Electrical Modeling and Design for 3D System Integration

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R. Abhari M. El-Hawary O. P. MalikJ. Anderson B-M. Haemmerli S. NahavandiG. W. Arnold M. Lanzerotti T. SamadF. Canavero D. Jacobson G. Zobrist

Kenneth Moore, Director of IEEE Book and Information Services (BIS)

Electrical Modeling and Design for 3D System Integration

3D Integrated Circuits and Packaging, Signal Integrity, Power Integrity and EMC

Er-Ping Li, BSc, MSc, PhD, IEEE FellowZhejiang UniversityHangzhou, China

A John Wiley & Sons, Inc., Publication

IEEE PRESS

Copyright © 2012 by Institute of Electrical and Electronics Engineers. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Li, Er-Ping. Electrical modeling and design for 3D system integration : 3D integrated circuits and packaging, signal integrity, power integrity and EMC / Er-Ping Li. p. cm. ISBN 978-0-470-62346-6 (hardback)1. Three-dimensional integrated circuits. I. Title. TK7874.893.L53 2011 621.3815–dc23 2011028946

Printed in the United States of America.

10 9 8 7 6 5 4 3 2 1

v

Contents

Foreword xiPreface xiii

1. Introduction 11.1 IntroductionofElectronicPackageIntegration, 11.2 ReviewofModelingTechnologies, 61.3 OrganizationoftheBook, 10

References, 11

2. MacromodelingofComplexInterconnectsin3DIntegration 162.1 Introduction, 16

2.1.1 Scopeofmacromodeling, 182.1.2 Macromodelinginthepictureofelectrical

modelingofinterconnects, 192.2 NetworkParameters:Impedance,Admittance,and

ScatteringMatrices, 192.2.1 Impedancematrix, 212.2.2 Admittancematrix, 222.2.3 Scatteringmatrix, 232.2.4 ConversionbetweenZ,Y,andSmatrices, 24

2.3 RationalFunctionApproximationwithPartialFractions, 252.3.1 Introduction, 252.3.2 Iterativeweightedlinearleast-squaresestimator, 27

2.4 VectorFitting(VF)Method, 292.4.1 Twostepsinvectorfittingmethod, 292.4.2 Fittingvectorswithcommonpoles, 352.4.3 Selectionofinitialpoles, 372.4.4 Enhancementtotheoriginalvectorfittingmethod, 38

2.5 MacromodelSynthesis, 412.5.1 Jordancanonicalmethodformacromodel

synthesis, 422.5.2 Equivalentcircuits, 46

vi  Contents

2.6 Stability,Causality,andPassivityofMacromodel, 482.6.1 Stability, 482.6.2 Causality, 502.6.3 Passivityassessment, 542.6.4 Passivityenforcement, 582.6.5 Otherissues, 78

2.7 MacromodelingAppliedtoHigh-SpeedInterconnectsandCircuits, 792.7.1 Alumpedcircuitwithnonlinearcomponents, 792.7.2 Verticallynaturalcapacitors(VNCAPs), 832.7.3 Stripline-to-microstriplinetransitionwithvias, 87

2.8 Conclusion, 91References, 92

3. 2.5DSimulationMethodfor3DIntegratedSystems 973.1 Introduction, 973.2 MultipleScatteringMethodforElectronicPackage

ModelingwithOpenBoundaryProblems, 983.2.1 Modalexpansionoffieldsinaparallel-plate

waveguide(PPWG), 983.2.2 Multiplescatteringcoefficientsamongcylindrical

PECandperfectmagneticconductor(PMC)vias, 1013.2.3 Excitationsourceandnetworkparameter

extraction, 1093.2.4 Implementationofeffectivematrix-vector

multiplication(MVM)inlinearequations, 1173.2.5 Numericalexamplesforsingle-layerpower-ground

planes, 1213.3 NovelBoundaryModelingMethodforSimulationof

Finite-DomainPower-GroundPlanes, 1273.3.1 Perfectmagneticconductor(PMC)boundary, 1283.3.2 Frequency-dependentcylinderlayer(FDCL), 1283.3.3 ValidationsofFDCL, 131

