Connections to External Power Supply(C4 Sites)

Circuit Blocks

Integrated Circuit

Figure 2.3: High level model of the power grid

2. Second, the currents flowing in signal and clock lines are easy to predict since they

are determined by charging and discharging of the capacitive loads in the circuit

which are usually known. Therefore, their reliability is relatively easy to estimate.

However, currents flowing in power grid lines are much harder to predict due to the

uncertainty about the underlying circuit activity and current requirements.

2.5.2 Power Grid Model

Because EM is a long-term cumulative failure mechanism, the changes in the current

waveforms on short time-scales are not very significant for EM degradation. In fact, the

standard approach to check for EM under time-varying current is to compute a constant

value called the effective-EM current, derived from the time-varying current waveform.

The value obtained represents the DC current that effectively gives the same lifetime

as the original waveform under the same conditions. As mentioned earlier, power grid

lines carry mostly-unidirectional currents for which, effective currents are chosen as the

average currents. Accordingly, it is sufficient to consider a DC model of the grid subject

to average current sources that model the currents drawn by the underlying logic blocks.

This is justified because the power grid is a linear system, and hence its average branch

currents can be obtained by subjecting it to average current sources.


i Vdd_+

gg

g g

g

g

g

g

g

g

g

g

g

ggg

g

g

g g g g

g

g

i

Figure 2.4: A resistive model of a power grid

Let the power grid consist of n+ q nodes, where nodes 1 . . . n have no voltage sources

attached, and the remaining nodes connect to ideal voltage sources that represent the

connections to external power supply, and let node 0 represent the ground node. Let

Ik be the current source connected to node k, where the direction of positive current is

from the node to ground. We assume that Ik ≥ 0 and that Ik is defined for every node

k = 1, . . . , n so that nodes with no current source attached have Ik = 0. Let I be the

vector of all Ik sources, k = 1, . . . , n. Let Uk(t) be the voltage at every node k, and let

U(t) be the vector of all Uk(t) values. Even though Uk is a DC value, we still introduce a

time dependence to reflect the changes that will occur when the grid lines start to fail due

to electromigration. Note that the nodes attached to vdd will not be explicitly included

in the system formulation below as their voltage levels are known (vdd).

Applying Kirchoff’s Current Law (KCL) at every node, k = 1, . . . , n, leads to the

following matrix formulation:

G(t)U(t) = −I +Gdd(t)Vdd (2.27)

where G(t) represents the conductance matrix of the grid resulting from the application

of modified nodal analysis (MNA), simplified by the fact that all the voltage sources are

to ground; Gdd(t) is another matrix consisting of conductance elements connected to the

vdd sources; Vdd is a constant vector each entry of which is equal to vdd. Again, the time

dependence in G(t) and Gdd(t) is there to reflect the changes in the grid structure and


conductance values as grid lines fail over time. If we set all sources Ik to zero in (2.27),

then U(t) = Vdd, and the equation becomes:

G(t)Vdd = Gdd(t)Vdd (2.28)

which allows us to rewrite (2.27) as:

G(t) [Vdd − U(t)] = I (2.29)

Define Vk(t) = vdd − Uk(t) to be the voltage drop at node k, and let V (t) be the vector

of all the voltage drops. The system equation becomes:

G(t)V (t) = I (2.30)

As long as the grid is connected, G(t) is known to be a diagonally-dominant symmetric

positive definite matrix with non-positive off-diagonal entries. Accordingly, G(t) can be

shown to be anM-matrix, so that G−1(t) exists and G−1(t) ≥ 0 [14].

Generally, G is formed using the MNA element stamping method as follows. Starting

with an n × n matrix of zeros, every conductance g in the grid connecting nodes i and

j (i, j ∈ {1, 2, . . . , n}), adds an n× n matrix ∆G to G such that ∆G contains all zeros

except that ∆Gii = ∆Gjj = −∆Gij = −∆Gji = g. If g connects node i to a voltage

supply (that is connected to ground, then ∆G has only one nonzero entry ∆Gii = g.

Notice that, in all cases, ∆G is a rank-1 matrix that can be written as an outer product

uuT with u being a vector of zeros except at positions i and j where ui = −uj = √g (All

outer products result in rank-1 matrices [15]). If g connects node i to a voltage supply,

then u is a vector of zero except at position i where ui =√g.

As an example, we will apply MNA to the circuit in figure 2.5. The ith resistor has a

conductance gi. The resulting conductance matrix is the following:

G =

g1 + g2 + g3 −g2 0 −g4 0 0

−g2 g2 + g3 + g5 −g3 0 −g5 0

0 −g3 g3 + g6 0 0 −g6−g4 0 0 g4 + g7 −g7 0

0 −g5 0 −g7 g5 + g7 + g8 −g80 0 −g6 0 −g8 g6 + g8 + g9

(2.31)


+_

+_dd

v

ddv

6

g g g

g g g

g g g

1 2 3

4 5 6

7 8 9i

i

4

3

1 2 3

4 5

Figure 2.5: A small resistive grid

With I being:

I =[

0 0 i3 i4 0 0]T

(2.32)

Because Black’s model depends on the current density through the metal line, branch

currents are needed. Let b be the number of branches in the grid, and let Ib,l(t) represent

the branch currents where l = 1, . . . , b, and let Ib(t) be the vector of all branch currents.

Relating all the branch currents to the voltage drops V (t) across them, we get:

Ib(t) = −R−1MTV (t) = −R−1MTG−1(t)I (2.33)

where R is a b × b diagonal matrix of the branch resistance values and M is an n × bincidence matrix whose elements are ±1 or 0 such that the term ±1 occurs in location

mkl of the matrix where node k is connected to the lth branch, else a 0 occurs. The signs

of the non-zero terms depend on the node under consideration. If the reference direction

for the current is away from the node, then the sign is positive, else it is negative.

Back to the example above, and based on the reference directions indicated in fig-

ure 2.5, the resulting R−1 and M are the following:

R−1 = diag(g1, g2, g3, g4, g5, g6, g7, g8, g9) (2.34)


M =

−1 1 0 1 0 0 0 0 0

0 −1 1 0 1 0 0 0 0

0 0 −1 0 0 1 0 0 0

0 0 0 −1 0 0 1 0 0

0 0 0 0 −1 0 −1 1 0

0 0 0 0 0 −1 0 −1 1

(2.35)

2.6 Sampling and Statistical Estimation

2.6.1 Overview

Sampling is the process of selecting a subset of individuals from the domain of a statistical

distribution to estimate certain characteristics of the whole population. As will be later

explained, the main parts of this research rely on sampling as well as mean and probability

estimation by random sampling. For that, Sections 2.6.2 and 2.6.3 show how to generate

samples from the standard normal and the lognormal distributions respectively, while

sections 2.6.4 and 2.6.5 focus on techniques for mean and probability estimation using

Monte Carlo.

2.6.2 Sampling from the Standard Normal

Many algorithms have been developed to sample from a given distribution. The Ziggurat

method is one of the most famous approaches developed in the early 1980’s by Marsaglia

and Tsang [16], which allows sampling from decreasing or symmetric unimodal proba-

bility density functions at high generation rates (meaning that the method is able to

generate a large number of samples efficiently, in a short amount of time). The method

was later improved in [17].

The general idea of sampling is to choose uniformly a point (x, y) under the curve of

the PDF, and return x as the required sample (Many software packages (C++, MATLAB,

etc.) have routines that return pseudo-random numbers from a uniform distribution). To

do that, the Ziggurat method covers the target density function with a set of horizontal

equal area rectangles, picks one of the rectangles randomly, and then samples a point

uniformly inside the chosen rectangle. If the point was found to be under the actual

PDF curve, then the corresponding horizontal coordinate is returned. Otherwise, another

point in the rectangle is sampled. When the sampling is to be done from the tail of the

distribution, a special expensive calculation is done using logarithms (see [18]) . Notice

that the accuracy of the method depends on the number of rectangles used to cover


the PDF. In [17], 255 rectangles were used and found to be sufficient for reliable and

fast sampling. Because the standard normal distribution is symmetric around zero, the

Ziggurat method was found to be very effective and easily implementable.

2.6.3 Sampling from the Lognormal

Because the PDF of the lognormal distribution is neither monotone nor symmetric, the

Ziggurat method cannot be used to sample from a lognormal distribution. However, it

is possible to obtain such a sample by proper modification of another sample obtained

from the standard normal. This, in fact, is easier and much more efficient.

Let T be a lognormally distributed random variable with µ = E[T], σ2 = Var(T),

µln = E[lnT], and σ2ln = Var(lnT). Because lnT is normally distributed, we know that

the RV Z = lnT−µln

σlnhas a standard normal distribution. Thus, we can write:

T = exp(µln + σlnZ) (2.36)

This means that, given a sample z from the standard normal distribution generated as ex-

plained in the previous section, we can derive a sample τ from the lognormal distribution

with the mean and variance above as follows:

τ = exp(µln + σlnz) (2.37)

In practice, µ instead of µln is usually known. An example of that is Black’s equation

that gives the MTF of a mortal line subject to electromigration. From (2.18), we can

write:

lnµ = µln +1

2σ2ln (2.38)

Hence, we can rewrite (2.37) as:

τ = exp

(

lnµ− 1

2σ2ln + σlnz

)

= µ exp

(

σlnz −1

2σ2ln

)

(2.39)

2.6.4 Mean Estimation by Random Sampling

Also known as the Monte-Carlo approach, mean estimation by random sampling refers

to iteratively selecting specific values from the domain of a distribution and computing

their arithmetic average as an estimate of the true mean of the distribution. Let X be


a continuous random variable (RV) with a density function f(x), and let µ = E[X],

and σ2 = Var(X). Also, Let X1,X2, . . . ,Xw be a set of independent and identically

distributed RVs with the same density function f(x) as X. This collection of RVs is

referred to as a random sample. Let Xw be the arithmetic average of all the Xi’s, then

Xw is an RV known as the sample mean, and is given by:

Xw =X1 +X2 + . . .+Xw

w=

1

w

w∑

i=1

Xi (2.40)

Clearly,

E[Xw] =1

w

w∑

i=1

E[Xi] =1

w

w∑

i=1

µ = µ (2.41)

and

Var(Xw) =w∑

i=1

Var

(

Xi

w

)

=w∑

i=1

σ2

w2=σ2

w(2.42)

Applying Chebyshev’s inequality with mean µ and variance σ2/w, we get:

Pr{

|Xw − µ| < ǫ}

≤ σ2

wǫ2(2.43)

which shows that the distribution of Xw tightens around the mean as w increases. When

w → +∞, Xw → µ with a probability 1. This is usually referred to as the law of large

numbers. In practice, one would like to know how large w should be in order to have

a certain confidence level that the obtained arithmetic average is within a certain small

interval around µ.

Sampling from a Normal

Assume in this section that X is known to be normal, so that Xw is also normal and the

random variable Z = Xw−µσ/

√w

has a standard normal distribution. Also, assume that xw =∑w

i=1xi

wis an observed value of Xw, corresponding to the observed values x1, x2, . . . , xw

of the RVs X1,X2, . . . ,Xw. For a given α ∈ [0, 1], we call zα/2 the (1−α/2)-percentile ofthe RV Z, i.e. the value that satisfies Pr{Z ≤ zα/2} = 1− α/2. Knowing α, zα/2 can be

obtained using statistical tables or using the erf() function available on most computer

systems. Due to symmetry, Pr{|Z| ≤ zα/2} = 1 − α, i.e. we can say with a confidence

(1− α) that:|xw − µ|σ/√w≤ zα/2 (2.44)


Dividing both sides by |xw| (assuming xw 6= 0), we get:

|xw − µ||xw|

≤ zα/2σ

|xw|√w

(2.45)

Hence, a sufficient condition to have an upper bound δ ∈ (0, 1) on the relative error |xw−µ||xw|

with a confidence (1− α), is to have:

zα/2σ

|xw|√w≤ δ (2.46)

which gives the following stopping criterion:

w ≥(

zα/2σ

|xw|δ

)2

(2.47)

Furthermore, it can be shown that if (2.46) is true, then we have:

|xw − µ||µ| ≤ δ

1− δ , ǫ (2.48)

which ensures an upper bound on the relative deviation from the true mean µ. For most

cases, ǫ is a better metric to use than δ. Clearly, δ = ǫ1+ǫ

, and hence the stopping criterion

becomes:

w ≥(

zα/2σ

|xw|ǫ/(1 + ǫ)

)2

(2.49)

One limitation of the above formula is that it requires the knowledge of σ which is

unavailable in most cases. A good way of overcoming this limitation is by using the

sample standard deviation given by:

sw =

√

√

√

√

1

w − 1

w∑

i=1

(xi − xw)2 (2.50)

With sw in place of σ, the RVT = Xw−µsw/

√wis known to have a Student’s t-distribution which

approaches the standard normal distribution for large w. Accordingly, for sufficiently

large w (typically w ≥ 30 as specified in [19]), the same stopping criterion above can be

used with sw used instead of σ.


Sampling from an Unknown Distribution

Studies have shown that the distribution of Xw−µsw/

√w

is, in most cases, fairly close to a

t-distribution even when X is not normal, and hence it approaches a standard normal

for sufficiently large w (typically w ≥ 30 as specified in [19]). In conclusion, when sam-

pling from an unknown distribution, the required stopping criteria to achieve a relative

deviation ǫ from the mean µ with a confidence level of (1− α), is:

w ≥(

zα/2sw|xw|ǫ/(1 + ǫ)

)2

for w ≥ 30 (2.51)

2.6.5 Probability Estimation by Random Sampling

Another application of Monte Carlo sampling is probability estimation. Consider an

experiment whose outcome is random and can be either of two possibilities: success and

failure. Such an experiment is referred to as a Bernoulli trial (or binomial trial). Let

p be the (unknown) probability of success. One way of estimating p is by performing a

sequence of w trials, and counting the number x of successes that are are observed. By

the law of large numbers :

limw→∞

x

w= p (2.52)

In practice, one would like to know how large w should be so that xwis fairly close to p.

