· Acknowledgements I would have been unable to complete this work without the contin-uous,...

Novel Measurement Systems for

Characterisation of Avalanche

Photo-Diodes

James Ernest Green

Department of Electronic and Electrical Engineering

University of Sheffield

A thesis submitted for the degree of

Doctor of Philosophy

August 2012

‘I am a Bear of Very Little Brain, and long words bother me.’

A. A. Milne

Acknowledgements

I would have been unable to complete this work without the contin-

uous, unwavering support of my partner Pippa and of my parents.

I would like to thank my supervisors Richard C. Tozer and John P.

R. David. Thanks is also due to the Engineering and Physical Sci-

ences Research Council for funding the research, the EEE department

technical staff and the EEE departmental administrators.

Thanks to Stanislav I. Soloviev, Jody A. Fronheiser and Peter M.

Sandvik at The Semiconductor Technology Laboratory, part of Gen-

eral Electric Global Research who generously provided the Silicon

Carbide semiconductor devices reported in Chapter 7. Thank you

also for a considerable amount of your time, the abridged device fab-

rication method and of course dinner at the restaurant–on–the–bridge

in Nurnberg.

Further thanks to Prabhat Agarwal, Chritelle, Ervin, Fred, Micha,

Ray and Bob formerly at Philips Semiconductor. For the Design of

the Si wafers and devices reported in Chapter 6.

I would also like to thank my office colleagues and co-authors Daniel S.

G. Ong, Rajiv Gomez, Andrew Marshall, Wei Sun Loh, Yu Ling Goh,

Simon Dimler, Lionel J. J. Tan, Siew Li Tan, Peter Vines, Syahrin

Idris, Wai Mun Soong, and Kan Yeep Choo.

I owe considerable gratitude to M. Bibby, a wise and loyal friend.

Without the influence of G. F. Green, S. P. Woolley, N. A. Jaques, C.

J. Bijon, R. Lapslie, E. S. Hart, R. Hart, R. Green and S. Burton, I

would not have found myself in a suitable position to begin.

I would like to acknowledge the work of Dave Massy, Kim Foung

Li, Beng Koon Ng and Chee Hing Tan on which this work is based.

iii

Further thanks are due to Asst. Prof. Ng for generously providing

the implementation of the RPL model used in Chapter 6. I would

also like to thank Sammual Goldwasser – of Sam’s Laser Guide – for

enlightening discussions on laser pathologies.

I would like to thank Dave Stone, for providing a pleasant change of

topic.

iv

Abstract

This work is concerned with the measurement and interpretation of

avalanche noise in avalanche photo-diodes (APDs). Two new excess

noise measurement systems are described. These systems are ca-

pable of measuring photo-multiplication and excess noise in APDs

with junction capacitance up to two orders of magnitude greater

than the prior Sheffield system, and greater than any other reported.

The systems use phase sensitive recovery techniques, which allow the

avalanche noise power and photo-current to be measured unambigu-

ously from the system noise and leakage current of the test device.

The use of two independent measurement systems provides an excel-

lent method of confirming the validity of the data. The design and

optimisation of these systems and associated material is presented in

detail.

Avalanche noise measurements have been performed on a range of Sili-

con Carbide nip and separate absorption and multiplication avalanche

photo-diodes at 244 nm and 325 nm. This is the first report of pure

hole injection in SiC. The devices have i region widths, w, of 2.7 µm

and 0.57 µm. The value of α – the electron ionisation coefficient –

has been experimentally verified over a range of lower electric fields

and the extracted is significantly lower than prior reports.

A systematic experimental analysis of the effect of the dead-space

in Silicon has been undertaken using the measurement systems de-

scribed. These devices have widths, w, ranging from 0.35 µm to

31 nm. The Silicon results show that, for w < 0.4 µm, the extracted

and non-local modelled impact ionisation coefficients begin to deviate

from the bulk coefficients. In a 31 nm device at 1 MV/cm both α and

β are approximately an order of magnitude lower than the bulk value.

v

Publications

Journal Papers

• James E. Green, Richard C. Tozer and John P. R. David, “A Transimpedance

Amplifier for Excess Noise Measurements of High Junction Capacitance

Avalanche Photodiodes”, Submitted Measurement Science and Technology.

• James E. Green, Richard C. Tozer, and John P. R. David, “Stability in

Small Signal Common Base Amplifiers”, Accepted IEEE Trans. Circuits

and Systems I: Regular Papers [Available] 10.1109/TCSI.2012.2209709.

• James E. Green, Richard C. Tozer and John P. R. David, “A Computation-

ally efficient circuit model of an Avalanche Photodiode”, In Prep for IEEE

Trans. Electron Devices

• James E. Green, Richard C. Tozer and John P. R. David, “An Opamp

Transimpedance Amplifier for Excess Noise Measurements of High Junction

Capacitance Avalanche Photodiodes”, In Prep. Measurement Science and

Technology

• James E. Green, Richard C. Tozer and John P. R. David, “Impact Ionization

and Dead Space in Silicon” Submitted IEEE Trans. Electron Devices.

• James E. Green, Wei Sun Loh, Andrew R. J. Marshall, Beng Koon Ng.

Richard C. Tozer, John P. R. David, Stanislav I. Soloviev and Peter M.

Sandvik, “Impact Ionization Coefficients in 4H-SiC by Ultra-Low Excess

Noise Measurement” IEEE Trans. Eletron Devices, Vol. 59, No. 4, pp.

1030 – 1036, April 2012 [Available] 10.1109/TED.2012.2185499

• S. L. Tan, W. M. Soong, J. E. Green, M. J. Steer, S. Y. Zhang, L. J. J. Tan,

Y. L. Goh, J. S. Ng, I. P. Marko, S. J. Sweeney, A. R. Adams, J. Allam, J.

vii

http://dx.doi.org/10.1109/TCSI.2012.2209709

http://dx.doi.org/10.1109/TED.2012.2185499

P. R. David, “Avalanche noise and breakdown characteristics in GaInNAs

diodes”, In Prep for Appl. Phys. Lett. or J. Appl. Phys.

Conference Papers

• James E. Green, W.S. Loh, J. P. R. David, R. C. Tozer, Stanislav I. Soloviev,

Peter M. Sandvik, “Characterisation of Low Noise 4H-SiC Avalanche Pho-

todiodes”, Materials Science Forum, 645-648 (2010) pp. 1081-1084, [Avail-

able] scientific.net

• M. J. Hobbs, S. D. Das, J. E. Green, J. P. David, C. H. Tan, S. Krishna, E.

Plis and J. Willmot, ”Uncooled GaSb/InAs type II superlattice photodiode

for radiation thermometry” in prep for IOP Photon12, Durham, UK

Book Chapters

• Daniel S. G. Ong and James E. Green (2011). “Avalanche Photodiodes in

High-Speed Receiver Systems”, in “Photodiodes - World Activities in 2011”,

Jeong-Woo Park (Ed.), ISBN: 978-953-307-530-3, InTech, 2011, [Available]

intechopen.com

viii

http://dx.doi.org/10.4028/www.scientific.net/MSF.645-648.1081

http://www.intechopen.com/articles/show/title/avalanche-photodiodes-in-high-speed-receiver-systems

Contents

Acknowledgements iii

Abstract v

Publications vii

Contents ix

1 Introduction 1

1.1 A Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Photo-multiplier Tube . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Impact Ionisation and Photo-Diodes . . . . . . . . . . . . . . . . 6

1.4 Non-Local Multiplication and Noise . . . . . . . . . . . . . . . . . 14

1.5 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.6 Extraction of Ionisation Coefficients . . . . . . . . . . . . . . . . . 22

2 Measurement Techniques 25

2.1 Current Voltage Characteristics . . . . . . . . . . . . . . . . . . . 25

2.2 Capacitance Voltage Characteristics . . . . . . . . . . . . . . . . . 28

2.3 Photo-Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4 Review of Noise Measurement Systems . . . . . . . . . . . . . . . 29

3 Transistorised High Capacitance Noise Measurement System 39

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Front-End Design Methodologies . . . . . . . . . . . . . . . . . . 40

3.3 Diode Small Signal Model . . . . . . . . . . . . . . . . . . . . . . 43

ix

CONTENTS

3.4 Noise Measurement System . . . . . . . . . . . . . . . . . . . . . 44

3.5 Transimpedance Amplifier . . . . . . . . . . . . . . . . . . . . . . 45

3.6 Opamp stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.7 Transistor Stage Biasing Feedback System . . . . . . . . . . . . . 49

3.8 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.9 Physical Construction . . . . . . . . . . . . . . . . . . . . . . . . 51

3.10 Noise Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.11 TIA Characterisation . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.12 Voltage Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.13 Noise Power Meter . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.14 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4 Small Signal Stability in Common Base Amplifiers 73

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2 Frequency Dependent Behaviour of a

Common Base Amplifier . . . . . . . . . . . . . . . . . . . . . . . 76

4.3 Addition of Parasitic Inductance in the Base Network . . . . . . . 77

4.4 The Addition of an Emitter Biasing Resistor . . . . . . . . . . . . 80

4.5 Addition of Emitter Capacitance . . . . . . . . . . . . . . . . . . 82

4.6 Addition of Load Capacitance and Collector - Base Junction Ca-

pacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.7 Colpitts Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.8 Analysis of the Common Base Amplifier from an Oscillator’s Per-

spective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.9 The Effectiveness of Increasing the

Base Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.10 Practical Methods of Assessing Oscillation . . . . . . . . . . . . . 89

4.11 High Frequency Techniques . . . . . . . . . . . . . . . . . . . . . 90

4.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5 1 MHz Opamp Based High Capacitance Noise Measurement Sys-

tem 93

x

CONTENTS

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2 Noise Measurement System . . . . . . . . . . . . . . . . . . . . . 94

5.3 Transimpedance Front End . . . . . . . . . . . . . . . . . . . . . . 96

5.4 Bandwidth Setting Filter and ENBW . . . . . . . . . . . . . . . . 103

5.5 Voltage Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.6 Noise Power Meter . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6 Excess Avalanche Noise Thin Silicon APDs 109

6.1 Review of Impact Ionisation in Silicon . . . . . . . . . . . . . . . 109

6.2 Application: On-Chip Photonic Data

Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.3 Application: Medical/Biological Imaging . . . . . . . . . . . . . . 114

6.4 Chapter Objective . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.5 Layer Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116



6.8 Photo-multiplication . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.9 Excess Noise Measurement . . . . . . . . . . . . . . . . . . . . . . 124

6.10 Local Ionisation Coefficient Comparison . . . . . . . . . . . . . . 127

6.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7 Excess Noise Measurements and Impact Ionisation Coefficients

in 4H-SiC 129

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.2 Review of Impact Ionisation in Silicon Carbide . . . . . . . . . . . 135

7.3 Chapter Objective . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.4 Structure Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 139



7.7 Photo-Multiplication Characteristics . . . . . . . . . . . . . . . . 148

7.8 Excess Noise Characteristics . . . . . . . . . . . . . . . . . . . . . 153

7.9 Ionisation Coefficient Extraction . . . . . . . . . . . . . . . . . . . 155

xi

CONTENTS

7.10 Comparison of GaN and SiC: GaN and AlGaN Detectors . . . . . 157

7.11 Comparison of GaN and SiC: GaN Power Electronics . . . . . . . 158

7.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

8 An APD SPICE Model for Circuit Simulation 161

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

8.2 Characteristics of the Model . . . . . . . . . . . . . . . . . . . . . 163

8.3 Parameters of the Model . . . . . . . . . . . . . . . . . . . . . . . 164

8.4 Examples of the Model . . . . . . . . . . . . . . . . . . . . . . . . 169

8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

9 Future Work 177

9.1 Microwave Small Signal Bandwidth Measurements . . . . . . . . . 177

9.2 Microwave Noise Measurements . . . . . . . . . . . . . . . . . . . 182

9.3 Laser Noise “Cancelling” Front End . . . . . . . . . . . . . . . . . 188

9.4 Improved Squaring Circuit . . . . . . . . . . . . . . . . . . . . . . 190

9.5 An On-Wafer SPAD Setup . . . . . . . . . . . . . . . . . . . . . . 191

9.6 NLOS UV Communications with SiC . . . . . . . . . . . . . . . . 193

10 Conclusion 195

References 197

A A One Dimensional Poisson Solver 227

A.1 Two Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

A.2 Three Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

A.3 Four Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

A.4 Five Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

B APD SPICE Model Source Code 235

C Phase-Sensitive Detection 239

D High Order Nature of the Common Base Amplifier 243

xii

Chapter 1

Introduction

This thesis concerns two related topics; firstly, the development of electro-optical

measurement systems, and secondly, the characterisation of Silicon and Silicon

Carbide avalanche photo-diodes (APDs). This chapter provides a historical per-

spective before introducing the impact ionisation process; specifically, the local

modelling of multiplication and excess noise. The timeliness of this work and

the present state of knowledge of impact ionisation in Silicon and Silicon Carbide

are discussed in their respective chapters, where a selection of example industrial

uses of APD detectors are given. Approximately half of the thesis is given over

to measurement system design and half to device characterisation.

A panoply of electronic devices are presently manufactured and accurate im-

pact ionisation coefficients are required for the design and simulation of any elec-

tronic device which is expected to develop a large electric field within its structure.

Accurate impact ionisation coefficients are doubly critical in the analysis of high

speed devices developing high electric fields; the feedback nature of impact ioni-

sation, which will be discussed shortly, is intrinsically quite a ‘slow’ process when

compared to the speed of modern microwave devices and systems. It is criti-

cal that high speed device designers can design their devices such that impact

ionisation will be avoided if it is undesirable.

1

CHAPTER 1

1.1 A Historical Perspective

In the thirty years surrounding the turn of the twentieth century physicists were

struggling with problems related to the light spectrum emitted a black body. One

of the problems is sometimes called the “ultraviolet catastrophe”. The question

was posed in terms of a Helmholtz oscillator (of which a black body is an example).

The particular difficulty is with the equipartition theorem, which relates the

average energy of each oscillator mode to the temperature of the oscillator.

Classical electromagnetism, which considers energy exchange to happen con-

tinuously, can be used to compute the energy emitted by a three dimensional

black body at a given wavelength. The calculation gives u as the number of

degrees of freedom in frequency × the average energy per degree of freedom.

The average energy per degree of freedom is given by,

E =

∫∞

0

E exp (−E/k T ) dE.∫

∞

0

exp (−E/k T ) dE.

(1.1)

which reduces to k T .

The number of degrees of freedom is given by (8 π v2)/(c3). Combining these

gives the the Rayleigh-Jeans formula,

u (v, T ) =8 π v2

c3k T (1.2)

As the frequency, v increases the energy, E, increases without limit, which

is in disagreement with empirical observation. Max Planck provided a radical

change of viewpoint which resolved disagreement, he postulated the quantisation

of energy. Energy transfer is then only allowed for specific amounts of energy.