3.4 NumericalSimulationsforFiniteStructures, 1333.4.1 Extendedscatteringmatrixmethod(SMM)

algorithmforfinitestructuresimulation, 1333.4.2 Modelingofarbitrarilyshapedboundary

structures, 139

Contents  vii

3.5 Modelingof3DElectronicPackageStructure, 1423.5.1 Modalexpansionsandboundaryconditions, 1433.5.2 ModematchinginPPWGs, 1503.5.3 GeneralizedT-matrixfortwo-layerproblem, 1583.5.4 Formulaesummaryfortwo-layerproblem, 1643.5.5 Formulaesummaryfor3Dstructureproblem, 1693.5.6 Numericalsimulationsformultilayered

power-groundplaneswithmultiplevias, 1763.6 Conclusion, 182

References, 183

4. HybridIntegralEquationModelingMethodsfor3DIntegration 1854.1 Introduction, 1854.2 2DIntegralEquationEquivalentCircuit(IEEC)

Method, 1864.2.1 Overviewofthealgorithm, 1864.2.2 Modaldecouplinginsidethepowerdistribution

network(PDN), 1874.2.3 2Dintegralequationsolutionofparallelplatemode

inpower-groundplanes(PGPs), 1894.2.4 Combinationsoftransmissionandparallelplate

modes, 1944.2.5 Cascadeconnectionsofequivalentnetworks, 2054.2.6 Simulationresults, 214

4.3 3DHybridIntegralEquationMethod, 2204.3.1 Overviewofthealgorithm, 2204.3.2 Equivalentelectromagneticcurrentsanddyadic

green’sfunctions, 2244.3.3 Simulationresults, 231

4.4 Conclusion, 238References, 238

5. SystematicMicrowaveNetworkAnalysisfor3DIntegratedSystems 2415.1 IntrinsicViaCircuitModelforMultipleViasinanIrregular

PlatePair, 2425.1.1 Introduction, 2425.1.2 Segmentationofviasandaplatepair, 245

viii  Contents

5.1.3 Anintrinsic3-portviacircuitmodel, 2485.1.4 Determinationofthevirtualviaboundary, 2635.1.5 Completemodelformultipleviasinanirregular

platepair, 2675.1.6 Validationandmeasurements, 2695.1.7 Conclusion, 280

5.2 ParallelPlanePairModel, 2815.2.1 Introduction, 2815.2.2 OverviewoftwoconventionalZppdefinitions, 2835.2.3 NewZppdefinitionusingthezero-orderparallel

platewaves, 2855.2.4 AnalyticalformulaforradialscatteringmatrixSpp

R inacircularplatepair, 290

5.2.5 BIEmethodtoevaluateSppR foranirregularplate

pair, 2925.2.6 Numericalexamplesandmeasurements, 2965.2.7 Conclusion, 303

5.3 CascadedMultiportNetworkAnalysisofMultilayerStructurewithMultipleVias, 3055.3.1 Introduction, 3055.3.2 MultilayerPCBwithviasanddecoupling

capacitors, 3075.3.3 Systematicmicrowavenetworkmethod, 3085.3.4 Validationsanddiscussion, 3165.3.5 Conclusion, 324Appendix:PropertiesoftheAuxiliaryFunctionWmn(x,y), 326References, 327

6. ModelingofThrough-SiliconVias(TSV)in3DIntegration 3316.1 Introduction, 331

6.1.1 OverviewofprocessandfabricationofTSV, 3326.1.2 ModelingofTSV, 335

6.2 EquivalentCircuitModelforTSV, 3366.2.1 Overview, 3376.2.2 Problemstatement:Two-TSVconfiguration, 3386.2.3 WidebandPi-typeequivalent-circuitmodel, 339

Contents  ix

6.2.4 Rigorousclosed-formformulaeforresistanceandinductance, 341

6.2.5 Scatteringparametersoftwo-TSVsystem, 3456.2.6 Resultsanddiscussion, 346

6.3 MOSCapacitanceEffectofTSV, 3516.3.1 MOScapacitanceeffect, 3516.3.2 Biasvoltage-dependentMOScapacitanceof

TSVs, 3516.3.3 Resultsandanalysis, 355

6.4 Conclusion, 356References, 358

Index 361

Foreword

Today, the modeling of electrical interconnects and packages is very important from both a practical and a theoretical point of view. High performance and high speed especially require a great deal of skill. An ever-increasing number of practical designs fall into this class.

