NOISE MODELLING OF SILICON GERMANIUM HETEROJUNCTION ...

NOISE MODELLING OF SILICON GERMANIUMHETEROJUNCTION BIPOLAR TRANSISTORS AT

MILLIMETRE-WAVE FREQUENCIES

BY

KENNETH HOI KAN YAU

A THESIS SUBMITTED IN CONFORMITY WITH THE REQUIREMENTS

FOR THE DEGREE OFMASTER OFAPPLIED SCIENCE

GRADUATE DEPARTMENT OFELECTRICAL AND COMPUTER ENGINEERING

UNIVERSITY OF TORONTO

c© KENNETH HOI KAN YAU , 2006

Noise Modelling of Silicon Germanium Heterojunction BipolarTransistors at Millimetre-Wave Frequencies

Kenneth Hoi Kan YauMaster of Applied Science, 2006

Graduate Department of Electrical and Computer EngineeringUniversity of Toronto

Abstract

Using 2D device simulations, it is predicted that the cutofffrequencies of SiGe HBTs can

be scaled beyond 500GHz. These devices have the potential toenable advanced millimetre-

wave circuits. However, shot noise correlation, which is captured through noise transit time,

becomes increasingly important as circuit designers continue to push the operating frequencies

of the circuits.

The technique for extracting the SiGe HBT noise parameters only from the measuredy-

parameters is extended to account for the presence of correlation. Unlike earlier publications,

this method does not need to fit the noise transit time to measured noise data. The technique

was validated using 2D device simulations and measured noise parameter data. It was found

that theNFMIN of SiGe HBTs withfT/fMAX of 160GHz is approximately 1.5dB lower at

60GHz when noise correlation is accounted for. However, forthese devices, noise correlation

proves to be insignificant below 18GHz.

iii

Acknowledgements

I would like to sincerely thank Prof. Sorin P. Voinigescu forhis invaluable advice and guidance.

It is my honour to have him as my advisor, my colleague and my friend. I would also like to

acknowledge my examination committee Prof. C. Andre T. Salama, Prof. Wai Tung Ng and

Prof. Amr Helmy. I have benefited significantly from their recommendations.

I am also grateful to my colleagues who have assisted me during the tenure of my Master’s

program. Particularly, I would like to thank Tod Dickson formanaging the lab equipment,

Alain Mangan for many consultations on IC-CAP and TheodorosChalvatzis for spending

countless hours with me to figure out how to use the noise parameter characterization system.

I would like to acknowledge the professors and supervisors Ihad in my undergraduate

program. The Engineering Physics program at the Universityof British Columbia, Vancou-

ver, Canada is truly outstanding and has prepared me well forthe challenges in graduate re-

search. Thanks to Prof. Jeff Young for maintaining the high standards of the program, and

to Prof. David Pulfrey for delivering an interesting coursein semiconductor devices. Thanks

also go to Prof. Walter Hardy, Prof. William McCutcheon, Prof. Irving Ozier, Prof. Anton Bui,

Prof. Brian Seymour and Prof. Gordon Slade, all of the University of British Columbia.

This thesis will not be possible without the support of my parents and my brother. I would

like to thank Janice, who has endured countless lonely weekends without me.

This project was supported by the Natural Science and Engineering Council of Canada

(NSERC).

v

Table of Contents

Abstract iii

Acknowledgements v

List of Tables xi

List of Figures xvi

List of Abbreviations xvii

1 Introduction 1

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

1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 3

2 Background 5

2.1 Noise Correlation Matrices . . . . . . . . . . . . . . . . . . . . . . . .. . . . 5

2.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Current Status of SiGe HBT Noise Modelling . . . . . . . . . .. . . . 7

2.2.2 An Existing Technique for the Extraction of Noise Parameters from

y-Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Device Figures of Merit 11

3.1 Device Gains and Unity Gain Frequencies . . . . . . . . . . . . . .. . . . . . 11

3.2 Series Resistances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 13

4 SiGe HBT Noise Modelling at Millimetre-Wave Frequencies 17

4.1 Noise Equivalent Circuit . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 17

vii

viii Table of Contents

4.1.1 Approximations and Assumptions in the Noise Equivalent Circuit . . . 19

4.1.2 Advantages of the Noise Equivalent Circuit . . . . . . . . .. . . . . . 20

4.2 Derivation of the Noise Parameter Equations . . . . . . . . . .. . . . . . . . . 21

4.2.1 Input Referred Noise Voltage . . . . . . . . . . . . . . . . . . . . .. . 21

4.2.1.1 Output Short-Circuit Current of SiGe HBT Noise Model . . . 21

4.2.1.2 Output Short-Circuit Current of the Chain Representation . . 23

4.2.1.3 Input Referred Noise Voltage Expression . . . . . . . . .. . 24

4.2.2 Input Referred Noise Current . . . . . . . . . . . . . . . . . . . . .. . 25

4.2.2.1 Output Open-Circuit Voltage of SiGe HBT Noise Model. . . 25

4.2.2.2 Output Open-Circuit Voltage of the Chain Representation . . 26

4.2.2.3 Input Referred Noise Current Expression . . . . . . . . .. . 27

4.3 SiGe HBT Noise Parameter Equations . . . . . . . . . . . . . . . . . .. . . . 27

4.4 Noise Transit Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30

5 Verification by Device Simulations 33

5.1 SiGe HBT Process Simulation . . . . . . . . . . . . . . . . . . . . . . . .. . 33

5.2 SiGe HBT Device Simulation . . . . . . . . . . . . . . . . . . . . . . . . .. . 37

5.2.1 Device Remeshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2.2 Simulation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2.3 Impedance Field Method for Noise Simulations . . . . . . .. . . . . . 39

5.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42

6 Experimental Procedure and De-embedding Techniques 45

6.1 SiGe HBT Test Structures . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45

6.2 Experimental Setup and Procedure . . . . . . . . . . . . . . . . . . .. . . . . 47

6.2.1 S-Parameter Experiment . . . . . . . . . . . . . . . . . . . . . . . . .47

6.2.2 Noise Parameter Experiment . . . . . . . . . . . . . . . . . . . . . .. 48

6.3 Modelling of Parasitic Elements . . . . . . . . . . . . . . . . . . . .. . . . . 52

6.3.1 S-Parameter De-embedding . . . . . . . . . . . . . . . . . . . . . . .53

6.3.2 Noise Parameter De-embedding . . . . . . . . . . . . . . . . . . . .. 55

7 Verification by Experiments 61

7.1 Model Extraction for the Pads . . . . . . . . . . . . . . . . . . . . . . .. . . 61

7.2 Device Parameter Extraction . . . . . . . . . . . . . . . . . . . . . . .. . . . 62

7.2.1 Unity Gain Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . 62

7.2.2 Emitter Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.2.3 Base Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Table of Contents ix

7.2.4 Noise Transit Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7.3 Noise Parameters vs. Bias . . . . . . . . . . . . . . . . . . . . . . . . . .. . 66

7.4 Noise Parameters vs. Frequency . . . . . . . . . . . . . . . . . . . . .. . . . 69

7.5 Impact of Correlation at Millimetre-Wave Frequencies .. . . . . . . . . . . . 70

8 Conclusion 71

8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

8.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

A Detailed Derivation of SiGe HBT Noise Parameter Equations 73

A.1 Input Referred Noise Voltage . . . . . . . . . . . . . . . . . . . . . . .. . . . 73

A.2 Input Referred Noise Current . . . . . . . . . . . . . . . . . . . . . . .. . . . 77

A.3 Transforming the Noise Power Spectral Densities to Extrinsic Y-Parameters . . 80

B Conversion Between Intrinsic and Extrinsic Y-Parameters 85

B.1 Converting from Intrinsic to Extrinsic Y-Parameters . .. . . . . . . . . . . . . 85

B.2 Converting from Extrinsic to Intrinsic Y-Parameters . .. . . . . . . . . . . . . 88

C The Selectively Implanted Collector 91

C.1 Unity Gain Frequency Revisited . . . . . . . . . . . . . . . . . . . . .. . . . 91

C.2 The Role of the SIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

D Simulation Decks 93

D.1 Process Simulation (ATHENA) . . . . . . . . . . . . . . . . . . . . . . .. . . 93

D.2 Device Simulations (ATLAS) . . . . . . . . . . . . . . . . . . . . . . . .. . . 97

D.2.1 DC Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

D.2.2 AC and Noise Simulations . . . . . . . . . . . . . . . . . . . . . . . . 98

Bibliography 100

x Table of Contents

List of Tables

6.1 List of Equipment for S-Parameter Characterization of SiGe HBTs . . . . . . . 47

6.2 List of equipment for noise parameter characterizationof SiGe HBTs. . . . . . 50

7.1 Parameter values for lumped pad model . . . . . . . . . . . . . . . .. . . . . 61

xi

xii List of Tables

List of Figures

1.1 fT, fMAX andNFMIN at 65 GHz of a 500-GHz SiGe HBT. . . . . . . . . . . . 2

2.1 Chain Representation of Noisy Two-Port . . . . . . . . . . . . . .. . . . . . . 6

2.2 Noise equivalent circuit in [1]. . . . . . . . . . . . . . . . . . . . .. . . . . . 8

3.1 Typicalh21 (f), MAG (f) andU (f) characteristics of a SiGe HBT . . . . . . 13

3.2 TypicalℜZ12 versus frequency characteristics of a SiGe HBT . . . . . . . . 14

3.3 Hybrid-π equivalent circuit forRBX andRBI extraction . . . . . . . . . . . . 14

4.1 Noise Equivalent Circuit Model . . . . . . . . . . . . . . . . . . . . .. . . . 18

4.2 Equivalent circuit representation of Ebers-Moll model. . . . . . . . . . . . . . 20

4.3 Noise Equivalent Circuit with Polarity of Noise Sources. . . . . . . . . . . . . 22

4.4 Schematic of Noise Equivalent Circuit Defining Symbols used in Derivingvn . 22

4.5 Schematic for Deriving Output Short-Circuit Current ofChain Representation

of Noisy Two-Port . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.6 Schematic for Deriving Output Open-Circuit Voltage of SiGe HBT Noise Equiv-

alent Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.7 Schematic for Deriving Output Open-Circuit Voltage of SiGe HBT Chain Rep-

resentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.8 A generalizedπ network for the extraction ofgm0 exp (−jωτn). . . . . . . . . 31

5.1 SiGe HBT Process Flow Employing Selective SiGe Base Epitaxy: (a) emitter

window, SIC implantation, (b) underetch, (c) selective SiGe base epitaxy and

(d) emitter formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34

5.2 Modified SiGe Process Flow: (a) SIC implantation, (b) SiGe Base, (c)p+-poly

extrinsic base and (d) emitter formation. . . . . . . . . . . . . . . .. . . . . . 35

5.3 Cross Section of the Simulated SiGe HBT . . . . . . . . . . . . . . .. . . . . 36

5.4 Doping Profile and Germanium Content of the Simulated SiGe HBT . . . . . . 37

xiii

xiv List of Figures

5.5 Cross section of the simulated SiGe HBT structure after remeshing. (a) full

view and (b) zoomed in view. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.6 Modelling current fluctuations using an impedance field.. . . . . . . . . . . . 39

5.7 fT andfMAX vs. collector current density (IC/AE). . . . . . . . . . . . . . . . 42

5.8 Phase ofgm (ω) vs. frequency at minimum noise bias. . . . . . . . . . . . . . . 42

5.9 NFMIN at 1.9GHz vs. collector current density (IC/AE). . . . . . . . . . . . . 43

5.10 NFMIN andRn vs. frequency at minimum noise bias. . . . . . . . . . . . . . . 44

5.11 Real and imaginary parts ofYOPT vs. frequency at minimum noise bias. . . . . 44

6.1 Typical layout of SiGe HBT test structures . . . . . . . . . . . .. . . . . . . . 46

6.2 Typical layout of SiGe HBT dummy structures. (a) a open structure and (b) a

short structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.3 SiGe HBT S-parameter characterization equipment setup. . . . . . . . . . . . . 47

6.4 Focus Microwaves Noise Parameter Measurement Setup . . .. . . . . . . . . 49

6.5 Lumped element model for the parasitic elements in the SiGe HBT test structures. 53

6.6 Lumped element model for the (a) open and (b) short de-embedding structures. 54

6.7 A model of SiGe HBT test structures for noise parameter analysis . . . . . . . 55

6.8 Lumped models for the (a) input and (b) output networks ofSiGe HBT test

structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.9 Building blocks of the input and output equivalent circuits . . . . . . . . . . . 59

6.10 Signal pad lumped element model . . . . . . . . . . . . . . . . . . . .. . . . 59

7.1 Signal pad lumped element model . . . . . . . . . . . . . . . . . . . . .. . . 61

7.2 Measured and Modelledℜ (y11 + y12) vs. frequency. . . . . . . . . . . . . . . 62

7.3 Measured and Modelledℑ (y11 + y12) vs. frequency. . . . . . . . . . . . . . . 62

7.4 fT andfMAX vs. collector current density atVCE = 1.5 V. . . . . . . . . . . . . 62

7.5 fT andfMAX vs. VCE at peakfT bias. . . . . . . . . . . . . . . . . . . . . . . 62

7.6 ℜz12 vs. frequency characteristics. . . . . . . . . . . . . . . . . . . . . . . 63

7.7 Extraction ofRE from ℜz12 by extrapolation. . . . . . . . . . . . . . . . . . 63

7.8 Extraction ofRBX from ℜz11 − z12. . . . . . . . . . . . . . . . . . . . . . 64

7.9 Extraction ofRBI using the modified impedance circle method. . . . . . . . . 64

7.10 Extracted base resistance vs. bias. . . . . . . . . . . . . . . . .. . . . . . . . 64

7.11 Phase ofgm (ω) at minimum noise bias. . . . . . . . . . . . . . . . . . . . . . 65

7.12 Comparison between measured and modelledNFMIN at 2 GHz vs. bias (with

pad parasitics). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7.13 Comparison between measured and modelledRn at 2 GHz vs. bias (with pad

parasitics). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

List of Figures xv

7.14 Comparison between measured and modelledℜYOPT at 2 GHz vs. bias

(with pad parasitics). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66

7.15 Comparison between measured and modelledℑYOPT at 2 GHz vs. bias



pad parasitics). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67


parasitics). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67






pad parasitics). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68


parasitics). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68





7.24 Comparison between measured and modelledNFMIN vs. frequency at mini-

mum noise bias (with pad parasitics). . . . . . . . . . . . . . . . . . . .. . . . 69

7.25 Comparison between measured and modelledRn vs. frequency at minimum

noise bias (with pad parasitics). . . . . . . . . . . . . . . . . . . . . . .. . . . 69

7.26 Comparison between measured and modelledℜYOPT vs. frequency at min-

imum noise bias (with pad parasitics). . . . . . . . . . . . . . . . . . .. . . . 69

7.27 Comparison between measured and modelledℑYOPT vs. frequency at min-

imum noise bias (with pad parasitics). . . . . . . . . . . . . . . . . . .. . . . 69

7.28 Comparison between modelledNFMIN with and without correlation at 60 GHz

(without pad parasitics). . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 70

A.1 Schematic of Noise Equivalent Circuit Defining Symbols used in Derivingvn . 74

A.2 Schematic for Deriving Output Short-Circuit Current ofChain Representation

of Noisy Two-Port . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

A.3 Schematic for Deriving Output Open-Circuit Voltage of SiGe HBT Noise Equiv-

alent Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

xvi List of Figures

A.4 Schematic for Deriving Output Open-Circuit Voltage of SiGe HBT Chain Rep-

resentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

B.1 Equivalent circuit relating intrinsic and extrinsicy-parameters of SiGe HBT . . 86

C.1 Current Flow in Modern Vertical SiGe HBTs. . . . . . . . . . . . .. . . . . . 92

List of Abbreviations

FMIN Minimum noise factor

fT Unity current gain frequency

fMAX Maximum oscillation frequency

HICUM High Current Model (Bipolar)

NFMIN Minimum noise figure

Rn Equivalent noise resistance

SGP Spice Gummel Poon bipolar transistor model

YS,OPT Optimum source admittance

KVL Kirchoff’s Voltage Law

KCL Kirchoff’s Current Law

LNA Low noise amplifier

VCO Voltage controlled oscillator

xvii

xviii List of Figures

1 Introduction

1.1. Motivation

SiGe BiCMOS technology has emerged as a viable option for realizing millimetre-wave inte-

grated circuits. Traditionally, III-V technologies such as gallium arsenide and indium phos-

phide have been used at these frequencies due to their superior material properties such as

carrier mobility and breakdown voltage compared to silicon. SiGe BiCMOS, however, has

the advantage that it is built upon the vast experience and knowledge of the mature silicon

processing and hence enjoys a higher integration density and lower costs over III-V processes.

The research interests in SiGe BiCMOS based millimetre-wave circuits are evident in the

recent publications in journals and conferences [2–4]. This has fuelled continual research

and development efforts in the performance of the SiGe HBTs.With the aggressive scaling

of CMOS lithography, recent publications indicated that the record unity gain and maximum

oscillation frequencies for successfully fabricated SiGeHBTs have exceeded 300 GHz [5].

As an investigation into future SiGe HBT scaling, plotted inFig. 1.1 is thefT, fMAX and

NFMIN (at 65 GHz) versus current density for a TCAD-simulated 500-GHz SiGe HBT, respec-

tively. These aggressively scaled SiGe HBTs have the potential of being applied to advanced

millimetre-wave applications such as the 77-GHz automotive radar and tera-hertz imaging cir-

cuits.

Noise performance is important for the radio receivers in the above radar and imaging

circuits. It is also of critical importance to other millimetre-wave circuits, such as voltage con-

trolled oscillators (VCOs). This in turn requires accuratemodelling of the transistors, including

the SiGe HBTs. This thesis is concerned about the modelling of the noise of SiGe HBTs at

millimetre-wave frequencies.

At millimetre-wave frequencies, shot noise due to current traversing semiconductor junc-

tions and thermal noise due to access resistances are the main contributions to the noise of

a bipolar transistor. There are two shot noise sources in a bipolar transistor, one due to the

1

2 Introduction

0

100

200

300

400

500

600

FR

EQ

UE

NC

Y (

GH

z)

fTfMAX

1 10 100COLLECTOR CURRENT DENSITY [J C/AE] (mA/ µm

2)

0

2

4

6

8

10

NF

MIN

(dB

)

NFMIN @ 65GHz

Fig. 1.1: fT, fMAX and NFMIN at 65 GHz of a 500-GHz SiGe HBT.

base current and one due to the collector current and these two noise sources are statistically

correlated. In the low frequency domain, the correlation may be ignored with minimal impact

on model accuracy. However, the importance of noise correlation increases with frequency. It

has been recognized that failure to include the correlationleads to an overestimate of the noise

of millimetre-wave circuits such as the noise figure of low-noise amplifiers (LNAs) [4] and the

phase noise of VCOs [3].

In existing literature, the correlation between the base and collector shot noise is captured

through the noise transit time parameter,τn, which is presently extracted by fitting to measured

noise parameter data [6, 7]. However, noise parameters are difficult to measure. Not only that

the measurements are time consuming, the data usually have significant scatter. On the other

hand, a method to calculate the noise parameters solely fromthey-parameters of the transistor

was developed and presented in [1]. The shortcoming of this method is that it does not account

for the correlation between the base and collector noise sources.

Since theS-parameters of the transistors can be measured relatively accurately compared

to noise parameters, it would be useful to extend they-parameter technique in [1] to account

for the correlation.

1.2. Objective

The objective of this thesis is to extend the technique for extracting noise parameters from the

y-parameters of bipolar devices in [1] to account for correlation between the base and collector

shot noise. A new technique is also proposed to extract the noise transit time solely from the

y-parameters of the device.

1.3 Organization of Thesis 3

1.3. Organization of Thesis

This thesis is organized as follows. First, an overview of the existing literature on bipolar

noise modelling and parameter extraction is presented in Chapters 2 and 3, respectively. The

derivation of a set of equations that relates the noise parameters of SiGe HBTs to theiry-

parameters while accounting for shot noise correlation is presented in Chapter 4.

Verification of the equations is provided by simulations andexperiments. In Chapter 5, the

derived equations are first verified by applying them to simulated data. Experimental verifica-

tion is provided on 160-GHz SiGe HBTs by conducting theS-parameter and noise parameter

experiments described in Chapter 6. Finally, the experimental results are summarized in Chap-

ter 7.