Kragh notes that Planck declared he turned to quantisation “in desperation”

having been unable to reconcile the classical approach with the observations [1].

The expression for the quantum of energy is,

ε = h v (1.3)

2

CHAPTER 1

where h is Planck’s constant and v is the frequency. The reduced Planck constant

~ = h / (2 π) is often used because it makes some expressions more elegant.

Energy, E, can only exist in integer multiples of the quantum,

E = nh v, where n = 0, 1, 2, . . . (1.4)

The quantised version of the black body expression is,

E =

∞∑

n=0

n ε exp (−n ε/k T )

∞∑

n=0

exp (−n ε/k T )

(1.5)

The ultraviolet catastrophe was not the only problem facing physics at the

beginning of the twentieth century, the observations of the interaction between

metal surfaces and light and the cathode rays that resulted - later named the

photoelectric effect - was difficult to explain using the continuous interpretation

of energy transfer. The photoelectric effect experiment requires a clean metal

surface (the cathode) which is exposed to light and yields electrons from its

surface. A second metal surface (the anode) which faces the cathode collects

electrons. Both electrodes are held in high vacuum. Light of different intensities

and wavelengths is shone onto the cathode. The cathode emits electrons (the

photoelectric current) which are collected on the anode.

Einstein’s 1921 Nobel prize was awarded “...especially for his discovery of the

law of the photoelectric effect”, which was published shortly before the, celebrated

special relativity paper of 1905. Einstein adopted a similar idea to Planck and

developed the idea that, in the photoelectric experiment, a photon (quantum of

electromagnetic energy) is absorbed by an electron which is bound to the surface

of the cathode metal. The energy imparted to the electron by the photon allows

the electron to escape the bound state and any extra energy that the photon

imparts is given to the electron as kinetic energy,

h v = Ek +W0 (1.6)

3

CHAPTER 1

where Ek is the free electron’s kinetic energy and W0 is the work function of the

metal. In the photoelectric effect experiment, a voltage may be applied between

the electrodes to cause the photoelectric current to cease. The magnitude of this

voltage must be,

e V = Ek (1.7)

An electron “cloud” may then form around the cathode. Equations 1.6 and 1.7

and may be solved for stopping voltage V ,

V =h

ev − W0

e(1.8)

If the stopping voltage is plotted as a function of frequency, a linear dependence

results where the constant of proportionality is h/e.

The photoelectric effect experiment shows that,

1. When light strikes a metal surface, a current flows immediately, irrespective

of the light intensity (assuming point 4 is satisfied).

2. For a particular frequency, the magnitude of the photoelectric current is

directly proportional to the intensity of the light.

3. The stopping potential V , and the maximum energy of the emitted elec-

trons, is a function of the frequency of the light and the work function of

the metal.

4. Each metal has a characteristic minimum frequency v0 such that

e V = h (v − vo)

5. The constant h is the same for all metals. It is the same constant used by

Planck in the black body derivation.

These experimental observations are easily explained if light is composed of

quanta. For example the second observation, is easy to explain from a quantum

paradigm. If the power of the light source incident on the surface is increased,

the number of photons incident on the cathode surface increases consequently

more electrons are released. In the classical approach, however, increasing the

intensity may be expected to increase the energy of each cathode ray as opposed

4

CHAPTER 1

to the number of cathode rays. The fourth observation is also dependent on a

quantum approach, in a classical approach cathode rays may be expected to leave

the cathode surface if light of any wavelength is incident on the cathode. The

optical energy incident on the surface may be integrated over time until enough

energy has been imparted to cause the emission of cathode rays. However this is

contrary to observation which shows that enough optical energy must be imparted

to the surface at one instant to free an electron.

1.2 Photo-multiplier Tube

Prior to the discovery of the useful optical detection capability of certain semi-

conductors, several types of thermionic valve were used to detect light including

the photomultiplier which fulfils similar application requirements to avalanche

photo-diodes. A schematic of a photomultiplier tube is shown in Fig. 1.1. It is an

evacuated glass envelope having an optically transparent faceplate behind which

is placed a photo-cathode.

The photo-cathode is a metal structure coated in an alkali metal for example

sodium-potassium-antimony or a III-V semiconductor such as Gallium Arsenide.

These materials are popular due to their low work function. Photons incident

on the photo-cathode cause the emission of electrons by the photo-electric effect.

An arrangement of focusing electrodes guide the free electrons towards a set of

dynodes. The dynodes have grids which attract the free electrons by electro-

static means. The dynodes are also metal (for example Nickel, Stainless Steel

or a Coper-Beryllium Alloy) structures which are coated in a secondary emissive

material such as Magnesium Oxide or Gallium Phosphide. A single dynode can

have a secondary emission ratio of approximately one hundred. Photo-multipliers

having up to nineteen dynodes have been commercially produced [2]. The sec-

ondary electrons travel from dynode to dynode, growing in number, until they

are collected in the last dynode behind the anode.

The key disadvantages of photo-multipliers are their bulk, and necessity for

high voltages. They are often said to be fragile, although several vidicon tubes

have visited the moon and returned pictures on more than one occasion. ‘Rugged’

photo-multipliers are also employed in oil exploration in down-hole applications.

5

CHAPTER 1

e−Directionof Light

Faceplate

Photocathode

︸︷︷︸

Dynodes Anode

Last DynodeSecondary ElectronFocusing Electrode

Vacuum ∼ 10−4 Pa

Pin

Figure 1.1 – Schematic of a Photo-multiplier tube. Redrawn from [2].

The cost of a commercial photo-multiplier and a commercial Silicon APD are

the same order of magnitude. Photo-multipliers require voltages between one

and two orders of magnitude greater than a Silicon APD when both devices are

biased to provide a similar gain.

1.3 Impact Ionisation and Photo-Diodes

In this section the generic three region device and separate absorption and mul-

tiplication layer avalanche photo diode (SAM-APD) detectors are discussed and

the impact ionisation process is qualitatively described with some simple exam-

ples. Various mathematical results relating to McIntyre’s local model [3] are also

derived.

1.3.1 Three Region Photo-Diode Detectors

A generic pin detector possessing an intrinsic region which is slightly n-type is

show schematically in Fig. 1.2 with an example electric field profile. Assuming the

device is homogeneous and uniform a one dimensional description of the device

does not represent a loss of generality. The device has been fabricated with ohmic

contacts and a precision controlled cell is applied in such a way that the device

is reverse biased. A window is etched into the top contact enabling light to enter

the p+ region.

6

CHAPTER 1

p+

n−

n+

|ξ|

x

Figure 1.2 – A generic pin / nip photo-diode with electric field shown to the right.

The reverse bias causes an electric field of greater magnitude than the diffusion

field to develop in the device. The characteristics of the field are a function

of the intrinsic region width and the doping densities in each region. At low

electric fields impact ionisation provides a negligible increase in total current.

Photons which are energetic with respect to the material band-gap may generate

carriers throughout the structure dependant on the absorption characteristics of

the material and the wavelength of the incident light. Assuming that the light is

entirely absorbed in the p+ region and all carriers are generated optically outside

of the depletion region, the minority electrons diffuse through the p+ material

until they recombine or reach the electric field. Once inside the electric field

region the electrons are swept through the device and induce a current in the

external circuit. The conditions necessary for impact ionisation are only that

the electric field be sufficient, given the material system under consideration,

that carriers drifting in an electric field can liberate further (secondary) carriers

by transferring some energy to those carriers, promoting them from the valence

band into the conduction band. Impact ionisation is shown schematically in

Fig. 1.3. The generation of the carrier is shown by the numbering. In this case

it is assumed that the primary carrier is an electron (blue) which has diffused

in to the high field region; the associate hole (red) is not shown. The primary

carrier ionises the lattice to generate a secondary e-h pair (2) the electron leaves

without taking further part. The secondary hole gains energy and ionises the

lattice producing a third generation (3) e-h pair. The second and third generation

holes leave the electric field region. The third generation electron ionises the

lattice creating a fourth generation e-h pair (4). The first three ionisation events

represent ‘carrier feedback’. This feedback is often undesirable as it causes the

7

CHAPTER 1

1

12

2

2

33

3

44

45

5

4

55

56

6

Figure 1.3 – Impact ionisation process schematically shown.

transient response of the device to increase with increasing multiplication. A

more preferable situation is one in which only electrons or holes are responsible

for impact ionisation. The fourth generation e-h pair ionises the lattice twice on

one transit of the high field region creating two fifth generation e-h pairs (5). The

hole created in the first of these fifth generation events goes on to ionise the lattice

(6) but none of the electrons from the fourth fifth or sixth generation carriers

ionise. The lack of carrier feedback results in a maximum device speed. In this

hypothetical device, in which carriers can be counted, the photo-multiplication is

seven. Seven electrons leave only one has entered. In this example the number of

holes leaving the high field region must be one less than the number of electrons,

because there is one primary carrier. If the primary carrier was a hole and the

colours of the carriers and the direction of the electric field reversed, holes would

be counted and the multiplication would be seven; one primary hole, and six

holes generated by impact ionisation. The situation described until now is that

8

CHAPTER 1

of pure injection, where all of the primary carriers are of one type and they

diffuse into the high field region. Pure injection is often desirable because it

yields the lowest possible excess noise, assuming the device is constructed such

that the more readily ionising carrier diffuses into the high field region. Use of

the less readily ionising carrier yields the largest possible excess noise. When

possible, pin and nip devices are often fabricated with identical region widths

and – as nearly as possible – doping densities such that the two devices will have

a similar electric field profile over a wide range of reverse bias. One device yields

pure electron initiated multiplication and excess noise, the other yields pure hole

initiated multiplication and excess noise. Possession of both pin and nip devices

allows cross-checking and confirmation of extracted results, including ionisation

coefficients.

Devices exhibit mixed injection when carrier pairs are photo-generated inside

the electric field region. Mixed injection may occur because the device is so

designed or by accident. Photons possessing energies only slightly larger than the

band-gap are weakly absorbed and tend to generate carriers far from the optical

window; photons that are energetic with respect to the band-gap are strongly

absorbed and tend to generate carriers close to the optical window. In the case of

mixed injection, shown in Fig. 1.4, the carrier pairs created in electron initiated

ionisation events may be thought of as possessing the multiplication and excess

noise characteristics due to pure electron injection, similar arguments apply to

hole initiated events. The multiplication and excess noise that a device yields

under mixed injection conditions is then a function of the ratio of number of

electron and hole initiated events. In the hypothetical example shown in Fig. 1.4,

β = α and there are the same number of each primary carrier type so the mixed

injection result would lay in the middle of the pure results, similar arguments

apply to noise. Note that the middle is not necessarily the mean. In the case

of β not equal to α the more readily ionising carrier type will have far more

events associated with it than the less readily ionising carrier type. When the

ionisation coefficients are disparate, special consideration should be given to the

case of very slightly mixed injection in which injection by diffusion of the less

readily ionising carrier dominates, but a few stray carriers of the more readily

ionising carrier type are also injected due to a small amount of generation inside

9

CHAPTER 1

1 1

1

2

2

1

2

2

2

3

3

2

3

3

Figure 1.4 – Mixed injection impact ionisation process shown schematically.

the e-field. In this case a very small multiplication and very high noise may be

expected. However, the very few more readily ionising carriers that do ionise the

lattice may have, on average, very long ionisation chains, whereas the very many

less readily ionising carriers may have an average chain length of only one or two

events. Hence the majority of the carrier population exists due to the few stray

more readily ionising carriers, despite the initial conditions, which may suggest

otherwise. This ensemble of carriers in which each pair of carriers may be thought

of as possessing multiplication and noise characteristics imbued to them by their

initiating carrier, would exhibit multiplication and noise characteristics similar

to that of the more readily ionising carrier. When measuring devices in which

α and β are widely separated, for example two orders of magnitude, it may be

very troublesome to produce a result exhibiting behaviour dominated by the less

readily ionising carrier type by anything other than perfect injection conditions.

1.3.2 SAM-APD

The five region structure shown in Fig. 1.5, is one possible implementation of

a separate absorption and multiplication region avalanche photo-diode (SAM-

APD). A plausible electric field profile is shown in the same figure. The operation

is fundamentally similar to the three region structure with the exception that two

further regions are added. Photons enter via the n+ layer which has an etched

10

CHAPTER 1

optical window. Assume that the majority of carriers are generated in the upper

n− region where a low electric field exists. Avalanche multiplication takes place

in the lower n− region. The benefits of a SAM structure include,

• Reduced transit time of the multiplication region. This reduces the effect

of carrier feedback over a similar width three region structure.

• Reduced breakdown voltage over a similar width three region structure.

This is a result of the high field being limited to the p+ n− and n+ charge

or grading layer. The device is approximately as thin as a three region

structure that is composed of only the p+ n− and n+ (charge) layers.

• Increased carrier collection and therefore quantum efficiency due to the

electric field reaching close to the device surface. More carriers maybe

collected by the field, as the number of carriers lost due to recombination

in the n+ cap layer is minimised.

• Increased velocity of carriers moving from the point of generation to the

high field region. In the three region structure carriers generated outside

the high field region must diffuse in. In the SAM structure the carriers

experience a low field in the absorption layer which sweeps them into the

multiplication layer.

• The possibility of pure injection is not negated by the electric filed reaching

close to the device surface. Carrier pairs generated in the absorption region

can contribute to pure injection if the carriers drifting in to the low field

towards the optical window cannot gain enough energy to ionise the lattice.

Under these conditions all the carriers entering into the multiplication re-

gion from the absorption region have been optically promoted and are not

the result of ionisation events. This is equivalent to optical carriers diffusing

into the high field region in a three region structure.

• A further benefit is the possibility of using two material systems to form

a heterostructure in which the absorption region is composed of a material

suited to absorbing light of the required wavelength and the multiplication

region is composed of a material that exhibits high gain, low dark current

11

CHAPTER 1

n+

n−

n+

n+

p+

n+

|ξ|

x

Figure 1.5 – A generic SAM-APD photo-diode with electric field shown to the right.

and bulk breakdown. An example of this is a Germanium absorber with

a Silicon multiplication region. Ge-Si SAM-APDs are in competition with

some III-V materials for use as detectors in the second and third generation

(1.3 µm and 1.55 µm) low loss fibre windows [4].

• A final advantage of the SAM structure is the lower capacitance of the device

due to the wider depletion region than in a three region structure with a

similar breakdown voltage. Assuming the device is not transit time limited,

lower capacitance translates directly into a higher bandwidth device.