The fact that we now have powerful design tools increases our ability to solve a larger number of real-world problems for many dif-ferent issues. This greatly helps solve most of the important problems for a large class of geometries. However, the ever-increasing perfor-mance of the technology requires a continuous evolution of the skills in modeling techniques. A key performance issue is the reduction in effort and computing time for very large problems. Clearly, better design tools and techniques lead to better designs. Over the years, we also could observe that the opposite is true, namely that the more chal-lenging problems lead to improved tools as well as better technical solutions. A consequence of this process is the continuous bootstrap-ping of the tools and techniques as well as the designers’ skills.

This book represents an educational tool for modelers as well as for tool designers. It offers an unusual combination of the latest tech-niques for the electromagnetic (EM) modeling of packages and signal interconnections, including the challenging via problems. In fact, it is much more detailed than some of the introductory texts which are available today on the subject. It considers all aspects such as the analysis methods for the construction of macromodels which are stable, causal, and passive. Such models are widely in use today, and the pas-sivity issue impacts the accuracy in both the frequency and time domains, while instability is unacceptable in the time domain. Also, key aspects of the modeling are the noise interactions between the multitude of wires and signal planes which are present in a typical design. All these aspects are considered in detail from an electromag-netic point of view, and sophisticated solution techniques are given. It is evident from this book that addressing the modern 3D packaging technology is an integral part of what makes the book relevant.

xi

xii  Foreword

We are fortunate to find in this book the contributions of an author who is both experienced and knowledgeable in this field. Dr. Er-Ping Li is an internationally well-known contributor to the field of electro-magnetic solutions in the area of interest. He has been a Principal Scientist and Director of Electronics and Photonics at A*STAR (Agency for Science, Technology and Research) Institute of High Performance Computing in Singapore. From 2010 he holds an appointment as Chair Professor in Zhejiang University, China. He is a Fellow of the IEEE and a Fellow of the Electromagnetic Academy, USA. He received numerous international awards and honors in recognition of his profes-sional work.

Albert E. Ruehli, PhD, Life Fellow of IEEEEmeritus, IBM T. J. Watson Research Center, Yorktown, NY, USA

Adjunct Professor, EMC Lab,Missouri University of Science and Technology, Rolla, MO, USA

xiii

Preface

The requirements of higher bandwidth and lower power consumption of electronic systems render the integration of circuits and electronic packages more and more complex. In particular, the introduction of three-dimensional (3D) structures based on through-silicon via (TSV) technology provides a potential solution to reduce the size and to increase the performance of these systems. As a consequence, the elec-tromagnetic compatibility (EMC) between circuits, signal integrity (SI), and power integrity (PI) in electronic integration are of vital importance. For this reason, the electronic circuits and packaging systems must be designed by taking into account the trade-offs between cost and perfor-mance. This requires ever more accurate modeling techniques and powerful simulation tools to achieve these goals. Incredible progress in electromagnetic field modeling has been achieved in the world. My research group has invested considerable efforts to develop novel simu-lation techniques over the last decade. Nevertheless, the present model-ing techniques may be still far from perfect; for example, the modeling of multiphysics relevant to 3D integration is still far behind the require-ments of the available technology.

This book presents the material that results from many years of our collective research work in the fields of modeling and simulation of SI, PI, and EMC in electronic package integration and multilayered printed circuit boards. It represents the state-of-the-art in electronic package integration and printed circuit board simulation and modeling technolo-gies. I hope this book can serve as a good basis for further progress in this field in both academic research and industrial applications. The book consists of six chapters: Chapter 1 is written by Er-Ping Li, Chapter 2 by Enxiao Liu and Er-Ping Li, Chapter 3 by Zaw-Zaw Oo and Er-Ping Li, Chapter 4 by Xingchang Wei and Er-Ping Li, Chapter 5 by Yaojiang Zhang, and Chapter 6 by En-Xiao Liu.