4 Introduction

2 Background

SILICON germanium (SiGe) heterojunction bipolar transistors (HBTs) are different from

conventional III-V HBTs. In a III-Vnpn HBT, the material used for the emitter has a

larger band gap than that of the base. This is done to minimisethe injection of holes from

the base into the emitter. However, instead of using two materials of different band gaps, a

graded germanium profile is introduced into the epitaxial base of the SiGe HBTs to create an

electric field in the base. This electric field, in turn, exerts a force on the electrons injected

from the emitter, thus reducing the base transit time. Because of the grading of the base band

gap, sometimes SiGe HBTs are referred to as “graded-base-bandgap transistors” [8].

Nowadays, SiGe HBTs are rarely found in a standalone processbut rather integrated with

conventional CMOS in a SiGe BiCMOS process [9]. Since SiGe HBTs’ critical dimensions

are determined by epitaxy rather than costly lithography, this opens the door to exciting oppor-

tunities to integrate analog and high speed circuitry with lower speed digital blocks on a same

chip. For example, millimetre-wave transceivers can take advantage of the speed of SiGe HBTs

without incurring the high costs of nano-scale CMOS while the lower speed digital functions

can be implemented in conventional coarser lithography MOSFETs.

This chapter is begins by introducing the concept of noise correlation matrices as presented

in [10,11]. Then, presented in the subsequent sections are the existing state-of-the-art of noise

modelling of SiGe HBTs and the technique to extract the noiseparameters of a bipolar transis-

tor from itsy-parameters.

2.1. Noise Correlation Matrices

The noise correlation matrix concept presented in [10, 11] is used in this thesis to derive the

SiGe HBT noise parameter equations. Any noisy two-port network can be represented by an

equivalent noiseless two-port network and two noise sources [12]. These two sources can be

voltage sources in series, current sources in parallel or a combination of the two at the input

and/or output ports [12]. A Hermitian matrix whose entries are the ensemble-averaged self-

5

6 Background

and cross-power spectral densities of the terminal noise sources,si andsj, is known as a noise

correlation matrix [10]. Mathematically,

C =1

2∆f

[

〈s1s∗1〉〈s

1s∗2〉

〈s∗1s2〉〈s

2s∗2〉

]

, (2.1)

wheres1 ands2 are used in place ofv1 andv2 to emphasize that thesi’s can represent voltages

or currents.〈·〉 denotes the average over an ensemble of identical random processes. All four

noise parameters,Rn, the real and imaginary parts ofYOPT andFMIN, can be determined from

the entries of the noise correlation matrix of the chain representation.Rn is known as the

equivalent noise resistance.YOPT, known as the optimum source admittance, is the source

admittance that produces the lowest possible noise,FMIN. The chain representation, shown in

Fig. 2.1, consists of a series noise voltage source and a parallel noise current source at the input

port.

vn

in

+

++

−

−−

V1V2

Noise-Free Block

Fig. 2.1: Chain Representation of Noisy Two-Port

The noise correlation matrix of the chain representation is

CA =1

2∆f

[

〈vnv∗n〉〈vni∗n〉

〈v∗nin〉〈ini∗n〉

]

. (2.2)

The expressions of the four noise parameters in terms of the entries of the correlation matrix

are [10,11]

Rn =CA11

2kBT(2.3)

YOPT =

√

CA22

CA11

−[

ℑ(

CA12

CA11

)]2

+ jℑ(

CA12

CA11

)

(2.4)

FMIN = 1 +CA12 + CA11Y

∗OPT

kBT(2.5)

2.2 State of the Art 7

wherekB is the Boltzmann’s constant andT is the absolute temperature in kelvin. It is now ob-

vious that if the expressions forvn andin are known, the expression for all the noise parameters

can be derived.

2.2. State of the Art

2.2.1. Current Status of SiGe HBT Noise Modelling

The impact of the correlation between the base and collectorshot noise current sources on

device noise modelling increases withf/fT, wheref is the frequency of interest andfT is

the unity current gain frequency of the transistor. However, the current versions of the bipolar

transistor models such as SGP and HICUM do not capture this aspect in noise simulations. This

leads to overly pessimistic noise simulation results on advanced millimetre-wave integrated

circuits, since the main effect of noise correlation is a reduction in the minimum noise figure of

the device. Recent publications on 60 GHz SiGe HBT circuits have indicated that the simulated

noise figure of LNAs [4] and the simulated phase noise of VCOs [3] are higher than their

measured values. Given that the circuits are implemented indifferent SiGe BiCMOS processes

from two different foundries, one can infer that the issue with pessimistic simulation results is

not related to model extraction, but would likely be explained by noise correlation not being

captured in the transistor models.

Several bipolar noise models that capture shot noise correlation have been published re-

cently [6, 7, 13–15]. However, they suffer from one or both ofthe problems outlined below.

First, the models in [13–15] require the extraction of either a complete hybrid-π or the T- small

signal equivalent circuit for the intrinsic transistor. The problem with this approach is that the

inevitable uncertainty in extracting the first tier parameters due to measurement uncertainty

and the extraction technique employed may cause unphysicalvalues to be extracted for the

subsequent parameters. Also, by incorporating a specific small signal equivalent circuit into

the noise equivalent circuit used in the derivation of noiseparameter equations, the model im-

plicitly inherits all the approximations in the small signal equivalent circuit. Hence, in addition

to being limited by the validity of the noise equivalent circuit, the model is also limited by the

validity of the small signal equivalent circuit and its parameter extraction methodology.

Second, [6, 7] rely on data fitting their model to measured noise spectrum to extract one

of their parameters, the noise transit time. In practice, this is met by one major difficulty.

Current noise parameter measurements techniques require elaborate characterization and de-

embedding of the contributions from the input and output text fixtures to obtain the DUT noise

parameters. Unfortunately, the measured DUT noise parameters include the contribution from

the pads and interconnects in addition to the actual transistor noise parameters. At radio fre-

8 Background

quencies, the contribution from the parasitics may simply be ignored and the measured noise

parameters are taken to be the noise parameters of the transistor, or simple lumped-element

approximations may be used to model and de-embed the contributions from the parasitics.

However, at millimetre-wave frequencies, the validity of the lumped-element approximations

of the parasitics is often questioned and more elaborate de-embedding techniques are required.

Due to the number of de-embedding steps required and the complexity of noise parameter mea-

surements, significant scatter is usually present in the measured noise parameter data, which

limits the accuracy of noise transit time extraction.

2.2.2. An Existing Technique for the Extraction of Noise Par ameters from y-

Parameters

A technique to extract the noise parameters of a bipolar transistor from itsy-parameters with-

out accounting for noise correlation was derived and presented in [1]. The derivation of the

technique was based on the noise equivalent circuit shown inFig. 2.2. Note that this equivalent

〈i2nC〉〈i2nB〉

RB

RE

〈v2nE〉

〈v2nB〉 YINT

Y

EE

CB

Fig. 2.2: Noise equivalent circuit in [1].

circuit is the same as the one that will be used in this thesis to derive the noise equations that

account for noise correlation. However, this work will not neglect the correlation between the

base and collector shot noise currents.

Since the power spectral densities of shot noise and thermalnoise are well known, the

expressions for the input referred noise voltage and noise current can be computed. The noise

parameters are then obtained using the noise correlation matrix as presented in section 2.1

2.2 State of the Art 9

as [1]

Rn =IC

2VT |y21|2+ RE + RB (2.6)

YOPT =

√

IB |y21|2 + IC |y11|2

2VT |y21|2 (RE + RB) + IC

−(

ICℑy112VT |y21|2 (RE + RB + IC)

)2

− jICℑy112

2VT |y21|2 (RE + RB) + IC

(2.7)

FMIN = 1 +IC

VT |y21|2×

ℜy11 +

√√√√

[

1 +2VT |y21|2 (RE + RB)

IC

][

|y11|2 +IB |y21|2

IC

− (ℑy11)2

]

,

(2.8)

whereyij are the extrinsicy-parameters of the SiGe HBT,IB andIC are the DC bias currents of

the base and collector, respectively andRE andRB are the emitter and base series resistances.

10 Background

3 Device Figures of Merit

T HIS CHAPTER presents an overview of the different SiGe HBT figures of merit relevant

for millimetre-wave circuit design and noise modelling. Various gains are defined and the

standard extraction techniques for the unity gain frequencies are presented in the first section.

The methodologies to extract the base, emitter and collector series resistances are presented in

the second section.

All parameters are extracted from the two-port electrical parameters (S/Y ) of the device

concerned. In simulations, the two-port parameters are calculated by the simulator while in

experiments, they are obtained from experimentalS-parameter data after the de-embedding of

the pad and interconnect parasitics. Since this work is mainly concerned with a SiGe HBT in

the common-emitter configuration, “port 1” and “port 2” of the two-port network parameters

are defined as the input (base) and the output (collector) ports, respectively.

3.1. Device Gains and Unity Gain Frequencies

The common-emitter unity current gain frequency,fT, and maximum oscillation frequency,

fMAX, are commonly used to characterise the high frequency performance of SiGe HBTs.fT

andfMAX, both of which are bias dependent, are defined as the frequencies where the small-

signal short-circuit current gain and the maximum available power gain (MAG), respectively,

drops to unity in magnitude. The maximum available gain [16]and the small-signal short-

circuit current gain, respectively, are defined as

MAG =

∣∣∣∣

s21

s12

∣∣∣∣

(

k −√

k2 − 1)

(3.1)

h21 ≡i2i1

∣∣∣∣v2=0

, (3.2)

11


whereh21 is the small-signal current gain,sij are theS-parameters of the transistor andk,

known as the stability factor, is defined as [16]

k =1 − |s11|2 − |s22|2 + |det [S]|

2 |s12| |s21|. (3.3)

Associated gain,GA, is also important as a figure of merit for a SiGe HBT. It is defined as

the maximum available power gain when the input is conjugately matched for minimum noise

figure,i.e.ZS = Z∗SOPT . It is analytically given by [17]

GA =

∣∣∣∣

y21

y11 + YOPT

∣∣∣∣

2 ℜYOPTGout

, (3.4)

where

Gout = ℜYout (3.5)

Yout = y22 −y12y21

y11 + YOPT

. (3.6)

In practice, equipment limitations forbid the measurementof theS-parameters up to the

unity gain frequencies of advanced SiGe HBTs. A commonly used practice is to extrapolate

the gains calculated from measuredS-parameters to determine the unity gain frequencies. As

shown in Fig. 3.1,20 log |h21 (f)| is linear versus log frequency in the high frequency domain

such thatfT can be determined by linear extrapolation. In contrast, as shown in Fig. 3.1,

MAG (f) does not have a constant slope, implyingfMAX cannot be extrapolated from MAG.

In fact, it is not defined for frequencies wherek < 1. This problem is solved by recognizing

the fact that bothMAG (f) and Mason’s unilateral power gain,U (f), defined as [16]

U =|(s21/s12) − 1|2

2k |s21/s12| − ℜ (s21/s12)(3.7)

cross 0 dB at the same frequency and that, as shown in Fig. 3.1,U (f) has a linear relationship

with log frequency in the high frequency range. Hence, instead of extrapolatingMAG (f),

fMAX is determined from thex-intercept of the extrapolatedU (f).

3.2 Series Resistances 13

35

30

25

20

15

10

5

CU

RR

EN

T G

AIN

(dB20)

91

2 3 4 5 6 7 8 910

2 3 4 5 6

FREQUENCY (GHz)

40

35

30

25

20

15

10

PO

WE

R G

AIN

(dB

10)

|h 21|

U MSG or MAG

Fig. 3.1: Typical h21 (f), MAG (f) and U (f) characteristics of a SiGe HBT

3.2. Series Resistances

All the series resistances have to be extracted in order to calculate the noise parameters of the

SiGe HBT concerned. For simplicity, the collector resistance RC is assumed to be constant

with respect to bias and is obtained from the foundry provided model for fabricated devices

and estimated from the sheet resistance of the buried subcollector layer for simulated devices.

A bias independent emitter resistanceRE is extracted from theℜZ12 versus1/IE char-

acteristics [18]. With reference to Fig. 3.2,ℜZ12 is averaged across the low frequency do-

main where it is relatively constant to minimize experimental uncertainty. From the averaged

ℜZ12 versus1/IE characteristics, whereIE is the emitter DC current,RE is extracted as the

y-intercept of the extrapolation from the points corresponding to low bias currents. At high bias

currents, theℜZ12 versusIE may deviate from a straight line and these points are discarded

during data fitting. To minimize self-heating effects, thez-parameters are obtained from aVCE

of 1 V [1].

Compared with emitter and collector resistances, the base resistance is the most difficult

to extract. It is composed of an intrinsic and an extrinsic portion with different extraction

methodologies. The extrinsic portion is estimated fromℜZ11 − Z12 at high frequencies.

Based on the hybrid-π equivalent circuit shown in Fig. 3.3, the justification is that at sufficiently

high frequencies where the capacitancesCBCX , Cµ andCπ are approximately short circuits,

the hybrid-π model may be approximated by a T-network of three series resistances,RBX , RE

andRC . In this limit, RBX is simply given byℜZ11 − Z12.


30

25

20

15

10

5

0

RE

AL

(Z1

2)

(W)

605040302010

FREQUENCY (GHz)

Fig. 3.2: Typical ℜZ12 versus frequency characteristics of a SiGe HBT

RBX RBI Cµ

Cµx

RE

RC

g′mv′

beRπ Cπ

YINT

Fig. 3.3: Hybrid-π equivalent circuit for RBX and RBI extraction

3.2 Series Resistances 15

The intrinsic base resistance is extracted using the modified impedance circle method [19],

which augments the original impedance circle method to account forCµx. The original method

considers only the intrinsic transistor and models it usingthe hybrid-π equivalent circuit as

shown inside the dashed box in Fig. 3.3, except thatCµx is omitted. Based on the equivalent

circuit, it can be shown that [19]

hINT

11 ≡ 1

yINT11

=gBI + gπ + jω (Cπ + Cµ)

gBI (gπ + jω [Cπ + Cµ]), (3.8)

wherehINT11 is theh-parameter of the intrinsic transistor andgi = R−1

i . When plotted on the

complex plane,hINT11

forms a semi-circle and the intrinsic base resistance is extracted as the

high frequency intercept with the real axis.

The modified method that is used in this thesis accounts forCµx, including it in the equiv-

alent circuit as shown inside the dashed box in Fig. 3.3. Since it still does not account for the

series resistances, they are first de-embedded from the extrinsic y-parameters using (B.22)–

(B.25). By definition1 [19],

Z ≡ 1

yINT11 + yINT

12

=gBI + gπ + jω (Cπ + Cµ)

gBI (gπ + jωCπ), (3.9)

whereyINT11

andyINT12

are the intrinsicy-parameters of the SiGe HBT andgi = R−1

i . With the

assumption thatCµ is much smaller thanCπ [19],

Z ≈ gBI + gπ + jωCπ

gBI (gπ + jωCπ)=

(gπ

g2π + ω2C2

π

+1

gBI

)

− jωCπ

g2π + ω2C2

π

. (3.10)

Since,(

ℜZ − 1

2gπ

− 1

gBI

)2

+ ℑ2 Z =1

4g2π

, (3.11)

ℜZ andℑZ satisfy the equation of a circle centred at(1/2gπ + 1/gBI , 0) with radius

1/2gπ. Sinceω is physically limited to positive values, a semi-circle is traced out in the clock-

wise direction on the lower half of the complex plane with increasing frequency. Since,

limω→∞

ℜZ =1

gBI

≡ RBI (3.12)

limω→∞

ℑZ = 0, (3.13)

RBI ≡ g−1

BI is extracted as the high frequency intercept ofZ with the real axis.

1This parameter is often denoted ash′

11in the literature. In this thesis,Z is used to avoid confusion, sinceh′

11

is not obtained from theh-parameters of the transistor.


The modified impedance circle technique is applied to they-parameters of each of the bias

points to obtainRBI (IC). In practice, at sufficiently high frequencies,Z often deviates from

a circular behaviour and the corresponding data points are discarded for the extraction of the

intrinsic base resistance [17].

The difference between the original impedance circle method and the modified one is very

subtle. Although with the assumption thatCµ may be neglected compared toCπ, equation (3.8)

reduces to equation (3.10), the difference lies in the fact that different equivalent circuits used

are different for the two methods. In other words, because ofthe differences in the equivalent

circuits, whenCµ is neglected compared toCπ, the original impedance circle method identifies

gBI + gπ + jωCπ

gBI (gπ + jωCπ + Cµ)(3.14)

with thehINT11

of the transistor. However, the modified method identifies the same expression

with Z ≡(yINT

11+ yINT

12

)−1. Hence,RBI is extracted by fitting to two different sets of data in

the two methods, producing different results. Interestingly, with the assumption thatCµ may

be neglected, the centre and the radius of the circle for the modified method presented above

are similar to those in [17], which considers the original impedance circle method.

4SiGe HBT Noise Modelling at

Millimetre-Wave Frequencies

T HE DERIVATION of a set of equations for the noise parameters of bipolar transistors

in the common-emitter configuration that accounts for base and collector shot noise cor-

relation is presented in this chapter. A noise equivalent circuit is used to describe the noise

behaviour of the SiGe HBTs. The expressions of the input referred noise sources are deter-

mined based on the equivalent circuit. All of the parametersin the equations can be extracted

entirely from the small-signal two-port parameters of the device. Hence, the noise figures of

merit of the devices can be readily obtained and the providedmodels can be verified without

performing elaborate noise parameter measurements.

This chapter first presents the noise equivalent circuit used to model the SiGe HBTs. Its

advantages and limitations are also discussed. Then, the expressions of the input referred

noise sources are derived. Finally, it will be shown that thefour noise parameters,Rn, YOPT

andFMIN, can be calculated based on the derived equations using the noise correlation matrix

technique.

4.1. Noise Equivalent Circuit

The derivation of a new set of noise parameter equations is based on a SiGe HBT noise equiv-

alent circuit model for the extrinsic transistor, including all device parasitics. The equivalent

circuit consists of a black box representing the intrinsic transistor by its two-port parameters

together with lumped resistive elements modelling the portion of emitter and base resistances,

RE andRB respectively, that contribute noise to the extrinsic transistor. The schematic repre-

sentation of the noise equivalent circuit is shown in Fig. 4.1. YINT andY are they-parameter

matrices of the intrinsic and extrinsic transistor, respectively. 〈i2nC〉 and 〈i2nB〉 represent the

collector and base shot noise current power spectral density, respectively, of the intrinsic tran-

sistor.〈v2nE〉 and〈v2

nB〉 are the thermal noise voltage power spectral densities due to the emitter

and base resistances, respectively. The base resistance term includes contributions from both

the intrinsic and extrinsic components of the base resistance. The expressions of the base and

17

18 SiGe HBT Noise Modelling at Millimetre-Wave Frequencies

〈i2nC〉〈i2nB〉

RB

RE

〈v2nE〉

〈v2nB〉 YINT

Y

EE

CB

Fig. 4.1: Noise Equivalent Circuit Model

collector shot noise power spectral densities are

⟨i2nB

⟩= 2qIB∆f (4.1)

⟨i2nC

⟩= 2qIC∆f. (4.2)

The cross-power spectral density between the base and collector shot noise can be expressed

as [20]

〈i∗nBinC〉 = 2qIC [exp (−jωτn) − 1] ∆f, (4.3)

whereq is the positive electron charge,IB andIC are the DC base and collector currents,ω is

the angular frequency andτn is the noise transit time, which models the time delay between

the base and collector shot noise currents. At lower frequencies,exp (−jωτn) − 1 ≈ 0 and

this correlation term may be ignored. However, its importance increases with frequency and

cannot be ignored at frequencies approachingfT. The noise power spectral densities for the

resistancesRB andRE are

⟨v2

nE

⟩= 4kBTRE∆f (4.4)

⟨v2

nB

⟩= 4kBTRE∆f, (4.5)

wherekB is the Boltzmann constant andT is the absolute temperature in kelvin. In this deriva-

tion, all noise sources except the base and collector shot noise currents are assumed to be

uncorrelated. Although the expressions of the noise power spectral densities are given above,

4.1 Noise Equivalent Circuit 19

the derivation shown in the following sections does not assume a particular expression of the

noise power spectral densities, making this approach applicable to other devices that can be

modelled by Fig. 4.1 as well, not just bipolar transistors.