This is generally quantified, as before, in terms of two small numbers α and ǫ as to say:

“we are (1− α)× 100% confident that∣

∣

xw− p∣

∣ < ǫ.”

In [20], three lower bounds on w were derived. These bounds are functions of α and

ǫ, and are found using the notion of confidence intervals from statistics [19].

The first bound corresponds to the case where p ∈ [0.1, 0.9], and is given by:

B1(α, ǫ) =(zα/2

2ǫ

)2

(2.53)

where zα/2 is as defined in the previous section. The second bound corresponds to the

case where p 6∈ [0.1, 0.9] and x is large (x > 15), and is found to be:

B2(α, ǫ) =

zα/2√2ǫ+ 0.1 +

√

(ǫ+ 0.1)z2α/2 + 3ǫ

2ǫ

(2.54)

and the third bound corresponds to the case where p 6∈ [0.1.0.9] and x is small (x ≤ 15),


and is found to be:

B3(α, ǫ) =(√

63 + zα/22√ǫ

)2

(2.55)

Ultimately, for a given error bound ǫ, and a confidence level (1 − α) × 100%, we can

determined the minimum number of patterns w to be applied by taking the maximum

of the three lower bounds predicted above:

w > max (B1(α, ǫ),B2(α, ǫ),B3(α, ǫ)) (2.56)

2.7 Notation

Throughout the rest of the thesis, we will be using the 1-norm and the infinity norm

defined as follows: given a vector x ∈ Rn with entries xi, i = 1 . . . n:

‖x‖1 ,n∑

i=1

|xi|

‖x‖∞ , maxi=1...n

|xi|

Also, we will be using the notation 1λ to denote a λ × 1 vector of ones, 0λ to denote a

λ× 1 vector of zeros, and eλ to denote the n× 1 vector containing 1 at the λth position

and zeros everywhere else (n is the number of nodes in a power grid and e0 = 0n)

Chapter 3

Vector-Based Power Grid

Electromigration Checking

3.1 Introduction

In this chapter, we describe a novel approach for power grid electromigration checking

based on a new failure model that is more realistic and more accurate than SEB. The

main drawback of SEB is that it applies overly conservative and pessimistic analysis.

Accordingly, and because SEB is still in use, design groups are suffering from a significant

loss of margins between the predicted EM stress and the allowed thresholds. Due to the

reduced margins, the designers are finding it very hard to meet the EM design rules and

to sigh-off on chip designs. In this chapter, we focus on reducing the pessimism of SEB

by improving the way system level reliability is obtained given the reliability of individual

lines. For that, we will assume (for now) that the currents drawn from the power grid

by the underlying logic blocks are known exactly. The issue of uncertainty about the

currents will be addressed in other chapters.

Recall that SEB relies on a series system assumption, where the power grid is deemed

to fail when any of its components fail. However, modern power grids are meshes, as

shown in figure 2.4, rather than the traditional comb structure. The mesh structure

allows multiple paths between any two nodes, and accordingly, the power grid will not

necessarily fail if one of its metal lines fails, but it can tolerate multiple failures as long

as the voltages at its nodes remain acceptable. This implies some level of redundancy

in the grid, which has largely been ignored in EM checking tools, both in academia

and industry. In this chapter, we develop a new model, referred to as the mesh model,

that factors in the redundancy of a power grid while estimating its MTF and reliability.

26

Chapter 3. Vector-Based Power Grid Electromigration Checking 27

Experimental results in Section 3.6 show that a grid can tolerate up to 50 or more line

failures before it truly fails, with 2-2.5X longer lifetimes than the series system.

3.2 The ‘Mesh’ Model

As explained earlier, the performance of a power grid is generally evaluated based on

how well the supply voltage vdd is conducted to grid nodes. In other words, for a grid

to function as intended, the voltage drop at each of its nodes should be smaller than a

certain threshold because otherwise, soft errors in the underlying logic may occur [12]. A

node is said to be safe when its voltage drop meets the corresponding threshold condition,

and unsafe otherwise. Let Vth be the vector of all the threshold values which are typically

user-specified, and assume that Vth > 0 to avoid trivial cases.

Because the currents drawn from the grid are known, the vector I in (2.30) and (2.33)

is a constant vector. We assume that at t = 0, the grid is connected, so that there is a

resistive path from any node to another that does not go through a vdd or ground node.

Also, we assume that the grid is safe at t = 0. That is, all the voltage drops at all the

nodes are below their corresponding threshold, i.e.:

V (0) = G−1(0)I ≤ Vth (3.1)

Notice that if this assumption is not true, the grid would be failing at t = 0, i.e. is unsafe

at production time.

As we move forward in time, the EM-susceptible lines start to fail in the order of

their failure times due to electromigration. Accordingly, the conductance matrix G(t)

of the grid changes and so does V (t). The grid is deemed to fail at the earliest time for

which the condition V (t) = G−1(t)I ≤ Vth is no longer true, meaning when any of the

grid nodes becomes unsafe. This new model is referred to as the mesh model, and will

be used to determine the failure time of the grid when the failure times of its lines are

known. Notice that if, for a particular vector I, the first failure in the grid causes the

condition V (t) ≤ Vth to be violated, then the mesh model reduces to the standard series

system model. Experimental data will show that a grid can actually tolerate more than

one failure.


3.3 Estimation Approach

3.3.1 MTF and Survival Probability Estimation

Let Tm be the random variable denoting the time-to-failure of the grid according to the

mesh model. In order to estimate the MTF of the power grid using the mesh model, i.e.

E[Tm], we perform Monte-Carlo analysis. In every iteration, we generate one sample of

the grid time-to-failure using the mesh model and we stop once the convergence criteria

of Monte Carlo is met (condition (2.51)).

Because I is known, one can find the branch currents in the grid using (2.33), and

then find the JL-product of every line. This allows filtering out the EM-immune lines.

The mean time-to-failure of all the other lines can then be found using Black’s equation.

For every Monte Carlo iteration, we choose time-to-failure samples from the lognormal

distribution for all the EM-susceptible lines (as in section 2.6.3). We then sort the samples

in increasing order and find the time at which the condition V (t) ≤ Vth is first violated

according to that particular order. This gives one grid time-to-failure sample.

We also use Monte-Carlo sampling to estimate the probability of survival of a grid

up to Y years, i.e. Pr (Tm > Y). For that, we repeat the same procedure in every Monte

Carlo iteration, and we try to figure out whether the grid has failed before t = Y or not.

Because this represents a Bernoulli trial, we use the bounds derived in section 2.6.5 to

determined how many trials are needed to have an error bound ǫ and a confidence level

(1− α)× 100%. If w trials were needed, and if the grid was found to be safe at t = Y in

x of those trials, then

Pr{Tg > Y} ≈x

w

3.3.2 Resistance Evolution

Because V (t) is needed to check if V (t) ≤ Vth, we need to model the resistance of grid lines

once they fail so that we know how G(t) evolves with time and compute V (t) accordingly

(recall, I is known). Extensive analysis has been done to model the evolution of resistance

of a metal line subject to electromigration. In [21], the authors show that for copper lines

from the 65 nm technology node, the resistance increases, due to void creation, by an

initial step Rstep at the failure time, and then continues to increase gradually (almost

linearly with a rate of change dRdt

= Rslope) afterwards as shown in figure 3.1a. Both Rstep

and Rslope seem to increase as the length of the line increases but are not affected by its

width. Other references such as [22] and [23] show similar observations and present a

similar model.


(a) Resistance evolution for copper lines fromthe 65 nm technology node (courtesy of [21])

R 0

TTFTime

Resistance

(b) Infinite resistance model; R0 is the initialresistance of the wire

Figure 3.1: Resistance evolution

In this work, we assume that the resistance of a line becomes infinite at its failure time

(see figure 3.1b). In effect, we are assuming that the failure is not gradual and is, in some

sense, quantized. This infinite resistance model leads to simple and conservative analysis

since in reality, lines continue to conduct current after failure but with high resistance,

and hence employing the infinite resistance model means we are assuming that the line

is more degraded than it actually is.

3.3.3 Generating Time-to-Failure Samples

As mentioned before, branch currents are needed to discover the EM-immune lines, and

to find the MTF of all the other lines using Black’s equation. Since the grid will be

changing over time due to the failure of its components, the branch currents will also

change. For simplicity, we will assume that the statistics of the lines can be determined

using the branch currents of the grid before the failure of any of its components. This

assumption means that, after the failure of a line, the MTFs of the other lines remain

the same even though the branch currents are changing. This will boost the speed of our

method and make it a lot simpler at the expense of some loss in accuracy. Please note

that the case of changing currents is fully detailed in [24].

If G0 is the conductance matrix of the original grid (i.e. G0 = G(0)), then the vector

of initial voltage drops can be written as V0 = V (0) = G−10 I. This allows writing:

Ib(0) = Ib = −R−1MTG−10 I


At t = 0, the current density of a line l with a cross sectional area al, length Ll, and

branch current Ib,l, can be written as:

Jl =|Ib,l|al

(3.2)

To know if line l is EM-susceptible, JlLl should be computed and compared to βc. If

JlLl < βc, then the line is EM-immune and should be discarded and removed from the

set of line that may fail and cause the grid to fail. Otherwise, its MTF µl should be

computed using Black’s equation which can be rewritten as follows:

µl =aη+1l

A|Ib,l|−η exp

(

Ea

kTm

)

(3.3)

For the purpose of Monte Carlo analysis, a Time-to-Failure (TTF) sample τl should be

assigned to every EM-susceptible line in every Monte Carlo iteration. This can be done

by sampling a real number ψl from the standard normal distribution N (0, 1), and then

applying the transformation presented in section 2.6.3:

τl = µl exp

(

ψlσln −1

2σ2ln

)

(3.4)

If bTl is the row of −R−1MTG−10 that corresponds to line l, then Ib,l = bTl I and hence,

given a sample ψl from the standard normal distribution, we can find a sample TTF τl

for every line l, using (3.4) and (3.3):

τl =aη+1l

A|bTl I|−η exp

(

Ea

kTm

)

exp

(

ψlσln −1

2σ2ln

)

(3.5)

Let

cl ,

[

aη+1l

Aexp

(

Ea

kTm

)

exp

(

ψlσln −1

2σ2ln

)]− 1η

bl

Then,

τl = |cTl I|−η (3.6)

3.4 Computing Voltage Drops

Checking if the grid is failed at a particular point in time requires checking the condition

V (t) ≤ Vth. Because the infinite resistance model is used, V (t) changes only when a line

fails, and remains the same between any two consecutive line failures. Therefore, V (t)


should be recomputed every time a line fails. One way of doing that is by updating G(t)

and then resolving V (t) = G−1(t)I using LU factorization of G(t) and backward and

forward solves. For LU factorization, G(t) is written as a product of a lower-triangular

matrix L(t) and an upper triangular matrix U(t):

G(t) = L(t)U(t), (3.7)

and (2.30) becomes:

L(t)U(t)V (t) = I. (3.8)

Define the vector Y (t) = U(t)V (t) so that (3.8) becomes:

L(t)Y (t) = I (3.9)

Because L(t) is lower triangular, a forward solve finds the values of the components

of Y (t) consecutively in O(n2) operations. Having solved for Y (t), a backward solve

calculates the values of the components of V (t) in reverse order, using the fact that

Y (t) = U(t)V (t) and that U(t) is upper triangular. The cost of the forward solve is

also O(n2), making the total cost of the forward/backward solves O(n2). Generally, the

complexity of the LU factorization itself is O(n3) for dense matrices, but since G(t) is

sparse, the complexity becomes around O(n1.5).

Unfortunately, we are required to solve for V (t) after every line failure until the

condition V (t) ≤ Vth is no longer true, and this procedure has to be repeated in every

Monte Carlo iteration. Thus, performing an LU factorization, from scratch, every time a

line fails is very expensive. But because we are modelling the failure of every line by an

open circuit, we can write the change inG corresponding to the kth line failure as a rank-1

matrix −∆Gk. This corresponds to the removal of a conductance from the conductance

matrix by reversing the element stamping procedure for that particular conductance.

Accordingly, ∆Gk is exactly as defined earlier (in section 2.5.2), and can be written as

∆Gk = ukuTk .

After the failure of k lines, let U be the n× k matrix such that:

U =[

u1 u2 . . . uk

]


Therefore,

UUT =[

u1 u2 . . . uk

]

uT1

uT2...

uTk

(3.10)

= u1uT1 + u2u

T2 + . . .+ uku

Tk =

k∑

j=1

ujuTj =

k∑

j=1

∆Gj (3.11)

This means we can write the vector of voltage drops Vk after the failure of k lines as:

Vk =

(

G0 −k∑

j=1

∆Gj

)−1

I =(

G0 −UUT)−1

I (3.12)

3.4.1 Sherman-Morrison-Woodbury Formula

Given the equation above and the initial vector of voltage drops V0, is it possible to obtain

Vk efficiently without computing the inverse of G0 −UUT ? The answer is yes, and for

that we employ the Sherman-Morrison-Woodbury formula [25]. In essence, the formula

asserts that the inverse of a rank-k correction of some invertible matrix can be computed

by doing a rank-k correction to the inverse of the original matrix. The formula is also

known as the matrix inversion lemma, and states the following: Given a nonsingular

matrix A ∈ Rn×n, and matrices P,Q ∈ R

n×k such that Ik + PTA−1Q is nonsingular,

then A+PQT is also nonsingular and:

(

A+PQT)−1

= A−1 −A−1P(Ik +QTA−1P)−1QTA−1 (3.13)

where Ik is the k × k identity matrix.