Chapter 1 provides a review of progress in modeling and simulation of SI, PI, and EMC scenarios; Chapter 2 focuses on the macromodeling technique used in the electrical and electromagnetic modeling and

xiv  Preface

simulation of complex interconnects in 3D integrated systems; Chapter 3 presents the semianalytical scattering matrix method (SMM) based on the N-body scattering theory for modeling of 3D electronic package and multilayered printed circuit boards with multiple vias. In Chapter 4, 2D and 3D integral equation methods are employed for the analysis of power distribution networks in 3D package integration. Chapter 5 describes the physics-based algorithm for extracting the equivalent circuit of a complex power distribution network in 3D integrated systems and printed circuit boards; Chapter 6 presents an equivalent-circuit model of through-silicon vias (TSV) and addresses the metal-oxide-semiconductor (MOS) capacitance effects of TSVs.

I gratefully acknowledge the technical reviewers of this book, Dr. Albert Ruehli, Emeritus of the IBM Watson Research Center, York-town, New York, USA; Prof. Wolfgang Hoefer, A*STAR, Singapore, and Prof. Zhongxiang Shen, Nanyang Technological University, Sin-gapore, who donated their time and effort to review the manuscript. Also acknowledged are the contributors of the book, Dr. Xingchang Wei, Dr. Enxiao Liu, Dr. Zaw Zaw OO, and Dr. Yaojiang Zhang, who did the really hard work. I also wish to express my gratitude to Mary Hatcher at Wiley/IEEE Press for her great help in keeping us on sched-ule. Finally, I am grateful to my wife and the contributors’ wives, for without their continuing support and understanding, this book would have never been published.

I hope that this book will serve as a valuable reference for engi-neers, researchers, and postgraduate students in electrical modeling and design of electronic packaging, 3D electronic integration, integrated circuits, and printed circuit boards. Even though much work has been accomplished in this field, I anticipate that many more exciting chal-lenges will arise in this area, particularly in 3D integrated circuits and systems.

Er-Ping LiWest Lake, Hangzhou, China

1

1.1 INTRODUCTION OF ELECTRONIC PACKAGE INTEGRATION

The rapid growth and convergence of digital computers and wireless communication have been driving semiconductor technology to con-tinue its evolution following Moore’s law in today’s nanometer regime. Future electronic systems require higher bandwidth with lower power consumption to handle the massive amount of data, especially for large memory systems, high-definition displays, and high-performance microprocessors. Electronic packaging is one of the key technologies to realize a wider bus architecture with high bandwidth operating at higher frequencies. Various packages have been developed toward a higher density structure. In particular, a three-dimensional (3D) integra-tion based on through-silicon via (TSV) [1] arrays technology provides a potential solution to reduce the size and to increase the performance of the systems. Furthermore, nano-interconnects to replace the Cu-based interconnects provides a promising solution for long-term application.

There is a great challenge for further increasing of the signal speed in electronic systems due to the serious electromagnetic compatibi-lity (EMC) problem. Figure 1.1 plots the technology trends versus actuals and survey, and Figure 1.2 shows the trends of microprocessors predicted by the International Technology Roadmap for Semiconduc-tors (ITRS) [2, 3]. From these figures one can see that

CHAPTER 1

Introduction

Electrical Modeling and Design for 3D System Integration: 3D Integrated Circuits and Packaging, Signal Integrity, Power Integrity and EMC, First Edition. Er-Ping Li. © 2012 Institute of Electrical and Electronics Engineers. Published 2012 by John Wiley & Sons, Inc.

2  CHAPTER 1 Introduction

Figure 1.1  2008 ITRS update—technology trends versus actuals and survey [2].

Jeff Butterbaugh/EEFGLphys Actuals(leading)

KwokNg/PIDSGLphys Survey(leading)

Glpr(nm) MPU (ITRS05-07)

2008Update

Glph(nm) MPU (ITRS05-07)

M1 Half Pitch(nm)MPU/ITRS 05-07[also BRAM M1 in2008 Update]

Poly Half Pitch(nm)Flash (ITRS 07)[Li1ho Driver after2007]

M1 Half Pitch(nm)DRAM (ITRS 05-07)

GLph Proposal 2008Update Final

2008Update

GLpr(nm) MPU HPProposal4 [FINAL-Li1ho Jul’08]

2007/08 ITRS: 2007-2022

1000

100

Nan

omet

ers

(1e-

9)

02000 2005 2010 2015

Year2020 2025

GLPhysical =~0.71x/3.8yrs

2.5-YearCycle

[5^(1/5yrs)]

“More Moo;e”Functronal

DensityComplementedby “Equivalent

Scaling”Performance/

Power Mgt[Copper IC;Strain Si.