4.1.1. Approximations and Assumptions in the Noise Equival ent Circuit

Three significant assumptions are introduced in using Fig. 4.1 to model the noise of the SiGe

HBT. First, the distributed base resistance and base-collector capacitance network is greatly

simplified. In reality, the distributed nature of the transistor demands a distributedR-C network

similar to Fig. 4.2. The small signal equivalent circuits inthe transistor models HICUM [21]

and SGP are even more complex. However, to make the analysis manageable, a simplifica-

tion is made as indicated by the dashed line in Fig. 4.2. It amounts to moving nodeA located

betweenRBX andRBI to A′ located betweenRBI andCdBCi, as indicated in the diagram.

The extrinsic and intrinsic base resistance,RBX andRBI , respectively, are lumped together

asRB = RBX + RBI . Therefore, the thermal noise〈v2nB〉 includes contributions from both

RBX andRBI . The intrinsic transistor is modelled as a black box described by itsy-parameter

matrix,YINT [1]. An equivalent circuit forYINT is not required, since the small signal char-

acteristics of the intrinsic transistor are completely captured by itsy-parameter matrix at each

bias and frequency point.

Second, the noise due to the intrinsic base resistance is notstrictly 4kBTRBI [17], al-

though this is usually assumed in the literature. This is because the intrinsic base resistance

is a lumped modelling element introduced to describe a number of distributive effects, such

as the distributive base current and current crowding phenomena. As such, the intrinsic base

resistance is bias dependent and it is extracted by fitting tosimulated or measured data, rather

than directly calculated from the sheet resistance of the SiGe layer. Since it is not completely

a resistive component, strictly speaking, its noise power spectrum is not necessarily given by

4kBTRBI [17]. However, no significant errors are introduced when this is assumed provided

that the transistor is biased in the low injection regime andwith the assumption of insignificant

current crowding [17]. Complicating the issue is that as in most cases of device modelling it

is difficult to quantify when high level of injection or significant current crowding occurred.

Therefore, it is assumed that for all bias points below the peakfT/fMAX bias, which is usually

the highest bias point used in millimetre-wave circuit design, the noise of the intrinsic base

resistance can simply be modelled as4kBTRBI without introducing significant errors.

Third, the collector resistance is neglected. The overall noise figure of a cascade ofN noisy

stages is

F = 1 +N−1∑

i=0

Fi − 1∏i−1

j=0Gj

, (4.6)


RBX RBI

RE

RC

CdBCx

CdCS

CdBE

CDE

CdBCi

ISFISF/β

YINT

E

B C

A

A′

Fig. 4.2: Equivalent circuit representation of Ebers-Moll model [8].

whereFi andGi are the noise factor and the power gain of thei-th stage. Located at the last

stage, the thermal noise of the collector resistance is divided by the power gain of the transistor

and will have a negligible contribution to the overall noiseof the transistor.

4.1.2. Advantages of the Noise Equivalent Circuit

Using a black box to represent the intrinsic transistor has several advantages. First, the validity

of the noise equivalent circuit shown in Fig. 4.1 is independent of the validity of a particular

small-signal equivalent circuit for the intrinsic transistor.

Second and perhaps most importantly is that the noise parameters of a bipolar transistor

can be extracted from its two-port electrical network parameters. Noise parameter measure-

ments are not only lengthy, but also prone to measurement scatter. In contrast,S-parameters

can be measured with great accuracy at a fraction of the time compared to noise parameter

measurements.

Third, the number of small-signal parameters that have to beextracted to compute the noise

parameters is minimized. At high frequencies, the bipolar transistor equivalent circuit needs

to be elaborate to properly capture the performance of the device. If such an equivalent circuit

is used, the derivation will be unmanageable due to the number of nodes present. Since it is

impractical to adopt an equivalent circuit as complicated as the ones in the HICUM and SGP

models, simplifications and assumptions have to be made on top of those already present in the

4.2 Derivation of the Noise Parameter Equations 21

models. These assumptions and simplifications will directly affect the validity and accuracy of

the results.

The fourth advantage is that the noise parameter equations can also be applied to MOSFETs

and other types of transistors. For example, MOSFETs in a common-source configuration can

be modelled using the noise parameter equations derived in this thesis withRB replaced by the

gate resistanceRG, andRE replaced by the source resistanceRS.

4.2. Derivation of the Noise Parameter Equations

Before solving forvn andin, the polarities of the noise sources have to be defined. In general,

the absolute polarity of the noise sources is unimportant. However, the relative polarity of the

two correlated noise sources is significant. This is true because the equations (2.3)–(2.5) de-

pend only on the average values of productsisj where thesi’s can represent either the noise

voltage or the noise current. When ensemble averaged, a product of two uncorrelated noise

sources is zero. The relative direction of the correlated noise sources with one another is sig-

nificant, however. If the relative polarity is incorrect, the sign of the correlation termssisj |i6=j

will also be incorrect. When the direction ofsi is flipped,si mathematically becomes−si. This

negative sign remains after− sisj |i6=jis ensemble averaged. For the purpose of deriving the

noise parameter equations, the noise equivalent circuit isadopted with the relative polarity of

the noise sources as indicated in Fig. 4.3. BothinB andinC are chosen to flow into the emitter

to be inline with the shot noise models in the literature. In the rare case where a shot noise

model withinB andinC pointing in opposite directions is to be used, the shot noisemodel must

be modified accordingly such that〈i∗nBinC〉 → −〈i∗nBinC〉.

4.2.1. Input Referred Noise Voltage

The expression for the input referred noise voltagevn is obtained by first short-circuiting the

input and output ports of the SiGe HBT noise equivalent circuit and of the chain representation

of a noisy two-port network which are shown in Fig. 4.3 and Fig. 2.1, respectively. Then, the

expressions of the short-circuit currents at the output ports are equated to obtain an equation

for vn. The derivation of the short-circuit current of the SiGe HBTnoise equivalent circuit is

presented first, followed by the derivation of the short-circuit current of the chain representation

of a noisy two-port network.

4.2.1.1. Output Short-Circuit Current of SiGe HBT Noise Model

All symbols relevant to the derivation of the output short-circuit current of the SiGe HBT noise

equivalent circuit are defined in Fig. 4.4.


inCinB

RB

RE

vnE

vnB YINT

Y

EE

CB

+

+

−

−

Fig. 4.3: Noise Equivalent Circuit with Polarity of Noise Sources

inCinB

RB

RE

vnE

vnB YINT

Y

EE

CB+ +

+

+

− −

−

−

I INT1

I INT2

IRE

ISC

L1

L2

vX

vINT1 vINT

2

Fig. 4.4: Schematic of Noise Equivalent Circuit Defining Symbols used in Deriving vn


First, the following two equations are obtained by applyingKCL at the output node and at

the nodevX .

ISC + I INT

2 + inC = 0 (4.7)

IRE− I INT

1− inB − I INT

2− inC = 0 (4.8)

Then, three additional equations are obtained by applying KVL around the loopsL1 andL2

and from nodevX to ground.

vnB −(I INT

1 + inB

)RB − vINT

1 − vX = 0 (4.9)

vINT

2 + vX = 0 (4.10)

vnE + IRERE = vX (4.11)

Finally, from the definition ofy-parameters,

[

I INT1

I INT2

]

=

[

yINT11

yINT12

yINT21 yINT

22

][

vINT1

vINT2

]

(4.12)

Observing that since seven equations (4.7)–(4.12) with seven unknowns(ISC , I INT1

, I INT2

, vINT1

,

vINT2

, IREand vX) are obtained, the quantity of interest,ISC can be solved by algebraic sub-

stitutions. The mathematical details are in appendix A.1. The short-circuit current of the SiGe

HBT noise equivalent circuit is

ISC = −I INT

2 − inC (4.13)

whereI INT2

is given by,

I INT

2=

1

ζ

vnB

(yINT

21− det [YINT] RE

)− vnE

(yINT

21+ yINT

22+ det [YINT]RB

)

− inC

(det [YINT] RBRE +

[yINT

21 + yINT

22

]RE

)

− inB

(yINT

21RB +

[yINT

21+ yINT

22

]RE

).

(4.14)

4.2.1.2. Output Short-Circuit Current of the Chain Representation

The next step in calculating the input referred noise voltage expression involves determining the

output short circuit current of the chain representation ofthe SiGe HBT noise equivalent circuit

with its input port short-circuited. The chain representation of a noisy two-port is shown in

Fig. 2.1. In this particular case, the “noise-free block” inFig. 2.1 is the noise-free representation

of everything enclosed by the dashed-box,Y, in the SiGe HBT noise equivalent circuit in


RB

RE

YINT

Y

EE

CB

+ −

ISC

vn

Fig. 4.5: Schematic for Deriving Output Short-Circuit Current of Chain Representation ofNoisy Two-Port

Fig. 4.1. With the input of the chain representation short-circuited, the input referred current

source is “shorted-out,” resulting in a circuit as shown in Fig. 4.5.

Fig. 4.5 is similar to the original noise equivalent circuitin Fig. 4.4. With the noise sources

inB, inC andvnE set to zero and the polarity ofvnB reversed, the original noise equivalent

circuit is transformed into Fig. 4.5. Hence, the expressionfor the output short-circuit current

can be obtained from equation (4.13) by settinginB, inC andvnE to zero and changingvnB to

−vn. The result is

ISC =vn

(yINT

21− det [YINT] RE

)

ζ. (4.15)

4.2.1.3. Input Referred Noise Voltage Expression

Equating equations (4.13) and (4.15), the expression forvn is obtained as

vn = vnB +1

C(DvnE + EinB + FinC) (4.16)

where

C = yINT

21− RE det [YINT] (4.17)

D = −yINT

21− yINT

22− RB det [YINT] (4.18)

E = −RByINT

21 − RE

(yINT

21 + yINT

22

)(4.19)

F = ζ − RBRE det [YINT] − RE

(yINT

21+ yINT

22

)(4.20)


4.2.2. Input Referred Noise Current

The expression of the input referred noise current,in, is derived by equating the expressions

representing the output open-circuit voltages of the SiGe HBT noise equivalent circuit and

the chain representation of a noisy two-port network in the case where the input ports of both

circuits are open-circuited. The expression for the SiGe HBT noise equivalent circuit is derived

first, followed by that of the chain representation.

4.2.2.1. Output Open-Circuit Voltage of SiGe HBT Noise Model

All of the symbols used in the derivation are defined in Fig. 4.6. First, since both the input and

the output ports are open-circuited, using KCL at the input and output nodes gives

I INT

1+ inB = 0 (4.21)

I INT

2+ inC = 0. (4.22)

Subsequently, another equation is obtained by applying KVLaround loopL1. Since that there

is no voltage drop acrossRE because no current flows across it due to the open-circuit condi-

tions imposed on the input and output ports, KVL gives

vo = vINT

2+ vnE . (4.23)

Finally, from the definition ofz-parameters

[

vINT1

vINT2

]

= ZINT

[

I INT1

I INT2

]

(4.24)

whereZINT is thez-parameter matrix of the intrinsic transistor’s black box,given by

ZINT = Y−1

INT(4.25)

=1

det [YINT]

[

yINT22 −yINT

12

−yINT21

yINT11

]

.

The mathematical details of solving the above equations forvo is included in appendix A.2.

The open-circuit voltage is given by

vo =1

det [YINT]

(yINT

21 inB − yINT

11 inC

)+ vnE. (4.26)


RB

RE

YINT

Y

E E

CB+ + ++

+

− −

−−

−

I INT1

I INT2

vINT1

vINT2

vo

vnB

L1

vnE

inB inC

Fig. 4.6: Schematic for Deriving Output Open-Circuit Voltage of SiGe HBT Noise Equiva-lent Circuit

4.2.2.2. Output Open-Circuit Voltage of the Chain Representation

Shown in Fig. 4.7 is a schematic of the noisy SiGe HBT chain representation with the input and

output ports open-circuited. The variables used in the derivation of the open-circuit voltage are

also defined in the figure. Open-circuiting the output port necessitates that

I INT

2= 0, (4.27)

while using KCL at the input node gives

I INT

1= −in. (4.28)

To satisfy KVL in loopL1,

vo = vINT

2 + vX , (4.29)

where

vX = −inRE = I INT

1RE . (4.30)

By the definition ofz-parameters,

vINT

2= z21I

INT

1+ z22I

INT

2

=1

det [YINT]

(−yINT

21 I INT

1 + yINT

11 I INT

2

), (4.31)

4.3 SiGe HBT Noise Parameter Equations 27

RB

RE

YINT

Y

E E

CB

+

−

I INT1

I INT2

voL1

invX

Fig. 4.7: Schematic for Deriving Output Open-Circuit Voltage of SiGe HBT Chain Repre-sentation

where the last step is obtained from equation (4.25). The details of solving the above equations

are summarized in appendix A.2. The open-circuit voltage isgiven by

vo =yINT

21in

det [YINT]− inRE . (4.32)

4.2.2.3. Input Referred Noise Current Expression

The expression for the input referred noise current is obtained by equating the open circuit

voltages in equations (4.26) and (4.32).

in =yINT

21inB − yINT

11inC + vnE det [YINT]

J(4.33)

where

J ≡ yINT

21− RE det [YINT] . (4.34)

4.3. SiGe HBT Noise Parameter Equations

The self- and cross-power spectral densities ofvn andin are calculated from equations (4.16)

and (4.33). Assuming that all noise sources except〈i2nB〉 and〈i2nC〉 are uncorrelated one obtains

⟨v2

n

⟩≡ 〈v∗

nvn〉

=⟨v2

nB

⟩+

∣∣∣∣

D

C

∣∣∣∣

2⟨v2

nE

⟩+

∣∣∣∣

E

C

∣∣∣∣

2⟨i2nB

⟩+

∣∣∣∣

F

C

∣∣∣∣

2⟨i2nC

⟩+

2

|C|2ℜ (EF ∗ 〈inBi∗nC〉) (4.35)


⟨i2n⟩≡ 〈i∗nin〉

=

∣∣∣∣

yINT21

J

∣∣∣∣

2⟨i2nB

⟩+

∣∣∣∣

yINT11

J

∣∣∣∣

2⟨i2nC

⟩+

∣∣∣∣

det [YINT]

J

∣∣∣∣

2⟨v2

nE

⟩(4.36)

− 2

|J |2ℜ(yINT

21

(yINT

11

)∗ 〈inBi∗nC〉)

〈v∗nin〉 =

det [YINT]

J

D∗

C∗

⟨v2

nE

⟩+

yINT21

J

E∗

C∗

⟨i2nB

⟩− yINT

11

J

E∗

C∗〈i∗nBinC〉 (4.37)

+yINT

21

J

F ∗

C∗〈inBi∗nC〉 −

yINT11

J

F ∗

C∗

⟨i2nC

⟩.

It is more convenient to express the above power spectral densities in terms of the extrinsic

y-parameters of the SiGe HBT. While the equations derived in appendix B can be used, it

is necessary to setRC = 0 to be consistent with the chosen noise equivalent circuit. The

simplified equations ofyINTij are

yINT

11=

y11 − det [Y] RE

1 − y11RB −∑ij yijRE + RBRE det [Y](4.38)

yINT

12 =y12 + det [Y]RE

1 − y11RB −∑

ij yijRE + RBRE det [Y](4.39)

yINT

21=

y21 + det [Y]RE

1 − y11RB −∑ij yijRE + RBRE det [Y](4.40)

yINT

22 =y22 − (RB + RE) det [Y]

1 − y11RB −∑

ij yijRE + RBRE det [Y]. (4.41)

The algebraic details of rewriting the noise power spectraldensities in terms of extrinsic

y-parameters using equations (4.38)–(4.41) are detailed inappendix A.3. The expressions of

the self- and cross-power spectral densities of the input referred noise sources for the SiGe

HBT noise equivalent circuit in terms of extrinsicy-parameters are

⟨v2

n

⟩=⟨v2

nB

⟩+

∣∣∣∣1 +

y22

y21

∣∣∣∣

2⟨v2

nE

⟩

+

∣∣∣∣RB + RE

[

1 +y22

y21

]∣∣∣∣

2⟨i2nB

⟩+

∣∣∣∣y−1

21− RE

[

1 +y22

y21

]∣∣∣∣

2⟨i2nC

⟩

− 2

|y21|2ℜ [(y21RB + RE [y21 + y22]) (1 − RE [y21 + y22])

∗ 〈inBi∗nC〉] (4.42)

⟨i2n⟩

=

∣∣∣∣1 +

RE det [Y]

y21

∣∣∣∣

2⟨i2nB

⟩+

∣∣∣∣

y11 − RE det [Y]

y21

∣∣∣∣

2⟨i2nC

⟩+

∣∣∣∣

det [Y]

y21

∣∣∣∣

2⟨v2

nE

⟩

− 2

|y21|2ℜ [(y21 + RE det [Y]) (y11 − RE det [Y])∗ 〈inBi∗nC〉] (4.43)

4.3 SiGe HBT Noise Parameter Equations 29

〈v∗nin〉 =

det [Y]

y21

(

1 +y22

y21

)∗⟨v2

nE

⟩+

(RE det [Y]

y21

+ 1

)(

RB + RE

[

1 +y22

y21

])∗⟨i2nB

⟩

−(

y11 − RE det [Y]

y21

)(

RB + RE

[

1 +y22

y21

])∗

〈i∗nBinC〉

−(

RE det [Y]

y21

+ 1

)(

y−1

21 − RE

[

1 +y22

y21

])∗

〈inBi∗nC〉

+

(y11 − RE det [Y]

y21

)(

y−1

21 − RE

[

1 +y22

y21

])∗⟨i2nC

⟩. (4.44)

By transforming the equations into extrinsicy-parameters, a subtle point is introduced.

Throughout the derivation, it is assumed that the series resistancesRB andRE and the intrin-

sic y-parameter matrixYINT are mathematically independent of each other. However, this is

evidently not the case with extrinsicy-parameters. From Fig. 4.1, the extrinsicy-parameters

represents all elements within the dashed box, which includes the intrinsicy-parameter black

box and the series resistancesRB andRE. Hence, when the power spectral densities are written

in terms of intrinsicy-parameters as in equations (4.35)–(4.37), the series resistances appear

only explicitly. However, in equations (4.42)–(4.44), theseries resistances appear both explic-

itly and implicitly through the extrinsicy-parameters.

In the case of SiGe HBTs, the noise terms can be expressed by equations (4.1)–(4.5), which

are repeated below for convenience.

⟨v2

nE

⟩= 4kBTRE∆f (4.45)

⟨v2

nB

⟩= 4kBTRE∆f (4.46)

⟨i2nB

⟩= 2qIB∆f (4.47)

⟨i2nC

⟩= 2qIC∆f (4.48)

〈i∗nBinC〉 = 〈inBi∗nC〉∗ = 2qIC [exp (−jωτn) − 1] ∆f. (4.49)

From the above equations of the power spectral densities andthe equations for the thermal

noise and shot noise, the entries of the noise correlation matrix of the chain representation of

the SiGe HBT noise equivalent circuit are computed as

CA11 =〈v2

n〉2∆f

(4.50)

CA22 =〈i2n〉2∆f

(4.51)

CA21 = C∗A12

=〈v∗

nin〉2∆f

(4.52)


The noise parameters are then numerically calculated from the entries of the correlation matrix

as [10,11]

Rn =CA11

2kBT(4.53)

YOPT =

√

CA22

CA11

−[

ℑ(

CA12

CA11

)]2

+ jℑ(

CA12

CA11

)

(4.54)

FMIN = 1 +CA12 + CA11Y

∗OPT

kBT. (4.55)

4.4. Noise Transit Time

At low to moderate injection levels, individual carriers crossingp-n junctions can modelled

as independent random events [22]. Assuming that this condition holds, considering only the

one-dimensional intrinsic transistor and neglecting reverse leakage currents and recombination

in the base, it can be shown that in the common-emitter configuration1 [22]

〈i∗nBinC〉 = 2kBT (gm (ω) − gm0) , (4.56)

wherekB is the Boltzmann constant,T is the temperature in kelvin,gm (ω) is the high fre-

quency transconductance andgm0 is the DC transconductance. At high injection levels, the

carriers may no longer be considered independent and additional correction factors are re-

quired [22].