Using (3.13), we can write the inverse of G0 −UUT as follows:

(

G0 −UUT)−1

= G−10 +G−1

0 U(Ik −UTG−10 U)−1UTG−1

0 (3.14)

This assumes that G0 is nonsingular (which we know because the grid is assumed to be

connected and safe at t = 0), and that Ik −UTG−10 U is also nonsingular. We will first

handle the case where Ik−UTG−10 U is nonsingular, and discuss the singularity case later

on.


Using (3.12) and (3.14), we have:

Vk = G−10 I +

[

G−10 U(Ik −UTG−1

0 U)−1UTG−10

]

I (3.15)

Define Zk = G−10 U = [G−1

0 u1 . . . G−10 uk]. Because G−1

0 I = V0, we can finally write:

Vk = V0 + ZkW−1k yk (3.16)

where

Wk = Ik −UTZ and yk = UTV0

The vector Vk must be computed using (3.16) for every k = 1, 2, . . . until the condition

Vk ≤ Vth is no longer true. Computing V0 should be done only once by doing an LU

factorization of G0 and forward/backward solves. For every k, Zk must be updated by

appending the column vector G−10 uk, which can be computed using forward/backward

substitutions. Finally, the inverse of the dense k× k matrix Wk must be computed. For

that, we notice that k is generally small, and hence we can factorize Wk for every k in

O(k3) time, which is cheap for small k. However, k can become large for large grids,

and hence computing the LU factorization of Wk may become expensive. To overcome

this limitation, we propose a further refinement based on the Banachiewicz-Schur form

so that the complexity is reduced to O(k2). To take full advantage of this technique,

we will always use the Banachiewicz-Schur form when updating the voltage drops (i.e.

∀k = 1, 2, . . .).

3.4.2 The Banachiewicz-Schur Form

Let M ∈ Rk×k be 2× 2 block matrix:

M =

[

A b

cT d

]

(3.17)

where A ∈ R(k−1)×(k−1), b ∈ R

k−1, c ∈ Rk−1, and d is a scalar. The Schur-complement of

A in M is the real number s given by:

s = d− cTA−1b (3.18)


If both M and A in (3.17) are non-singular, then s 6= 0. This allows writing, M as:

M =

[

Ik−1 0

cTA−1 1

][

A 0

0 s

][

Ik−1 A−1b

0 1

]

(3.19)

where Ik−1 is the identity matrix of size (k − 1) × (k − 1). The expression above can

be verified by performing the multiplication of the three matrices shown. The inverse of

M as given in the form above can be found by inverting each of the three matrices, and

reversing the order of their multiplication. The inverse obtained is [26]:

M−1 =

[

Ik−1 −A−1b

0 1

]

A−1 0

01

s

[

Ik−1 0

−cTA−1 1

]

(3.20)

which can be reduced to:

M−1 =

A−1 +A−1bcTA−1

s−A−1bT

s

−cTA−1

s

1

s

(3.21)

Equation (3.21) is known as the Banachiewicz-Schur form. It expresses M−1 in terms of

A−1, b, c, and d.

Back to (3.16), we observe that Wk can be written as:

Wk = Ik −UTZk = Ik −UTG−10 U

= Ik −

uT1...

uTk−1

uTk

G−10 [u1 . . . uk−1 uk]

Therefore,

Wk =

1− uT1G−10 u1 . . . −uT1G−1

0 uk−1 −uT1G−10 uk

.... . .

......

−uTk−1G−10 u1 . . . 1− uTk−1G

−10 uk−1 −uTk−1G

−10 uk

−uTkG−10 u1 . . . −uTkG−1

0 uk−1 1− uTkG−10 uk

(3.22)


From (3.22), and because for every j ∈ {1, . . . , k},

uTkG−10 uj =

(

uTkG−10 uj

)T= uTj

(

G−10

)Tuk = uTj G

−10 uk

we can write Wk in terms of Wk−1 (from the previous iteration) as:

Wk =

[

Wk−1 bk

bTk dk

]

(3.23)

where

bk = [−uT1G−10 uk . . . − uTk−1G

−10 uk]

T ∈ Rk−1 (3.24)

dk = 1− uTkG−10 uk ∈ R (3.25)

Hence, using the Banachiewicz-schur form, we can express W−1k in terms of W−1

k−1 as:

W−1k =

W−1k−1 +

W−1k−1bkb

TkW

−1k−1

sk−W−1

k−1bk

sk

−bTkW

−1k−1

sk

1

sk

(3.26)

where sk is the schur complement of Wk−1 in Wk. Notice that sk 6= 0 because Wk is

assumed to be invertible for now. Using (3.18), we can write:

sk = dk − bTkW−1k−1bk (3.27)

Also, by construction, we know that after k interconnect failures:

yk = UTV0 =

uT1...

uTk

V0 =

uT1 V0...

uTk V0

=

[

yk−1

uTk V0

]

(3.28)

Thus, we can update yk from yk−1 by appending pk = uTk V0 at the end. Now, we can


write W−1k yk as:

W−1k yk =

W−1k−1 +

W−1k−1bkb

TkW

−1k−1

sk−W−1

k−1bk

sk

−bTkW

−1k−1

sk

1

sk

[

yk−1

pk

]

=

W−1k−1yk−1 +

W−1k−1bkb

TkW

−1k−1yk−1

sk−W−1

k−1bk

skpk

−bTkW

−1k−1yk−1

sk+pksk

(3.29)

But, the previous solution xk−1 = W−1k−1yk−1 is known from the previous iteration, there-

fore:

xk =

xk−1 +W−1

k−1bkbTk xk−1

sk−W−1

k−1bk

skpk

−bTk xk−1

sk+pksk

(3.30)

Define ak =bTk xk−1 − pk

sk. Now, we can rewrite (3.30) as:

xk =

[

xk−1 + akW−1k−1bk

−ak

]

(3.31)

We can use (3.26) and (3.31) to directly update W−1k and xk from their previous values.

Notice thatW−1k is required because, in the next iteration, W−1

k bk+1 is needed to compute

xk+1 using (3.31). The implementation requires a single matrix-vector product (O(k2))and O(k2) additions and divisions.

3.4.3 Case of Singularity

Recall that G0 −UUT is invertible if and only if Wk is invertible, which is invertible if

and only if sk 6= 0. Therefore, if for some k, sk is found to be zero, then we know that

both Wk and G0 −UUT are singular. In this case, Vk cannot be computed. Physically,

because we are modelling the failure of a line by an open circuit, it is possible for a

node to become isolated making the conductance matrix non-invertible. Accordingly,

the condition V (t) ≤ Vth is automatically violated because an isolated node represents

a high impedance with an unknown voltage level. Overall, the grid is deemed to fail

at the earliest time for which the condition V (t) ≤ Vth is no longer true or when the


conductance matrix of the grid becomes singular (which can be detected by checking if

sk is zero).

3.5 Implementation

Algorithm 1 FIND GRID TTF

Input: V0, G0, LOutput: τs, τm1: Assign TTF samples to all the lines in list L (Assign a TTF of ∞ for immortal lines)2: Find the line in L with lowest TTF and assign its TTF to τs.3: Z0 ← [ ],W−1

0 ← [ ], x0 ← [ ], y0 ← [ ], grid singular ← 0, k ← 14: while (Vk ≤ Vth and grid singular = 0) do5: Find line lk ∈ L with lowest TTF and its conductance stamp ∆Gk

6: Find uk such that ∆Gk = uTk uk.7: (Vk,Zk,W

−1k , xk, yk, grid singular) ← FIND VK (V0,G0, uk,Zk−1,W

−1k−1, xk−1, yk−1, k)

8: L ← L− lk9: k ← k + 1

10: end while

11: Assign to τm the TTF of line lk.12: return τm

The overall flow for obtaining a sample of power grid TTF using both the series and the

mesh model is given in Algorithm 1. The algorithm requires G0, V0 which can be computed,

once for all Monte Carlo iterations, using LU factorization, and L, a list containing all the

lines in the grid. We start by assigning TTF samples to all the resistors in the power grid as

described in section 3.3.3 (We assign a TTF of ∞ for EM-immune lines). If the grid is viewed

as a series system, the failure of the first resistor causes the grid to fail, i.e. the series system

TTF τs is assigned to the TTF sample of the first failing resistor in L. The algorithm then

continues to compute the mesh model TTF τm by failing grid lines and computing the vector of

voltage drops Vk using Procedure 1 which employs both Sherman-Morrison-Woodbury formula

and the Banachiewicz-Schur form. The algorithm exits once the grid becomes singular (flagged

by grid singular generated by Procedure 1), or when the condition Vk ≤ Vth is violated. The

sample τm is assigned the TTF sample of the last line that caused the grid to fail. To further

clarify the procedure, we present the flow chart of figure 3.2 to explain the steps to follow to

estimate the grid MTF using the mesh model.

To estimate the MTF of the grid, Algorithm 1 is run w times to generate w grid TTF

samples. As mentioned earlier, the number of iterations w is determined using (2.51). The

MTF of the grid is then estimated by the arithmetic mean of all the samples obtained. Also,

Algorithm 1 can be used to find the probability of survival of a grid up to a period of Y years.

Recall that computing the survival probability requires a number w of trials for which we count


signular?Is the grid

Are all the nodes safe?

Did Monte Carlo converge?

Start

Return grid MTF obtained

End

Yes

Yes

Yes

NoNo

No

Select TTF samples for all the

respective distributionsEM-susceptible lines from their

Remove the surviving line with lowest TTF

voltage drops, filter theEM-immune lines, andcompue the MTF of the

remaining lines

Find vector of initial

sample and update theMTF estimate of the grid

Obtain a grid TTF Compute the new vector ofvoltage drops using Woodbury

formformula and Banachiewicz-Schur

Figure 3.2: Mesh model MTF estimation


Procedure 1 FIND VK

Input: V0, G0, uk, Zk−1, W−1k−1, xk−1, yk−1, k

Output: Vk, Zk, W−1k , xk, yk, grid singular

1: zk ← G−10 uk using LU factorization followed by backward forward substitutions.

2: pk ← uTk V03: if k = 1 then

4: Zk ← [zk]5: yk ← [pk]6: W−1

k ← 11−uT

kzk

7: xk ←W−1k yk

8: else

9: Zk ← [Zk−1 zk]10: yk ← [yTk−1 pk]

T

11: Find bk and dk as given in (3.24) and (3.25)12: Wb ←W−1

k−1bk13: sk ← dk − bTkWb

14: if sk = 0 then

15: grid singular ← 116: else

17: ak ← bTkxk−1−pk

sk18: Find xk using (3.31)19: Find Wk using (3.26)20: end if

21: end if

22: Vk ← V0 − Zkxk

the number of times the grid is found to survive up to t = Y (success). For a particular trial,

checking for a success can be done easily using Algorithm 1. In fact, computing the survival

probability for different values of Y allows us to derive the reliability function as well as other

statistical measures (PDF, CDF, and failure rate).

3.6 Experimental Results

Algorithms 1 and Procedure 1 have been implemented in C++. We carried out several ex-

periments using 5 different power grids generated as per user specifications, including grid

dimensions, metal layers, pitch and width per layer. Supply voltages and current sources were

randomly placed on the grid which is assumed to have Aluminum interconnects. The param-

eters of the grids are consistent with 1.1V 65nm CMOS technology. As for the EM model

employed, and because Aluminum is assumed, we use an activation energy of 0.9eV , a current

exponent η = 1 (we assume that the lifetime is dominated by the time taken by the void to

grow), a nominal temperature Tm = 373K (The user can provide any temperature profile for

the grid lines; we use Tm = 373 as an average temperature throughout the chip), a critical Blech


Table 3.1: Comparison of power grid MTF estimated using the series model and themesh model

Power Grid ‖V0‖∞vdd

Series MeshGainRatioName Nodes C4’s Sources

mean mean Avg CPU(yrs) (yrs) fails Time

DC1 50K 870 3.2K 4.29% 4.48 11.27 51 2 min 2.52DC2 100K 1.7K 6.3K 4.12% 4.64 11.72 92 9 min 2.53DC3 200K 3.4K 12.7K 4.33% 4.12 9.86 115 27 min 2.40DC4 450K 7.5K 28.1K 4.16% 3.98 9.96 215 2.2 h 2.50DC5 1M 16.8K 63K 4.61% 4.05 9.01 251 6.7 h 2.22

Table 3.2: Survival probability estimation

Power Grid ‖V0‖∞vdd

Y (yrs) Ps Pm CPUName Nodes Time

DC1 50K 4.29% 5 0.27 1 45 secDC2 100K 4.33% 5 0.26 1 1.93 minDC3 200K 4.16% 5 0.05 1 8.25 minDC4 450K 4.12% 5 0.02 1 30.27 minDC5 1M 4.61% 5 0.002 1 1.73 h

product βc = 3000A/cm. The lognormal standard deviation we use is σln = 0.3 as in [6]. All

the experiments were carried out on a 2.6GHz Linux machine with 24GB of RAM. To assess the

quality of our results, we computed the mean time-to-failure using both the series and the mesh

model together with the required CPU time for every grid when the mesh model is employed.

The Monte Carlo parameters we use for that are ǫ = 0.05 and α = 0.05 for which zα/2 = 1.96.