Metal Gate/Hi-K

UTB/FDSOI;MUG, etc]

GLPrinted =[DecreasingEtch Ratio]1.6818/2007Then Parallel

to MPU/DRAM

3-YearCycle

[5^(1/6yrs)]

Figure 1.2  The trends of microprocessor predicted by the International Technology Roadmap for Semiconductors (ITRS).

2010 2012 2014 2016 2018 2020 2022 2010 2012 2014 2016 2018 2020 202210

20

30

40

50Interconnect Pitch (nm)On-Chip Clock (GHz)

Voltage Supply (V)Maximum Power Density (Watts/mm2)

Year Year

Inte

rcon

nect

Pitc

h (n

m)

5

7

9

11

13

15

On-

Chi

p C

lock

(G

Hz)

0.5

1.0

1.5

2.0

Vol

tage

Sup

ply

(V)

0.5

1.0

1.5

2.0

Max

imum

Pow

er D

ensi

ty(W

atts

/mm

2)

(a) (b)

1.1 Introduction of Electronic Package Integration  3

• Interconnect pitch will continue to decrease to 11.3 nanometer, while the on-chip clock frequency will be increased to 14.3 GHz by 2022. Due to the reduction of the feature size and pitch, more and more circuits are integrated into one electronic package, such as the system in package (SIP) and the 3D integration. This results in a complex and high-density environment inside the electronic systems. At the same time, with the ever-increasing clock frequency (also its high-frequency harmonics), the physical size of the small electronic package becomes electrically large, and so the electromagnetic wave propagation inside such a small structure must be considered.

• Until 2011, the voltage supply of the microprocessor is continu-ally reduced with an increased power density. The electromag-netic noise will be pronounced due to the increased power density, which then makes the decreased voltage supply unstable. To design a high-speed and stable electronic system, we need better understand the electromagnetic interactions and the EMC issues inside the electronic package.

The EMC researches related to the high-speed circuit systems have a long history, which can be classified into different levels according to the size of the interested objects, which includes the system level, printed circuit board (PCB) level, electronic package level, and com-ponent level. The increasing clock frequency makes the size of tiny structures on the chip be comparable with the wavelength of interest. The fluctuation of electromagnetic wave cannot be ignored any more. Therefore, we must accurately model the electromagnetic wave behav-ior for all scales of the high-speed circuit systems. In the near future, the nanoscale integrated circuits (ICs) will be characterized by using the electric and magnetic fields instead of the conventional voltage and current. EM in micro-E is becoming a hot topic in both academic com-munity and industrial applications.

The EMC analysis for high-speed electronics includes lots of issues, such as the ground bounce, cross talk, conducted emission, radi-ated emission, conducted immunity, and radiated immunity. The inter-action between on-board capacitance and on-chip capacitance causes an antiresonance which induces a peak in the total power distribution network (PDN) impedance as shown in Figure 1.3. Figure 1.4 shows a typical multilayered advanced electronic package which consists of two

4  CHAPTER 1 Introduction

main electrically functional systems: the PDN and the signal distribu-tion network (SDN). The passive structures are composed of three main categories: (1) traces or transmission lines, typically microstrip lines or striplines, (2) vias used as vertical interconnections, and (3) conductor plates serving as power or ground planes. Because of the complexity of an advanced package, it is difficult to model the entire SDN or PDN simultaneously. Yet, we need to consider the impact of the PDN on the SDN in order to characterize the SDN more accurately. Many research-ers have proposed various approaches to study the electrical properties of the above passive structures [4–44].

A typical EMC problem residing in this PDN of the electronic package is illustrated in Figure 1.5. In Figure 1.5, the power and ground planes are used to supply DC power for the circuits integrated in the electronic package. The signal traces are often laid out in different

Figure 1.3  Example of antiresonances in total PDN impedances for various on-chip capacitance values [3]. (See color insert.)