When equation (4.56) is rewritten as

〈i∗nBinC〉 = 2kBT (|gm (ω)| exp (−jθ) − gm0) , (4.57)

it can be seen that the variation in〈i∗nBinC〉 with frequency is mainly captured in the phase

of gm (ω). Recently published compact shot noise models introduced the noise transit time

parameterτn and expressed〈i∗nBinC〉 throughτn as [20]

〈i∗nBinC〉 = 2qIC [exp (−jωτn) − 1] , (4.58)

which is used in this thesis. The above equation assumes that|gm (ω)| = gm0 and the phase

of gm (ω) is given byωτn. Applicable only in low to moderate injection levels,τn is assumed

to be bias and frequency independent. At present,τn is extracted by fitting the noise model

1Low to moderate of injection is also assumed in the shot noiseequations for⟨i2nB

⟩and

⟨i2nC

⟩presented as

equations (4.1) and (4.2) which are repeated in (4.47) and (4.48).

4.4 Noise Transit Time 31

equations to the measured noise parameters as in [6]. This prevents noise parameters that

account for noise correlation be calculated solely fromS-parameters andτn has to be extracted

by fitting to measured high frequency noise parameters, which are more noisy compared to

S-parameters measured at the same frequency.

High injection effects are assumed to be insignificant up to the peakfT/fMAX bias. It is

proposed thatτn be extracted fromS-parameters as follows. The series resistancesRBX (IC),

RE andRC are extracted and de-embedded to obtain the intrinsicy-parameters using equa-

tions (B.22)–(B.25). It is assumed that the resulting intrinsic transistor can be adequately de-

scribed by a generalizedπ network as shown in Fig. 4.8. The general impedancesZ1, Z2 and

Z3 can be complex and frequency dependent. It can be shown that

yINT

21= gm0 exp (−jωτn) − 1

Z2

(4.59)

yINT

12= − 1

Z2

, (4.60)

whereyINTij are they-parameters of theπ-network shown in Fig. 4.8. The high frequency

transconductance is extracted asyINT21

− yINT12

and the noise transit time is extracted from the

y-parameters corresponding to peakfMAX bias as

τn = − ∂

∂ωphase

(yINT

21− yINT

12

)(4.61)

in the high frequency domain where the phase of the transconductance is linear. At first glance,

the proposed method has the same problem as equation (4.56) as τn is computed the phase

of gm whose high pass characteristic is problematic in practice.However, sinceτn is now

assumed to be frequency and bias independent, linear regression is performed in the domain

Z1

Z2

Z3vbe gm0e−jωτnvbe

+

−

Fig. 4.8: A generalized π network for the extraction of gm0 exp (−jωτn).


where the phase is linear with frequency andτn is then extracted as the negative of the slope of

the regression line. Measurement error is minimized through the linear regression technique.

5 Verification by Device

Simulations

V ERIFICATION of the derived noise parameter equations is first provided by device sim-

ulations. A 2-D SiGe HBT structure is constructed using a TCAD process simulator. Its

y-parameters are obtained from a device simulator. The derived noise equations and noise tran-

sit time extraction technique are applied to the simulated data to calculate the noise parameters.

These values are then compared with those that are directly calculated by the noise simulation

module in the device simulator using the impedance field method.

Note that the 2-D SiGe HBT simulated does not correspond exactly to fabricated device

that will be discussed in Chapters 6 and 7. This is because theexact geometry and doping

profiles of a SiGe HBT are considered proprietary information and hence unavailable. However

sometimes partial information such as the integrated dose in the base and germanium profiles

are available in the literature, such as in [23, 24]. Based onthese publications, the 2-D SiGe

HBT structure is adjusted to have similarfT/fMAX characteristics as the fabricated device.

5.1. SiGe HBT Process Simulation

This section describes the simulation of the process flow of a160 GHz SiGe HBT using Athena,

the process simulator within the Silvaco TCAD simulators suite. The process started with the

formation of then+ buried layer by ion implantation of Arsenic ions at50 keV at a dose of

7×17 cm−3. This was followed by a diffusion drive-in at1100C for two minutes. An Arsenic

doped,0.1 µm thick collector epitaxial layer was formed next, followed by the formation of the

shallow trench isolation (STI) regions and then+ collector reach through by ion implantation

using Arsenic. Geometrical etching was used to form the STI regions, which implied that the

silicon etched was merely removed from the structure [25]. In addition, geometrical etching,

which is simulated as a low-temperature process, ignores impurity redistribution [25]. Note

that geometrical etching was used to simulate all etching steps in this SiGe HBT fabrication

process.

33

34 Verification by Device Simulations

The base region was fabricated next. One common way of forming the base region in

commercial processes is as follows [26]. The extrinsicp+ base polysilicon and appropriate etch

stop layers are deposited and the emitter window is opened. Next, the selectively implanted

collector (SIC) is fabricated using the extrinsic base polysilicon as a self-aligned mask. The

purpose of the SIC is described in detail in Appendix C. An underetch is performed to remove

a thin layer of silicon under the emitter window. Finally, this void is filled by the intrinsic SiGe

base, which is in-situ doped and formed by selective epitaxy. Fig. 5.1 summarizes this SiGe

HBT fabrication method.

Fig. 5.1: SiGe HBT Process Flow Employing Selective SiGe Base Epitaxy: (a) emitterwindow, SIC implantation, (b) underetch, (c) selective SiGe base epitaxy and (d) emitterformation [26].

The problem in implementing this sequence in the Athena simulator is mainly in the SiGe

base epitaxial step to fill the void left by the underetch. In Athena, all deposition steps are

conformal, meaning that the same thickness is deposited on all surfaces, including the regions

underneath the base poly that are exposed by the underetch step. This often causes convergence

problems in subsequent simulations.

Therefore, the base processing steps were modified as follow. After the collector reach

through and the STI regions were formed, the SIC was implanted using an oxide layer as a

mask. The intrinsic SiGe base was then deposited, followed by the deposition of the extrin-

sic base polysilicon and the opening of the emitter window. This process flow avoided an

5.1 SiGe HBT Process Simulation 35

underetching step and hence eliminated the associated numerical convergence problems. The

modified flow is summarized in Fig. 5.2.

Fig. 5.2: Modified SiGe Process Flow: (a) SIC implantation, (b) SiGe Base, (c) p+-polyextrinsic base and (d) emitter formation.

The intrinsic base had a thickness of 20 nm and a uniform borondoping profile of1 ×19 cm−3. The germanium content was graded from 10% at the emitter side to 30% at the

collector side. To reduce boron out diffusion during subsequent thermal cycles, a boxed carbon

profile of 2 × 20 cm−3, which corresponds to 0.4% concentration, was also incorporated into

the SiGe base. The base doping, carbon content, germanium profile and thickness are similar to

values published in recent literature on SiGe HBTs that haveafT of about 230 GHz [23,24]. To

account for the out diffusion of boron in the presence of carbon in the SiGe layer, an empirical

boron diffusion model provided by Athena [25] was used in thesimulations.

The emitter polysilicon was deposited followed by a thermalanneal. Out diffusion of

arsenic from the emitter polysilicon formed the mono-emitter region. Note that in latest fabri-

cation processes, both the emitter and base polysilicon layers are silicided with titanium, cobalt

or nickel to reduce their sheet resistivity [27]. Although the silicidation process can be simu-

lated in Athena, the device simulator Atlas regards silicided layers as metals unless the sheet

resistance or resistivity of the layers are specified explicitly. Since the sheet resistance of these


polysilicon layers is neither measured experimentally norprovided by the foundry that fabri-

cated the 160 GHz SiGe HBTs used to experimentally verify thederived noise equations and

parameter extraction technique, therefore, all polysilicon layers simulated were not silicided to

avoid overly optimistic results. The cross section of the constructed device is shown in Fig. 5.3.

The doping profile and the germanium content along a line through the centre of the emitter is

shown in Fig. 5.4.

-2-1012

-0.4

-0.2

0

0.2

0.4

base base

collector collector

emitt

er

MaterialsSiliconSiO2PolysiliconAluminumSiGeElectrodes

DIS

TAN

CE

(µm

)

DISTANCE (µm)

Fig. 5.3: Cross Section of the Simulated SiGe HBT

Note that the 2-D SiGe HBT structure constructed is fairly ideal. Although dopant diffusion

due to annealing is captured by the process simulator, the geometry of individual components

such as the base polysilion are very regular. This is becauseof the geometrical etching used

in the simulations. A real device, however, does not have very regular geometries. As a

comparison, scanning electron microscopy (SEM) images of actual SiGe HBTs are available

in the literature, such as [28].

5.2 SiGe HBT Device Simulation 37

1016

1017

1018

1019

1020

1021

NE

T D

OP

ING

(c

m-3

)

0.400.350.300.250.200.15

VERTICAL DEPTH (mm)

35

30

25

20

15

10

5

GE

RM

AN

IUM

CO

NT

EN

T (%

)

NET DOPING GERMANIUM CONTENT

Fig. 5.4: Doping Profile and Germanium Content of the Simulated SiGe HBT

5.2. SiGe HBT Device Simulation

Verification of equations (4.42)–(4.44) and the new noise transit time extraction technique is

provided initially by device simulations. This has the advantage that experimental uncertainties

are eliminated from the verification. They-parameters of the SiGe HBT structure constructed

in Athena were obtained up to 100 GHz for different bias points. The equations and extraction

technique were applied to the simulatedy-parameters to calculate the noise parameters of the

device. The calculated noise parameters were then comparedto those calculated directly by the

Silvaco TCAD device simulator, Atlas, using the impedance field method [29]. Note that since

Atlas noise simulations did not involve a noise equivalent circuit, all approximations and sim-

plifications introduced into the derived equations were eliminated from the results calculated

by Atlas.

The details of the impedance field method are further described in section 5.2.3 following

the presentation of the remeshing of the SiGe HBT structure and the device simulation models

in the following two sections.

5.2.1. Device Remeshing

Quite often, the mesh used in process simulations is not suitable for device simulations. In

this particular case of SiGe HBTs, for example, a fine mesh is required to properly account for

the implantation and the subsequent dopant out-diffusion due to annealling of the subcollector

region. Also, a fine mesh is needed to properly resolve the implantation of the SIC region.

However, from the device simulation point of view, a fine meshis generally only required for


the emitter and the base regions. Since in most cases the simulation time grows geometrically

with the number of nodes, having a mesh that is too fine unnecessarily increases the simulation

time [30].

DevEdit, which is a software program part of the Silvaco simulation software suite, is used

to remesh the structure obtained from Athena for the device simulations. The following list

summarizes the important constraints used in remeshing thedevice.

1. A minimum of 20 and 10 points (vertically) should allocated for the base and emitter

regions, respectively.

2. A minimum of 10 points (vertically) are allocated for the collector.

3. The adaptive meshing algorithm is activated. This algorithm increases the number of

points in regions where the doping concentration is changing rapidly.

4. The maximum ratio between the lengths of the longest and shortest edge of a mesh

triangle is limited to 50. A triangle that is too thin may cause numerical problems.

Based on the above constraints, the program generates a new mesh for the structure. It will

also ensure a gradual change in the mesh from the coarse regions to the fine regions, which is

also important for numerical convergence. The final structure showing the new mesh is shown

in Fig. 5.5.

-2 -1 0 1 2

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

basebasecollectorcollector

emitt

er

substrate


Dis

tanc

e(µ

m)

Distance (µm)(a)

-1.2 -0.8 -0.4 0 0.4 0.8 1.2

-0.2

-0.1

0

0.1

basebase

collector

emitt

er


Dis

tanc

e(µ

m)

Distance (µm)(b)

Fig. 5.5: Cross section of the simulated SiGe HBT structure after remeshing. (a) full viewand (b) zoomed in view.


5.2.2. Simulation Models

The following models and options are enabled in the device simulations: drift-diffusion equa-

tions, lattice heating, Auger recombination, concentration dependent Shockley-Read-Hall re-

combination, concentration dependent mobility, field-dependent mobility, band-to-band tun-

nelling, band gap narrowing and Fermi-Dirac statistics. Representative simulation decks are

included in Appendix D.

5.2.3. Impedance Field Method for Noise Simulations

The noise simulation module in Atlas employs the impedance field method first derived by

Shockleyet. al. in [31]. This section will concisely describe this method. Additional details

may be found in [31].

For simplicity, consider a two-terminal active device connected as shown in Fig. 5.6, where

one of the terminals is grounded. Noise can be regarded as a superposition of small fluctuations

and an ideal noiseless signal. The noise due to a small volumedV can be modelled as extracting

electrons at a rate ofδIα/q and injecting the same number of electrons into another volumedV ,

as shown in Fig. 5.6. The subscriptα labels the location ofdV .

The noise voltage on the terminalN due to a small current injectionδIα at location~rα is

defined to be

δVN = ZNαδIα, (5.1)

whereδVN , ZNα andδIα are considered as complex quantities [31]. The real currentis given

by the real part ofδIα. Now, by superposition, the resultant noise voltage due to extraction of

current at~r′α and injection at~rα is given by

δVN = [ZNα (~rα) − ZNα (~r′α)] δIα (5.2)

= ∇ZNr · δ~r δIα = ∇ZNr · δPα, (5.3)

Fig. 5.6: Modelling current fluctuations using an impedance field.


where∇ZNr is known as the vector impedance field andPα is the dipole current vector. The

total noise voltage at nodeN is then given by the sum over all the small volume elements of

the device as

δVN =∑

α

∇ZNr · δPα. (5.4)

The quantity of interest for the purpose of characterising the noise performance of a device

is the power spectral densities of the noise voltageδVN . This can be evaluated using the

additivity theorem for spectral densities that is proved in[31]. The additivity theorem says if

δN =∑

j

MjδFj (5.5)

whereδFj is some random fluctuation andMj is some frequency dependent complex factor,

then the spectral density ofδN is given by

S (δN, ω) =∑

j

|Mj (ω)|2 S (δFj, ω) . (5.6)

whereS (·) denotes the power spectral density. The power spectral density is defined as

S (δN, ω) =

⟨(δN in ∆f aboutω)2

⟩

∆f. (5.7)

Using the additivity theorem, it can be concluded that

S (δVN , ω) =∑

α

|ZNr|2 S(

δPα, ω)

, (5.8)

whereS(

δPα, ω)

is the power spectral density of of the individual fluctuations in the small

volume elements. If the volume element is small, the the summation carries over to a volume

integral.

S (δVN , ω) =

∫

|∇ZNr|2 S(

δP , ω)

dV (5.9)

These small current fluctuations can be evaluated for different types of noise. For example,

in the case of diffusion noise,

S(

δP , ω)

= 4q2n (~r)D (ω) , (5.10)


wheren (~r) is the conduction carrier concentration at~r andD (ω) is the Einstein’s diffusion

constant [31], which is given by

D (ω) =kBT

qµ. (5.11)

The models used by the Atlas’ noise simulation module for other types of noise such as

generation-recombination noise are more complicated due to the number of parameters re-

quired. Details of these models can be found in [29].

Before proceeding further, one comment has to be made regarding the size of these volume

elements. The impedance field method as derived in [31] and implemented in Atlas’ noise

module does not account for correlation between the small fluctuations at different volume

elements. In other words, the volume elements are assumed tobe large enough such that the

fluctuationsδI at different sites can be assumed to be statistically independent. The correlation

between the base and collector terminal noise sources is captured, however, subject to the above

assumption. This subtlety is because the term “shot noise” refers to the noise power seen at

a terminal, which include noise contributions from all points in the device. The noise does

not happen just at the terminal but is rather distributed throughout the device. The statistical

correlation between the base and collector shot noise is fundamentally due to the fact that

these noise sources are caused by the same electrons. Hence,the correlation between them

is captured by the impedance field method and the simulated data can be used to verify the

derived noise equations and noise transit time extraction technique.

The derivation so far is limited to two-terminal devices. The impedance field method can

be extended to three-terminal devices. For these devices, there are four impedance fields:Z1n,

Z1p, Z2n andZ2p associated with the two ports of the device [29]. They account for the effect

of electron (n) and hole (p) fluctuations on the two ports. The spectral density for the case of

two-port devices is given by [29]

Sij (ω) =

∫

∇Zi (~r, ω)S (~r, ω) · (∇Zj (~r, ω))∗ dV, (5.12)

whereS (~r, ω) is the spectral density of the small current fluctuations at~r andi, j = 1, 2 label

the ports of the device.

Once the power spectral densities at both terminals are determined, they can be arranged in

a correlation matrix and then transformed into the chain representation (Fig. 2.1) using the ma-

trix equations in [10]. The noise parameters of the SiGe HBT,FMIN, YOPT andRn, can be cal-

culated directly from the correlation matrix for the chain representation using equations (2.3)–

(2.5) [10,11].


5.3. Simulation Results

fT andfMAX are extracted from the simulatedS-parameters of the SiGe HBT. The results are

plotted in Fig. 5.7. These results indicate that the structure has peakfT andfMAX values of

approximately 160 GHz.

160

140

120

100

80

60

40

FR

EQ

UE

NC

Y (

GH

z)

6 7 8 9

12 3 4 5 6 7 8 9

10CURRENT DENSITY [IC/AE] (mA/mm

2)

fT

fMAX

Fig. 5.7: fT and fMAX vs. collector current density (IC/AE).

The negative of the phase of thegm (ω) of the device at the simulated minimum noise bias

is plotted against the frequency in Fig. 5.8. Using the new technique presented in this thesis, a

value of 0.37 ps is extracted for the noise transit time,τn.

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-PH

AS

E(G

M)

(RA

DIA

NS

)

1009080706050403020100

FREQUENCY (GHz)

tn = 0.37ps

SIMULATED DATA LINEAR REGRESSION

Fig. 5.8: Phase of gm (ω) vs. frequency at minimum noise bias.

5.3 Simulation Results 43

The first requirement from the derived noise equations andτn extraction technique is that

the noise parameters calculated with correlation approachthose calculated without in the low

frequency range. Plotted in Fig. 5.9 are theNFMIN at 1.9 GHz calculated with and without

noise correlation from the derived equations and those calculated directly by the device simu-

lator. As expected, the values calculated with noise correlation are slightly lower than those cal-

1.0

0.8

0.6

0.4

0.2

0.0

NF

MIN

(d

B)

6 7 8 9

12 3 4 5 6 7 8 9

10

CURRENT DENSITY [IC/AE] (mA/mm2)

SILVACO SIMULATED WITH CORRELATION WITHOUT CORRELATION

Fig. 5.9: NFMIN at 1.9GHz vs. collector current density (IC/AE).

culated without. However, the difference between the two cases is very small (approximately

0.1 dB) at this frequency. Also, the difference between the values calculated with correlation

and those calculated by the simulator is approximately 0.05dB at minimum noise bias.

Following the low frequency verification, the equations areverified at millimetre-wave

frequencies. Plotted in Fig. 5.10 and 5.11 are the noise parameters versus frequency at the

minimum noise bias. It can be seen that at low frequencies, the calculated noise parameters

with and without correlation and those obtained directly from the device simulator match, in

agreement with the low frequency verification discussed above. However, as the frequency

increases, failure to account for noise correlation introduces appreciable error in almost all

cases. The imaginary part ofYOPT seems to be the exception. The values predicted with and

without correlation match the values calculated by the simulator with similar accuracy.


5

4

3

2

1

0

NF

MIN

(d

B)

100806040200

FREQUENCY (GHz)

1400

1200

1000

800

600

400

Rn (W

) SILVACO SIMULATED

EXTRACTED WITH CORRELATION EXTRACTED WITHOUT CORRELATION

Fig. 5.10: NFMIN and Rn vs. frequency at minimum noise bias.

2.0

1.5

1.0

0.5

0.0

Real(

YO

PT)

(mS

)

100806040200

FREQUENCY (GHz)

-2.0

-1.5

-1.0

-0.5

0.0

Imag

(Y O

PT ) (m

S)

SIMULATED

EXTRACTED WITH CORRELATION EXTRACTED WITHOUT CORRELATION

Fig. 5.11: Real and imaginary parts of YOPT vs. frequency at minimum noise bias.

6Experimental Procedure and

De-embedding Techniques

T HE VALIDITY of the derived SiGe HBT noise parameter equations is verified on a SiGe

HBT from an existing commercial SiGe BiCMOS process. The design of the SiGe HBT

test structures is briefly discussed, followed by the presentation of the experiments performed

and the analysis technique used to obtain the high frequencyfigures of merit and the noise

parameters of the transistors.

6.1. SiGe HBT Test Structures

SiGe HBT test structures were fabricated in Jazz Semiconductor’s 0.18µm SBC18HX SiGe

BiCMOS process, featuring SiGe HBTs with a peakfT of 155 GHz and aBVCEO of 2.2 V [32].