Using these parameters, the number of Monte Carlo iterations that were required was between

30 and 40 for all the test grids. We also compute the probability of the grids surviving up to 5

years using both models. By choosing ǫ = α = 0.05, and by applying (2.56), the total number of

iterations required is 489. Table 3.1 compares the power grid MTF as estimated using the series

model and the mesh model. We notice a gain ratio that ranges between 2.22 and 2.53. Table 3.2

compares the survival probability up to Y = 5 years. It is seen that by taking redundancies

into account, the mesh model consistently predicts a higher survival probability as well as a

higher MTF as compared to the series model. For a given grid, the time required to estimate

the survival probability using the mesh model increases with increase in Y, but it also enables

us to estimate the reliability of the grid ∀t ≤ Y. For a complete overview, figures 3.4a and 3.4b

plot the reliability function and the probability density function (PDF) of DC3 as estimated

using both the series and the mesh model. Clearly, the series model gives a pessimistic estimate

of power grid TTF statistics.


0 100K 200K 300K 400K 500K 600K 700K 800K 900K 1M0

1

2

3

4

5

6

7

Number of Nodes

Tot

al C

PU

Tim

e (h

rs)

~O(n1.4)

Figure 3.3: CPU time of MTF estimation using the mesh model

Figure 3.3 shows the CPU time taken by the algorithm proposed. We can see that the

approach is scalable since the run time is slightly super-linear ( O(n1.4)). Moreover, due to the

inherent independence of Monte-Carlo iterations, the algorithm is highly parallelizable.

3.7 Conclusion

We described a novel approach for power grid electromigration checking based on a new failure

model that is more realistic and more accurate than SEB. The proposed approach is useful in

the case where the patterns of the currents drawn from the power grid are known exactly. As

showed in our results, the mesh model guarantees less pessimistic lifetimes while taking into

account the redundancies in modern power grids resulting from their many parallel paths.


0 2 4 6 8 10 12 14 160

0.2

0.4

0.6

0.8

1

Time (years)

Rel

iabi

lity

Series ModelMesh Model

(a) Reliability function of grid DC3

0 2 4 6 8 10 12 14 160

0.02

0.04

0.06

0.08

0.1

Time (years)

Pro

babi

lity

Series ModelMesh Model

(b) Probability distribution function for TTFs of grid DC3

Figure 3.4: Estimated statistics for grid DC3 (200K nodes)

Chapter 4

Vectorless Power Grid


4.1 Introduction

In chapter 3, we have seen how to estimate the reliability statistics of a power grid under the

effect of electromigration using the newly introduced mesh model. Recall that the currents

drawn from the power grid by the underlying logic blocks were assumed to be known exactly.

Clearly, if these currents change, the rate of EM degradation also changes. In practice, it is

not realistic to assume that the exact current waveforms are available for all the chip workload

scenarios, since this would require the simulation of the chip for millions of clock cycles at a low

enough level of abstraction that would provide the current waveforms. Moreover, one might

need to verify and check the grid early in the design flow, before fully designing the underlying

circuit. Therefore, one would like a vectorless approach that can deal with the uncertainty

about the underlying circuit currents. In this chapter, we present a vectorless extension of the

mesh model presented in the previous chapter. We show how the reliability statistics can be

obtained when only partial information about the power budget and the workload activity is

known.

4.2 Problem Definition

A constraint-based Vectorless power grid verification framework was first introduced in [27].

This framework defines a feasible space for currents in the form of current bounds, the idea

behind which is to capture circuit uncertainty via design specs or power budgets known in

the early design stages. Two types of constraints were defined: local constraints and global

constraints which respectively express bounds on the currents drawn by individual current

sources and by groups of current sources concurrently.

43

Chapter 4. Vectorless Power Grid Electromigration Checking 44

4.2.1 Modal Probabilities

Modern integrated circuits have complex multi-modal behavior, where major blocks of the chip

have different modes of operation (such as stand-by, low power, high performance, etc.). Spec-

ifying the block power dissipation requires knowledge of how often these modes are exercised.

For every circuit block j, let k = 1 . . . r enumerate the different modes of operation and Ijk

denote the block average supply current in that mode. The overall average supply current of

that block is given by Ij =r∑

k=1

αjkIjk, where 0 ≤ αjk ≤ 1 represent the probability of being in

different modes with the constraint thatr∑

k=1

αjk = 1. We propose that it is reasonable to expect

the user to specify the currents Ijk using the average power dissipation of each block in every

power mode. The mode probabilities αjk are generally harder to assess, but users are expected

to be able to specify values for some of them, or narrow ranges for others. If α denotes the

nr× 1 vector of all the mode probabilities (considering all the n blocks connected to the n grid

ndoes, having r modes of operation each), then we can write:

αmin ≤ α ≤ αmax (4.1)

where αmin and αmax have entries between 0 and 1, and contain any information the user may

have about the modes of operation. For a node with no current source attached, the lower and

upper bounds on its corresponding mode probabilities are set to zero.

The user can also specify bounds on the average current of every block, if available. This

allows us to infer other constraints on α in the form:

Iℓ,min ≤ Lα ≤ Iℓ,max (4.2)

where L is an n× nr matrix such that I = Lα. The matrix L contains information about the

currents drawn by the circuit blocks in each power mode.

Since chip components rarely draw their maximum currents simultaneously, global con-

straints are also used. For instance, if a certain limit is specified on the average power dissipa-

tion of the chip, then one may say that the sum of all the current sources is no more than a

certain upper bound. In general, the same concept can be applied for groups of current sources

forming functional blocks with known upper and lower bounds on their average power [27]. If

m is the total number of global constraints, then we can write:

Ig,min ≤ SLα ≤ Ig,max (4.3)

where S is an m × n matrix that only contains 0s and 1s and indicates which current sources

are present in each global constraint. The matrix contains a 1 at the kth entry of the ith row if


the kth circuit block (current sources) is present in the ith global constraint.

One last set of constraints should be added to guarantee that

r∑

k=1

αjk = 1 for every block j:

Bα = 1n (4.4)

where B is an n×nr matrix containing only 1s and 0s such that the vector Bα contains the sum

of mode probabilities per block in each of its entries. Together, all the constraints presented

above define a feasible space of mode probabilities, denoted by Fα, such that α ∈ Fα if and

only if, α satisfies (4.1) , (4.2), (4.3), and (4.4).

For example, consider a circuit having three blocks with two modes of operation each: high

performance and low power. Assume that the blocks draw respectively 0.2A, 0.3A, and 0.25A on

average in high performance mode, and 0.1A, 0.2A, and 0.15A in low power mode. Also, let α11,

α21, and α31 denote the probabilities of the blocks being in high performance mode, and α12,

α22, and α32 the probabilities of being in low power mode. If the average currents of the blocks

are I1, I2, and I3, and if I =[

I1 I2 I3

]Tand α =

[

α11 α12 α21 α22 α31 α32

]T

then we can write:

I = Lα =

0.2 0.1 0 0 0 0

0 0 0.3 0.2 0 0

0 0 0 0 0.25 0.15

α

The following is a possible set of constraints that a user can specify:

0.1

0.2

0.2

0.3

0.6

0.1

≤ α ≤

0.7

0.6

0.5

0.9

0.9

0.9

0.11

0.21

0.17

≤ Lα ≤

0.18

0.29

0.24

[

0.35

0.4

]

≤ SLα =

[

1 1 0

0 1 1

]

Lα ≤[

0.41

0.48

]

Bα =

1 1 0 0 0 0

0 0 1 1 0 0

0 0 0 0 1 1

α =

1

1

1

For every feasible setting of α, the overall block average currents are different, and the

reliability of the power grid is correspondingly different.


4.2.2 Current Feasible Space

As a first step towards finding the worst-case TTF of the grid, we transform the feasible space

Fα to the current domain. This helps reduce the number of variables from nr to n, as well as

the number of constraints. It is easy to see that replacing Lα by I in (4.2) and (4.3) results in

the first set of constraints defining the feasible space of currents:

Iℓ,min ≤ I ≤ Iℓ,max (4.5)

Ig,min ≤ SI ≤ Ig,max (4.6)

On the other hand, given the constraints on the individual α’s for every current source, we can

find lower and upper bounds for all the sources, as follows. Recall that every current source Ij

can be written as Ij =∑r

k=1 αjkIjk, and let αj denote the vector of all the mode probabilities

corresponding to Ij , then due to (4.1) we can write:

αj,min ≤ αj ≤ αj,max

where αj,min and αj,max contain the upper and lower bounds on the entries of αj as specified

in (4.1). Due to (4.4), we can write:∑r

k=1 αjk = 1, and hence, we can find bounds Ij,min and

Ij,max on Ij by solving the following two linear programs (LP):

Min/Maxr∑

k=1

αjkIjk

subject to αj,min ≤ αj ≤ αj,maxr∑

k=1

αjk = 1

(4.7)

The LPs above should be solved for every current source in the power grid. If any of the

LPs turns out to be infeasible, then the user specifications are not consistent. Notice that

due to the structure of the LPs above, we do not need to use any of the classical LP solving

methods (simplex or interior point). In fact, the claim below shows how compute the solutions

directly. Assume, without loss of generality, that the modes of operation of block j are sorted

in decreasing order of their power consumption, i.e. Ij1 ≥ Ij2 ≥ . . . ≥ Ijr. Also, call αjk,min

and αjk,max, k = 1, . . . , r, the entries of the vectors αj,min and αj,max respectively.

Claim 1. Consider the largest h ≤ r for whichh−1∑

k=1

αjk,max ≤ 1. Then, the solution to the


maximization problem in (4.7) is:

αjk =

αjk,max for k = 1, . . . , h− 1

1−h−1∑

k=1

αjk for k = h

αjk,min for k = h+ 1, . . . , r

Proof. To see why this works, notice that the problem is infeasible if∑r

k=1 αjk,min > 1 or∑r

k=1 αjk,max < 1. Assuming that the problem is feasible, we notice that we can replace the

last equality constraint by the inequality constraint∑r

k=1 αjk ≤ 1 without changing the optimal

solution. The reason is that if we were able to fit all the α’s without reaching equality, then∑r

k=1 αjk,max < 1, making the original problem infeasible, which contradicts our assumption.

Accordingly, we want to show that the greedy approach explained above solve the problem

below:

Maximize

r∑

k=1

αjkIjk

subject to αj,min ≤ αj ≤ αj,maxr∑

k=1

αjk ≤ 1

Consider the following transformation of variables:

wk =αjk − αjk,min

αjk,max − αjk,min, for k = 1, . . . , r

In the space of w, the problem becomes:

Maximizer∑

k=1

ckwk +r∑

k=1

Ijkαjk,min

subject to 0 ≤ wk ≤ 1, k = 1, . . . , rr∑

k=1

bkwk ≤ d

(4.8)

where ck = Ijk(αjk,max −αjk,min), bk = (αjk,max −αjk,min), and d = 1−∑rk=1 αjk,min. Because

the original problem is assumed to be feasible, we have d ≥ 0. Also, we notice that ck ≥ 0 and

bk ≥ 0 for every k. Ignoring the constant term∑r

k=1 Ijkαjk,min in the objective function, (4.8)

becomes an LP relaxation of the well known 0-1 Knapsack problem [28] for which the optimal

solution can be found using a greedy approach. If c1b1≥ c2

b2≥ . . . ≥ cr

br(which is true because

ck

bk= Ijk and the Ijk’s are assumed to be sorted in this order), then the optimal solution can be

found as follows: set w1 = w2 = . . . = wh−1 = 1, wh = d−∑h−1k=1 bk, and wh+2 = . . . = wr = 0,

where h is the largest possible. Transforming this solution back into the α space gives the

solution described earlier.


Claim 2. Consider the smallest g ≥ 1 for which

r∑

k=g+1

αjk,max ≤ 1. Then, the solution to the

minimization problem in (4.7) is:

αjk =

αjk,max for k = g + 1, . . . , r

1−r∑

k=g+1

αjk for k = g

αjk,min for k = 1, . . . , g − 1

The proof the claim 2 is similar to the proof of claim 1.

Ultimately if all the LPs turn out to be feasible, we obtain a lower and an upper bound on

every current source. However, (4.5) also provides similar bounds, hence, all the bounds should

be combined to obtain:

Imin ≤ I ≤ Imax (4.9)

Overall, we obtain a new feasible space of currents, that we call F , such that I ∈ F if and only

if, I satisfies (4.9) and (4.6).

Back to the example in the previous section, the resulting reduced set of constraints in the

current domain would be:

0.14

0.22

0.21

≤ I ≤

0.17

0.25

0.24

[

0.35

0.4

]

≤ SI =

[

1 1 0

0 1 1

]

I ≤[

0.41

0.48

]

Our goal is to look for the worst-case reliability of the power grid given all the possible feasible

combinations of I. For that, one can look into finding the worst-case MTF of the grid, or

the average worst-case TTF. Because our original vector-based engine uses Monte Carlo and

computes one grid TTF at a time, we will follow this up by also using a Monte Carlo approach

while computing a worst-case TTF of the grid, in every iteration, given all the constraints,

and finally report the average of all the obtained TTFs. This approach makes sense because

one would want to look into several samples of the grid and obtain the worst-case TTF for

each sample, and finally generate an average of all the minimums obtained. This leads to a

framework that allows vectorless EM checking while imposing reasonable and minimal demands

on the user.

4.3 Optimization

Over all the feasible vectors I ∈ F , we would like to find the average worst-case TTF of the grid,

which we do by performing a Monte Carlo analysis as before. In every iteration, we choose a

sample from the standard normal distribution for every line in the grid, and we find the smallest


grid TTF that can be obtained using the mesh model given any I ∈ F , and the set of samples

chosen for the lines. Recall that these samples are used to sample failure times for the lines

using equation (3.6) which in this case yields an expression for every TTF since I is not fixed.