1E6 1E7 1E8 1E9 8E90.001

100 nF

10 nF1 nF

100 pF10 pF

100

10

1

0.1

0.01Impe

danc

e (o

hms)

Frequency (Hz)

Figure 1.4  A schematic diagram of a multilayered electronic package [37].

SignalP/G

G/PSignal

1.1 Introduction of Electronic Package Integration  5

layers of power-ground planes. Their return currents flow on the power- ground planes just below them. When the traces pass through different layers, their return currents also exchange from one plane to another plane, as shown in Figure 1.5. Accordingly, a vertical displacement current is induced between different planes for the continuity of the return currents. This displacement current will excite electromagnetic field noise, which then propagates inside the power-ground planes and couples to other signal traces passing through the same layer. At the same time, this noise also leaks to the surrounding area of the electronic package through the periphery and gaps of the power-ground planes. These interferences will be further amplified if the noise’s spectrum covers any inherent resonant frequency of the cavity-like power-ground planes.

To achieve first-pass design success, we must employ an advanced modeling and simulation technique to analyze the electrical perfor-mance of the 3D electronic packages, PCB, and chips at the system level. However, both industry and academia communities face the great challenges in developing the electrical design and simulation tools due to the multiscale nature of the problem, the strong local and global electromagnetic coupling, and the complexity of 3D integration systems. ITRS has summarized the state of the art of current semiconductor industry development, where the major challenges for simulation and modeling are listed as [2] mixed-signal co-design and simulation envi-ronment, rapid turnaround modeling and simulation, electrical (power disturbs, electromagnetic interference (EMI), signal and power integ-rity associated with higher frequency/current and lower voltage switch-ing), system-level co-design, electronic design automation (EDA) for “native” area array to meet the roadmap projections, and models for

Figure 1.5  Noise coupling inside and emission from the power-ground planes.

signal trace

ground plane

power plane

via antipadsignal current

return current

6  CHAPTER 1 Introduction

reliability prediction. Therefore, advanced modeling techniques, which stand up to the challenges imposed by the complexity of nanoscale silicon chips and their interconnections including 3D ICs, 3D packag-ing, and PCB [45–47], are in great demand.

1.2 REVIEW OF MODELING TECHNOLOGIES

Modeling of transmission lines has a long history and is well docu-mented in many textbooks [4]. So in the following, we will mainly review the modeling of vias and power-ground planes for electronic packaging and PCBs. Such modeling methods can be roughly classified into three categories: (1) lumped circuit approaches, (2) full-wave approaches, and (3) hybrid circuit coupled full-wave approaches.

For its simplification and ease of understanding, at the beginning of the research, lumped circuit approaches have been used for the elec-trical modeling of electronic packages. Such examples are shown in Figures 1.6 and 1.7. Empirical and analytical formulae for via capaci-tance and inductance can be easily found in many handbooks. Quasi-static numerical methods have also been introduced to calculate the

Figure 1.6  A typical transmission line model on a printed circuit board [3]. PRBS: pseudo-random binary sequence. ODT: on die termination.

PCB trace

R CL G

R CL G

S dd11S dd21

R CL G

R CL G –350 5 10 15 20 25

–30–25–20–15–10

–50

Frequency [GHz]

Sdd11Sdd21

PCB trace

Packagetrace

Driver IC (Tx)

ODT (50Ω)

Receiver IC (Rx)

Eye diagram

PRBSsignal

Ci

Packagetrace

S parameters by 3D field solverS parameters by 3D field solverW-element model by 2D field solver W-element model by 2D field solver

Transmission line model Via model

R CL

R CL

G

G

1.2 Review of Modeling Technologies  7

lumped circuit values in T or PI types of via models [5, 6]. These methods allow rapid computation, but often suffer from accuracy prob-lems. The authors in Reference 7 proposed a model of a magnetic-frill array and utilized microwave network theory for analysis of vias in multilayered packages. But it is a single via model which is difficult to be generalized to multiple vias. The equivalent circuits of much com-plex via array can be extracted by using de-embedding method [8]. Distributed circuit approaches have also been widely used for pack-age modeling, such as the partial element equivalent circuit (PEEC) method [9, 10].