Each SiGe HBT test structure is composed of two100µm-pitch1 ground-signal-ground (GSG)

pads in the east-west configuration to provide connection tothe common-emitter SiGe device.

A typical layout is shown in Fig. 6.1. Each test structure is also accompanied by a dummy

open and a dummy short structure forS-parameter de-embedding purposes [33]. Shown in

Fig. 6.2, the open structure is obtained by removing the transistor from the test structure while

the short structure is formed by replacing the transistor with a ground connection. Both the

dummy structures use the same pad frame as the transistor test structure.

A general test structure design rule is to minimise parasitics by using good layout tech-

niques such that the devices may be characterized as accurately as possible. The following list

summarizes the key layout techniques used in the SiGe HBT test structures.

1. Optimize the spacing of the two GSG pads to limit the amountof cross talk between

them and to minimise the length of the interconnects from thepads to the transistor.

2. Minimize the size of the signal pads. The area of the signalpads are48 × 48µm2. The

pad capacitance is also minimised by employing only the top metal layer.

1The pitch is defined as the centre-to-centre separation between the ground and the signal pads.

45

46 Experimental Procedure and De-embedding Techniques

GND

S

GND

ALL DIMENSIONS IN MICRONS

GND

S

GND

48 40

4840

8880

100

68 60

129.26

PLANEM1 GROUND

Fig. 6.1: Typical layout of SiGe HBT test structures

S

GND

S

GND

GND

GND

S

GND

S

GND

GND

GNDCONNECTIONTO GROUND

(a) (b)

Fig. 6.2: Typical layout of SiGe HBT dummy structures. (a) a open structure and (b) ashort structure

3. Implement the interconnects from the signal pads to the transistor in the top metal layer.

4. Use metal 1 as a ground plane as shown in Fig. 6.1 to minimizeground inductance and

to avoid frequency dependent substrate loss.

5. Around the SiGe HBT, avoid crossing the metal lines, especially the base and collector

metals, to minimise parasitic capacitances. In addition, the interconnects are routed on

top of the oxide-filled isolation trenches around the deviceto reduce parasitic capacitance

to ground.

6.2 Experimental Setup and Procedure 47

6.2. Experimental Setup and Procedure

The two experiments performed on the SiGe HBTs to characterize their performance are de-

scribed in this section. The experimental procedures are defined below.

6.2.1. S-Parameter Experiment

TheS-parameters of the SiGe HBT test structures and the dummy structures were measured

up to 65 GHz. A diagram of the experimental setup is shown in Fig. 6.3 and the required

equipment is listed in Table 6.1.

G

S

G

WILTRON 360B

ANALYZERVECTOR NETWORK

PORT 1 PORT 2

HP4145 TRIAXIAL TO COAXIALADAPTER (CUSTOM)

HP4145SEMICONDUCTOR PARAMETERANALYZER

BIA

S, P

OR

T 1

INT

ER

NA

LT

O V

NA

INT

ER

NA

LT

O V

NA

BIA

S, P

OR

T 2

FR

OM

SM

U

FR

OM

SM

U

GPIB CONTROL

PROBE STATION

G

S

G

G

S

G

Fig. 6.3: SiGe HBT S-parameter characterization equipment setup.

Table 6.1: List of Equipment for S-Parameter Characterization of SiGe HBTsDescription EquipmentVector network analyzer Wiltron 360B VNADC bias supply HP4145B Semiconductor Parameter AnalyzerProbes Cascade Microtech 67 GHz,100µm pitch GSG Infinity

probesCalibration substrate Cascade Microtech impedance standard substrate, model

101-190


The following outlines the experimental procedure of the characterization, which is auto-

mated using a custom LabVIEW program. First, the VNA is calibrated at the reference planes

located at the probe tips using the network analyzer’s built-in LRM routine with the reflect

device being an open circuit. The open circuit condition is realised by lifting the probes at least

250µm above the calibration substrate in the air [34]. The qualityof the calibration is assured

by requiring the calibrated S-parameters of the thru line standard to exhibit

20 log |s11| ≤ −40 (6.1)

20 log |s22| ≤ −40 (6.2)

20 log |s12| ≤ 0.1 (6.3)

20 log |s21| ≤ 0.1. (6.4)

at all frequencies up to 65 GHz. TheS-parameters of the dummy de-embedding structures are

assumed to be bias independent. The DC bias of the SiGe HBT test structures is provided by

the HP4145B through the internal bias-tees of the network analyzer. At a fixedVCE of 1.5 V,

the appliedVBE is swept to obtain theS-parameters as a function of the collector bias current

and the frequency. The collector current is used as the independent variable, since the devices

are sensitive to temperature changes andVBE variations due to ohmic losses in the cables. A

second set ofS-parameters are obtained versusVCE with IC set at the peakfT/fMAX bias.

6.2.2. Noise Parameter Experiment

The four noise parameters of the SiGe HBT test structures,Rn, the real and imaginary parts of

YOPT andFMIN, were measured up to 18 GHz using the setup shown in Fig. 6.4. This upper

frequency limit is due to the limitation of the equipment. Table 6.2 summarises the equipment

used in the noise parameter experiment.

Performed using the cold-noise technique in an unshielded environment, the measurement

uses the noise source only to calibrate the noise receiver [35]. In actual measurements, the

thermal noise generated by the test setup at the input to the test structure is used as a noise

source and the noise figure is calculated from the noise powermeasured by the noise figure

meter. The input tuner changes the impedance seen by the teststructure at its input port. All

four noise parameters can be determined by fitting the measured noise figure at different tuner

settings, corresponding to different source admittances,to

F (YS) = FMIN +Rn

ℜ (YS)|YS − YOPT|2 . (6.5)

WinNoise, the piece of software provided by Focus Microwaves to control all the components

6.2E

xperimentalS

etupand

Procedure

49

G

S

G

WILTRON 360B

ANALYZERVECTOR NETWORK

PORT 1 PORT 2

AGILENT 4352B

ANALYZERVCO/PLL SIGNAL

DC CONTROL DC POWER

HP83650BSERIES SWEPTSIGNAL GENERATOR

RF OUT

HP8971CDOWN CONVERTMIXER

RF IN

HP8970BNOISE FIGURE METER

DRIVENOISE SOURCE

NOISE COM0.01−18GHzNOISE SOURCE

NOISE SOURCEREFLECTION COEFFICIENT

KMM

FF

FF

FF

KMM

FF

TU

NE

R C

ON

TR

OL

IEEE 488 (GPIB)

SWITCH CONTROLSWITCH CONTROL

HP

8970

B S

YS

TE

M IN

TE

RF

AC

E B

US

N34

6C

NOISE FIGURE TEST SET

APC7−SMAAPC7

K−MALE

PROGRAMMABLETUNER

COMPUTER

DC BIAS, VDC BIAS, V BE CE

BIAS−TEE BIAS−TEESW SW

LEGEND:KMM − K MALE−TO−MALE ADAPTERFF − 3.5mm FEMALE−TO−FEMALE ADAPTERAPC7−SMA − APC 7 TO SMA ADAPTERSW − AGILENT DC−40GHz TRANSFER SWITCH

INPUT STAGE

G

S

G

G

S

G

Fig. 6.4: Focus Microwaves Noise Parameter Measurement Setup


Table 6.2: List of equipment for noise parameter characterization of SiGe HBTs.Description EquipmentVector network analyzer Wiltron 360B network analyzerDC bias supply Agilent 4352B VCO/PLL signal analyzerTuner Focus Microwaves 0.8–18 GHz computer controlled

mechanical tunerCoaxial switches Agilent 8765D OPT 010-292 DC to 40 GHz coaxial

SPDT switches (×2)Bias-tees 40 GHz bias-teesNoise receiver HP8970B noise figure meter + HP8971C down con-

version mixerMixer LO supply HP83650B 0.01–50 GHz series swept signal genera-

torNoise source NoiseCom 346B 0.01–18 GHz coaxial noise sourceProbes GGB Industries 40 GHz,100µm pitch GSG probesOn-wafer calibration substrateCascade Microtech impedance standard substrate,

model 101-190Coaxial calibration standard Anritsu K calibration kit, model 3652

except the DC bias supply2 and to determine the noise parameters by data fitting, does not

directly use equation (6.5) [35]. The reason is mainly because the equation is nonlinear with

respect to the noise parameters. Instead, it uses an alternate form of equation (6.5) [36]. In this

alternate form, four new noise parametersA, B, C andD are defined and

F (YS) = A + Bℜ (YS) +C + Bℑ2 (YS) + Dℑ2 (YS)

ℜ (YS). (6.6)

In the above equation, the unknown parameters can be easily found by linear least squares

fitting. Rn, the real and imaginary parts ofYOPT andFMIN are determined as [36]

Rn = B (6.7)

YOPT =

√4BC − D2

2B− j

D

2B(6.8)

FMIN = A +√

4BC − D2. (6.9)

Using the low noise DC power supply in Agilent 4352B to bias the SiGe HBT test structures

reduces error in the measured noise figure by minimising the contribution of power supply

noise. Any noise from the power supply is not accounted for inthe noise meter calibration. The

DC power and DC control outputs of the Agilent 4352B can supply up to 50 mA and 20 mA,

2The Agilent 4352B is not supported by WinNoise.

6.2 Experimental Setup and Procedure 51

respectively [37], which is sufficient for the biasing of theSiGe HBTs. On the other hand, the

use of the switches in the test setup improves the repeatability of the measurements and reduces

the wear on the equipment. The switches allow the system to switch between the VNA and the

noise measurement system without any disassembly of the cables. This is especially important

since theS-parameters of the test setup have to be measured in the calibration process.

The following list briefly describes the procedure used in the noise parameter experiment.

Refer to [35] for detailed information on the noise parameter measurement system.

1. Perform a coaxial SOLT calibration using the K calibration kit on the Wiltron 360B

network analyzer. The calibration is performed at the frequency points where the noise

parameters will be measured and the calibration reference planes are located at the ends

of the two coaxial cables connected to the network analyzer.

2. Measure the reflection coefficient of the coaxial NoiseCom346B noise source in its off

state by performing a one-port S-parameter measurement.

3. Measure the one-port S-parameters of three coaxial standards, an open, a short and a

50 Ω match. These coaxial standards are used in a later step to determine the loss in the

experimental setup between the output of the noise source and the tip of the input probe.

4. Reassemble the system as shown in Fig. 6.4.

5. Recalibrate the Wiltron 360B network analyzer at the frequency points where the noise

parameters will be measured using a on-wafer LRM calibration technique with the ref-

erence planes located at the probe tips.

6. Measure the loss of the setup between the output of the noise source and the input probe

tip. A thru line on the calibration substrate is connected between the two GSG probes.

The noise source is disconnected, the input switch is shifted to the path of the noise

source and the three standards measured in step 3 are connected one at a time in place of

the noise source to the measurement setup. The two-portS-parameters of the network

are determined by measuring theS22 for each of the three cases. TheS-parameters of the

input network excluding the noise source are determined from the three sets of measured

S-parameters.

7. Calibrate the tuner with the noise source reconnected to the setup and the probes landed

on a thru line on the calibration substrate. At each of the measurement frequencies, the

tuner is tuned to 200 different impedance states that are evenly distributed on the Smith

chart. The reflection coefficient as the input probe tip is measured for each of the states.


The maximum attainable magnitude of the reflection coefficient is limited by the tuner

and is different for different frequencies.

8. Out of the 200 impedance states, about 25–30 where the noise figure of the test structure

will be measured are selected. For subsequent measurements, the previous step may be

skipped and the tuner only recalibrated at the selected points. To improve the accuracy

of the measurements, select the states closest to the expectedYOPT. In addition, avoid

the points inside the stability circles, which correspond to the source reflection coeffi-

cients that causes the stability factor (k) of the SiGe HBT test structure to be less than

unity. With k < 1, the SiGe HBT test structure is unstable and may oscillate, causing

measurement errors.

The choice of states is based on experience and the initial guess ofYOPT may be obtained

from circuit level simulations. The stability circles of the SiGe HBT test structure are

plotted after itsS-parameters are measured in step 11.

9. Measure the input impedance of the noise receiver stage asan one-portS-parameter

measurement with the probes remaining on the calibration substrate’s thru line.

10. Calibrate the noise receiver at different source impedances using the excess noise ratio

data of the noise source. Extract the noise parameters of thenoise receiver.

11. Measure theS-parameters of the DUT at all the frequencies and bias points. The states

where the noise figure will be measured, as selected in step 8,are verified to be outside

the stability circles.

12. Measure the noise figure of the test structure at the selected tuner settings. Instead of

directly measuring the noise figure of the SiGe HBT test structure, the system measures

the noise figure of the test structure and the noise receiver as a cascade. TheS-parameters

of the test structure and the input impedance of the noise receiver are used to de-embed

the contribution from the measurement setup and to account for the reflections at the

DUT input and output, since the test structures are neither input nor output matched.

The noise parameters of the SiGe HBT test structures are extracted from its measured

noise figure at different source impedances.

6.3. Modelling of Parasitic Elements

The electrical effects of the test structure pads and the interconnects leading from them have

to be accounted for such that the SiGe HBTs can be accurately characterized. This section

6.3 Modelling of Parasitic Elements 53

outlines the techniques to account for the parasitic elements in theS-parameter and noise pa-

rameter experiments.

6.3.1. S-Parameter De-embedding

Commonly referred to as the “open-short” method, the technique used to mathematically re-

move the effects of the parasitic elements from the measuredS-parameters of the SiGe HBT

test structures was first presented in [33]. The parasitic elements in the SiGe HBT test struc-

tures are modelled as shown in Fig. 6.5. This technique ignores the distributed nature of the

SiGeHBT

ZP1 ZP2

ZP3

ZS1 ZS2

ZS3

Fig. 6.5: Lumped element model for the parasitic elements in the SiGe HBT test struc-tures [33].

parasitics and model them as lumped elements. The signal pads are assumed to be described

by ZP1 andZP2 while the interconnects from the pads are described by the series elementsZS1

andZS2. ZP3 models the coupling between the input and output ports andZS3 describes the

ground connection from the SiGe HBT. Although lumped element approximations are used, a

specific equivalent circuit is not assumed for any of the lumped elements. In fact, their charac-

teristics may be frequency dependent.

Two dummy structures are used to obtain the complex impedances of the lumped ele-

ments [33]. The lumped element equivalent circuit for the dummy open and the dummy short

structures are shown in Fig. 6.6. In general, the open and short structures capture the pad and

the interconnect parasitics, respectively.

The parasitic elements are mathematically removed from themeasured S-parameters to

obtain the two-port parameters of the SiGe HBTs. At each frequency and bias point

1. Convert the measuredS-parameters of the open and short structures toy-parameters,

denoted byYOPEN andYSHORT , respectively.

2. RemoveZP1, ZP2 andZP3 from the short structure to obtain the two-port parameters

of the T-network made up ofZS1, ZS2 andZS3. This is done by recognizing that the


(b)(a)

ZP1ZP1 ZP2ZP2

ZP3ZP3

ZS1 ZS2

ZS3

[ZS−O

]

Fig. 6.6: Lumped element model for the (a) open and (b) short de-embedding struc-tures [33].

elements to be removed form a network that is connected in parallel to the T-network

and hence can be achieved byYSHORT − YOPEN .

3. Convert the measuredS-parameters of the SiGe HBT test structures toy-parameters,

representing the result asYTS.

4. RemoveZP1, ZP2 andZP3 from YTS by YTS − YOPEN .

5. T-network is de-embedded next from the SiGe HBT test structure. This is achieved

by first converting the matrix obtained in the previous step to z-parameters and then

subtracting thez-parameters of the T-network.

The de-embedding technique can be elegantly summarized as

ZHBT = (YTS − YOPEN)−1 − (YSHORT − YOPEN)−1 (6.10)

whereZHBT is thez-parameter matrix of the SiGe HBT obtained by de-embedding the para-

sitic elements.

Although [33] is simple and easy to implement, it is not without its shortcomings. The

most important one lies in its use of lumped elements to modeland de-embed the parasitics. In

the millimetre-wave regime, the lumped element approximation starts to fail, directly affecting

the validity of [33]. Advances have been made such as in [38] and [39] to attempt to resolve

this problem. However, the required de-embedding structures were not included in this work.


6.3.2. Noise Parameter De-embedding

In contrast toS-parameters, noise parameters are more complicated to de-embed. Rather than

de-embedding the measured data, the parasitics are embedded into the noise parameter equa-

tions and the results are compared with the experimental data. This is opposite to the method

employed in theS-parameter experiment where the parasitics are mathematically removed

from the measured values. The approach taken is a direct application of [10, 11]. The follow-

ing shall first present a general methodology to embed parasitics. A specific case applicable to

this work is presented afterwards.

For noise parameter analysis purposes, the SiGe HBT test structures are modelled as shown

in Fig. 6.7.〈v2n〉 and〈i2n〉, which may be calculated from equations (4.42)–(4.44), arethe power

SiGe HBT OutputNetwork

Input Network

[ASiGe]

〈v2n〉

〈i2n〉〈i2IN1〉〈i2IN2

〉〈i2OUT1〉〈i2OUT2

〉[YIN ] [YOUT ]

︸︷︷︸

CA1−2, A1−2

︸︷︷︸

CA, A

Fig. 6.7: A model of SiGe HBT test structures for noise parameter analysis

spectral densities of the input referred noise voltage and current, respectively, of the SiGe HBT.

Represented by theiry-parameter matrix,YIN andYOUT , respectively, the input and output

network are assumed to be passive and non-interacting except through the SiGe HBT. The noise

properties of each of the two networks are modelled in admittance formalism by two noise

current sources. The case where the input and output networks interact is solved in [38,40] by

performing four-port network analysis. In the following, the two equations which serve as the

foundation of the embedding technique are presented.

The first equation relates the noise correlation matrix of a passive network to its two-port

network parameters. Derived from thermodynamics, the admittance correlation matrix whose

elements are the ensemble averages of the terminal noise currents of a passive network is given

by [41]

CY = kBT(Y + Y

†), (6.11)

wherekB is the Boltzmann’s constant,T is the absolute temperature in kelvin,Y is they-

parameter matrix of the passive two port andY† is the adjoint ofY. The second equation gives


the noise correlation matrix in the chain representation ofa cascade of two noisy two-port

networks as [10]

CA = A1CA2A†1+ CA1 (6.12)

whereCA is the overall chain noise correlation matrix,A1 is the ABCD matrix of the first

two-port andCA1 andCA2 are the chain correlation matrix of the first and second two-port

networks, respectively.

The chain noise correlation matrix of the overall cascade isobtained by utilizing the equa-

tions (6.11) and (6.12) as shown below. From equation (6.11), the noise correlation matrix in

admittance representation of the input network is

CY IN = kBT(

YIN + Y†IN

)

. (6.13)

From the admittance representation, the matrix may be converted to its chain representation as

required by equation (6.12) using [10]

CAIN = TCY INT†, (6.14)

whereT is the matrix that transformsCY IN from the admittance representation to the chain

representation. It is given by [10]

T =

[

0 a12

1 a22

]

, (6.15)

wherea12 anda22 are the elements of the ABCD matrix of the input network. The enumeration

of the matrix elements follows the row-column convention. The chain noise correlation matrix,

CA1−2, and the ABCD matrix,A1−2 of the input and SiGe HBT cascade are given by

CA1−2 = AINCA,SiGeA†IN + CAIN (6.16)

A1−2 = AINASiGe (6.17)

whereCA,SiGe is the chain correlation matrix of the SiGe HBT, obtainable from the noise

parameters as

CA,SiGe = 2kBT

[

RnFMIN − 1

2 − RnY ∗OPT

FMIN − 12 − RnYOPT Rn |YOPT|2

]

, (6.18)

whereRn, YOPT andFMIN are the noise parameters of the SiGe HBT as calculated from the

derived equations [10,11]. The above procedure is then repeated to cascade the input and SiGe


HBT networks with the test structure’s output network, resulting in

CA = A1−2CAOUTA†1−2

+ CA1−2

= AINASiGeCAOUTA†SiGeA

†IN + AINCA,SiGeA

†IN + CAIN (6.19)

A = A1−2AOUT

= AINASiGeAOUT , (6.20)

whereAOUT is the ABCD matrix of the output network.CAOUT is the noise correlation

matrix of the output network obtained by first calculating itin admittance representation by

equation (6.11) and then converting the result to chain representation by equation (6.14). The

noise parameters of the SiGe HBT test structure are calculated from its noise correlation matrix

derived above using equations (2.3)–(2.5).