Define Ψ to be the vector containing the samples ψl, l = 1 . . . b, and let T (Ψ, I) be a function

defined on F such that for every vector I ∈ F , T (Ψ, I) is the grid failure time corresponding

to the set of samples in Ψ and subject to the vector of source currents I. If Ψi represents the

vector containing the samples chosen at Monte Carlo iteration i, then the goal is to solve the

following set of optimization problems:

While (Monte-Carlo has not converged) :

Minimize: T (Ψi, I)

subject to: I ∈ F(4.10)

In the following, we discuss how to solve every minimization problem in the loop above given

a fixed vector Ψ. We do that by partitioning the feasible space into small subsets in which we

perform local optimizations. The global optimum will be the smallest of all the local optimums.

We first explain the local optimization in the first subset given a starting point in the feasible

space, and then show how to move into the other subsets.

4.3.1 Local Optimization

In this section, we will refer to T (Ψ, I) by T (I) for convenience. Let I(1) be a given point in

F and T (I(1)) be the corresponding grid time-to-failure. As will be explained later, we need

several initial points in F to solve every iteration in (4.10), and every initial point will lead to

a subset of F in which a local optimization will be performed. Therefore, a superscript is used

to index the initial points that will be chosen, as well as the corresponding subsets.

In order to compute T (I(1)), we need to compute the JL product of every line, filter

out all the lines that turn out to be EM-immune, and sort the other lines according to their

time-to-failure. Let l(1)1 , l

(1)2 , . . . , l

(1)ζ be the resulting sorted list of EM-susceptible lines, and

τ(1)1 , τ

(1)2 , . . . , τ

(1)ζ the corresponding list of TTFs such that:

τ(1)1 ≤ τ (1)2 ≤ . . . ≤ τ (1)ζ (4.11)

Also, let l(1)ζ+1, l

(1)ζ+2, . . . , l

(1)b be the list of all the other lines, i.e. the EM-immune ones.

Assume that, according to the order in (4.11), the grid fails for the first time after the

failure of the first p EM-susceptible lines. In other words, the grid is safe if l(1)1 , l

(1)2 , . . . , l

(1)p−1

fail, but fails when l(1)p fails. This implies that T (I(1)) = τ

(1)p as explained in the previous

chapter. Throughout the rest of this chapter, we assume that p < ζ, because otherwise, the

grid becomes immortal indefinitely which is unrealistic.


General Case

Definition 3. We define S(1) to be the subset of F corresponding to I(1) such that, at every

I ∈ S(1), the set of lines that fail until the failure time of the grid, their branch current directions,

the line that ultimately causes the grid to fail, and the EM-immune lines, are all the same as

those at I(1).

If, at any given point I ∈ S(1), the list of lines that fail before the failure of the grid is

l(I)1 , . . . , l

(I)p (in this order), the set of lines that are EM-susceptible but do not fail before the

failure of the grid is {l(I)p+1, . . . , l(I)ζ }, and the set of EM-immune lines is {l(I)ζ+1, . . . , l

(I)b }, then

we know, from the definition, that {l(I)1 , . . . , l(I)p } = {l(1)1 , . . . , l

(1)p }, l(I)p = l

(1)p , {l(I)p+1, . . . , l

(I)ζ } =

{l(1)p+1, . . . , l(1)ζ }, and {l

(I)ζ+1, . . . , l

(I)b } = {l

(1)ζ+1, . . . , l

(1)b }. Clearly, I(1) belongs to S(1) as all of the

conditions explained in the definition are satisfied.

For I ∈ F , assume that the TTF of line l(I)i , i ∈ {1, . . . , ζ}, can be written using (3.6) as:

τi(I) = |cTi I|−η

Let ξi = ±1 denote the sign of cTi I(1) for i ∈ {1, . . . , p}, i.e. ξi =

cTi I(1)

|cTi I(1)|, which implies

ξicTi I

(1) ≥ 0.

Claim 3. For every I ∈ S(1), T (I) = τp(I)

Proof. For any I ∈ S(1), l(I)p fails right after the failure of the set {l(I)1 , . . . , l(I)p−1} and is the first

to cause the failure of the grid. Therefore, T (I) = τp(I).

Claim 3 shows how to write T (I) as a closed form expression inside S(1) which is a non-empty

subset because I(1) ∈ S(1). To minimize T in F , we start by performing a local optimization in

a subset of F where T can be explicitly defined, namely S(1). In other words, we are interested

in solving the following optimization problem:

Minimize τp(I)

subject to I ∈ S(1)(4.12)

To solve (4.12), we introduce a new function νp on S(1) defined by νp(I) = (τp(I))− 1

η . Because

ξpcTp I

(1) ≥ 0, and because the direction of the current in line l(I)p (= l

(1)p ) is the same at both

I(1) and any I ∈ S(1), we have that ξpcTp I ≥ 0, ∀I ∈ S(1). Therefore, we can write νp(I) =

(τp(I))− 1

η =[

∣

∣cTp I∣

∣

−η]− 1

η=∣

∣cTp I∣

∣ = ξpcTp I, meaning, νp(I) is a linear function.

Lemma 1. A point Iopt is a solution for (4.12) if, and only if, νp(Iopt) is a maximum for νp(I)

in S(1).


Proof. Since η > 0 and τp(I) is a positive function for I ∈ S(1) (meaning νp(I) is a positive

function in S(1) as well), then:

Iopt is a solution for (4.12)⇔ τp(I) ≥ τp(Iopt), ∀I ∈ S(1)

⇔ (νp(I))−η ≥ (νp(Iopt))

−η, ∀I ∈ S(1)

⇔[

(νp(I))−η]− 1

η ≤[

(νp(Iopt))−η]− 1

η , ∀I ∈ S(1)

⇔ νp(I) ≤ νp(Iopt), ∀I ∈ S(1)

which proves the lemma.

Lemma 1 implies that in order to solve (4.12), we can solve instead the following maximiza-

tion problem

Maximize ξpcTp I

subject to I ∈ S(1)(4.13)

to get Iopt. The solution to (4.12) would simply be (ξpcTp Iopt)

−η.

For i ∈ {1, . . . , b}, let bi denotes the row of −R−1MTG−10 (from equation (2.33)) that

corresponds to line l(I)i , and let ai and Li denote the cross sectional area and length of l

(I)i

respectively. Also, define Gp to be the conductance matrix of the grid after the failure of

l(I)1 , . . . , l

(I)p , and Gp−1 the conductance matrix after the failure of l

(I)1 , . . . , l

(I)p−1 only. For now,

we assume that the failure occurs due to the violation of the voltage drop condition. The case

of failure by singularity is discussed later.

Theorem 1. I ∈ S(1) if, and only if, the following constraints are satisfied:

Li

aiξib

Ti I ≥ βc for i ∈ {1, . . . , p} (4.14)

Li

ai|bTi I| < βc for i ∈ {ζ + 1, . . . , b} (4.15)

ξpcTp I − ξicTi I ≤ 0 for i ∈ {1, . . . , p− 1} (4.16)

|cTi I| − ξpcTp I ≤ 0 for i ∈ {p+ 1, . . . , ζ} (4.17)

G−1p−1I ≤ Vth (4.18)

G−1p I 6≤ Vth (4.19)

Proof. Using definition 3, we know that for every I ∈ S(1), lines l(I)1 , . . . , l(I)p have the same

direction they have at I(1), therefore:

ξibTi I ≥ 0 and ξic

Ti I ≥ 0 for j ∈ {1, . . . , p} (4.20)

Also, lines l(I)1 , . . . , l

(I)p are EM-susceptible, meaning their JL products are greater than βc.


Using (3.2), this can be written as:

|bTi I|ai

Li ≥ βc, i = 1, . . . , p (4.21)

Similarly, lines l(I)ζ+1, . . . , l

(I)b are EM-immune, meaning we can write:

|bTi I|ai

Li < βc, i = ζ + 1, . . . , b

which is identical to (4.15).

Moreover, l(I)p fails after lines {l(I)1 , . . . , l

(I)p−1}, and before all the other lines. This is equivalent

to:

maxj∈{1,...,p−1}

τj(I) ≤ τp(I) ≤ mink∈{p+1,...,ζ}

τk(I)

which is also equivalent to:

maxj∈{1,...,p−1}

|cTj I|−η ≤ |cTp I|−η ≤ mink∈{p+1,...,ζ}

|cTk I|−η (4.22)

The grid is safe after the failure of the lines in the set {l(I)1 , . . . , l(I)p−1}. This is equivalent to (4.18)

because Gp−1 is defined to be the conductance matrix of the grid after the failure of those lines.

The grid fails after the failure of the lines in the set {l(I)1 , . . . , l(I)p }. This is equivalent to (4.19)

because Gp is defined to be the conductance matrix of the grid after the failure of those lines.

It remains to show that:

(4.20), (4.21), and (4.22)⇔ (4.14), (4.16), and (4.17)

We do this using a two way proof. Assume (4.20), (4.21), and (4.22) are true, then (4.21)

implies (4.14) because |bTi I| = ξibTi I for i = 1, . . . , p. Also, (4.22) implies:

maxj∈{1,...,p−1}

(

ξjcTj I)−η ≤

(

ξpcTp I)−η ≤ min

k∈{p+1,...,ζ}|cTk I|−η (4.23)

By taking the(

− 1η

)thpower of all the terms of (4.23), we can write

maxk∈{p+1,...,ζ}

|cTk I| ≤ ξpcTp I ≤ minj∈{1,...,p−1}

ξjcTj I (4.24)

which implies (4.16) and (4.17). This is true because − 1η < 0 and hence, taking the

(

− 1η

)th

power reverses all the inequalities in which case, the min operator becomes a max, and vice

versa.

On the other hand, assume (4.14), (4.16), and (4.17) are true, then (4.24) is true. Since


maxk∈{p+1,...,ζ}

|cTk I| ≥ 0, then

0 ≤ ξpcTp I ≤ minj∈{1,...,p−1}

ξjcTj I

and hence, (4.20) is true. In addition, we can take the (−η)th power of all the terms in (4.24)

to get (4.23) because −η < 0. Now, we can easily get (4.22) from (4.23) because (4.20) is true.

Finally, (4.21) is also true because of (4.20) and (4.14).

Notice that (4.15) is equivalent to:

Li

aibTi I < βc and − Li

aibTi I < βc for i ∈ {ζ + 1, . . . , b} (4.25)

which can be written in matrix form as

H1I < h1 (4.26)

where H1 is a 2(b− ζ)× n matrix and whose rows are the row vectors

Li

aibTi and − Li

aibTi , i ∈ {ζ + 1, . . . , b}

Call γ1 the number of rows in H1 (γ1 = 2(b − ζ)). The vector h1 is the vector of size γ1

containing βc in all its entries.

Similarly, the inequalities in (4.14), (4.16), (4.17), and (4.18) can be combined in matrix

form as

H2I ≤ h2 (4.27)

with H2 being the (n+ 2ζ − 1)× n matrix whose rows are the row vectors

−Li

aiξib

Ti , i ∈ {1, . . . , p}

ξpcTp − ξicTi , i ∈ {1, . . . , p− 1}

cTi − ξpcTp and − cTi − ξpcTp , i ∈ {p+ 1, . . . , ζ}

and the rows of G−1p . Call γ2 the number of rows in H2 (γ2 = n+ 2ζ − 1). The vector h2 is of

size γ2 and is the following:

h2 =

−βc1p02ζ−p−1

Vth


Ultimately, S(1) can be redefined using the following set of constraints:

I ∈ FH1I < h1

H2I ≤ h2

G−1p I 6≤ Vth

All the constraints presented above are linear and define a convex polytope (or the interior of

a convex polytope) except G−1p I 6≤ Vth which consists of a disjunction of constraints where at

least one entry of G−1p I has to be greater than its corresponding entry in Vth. We deal with

that using a theorem that will be presented shortly.

For any strictly positive number δ define a fixed real number d = ‖Vth‖∞(1 + δ).

Theorem 2. For any I, G−1p I 6≤ Vth if and only if, ∃y ∈ {0, 1}n with ‖y‖1 ≤ n − 1 such that

G−1p I > Vth − dy.

Proof. If G−1p I 6≤ Vth, then there exists a non-empty set of indices K ⊆ {1, . . . , n} such that

eTkG−1p I > eTk Vth for every k ∈ K. If we let y = [y1 . . . yn]

T with yk = 0 for k ∈ K, and yk = 1

otherwise, then clearly ‖y‖1 ≤ n− 1, and G−1p I > Vth− dy because d > ‖Vth‖∞ and the entries

of the vector G−1p I are always positive.

On the other hand, if there exists y ∈ {0, 1}n with ‖y‖1 ≤ n − 1, and G−1p I > Vth − dy,

then ∃k such that yk = 0, and eTkG−1p I > eTk Vth. Therefore, G

−1p I 6≤ Vth.

The theorem above allows rewriting (4.13) as follows:

Maximize ξpcTp I

subject to I ∈ FH1I < h1

H2I ≤ h2

G−1p I > Vth − dy‖y‖1 ≤ n− 1

y ∈ {0, 1}n

(4.28)

The problem above is an integer linear program (ILP) because it has a linear objective function,

linear constraints, and some integer variables, namely the entries of y; solving this ILP would

solve (4.12) as explained before, and would minimize T inside S(1) ⊂ F .