Second, full-wave methods both in the time and frequency domains have been employed to study the packaging problems. The commonly used full-wave commercial simulators include Ansoft HFSS and CST Microwave Studio, which are based on finite element method (FEM) [11, 12] and finite integral technique, respectively. Other full-wave algorithm includes the finite-difference time-domain (FDTD) method [13] and the transmission line matrix method (TLM) [14]. Recently, the integral equation-based full-wave method begins to attract more attention and had been employed. The advantage of the integral equa-tion method is that it can use the suitable Green’s function to present

Figure 1.7  Power distribution network [3].

Packageinductance

CMOScircuit

Board

Power supplynoise

v(t),V(w)

Vddplane

GND plane

On-boarddecouplingcapacitor

On-chipdecouplingcapacitor

VRM

Switching currenti(t),l(w)

p-MOS

n-MOS

Power supply impedanceof PCB:Zpcb(w)

Total power supplyimpedance Z(w)

8  CHAPTER 1 Introduction

the effect of the complex environment, so that the unknowns are only placed on discontinuities inside the PDN. This can give an efficient simulation. According to the different Green’s functions used, the inte-gral equation methods can be classified into (a) two-dimensional (2D) integral equations, including 2D mode method [15] and image method for rectangular power and ground planes, and 2D transverse magnetic (TM) integral equation for arbitrarily shaped power and ground planes [16–18]; (b) 3D integral equations, including 3D cavity mode method for rectangular power and ground planes [19] and parallel plate mode method for arbitrarily shaped power and ground planes [20, 21]. For most real applications the parallel plates have regular shapes, such as rectangles, circles, or triangles, a closed form of the Green’s functions can be formulated which results in an impedance formula in terms of the summation of infinite number of resonant modes [22–24]. This 2D integral equation method is sometimes called the cavity resonator method. Segmentation techniques may be applied to extend the cavity resonator method for parallel plates with irregular shapes.

These full-wave methods are versatile and able to solve a wide range of problems, but at the expense of large memory usage and long CPU time, especially for those 3D full-wave methods. Although the overall size of the electronic packages is small enough to apply these full-wave methods, the high aspect ratio of the power and ground planes and the tiny structures, such as the signal traces and narrow slots, result in a huge number of meshing. This makes these full-wave methods very expensive in terms of computing time and memory requirement.

Third, to avoid the computational cost of these full-wave methods and the geometrical limitations of the analytical methods, a more effi-cient approach is to combine both methods together, so that we can benefit from both analytic and numerical techniques. The coupled circuit-field approaches are also widely used to model the electronic packages in order to leverage the advantages of both circuit and full-wave approaches. An important approach under this category is rooted in the theory of modal decomposition and the salient features of elec-tronic packages. The transmission lines and power-ground planes in an electronic package convey different modes, that is, transmission line modes and parallel plate modes. Modal decomposition can be used to decouple these two modes, which are then solved independently. These two modes are finally recombined to reflect the original problem. The

1.2 Review of Modeling Technologies  9

coupling between the transmission line mode and the parallel-plate mode often occurs due to the vias. The current flowing in the via excites the parallel-plate mode field, while the transmission line experiences the loading effect of the power-ground plane in the presence of the via.

The complete modal decomposition and recombination approach has been demonstrated by several researchers. Current or voltage con-trolled sources are used to link these two modes. A general modal recombination approach was presented in Reference 26 for coupled striplines and nonideal power-ground planes, while the parallel-plate mode associated with the power-ground planes has been studied by many researchers. 2D full-wave methods have been extensively employed in the literature to model the power-ground planes. The 2D integral equation method is also called the contour integral method and has been used in Reference 16 to study general parallel-plate structures with arbitrary shapes. Another 2D approach, called the 2D FDTD, has also been used to model parallel plates [27]. Discretization of the metal plates by the finite-difference method was interpreted as a 2D distrib-uted LC circuit, and a rigorous derivation is given in Reference [27]. The 2D distributed RLCG (resistance, inductance, capacitance, and conductance) circuit network, which is widely used in the literature to represent the power and ground planes, can be considered as an exten-sion of the LC network derived from the finite-difference method. Instead of using Simulation Program with Integrated Circuit Emphasis (SPICE)-like solvers to simulate the large equivalent circuit network of power-ground planes, the latency insertion method is proposed in Reference 28 to perform fast transient simulation of large RLC net-works. Moreover, a transmission matrix method reported in Reference 29 divides the 2D distributed RLCG circuit network into many inter-dependent blocks, and each block is formulated as a transmission (ABCD) matrix. Cascading those transmission matrices produces a fast way to obtain the desired impedance of the power-ground plane. A multilayered finite-difference method (MFDM) was recently proposed in Reference 26. The 2D finite element method (2D FEM) is also used to simulate power-ground planes [30] and had been integrated into the commercial software Ansoft SIWave. In addition, the radial transmis-sion line theory has been applied to derive an admittance matrix to account for the effect of the parallel plates [31]. However, image theory [32] is needed to model the reflection from the edges of finite-sized substrates. Image theory is elegant for modeling the boundary with a