1.3.3 Local Model Multiplication and Excess Noise

The ‘local’ model of photo-multiplication and excess noise in avalanche photo-

diodes is principally due to two parties. The multiplication expressions were

reported by the early Silicon workers including [5–12]. The local excess noise

model was derived by McIntyre [3]. The model is elegant, and may be solved with

closed form for some simple cases making it computationally efficient. Solutions

may be found for quite complex cases of mixed injection profile and continuous or

piecewise continuous electric fields for any number of regions. The expressions can

be evaluated by numerical approximation using an efficient quadrature algorithm

such as Simpson’s Rule or Gauss Legendre quadrature [13]. Evaluating symbolic

solutions is of limited value in all but the simplest cases where a constant electric

field is assumed. In this case the symbolic solutions may be used to extract α

and β from measurements of photo-multiplication and excess noise factor. The

12

CHAPTER 1

governing approximation of the model is that the likelihood of lattice ionisation

by any given carrier is a function of the electric filed in the vicinity of the carrier,

and is independent of the energy history of the carrier.

When invoking the local model, the ratio of ionisation coefficients is labelled k

and it is often assumed constant for a particular device under common conditions

such as pure electron and pure hole injection. k is however a function of α and β

and so changes across the depletion distance. k is sometimes quoted for a device,

this is the “effective k” and is particular to the device. Effective k is formed

from an ensemble of the position dependent k over all positions. It is however

not possible to assign a particular value of effective k to a material system as

the effective k is a function of device structure. Thicker devices are generally

less noisy than thinner ones. This generalisation is based on the premise that, in

many material systems, α and β diverge as electric field diminishes. In a thicker

device, which will break down at a lower electric field than an arbitrarily chosen

thinner device, the effective k will be lower than in the thinner structure biased

such that it produces the same ensemble multiplication.

1.3.3.1 Applicability and Limit

Dead space is a term for the distance a carrier must travel through the lattice

before the carrier can possess a non-zero probability of ionising the lattice. The

dead space is indirectly measured by expressing the energy in electron Volts

required to overcome the dead space. During the traversal of the dead space

the carrier gains energy from the electric field. It is common to assume that

carriers drift at the saturation velocity and the energy gain is in the form of

momentum. The accuracy of the local model decreases as the dead space length

becomes significant with respect to the high field region width. Put another way,

as the device is made thinner – intrinsic width is reduced – the dead space length

is a larger proportion of the intrinsic width. Under these conditions the prior

interactions between the carrier and the lattice, and the carrier and the electric

field influence the probability of ionisation in a particular region under scrutiny.

As this influence increases, the local model becomes less accurate.

In the limit, α or β → 0, single carrier multiplication exists. If the multipli-

13

CHAPTER 1

cation is large, the average multiplication chain length must be long. Since there

is little or no carrier feedback, an average carrier must ionise many times in one

pass of the high field region to produce the ensemble multiplication. There will

be a spread of multiplication chain lengths, some carriers will ionise the lattice

many times and others will will pass out of the high field region without ionising

the lattice at all. In this case a more general relationship between the average

multiplication and the dead-space is unveiled, where the parameters of interest

are the multiplication, dead-space length, and device width. The product of the

former two can not be greater than the latter. This implies that in single carrier

multiplication an upper boundary of multiplication exists for a given magnitude

of electric field. Under favourable conditions secondary carriers may be gener-

ated in spatially confined regions and then travel through the device in quasi two

dimensional ‘sheets’. This collecting of carriers – which is essentially a reduction

in the variance of multiplication values for the population of carriers – causes the

device to exhibit less noise than is supposed by the local model. Devices exhibit-

ing less noise than the minimum local model noise have been reported in InAs [14]

and HgCdTe [15] material systems. Marshall [16] has proposed an excess noise

reduction factor which provides a figure of merit proportional to the reduction of

excess noise due to the spatial confinement of the ionisation events.

1.4 Non-Local Multiplication and Noise

In this work the non local RPL model reported by Ong et al. [17] is used. The

program used to perform the required modelling is due to B. K. Ng [18], and is

used with permission. The RPL model operates by sequentially passing many

carriers into a device structure of known region widths and doping densities. The

likelihood of a carrier reaching a given distance from its prior ionisation location

(or generation point etc.) without ionising is given by a probability density

function. This is often composed of the forbidden ‘dead space’ followed by an

exponential decay. Each carrier is ‘followed’ through the device; each carrier

pair that is promoted by the original carrier is then followed. The following

of carriers continues until all the carriers produced in a particular ionisation

chain have left the electric field region. A new carrier is then inserted, and the

14

CHAPTER 1

p+ n− n+

hv hv

Figure 1.6 – Pure injection in a three region device. Both electron and hole injection areshown.

process is repeated. To provide a statistically representative result a large number

of carriers, perhaps as many as a million, are used. This modelling method

computationally intensive compared to the local model. However RPL is more

capable of accurately representing measured device behaviour under conditions

where the stochastic nature of the scattering process is limited by the prevailing

device conditions.

1.5 Derivations

In this section the local model expressions required to produce a mixed injection

model are derived. These expressions are often used with a piecewise contin-

uous approximation to the electric field, and a Poisson solver under which the

depleation approximation is observed. A suitable Poisson solver is derived in

Appendix A.

1.5.0.2 Pure Injection

The pure injection cases are simplifications of the mixed injection case, the deriva-

tion of the local model can be found in the literature [3]. Light enters the p+ or

n+ layer of a three region structure, shown in Fig. 1.6. All carriers are generated

outside the electric field.

15

CHAPTER 1

Pure Electron Injection The expression for photo multiplication is a simpli-

fication of the mixed injection case given by McIntyre [3] equations 1 - 5.

Me =1

1−(∫ w

0

α(x) exp

(

−∫ x

0

α(x)− β(x) dx.

)

dx.

) (1.9)

Note that the multiplication will become undefined when the integral in the de-

nominator reaches unity, this is regarded as a definition of breakdown. From a

perspective of considering carriers in a device, the requirement is that a single

carrier can promote an unlimited number of secondary carriers. This has destruc-

tive implications for a real device unless steps are taken to limit the current flow

and power dissipation. Breakdown is independent of the injection conditions and

incident photon energies. pin and nip devices possessing identical region widths

and doping densities and being made from the same material will break down at

the same reverse bias voltage.

The noise due to pure electron injection may be given by,

Fe = k ·Me + (1− k)

(

2− 1

Me

)

(1.10)

where k = α/β and may be considered constant over a suitably small range of

electric fields. Without invoking k it may be shown that the noise is also given by

considering the noise spectral density in a device. This expression can be derived

from [3] equation 13. Note that to arrive at this equation we have considered the

increase in hole current in a distance dx whereas McIntyre chose to consider the

increase in electron current in a small region dx. This does not represent a loss

of generality.

The noise spectral density for the pure electron case,

φ = 2 · q(

2 · Ie ·M2e + It

(

2

∫ w

0

β(x)M(x)2 dx.−M2h

))

(1.11)

The total current in the device,

It = Ie ·Me (1.12)

16

CHAPTER 1

substituting,

φ = 2 · q(

2 · Ie ·M2e + Ie ·Me

(

2

∫ w

0


))

(1.13)

The excess noise in terms of noise power spectral density,

F =φ

2 · q · It ·M2e

(1.14)

substituting,

F =

2 · Ie ·M2e + Ie ·Me

(

2

∫ w

0


)

Ie ·M2e

(1.15)

Pure Hole Injection For pure hole injection multiplication similar arguments

apply as for electrons. The expression is,

Mh =1

1−(∫ w

0

β(x) exp

(

−∫ w

x

α(x)− β(x) dx.

)

dx.

) (1.16)

The noise due to pure hole multiplication while defining k as αβis,

Fh =1

k·Mh +

(

1− 1

k

)(

2− 1

Mh

)

(1.17)

Similar arguments apply as in the prior section where the noise under pure hole

injection conditions is given by,

F =

2 · Ih ·M2h + Ih ·Mh

(

2

∫ w

0


)

Ih ·M2h

(1.18)

1.5.0.3 Generation of Carriers

A simple model of distributed injection has been reported by Tan et al. [19].

Their model has been used in Chapter 7. The operation of the model is shown

for p+ side injection in Fig. 1.7. Laser light is incident on the p+ surface of the

17

CHAPTER 1

p+ n− n+

G(x) = A0e−α·x

hv

Xp w XnA B

Figure 1.7 – This figure shows the relevant distances and expressions for p+ side injection.

device, and the n+ surface is backed by a substrate. The electric field lies in the

region marked by w, Xp and Xn are the distances to the layer edges from the

electric field edge. The minority electrons in the p+ layer traverse from left to

right and minority holes in the n regions traverse from right to left. Any carriers

generated outside the electric field will be captured by it. If it is desirable to

implement an approximation to the recombination of some optically generated

carriers in the p+ and n+ regions then an arbitrarily complex function of distance

may be developed to determine which carriers are collected and which recombine.

The simplest of these, in terms of implementation, is a hard threshold at a fixed

distance from A and B, all carriers generated within the threshold are collected

all others are not.

The quantity of carriers that diffuse into the p side of the electric field is given

by,

Ap = A0 (1− exp (−ϕ ·Xp)) (1.19)

where A0 is the number of carriers incident on the p+ surface per unit area per

unit time, ϕ is the absorption coefficient at the wavelength of interest and Xp is

the distance of undepleated material between the electric field and the p+ surface.

Similarly the quantity of carriers that diffuse into the n side of the electric

18

CHAPTER 1

field is given by,

An = A0 exp (−ϕ (Xp + w)) (1− exp (−ϕ ·Xn)) (1.20)

where w is the electric field distance and Xn is the distance of undepleated ma-

terial between the n side of the electric field and the n side of the device.

The total carriers generated in the device is given by,

At = Ap ·M(0)+An ·M(w)+

∫ w

0

M(x) ·A0 ·ϕ exp (−ϕXp) exp (−ϕx) dx. (1.21)

assuming that x is measured from the electric field edge at A, i.e. x = 0 at A,

x = w at B. The total quantity of optically generated carriers is,

Ag = A0 (1− exp (−ϕ (Xp + w +Xn))) (1.22)

and the ensemble multiplication for the device is given by M = At

Ag,

M =

Ap ·M(0) + An ·M(w) +

∫ w

0

M(x) · A0 · ϕ exp (−ϕXp) exp (−ϕx) dx.

A0 (1− exp (−ϕ (Xp + w +Xn)))(1.23)

The shot noise associated with the photo-current generated at x is ideally

multiplied by M2(x). Hence the shot noise generated by the multiplied primary

photo-current is given by,

Nshot = 2 · q · Ag ·M2 (1.24)

and the total device noise is,

Nt = 2 · q2 · Ap ·M(0)2 · F (0) + 2 · q2 · An ·M(w)2 · F (w)

+ 2 · q2(∫ w

0

M(x)2 · F (x) · A0 · φ exp (−φXp) exp (−φx) dx.

)

(1.25)

where,

F (x) = k ·M(x) +

(

2− 1

M(x)

)

(1− k) (1.26)

19

CHAPTER 1

p+ n− n+

G(x) = A0eα·x

hv

Xp w XnA B CZ

Figure 1.8 – Figure showing the relevant distances and expressions for n+ side injection.

and k has the usual meaning αβ.

The excess noise factor F can be expressed as F = Ntotal

Nshot,

F =1

M2 · Ag

(

Ap ·M(0)2 · F (0) + An ·M(w)2 · F (w)

+

∫ w

0

M(x)2 · F (x) · A0 · φ exp (−φXp) exp (−φx) dx.

)

(1.27)

This expression for F provides equivalent numerical solutions to an expression

for the mixed injection excess noise factor, F , which is derived without invoking

k. M as a function of x is given by the following expression (see [3] equation 5).

M(x) =

exp

(

−∫ w

x

(α(x)− β(x)) dx.

)

1−∫ w

0

α(x) · exp(

−∫ w

x

α(x)− β(x) dx.

)

dx.

(1.28)

For injection of carriers by photons entering from the n+ side,

Making the same original assumptions as Tan et al. [19],

An =

∫ C

B

A0 · exp (ϕ · x) dx. = −−A0 (exp (ϕB)− exp (ϕC))

ϕ(1.29)

20

CHAPTER 1

Ap =

∫ A

Z

A0 · exp (ϕ · x) dx. = −−A0 (exp (ϕZ)− exp (ϕA))

ϕ(1.30)

At = M(A) · Ap +M(B) · An +

∫ B

A

M(x) · A0 · exp (ϕA) · exp (ϕx) dx. (1.31)

Depending on the choice of origin in the diagram Z or A is often zero. In this

case A = 0, B = w.

1.5.0.4 Mixed Injection Noise

The noise power spectral density for the mixed injection case (1.32) is derived

by McIntyre [3]. Note that to arrive at this expression we have considered the

increase in hole current in a small distance dx whereas McIntyre chose to consider

the increase in electron current in a small region dx. This does not represent a

loss of generality.

The noise power spectral density is given by,

φ = 2 · q(

2 · Ie ·M2e + 2 · Ih ·M2

h + 2

(∫ w

0

G(x) ·M(x)2 dx.

)

+It

(

2

(∫ w

0

β(x) ·M(x)2 dx.

)

−M2h

))

(1.32)

The total current in the device,

It = Ie ·Me + Ih ·Mh +

∫ w

0

G(x) ·M(x) dx. (1.33)

substituting,

φ = 2 · q(

2 · Ie ·M2e + 2 · Ih ·M2

h + 2

(∫ w

0

G(x) ·M(x)2 dx.

)

+

(

Ie ·Me + Ih ·Mh +

∫ w

0

G(x) ·M(x) dx.

)

·(

2

(∫ w

0

β(x) ·M(x)2 dx.

)

−M2h

))

(1.34)

21

CHAPTER 1

The following expression represents total photo-current multiplied by M2 in

the noise equation,

It ·M2 = Ie ·M2e + Ih ·Mh2 +

∫ w

0

G(x) ·M(x)2 dx. (1.35)

The excess noise in terms of noise power spectral density,

F =φ

2 · q · It ·M2(1.36)

substitution of equations 1.34 and 1.35 into equation 1.36 yields the final form.

The generation profile is a relative measure of generated carriers as a function

of distance. Therefore Ie can be replaced with G(0) and Ih with G(w) assuming

the origin is the point where electrons diffuse into the high field and w to be a

similar point for holes. Me can be replaced with M(0) where M(x) is the position

dependant multiplication expression (1.28) and Mh is replaced by M(w). If one

chooses to consider the increase in electron current in a small region dx, a slightly

different form of (1.34) results, however they provide identical numerical results.