Remaining to be determined are the ABCD matrices of the inputand output networks,

AIN andAOUT , respectively. However, because structures that allowAIN andAOUT to be

measured directly, such as those in [42], are unavailable, further assumptions have to be made

about the structure of the input and output networks.

It is assumed that the input and output network can be adequately described up to 18 GHz,

which is the upper limit on the noise parameter experiment, by the equivalent circuits shown

in Fig. 6.8, where the parasitics in each of the networks are lumped into two general impedances.

The idea is that the shunt elements capture the parasitics between the signal pads and the sub-

(a) (b)

ZSI

ZPI

ZSO

ZPO

Fig. 6.8: Lumped models for the (a) input and (b) output networks of SiGe HBT teststructures

strate while the series elements capture the resistance andinductance of the interconnects from

the signal pads to the SiGe HBT device. With reference to Fig.6.6, the four impedances are

obtained as

ZPI = ZP1 =(yOPEN

11+ yOPEN

12

)−1(6.21)

ZSI = ZS1 = zS−O11 − zS−O

12 (6.22)


ZPO = ZP2 =(yOPEN

22+ yOPEN

21

)−1(6.23)

ZSO = ZS2 = zS−O22 − zS−O

21 . (6.24)

To obtain the ABCD matrices of the equivalent circuits in Fig. 6.8, first consider the prob-

lem of determining the ABCD matrices of Fig. 6.9. By applyingthe definition of the ABCD

parameters, it can be shown that the matrices are

AP (ZP ) =

[

1 01

ZP1

]

(6.25)

AS (ZS) =

[

1 ZS

0 1

]

(6.26)

whereAP andAS are the ABCD matrices of the circuit in Fig. 6.9(a) and Fig. 6.9(b), respec-

tively. The ABCD matrices of the input and output equivalentcircuits are then obtained as

AIN = AP (ZPI)AS (ZSI) =

[

1 ZSI

1ZPI

ZSI

ZPI+ 1

]

(6.27)

AOUT = AS (ZSO)AP (ZPO) =

1 + ZSO

ZPOZSO

1ZPO

1

. (6.28)

In reality, due to measurement errors, the admittance of thepads,Z−1

PI andZ−1

PO, may ex-

hibit a small negative real part at a few frequency points. This is due to the real part of the

admittances of the dummy open structures being close to zero, especially at low frequencies.

Since the noise correlation matrix of a passive two-port network is given by equation (6.11),

an unphysical negative real admittance implies that the two-port enhances the output signal-to-

noise ratio. This problem is resolved by data fitting a pad lumped element model, as shown in

Fig. 6.10, to the data obtained from equations (6.21) and (6.23). The admittance across the pad

model can be written as

YPAD =

[(

jωCSUB +1

RSUB

)−1

+1

jωCPAD

]−1

=ω2C2

PADRSUB

1 + ω2R2

SUB (CSUB + CPAD)2+ jωCPAD

[

1 − ω2R2SUBCPAD (CPAD + CSUB)

1 + ω2R2

SUB (CSUB + CPAD)2

]

.

(6.29)

In the frequency range of interest,ω is on the order of1010 rad/s and for these test structures,

the capacitancesCPAD, CSUB are on the order of 10 fF.RSUB is on the order of10 Ω. With


(a) (b)

ZS

ZP

Fig. 6.9: Building blocks of the input and output equivalent circuits

these assumptionsω2R2

SUB (CSUB + CPAD)2 ∼ 10−6. Hence,

1 + ω2R2

SUB (CSUB + CPAD)2 ≈ 1. (6.30)

Similarly,ω2R2

SUBCPAD (CPAD + CSUB) ∼ 10−6. Therefore, the admittanceYPAD is approx-

imately

YPAD ≈ ω2C2

PADRSUB + jωCPAD. (6.31)

The above equation suggests that the three parametersCPAD, CSUB andRSUB may be found

as follows:

1. CalculateZ−1

PI andZ−1

PO from equations (6.21) and (6.23), respectively.

2. ObtainCPAD from the slope of the measured admittances versus angular frequency.

3. Plot the real part of the measured admittances versus angular frequency. ObtainRSUB

by fitting to the curvature of the parabola.

CPAD

CSUBRSUB

Fig. 6.10: Signal pad lumped element model


4. UseCSUB to obtain a better fit at high frequencies.

This approach however is not without limitations. First, athigh enough frequencies, the

lumped element approach taken in Fig. 6.8 may not be valid. Second, a more involved model

or equivalent circuit is required to properly model the substrate noise as frequency increases.

Both of these limitations point to a need for a better characterisation technique for the parasitic

input and output networks.

7 Verification by Experiments

T HE experimental verification results of the derived equations and noise transit time ex-

traction technique is presented in this chapter. Both theS-parameters and noise param-

eters of a SiGe HBT were measured using the experiments described in the previous chapter

to verify the developed technique. The transistor characterized had 2-emitter, 3-base and 2-

collector contacts and the emitter length was5.46µm long. The derived equations were applied

to the measuredS-parameters of the device to calculate its noise parameters. The calculated

results were compared with those measured by the noise parameter experiment.

The organization of this chapter is as follows. The extracted lumped element model for the

pads as described in section 6.3.2 is presented first. Then, acomparison is made between the

measured and calculated noise parameters of the SiGe HBT.

7.1. Model Extraction for the Pads

Using the methodology derived in section 6.3.2, the parameters required by the lumped pad

model are extracted and summarized in Table 7.1. The measured and modelled values are

CPAD

CSUBRSUB

Fig. 7.1: Signal pad lumped elementmodel

Table 7.1: Parameter values for lumpedpad model

Parameter Value

CPAD 10 fFCSUB 5 fFRSUB 60Ω

61


plotted in Figs. 7.2 and 7.3.

800

600

400

200

0

RE

AL

(Y11+

Y12)

(mS

)

605040302010FREQUENCY (GHz)

Measured

Modelled

Fig. 7.2: Measured and Modelledℜ (y11 + y12) vs. frequency.

3.5

3.0

2.5

2.0

1.5

1.0

0.5

IMA

G(Y

11+

Y12)

(mS

)

605040302010

FREQUENCY (GHz)

Measured Modelled

Fig. 7.3: Measured and Modelledℑ (y11 + y12) vs. frequency.

7.2. Device Parameter Extraction

7.2.1. Unity Gain Frequencies

From the de-embeddedS-parameters up to 65 GHz, thefT and fMAX of the transistor are

extracted and plotted in Fig. 7.4 versusJC = IC/AE, whereAE is the total emitter area.fT

andfMAX versusVCE at the peakfT bias are plotted in Fig. 7.5.

10-1

100 10

1

COLLECTOR CURRENT DENSITY [I C/AE] (mA/ µm2)

0

50

100

150

200

FR

EQ

UE

NC

Y (

GH

z)

fTfMAX

Fig. 7.4: fT and fMAX vs. collector cur-rent density at VCE = 1.5 V.

0.5 1.0 1.5COLLECTOR-EMITTER VOLTAGE (V CE)

0

50

100

150

200

FR

EQ

UE

NC

Y (

GH

z)

fTfMAX

Fig. 7.5: fT and fMAX vs. VCE at peakfT bias.

7.2 Device Parameter Extraction 63

7.2.2. Emitter Resistance

Fig. 7.6 plotsℜz12 versus frequency for all bias points measured. To avoid self-heating

effects, the emitter resistance is extracted atVCE = 1 V [1]. The low frequency values of

ℜz12 are averaged and plotted against the DC emitter bias currentin Fig. 7.7. The bias-

independent value forRE is extracted as they-intercept of the extrapolation to be2.08Ω, which

is reasonably close to the value extracted by the foundry.

30

25

20

15

10

5

0

RE

AL

(Z1

2)

(W)

605040302010

FREQUENCY (GHz)

Fig. 7.6: ℜz12 vs. frequency charac-teristics.

0.00 0.10 0.20 0.30 0.40IE

-1(mA

-1)

0

5

10

15

Re(

Z 12)

(Ω)

Slope: 25.6 mV

Intercept (R E) = 2.08Ω

Fig. 7.7: Extraction of RE from ℜz12by extrapolation.

7.2.3. Base Resistance

Theℜz11 − z12 characteristics of the transistor is plotted versus frequency in Fig. 7.8 for

all the bias points measured. In the high frequency domain, abias-dependentRBX is ex-

tracted as presented in section 3.2. The intrinsic base resistance is extracted using the modified

impedance circle method. A representative plot of[yINT

11+ yINT

12

]−1of one of the bias points

on the complex plane is shown in Fig. 7.9. The extractedRBX andRBI versus bias are plotted

in Fig. 7.10.

Upon comparison with the foundry values, unfortunately, ithas been found that the ex-

tracted intrinsic base resistance values experience a large error while the extrinsic base resis-

tance is relatively close. This is because the extrinsic base resistance is mostly due to the sheet

resistance of the base polysilicon and the base contacts andit is captured relatively well by the

extraction method used. However, the extraction of the intrinsic base resistance is more com-

plicated, as it captures various effects such as the distributive base current and emitter current

crowding. An alternate extraction methodology is available in [43]. However, it uses a separate

test structure that is not available in this work to measure the sheet resistance of the SiGe base.


50

40

30

20

10

RE

AL

(Z1

1-Z

12)

(W)

605040302010

FREQUENCY (GHz)

Fig. 7.8: Extraction of RBX fromℜz11 − z12.

-160

-140

-120

-100

-80

-60

-40

-20

IMA

G([

y1

1+

y1

2]-1

) (W

)

250200150100500

REAL([y11+y12]-1

) (W)

FITTED RESULTS MEASURED RESULTS

Fig. 7.9: Extraction of RBI using themodified impedance circle method.

25

20

15

10

5

0

RE

SIS

TA

NC

E (W

)

5 6 7

12 3 4 5 6 7

102 3 4 5

CURRENT DENSITY [IC/AE] (mA/mm2)

RBX

RBI

Fig. 7.10: Extracted base resistance vs. bias.

The suspicion that the intrinsic base resistance plotted inFig. 7.10 experiences a large error

is confirmed by noise parameter measurements results shown below. Unless the values from

the foundry are used, the measured and modelled equivalent noise resistanceRn experience

a significant deviation from one other, even in the low frequencies where correlation may be

7.2 Device Parameter Extraction 65

ignored. Ignoring correlation,Rn may be approximated by [1]

Rn =IC

2VT |y21|2+ RE + RB, (7.1)

whereRB = RBX + RBI and VT is the thermal voltage of the device. SinceIC may be

accurately measured and the extractedRE andRBX are verified against the model file, it is

reasonable to assume the deviation between measured and modelled values inRn at low fre-

quencies is due to the inaccuracy in extractingRBI .

Therefore, only the extractedRBX values are used in this work.RBI is taken from the

model file provided by the foundry under the assumption that those models are verified against

measured data.

7.2.4. Noise Transit Time

The noise transit timeτn is extracted from the negative of the phase of the high frequency

transconductance of the device. The negative phase ofgm at the minimum noise bias is plotted

against the frequency in Fig. 7.11. From the high frequency domain, a value of 0.28 ps is

extracted forτn.

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-PH

AS

E(G

M)

(RA

DIA

NS

)

6050403020100

FREQUENCY (GHz)

MEASURED LINEAR REGRESSION

tn=0.28ps

Fig. 7.11: Phase of gm (ω) at minimum noise bias.


7.3. Noise Parameters vs. Bias

The noise parameters with and without accounting for noise correlation are compared against

the measured values at 3 different frequencies, 2, 10 and 18 GHz, as functions of bias. In

the plots below, the results indicate that noise correlation is insignificant for these 160-GHz

SiGe HBTs at frequencies below 18 GHz for all bias points. Qualitatively, the scatter in the

measured data is larger than the difference between the values calculated with and without

noise correlation.

2 GHz5

4

3

2

1

0

NF

MIN

(dB

)

0.1 1 10

COLLECTOR CURRENT DENSITY [IC/AE] (mA/mm2)

WITHOUT CORRELATION WITH CORRELATION MEASURED

Fig. 7.12: Comparison between mea-sured and modelled NFMIN at 2 GHz vs.bias (with pad parasitics).

120

100

80

60

40

20

Rn(W

)

0.12 3 4 5 6 7

12 3 4 5 6 7

10



Fig. 7.13: Comparison between mea-sured and modelled Rn at 2 GHz vs. bias(with pad parasitics).

20

15

10

5

0

RE

AL

(YO

PT)

(mS

)

0.12 3 4 5 6 7

12 3 4 5 6 7

10


MEASURED WITHOUT CORRELATION WITH CORRELATION

Fig. 7.14: Comparison between mea-sured and modelled ℜYOPT at 2 GHzvs. bias (with pad parasitics).

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

IMA

G(Y

OP

T)

(mS

)

0.12 3 4 5 6 7

12 3 4 5 6 7

10


MEASURED

WITHOUT CORRELATION WITH CORRELATION

Fig. 7.15: Comparison between mea-sured and modelled ℑYOPT at 2 GHzvs. bias (with pad parasitics).

7.3 Noise Parameters vs. Bias 67

10 GHz8

6

4

2

0

NF

MIN

(dB

)

0.1 1 10



Fig. 7.16: Comparison between mea-sured and modelled NFMIN at 10 GHzvs. bias (with pad parasitics).

120

100

80

60

40

20R

n(W

)0.1

2 3 4 5 6 7

12 3 4 5 6 7

10



Fig. 7.17: Comparison between mea-sured and modelled Rn at 10 GHz vs.bias (with pad parasitics).

20

15

10

5

0

RE

AL

(YO

PT)

(mS

)

0.12 3 4 5 6 7

12 3 4 5 6 7

10




-8

-6

-4

-2

0

2

IMA

G(Y

OP

T)

(mS

)

0.12 3 4 5 6 7

12 3 4 5 6 7

10





18 GHz8

6

4

2

0

NF

MIN

(dB

)

0.1 1 10



Fig. 7.20: Comparison between mea-sured and modelled NFMIN at 18 GHzvs. bias (with pad parasitics).

140

120

100

80

60

40

20

Rn(W

)

0.12 3 4 5 6 7

12 3 4 5 6 7

10



Fig. 7.21: Comparison between mea-sured and modelled Rn at 18 GHz vs.bias (with pad parasitics).

20

15

10

5

0

RE

AL

(YO

PT)

(mS

)

0.12 3 4 5 6 7

12 3 4 5 6 7

10




-8

-6

-4

-2

0

2

4

6

8

IMA

G(Y

OP

T)

(mS

)

0.12 3 4 5 6 7

12 3 4 5 6 7

10




7.4 Noise Parameters vs. Frequency 69

7.4. Noise Parameters vs. Frequency

Another verification is performed by comparing the noise parameters at the minimum noise

bias of JC = 1.34 mA/µm2. Plotted in the figures below are the noise parameters versus

frequency at this bias point. These results also indicate that noise correlation is insignificant up

to 18 GHz compared to the scatter present in the measured data, in agreement with the results

presented in the previous section.

8

6

4

2

0

NF

MIN

(dB

)

18161412108642

FREQUENCY (GHz)


Fig. 7.24: Comparison between mea-sured and modelled NFMIN vs. fre-quency at minimum noise bias (with padparasitics).

70

60

50

40

30

Rn(W

)

18161412108642

FREQUENCY (GHz)


Fig. 7.25: Comparison between mea-sured and modelled Rn vs. frequency atminimum noise bias (with pad parasitics).

8

6

4

2

0

RE

AL

(YO

PT)

(mS

)

18161412108642

FREQUENCY (GHz)


Fig. 7.26: Comparison between mea-sured and modelled ℜYOPT vs. fre-quency at minimum noise bias (with padparasitics).

-4

-3

-2

-1

0

IMA

G(Y

OP

T)

(mS

)

18161412108642

FREQUENCY (GHz)


Fig. 7.27: Comparison between mea-sured and modelled ℑYOPT vs. fre-quency at minimum noise bias (with padparasitics).


7.5. Impact of Correlation at Millimetre-Wave Frequencies

Having verified the derived equations and noise transit timeextract technique, the equations

are applied to measured and de-embeddedS-parameters up to 65 GHz to predict the noise

parameters at millimetre-wave frequencies. Fig. 7.28 is the NFMIN as a function ofJC at

60 GHz.

10

9

8

7

6

5

4

3

2

NF

MIN

(d

B)

0.12 4 6 8

12 4 6 8

102


WITHOUT CORRELATION WITH CORRELATION

Fig. 7.28: Comparison between modelled NFMIN with and without correlation at 60 GHz(without pad parasitics).

The technique predicts a minimumNFMIN that is approximately 1.5 dB lower when noise

correlation is accounted for. Because of equipment limitations, this prediction cannot be ver-

ified directly by experiments. However, recent publications on 60-GHz SiGe HBT circuits

reported that the measured phase noise of VCOs [3] and noise figure of LNAs [4] are system-

atically lower than simulated values.

Especially significant are the results reported on VCOs in [3]. The measured data are obtain

from millimetre-wave VCOs fabricated using the same SiGe BiCMOS technology on the same

tapeout. Since VCO phase noise is directly proportional to the noise figure of the transistors, a

lower device noise figure translates directly to lower VCO phase noise. These results reported

by others indicate qualitatively that noise correlation issignificant at 60 GHz, consistent with

Fig. 7.28.

8 Conclusion

8.1. Summary

The focus of this thesis is to extend the originaly-parameter based technique in [1] to extract

the noise parameters of bipolar transistors to account for the correlation between the base and

collector shot noise currents. The contributions are a set of noise equations that account for

the correlation between base and collector shot noise currents and a new technique to extract

the noise transit timeτn from measuredy-parameters. Verification of the equations andτn

extraction technique is provided by device simulations andmeasurements up to 18 GHz.

Based on a noise equivalent circuit that considers the intrinsic transistor as a black box, a

set of new equations for the power spectral densities of the input-referred noise sources of a

bipolar transistor is systematically derived in Chapter 4.A technique has also been developed

to extractτn from the high frequency transconductance of the transistor, without fitting to noise

data.

Verification of the derived equations andτn extraction technique is provided initially by

device simulations using 2-D TCAD simulations. The technique is applied to the simulated

y-parameters of a two-dimensional SiGe HBT to calculate its noise parameters. The values

calculated from these equations are then compared with those calculated directly by the sim-

ulator using the impedance field method. From the results presented in Chapter 5, it can be

seen that the equations have reasonable agreement with the noise parameters calculated by the

simulator, even at millimetre-wave frequencies, when noise correlation is accounted for.

Further verification of the technique is provided experimentally as described in Chapters 6

and 7. TheS-parameters and noise parameters of a commercial 160-GHz SiGe HBT are mea-

sured up to 65 GHz and 18 GHz, respectively. The noise parameters calculated by applying the

equations to the measuredS-parameters of the device are compared with those measured using

the noise parameter measurement system. Experimental results indicate that noise correlation

is insignificant up to 18 GHz for these devices. Finally, thistechnique predicts a minimum

71

72 Conclusion

NFMIN at 60 GHz that is approximately 1.5 dB lower when correlationis accounted for, in

agreement with published VCO and LNA results.

8.2. Future Work

Future work can be grouped into two major areas: experimental verification and development.

On the verification side, it is necessary to provide experimental proof that this technique is

able to provide accurate results at millimetre-wave frequencies in addition to the simulation

results presented. This points to a need to perform noise parameter characterization at these

frequencies. The required equipment, however, is not currently available at the University

of Toronto. Although there is commercially available noisecharacterization equipment up to

110 GHz, the experimental uncertainly may be larger than thedifference between uncorrelated

and correlated values. This is especially true for device noise characterization, as the test

structures are impedance mismatched to 50Ω and most of the noise power is reflected at the

interfaces. One possible solution is to develop a mathematical link between the device noise

figure and some figure of merit of simple millimetre-wave circuits, such as the phase noise of

a VCO.

Also, the accuracy of verification may be improved through the use of more advanced

de-embedding techniques. In particular, the transmissionline based de-embedding technique

that was developed and presented in [39] is a suitable candidate. Unlike the lumped-element

based open-short de-embedding used in this work, the transmission-line based method ac-

counts for the distributive nature of the parasitics, whichis important at the millimetre-wave

regime. This technique can also be applied effectively to de-embed the noise parameters, as

the line itself is a passive device and hence its noise is completely characterized by itsY/Z-

parameters [41]. Since the noise contributed by the transmission lines leading from the pads to

the device is known, it can be de-embedded from the measured noise parameters using matrix

techniques [10,11].