10  CHAPTER 1 Introduction

regular shape but is cumbersome for modeling arbitrary shapes of the edges of PCBs or packages. In the model decomposition and recom-bination approach, a single via can be represented by a PI type of equivalent circuit. The capacitance and inductance in the PI circuit are usually computed by analytical formulae or quasi-static solvers. Recently, an elegant analytical formula was derived for the via barrel-plate capacitance [6].

1.3 ORGANIZATION OF THE BOOK

This book is organized in six chapters. Chapter 1 provides an overview of the state-of-art of electrical modeling and simulation techniques for electrical packaging systems. Chapter 2 focuses on the macromodeling technique widely used in the electrical and electromagnetic modeling and simulation of complex interconnects in 3D integrated systems. Macromodels are generated by employing the vector fitting (VF) method to perform rational-function approximation of scattering or admittance network parameters of high-speed complex interconnects and passive circuits. Subsequently, the macromodel can be synthesized as an equivalent circuit, which is compatible with the SPICE circuit simulator and can be combined with other external linear or nonlinear circuits to perform signal and power integrity analysis or other electri-cal performance analysis of electronic systems. The stability, causality, and passivity assessment and enforcement of the macromodel are also discussed in detail. Finally, numerical examples of macromodeling are presented and discussed.

In Chapter 3, the semianalytical scattering matrix method (SMM) based on the N-body scattering theory is presented for modeling of 3D electronic package and multilayered PCBs with multiple vias. Using the modal expansion of fields in a parallel-plate waveguide, the formula derivation of the SMM is presented in detail. In the conventional SMM, the power-ground planes are assumed to be infinitely large so they cannot capture the resonant behavior of the real-world packages. In particular, the SMM method has been extended to solve the finite domain of power-ground planes in coupling with a novel boundary modeling method proposed by the author’s group. This method has demonstrated its unique features which is capable to efficiently handle the complex real-world 3D package integration and PCB structures.

References  11

In Chapter 4, 2D and 3D integral equation methods are employed for the analysis of PDN in 3D package integration. The 2D integral equation method provides a comprehensive way for one to quickly extract the equivalent circuits of the PDN, and then substitute them into a SPICE-like simulator to perform the signal and power integrity analy-sis. The 3D integral equation method provides a more accurate solution for both the emission and susceptibility issues of the PDN. Both of the 2D and 3D integral equation methods are optimized by making a full use of the structural features of the PDN.

Chapter 5 is based on the physical-based algorithm to extract the equivalent circuit of the complex PDN in 3D integrated systems and PCBs. An intrinsic via circuit model is first derived through rigorous electromagnetic analysis for an irregular plate pair with multiple vias in a PCB. The derivation of the intrinsic via circuit model naturally leads to a new impedance definition of plate pair or power-bus, which is expressed in terms of cylindrical waves. The new plate pair imped-ance has clear physical meaning and makes possible signal/power integrity co-simulations. Numerical and measurement examples have indicated that while the new impedance gives almost the same results to the conventional one in a plate pair with few vias, it can correctly predict the resonant frequency shift in the case of a plate pair with a large amount of vias.

Chapter 6 presents a compact wideband equivalent-circuit model for electrical modeling of TSVs and addresses the metal-oxide-semi-conductor (MOS) capacitance effects of TSVs.

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12  CHAPTER 1 Introduction

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