1.6 Extraction of Ionisation Coefficients

In some cases multiplication and noise due to one carrier type may not be easily

measurable. In this case the values of α and β can be extracted from measure-

ments of the other carrier type, provided both multiplication and excess noise

data are available. In the constant electric field approximation α and β are sin-

gle valued and independent of distance. An algebraic analysis is possible. The

expression for Mh (1.16) reduces to,

Mh = − α− β

β exp ((α− β)w)(1.37)

and the relation between Fh and Mh while invoking k is given in (1.17) substi-

tuting k = α/β,

Fh =α ·Mh

β+

(

2− 1

Mh

)(

1− α

β

)

(1.38)

22

CHAPTER 1

α and β may be expressed as a functions of Mh and Fh only by simultaneous

solution of (1.37) and (1.38),

α =

ln

(

− Fh −Mh

M2h − 2 ·Mh + 1

)

(Fh ·Mh − 2 ·Mh + 1)

Mh · w (Fh −Mh)(1.39)

β =

ln

(

− Fh −Mh

M2h − 2 ·Mh + 1

)

(M2h − 2 ·Mh + 1)

Mh · w (Fh −Mh)(1.40)

If Mh, Fh and the depletion width, w, are known for a particular reverse bias

voltage, or range thereof, β and α as functions of electric field can be found

directly by substitution.

23

CHAPTER 1

24

Chapter 2

Measurement Techniques

In this chapter some standard measurements are discussed including current –

voltage, capacitance – voltage and photo-multiplication measurements. These

measurements are routinely performed when a new diode sample becomes avail-

able. The excess noise measurement system after Li [20] is included. The chapter

concludes with a description and evaluation of other noise measurement systems

reported in the literature.

2.1 Current Voltage Characteristics

Forward and reverse current – voltage characteristics are one of a number of

tests performed shortly after devices are manufactured. Forward current voltage

characteristics reveal whether the structure has an exponential characteristic and

therefore obeys the Shockley equation (2.1) [21].

Ia = Is

(

expq (Va − IaRs)

n k T− 1

)

(2.1)

where I is the total current, Is represents the saturation current, q is the

electron charge, k is Boltzmann’s constant, T is the absolute temperature, and Va

is the terminal voltage, n is the ideality factor and Rs is the series resistance. The

ideality factor gives insight to the nature of the current flow across the transition

25

CHAPTER 2

Figure 2.1 – Figure showing the effect of series resistance on the characteristics of a forwardbiased diode. Is = 0.1×10−27 A; n = 2; k = 1.38×10−23 JK−1; q = 1.6×10−19 J; T = 300 K;Rs = 1 Ω to 1 MΩ in decade steps.

region. Ideality factors close to unity represent dominance of diffusion of carriers

into the transition region. Ideality factors close to two represent the opposite

situation where the current flow across the junction is dominated by generation

– recombination within the transition region. The ideality factor can be treated

as a figure of merit related to the crystal quality. The forward characteristics will

also reveal any series resistance, which may ultimately limit the device current.

Attempting a direct solution of the Shockley equation by collecting the terms

in Ia requires the use of Lambert’s Omega, and takes the form,

Ia = −−n k T W

(q Rs Is exp( q (Va+Rs Is)

n k T )nk T

)

+ q Rs Is

q Rs(2.2)

This expression can be solved numerically to produce Fig. 2.1 which demonstrates

the effect of series resistance, but (2.2) is cumbersome. The W function can be

avoided by solving for Rs and not attempting to collect terms in Ia. The resulting

expression is,

Rs = −ln( Ia+Is

Is)n k T

q− Va

Ia(2.3)

26

CHAPTER 2

The practical use of this expression is possible, by inserting a value of anode –

cathode voltage and anode current and obtaining a value for Rs. It is important

to choose a voltage at which the series resistance is apparent otherwise accuracy

is compromised. It is also necessary first to know the value of Is, which may be

found by fitting (2.1) to the data (assuming Rs = 0) or by inserting values into,

Is = − Ia

− exp(

q Va

nk T

)+ 1

(2.4)

The values of Va and Ia substituted into (2.4) must come from the region where

the series resistance does not dominate. Substitution of Is into (2.3) is the next

step, but an estimate of the ideality factor is also required. It is very much quicker

to use a trial and correction approach to find Is and Rs using only (2.1). Using a

graphical method the quality of the fitting can be grasped instantly and corrected

based on all of the data, not just a single point, without difficulty.

The reverse bias characteristics show the breakdown voltage, which can be

used in conjunction with capacitance voltage characteristics to assist modelling

the device region widths or doping densities. The nature of the breakdown may

also be apparent if several devices sizes are available, bulk breakdown is preferable.

This occurs when the maximum electric field that the material type can support

is exceeded. Ideally the current flowing when the device is reverse biased would

be the saturation current only. However in real devices the reverse leakage current

is significantly higher and is composed of the diffusion current from outside the

transition region and the generation current. There may also be an element

of surface leakage and tunnelling. Reverse IV characteristics can also exhibit

behaviour caused by other aspects of the diode structure. For example a very low

leakage current diode may exhibit a linear relationship on a log–log plot for the

first 75% of the reverse bias voltage, before undergoing bulk avalanche breakdown.

In this instance one plausible explanation is that the surface passivation has a

resistance which is constant and dominates when the diode is only exhibiting the

saturation current.

If forward and reverse characteristics can be obtained from devices having

a range of areas, the nature of the current flow may be examined. In devices

where current scales with the device perimeter, it is likely that the majority of

27

CHAPTER 2

the current flow occurs down the surface of the mesa. However a result that scales

with area is indicative of bulk current flow. If neither bulk nor surface processes

dominate the conduction, current density scales poorly with both perimeter and

area.

2.2 Capacitance Voltage Characteristics

The forward capacitance voltage (CV) characteristic yields the diffusion potential

for the device. For a one sided abrupt junction, as larger forward bias is applied

the space charge region shrinks in proportion to the square root of the inverse of

the applied voltage. The parallel plate capacitance is given by,

C = A

(e ε0 εr Nd

2 (VB + V )

) 12

(2.5)

Where e is the electron charge, ε0 is the permittivity of free space, εr is the

relative permittivity of the semiconductor material, Nd is the doping density, VB

is the diffusion or ‘built in’ voltage, and V is the applied terminal voltage.

The built in voltage may be estimated from a plot of 1/C2 versus bias. It is

given by the intersection of an extrapolation of the data and the bias axis. As

the device begins to turn on the incremental resistance reduces significantly, and

the measurement accuracy deteriorates rapidly.

Reverse capacitance voltage characteristics, are used to determine the deple-

tion width or doping density in conjunction with data provided by other experi-

ments such as secondary ion mass spectroscopy (SIMS) and the use of a Poisson

solver (see Appendix A). The depletion width and doping densities may be ad-

justed to fit a device CV profile. Breakdown voltage is also used as a guide in

determining the intrinsic region width. The reverse CV profile is required to

calibrate the new excess noise measurement systems described herein and to cal-

ibrate the excess noise measurement system after Li [20], which will be discussed

shortly. Reverse CV measurements should scale with device area.

28

CHAPTER 2

2.3 Photo-Multiplication

The simplest form of photo-multiplication measurement is similar to reverse IV

measurements with the exception that the diode is illuminated. This DC method

can not separate the dark current from the photo-current. The DC experiment

may be considered accurate when the dark current is at least two orders of mag-

nitude lower than the photo-current. In this work several combined AC multipli-

cation and excess noise measurement systems are used, these are used to obtain

multiplication and excess noise simultaneously. However from a multiplication

perspective they may be treated as any other AC method. As the reverse bias

across a pin / nip is increased, the depletion region extends into the highly doped

p+ and n+ layers on each side of the intrinsic region. As a result, minority car-

riers, generated outside the depletion region, have a greater chance of diffusing

into the depletion region. This increase in collection efficiency is manifested by an

approximately linear increase of photo-current with bias before the onset of any

real multiplication. After multiplication begins the collection efficiency continues

to increase; however, it is masked by the multiplication process. The collection

efficiency has a significant effect on the primary photo-current. It is critical to

account for this effect by using the non multiplying region data to estimate the

primary photo-current in the region where impaction ionisation does occur. The

value of multiplication is obtained from point by point division of the multiplied

photo-current by the (regressed) primary photo-current. Many data points are

taken in the non-multiplying bias region in order to yield sufficient accuracy.

2.4 Review of Noise Measurement Systems

Several excess noise measurement systems have been reported in the literature

and comparisons between the circuits described in this thesis and those previously

reported may be drawn. The figures of comparison are,

• The system signal to noise ratio, where the signal is defined as full shot

noise exhibited by 1µA

• The maximum permissible junction capacitance. In the case of multi-

29

CHAPTER 2

Laser Chopper

Chopper Driver LIA

R1

C1

Bias Tee DC Bias

AIL 13680Pre-Amplifier

AIL 1361030MHz Reciever

LIA Chart Recorder

Figure 2.2 – Measurement system after Bulman.

frequency systems the lowest available frequency is used; this produces the

most favourable result. It is assumed that the system input impedance and

diode junction capacitance form a first order low pass network.

The first reported noise measurements on photo-diodes was by Baertsch [22].

Insufficient information is provided to estimate this system’s figures of merit so

it is excluded from the comparison. Xie et al. [23] proposed a measurement

system that was substantially similar to Toivonen et al. [24]. The Xie et al.

system represents both. Bulman [25], Ando and Kanabe [26] and Lau et al. [27]

presented systems based on phase sensitive detection. Xie and Toivonen used a

DC approach.

2.4.1 Bulman’s System

Figure 2.2 shows the system reported by Bulman. It is a phase sensitive detection

(PSD) based system in which photo-current and excess noise are extracted and

read out using two lock in amplifiers. The APD is loaded by the AIL13680

through the bias tee. It may be assumed that the APD experiences a 50 Ω load.

The pre-amplifier output is fed to a receiver having a calibrated bandwidth. The

30

CHAPTER 2

resulting signal proportional to the noise power contained within the calibrated

bandwidth is passed to a lock-in-amplifier.

Bulman proposes two methods to quantify the absolute noise power mea-

sured. Firstly a pin detector is illuminated under unity gain conditions. It is

assumed that under these conditions the system will measure full shot noise. A

second calibration method is proposed in which a calibrated oscillator is used

in place of the APD. This allows the experimenter to set the power which will

be measured. Adjusting the power supplied by the oscillator permits the linear

range of the system to be estimated. Bulman reports 30 dB linear range between

−140 dBm (10−17 W) and −110 dBm (10−14 W). In several III–V semiconduc-

tors the impact ionisation coefficients are nearly equal. Under this assumption

Tager [28] has shown that excess noise is proportional to M3, when the local

model holds. Assuming Bulman’s system is used from unity gain with a noise

power of −140 dBm the maximum multiplication prior to the limit of linearity

is 10. Bulman’s report lacks some information regarding the front end amplifier.

An Analog Devices AD9618 low noise opamp in non-inverting mode is used as

a model. It achieves a gain of 100 V/V and a bandwidth of 80 MHz with 50 Ω

input impedance. The equivalent input noise voltage is 1.94 nV/√Hz. Using

this model as an approximation the signal to noise ratio for Bulmans system is,

S

N=

((100 · 50)2 · 2 · q · iph

)

(1.94× 10−9 · 100)2= −36.73 dB (2.6)

The capacitance of the diode which can be tolerated by Bulman’s system is given

to a first approximation by considering the system input impedance and the

junction capacitance as a first order RC network and assuming that all other

parasitic effects (for example diode series resistance) are negligible in comparison.

Applying the commonly known expression for the corner frequency of a first order

network RC,

C =1

2 π f R, and using f = 30 MHz and R = 50 Ω gives C = 106 pF. (2.7)

Larger capacitance will attenuate the components of noise power at the 30 MHz

measurement frequency.

31

CHAPTER 2

Light Source Chopper

Chopper Driver

A

V

Sig. Gen.

Power Meter

IF Amplifier

LIA

Bias Tee

Figure 2.3 – Measurement system after Ando and Kanbe.

2.4.2 Ando and Kanbe

Ando and Kanbe [26] reported the system shown in Fig. 2.3. It is a PSD system in

which the APD is loaded by 50 Ω due to the input impedance of the IF amplifier.

The measurement system bandwidth is defined by the IF amplifier and is 1 MHz

centred on 30 MHz. The APD is biased using a bias tee. The noise power is

read out from a lock in amplifier. A power meter and signal generator with its

output passed through a calibrated attenuator provides a means of relating the

absolute signal power to the value measured by the IF amplifier. Photo-current is

extracted by DC measurement. The measurement of devices exhibiting high dark

current is therefore difficult with this system. The various connections required

to calibrate the system and perform measurements are made using relays.

Ando and Kanbe did not report any attempts to measure shot noise on their

system. They also do not give information regarding the model numbers or

manufactures of their system components. No noise specifications for the instru-

mentation are given. Assuming that their system adds no noise other than the

32

CHAPTER 2

thermal noise of the 50 Ω input impedance then the signal to noise ratio can be

computed using,S

N=

2 q iphBR

(4 k T BR

)= −23.98 dB (2.8)

where R = 50 Ω, B = 1 MHz, T = 300 K and iph = 1 µA. The junction

capacitance which can be tolerated by Ando and Kanbe’s system is calculated in

a similar way to Bulman’s system and produces the same result C = 106 pF. The

authors claim that noise power as low as -130 dBm can be measured with 0.5 dB

accuracy. This represents a current of 6.25 µA developing full shot noise.

2.4.3 Xie

The system proposed by Xie [23] is similar to that proposed by Toivonen [24].

Figure 2.4 shows the APD is connected to a micro-strip line and DC voltage is

applied via a bias tee. The measurement is made using a CW light source and

a noise figure meter such as the Hewlett Packard 8970A. The system has two

significant advantages over PSD systems such as those of Bulman and Li. Firstly

multiple frequencies are available up to the limit of the circuits or the analyser.

Presently Agilent Technology manufactures noise figure meters capable of mea-

suring 10 MHz to 26 GHz with variable effective measurement bandwidth. Xie’s

system was limited to a maximum frequency of 1.3 GHz and 4 MHz bandwidth.

Secondly the measurement is in principle quicker than a PSD system. The oper-

ation of PSD is discussed in Appendix C but in general the time constant of a

PSD measurement may be expected to be longer than of a noise figure meter. DC

measurements have several disadvantages for example the lowest practically mea-

surable photo-current is higher than in some PSD systems using transimpedance

based amplifiers Li [20] has shown that the transimpedance amplifier reported

by Lau [27] has a more favourable noise signal to noise ratio than is theoretically

possible with a 50 Ω measurement system. The noise without illumination should

be periodically measured in order to maintain consistency. The noise without il-

lumination should be stable and sufficiently small compared to the noise with

illumination that the total noise is dominated by the photo-generated noise. If

this condition is not met the confidence of the measurement is compromised.

33

CHAPTER 2

ZT

C1

L1

−+

VB

C2

zin

Vout

Figure 2.4 – Measurement system after Xie et al.