On the development side, the error introduced by ignoring the distributed nature of the base

resistance should be investigated. This involves derivinganother set of equations that splits

RB into RBX andRBI . The equations derived in this thesis may overestimate the noise of the

device, since at sufficiently high frequencies, the noise contributed byRBI may be shunted out

by the extrinsic base-collector capacitanceCBCX and therefore reducing the noise figure of the

device.

ADetailed Derivation of SiGe

HBT Noise Parameter

Equations

A.1. Input Referred Noise Voltage

In section 4.2.1, it was shown that a system of seven equations with seven unknowns has to

be solved in order to obtain an expression for the output short-circuit current of the SiGe HBT

noise equivalent circuit. The algebraic details omitted inthat section are present here.

For completeness, the SiGe HBT noise equivalent circuit is reproduced in Fig. A.1. The

system of seven equations are

ISC + I INT

2+ inC = 0 (A.1)

IRE− I INT

1− inB − I INT

2− inC = 0 (A.2)

vnB −(I INT

1 + inB

)RB − vINT

1 − vX = 0 (A.3)

vINT

2+ vX = 0 (A.4)

vnE + IRERE = vX (A.5)

[

I INT1

I INT2

]

=

[

yINT11

yINT12

yINT21

yINT22

][

vINT1

vINT2

]

. (A.6)

The expression for the input referred noise voltage,vn, is obtained by equating the output

short-circuit currents of the SiGe HBT noise equivalent circuit and the chain representation of

a noisy two-port network.

The strategy for solvingISC from the seven equations is as follows:

1. Solve equation (A.2) forIRE.

2. Substitute the result from above into equation (A.5) to obtainvX = f(I INT1

, I INT2

).

3. SubstitutevX = f(I INT1

, I INT2

)into equations (A.3) and (A.4) to obtain two equations,

vINT1 = f1

(I INT1 , I INT

2

)andvINT

2 = f2

(I INT1 , I INT

2

).

73

74 Detailed Derivation of SiGe HBT Noise Parameter Equations

inCinB

RB

RE

vnE

vnB YINT

Y

EE

CB+ +

+

+

− −

−

−

I INT1

I INT2

IRE

ISC

L1

L2

vX

vINT1

vINT2

Fig. A.1: Schematic of Noise Equivalent Circuit Defining Symbols used in Deriving vn

4. SubstitutevINT1

= f1

(I INT1

, I INT2

)andvINT

2= f2

(I INT1

, I INT2

)into the first matrix equa-

tion of (A.6) and obtainI INT1 = f3

(I INT2

).

5. Repeat the above step for the second matrix equations of (A.6) and obtainI INT1

=

f(I INT2

).

6. Equate the equations obtained from steps 4 and 5 to obtain an expression forI INT2 .

7. Using the above expression, obtain an equation forISC from equation (A.1).

Substituting the expression forIREobtained from equation (A.2) into equation (A.5), the

equation forvX is

vX = vnE +(I INT

1+ I INT

2+ inB + inC

)RE . (A.7)

vINT1

= f(I INT1

, I INT2

)is obtained by solving equation (A.3) forvINT

1and substituting the

expression ofvX from above.

vINT

1= vnB −

(I INT

1+ inB

)RB − vnE −

(I INT

1+ I INT

2+ inB + inC

)RE (A.8)

From the first matrix equation (A.6), upon substituting expressions forvINT1

, vINT2

and using

equation (A.4),I INT1

is obtained as

I INT

1= yINT

11

[vnB −

(I INT

1+ inB

)RB − vnE −

(I INT

1+ I INT

2+ inB + inC

)RE

]

+ yINT

12

[−vnE −

(I INT

1 + I INT

2 + inB + inC

)RE

].

(A.9)

A.1 Input Referred Noise Voltage 75

IsolatingI INT1

in the above equation gives

I INT

1

[1 + yINT

11 (RB + RE) + yINT

12 RE

]= I INT

2

(−yINT

11 RE − yINT

12 RE

)

+ yINT

11 (vnB − inBRB − vnE − (inB + inC)RE) − yINT

12 (vnE + (inB + inC)RE) .

(A.10)

Similarly, by using the second equation from the matrix equation (A.6) and substituting expres-

sions forvINT1 andvINT

2 from, equations (A.8) and (A.4), respectively, gives a second relation

betweenI INT1

andI INT2

.

I INT

2= yINT

21

(vnB −

(I INT

1+ I INT

2

)RB − vnE −

(I INT

1+ I INT

2+ inB + inC

)RE

)

− yINT

22

(vnE +

(I INT

1+ I INT

2+ inB + inC

)RE

) (A.11)

SeparatingI INT1 andI INT

2 in the above equation gives

I INT

2

(1 + yINT

21RE + yINT

22RE

)= I INT

1

(−yINT

21RB − yINT

21RE − yINT

22RE

)

+ yINT

21(vnB − inBRB − vnE − (inB + inC)RE) − yINT

22(vnE + (inB + inC)RE) .

(A.12)

Substituting the expression forI INT1

from equation (A.10) results in

I INT

2

(1 +

[yINT

21+ yINT

22

]RE

)= yINT

21(vnB − inBRB − vnE − (inB + inC)RE)

− yINT

22 (vnE + (inB + inC) RE) −(

yINT21

RB +(yINT

21+ yINT

22

)RE

1 + yINT11

(RB + RE) + yINT12

RE

)

×[−I INT

2

(yINT

11+ yINT

12

)RE + yINT

11(vnB − inBRB − vnE − (inB + inC) RE)

−yINT

12(vnE + (inB + inC) RE)

].

(A.13)

FactoringI INT2 in the above equation gives

I INT

2

1 +[yINT

21 + yINT

22

]RE −

(

yINT21 RB +

(yINT

21 + yINT22

)RE

1 + yINT11

(RB + RE) + yINT12

RE

)

(yINT

11 + yINT

12

)RE

= − yINT21

RB +(yINT

21+ yINT

22

)RE

1 + yINT11

(RB + RE) + yINT12

RE

[yINT

11 (vnB − inBRB − vnE − (inB + inC) RE)

− yINT

12(vnE + (inB + inC)RE)

]+ yINT

21(vnB − inBRB − vnE − (inB + inC) RE)

− yINT

22(vnE + (inB + inC)RB) .

(A.14)


It would be more convenient at this point to assign the factormultiplying I INT2

in the above

equation to a variableΓ and simplify it separately.

Γ = 1 +[yINT

21 + yINT

22

]RE −

(

yINT21

RB +(yINT

21+ yINT

22

)RE

1 + yINT11

(RB + RE) + yINT12

RE

)

(yINT

11 + yINT

12

)RE

=

(1 + yINT

11RB

) (1 +

(yINT

21+ yINT

22

)RE

)+(yINT

11+ yINT

12

) (1 − yINT

21RB

)RE

1 + yINT11 RB + (yINT

11 + yINT12 ) RE

(A.15)

The above expression forΓ can then be substituted back into equation (A.14) to obtain another

expression forI INT2 . ζ is defined to be the numerator ofΓ and simplified as

ζ =(1 + yINT

11 RB

) (1 +

(yINT

21 + yINT

22

)RE

)+(yINT

11 + yINT

12

) (1 − yINT

21 RB

)RE (A.16)

= 1 + yINT

11 RB + RE

∑

i,j

yINT

ij + RBRE det [YINT] .

ζI INT

2= −yINT

22(vnE + (inB + inC)RE)

(1 + yINT

11RB +

(yINT

11+ yINT

12

)RE

)

+ yINT

21 (vnB − inBRB − vnE − (inB + inC) RE)(1 + yINT

11 RB +(yINT

11 + yINT

12

)RE

)

−(yINT

21 RB +(yINT

21 + yINT

22

)RE

) [yINT

11 (vnB − inBRB − vnE − (inB + inC) RE)

− yINT

12(vnE + (inB + inC) RE)

]

(A.17)

The above expression can be rearranged and simplified to obtain the final expression forI INT2

.

Finally, by equation (A.1), the output short-circuit current is given by

ISC = −I INT

2 − inC . (A.18)

The equivalent circuit for the chain representation is reproduced in Fig. A.2. It was shown in

section 4.2.1 that the output short-circuit current can be obtained directly from equation (A.18)

by takingvnB → vn and setting all other noise sources to zero. The result is

ISC =vn

(yINT

21 − det [YINT] RE

)

ζ. (A.19)

By equating equations (A.18) and (A.19), the expression forvn is obtained as

vn = vnB +1

C(DvnE + EinB + FinC) (A.20)

A.2 Input Referred Noise Current 77

RB

RE

YINT

Y

EE

CB

+ −

ISC

vn

Fig. A.2: Schematic for Deriving Output Short-Circuit Current of Chain Representation ofNoisy Two-Port

where

C = yINT

21− RE det [YINT] (A.21)

D = −yINT

21− yINT

22− RB det [YINT] (A.22)

E = −RByINT

21 − RE

(yINT

21 + yINT

22

)(A.23)


(yINT

21+ yINT

22

). (A.24)

A.2. Input Referred Noise Current

The output open-circuit voltage of the SiGe HBT noise equivalent circuit is obtained by solving

a system of equations, as shown in section 4.2.2. Following is the algebraic details that were

omitted in that section.

Reproduced in Fig. A.3 is the open-circuited noise equivalent circuit. The equations to be

solved are

I INT

1+ inB = 0 (A.25)

I INT

2 + inC = 0 (A.26)

vo = vINT

2 + vnE (A.27)[

vINT1

vINT2

]

= ZINT

[

I INT1

I INT2

]

(A.28)


RB

RE

YINT

Y

E E

CB+ + ++

+

− −

−−

−

I INT1

I INT2

vINT1

vINT2

vo

vnB

L1

vnE

inB inC

Fig. A.3: Schematic for Deriving Output Open-Circuit Voltage of SiGe HBT Noise Equiva-lent Circuit

where,

ZINT =1

det [YINT]

[

yINT22

−yINT12

−yINT21 yINT

11

]

. (A.29)

Solving the above equations forin is considerably simpler than in thevn case. From equa-

tions (A.28) and (A.29),

vINT

2=

1

det [YINT]

(−yINT

21I INT

1+ yINT

11I INT

2

)(A.30)

=1

det [YINT]

(yINT

21inB − yINT

11inC

).

The last step is obtained by applying equations (A.25) and (A.26). Hence, from equation (A.27),

vo =1

det [YINT]

(yINT

21inB − yINT

11inC

)+ vnE. (A.31)

Fig. A.4 reproduces the equivalent circuit of the chain representation with its inputs and

outputs open-circuited for completeness. The equations that need to be solved to obtain the

open-circuit voltage are

I INT

2= 0 (A.32)

I INT

1= −in (A.33)

vo = vINT

2 + vX (A.34)

A.2 Input Referred Noise Current 79

RB

RE

YINT

Y

E E

CB

+

−

I INT1

I INT2

voL1

invX

Fig. A.4: Schematic for Deriving Output Open-Circuit Voltage of SiGe HBT Chain Repre-sentation

vX = −inRE = I INT

1RE (A.35)

vINT

2 =1

det [YINT]

(−yINT

21 I INT

1 + yINT

11 I INT

2

). (A.36)

By using equations (A.32) and (A.33),

vINT

2=

yINT21 in

det [YINT]. (A.37)

Finally, from equations (A.34) and (A.35),

vo =yINT

21in

det [YINT]− inRE . (A.38)

By equating equation (A.31) with (A.38), the input-referred noise current is given by

in

(yINT

21

det [YINT]− RE

)

=yINT

21inB − yINT

11inC

det [YINT]+ vnE (A.39)

in =yINT

21inB − yINT

11inC + vnE det [YINT]

J(A.40)

where

J ≡ yINT

21 − RE det [YINT] . (A.41)


A.3. Transforming the Noise Power Spectral Densities to Ext rinsic

Y-Parameters

In section 4.3 that the input referred noise power spectral densities are given by

⟨v2

n

⟩≡ 〈v∗

nvn〉

=⟨v2

nB

⟩+

∣∣∣∣

D

C

∣∣∣∣

2⟨v2

nE

⟩+

∣∣∣∣

E

C

∣∣∣∣

2⟨i2nB

⟩+

∣∣∣∣

F

C

∣∣∣∣

2⟨i2nC

⟩+

2

|C|2ℜ (EF ∗ 〈inBi∗nC〉)

(A.42)⟨i2n⟩≡ 〈i∗nin〉

=

∣∣∣∣

yINT21

J

∣∣∣∣

2⟨i2nB

⟩+

∣∣∣∣

yINT11

J

∣∣∣∣

2⟨i2nC

⟩+

∣∣∣∣

det [YINT]

J

∣∣∣∣

2⟨v2

nE

⟩(A.43)

− 2

|J |2ℜ(yINT

21

(yINT

11

)∗ 〈inBi∗nC〉)

〈v∗nin〉 =

det [YINT]

J

D∗

C∗

⟨v2

nE

⟩+

yINT21

J

E∗

C∗

⟨i2nB

⟩− yINT

11

J

E∗

C∗〈i∗nBinC〉 (A.44)

+yINT

21

J

F ∗

C∗〈inBi∗nC〉 −

yINT11

J

F ∗

C∗

⟨i2nC

⟩.

In this section, the above equations will be rewritten in terms of the extrinsicy-parameters of

the SiGe HBT. From appendix B, the intrinsicy-parameters of the SiGe HBT are related to its

extrinsicy-parameters by

yINT

11=

y11 − det [Y] RE

1 − y11RB −∑

ij yijRE + RBRE det [Y](A.45)

yINT

12=

y12 + det [Y]RE

1 − y11RB −∑ij yijRE + RBRE det [Y](A.46)

yINT

21 =y21 + det [Y]RE

1 − y11RB −∑


yINT

22=

y22 − (RB + RE) det [Y]

1 − y11RB −∑

ij yijRE + RBRE det [Y]. (A.48)

The determinantdet [YINT] is first compute as follows. Since

yINT

11 yINT

22 =y11y22 − det [Y] [y11 (RE + RB) + y22RE] + RE (RE + RB) det2 [Y]

(

1 − y11RB −∑

ij yijRE + RBRE det [Y])2

(A.49)

yINT

12yINT

21=

y12y21 + y12RE det [Y] + y21RE det [Y] + R2

E det2 [Y](

1 − y11RB −∑

ij yijRE + RBRE det [Y])2

, (A.50)

A.3 Transforming the Noise Power Spectral Densities to Extrinsic Y-Parameters 81

then

det [YINT] =det [Y] − det [Y] y11RB − det [Y] RE (y11 + y22 + y12 + y21) + RERB det2 [Y]

(1 − y11RB −∑

yijRE + RBRE det [Y])2

=det [Y]

1 − y11RB −∑ yijRE + RBRE det [Y]. (A.51)

〈v2n〉 is expressed as a function of the extrinsicy-parameters first. The variablesC, D and

E are expressed in terms of extrinsicy-parameters as detailed below.

C ≡ yINT

21− RE det [YINT]

=y21 + det [Y] RE

1 − y11RB −∑

yijRE + RBRE det [Y]− RE det [Y]

1 − y11RB − RE

∑yij + RBRE det [Y]

=y21

1 − y11RB − RE


(A.52)

D ≡ −RB det [YINT] − yINT

21− yINT

22

= − RB det [Y]

1 − y11RB − RE


− y21 + RE det [Y]

1 − y11RB − RE


− y22 − (RB + RE) det [Y]

1 − y11RB − RE


= − y21 + y22

1 − y11RB − RE

∑yij + RERB det [Y]

(A.53)

E ≡ −yINT

21 RB − RE

(yINT

21 + yINT

22

)

= − (y21 + RE det [Y]) RB

1 − y11RB − RE


− RE

y21 + RE det [Y] + y22 − (RB + RE) det [Y]

1 − y11RB − RE


= − y21RB + RE (y21 + y22)

1 − y11RB − RE


(A.54)

The summation∑

ij yINTij can be expressed in terms of the extrinsicy-parameters as

∑

ij

yINT

ij =1

1 − y11RB −∑

ij yijRE + RBRE det [Y]×

y11 − RE det [Y] + y12 + RE det [Y]

+y21 + RE det [Y] + y22 − (RB + RE) det [Y]

=

∑

ij yij − RB det [Y]

1 − y11RB −∑

ij yijRE + RBRE det [Y]. (A.55)


This result is used in conjunction with equations (A.45)– (A.48) to simplifyζ = 1+RE

∑

ij yINTij +

RBy11 + RBRE det [YINT].

ζ = 1 + RE

∑

ij yij − RB det [Y]

1 − y11RB − RE

∑

ij yij + RBRE det [Y]

+ RB

y11 − RE det [Y]

1 − y11RB − RE

∑


+ RBRE

det [Y]

1 − y11RB −∑

ij yijRE + RBRE det [Y]

= 1 −1 − y11RB − RE

∑

ij yij + RBRE det [Y] − 1

1 − y11RB −∑

ij yijRE + RBRE det [Y]

=1

1 − y11RB −∑


F is then expressed in terms of the extrinsicy-parameters as


(yINT

21+ yINT

22

)

=1

1 − y11RB − RE

∑

ij yij + RBRE det [Y]− RBRE det [Y]

1 − y11RB − RE

∑


− RE

y21 + RE det [Y] + y22 − (RB + RE) det [Y]

1 − y11RB − RE

∑


=1 − RE (y21 + y22)

1 − y11RB − RE

∑

ij yij + RBRE det [Y]. (A.57)

The ratiosD/C, E/C andF/C and the productEF ∗ are computed in terms of extrinsicy-

parameters.

D

C= −y21 + y22

y21

= −(

1 +y22

y21

)

(A.58)

E

C= −y21RB + RE (y21 + y22)

y21

= −(

RB + RE

[

1 +y22

y21

])

(A.59)

F

C=

1 − RE (y21 + y22)

y21

=1

y21

− RE

[

1 +y22

y21

]

(A.60)

EF ∗ = −(y21RB + RE [y21 + y22]) (1 − RE [y21 + y22])∗

∣∣∣1 − y11RB − RE

∑

ij yij + RBRE det [Y]∣∣∣

2(A.61)

A.3 Transforming the Noise Power Spectral Densities to Extrinsic Y-Parameters 83

Using the above results,〈v2n〉 as a function of the extrinsicy-parameters of the SiGe HBT is

⟨v2

n

⟩=⟨v2

nB

⟩+

∣∣∣∣1 +

y22

y21

∣∣∣∣

2⟨v2

nE

⟩

+

∣∣∣∣RB + RE

[

1 +y22

y21

]∣∣∣∣

2⟨i2nB

⟩+

∣∣∣∣y−1

21− RE

[

1 +y22

y21

]∣∣∣∣

2⟨i2nC

⟩

− 2

|y21|2ℜ [(y21RB + RE [y21 + y22]) (1 − RE [y21 + y22])

∗ 〈inBi∗nC〉] .

(A.62)

Following the transformation of〈v2n〉, the power spectral density of the input referred noise

current, 〈i2n〉 is expressed in terms of the extrinsicy-parameters. First, equation (A.51) is

applied to simplifyJ as follows.