Singular Case

If at I(1) the grid fails by singularity, then the same analysis as above can be done. The only

difference is that the constraint G−1p I 6≤ Vth cannot be added, and is in fact redundant because


(2)

F

S(1)

I

I

(1)

Figure 4.1: Choosing the next starting point I(2)

Gp is known to be singular in this case, and there is no need to add that as one of the constraints

defining S(1). Notice that in this case, (4.13) becomes a linear program:

Maximize ξpcTp I

subject to I ∈ FH1I < h1

H2I ≤ h2

(4.29)

In the following, we show how to globally minimize T in F by performing a set of local

optimizations as above until F is fully explored.

4.3.2 Exact Global Optimization

Similarly to S(1), every local optimization requires a starting point in F . In order to create S(2),we need a point I(2) ∈ F at which we compute the TTF of the grid and then follow a similar

procedure to the one explained in the previous section. Note that finding I(1) can be done

by solving a linear feasibility problem in the set F . However, we cannot do the same for I(2)

because if we choose I(2) in F without other restrictions and it turns out that it belongs to S(1),then S(2) becomes identical to S(1), which adds redundancy to our approach and accordingly,

the global optimization may or may not terminate. In short, I(2) should be chosen in the set

F −S(1) (See figure 4.1). Using the constraints in theorem 1, we can infer the set of conditions


required for I(2) to be in outside S(1) (in the general case) as follows:

H1I 6< h1

or H2I 6≤ h2

or G−1p I ≤ Vth

(4.30)

For any strictly positive number δ, define

d = (1 + δ)max(

‖h1 −H1I‖∞, ‖h2 −H2I‖∞, ‖G−1p I − Vth‖∞

)

Theorem 3. (4.30) is true if and only if, there exists x ∈ {0, 1}γ1, y ∈ {0, 1}γ2, and z ∈ {0, 1}with

‖x‖1 + ‖y‖1 + z ≤ γ1 + γ2 (4.31)

such that:H1I ≥ h1 − dxH2I > h2 − dy

G−1p I ≤ Vth + dz1n

(4.32)

Proof. The proof is similar to that of theorem 2. Assume (4.30) is true, and define x =

[x1 . . . xγ1 ]T , y = [y1 . . . yγ2 ]

T , and z, such that:

xk = 0 ⇐⇒ eTkH1I ≥ eTk h1, for k = 1, . . . , γ1

yk = 0 ⇐⇒ eTkH2I > eTk h2, for k = 1, . . . , γ2

z = 0 ⇐⇒ G−1p I ≤ Vth

Also, let v = [xT yT z]T . Because (4.30) is true, then at least one entry in v is zero, since

otherwise, we have that H1I < h1, H2I ≤ h2, and G−1p 6≤ Vth, which makes (4.30) false. Having

at least one zero entry in v means that there exists a nonempty set of indices K ⊆ {1, . . . , γ1 +γ2+1} such that vk = 0 for k ∈ K. Clearly, (4.31) is true because ‖x‖1+‖y‖1+z = ‖v‖1 ≤ γ1+γ2.We still have to check that (4.32) is true.

1. Define K1 = K ∩ {1, . . . , γ1}. We will first show that

H1I ≥ h1 − dx (4.33)

i.e. eTkH1I ≥ eTk h1 − dxk for k ∈ {1, . . . , γ1}. If k ∈ K1, then we know eTkH1I > eTk h1,

meaning eTkH1I > eTk h1 − dxk (because xk = 0). If k ∈ {1, . . . , γ1} − K1, then eTkH1I >

eTk h1 − dxk because xk = 1, and d > ‖h1 −H1I‖∞ (i.e. d ≥ eTk (h1 −H1I) for every k).

2. Define K′2 = K ∩ {γ1 + 1, . . . , γ1 + γ2}, and K2 = {k − γ1 : k ∈ K′

2}. We will now show

that

H2I > h2 − dy (4.34)


i.e. eTkH2I > eTk h2 − dyk for k ∈ {1, . . . , γ2}. If k ∈ K2, then we know eTkH2I > eTk h2,

meaning eTkH2I > eTk h2 − dyk (because yk = 0). If k ∈ {1, . . . , γ2} − K2, then eTkH2I >

eTk h2 − dyk because yk = 1, and d > ‖h2 −H2I‖∞ (i.e. d > eTk (h2 −H2I) for every k).

3. We will finally show that:

G−1p I ≤ Vth + dz1n (4.35)

i.e. eTkG−1p I ≤ eTk Vth + dz for k ∈ {1, . . . , n}. If z = 0, then we know G−1

p I ≤ Vth which

means that (4.35) is automatically true. If z = 1, then eTkG−1p I ≤ eTk Vth+ dz is true (and

so is (4.35)) because d > ‖G−1p I − Vth‖∞ (i.e. d ≥ eTk (G−1

p I − Vth) for every k).

We now prove the other direction of the theorem. Assume that there exists x ∈ {0, 1}γ1 ,y ∈ {0, 1}γ2 , and z ∈ {0, 1} with (4.31) and (4.32) being true, then either z = 0, or ∃k such that

xk = 0 or ∃k such that yk = 0. If z = 0, then G−1p I ≤ Vth, and if xk = 0, then eTkH1I > eTk h1

and if yk = 0 then eTkH2I > eTk h2, which basically implies (4.30).

In the singular case, the last set of constraints in (4.32) as well as the binary variable z are

not needed. Theorem 3 implies that finding I(2) requires solving a feasibility problem in the

following space:

I ∈ FH1I ≥ h1 − dxH2I > h2 − dy

G−1p I ≤ Vth + dz1n

‖x‖1 + ‖y‖1 + z ≤ γ1 + γ2

x ∈ {0, 1}γ1 , y ∈ {0, 1}γ2 , z ∈ {0, 1}

which can be done using an ILP. The same approach should be used to find the ith starting

point corresponding to subset S(j): I(i) should be chosen as to satisfy the constraints I ∈ Fand I 6∈ S(j), j = 1, . . . , i− 1. This can also be done using an ILP similarly to I(2). When such

point cannot be found, i.e. when F −⋃i−1j=1 S(j) becomes empty, we infer that the feasible space

F is fully explored and the algorithm terminates while returning the best local minimum found.

The result is one sample TTF for the grid which should be added to the other samples found

in other Monte Carlo iterations. Algorithm 2 shows how to compute exact global minimum of

the grid TTF given a set of normal samples Ψ using the proposed approach.


Algorithm 2 has been implemented in C++. The algorithm use the Mosek optimization pack-

age [29] to solve the required LPs and ILPs. An approximate sparse inverse of G0 is found

using SPAI [30], and all the other required inverses are found using the Woodbury formula,

i.e. (3.14). We carried out several experiments using 5 different power grids generated as per


Algorithm 2 EXACT GLOBAL MINIMIZATION

Input: ΨOutput: Global Minimum of T (Ψ, I)1: Find I(1) in F using an LP2: Set solved← false and i← 23: while (solved = false) do4: Find T (Ψ, I(i−1)) using Algorithm 15: Build the constraints defining S(i−1) as outlined in section 4.3.16: Solve maxI∈S(i−1) T (I) using an ILP

7: Solve a feasibility problem in F −⋃i−1j=1 S(j) to get I(i) as outlined in section 4.3.2

8: if (I(i) cannot be found) then9: solved← true

10: end if

11: i← i+ 112: end while

13: Return the smallest local minimum found

Table 4.1: Exact average minimum TTF computation

Power Grid Proposed Approach

Name Nodes C4’s SourcesAvg Min CPUTTF (yrs) Time

G1 141 12 12 7.50 3.21 minG2 177 12 12 9.11 5.94 minG3 237 12 20 10.38 25.10 minG4 392 24 30 8.01 49.24 minG5 586 12 42 9.79 4.42 h

user specifications, including grid dimensions, metal layers, pitch and width per layer. Supply

voltages and current sources were randomly placed on the grid. The parameters of the grids

are consistent with 1.1V 65nm CMOS technology. As for the EM model employed, we use,

as before, an activation energy of 0.9eV , a current exponent η = 1, a nominal temperature

Tm = 373K, a critical Blech product βc = 3000A/cm, and a standard deviation σln = 0.3.

All the experiments were carried out on a 2.6GHz Linux machine with 24GB of RAM. We

compute the average minimum time-to-failure of the grid and report the required CPU time

for every grid. The Monte Carlo parameters we use for that are ǫ = 0.1 and α = 0.05 for which

zα/2 = 1.96. Table 4.1 shows the test grids with the number of nodes, the number of voltage

sources (C4s), and the number of current sources indicated in each case. The obtained average

minimum grid time-to-failure for each grid are reported. The CPU time required is reported

as well and is shown in figure 4.2. By investigating the runtime of the different parts of our

algorithm, it turns out that most of the total CPU time is spent on selecting starting points

required to generate the subsets. It should also be noted that the number of subsets that were


100 150 200 250 300 350 400 450 500 550 6000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Number of Nodes

Tot

al C

PU

Tim

e (h

rs)

Figure 4.2: CPU time of the exact approach versus the number of grid nodes

found in F is on average, between 20 and 40 subsets depending on the structure of the feasible

space as well as on the random seeds used for selecting the TTFs of the grid lines.

4.5 Conclusion

We described an early vectorless approach for power grid electromigration checking under a

constraint-based framework to capture workload uncertainty. We presented an exact, theo-

retically interesting approach which requires solving several ILPs. With proper parallelization

the exact optimization may become practical to check for electromigration in the main feeder

network or in parts of the whole grid, but for all practical purposes, the method is of theoretical

interest only and is not scalable.

Chapter 5

Simulated Annealing Based


5.1 Introduction

As mentioned in chapter 4, vectorless electromigration checking is required to manage user

uncertainties about the chip workload. The approach we developed is exact, and is useful for

small grids. However, for large grids containing hundreds of thousands of nodes, the method

becomes impractical to use as it requires solving several ILPs in every Monte Carlo iteration.

In this chapter, we present three approximate approaches for constraint-based power grid elec-

tromigration checking. All the methods are based on Simulated Annealing, a heuristic based

global optimization technique. The first method uses the TTF estimation technique developed

in chapter 3 as well as the local optimizer developed in chapter 4. The second method uses the

TTF estimation technique developed in [24], which is an extension to our original estimation

technique (from chapter 3) and that takes care of changing branch currents as grid lines start to

fail. The third method uses an extension to the TTF estimation technique of [24], and explores

locality in the grid.

5.2 Simulated Annealing for Continuous Problems

5.2.1 Overview

Simulated Annealing (SA) is a random-search global optimization techniques that was first

developed by Kirkpatrick et al [31] and Cerny [32]. SA was found to be useful to solve many

VLSI CAD problems. SA is often used when the search space is discrete but can also be used for

continuous global optimization problems. In this chapter, we are concerned with optimization

over continuous variables as our feasible space is a continuous domain of currents, i.e. we are

60

Chapter 5. Simulated Annealing Based Electromigration Checking 61

concerned with problems having the following form:

f∗ = minx∈X

f(x) (5.1)

where X ⊆ Rn is a continuous compact domain. SA algorithms are based on an analogy with

annealing in metallurgy, a technique that involves cooling and heating a material as to increase

the size of its crystals and reduce their defects. If the temperature is slowly decreased, the

thermal mobility of the molecules is reduced and they form a pure crystal that corresponds to

the state of minimum energy. If the temperature is decreased quickly, a liquid metal rather

ends up in a polycrystalline or amorphous state with a higher energy and not in a pure crystal.

In [33], a Metropolis Monte Carlo method was proposed to simulate the physical annealing

process. Later, SA algorithms for combinatorial optimization problems were developed by

observing the analogies between the configurations of a physical system and the feasible points,

and between the energy function and the objective function. The approach has been later

extended to continuous global optimization problems.

5.2.2 Main Algorithm

At each iteration, SA algorithms generate a candidate point and decide whether to move to it

or to stay at the current point based on a random mechanism controlled by a parameter called

temperature. The flow of the algorithm is as follows:

• Step 0: Choose x0 ∈ X and let z0 = {x0} and k = 0.

• Step 1: Sample a point yk+1 from the next candidate distribution D(zk).

• Step 2: Sample a number p between 0 and 1 from the uniform distribution and set:

xk+1 =

{

yk+1 if p ≤ A(xk, yk+1, Tk)

xk otherwise(5.2)

where A is called the acceptance function and has values in [0, 1], and Tk is a parameter

called the temperature at iteration k.

• Step 3: Set zk+1 = zk ∪{yk+1}. The set zk contains all the information collected up to

iteration k, i.e. all the points observed up to this iteration.

• Step 4: Set Tk+1 = U(zk+1, T0), where U is called the cooling schedule and is a function

with nonnegative values.

• Step 5: Check a stopping criterion and if it fails, set k ← k + 1 and go back to step 1.


In order to define a complete SA algorithm, one should appropriately choose the distribution

D of the next candidate point, the acceptance function A, the cooling schedule U , and the

stopping criterion. Below, we present a short discussion about each component separately.

5.2.3 The Acceptance Function

Very few acceptance functions have been employed in the existing literature about SA for

continuous optimization problems. The acceptance function used in most cases is the Metropolis

function presented below:

A(x, y, T ) = min

{

1, exp

(

−f(y)− f(x)T

)}

(5.3)

Notice that the condition p ≤ A(xk, yk+1, Tk) in step 2 of the SA algorithm is automatically

satisfied if f(yk+1) ≤ f(xk) because in this case, A(xk, yk+1, Tk) = 1. Accordingly, xk+1 = yk+1.

In the case where f(yk+1) ≥ f(xk), the Metropolis function generates a number between 0 and

1 representing the probability of accepting the next candidate point yk+1. This probability

depends on the temperature parameter Tk and on how large is the gap |f(yk+1) − f(xk)|.Notice that, a large gap or a low temperature result in a low acceptance probability. Accepting

an ascent step from f(xk) to f(yk+1) is sometimes necessary to avoid being trapped at a local

minimum, and is called hill-climbing.