Xie et al. reported measuring noise power as low as −190 dBmHz−1 without

difficulty using this measurement system. In a 50 Ω system this is equivalent

to the full shot noise generated by 5 µA photo-current. The capacitance which

can be tolerated by this measurement system is computed at the lowest usable

frequency, as this produces the most favourable result. By the same first order

approximation used in Bulman’s and Ando and Kanbe’s systems Xie’s system

will exhibit a -3 dB bandwidth of 10 MHz when the input is loaded with 636 pF.

In Chapter 9 an improved noise figure meter based measurement system is

proposed [29].

2.4.4 Li’s System

The system after Li [20,27] employs phase sensitive detection and a transimpedance

amplifier. A schematic diagram is shown in Fig. 2.5. The laser is chopped by

mechanical means at 180 Hz and is presented to the diode via a system of optics.

The TIA is used to convert the diode current into a voltage. This voltage is

amplified using a commercial low noise wide band amplifier module (Minicircuits

ZFL-500). A precision stepped attenuator (HP-355D) is used to vary the system

gain permitting measurement of high and low noise devices. The noise signal is

separated from the low frequency component of the photo-current by a Minicir-

cuits SBP-10.7+ LC ladder filter. After filtration, the signal resembles an am-

plitude modulated noise waveform, where periods of diode illumination produce

greater noise amplitude than periods of darkness. Further amplification follows,

34

CHAPTER 2

Chopper Driver Lock-in-amplifier

Stanford SR830

Lock-in-amplifier

Stanford SR830

Laser

He-Ne

2.5mW, 633nm

Chopper

Bias Source

Keithley 236

TIAVoltage Amp

Minicircuits

ZFL-500LN+ HP 355D Minicircuits

SBP-10.7+

Voltage Amp

ZFL-500LN+

Power Meter

VoltageAmplifier

Figure 2.5 – Measurement system after Li.

prior to a wide band squaring and averaging circuit. The output of the squaring

and averaging circuit is an approximately square voltage signal. The fundamen-

tal frequency of the signal is 180 Hz; the amplitude is proportional to the noise

power contained within the pass band of the SBP-10.7+ filter. The squaring cir-

cuit is based on an Analogue Devices AD834 analogue multiplier. The averaging

circuit is a first order RC filter with a time constant of approximately 100 µs.

The output from the squaring and averaging circuit is measured using a lock-in-

amplifier. The photo-current signal is taken from the output of the TIA where

the amplitude of the 180 Hz square wave is proportional to the photo-current.

The photo-current signal is measured on a second lock-in-amplifier.

The system after Li is superior in noise performance to prior reported systems.

The transimpedance amplifier provides a signal to noise ratio which is superior to

that theoretically possible in a 50 Ω system. Consider the connection of a photo-

diode and a 50 Ω resistor. Assume that full shot noise generated by 1 µA flows

through the resistor which exhibits thermal noise at 300 K. The noise signal to

noise ratio is then,

20 log

(√2 q iph

)

(√4 k T 50

)

= −30.15 dB (2.9)

The noise signal to noise ratio (also considering 1 µA photo-current) of Li’s

system is -25.7 dB [20]. A first order approximation considering only the volt-

35

CHAPTER 2

age noise of the Analog Devices AD9631 (7 nVHz−1/2) [30] operational amplifier

based TIA yields a signal to noise ratio of approximately -21 dB; this simple

calculation demonstrates the advantage of transimpedance amplifiers.

The dynamic range of this system is limited at the lower bound by the abil-

ity of the lock in amplifier to extract the in-phase signal from the system noise.

Practical experimentation has shown that full shot noise developed by 1 µA is

approaching the limit and the shot noise from 0.1 µA is not measurable. The

precise limit is difficult to quantify because it is affected by the prevailing electro-

magnetic conditions both passing through the experiment area and on the mains

power supply. At the upper bound the maximum attenuation of the stepped at-

tenuator provides a limitation however more attenuation could be added without

difficulty. The linearity of the transimpedance amplifier at high input current is

a second limit. Because the relationship between excess noise factor and photo-

multiplication varies between material systems it is unwise to speculate on the

maximum multiplication which can be used. Furthermore if a device is avail-

able which can be operated with a very large gain the optical illumination may

be reduced in order to reduce the multiplied photo-current and the excess noise

power. In this way higher multiplication values may be measured. In order to

measure lower multiplication values a larger primary photo-current is required.

By performing two or more measurements with differing primary photo-currents

it is possible, assuming the APD is sufficiently robust, to measure multiplication

and excess noise power over any desirable range.

The capacitance tolerated by Li’s transimpedance amplifier is lower than all

of the other systems. The interaction of the APD junction capacitance, the feed-

back resistance and the open loop gain of the operational amplifier permit the

existence of resonance in the frequency response of the transimpedance amplifier.

When the capacitance is sufficiently large oscillation breaks out and the mea-

surement system is saturated. The limit of measurable junction capacitance is

however not governed by the presence of oscillation. A result of the interaction of

the diode junction capacitance and the transimpedance amplifier is a dependence

of the effective noise power bandwidth of the system on the diode junction ca-

pacitance (which is itself dependant on the applied DC bias voltage). As a result

a correction must be applied when post processing the measurement data. The

36

CHAPTER 2

limitation of the capacitance is governed by the quality of the correction which

can be achieved. While it is known that up to 56 pF does not cause oscillation,

Li placed the limit at 28 pF [20]. This limit was obtained by calibrating the

bandwidth of the transimpedance amplifier with several values of capacitance.

Having performed the calibration, shot noise due to photo-generated carriers was

measured using a silicon photo-diode. A second data set was gathered in which

extra capacitance was placed in parallel with the photo-diode to simulate a diode

of greater capacitance. The simulated higher capacitance shot noise data was

processed using the original calibration and the quality of the fitting of the stan-

dard photo-diode shot noise and the simulated extra capacitance shot noise data

was used as a basis for defining the upper limit of the junction capacitance.

2.4.5 Summary of Prior noise measurement systems

Table 2.1 – Comparison of published noise measurement systems calculated for 1 µA signalcurrent.

AuthorNSNR [dB](cj = 1pF)

Maximum cj [pF]

Bulman et al. -36.72 106Xie et al. -31.58 636

Ando and Kanbe -23.98 106Li et al. -25.70 50

37

CHAPTER 2

38

Chapter 3

Transistorised High Capacitance

Noise Measurement System

In this chapter a versatile system for measuring excess noise and multiplication in

avalanche photo-diodes, using a bipolar junction transistor based transimpedance

amplifier front-end is reported. This system is based on phase-sensitive detection,

which permits accurate measurement in the presence of a high dark current. The

system can reliably measure the excess noise factor of devices with capacitance

up to 5 nF. It has been used to measure the thin, large area Si pin APDs which

are reported in Chapter 6. These data are in good agreement with measurements

of the same devices obtained from a second noise measurement system which is

reported in Chapter 5.

3.1 Introduction

Avalanche photo-diodes exhibit an internal gain mechanism whereby secondary

carriers are generated by impact ionisation [21]. Impact ionisation is often ex-

ploited to enhance the signal-to-noise ratio of electro-optical systems in commu-

nications, medical and military applications [31–33]. A popular model that de-

scribes avalanche multiplication was proposed by McIntyre [3]. The electron (α)

and hole (β) ionisation coefficients are usually reported as a function of electric

field. α and β are often extracted from measurements of the photo-multiplication

39

CHAPTER 3

R1

−+

VB

Voutcp

C1

Figure 3.1 – High impedance amplifier.

and excess noise factor of a particular APD structure, fabricated from a certain

material; for example in [34–36]. The ratio keff = α/β may be used as a relative

figure of merit when comparing two or more competing material systems.

Several measurement systems have been reported which evaluate the excess

noise associated with impact ionisation mechanism [23, 25–27]. Each of these

systems has some limitations with respect to the measurement bandwidth, the

minimum detectable photo-generated noise, and the maximum permissible device

junction capacitance. The merits of each system are discussed in Chapter 2.

The measurement system and front end reported in this chapter enable mea-

surements of photo-multiplication and excess noise on devices with junction ca-

pacitance, cj , up to 5 nF. For the first time wide band excess noise measurements

on thin, large area devices including forward biased devices and photovoltaic cells

are possible. It has been suggested that the noise magnitude of a solar cell is pro-

portional to the density of material defects [37]. The defect density is linked to

cell quality [38]. Noise measurements may be used to grade solar cells at the time

of production. However the measurement system is non-specific and any current

mode device or circuit – one that can be approximated by a Norton source (see

Fig. 3.4) – may be measured.

3.2 Front-End Design Methodologies

Ignoring distributed techniques, amplifiers that interface electronic systems with

electro-optical detectors can be split into three groups based on their input

impedance [39]. Each of these will be discussed briefly in order to illuminate

their relative merits. They are,

40

CHAPTER 3

ZT

C1

L1

−+

VB

C2

zin

Vout

Figure 3.2 – Impedance matched amplifier.

• High impedance (voltage) amplifiers

• Impedance matched (power) amplifiers

• Transimpedance amplifiers or current to voltage converters

3.2.1 High Impedance Amplifiers

A generic high impedance amplifier is shown in Fig. 3.1. A detector, such as a

pin diode, is connected in series with a resistance, R1, and is reverse biased. The

device is illuminated; a photo-current flows in the diode-resistor combination.

A voltage proportional to the photo-current appears across R1; it is presented

to an amplifier having a sufficiently high input impedance that it does not load

the diode – resistor network. While this topology is potentially the simplest

to design, it has considerable practical problems [40]. The bandwidth of the

system is often limited by the time constant created by the resistor, R1, and the

parasitic capacitance associated with the input node, cp. The noise performance

of the system is often dominated by the value of R1.

3.2.2 Impedance Matched Amplifiers

The impedance matched amplifier shown in Fig. 3.2 is designed with a specified

input impedance, zin, which is usually 50 Ω or 75 Ω. This topology is particularly

suited to microwave noise measurements. The measurement system, and in some

cases, the APD impedance, must be carefully designed to ensure that nearly all

41

CHAPTER 3

−+

VB C1 −

+

RF

CF

Vout

Figure 3.3 – Generic opamp transimpedance amplifier.

of the noise power generated in the detector enters the measurement system.

Xie et al. have used a matched approach to noise measurements [23]. In their

system the diode is terminated for AC signals by ZT = 50 Ω such that both ends

of the microstrip line are terminated. Assuming the APD has a large dynamic

resistance, rd, and terminating both ends of the transmission line, the necessity

of tightly controlling the APD resistance is removed. The noise power generated

by the diode is divided equally between the termination, ZT , impedance and the

measurement system input impedance, zin.

In the context of the present work, a diode capacitance, cj , of 1 nF and 50 Ω

input impedance would exhibit an input time constant of 50 ns, which corresponds

to a -3 dB frequency of 3.2 MHz. Attaining wide bandwidths from devices with

high junction capacitance requires a different approach.

3.2.3 Transimpedance Amplifiers

A generic opamp transimpedance amplifier is shown in Fig. 3.3. The shunt–

shunt feedback connection causes the circuit to present low input and output

impedance. The circuit converts the diode current into a voltage. From DC to

HF frequencies operational amplifiers are often well suited to this task, requiring

only a few external components [41]. The Analog Devices AD9631 [30] and Texas

Instruments OPA129 [42] are specifically designed for this application. Several

transimpedance amplifier topologies are reviewed by Sackinger [43]. TIAs appear

frequently in optical communications literature [44,45] and physical measurement

applications in single ended [40,46–50] and differential forms [51]. Excluding the

42

CHAPTER 3

iph rd cj

Figure 3.4 – APD small signal model.

optical communications TIAs, which generally make use of integrated circuit

technology, most measurement TIAs are developed to fulfil a specific instrumen-

tation requirement. Consequently, there is considerable application-dependent

topological variation of TIAs in the literature. At HF frequencies, operational

amplifier based TIAs are not suitable for measuring APDs with high junction

capacitances. It may be shown that an approximation of the input impedance of

an opamp based TIA, when the opamp open loop gain is much greater than unity,

is zin = RF/Av. Open loop gain, Av, decreases with increasing frequency and so

input impedance rises. Opamp based transimpedance amplifiers are also prone

to instability when the input node is loaded with significant capacitance [27].

Consider the input impedance of a transimpedance amplifier designed using an

Analog Devices AD9631 at 10 MHz with a 2.2 kΩ feedback resistor. The open

loop gain at 10 MHz is approximately 22 dB = 12.59 [30], the resulting input

impedance is approximately 175 Ω. The system reported by Lau et al. [27], which

has similar bandwidth and gain requirements to the present work, is unstable with

junction capacitance in excess of ∼50 pF. This chapter is concerned with a new

transimpedance amplifier designed using a bipolar transistor based front end.

3.3 Diode Small Signal Model

A small signal model of a reverse biased pin / nip diode is shown in Fig. 3.4.

The model is comprised of a Norton source iph and rd with a parallel capacitance,

cj . rd is the small signal dynamic resistance of the APD. cj is proportional

to the depletion width, the area of the APD and the relative permittivity of

the material. This model is used throughout the present work. It has proved

satisfactory from DC to HF frequencies. Series resistance may be added to either

terminal if necessary.

43

CHAPTER 3

Chopper Driver

Scitec Inst. 300C

PhotocurrentLock-in-amplifier

Stanford SR830

Excess NoiseLock-in-amplifier

Stanford SR830

Reference Signal

Laser

He-Ne

2.5mW, 633nm

Chopper

Bias Source

Keithley 236

TIAVoltage Amp

HP 355D

Minicircuits

SBP-10.7+

Voltage Amp

Power Meter

Figure 3.5 – Measurement System Block Diagram.

3.4 Noise Measurement System

A block diagram of the noise measurement system is shown in Fig. 3.5. The

laser is chopped by mechanical means at 180 Hz and is presented to the diode

via a system of optics. The TIA is used to convert the diode current into a

voltage. This voltage is amplified using a series of operational amplifiers with a

total terminated gain of ∼13. A precision stepped attenuator is used to vary the

system gain permitting measurement of high and low noise devices. The noise

signal is separated from the low frequency component of the photo-current by

a Minicircuits SBP-10.7+ LC ladder filter. After filtration, the signal resembles

an amplitude modulated noise waveform, where periods of diode illumination

produce greater noise amplitude than periods of darkness. Voltage gain (∼41

terminated) provided by operational amplifiers follows, prior to a wide band,

250 MHz, squaring and averaging circuit which acts as a power meter. The

squared signal is further amplified sixteen times. The output of the squaring and

averaging circuit is an approximately square voltage signal. The fundamental

frequency of the signal is 180 Hz; the amplitude is proportional to the noise power

contained within the pass band of the SBP-10.7+ filter. The squaring circuit is

based on an Analogue Devices AD835 analogue multiplier. The averaging circuit

is a first order RC filter with a time constant of approximately 100 µs. The output

from the squaring and averaging circuit is measured using a lock-in-amplifier. The

44

CHAPTER 3

photo-current signal is taken from the output of the TIA where the amplitude of

the 180 Hz square wave is proportional to the photo-current. The photo-current

signal is measured on a second lock-in-amplifier.