J ≡ yINT

21− RE det [YINT]

=y21 + det [Y] RE

1 − y11RB −∑

yijRE + RBRE det [Y]− RE det [Y]

1 − y11RB −∑

yijRE + RBRE det [Y]

=y21

1 − y11RB −∑ yijRE + RBRE det [Y]. (A.63)

The multiplying factors in the equation of〈i2n〉, namelyyINT21 /J, yINT

11 /J anddet [YINT] /J , are

transformed into extrinsicy-parameters. Using the above equation in conjunction with (A.45)

and (A.51) yields

yINT11

J=

y11 − det [Y]RE

1 − y11RB −∑ yijRE + RBRE det [Y]× 1 − y11RB −

∑yijRE + RBRE det [Y]

y21

=y11 − RE det [Y]

y21

(A.64)

det [YINT]

J=

det [Y]

1 − y11RB −∑ yijRE + RBRE det [Y]

× 1 − y11RB −∑

yijRE + RBRE det [Y]

y21

=det [Y]

y21

. (A.65)

Re-arranging the terms in

J ≡ yINT

21− RE det [YINT] (A.66)

asyINT

21

J= 1 +

RE det [YINT]

J(A.67)


and observing that

RE det [YINT]

J=

RE det [Y]

1 − y11RB − RE

∑

ij yij + RERB det [Y]

×1 − y11RB − RE

∑

ij yij + RERB det [Y]

y21

=RE det [Y]

y21

,

(A.68)

therefore,yINT

21

J= 1 +

RE det [Y]

y21

. (A.69)

By using the above equations,〈i2n〉 as a function of extrinsicy-parameters is

⟨i2n⟩

=

∣∣∣∣1 +

RE det [Y]

y21

∣∣∣∣

2⟨i2nB

⟩+

∣∣∣∣

y11 − RE det [Y]

y21

∣∣∣∣

2⟨i2nC

⟩+

∣∣∣∣

det [Y]

y21

∣∣∣∣

2⟨v2

nE

⟩

− 2

|y21|2ℜ [(y21 + RE det [Y]) (y11 − RE det [Y])∗ 〈inBi∗nC〉]

(A.70)

and the cross power spectral density is given by

〈v∗nin〉 =

det [Y]

y21

(

1 +y22

y21

)∗⟨v2

nE

⟩+

(RE det [Y]

y21

+ 1

)(

RB + RE

[

1 +y22

y21

])∗⟨i2nB

⟩

−(

y11 − RE det [Y]

y21

)(

RB + RE

[

1 +y22

y21

])∗

〈i∗nBinC〉

−(

RE det [Y]

y21

+ 1

)(

y−1

21 − RE

[

1 +y22

y21

])∗

〈inBi∗nC〉

+

(y11 − RE det [Y]

y21

)(

y−1

21 − RE

[

1 +y22

y21

])∗⟨i2nC

⟩. (A.71)

BConversion Between

Intrinsic and Extrinsic

Y-Parameters

W HEN measuring or simulating the high frequency performanceof a SiGe HBT the

extrinsic two-port parameters of the device for each bias point are readily available1.

It is therefore convenient to express the input referred noise voltage and noise current in terms

of the extrinsicy-parameters of the device instead of intrinsicy-parameters.

Equations are derived and presented in this appendix for theconversion between intrinsic

and extrinsicy-parameters, as they are defined in Fig. B.1. These equationsare employed to

convert the expressions of the input referred noise voltageand noise current in terms of intrinsic

y-parameters to extrinsicy-parameters.

B.1. Converting from Intrinsic to Extrinsic Y-Parameters

Intrinsic y-parameters,yINTij , as functions of the extrinsicy-parameters,yij, are derived first.

By definition,

I1 = y11v1 + y12v2 (B.1)

I2 = y21v1 + y22v2. (B.2)

Also, by observation,

I1 = I INT

1 (B.3)

I2 = I INT

2. (B.4)

1Technically speaking, some form of parasitic de-embeedingis required for the experimental data to removethe effects of the pad and interconnect parasitics. De-embedding techniques are presented in section 6.3.1. Toavoid confusion, in this thesis, the term “extrinsicy-parameters” refers to they-parameters of the device, excludingthe effects of pads and interconnects.

85

86 Conversion Between Intrinsic and Extrinsic Y-Parameters

++ ++

−

− −

−

RB

RE

RCYINT

Y

vINT1

vINT2

v1 v2

I INT1

I INT2I1 I2

EE

B C

vX

Fig. B.1: Equivalent circuit relating intrinsic and extrinsic y-parameters of SiGe HBT

From KVL,

v1 − I1RB − vINT

1 − vX = 0 (B.5)

v2 − I2RC − vINT

2− vX = 0 (B.6)

vX − (I1 + I2) RE = 0. (B.7)

Using equations (B.3)–(B.7), equations (B.1) and (B.2) arerewritten as

I INT

1= y11

(I INT

1RB + vINT

1+[I INT

1+ I INT

2

]RE

)

+ y12

(I INT

2 RC + vINT

2 +[I INT

1 + I INT

2

]RE

)(B.8)

I INT

2= y21

(I INT

1RB + vINT

1+[I INT

1+ I INT

2

]RE

)

+ y22

(I INT

2RC + vINT

2+[I INT

1+ I INT

2

]RE

). (B.9)

Solving forI INT1

in both of the above equations,

I INT

1=

y11

(vINT1 + I INT

2 RE

)

1 − y11RB − y11RE − y12RE

+y12

(I INT2 RC + vINT

2 + I INT2 RE

)

1 − y11RB − y11RE − y12RE

(B.10)

I INT

1 =I INT2

y21RB + y21RE + y22RE

− y21

(vINT1

+ I INT2

RE

)


− y22

(I INT2

RC + vINT2

+ I INT2

RE

)


. (B.11)

B.1 Converting from Intrinsic to Extrinsic Y-Parameters 87

I INT1

is then eliminated by equating the two equations above. Rearranging the result yields

I INT

1Γ = vINT

1(y11 [y21RB + y21RE + y22RE] + y21 [1 − y11RB − y11RE − y12RE ])

+ vINT

2 (y12 [y21RB + y21RE + y22RE ] + y22 [1 − y11RB − y11RE − y12RE ]) ,

(B.12)

where

Γ = − (y11RE + y12 [RC + RE ]) (y21RB + y21RE + y22RE)

− (y21RE + y22 [RC + RE ] − 1) (1 − y11RB − y11RE − y12RE) . (B.13)

Upon simplification,

I INT

2= vINT

1

[

y21 + RE det [Y]

1 − y11RB − y22RC − RE

∑

ij yij + (RBRC + RBRE + RERC) det [Y]

]

+ vINT

2

[

y22 − (RB + RE) det [Y]

1 − y11RB − y22RC − RE

∑


]

,

(B.14)

where the summation is to be taken over the entire range of indices. From the above equation,

it can be deduced that

yINT

21 =y21 + RE det [Y]

1 − y11RB − y22RC − RE

∑

ij yij + (RBRC + RBRE + RERC) det [Y](B.15)

yINT

22=

y22 − (RB + RE) det [Y]

1 − y11RB − y22RC − RE

∑

ij yij + (RBRC + RBRE + RERC) det [Y]. (B.16)

The equations ofyINT11

andyINT12

are obtained using the same procedure as above, but equa-

tions (B.8) and (B.9) are solved forI INT2

instead ofI INT1

. The following is obtained when the

two resultant equations are equated to eliminateI INT2 .

I INT

1Γ = vINT

1

[y11

y11RE + y12RC + y12RE

+y21

1 − y21RE − y22RC − y22RE

]

+ vINT

2

[y12


+y22

1 − y21RE − y22RC − y22RE

]

, (B.17)

where

Γ = −y11RB + y11RE + y12RE − 1


− y21 (RB + RE) + y22RE

1 − y21RE − y22RC − y22RE

. (B.18)


Upon simplification,

I INT

1= vINT

1

(

y11 − det [Y] (RE + RC)

1 − y11RB − y22RC − RE

∑


)

+ vINT

2

(

y12 + RE det [Y]

1 − y11RB − y22RC − RE

∑


)

,

(B.19)

from whichyINT11

andyINT12

can be deduced as

yINT

11 =y11 − det [Y] (RE + RC)

1 − y11RB − y22RC − RE

∑


yINT

12=

y12 + RE det [Y]

1 − y11RB − y22RC − RE

∑

ij yij + (RBRC + RBRE + RERC) det [Y], (B.21)

where the summation is to be taken over the entire range of indices.

To summarize, it has been shown that for the equivalent circuit in Fig. B.1, the intrinsic and

extrinsicy-parameters are related by

yINT

11 =y11 − det [Y] (RE + RC)

1 − y11RB − y22RC − RE

∑


yINT

12=

y12 + RE det [Y]

1 − y11RB − y22RC − RE

∑


yINT

21=

y21 + RE det [Y]

1 − y11RB − y22RC − RE

∑


yINT

22 =y22 − (RB + RE) det [Y]

1 − y11RB − y22RC − RE

∑

ij yij + (RBRC + RBRE + RERC) det [Y]. (B.25)

B.2. Converting from Extrinsic to Intrinsic Y-Parameters

Following a procedure similar to the previous section, equations are derived relating the extrin-

sic y-parameters,yij, in terms of the intrinsicy-parameters,yINTij , and the series resistances,

RB, RE andRC . In terms ofyINTij ,

I INT

1 = yINT

11 vINT

1 + yINT

12 vINT

2 (B.26)

I INT

2= yINT

21vINT

1+ yINT

22vINT

2. (B.27)

B.2 Converting from Extrinsic to Intrinsic Y-Parameters 89

Equations (B.5)–(B.7) are then used to eliminate the currents and voltages of the intrinsic

device to arrive at the following two equations.

I1 = yINT

11 (v1 − I1RB − [I1 + I2] RE) + yINT

12 (v2 − I2RC − [I1 + I2] RE) (B.28)

I2 = yINT

21(v1 − I1RB − [I1 + I2] RE) + yINT

22(v2 − I2RC − [I1 + I2] RE) (B.29)

The same procedure as in the derivation ofyINTij in terms ofyij is used to eliminateI1 andI2

one at a time to determine the expressions ofyij. The results are

y11 =yINT

11 + (RC + RE) det [YINT]

Φ(B.30)

y12 =yINT

12− RE det [YINT]

Φ(B.31)

y21 =yINT

21 − RE det [YINT]

Φ(B.32)

y22 =yINT

22+ (RB + RE) det [YINT]

Φ(B.33)

where

Φ = 1 + yINT

11 RB + yINT

22 RC + RE

∑

yINT

ij + (RCRB + RCRE + RERB) det [YINT] (B.34)

C The Selectively Implanted

Collector

T HE selectively implanted collector (SIC) is a region of the collector that is engineered

to optimise the performance of high-speed SiGe HBTs. The role of the SIC in the op-

timization of the high frequency performance of SiGe HBTs isqualitatively described in this

appendix.

C.1. Unity Gain Frequency Revisited

It can be shown that the unity gain frequency of a bipolar transistor is given by

1

2πfT

= τF +kBT

qIC

(CdBE + CdBC) + CdBC (RE + RC) , (C.1)

whereCdBE andCdBC are the base-emitter and base-collector depletion capacitances andRE

andRC are the emitter and collector series resistances [8].

ThefT of a transistor is increased when the forward transit timeτF is decreased. This can

be achieved by device scaling, which reduces the carrier transit time by reducing the device

dimensions. However,τF is not the only component in the above equation. In modern high-

speed SiGe HBTs, SICs are introduced to reduce theCdBCRC product to further increase the

fT’s of the transistors.

C.2. The Role of the SIC

One way to reduce theCdBCRC product is to reduce the collector resistance by increasingthe

collector doping. As with any design problems, there are tradeoffs that have to be considered.

There are two disadvantages to increasing the collector doping. First, as the doping of the

base region cannot be reduced without degrading the base resistance, increasing the collector

doping increases the base-collector depletion capacitance CdBC , as the capacitance per unit

area is inversely proportional to the depletion layer width[44]. For transistors in the common

91

92 The Selectively Implanted Collector

emitter configuration, this capacitance is multiplied by the Miller effect, which further degrades

the gain at high frequencies.

The second disadvantage is that if both the base and collector regions are highly doped, the

breakdown voltage decreases. The electric field within the base-collector space-charge region

is increased since the width of the depletion layer is reduced.

The current flow in present SiGe HBTs are predominantly vertical and directly under the

emitter, as indicated in Fig. C.1. The collector region outside the shaded arrow does not carry

significant current. Increasing the collector doping outside the emitter window does little in

reducingRC , but increases theCdBC .

Fig. C.1: Current Flow in Modern Vertical SiGe HBTs.

In modern high-speed SiGe HBTs, the collector region directly under the emitter has a

retrograded doping profile, similar to Fig. 5.4. The retrograded doping profile reduces the

electric field in the base-collector space charge region andhence improves the breakdown

voltage of the device. This region of increased doping concentration is known as the selectively

implanted collector (SIC). Since the SIC is directly located underneath the emitter, it can be

implanted using the emitter opening as a self-aligned mask,as shown in Fig. 5.1.

D Simulation Decks

D.1. Process Simulation (ATHENA)# This file simulates a 2-dimensional npn122 device with# collector reach through

go athena

method model.sigec model.sige back=6 min.temp=600

# create half device# put a vertical line at each location where etching will# occur in subsequent stepsline x loc=0.00 spac=0.01line x loc=0.10 spac=0.01line x loc=0.185 spac=0.025line x loc=0.435 spac=0.05line x loc=0.55 spac=0.05line x loc=0.77 spac=0.05line x loc=0.87 spac=0.05line x loc=1.17 spac=0.05line x loc=2.36 spac=0.05line x loc=2.5 spac=0.05

line y loc=0.00 spac=0.005line y loc=0.10 spac=0.005line y loc=0.4 spac=0.01line y loc=1.8 spac=0.2

init silicon c.boron=1e15 orientation=100 two.d

# subcollector implantimplant arsenic dose=7e17 energy=50 gauss tilt=0 rotation=0 crystaldiffuse time=2 minutes temperature=1100 nitrogen pressure=1e-10

93

94 Simulation Decks

extract name="burried_layer_sh_res" sheet.res material="Silicon"\mat.occno=1 x.val=0 region.occno=1 semi.poly

# collector epi layer growthdeposit silicon thick=0.1 c.arsenic=1e14 divisions=40

# shallow trench isolationetch silicon start x=0.435 y=-0.1etch cont x=0.435 y=0etch cont x=1.17 y=0etch done x=1.17 y=-0.1

etch silicon start x=2.36 y=-0.1etch cont x=2.36 y=0etch cont x=2.5 y=0etch done x=2.5 y=-0.1

deposit oxide thick=0.5 divisions=200

# collector reach through implantetch oxide start x=1.17 y=-0.6etch cont x=1.17 y=-0.1etch cont x=2.36 y=-0.1etch done x=2.36 y=-0.6

implant arsenic dose=1e17 energy=80 gauss tilt=0 rotation=0 crystaletch oxide above p1.y=-0.1diffuse time=1 minutes temperature=900 nitrogen pressure=1e-10

# Base poly, SiGe base formationdeposit oxide thick=1 divisions=50etch oxide left p1.x=0.1

# SIC self-aligned implant formationimplant arsenic dose=1e14 energy=200 gauss tilt=0 rotation=0 crystaldeposit oxide thick=0.05 divisions=10diffuse time=1 minutes temperature=900 nitrogen pressure=1e-10etch above p1.y=-0.1deposit oxide thick=0.05 divisions=10etch oxide left p1.x=0.435

# deposit sige base# deposit undoped sige layerdeposit silicon thick=0.001 divisions=5 \

c.germanium=1.5e22 f.germanium=1.45e22 c.carbon=2e21 si_to_poly

D.1 Process Simulation (ATHENA) 95

# deposit boron-spikedeposit silicon thick=0.018 divisions=10 \

c.boron=1e19 c.germanium=1.45e22 f.germanium=5.5e21 \c.carbon=2e20 si_to_poly

# deposit another undoped sige layerdeposit silicon thick=0.001 divisions=5 \

c.germanium=5.5e21 f.germanium=5e21 c.carbon=2e21 si_to_poly

struct outfile=hbt_after_sige_base.str

# deposit base polyetch poly above p1.y=0etch above p1.y=-0.12deposit poly thick=0.1 c.indium=1e21 divisions=20etch poly left p1.x=0.185etch poly right p1.x=0.87deposit oxide thick=0.01 divisions=10diffuse time=60 temp=600 nitrogenetch oxide left p1.x=0.185etch oxide above p1.y=-0.2etch oxide start x=0.87 y=-0.2etch cont x=0.87 y=-0.12etch cont x=2.5 y=-0.12etch done x=2.5 y=-0.2

# Deposit oxide spacerdeposit oxide thick=0.1 divisions=20

# Etch contact holes# emitteretch oxide left p1.x=0.1

# deposit poly-emitterdeposit silicon thick=0.01 divisions=10etch silicon above p1.y=-0.13deposit poly thick=0.04 c.arsenic=1e21 divisions=8etch poly above p1.y=-0.17

# base contact holesetch oxide start x=0.55 y=-0.32etch cont x=0.55 y=-0.22etch cont x=0.77 y=-0.22etch done x=0.77 y=-0.32

# collector contact holesetch oxide start x=1.17 y=-0.22

96 Simulation Decks

etch cont x=1.17 y=-0.1etch cont x=2.36 y=-0.1etch done x=2.36 y=-0.22

etch above p1.y=-0.26

structure outf=hbt122_no_contact.str

# Deposit metal contactsdeposit aluminum thick=1 divisions=75etch aluminum above p1.y=-0.25etch aluminum start x=0.9 y=-0.26etch cont x=0.9 y=-0.2etch cont x=2.5 y=-0.2etch done x=2.5 y=-0.26

# Annealing to active base dopants and form intrinsic emitterdiffus time=0.5 temp=900 nitro

struct outf=hbt122_with_al.str

# Define electrodeselectrode name=emitter x=0electrode name=base x=0.66electrode name=collector x=1.6electrode name=substrate backside

structure outf=sige_device_half.strstructure outf=sige_device_half.atlas.str sige.conv

init inf=hbt122_with_al.strstructure mirror left

electrode name=emitter x=0electrode name=base x=0.66electrode name=base x=-0.66electrode name=collector x=1.6electrode name=collector x=-1.6electrode name=substrate backside

structure outf=sige_device.strstructure outf=sige_device.atlas.str sige.conv

D.2 Device Simulations (ATLAS) 97

D.2. Device Simulations (ATLAS)

D.2.1. DC Simulations

go atlas

mesh inf=sige_device_remeshed.atlas.str

# Simulation modelsmodels auger consrh conmob fldmob b.electrons=2 b.holes=1 evsatmod=0 \

hvsatmod=0 bbt.kl fermi bgn print numcarr=2 temperature=300 \lat.temp

thermcontact num=1 elec.num=6 ext.temp=300

# Specify what to calculateoutput \

band.param \band.temp \charge \con.band \efield \e.lines \e.mobility \e.temp \e.velocity \h.mobility \h.temp \h.velocity \impact \j.conduc \j.disp \j.electron \j.hole \j.total \noise.all

# Specify the solver methodmethod block newton trap autonr

# Solve Gummel at Vce=1.5V, first ramp up to Vce=1.5solve initsolve vcollector=0.05solve prev vcollector=0.1solve proj vcollector=0.15 vstep=0.05 name=collector vfinal=1.5

# Specify what to include in log file

98 Simulation Decks

log outf=hbt122_device_dc.log \master \j.disp \j.electron \j.hole \s.param \impedance =50 \gains \inport = base \outport = collector \noise.all \width=1

# Start ramping base electrodesolve prev vbase=0.05solve prev vbase=0.1solve proj vbase=0.15 vstep=0.05 name=base vfinal=1

log offquit

D.2.2. AC and Noise Simulations

go atlas

mesh inf=sige_device_remeshed.atlas.str

# Set model parametersmodels auger consrh conmob fldmob b.electrons=2 b.holes=1 evsatmod=0 \

hvsatmod=0 bbt.kl fermi bgn print numcarr=2 temperature=300 \lat.temp

thermcontact num=1 elec.number=6 ext.temp=300

# Specify what to calculateoutput \

band.param \band.temp \charge \con.band \efield \e.lines \e.mobility \e.temp \e.velocity \h.mobility \h.temp \h.velocity \

D.2 Device Simulations (ATLAS) 99

impact \j.conduc \j.disp \j.electron \j.hole \j.total \noise.all

# Specify the solver methodmethod block newton itlimit=50 trap autonr \

carriers=2 nblockit=50

# Ramp Vce up to 1.5Vsolve initsolve vcollector=0.05solve prev vcollector=0.1solve proj vcollector=0.15 vstep=0.05 name=collector vfinal=1.5

# Ramp Vbe up to about 0.7Vsolve vbase=0.05solve vbase=0.1solve vbase=0.15 vstep=0.05 name=base vfinal=0.76

# Specify what to include in log filelog outf=hbt122_device_rf_vce1p5.log \

master \j.disp \j.electron \j.hole \s.param \impedance =50 \gains \inport = base \outport = collector \noise.all \width=1

solve proj vbase=0.78 vstep=0.005 name=base vfinal=0.89 \ac noise.ss freq=5e9 auto fstep=10e9 nfstep=20 vss=0.005 \outf=sige_device_noise_vce1p5_a master

quit

100 Simulation Decks

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