Another possible acceptance function is the Barker function:

A(x, y, T ) = 1

1 + exp(

f(y)−f(x)T

) (5.4)

Notice that the Barker function may not accept descent steps if they don’t improve the function

value by a significant amount. Nonetheless, at low T , descent steps are most likely accepted.

5.2.4 Cooling Schedule

An appropriate choice of the cooling schedule U is critical for a well performing SA algorithm.

Good cooling schedules generally depend on the value of f∗ (the optimal objective), in which

case the purpose of SA is finding a point x∗ ∈ X having a (near) optimal objective. In [34], the

cooling schedule is defined as follows:

U(zk, T0) = β [f(xk)− f∗]g1 (5.5)

where β, g1 > 0 are constants. If f∗ is not known, it is suggested to employ an estimate f of it

which is updated every time a function value lower than the estimate is found. The basic idea

of the schedule above is that ascent steps should be accepted with a low probability when the


function value at the current point xk is close to the global optimum. Instead, if the function

value at xk is much greater than the global optimum, the temperature is high and ascent steps

are accepted to prevent the algorithm from getting trapped away from the global optimum.

In the case where no estimate f can be obtained, one can simply use:

Tk+1 = a⌊ kM ⌋T0 (5.6)

where T0 is the initial temperature, a is a real number between 0.8 and 0.99, and M is an

integer. This cooling schedule allows the temperature to decrease by a factor of a after each

group of M iterations.

Notice that any cooling schedule must take into account the scale of the function in consid-

eration. For example, assume that Tk = 1, f(xk) = 0, and f(yk+1) = 0.1, then the probability

that xk+1 = yk+1 is equal to e−0.1 ≈ 0.9 (according to the Metropolis function). For another

function g(x) = 1000 × f(x), g(xk) = 0, and g(yk+1) = 100, and hence the probability that

xk+1 = yk+1 is equal to e−100 ≈ 0. Therefore, we notice how poorly the SA algorithm behaves

when the temperature does not take into account the scale of the function.

To solve this problem, one can compute the objective function at one or more points in

the feasible space. A good starting temperature would be 5-10 times larger than the observed

function values.

5.2.5 Next Candidate Distribution

Because the feasible space is a continuous domain, there are infinitely many possible next

candidate points. The point starting point x0 can be found by solving a feasibility problem

inside the space X (which can be done by solving an LP in the case where X is a convex

polytope). Choosing other points in X, however, must be done in a random and more efficient

way. In [34] the next candidate point yk+1 is generated as follows:

yk+1 = xk +∆rθk (5.7)

where θk is a random direction in Rn with ‖θk‖2 = 1, and ∆r is a fixed step size. If the point

obtained is outside X, another point is generated using the same procedure. The choice of ∆r

usually depends on the objective function f and on the volume of the feasible space, and bad

choices may lead to a deterioration of the performance of the algorithm. Moreover, when the

dimension n is high, the probability of obtaining a point inside the feasible space becomes low,

and one might need to repeat the procedure above many times before obtaining a point inside

X.

A better approach explained in [35] generates yk+1 by first choosing θk as above, and then


generating a random point λk in the set:

Λk = Λk(θk) = {λ : xk + λθk ∈ X} (5.8)

The next candidate point yk+1 is then obtained as follows:

yk+1 = xk + λkθk (5.9)

Such yk+1 is guaranteed to be inside X. This two-phase generation approach has the advantage

that it does not need an acceptance/rejection mechanism as before. The only drawback is that

we are required to find Λk, and this might be a difficult task depending on the structure of X.

In the simple case where X is a convex polytope (which is exactly the case of our feasible space

F of currents), Λk can be computed exactly. Assume that X can be written as follows:

X = {x : aTi x ≤ bi, i = 1, 2, . . . ,m} (5.10)

and assume that we are given a point xk ∈ X and a unit vector θk. We are after the values

of λ for which xk + λθk ∈ X. For that, we compute the intersection between the line formed

by the set of points xk + λθk, and the boundary of X, i.e. the hyperplanes aTi x = bi, i =

1, 2, . . . ,m}. To find the intersection with the hyperplane aTi x = bi, we simply find λi for which

aTi (xk + λiθk) = bi, i.e. aTi xk + λia

Ti θk = bi. This gives:

λi =bi − aTi xkaTi θk

(5.11)

Because xk ∈ X, we know that bi − aTi xk ≥ 0, and hence the sign of λi depends on the sign of

the dot product aTi θk:

• If aTi θk ≥ 0, then λi ≥ 0, and hence, for any λ ≤ λi, we have λaTi θk ≤ λiaTi θk, i.e.

aTi (xk+λθk) ≤ aTi (xk+λiθk) = bi, meaning xk+λθk belongs to the halfspace {x : aTi x ≤bi}

• If aTi θk ≤ 0, then λi ≤ 0, and hence, for any λ ≥ λi, we have λaTi θk ≤ λiaTi θk, i.e.

aTi (xk+λθk) ≤ aTi (xk+λiθk) = bi, meaning xk+λθk belongs to the halfspace {x : aTi x ≤bi}

• If aTi θk = 0, then the hyperplane {x : aTi x = bi} should be discarded because it is parallel

to the direction θk.

Let λmin = max{λi : λi ≤ 0, i = 1, 2, . . . ,m} and λmax = min{λi : λi ≥ 0, i = 1, 2, . . . ,m}.Following the reasoning above, any λ between λmin and λmax must generate a point yk+1 that

belongs to X. Notice that we must have λmin ≤ 0 ≤ λmax because λ = 0 leads to yk+1 = xk


| |X

λ

λ

xk

θk

max

min

λ

yk+1

|| |

Figure 5.1: Generating lambda

which belongs to X. Ultimately, the set Λk can be described as follows:

Λk = {λ : λmin ≤ λ ≤ λmax} (5.12)

The procedure above is illustrated in figure 5.1. Even though this two-phase mechanism

presents a huge advantage over the first approach, it still presents a major drawback known as

the jamming problem. Jamming occurs when the point xk is very close to a boundary of the

feasible region. In this case, the set Λk is very small along many directions θk. To see why,

consider a hypercube in Rn and assume that xk is very close to one of its corners. The fraction

of the whole set of directions leading to small set Λk is about 1− 12n (The only viable directions

are the ones away from the corner, and these roughly represent 12n of all the directions). When

the set Λk is small, the next candidate point will be very close to xk, and hence a small progress

will be observed in the algorithm. To solve this problem, the concept of reflection is introduced

in [36]. The basic idea of reflection consists of generating points outside X and reflect them

back into X. Let X be some compact set containing X. We define the set Λk as follows:

Λk = Λk(θk) = {λ : xk + λθk ∈ X} (5.13)

We then sample a uniform random point λk from Λk, and obtain the point:

yk+1 = xk + λkθk (5.14)


θθ

k

k

xk

yk +1

yk +1

X

X

y+1k

~

Figure 5.2: One way of reflecting yk+1 back into X to obtain yk+1 = yk+1

If yk+1 ∈ X, then we simply set yk+1 = yk+1 to obtain our next candidate point. Otherwise, we

reflect yk+1 back into X by first finding the point yk+1, intersection between the line segment

[xk, yk+1] and the boundary of X (which can be easily done as before when X is a convex

polytope), and then computing the point y′k+1 as follows:

y′k+1 = yk+1 + ‖yk+1 − yk+1‖θk (5.15)

where θk is a random vector in Rn with ‖θk‖ = 1. If y′k+1 ∈ X, then we set yk+1 = y′k+1,

otherwise, we reflect y′k+1 back into X. Figure 5.2 illustrates the procedure explained.

5.2.6 Stopping Criterion

Due to the difficult nature of the problems solved by SA algorithms, it is hard to define a

stopping criterion which guarantees a global optimum within a given accuracy. Typically, one

of the following rules is applied:

• A given minimum temperature has been reached.

• A certain number of iterations has passed without accepting a new solution.

• A specific number of total iterations has been executed.


5.3 Simulated Annealing with Local Optimization

As our first approximate approach, we propose using SA in the context of a Monte Carlo simu-

lation to find the average minimum grid TTF of the grid. We use SA to minimize the function

T (Ψi, I) (defined in section 4.3) over the feasible space F in every Monte-Carlo iteration. We

use the Metropolis function as our acceptance function, and use the cooling schedule in (5.6).

To sample random points in F , we use the two phase generation mechanism presented earlier

as well as reflection for a better space exploration. Our SA algorithm converges once the tem-

perature reaches a certain minimum value Tǫ. Once SA terminates, we run the local optimizer

we developed in section 4.3.1 at the best point found. However, instead of solving an ILP, we

solve a relaxed version by allowing the entries of the vector y (in (4.28)) to be in the range

[0, 1] instead of the set {0, 1}. The result is an estimate sample for the minimum grid TTF. As

before, enough samples should be collected until Monte Carlo converges. Algorithm 3 shows

the details of our first approximate approach to minimize the grid TTF given a set of normal

samples Ψ using SA and the local minimizer.

Algorithm 3 SIMULATED ANNEALING WITH LOCAL OPTIMIZATION

Input: ΨOutput: Global Minimum of T (Ψ, I)1: Find a starting point I0 in F using an LP2: Compute T (Ψ, I0) using Algorithm 13: Set k ← 0, and choose an initial temperature T0.4: while (Tk+1 ≥ ǫ) do5: Sample a new point I ′k+1 in F as explained in section 5.2.56: Find T (Ψ, I ′k+1) using Algorithm 17: Find Ik+1 based on the acceptance function (5.3) as in (5.2)8: Find Tk+1 using (5.6)9: Set k ← k + 1

10: end while

11: Solve a local minimization around the best point found using a convex relaxation of theILP and return the result (procedure of section 4.3.1).

5.4 Optimization with Changing Currents

5.4.1 Estimating EM Statistics for Step Currents

Recall (from section 3.3.3) that when developing the approach for estimating the EM statistics

of the power grid when the chip workload is known exactly, we assumed that the statistics of

the individual lines can be determined using the branch currents of the grid before the failure

of any of its components. This basically means that, as the grid lines start to fail, we ignore

the changes in the branch currents and the effect of these changes on the time-to-failure of


the lines. In [24], the authors developed an extension to this approach in which the change in

failure statistics is estimated when the current densities in the lines change over time. Below,

we summarize the key points in their approach.

Consider a specific metal line of length L in the power grid subject to the following current

density profile:

J(t) =

J0 for t0 ≤ t ≤ t1J1 for t1 < t ≤ t2...

Jk for tk < t ≤ tk+1

...

Jp for tp < t < tp+1

(5.16)

where Jk−1 6= Jk ∀k > 0, t0 = 0, and tp+1 = ∞. It is interesting to note that (5.16) is the

typical current density profile of a surviving interconnect in the power grid, where the kth

failing interconnect has τ = tk. Let Tk represent the random variable describing the statistics

of the line for the time span tk < t ≤ tk+1. The RV Tk is defined over [0,∞] but describes the

statistics of line only for t ∈ (tk, tk+1]. Let µT,k = E[Tk], and let τk represent the time-to-failure

sample of the line for the same time span tk < t ≤ tk+1. Under certain mild assumptions, and

ignoring the Blech effect, the authors show that the following holds:

(

µT,k

µT,k-1

)

=

(

Jk−1

Jk

)η

(5.17)

which then leads to the following formula to update the TTF sample of the line:

τk = tk + (τk−1 − tk−1)

(

Jk−1

Jk

)η

(5.18)

If the Blech effect is considered, then two cases arise. If JkL ≤ βc, then τk = ∞. Otherwise,

the formula to update the TTF sample becomes as follows:

τk = tk + (τk−q − tk−q+1)

(

Jk−q

Jk

)η

(5.19)

where q is such that Jk−qL > βc, JkL > βc, and JiL ≤ βc for i ∈ {k − q + 1, . . . , k − 1}.

In order to find the TTF of a grid using the changing currents model, the authors follow

the steps of Algorithm 4. They start with a list L of all the grid lines to which they assign TTF

samples (they assign ∞ to immortal lines). Every time a line fails, the TTF of the remaining

lines in L are updated as explained above. Procedure 1 of chapter 3 is also used to update the

voltage drops as before (Woodbury formula and Banachiewicz-Schur).


Algorithm 4 FIND GRID TTF WITH CHANGING CURRENTS

Input: V0, G0, LOutput: τm1: Z0 ← [ ],W−1

0 ← [ ], x0 ← [ ], y0 ← [ ], grid singular ← 0, k ← 12: while (Vk ≤ Vth and grid singular = 0) do3: Find line lk ∈ L with lowest TTF and its conductance stamp ∆Gk

4: Find uk such that ∆Gk = uTk uk.5: (Vk,Zk,W

−1k , xk, yk, grid singular) ← FIND VK (V0,G0, uk,Zk−1,W

−1k−1, xk−1, yk−1, k)

6: L ← L− lk7: Update the TTFs of the lines in the set L as outlined in section 5.4.1.8: k ← k + 19: end while

10: Assign to τm the TTF of line lk.11: return τm

5.4.2 Optimization

Let Tc(Ψ, I) be a function defined on F such that for every vector I ∈ F , T (Ψ, I) is the

grid failure time, computed using Algorithm 4, corresponding to the set of samples in Ψ and

subject to the vector of source currents I. As our second approximate approach to finding the

average minimum grid TTF, we propose using SA again by minimizing the function Tc insteadof the function T . We use the same acceptance function, cooling schedule, next candidate

distribution, and stopping criterion as before. Every minimization leads to an estimate sample

for the minimum grid TTF. Enough samples must be collected until Monte Carlo converges.