3.5 Transimpedance Amplifier

A simplified circuit diagram of the TIA first stage and biasing circuit is shown

in Fig. 3.6. Some circuit elements have been removed or combined to make the

circuit operation clear. There are two transistor stages which are electrically

similar and are constructed to the same physical layout. One transistor stage,

composed of T1−3, R1, R2, R3, R4 and C1, is ‘active’ and is presented with the

APD. The other, T4−6, R5, R6, R7, R8 and C2, is ‘passive’ and has only quiescent

(DC) conditions. The objective of the active transistor stage is to present a low

impedance to the APD and to convert the photo-current and noise signal into a

voltage. It is preferable to maintain the input node at ground. To enable this,

one of the power supply rails, which supplies both transistor stages, is controlled

by a feedback system composed of, R10, C3, A1 and T7. The advantages this

brings will be discussed in Section 3.7. The outputs of the two transistor stages

are subtracted using an opamp circuit to remove, as far as possible, the DC

conditions of the transistor stages from the noisy photo-current signal. This is

shown in Fig. 3.7. Further gain is provided by several more opamp stages. Output

DC offset adjustment circuitry; similar to an opamp offset null is also provided

by R11.

3.5.1 Transistor Stages

The transimpedance function of the first stage is realised by a single transistor

common base amplifier. The operation of this configuration can be thought of as

a ‘low impedance current amplifier’ [52]. It is the load resistor which converts the

current to a voltage. The common base small signal current gain (α) is slightly

less than unity; therefore the small signal current flowing in the load resistor, R1,

is approximately equal to the small signal input current. It can be shown without

difficulty that the transimpedance gain in the mid band is set by the load resistor.

45

CHAPTER 3

R2, R3 and R4 bias the transistor in a way that provides voltage and thermal

stability. The capacitances C1 and C2 provide ground at the base for AC signals.

C5 is required to make the source measure unit (SMU) a signal ground from the

viewpoint of the APD, and to significantly lessen the high frequency noise that

the SMU injects into the measurement system. The SMUs which have been used

throughout this work are unstable when the output is loaded with more than

20 nF. A small resistance R12 is added in series to isolate it from C5.

3.5.1.1 Design Requirements

The design requirements for the transistor TIA are partially derived from the

existing Li [20] measurement system.

• Transimpedance gain of approximately 2200 V/A (1100 V/A terminated).

• -3dB bandwidth of at least 10 MHz with maximum device junction capac-

itance of 5 nF.

• Lowest obtainable noise.

• DC coupling of the APD to the TIA.

• DC coupling of the output to the rest of the front end.

• Guaranteed stability, irrespective of APD capacitance.

• Operate from +/-15 V supplies.

The first three points are bound to the transistor operating conditions. The

remainder are system considerations. Design proceeds with the solution to the

bandwidth, capacitance and gain problem.

3.5.1.2 Design Process

Using a simplified hybrid–π model the common base stage may be represented

by two real poles [43]. The lower frequency pole is formed at the emitter node by

the APD junction capacitance, cj, and the impedance looking into the emitter,

re. The higher frequency pole is formed at the collector node by the load resistor

46

CHAPTER 3

50 Ω

T1−3

BF840

R1

500 Ω

R2

680 Ω

R3

2.2 k

R4

2.7 k

C1

220 µF

C5

1 µF

R12

100 Ω

VB

C2

220 µF

R7

2.2 k

R8

2.7 k

50 Ω

T4−6

BF840

R5

500 Ω

R6

680 Ω

C3

22 µF

−

+A1

TL071

R10

100 k

R11

10 k

+15V

−15V

+15V

−15V

T7

TIP32C

−15V

+15V

C4

220 µF

− +

A2AD829

C6

68 pF

R13

100 Ω

OP2

−+

A3AD829

C7

68 pF

R14

100 Ω

OP1

+15V −15V −15V +15V

Figure 3.6 – Simplified bipolar transistor transimpedance amplifier.

R1 and all stray capacitances from the collector, including the input capacitance

of the buffer opamp, which is approximately 5 pF for the Analog Devices AD829.

Assuming that the transistor stage can be approximated by a first order low

pass system, composed of only the input pole, the required time constant (τ) to

provide a -3 dB bandwidth of 10 MHz is (1/(2 π)) × 10−7 s is formed by cj =

10 nF and the small signal resistance looking into the emitter, re. The maximum

permissible value of re is approximately 1.6 Ω. re does not appear explicitly in

the hybrid–π model but it can be shown that re ≈ 1/gm if β0 >> 1. The collector

current required to achieve re = 1.6 Ω is given by IC = (k T )/(e re) ≈ 16 mA.

Where k is Boltzmann’s constant, T is the absolute temperature and e is the

electron charge.

The aim is to design an input stage with the same gain as the one used

by Li [20]. It is not possible to obtain all of the required gain from the first

(common base) stage for two reasons. Firstly, the collector to base junction

capacitance, 0.3 pF, and the input capacitance of the opamp buffer, 5 pF, form

a pole with the load resistance at the collector node. Using a load resistance of

47

CHAPTER 3

R3

1kΩ

−

+

A1AD829

R4

1kΩ

OP1

R2

1kΩ

R1

1kΩOP2

C1

68 pF

+15V

−15V

−

+

A2AD829

−15V

+15V

R5

1kΩ

R6

1kΩ

C2

68 pF

−

+

A3LM7171

−15V

+15V

R7

470Ω

R8

470Ω

R9

50ΩC3

220 nF

Noise O/P

−

+

A4TLE2141

−15V

+15V

Photocurrent O/P

Figure 3.7 – Post common base operational amplifier stages.

2.2 kΩ produces a pole at approximately 14 MHz. Stray capacitance will increase

the datasheet capacitance values, further reducing the pole frequency. Secondly,

given the collector current requirement, the DC voltage drop across 2.2 kΩ would

require the use of a supply voltage greater than +/- 15V. This is undesirable

because the circuit is used in a measurement system where +/- 15V is already

available. A 500 Ω load resistance provides an acceptable frequency response, first

stage gain and DC conditions at the collector. Additional voltage amplification

is required and is supplied using operational amplifiers after the transistor stage.

These are shown in Fig. 3.7. The transistor stage supplies approximately one

quarter of the required gain.

3.6 Opamp stages

It is preferable to DC couple the output of the common base stage as well as the

input so that the output of the front end system (Figs. 3.6 and 3.7) contains total

current as well as photo-current and noise information. Using the two common

base stages, with nearly identical operating conditions, the DC voltage added to

the output by the common base biasing arrangement can be largely cancelled by

subtraction to leave only an offset caused by the imbalance between the two halves

of the input stage. This subtraction is accomplished by the operational amplifier

based subtraction circuit shown in in Fig. 3.7. The photo-current output is a

48

CHAPTER 3

buffered version of the noise output. The buffer opamp has a lower bandwidth

than the noise measurement frequency.

3.7 Transistor Stage Biasing Feedback System

The biasing feedback system is designed to permit DC coupling of the APD to the

TIA by maintaining the active transistor stage emitter node at ground potential.

All of the components of the diode current flow into the TIA (assuming rd >> re).

Unlike AC coupling, no separate DC path is required through which noise power

could pass without entering the measurement system. The DC components of

the diode current produce a DC offset at the photo-current output (Fig. 3.7).

The bias voltage displayed on the APD biasing equipment is then equal to the

bias appearing across the device.

The feedback system, which is shown in Fig. 3.6, allows the TIA to be DC

coupled by modifying the lower supply rail voltage of both transistor stages in

order to maintain the potential of the emitters at a level close to ground potential.

This is implemented by using the second transistor stage which possesses, as

closely as practically possible, the same biasing conditions as the first, but without

an APD connected. The quiescent conditions of this ‘passive’ stage are measured

electronically and the lower supply rail of both transistor stages is adjusted in

order to bring the emitter nodes to ground potential. The opamp, A1, changes

the voltage on the lower supply rail in order to maintain the inverting input

and the non inverting input at the same potential (ground). The opamp, A1,

and the RC network of C3 and R10 form an integrator, – the capacitor is in the

feedback loop of the opamp – the time constant is a few seconds. R10 is large

in order to avoid loading the emitter node of T4−6 and in order that a long time

constant should be formed by a reasonably small value electrolytic capacitor. The

potentiometer, R11 may be used to introduce an extra DC current to the passive

front end, modifying its DC conditions slightly in order to produce an offset null

control.

49

CHAPTER 3

3.8 Stability

Unlike TIAs with feedback around more than one active device, there is no inten-

tional negative feedback in the single transistor stage. Consequently, regeneration

due to excessive phase shift, arising from useful components, is not a concern.

However, any physical realisation of a circuit possessing gain may oscillate under

sympathetic conditions. Both emitter follower and common base transistor cir-

cuits suffer from a potential stability problem when the emitter node is loaded

with capacitance [40]. SPICE simulation of the input impedance of the TIA

while connected to the APD small signal model shown in Fig. 3.4 or the model

described in Chapter 8 can provide an estimation of the likelihood of oscillation.

For the common base transistor, oscillation is likely when the real part of the

impedance looking into the emitter, ℜ(zin) is negative for any frequency below

the transistor’s transition frequency. The question of stability in the common

base amplifier is addressed from an analytical perspective in Chapter 4. In this

circuit two avenues of attack are open to combat oscillation problems both at-

tempt to prevent oscillation by affecting the transconductance of the device either

directly or by adjusting the transconductance required for oscillation to begin,

both avenues will be discussed now.

3.8.1 Parallel Transistors

Several transistors may be connected in parallel such that the total terminal

currents are shared approximately equally between the devices. Each transistor

then has lower transconductance than if a single device is used. Consequently,

the impedance looking into each transistor’s emitter is greater than for an equiv-

alently biased single transistor. The new operating conditions, under which each

transistor has an decreased gm, are less likely to be sympathetic to oscillation

for a given cj than a similarly biased stage composed of a single transistor. The

ensemble transconductance and input impedance remain unchanged. The base

spreading resistance of each transistor appears in parallel, so the ensemble rb is

reduced. For every doubling of the number of paralleled transistors, the noise due

to the base spreading resistance will be reduced by√2. The shot noise processes

are unaffected.

50

CHAPTER 3

3.8.2 Increased base resistance

A small resistance may be connected in series with the transistor base lead. A

resistance in this location worsens the effect of gain peaking and increases the

stage noise. If the resistance is made too large, it will affect the frequency response

and frequency independent gain, k, of the stage. Observation of the time constant,

τ , and frequency independent gain of the transimpedance gain in (3.1) will show

this. The use of both mechanisms provides a reasonable compromise between

noise, stability, ease of construction and frequency response.

3.9 Physical Construction

The transistors T1−3 and T4−6 which are in SOT-23 packages are placed on top

of each other. Vertically placed wires connect their electrodes together. The

transistor closest to the circuit board is soldered to the copper tracks. No other

sufficiently compact layout has been found. This layout also provides a degree

of isothermal matching. The PCB is a standard double sided 1.6 mm FR4 glass

fibre type, coated with 1 oz copper on both sides. Several board designs were

developed in order to explore the layout options and noise reducing effect of par-

alleling transistors. Up to fifteen paralleled transistors were used in various layout

implementations. The resulting conclusion is that the distance between emitter

terminals in parallel transistor front ends must be as small as is practically possi-

ble when the impedance looking into the input is low. Any parasitic components

formed between the individual transistor terminals have deleterious effects on the

frequency response, rendering the layout unusable. Similar rules apply to the

distance between the APD and the first transistor stack.

3.10 Noise Analysis

Analytical, SPICE and experimental methods which complement each other are

used to assess the front end circuit noise. This section is split into two parts,

1. Analytical noise analysis and possible optimisations

51

CHAPTER 3

2. Comparison of the analytical model with SPICE

Analytical methods are used to isolate the dominant noise source in the tran-

sistor stages. A first order analytical model shown in Fig. 3.8 has been developed

by ignoring the base emitter capacitance, cbe, and the base collector capacitance,

ccb, and the load capacitance, CL. It will be shown in the second part of this

section that the error introduced by the simplifications does not significantly af-

fect the results at the frequencies of interest. The agreement between the SPICE

noise prediction and the measured noise of the circuit is shown in section 3.11.

Optimisation of the circuit noise by component selection is explored within the

limitations of the specifications listed in section 3.5.1.1.

The APD junction capacitance, cj , acts to increase the noise contribution of

the active transistor stage as shown in Fig. 3.22 which shows measured data.

Under normal operating conditions, only the active transistor stage’s noise con-

tribution is significant.

An introduction to noise analysis techniques can be found in Leach [53] and

Motchenbacher and Connelly [54].

3.10.1 Common Base Analytical Noise Analysis

The circuit noise is most critical in the transistor stage, the signal to noise ratio

is heavily dominated by this stage. To assess the scope for improving the noise

characteristics of first transistor stage, an analysis of the relative magnitude of the

contributing noise sources, and the transfer ratio of each source to the output node

– the noise gain – has been analytically derived. An analytic solution of the noise

transfer ratios is preferable as it permits examination of the circuit components

that contribute to the frequency independent gain and time constants of the

transfer ratios. This knowledge may be used to assist the selection of component

values which minimise the circuit noise without compromising the specifications

in Section 3.5.1.1. In order to make the task tractable the frequency dependence

of the transistor β and several inter-electrode capacitances are ignored.

The small-signal model used in this analysis is shown in Fig. 3.8. Table 3.1 lists

the noise generators and their associated physical sources and Table 3.2 lists the

expressions required to compute the magnitude of the noise sources. Equations

52

CHAPTER 3

Enrb

rbInb rbe gmvbe Inc

RL

EnRL

RE

EnRE

iin cj

Figure 3.8 – Small signal noise model of common base amplifier. Definitions of the terms aregiven in Table. 3.1. The magnitudes of the noise generators are given in Table. 3.2.

3.1 – 3.6 list the noise gain of each noise generator in the noise model. The noise

gain of each noise contributor can be expressed as either a low pass or a pole-zero

form, except for the contribution of the load resistor which appears directly at

the output.

In order to ensure that the analytical expressions (3.1 – 3.6) have been derived

without error they are compared with a SPICE .AC simulation of the small signal

model (Fig. 3.8). The result of this simulation is shown by the red lines in Fig. 3.9.

From this figure it is easily seen that the base spreading resistance thermal noise

has the highest noise gain as frequency increases. The small signal parameters

are given in the figure caption.

The noise contribution at the collector node for each source is shown in

Fig. 3.10 for three parallel transistors and for twenty five parallel transistors.