Algorithm 5 shows the details of our second approximate approach to minimize the grid TTF

given a set of normal samples Ψ using SA. Notice that we do not perform an additional local

optimization step as in the previous method because the TTF of the lines are no longer fixed

as before and hence it is not easy to capture their order in the form of linear constraints.

Algorithm 5 SIMULATED ANNEALING WITH CHANGING CURRENTS

Input: ΨOutput: Global Minimum of Tc(Ψ, I)1: Find a starting point I0 in F using an LP2: Compute Tc(Ψ, I0) using Algorithm 43: Set k ← 0, and choose an initial temperature T0.4: while (Tk+1 ≥ Tǫ) do5: Sample a new point I ′k+1 in F as in section 5.2.56: Find Tc(Ψ, I ′k+1) using Algorithm 47: Find Ik+1 based on the acceptance function (5.3) as in (5.2)8: Find Tk+1 using (5.6)9: Set k ← k + 1

10: end while

11: Return the best grid TTF found.


Table 5.1: Speed and accuracy comparison between the first Simulated Annealing basedmethod and the exact solution of chapter 4

Power Grid Exact Solution Simulated Annealing

ErrorName Nodes

Avg Min CPU Avg Min CPUTTF (yrs) Time TTF (yrs) Time

G1 141 7.50 3.21 min 7.39 2.12 min -1.47%G2 177 9.11 5.94 min 9.54 2.13 min 4.72%G3 237 10.38 25.10 min 9.91 2.44 min -4.53%G4 392 8.01 49.24 min 7.93 2.69 min -1.00%G5 586 9.79 4.42 h 10.05 2.58 min 2.66%

5.5 Optimization with Selective Updates

When estimating the EM statistics for step currents, the most computationally expensive steps

are updating the voltage drops and updating the TTFs after the failure of each line. In this

section, we describe a new approach for grid TTF estimation based on locality. The idea is

fully explained in [37] and emerges from observing that not all the nodes are equally impacted

when a line in the grid fails. In fact, only the nodes located in the immediate neighborhood of

the failing line are significantly impacted. This locality can be exploited to update the TTFs

of only a selection of the grid lines in order to speed-up the TTF estimation process at the cost

of some loss in accuracy.

Let N = {1, 2, . . . , n} denotes the set of all the nodes in the power grid. Also, let∂V[k]

∂Rdenotes the change in the voltage drop of node k with respect to the failure of interconnect R.

Define the set NR as follows:

NR =

{

k ∈ N :∂V[k]

∂R> δv

}

(5.20)

where δv is a user defined threshold. Finding NR can be done by checking which nodes in the

grid presented a change in their voltage drop larger than δv after the failure of R. The proposed

algorithm updates only the TTFs of the lines connected to the nodes in NR. A smaller value of

δv improves the accuracy of the estimated TTF at the cost of reduction in speed, and vice-versa.

As before, let Tl(Ψ, I) be a function defined on F such that for every vector I ∈ F , Tl(Ψ, I)is the grid failure time computed using Algorithm 4 but modified as to perform a selective TTF

update as explained above, corresponding to the set of samples in Ψ, and subject to the vector

of source currents I. As our third approximate approach to finding the average minimum grid

TTF, we use SA by minimizing the function Tl. Algorithm 6 shows the details of the proposed

approach. No additional local optimization is done here as well for the same reasons explained

in the previous section.


Algorithm 6 SIMULATED ANNEALING WITH SELECTIVE UPDATES

Input: ΨOutput: Global Minimum of Tc(Ψ, I)1: Find a starting point I0 in F using an LP2: Compute Tc(Ψ, I0) using Algorithm 43: Set k ← 0, and choose an initial temperature T0.4: while (Tk+1 ≥ Tǫ) do5: Sample a new point I ′k+1 in F as in section 5.2.56: Find Tc(Ψ, I ′k+1) using Algorithm 4 but modified as to perform a selective TTF update

as explained in section 5.57: Find Ik+1 based on the acceptance function (5.3) as in (5.2)8: Find Tk+1 using (5.6)9: Set k ← k + 1

10: end while

11: Return the best grid TTF found.

Table 5.2: Comparison of power grid average minimum TTF and CPU time for thethree Simulated Annealing based methods

Power SA with SA with SA with

Grid Local Opt Changing Currents Locality

Name NodesAvg.

CPU TimeAvg.

CPU TimeAvg.

CPU TimeMin TTF Min TTF Min TTF

G6 9K 15.14 yrs 17.4 min 15.05 yrs 9.6 min 15.26 yrs 8.0 minG7 19K 14.20 yrs 66.0 min 13.37 yrs 57.6 min 14.13 yrs 36.8 minG8 33K 12.19 yrs 2.3 hrs 12.18 yrs 2.5 hrs 12.18 yrs 1.5 hrsG9 51K 12.67 yrs 4.8 hrs 14.10 yrs 4.0 hrs 14.19 yrs 2.9 hrsG10 73K 12.48 yrs 7.2 hrs 13.89 yrs 6.6 hrs 13.98 yrs 3.9 hrsG11 132K 12.70 yrs 21.0 hrs 14.46 yrs 22.1 hrs 14.59 yrs 15.0 hrs


Algorithms 3, 4, 5, and 6 have been implemented in C++. To solve the required linear programs

in algorithm 3, we again used the Mosek optimization package [29]. We use SPAI [30] as well to

compute the sparse inverse of G0. We carried out experiments on a set of randomly-generated

power grids, using a 2.6 GHz Linux machine with 24 GB of RAM. The grids are generated

based on user specifications, including grid dimensions, metal layers, pitch and width per layer,

and C4 and current source distribution. Moreover, all experiments were performed on grids

with up to ten global constraints (on groups of current sources). The parameters of the grids

are consistent with 1.1V 65nm CMOS technology. Notice that the first five grids (G1-G5) are

the same grids used in chapter 4. As for the EM model employed, we use similar parameters as

in the previous chapters: an activation energy of 0.9eV , a current exponent η = 1, a nominal

temperature Tm = 373K, a critical Blech product β = 3000A/cm, and a standard deviation


σln = 0.3. To asses the quality of our results, we computed the average worst-case grid TTF

using the SA with local optimization approach together with the required CPU time for every

grid. Table 5.1 shows the speed and accuracy of the simulated annealing approach by comparing

the obtained averages with the results of the exact approach of chapter 4. we can see that the

error is always less than ±5% while the run time is much less for the SA approach. The exact

approach required solving several ILPs and for that reason, the required CPU time is much

larger than that of the first SA based method. The observed accuracy for the small grids shows

that SA is able to explore the feasible space relatively well while reaching points very close to

the exact global optimum.

Table 5.2 shows the average minimum grid TTF obtained using the three Simulated Anneal-

ing based methods proposed for larger grids (G6-G11). As observed, the first method generally

generates a lower average minimum grid TTF than the other two methods because it includes

an additional local optimization step. The runtime of the second method is comparable to the

first because the TTF estimation in the second method requires updating the TTF of all the

lines in the grid every time a line fails, which is expensive. This problem is solved in the third

method where we see that the average minimum TTF obtained is very close to that of the

second method, while the run time is, on average, 1.5X better. The number of Monte Carlo

iterations that were required for convergence were between 30 and 40 for all the test grids.

Figure 5.3 shows the CPU time of all three methods versus the number of nodes in the grid.

Complexity analysis shows that the first two methods have around O(n1.6) empirical complexity

while the third have around O(n1.4) empirical complexity. Even though the first two methods

use different TTF estimators, the runtime for both methods is similar, and the reason is that

the second method uses a more expensive TTF estimator (the one developed in [37]) but does

not have the extra local optimization step. The average minimum TTF obtained are shown

in figure 5.4, where we can that the results of the second and third method almost overlap.

To better show the effectiveness of Simulated Annealing in finding the worst-case grid TTF,

figure 5.5 shows the progress of SA for all three methods for a particular Monte Carlo iteration

(33K nodes grid). We can see that SA is able to reduce the TTF estimated from around 21

years to 12 years. The local optimizer is able to provide a further reduction of 1 year (this

reduction always seems to be between 0.5 and 3 years for all the grids). In addition, the figure

shows that there is in fact a large separation between the TTF of the grid at different points

in space, i.e. the TTF of the grid is highly sensitive to the change in currents. This basically

means that it is not enough to compute the TTF of the grid at an arbitrary feasible point:

computing the minimum is necessary.

Lastly, and to check how sensitive SA is to the random seed used to travel the feasible

space, we minimized the TTF of grid G6 (using the optimization method of section 5.5) 50

times while keeping the same TTF samples for the grid lines and while changing the random

seed that controls how the next candidate points are being chosen. We obtained 50 different


0 20K 40K 60K 80K 100K 120K 140K0

5

10

15

20

25

Number of Nodes

Tot

al C

PU

Tim

e (h

rs)

SA with Local OptimizationSA with Changing CurrentsSA with Changing Currents and Locality

Figure 5.3: CPU time of the Simulated Annealing based methods versus the number ofgrid nodes

minimums having a mean of 11.74 years and a standard deviation of 0.68 years, leading to a

mean to standard deviation ratio of 0.058. This shows that SA is not very sensitive to the

change in the random seed, and hence, the result of SA is, up to certain extent, a good estimate

of the minimum grid TTF.

The proposed approaches are important because they check for EM safety using a less pes-

simistic model (mesh model) in a truly vectorless framework, which justifies a slower algorithm

than typical EM checking tools. Besides, all the methods proposed are highly parallelizable due

to the inherent independence of Monte Carlo iterations.


0 20K 40K 60K 80K 100K 120K 140K0

2

4

6

8

10

12

14

16

18

20

Number of Nodes

Est

imat

ed A

vera

ge M

inim

um T

TF

(yr

s)


Figure 5.4: Average minimum TTF estimated for the three Simulated Annealing basedmethods versus the number of grid nodes

5.7 Conclusion

We described three early vectorless approaches for power grid electromigration checking under

a constraint-based framework. The approaches employ the well known Simulated Annealing al-

gorithm. We show the accuracy of the methods as compared to the exact approach of chapter 4,

and show that the methods are very scalable as the complexity is slightly super linear.


0 20 40 60 80 100 120 140 16010

12

14

16

18

20

22

SA iterations

Est

imat

ed T

TF


Start of LocalOptimization(Red Line)

Figure 5.5: Simulated Annealing progress for a particular TTF sample using all thethree proposed methods (33K grid)

Chapter 6

Conclusion and Future Work

The latest trends toward low power and high performance semiconductor manufacturing has

emphasized the need for a robust design of the power delivery network. Every node in the

power grid must behave as a reliable source of supply voltage and must behave as such for a

certain number of years before failing. Timing violations and logic failures are bound to happen

when large voltage drops occur at grid nodes.

Under the effect of electromigration, metal line resistance increases as a line approaches

failure and starts to deform due to void creation. This affects the distribution of power among

grid nodes, and in most cases, leads to large voltage fluctuations. Accordingly, power grid

electromigration checking involves computing the mean time-to-failure of the power grid. Such

a metric gives the designer an idea about the robustness of the grid and whether it needs to be

redesigned or not.

Existing electromigration checking techniques and commercial tools assume the grid to be

a series system. As a result, the predicted EM stress is much worse than it actually is because

the grid is deemed to fail when any of its lines fail. This is leading to minimal margins between

the predicted EM stress and the EM design rules, making it very hard to sign-off chip designs

using traditional EM checking approaches. Therefore, there is a need to reconsider the existing

tools and to look with suspicion at the pessimism built into traditional EM checking methods.

Another critical aspect in EM checking, that which was the other focus of this thesis, is

the imprecise characterization of circuit currents. To capture this uncertainty, we built on the

framework of currents constraints as well as the constraints on the activity of different power

modes in every block. Verifying the power grids becomes a question of finding the average

worst-case time-to-failure of the power grid over all the possible chip workload scenarios. The

benefit of such a systematic framework is that it helps the user to manage design uncertainties

especially early in the design flow.

In chapter 3, we developed a new power grid MTF estimation technique to capture the

inherent redundancy in the grid. The method was verified using vector-based simulation where

76

Chapter 6. Conclusion and Future Work 77

we have shown that the MTF estimated is around 2-2.5X greater than the MTF predicted by

SEB. Besides, the method was shown to be runtime-efficient, scalable, and easily parallelizable.

In chapter 4, we extended the approach to the vectorless case, where an exact average worst-

case TTF of the grid was computed over all the possible chip workload scenarios using a set of

linear and mixed integer optimization problems. An important drawback of the exact approach

is that its runtime is too prohibitive for it to be scalable or useful except for small grids or

small islands in large grids. For that, chapter 5 introduces three more methods that are based

on the well known Simulated Annealing algorithm. The approaches are based on different TTF

estimation engines and were shown to be fairly accurate and relatively scalable. Collectively,

all the techniques presented can fill real and diverse design needs.

The research presented in this thesis seems to have raised many new questions with the

introduction of the mesh model. One might ask about the applicability of the model in modern

power grids where the lifetime of individual metal lines is governed by more complicated prob-

abilistic models than Black’s model. In addition, the approach must be extended to consider

better resistance evolution models for the failing lines. The infinite resistance was a simplifying

assumption which led to conservative yet possibly inaccurate results. On the other hand, a

Monte Carlo approach will always be relatively expensive and hence, one might want to look

into a direct approach for computing the MTF of the grid without the need for sampling. This

would also help the vectorless part of the work as it would make the Simulated Annealing based

algorithms much faster. Finally, it might be viable to explore existing model reduction schemes

which might allow hierarchical and incremental electromigration checking.

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