In the twenty five transistor case it is assumed that the transconductance of each

transistor will be sufficiently reduced that the internal base spreading resistance

of each transistor is sufficient to make oscillation unlikely. In the three transistor

case 50 Ω is added in series as shown in Fig. 3.6. The question of stability is dis-

cussed in fully in the next chapter. The bottom right graph in Fig. 3.10 shows the

comparison of each noise source and the total noise at the collector node for three

and twenty five transistors. The solid lines represent the three transistor case,

dashed lines represent the twenty five transistor case. Note that in the twenty

five transistor case the two dominant noise mechanisms, the collector – base shot

noise and the base spreading resistance thermal noise are practically equal, and

53

CHAPTER 3

Table 3.1 – Table of analytical noise model components.

Name Description Component of

Enrb Voltage noise sourceBase spreading resistance

rb Physical resistance

Inb Current noise sourceTransistor Base – Emitter

junction

Inc Current noise sourceTransistor Base – Collector

junctionRE Physical resistance

Emitter biasing resistanceEnRE Voltage noise sourceIin AC signal current

APD small signal modelcj Physical capacitance

gmvbeVoltage controlledcurrent source

The model oftransistor action

rbe Model resistanceRL Physical resistance

Load resistanceEnRL Voltage noise source

all of the other noise sources are considerably less influential than these two. This

is a preferable situation, as uncorrelated noise combine as the root of the sum

of the squared values. Therefore having the two most dominant sources equal

is a minimum condition. The choice of twenty five transistors as an example of

‘many’ was chosen specifically because it fulfils this condition for the particular

set of small signal parameters given in the figure caption. Unfortunately it is not

possible to use so many discrete transistors. The imperfections associated with

physical circuit manifestations prevent the circuit from operating as designed. In

an integrated circuit environment it may be possible.

It is possible to enhance the noise performance of the front end by selecting

specific component values rather than attempting to adjust the noise performance

by paralleling active devices. The effects of the load resistance noise, the emitter

biasing resistance noise and the noise due to the quiescent collector current are

described now.

54

CHAPTER 3

For the input current iin

vciin

= k · 1

1 + s · τ k =RL · rbe ·RE · gm

rbe + rb +RE + gm · rbe · RE

τ =cj · RE · (rbe + rb)

rbe + rb +RE + gm · rbe · RE

(3.1)

For the load resistance EnRL

vcEnRL

= 1 (3.2)

For the Base – Collector shot noise Inc

vcInc

= k · 1 + s · τ11 + s · τ2

k =RL · (RE + rbe + rb)

RE + gm · rbe · RE + rbe + rb

τ1 =cj · RE · (rbe + rb)

RE + rbe + rbτ2 =

cj · RE · (rbe + rb)

RE + gm · rbe · RE + rbe + rb(3.3)

For the Base – Emitter shot noise Inb

vcInb

= k · 1 + s · τ11 + s · τ2

k = − RL · rbe · gm · (rb + rbe)

gm · rbe · RE +RE + rbe + rb

τ1 =cj · RE · rbrb +RE

τ2 =cj · RE · (rbe + rb)

gm · rbe · RE +RE + rbe + rb(3.4)

For the base spreading resistance, Enrb

vcEnrb

= k · 1 + s · τ11 + s · τ2

k = − RL · gm · rbegm · rbe · RE +RE + rb + rbe

τ1 = cj · RE τ2 =cj · RE · (rbe + rb)

gm · rbe · RE +RE + rbe + rb(3.5)

For the emitter biasing resistance EnRE

vcEnRE

= k · 1

1 + s · τ k =RL · rbe · gm

RE + gm · rbe ·RE + rb + rbe

τ =cj · RE · (rb + rbe)

gm · rbe · RE +RE + rb + rbe(3.6)

55

CHAPTER 3

Figure 3.9 – Transfer ratios for each noise source in the analytical model. Top left shows thenoise gain for the base spreading resistance thermal noise. Top right, base – collector shot noise,middle left, base – emitter shot noise, middle right, emitter biasing resistor thermal noise andthe bottom graph shows the transimpedance gain. Black open circles represent (3.1 – 3.6). Redsolid lines represent the SPICE .AC analysis of the small signal equivalent circuit of Fig. 3.8 inorder to check that the analytical noise equations have been derived without error. The smallsignal parameters are, rb = 20 Ω, RE = 90 Ω, gm = 0.56 A/V, rbe = 244 Ω, cj = 1 nF andRL = 500 Ω.

56

CHAPTER 3

Figure 3.10 – Collector node noise voltage due to each noise generator. From top left tobottom right the graphs represent the base spreading resistance thermal noise, the base –collector shot noise, the base – emitter shot noise, the emitter biasing resistance thermal noise,the load resistance thermal noise and all of the individual sources with the total (the rootedsum of the squared contributions) output noise voltage also. Black solid lines represent a threetransistor front end with an additional base resistance of 50 Ω. Red solid lines represent theparalleling of twenty-five transistors. In the bottom right graph solid lines represent the threetransistor front end, dashed lines represent the twenty-five transistor front end.

57

CHAPTER 3

Table 3.2 – Table of noise source expressions.

Source Description Expression [ V2

Hz, A2

Hz]

EnrbThermal noise of the base

spreading resistor4 · k · T · rb

Inb Shot noise in the B–C junction 2 · e · ICInc Shot noise in the B–E junction 2 · e · IB

EnREThermal noise of the emitter

biasing resistor4 · k · T · RE

EnRLThermal noise of the load

resistor4 · k · T ·RL

Where k is Boltzmann’s constant, e is the electron charge and T is the absolutetemperature.

3.10.2 Quiescent Collector Current

The quiescent collector current is closely bound to the frequency response of

the amplifier. From a noise perspective, an optimum quiescent collector current

exists. It is interesting to find out how far from the optimum the circuit will

operate, and if any optimisation is possible. Fig. 3.11 shows the effect on NSNR

at 10 MHz of varying the quiescent collector current. The solid line represents

RE of 680 Ω, the dashed line RE of 100 Ω. From this graph it is evident that

an optimum current exists and that the amplifier may be designed such that the

optimum can be moved towards the value of collector current required to produce

the desired input impedance. Unfortunately reducing the value of RE reduces the

NSNR so while the amplifier can operate nearer to the optimum collector current

it would do so with worse overall performance.

3.10.3 Load Resistance

The case for increasing the load resistance is relatively simple, the larger the

load resistance, the more gain the stage has. NSNR should increase with RL.

Figure 3.12 shows the effect of increasing RL on the noise signal to noise ratio

while holding all other parameters constant. The effect of the load resistance is

not to increase the NSNR without limit however as the load resistance also has

a hand in setting the noise gain of all the noise generators and after a certain

58

CHAPTER 3

Figure 3.11 – NSNR of the front-end at 10 MHz with variation in quiescent collector currenthaving RE as a parameter. RE = 680 Ω (Solid line), RE = 100 Ω (Dashed).

frequency increasing the load resistance tends to increase the transimpedance gain

and the noise at the collector similarly such that there is no further improvement

in NSNR. The practicality of passing a quiescent current of 15 mA through 10 kΩ

while maintaining power supplies of ± 15 V is neglected as is the time constant

caused by the collector node capacitance and the load resistance.

3.10.4 Emitter Biasing Resistance

Increasing the emitter biasing resistance increases NSNR (with certain small sig-

nal parameters). This may be seen by observing (3.1 – 3.6). However it is not

possible to meaningfully improve the performance of the front-end under all situa-

tions. The upper left graph in Fig. 3.13 shows the effect on NSNR with frequency

as RE is increased from 100 Ω (dashed line) to 680 Ω (solid line) while cj is 1 nF.

At 10 MHz the benefit obtained at lower frequencies by increasing RE is lost

because the higher value of RE moves the pole down in frequency. Similarly the

upper right hand graph in Fig. 3.13 shows the collector node noise voltage due

to the base spreading resistance thermal noise, Enrb, as a function of frequency

where RE is a parameter. A zero moves down in frequency by an amount just

59

CHAPTER 3

Figure 3.12 – NSNR of the front-end at 10 MHz with variation in load resistance, RL.

sufficient to negate any advantage that was obtained at low frequencies. One

example where an improvement can be obtained is in the noise voltage appearing

at the collector due to the thermal noise of the emitter biasing resistance EnRE .

This is shown in the lower graph in Fig. 3.13 as a function of frequency where

RE is a parameter. The pole still moves down in frequency but does not interfere

with the measurement bandwidth while cj is 1 nF. This is little consolation how-

ever as observation of the upper left graph in Fig. 3.13 or the lower right hand

graph in Fig. 3.10 shows that the noise due to the emitter biasing resistance is

not dominant and so no meaningful improvement in NSNR is obtained.

3.10.5 APD Junction Capacitance

The APD junction capacitance forms a pole with a combination of the emitter

biasing resistance and the input resistance of the transistor. It is therefore ex-

pected that the NSNR will worsen with increasing cj . RE is always much bigger

than the input resistance but the two appear in parallel from the perspective of

the input signal. Therefore having a large value of RE does not materially worsen

the frequency response. Under conditions where cj is small (< 100 pF, black line

in Fig. 3.14) the large value of RE does produce a material benefit in NSNR,

60

CHAPTER 3

Figure 3.13 – Exploration of the possible noise optimisation of the emitter biasing resistance.The constant small signal parameters are, rb = 56.53 Ω, RE = 680 (solid), 100 (dashed) Ω,IC = 15 mA, β = 105.8, cj = 1 nF and RL = 500 Ω.

however under the majority of conditions for which the circuit has been designed

(cj > 100 pF) the value of RE does not greatly affect the overall noise perfor-

mance (see Fig. 3.13, upper left graph). The effect cj has on NSNR is shown in

Fig. 3.14.

3.10.6 Noise Analysis Summery

The following conclusions may be drawn from the noise analysis,

• Around 10 MHz, the dominant noise contributor is the thermal noise asso-

ciated with the transistor’s base spreading resistance and the external base

resistor, Enrb,. The collector – emitter shot noise, Inc, and collector–base

shot noise, Inb, are the second and third largest contributors respectively.

Uncorrelated sources sum as the root of the contributions squared; the base

61

CHAPTER 3

Figure 3.14 – Effect on NSNR of increasing cj , all other parameters held constant: rb =56.53 Ω, RE = 680 Ω, IC = 15 mA, β = 105.8, and RL = 500 Ω. cj = 10, 100, 500 pF, 1, 2, 3,4 and 5 nF.

spreading resistance thermal noise is responsible for almost all of the output

noise.

• Paralleling transistors reduces the base spreading resistance thermal noise

contribution. Noise can be reduced without limit using this method because

the amplifier noise is greater than the source noise [55]. The diode noise

signal to front-end noise ratio is often negative (Table. 2.1). The practical

problems of layout are the limiting factor in this respect.

• Increasing the value of the emitter biasing resistor, RE , acts to reduce the

noise gain of the emitter biasing resistor’s thermal noise, EnRE , to the

collector node.

• Decreasing the time constant of RE and cj will increase the frequency at

which the noise gain of the base spreading resistance, Enrb, the dominant

noise source, begins to rise. However, decreasing the value of RE in order

to move the zero up in frequency will increase the frequency independent

gain, k, for the EnRE generator by an amount which is sufficient to negate

any improvement in noise performance.

62

CHAPTER 3

Figure 3.15 – Comparison of analytical and SPICE noise prediction. The analytical model(circles) is a superposition of the transistor noise sources listed in Table 3.2. The SPICE analysisis the solid line. The small signal parameters for both analysis are IC = 15 mA, β = 105.8,rb = 19 Ω, RE = 90 Ω and cj = 1 nF.

In general the problem of noise analysis in this case is multifaceted, it is

likely that an optimum NSNR exists which is greater than that possessed by

the implementation shown in Fig. 3.6 but noise is not the only parameter of

importance so the final circuit will always be a compromise. This circuit has

been successfully used to gather the data presented later in this thesis. A full

optimisation of this circuit may be undertaken in the future by means of a genetic

algorithm or another suitable computerised process that operates by attempting

to minimise cost while maintaining certain parameters.

3.10.7 Comparison with SPICE

To assess the usefulness of the fist order analytical model over the frequency

range of interest, it was compared with a SPICE model using identical small

signal parameters. The comparison is given in Fig. 3.15. The deviation of the

analytical model from the SPICE model occurs between 50 MHz and 60 MHz.

The error introduced by has been investigated by adding components to the

analytical model and using it to produce numerical results. These results were

then compared with SPICE analyses. A purely analytical route quickly becomes

impractical due to the large number of terms appearing in the equations as the

63

CHAPTER 3

VS

50Ω

50Ω

33kΩ

33kΩ

33kΩ

33kΩ

33kΩ

33kΩ

33kΩ

cj

it

33kΩ

33kΩ

Figure 3.16 – Circuit approximation of an APD.

circuit becomes higher order. The high frequency discrepancy between SPICE

and the first order analytical model is principally due to the effect of the Collector

– Base junction capacitance, ccb. The load capacitance, CL, and the base emitter

junction capacitance, cbe, also play some part but their effect is less dominant in

this particular design. Since the noise measurement is performed at 10 MHz, the

first order approximation does not represent a significant loss of accuracy. The

mid-band error between the SPICE transistor model and the first order analytical

model is 84 pVHz−1/2 with the analytical model producing an over estimation of

noise.

3.11 TIA Characterisation

The circuit in Fig. 3.16 is used to facilitate direct measurement of the tran-

simpedance amplifier’s frequency response and to simulate the APD capacitance.

The impedance of the resistor chain is constant within 3 dB from DC to 20 MHz.

As frequency increases, the chain impedance falls as the resistors parasitic capac-

itance becomes more significant.

3.11.1 DC Linearity

The DC linearity of the front-end should be excellent over a wide range of to-

tal APD current. It may be presumed that this measurement system will be

required to measure small photo-current (and photo-generated noise) in the pres-

ence of much larger leakage or tunnelling currents especially in very thin struc-

tures and narrow band-gap materials. The DC linearity of the front-end is shown

in Fig. 3.17. The measurement system may not be usable at 10 MHz over the full

range of input currents shown. As DC current is injected into the input it acts

64

CHAPTER 3

Figure 3.17 – The DC transfer function of the front end, including the operational amplifierswhich follow the common base stage. This measurement is performed by connecting a precisioncontrolled DC current source to the input and monitoring the DC output voltage using a DMM.

to reduce the quiescent current of the input stage, increasing re. It is therefore

more taxing to measure a high capacitance, high dark current nip device than an

equivalent pin device. In principal a device could have the position of the anode

and cathode exchanged and the biasing voltage polarity reversed. This is trivial

when measuring a packaged device, diced wafers which present p-type material

closest to the back contact may benefit from packaging.

3.11.2 Frequency Response

SPICE simulation and experimental measurements shown in Fig. 3.18 in which cj

is stepped from 0 nF to 10 nF in 1 nF steps results in a resonance phenomenon in

the amplitude response, near to the noise measurement frequency. Grey et al. [52]

have noted the necessity of considering the effect of the base spreading resistance

rb when the quiescent conditions give rise to rbe ≈ rb. The input and output

poles are weakly interacting provided rbe >> rb (Appendix D shows this). While

rbe >> rb, the transistor’s internal base node is close ground potential. If rbe is

approximately equal to rb, the transistor’s internal base node, which joins rbe and

rb, will deviate significantly from ground, and energy transfer between the emitter

and base circuits will be possible. Further analysis of the frequency response and

65

CHAPTER 3

stability of the common base circuit when rbe ≈ rb is given in Chapter 4.

It is desirable to minimise the peaking effect caused by the change in cj . The

available circuit variables that impact the magnitude of the peaking include,

• Base spreading resistance, rb.

• Collector – Base depletion capacitance, ccb.

• Base – Emitter depletion capacitance, cbe.

• Base – Emitter resistance, rbe.

Unfortunately none of these variables can be altered sufficiently to make the peak-

ing effect negligible while maintaining the gain and bandwidth requirements. It

is neither possible nor desirable to make rb zero. All of the available parame-

ters are ultimately bound to the transistor’s quiescent conditions. In this design

problem, where rb is a significant fraction of rbe, the dependence of the frequency

response on the APD junction capacitance appears to be insoluble. All transis-

tors have inter-electrode capacitances and non-zero rb, and will exhibit peaking.

Any transistor with very low rb would be unsuitable because it would oscillate.

The necessity of rb to maintain stability will be described shortly. A calibration

procedure similar to that reported by Lau [27] is used to mitigate the peaking

effect.

3.11.3 Effective Noise Power Bandwidth

The effective noise power bandwidth is obtained by connection of Fig. 3.16 to the

front end and the use of a vector network analyser with a resistive power splitter to

provide appropriate impedance matching. cj in Fig. 3.16 is incremented through a

range of values and the transimpedance gain as a function of frequency is recorded

for each capacitance. This produces a graph similar to Fig. 3.19. The area under

each curve is integrated numerically and normalised to the centre frequency,

Beff(cj) =1

G20

∫ f2

f1

|G(f, cj)|2 df. (3.7)

66

CHAPTER 3

Figure 3.18 – Measured TIA frequency response showing the variation in relative responsewith increasing APD junction capacitance. cj = 0 nF to 5 nF in 0.5 nF steps. The black solidline represents no additional capacitance. The dashed upper dashed line represents 0.5 nF.

where Beff is the ENBW, G0 is the gain at the centre frequency, G(f, cj) is the

frequency and APD capacitance dependent TIA and filter gain. f1 and f2 are

placed sufficiently far from the centre frequency, f0, that practically all of the area

under the curve in Fig. 3.19 is integrated. The result of the integration is one

value of effective noise power bandwidth per test capacitance. These values can

be visualised by plotting ENBW versus test capacitance as shown in Fig. 3.20. A

regression of this data is possible such that any value of junction capacitance can

be corrected. In the case of very high capacitance diodes the junction capacitance

tends not to change appreciably over the range of voltages for which excess noise

is measured. In most cases a constant value of junction capacitance is sufficient

to correct the measurement bandwidth.

3.11.4 Shot Noise Measurement

In practice, once a measurement of multiplication as a function of bias, and excess

noise as a function of multiplication has been recorded the computation of excess

noise is performed using,

F =Napd

NshotM· Bapd

Bcal(3.8)

67

CHAPTER 3

Figure 3.19 – Magnitude response as a function of frequency where APD junction capacitanceis a parameter. The junction capacitance is ‘simulated’ by a 1206 SMT resistor. cj = 0 to 5 nFin 0.5 nF steps. Three sets of data for 5 nF are shown to check consistency, they all lay overeach other.

where F is the excess noise factor, Napd is the value (arbitrary units) represent-

ing the noise power generated by the diode at a particular multiplication, M .

Nshot is the shot noise power (arbitrary units) generated by the primary photo-

current. Bapd and Bcal are the effective noise power bandwidths of the APD and

the calibration detector at the specific value of M , which is a function of bias

voltage. It is necessary to measure the shot noise generated by a range of photo-

currents. The result of this measurement is shown in Fig. 3.21. All unity gain

detectors should produce the same result once correction is made for the junction

capacitance. When this result is plotted on log-log axes the expected gradient is

unity. This results from the shot process and is a reasonable indicator that the

measurement system is behaving as expected.

3.11.5 Measurement of Circuit Noise

In Section 3.10 it was shown that a SPICE simulation of the common base noise

agreed with the first order analytical model for a wide range of frequency. Mea-

surement of the circuit noise produces a similar result to the SPICE analysis thus

68

CHAPTER 3

Figure 3.20 – TIA effective noise power bandwidth as a function of device capacitance. Theeffective noise power bandwidth is used in the calibration process [27]. It is given by thenormalised area under the transfer response within the bandwidth of the filter: ENBW (cj) =1G2

0

·∫∞

0 |A(f, cj)|2 · df where A is the transimpedance gain at a frequency f with a particular

value of cj . A0 is the transimpedance gain at the centre frequency of the SBP-10.7+. In practicethis integration is performed numerically on data obtained by connecting the circuit of Fig. 3.16to the TIA and recording the transfer response using a vector network analyser.

supporting the validity of the SPICE analysis and the analytical analysis are

valid. A measurement of the circuit noise as a function of frequency with junc-

tion capacitance as a parameter is shown in Fig. 3.22. Details of the measurement

are given in the caption.

3.12 Voltage Amplifiers

The voltage amplifiers required to replicate the measurement system are similar

to those shown in Fig. 5.13. The objective of this amplification is to scale the

noise signal to a suitable value to generate an output from the noise power meter

with minimal ambiguity. The gain of each set of amplifiers is given in the text

which describes the block diagram (Fig. 3.5).

The noise signal amplitude is unpredictable. The instantaneous value could

lay in the range -∞ to ∞, however practical considerations particular to this

circuit limit the range to less than the power supply voltage. An assumption

69

CHAPTER 3

Figure 3.21 – Reference data from two Si pin devices.

must be made with respect to the noise voltage values entering the power meter

so that a gain which will make the noise voltage sufficiently large to be easily

detected can be selected. The assumption places a limit on the headroom in the

power meter. Because the NSNR of the system is negative the system noise is

sufficient for calculations. If the APD noise is large enough to make the NSNR

positive, a significant attenuation will already have been applied and the system

noise will have been attenuated accordingly. Setting the maximum permissible

input voltage to the power meter at 0.7 V RMS permits a crest factor of 2. The

system noise will exceed the power meter input range approximately 4.6% of the

time. It must be noted that the circuit noise of the front end increases so greatly

with cj that the 0 dB attenuation setting is used rarely and that as attenuation is

increased the system noise drops commensurately with the attenuation because

the dominant system noise generators are placed prior to the attenuator. The

10 dB attenuation setting yields a crest factor of 6.42 and the percentage of the

waveform clipping the power meter input is negligible.

Calibration measurements on the 0 dB attenuation, shown in Fig. 3.21, using

a BPX65 (cj ∼ 4 pF) photo-diode show output noise power increases linearly

with photo-current over a range of photo-currents from 0 to 50 µA indicating

that full shot noise has been measured. Further evidence for the acceptability of

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CHAPTER 3

Figure 3.22 – Transistor TIA noise measured using a HP4396A spectrum analyser. The TIAis connected in series with the SBP-10.7+ Filter in order to show the extent of the pass band.cj = 0 nF to 5 nF in 0.5 nF steps. SPICE simulation results for the noise voltage at the outputare also shown. The insertion loss of the filter is not more than 1.5 dB [56]; the total errorbetween SPICE and the measured performance is 2.5 dBV -approximately 13% - with SPICEunderestimating the noise.

a low crest factor is the agreement of the data gathered using this system and

those gathered using the 1 MHz system reported in Chapter 5. These data are

presented in Chapter 6. Yet more evidence in favour of proper operation is the

change in output noise voltage by one order of magnitude when the attenuator

is increased by 10 dB (the attenuator is graduated in dBV), and the squaring

function causes this to be converted into dBW.

The linearity of the system up to the point of the power meter input is exam-

ined by measuring the noise voltage using a spectrum analyser while driving the

system with a pseudo-random noise generator or by connecting and biasing an

APD into avalanche multiplication (dark or photo-generated current is accept-

able). Harmonic distortion manifests itself as a band of higher-than-background

noise around integer multiples of 10 MHz. Harmonic distortion is problematic

in this situation despite laying outside the filter pass-band because the power

meter is wide band (250 MHz) and so will act to square the unwanted extra noise

voltage caused by the distortion. Higher amplifications were used to elicit this

effect so that it could be observed. Clipping of the system noise on the 0 dB

71

CHAPTER 3

attenuation setting is not present in the reported design. It is however possible

to saturate the power meter by improper choice of attenuation.

Saturation (if it is not obvious) may be detected while performing noise mea-

surements by adding a further 10 dB attenuation and observing the power meter

output. The power meter output must reduce by an order of magnitude, if it does

not, the experiment must return to a lower photo-current (by reducing the bias

on the APD) where the required effect is observed. A proper attenuation should

then be selected and the measurement continued from this value of multiplication.

All previous higher multiplication and noise data should be discarded.

3.13 Noise Power Meter

The noise power meter used in this design is due to B. K. Ng and is described

fully in Chapter 5.

3.14 Conclusion

This chapter has presented a new transimpedance amplifier for use in measure-

ments of excess noise in avalanche photo-diodes. This measurement system has

found use in situations where the device being measured has high junction capac-

itance, up to 5 nF, and where the device has lower than usual dynamic resistance,

rd. It has been used to collect half of the data shown in Chapter 6. The measure-

ment of a much wider range of devices is now possible. This circuit design may

be used to perform measurements in modes of APD operation which increase

the capacitance and decrease the dynamic resistance such as forward bias and

photovoltaic mode. Noise measurements may yield novel methods of grading the

quality of solar cells.

72

Chapter 4

Small Signal Stability in

Common Base Amplifiers

During the design and testing of the circuitry described in Chapter 3 a persis-

tent oscillation was observed. After the circuitry was corrected, by the addition

of a judiciously placed resistance, the nature of the oscillation was unresolved.

That is to say, it was unclear which component or set of components was to

blame. Furthermore the method by which the oscillation was resolved could not

be sufficiently explained without a lot of ‘arm waving’. The analysis presented

in this chapter aims to set out why some single transistor amplifiers oscillate and

why some single transistor oscillators amplify! This chapter concentrates on the

common base connection of bipolar devices, however, similar arguments apply to

other three terminal amplifying devices and in other connections, with suitable

mathematical adjustments.

Nomenclature

In this chapter small signal parameters are lower case. Large signal parameters,

or parameters which exist in large signal and small signal scenarios, are upper

case.

IC The quiescent collector current of a transistor.

gm The conductance (transconductance) which partially models the operation

73

CHAPTER 4

of transistor. It is related to the quiescent collector current by gm =

(e IC)/(k T ) in a BJT and by gm ∝√ID in a field effect transistor un-

der strong inversion conditions.

rbe The dynamic resistance which partially models the operation of a bipolar

transistor.

cbe The capacitance which models the frequency dependence of the transistor

operation.

rb The base spreading resistance of a transistor. Both extrinsic and intrinsic

parts are lumped.

ccb The capacitance of the reverse biased collector - base junction. It is assumed

that all of this capacitance is connected to the internal base node; that is,

at the node connecting rbe and rb.

RL The load resistance; generally any resistance in the collector circuit across

which the output voltage is developed.

CL The load capacitance; generally any parasitic capacitance to ground asso-

ciated with the collector node or the circuits which load the collector.

RE The emitter biasing resistance.

CE The capacitance to ground associated with the emitter node.

LP The parasitic series inductance in the base network.

β The frequency dependent small signal relationship between ic and ib in a

bipolar transistor.

β0 The frequency independent small signal relationship between ic and ib in a

bipolar transistor.

α The frequency dependent small signal relationship between ie and ic.

α0 The frequency independent small signal relationship between ie and ic.

74

CHAPTER 4

τT The time constant associated with the unity current gain frequency, fT .

This parameter could be called τα as it is also the -3 dB point of α.

τβ The time constant associated with the -3 dB point of β.

τE The time constant of CE and RE .

τC The time constant of the load resistance, RL and the load capacitance, CL.

τL The time constant of Lp and rb.

4.1 Introduction

Several reports of high frequency stability limitations in common collector circuits

exist including [57–59]. There is comparatively less reported analysis on the

common base connection [60, 61]. This chapter considers the stability of small

signal common base bipolar transistor amplifiers in the frequency domain. This

chapter also views an unstable amplifier as a potential oscillator, and provides

a method of determining the conditions for stability by assessing the conditions

necessary for instability. Some practical guidance on the physical implementation

of video bandwidth circuits is also provided. This information is often neglected

in formal publications, or is simply addressed as ‘high frequency techniques’ with

little explanation of what is required. A particular HF technique which is well

known, but not often analysed mathematically, is the addition of base or gate

stopper resistors. Section 4.9 provides an illuminating treatment.

The arguments in this chapter are presented using a bipolar transistor because

the impetus for the work was found in a bipolar device; similar arguments apply

to FETs and comparable devices.

The hybrid-π model of the bipolar transistor is used and the usual limitation of

validity (fT )(2) is assumed throughout [62]. This chapter is divided into sections;

each section contains analysis which builds on the prior section such that the

modification to the performance of the circuit due to each addition is evident. For

each component addition the conditions sufficient for negative resistance looking

into the emitter are developed. The boundary conditions necessary for stability

are explored analytically and in terms of design rules.

75

CHAPTER 4

iin

gmvbe

RL

cberbe

rb

Figure 4.1 – Common base amplifier with base spreading resistance and frequency dependence.

Oscillation may occur in a single transistor circuit when a number of electrodes

exhibit a negative resistance over a range of frequencies. Negative resistance

may be generated due to the impedance transforming property of the transistor.

A suitable parasitic, or intentionally designed, resonant circuit must exist to

permit oscillation and to set the oscillation frequency. The oscillation described

is assumed to be unwanted or parasitic; however, several popular circuits operate

by the negative resistance method [63–67]. The negative resistance arises partly

due to passive circuit elements and partly due to the transistors intrinsic frequency

response. Analysis begins with analysis of the intrinsic frequency response of a

transistor in the common base connection.

4.2 Frequency Dependent Behaviour of a

Common Base Amplifier

The circuit of Fig. 4.1 shows a partial hybrid-π small signal equivalent circuit of a

common base transimpedance amplifier. It can be shown that a transistor in the

common base connection possesses limited bandwidth and that the limitation is

intrinsic to the transistor.

There is often a preference for the use of current gain, β, or transconductance,

gm, when performing analysis on small signal models of bipolar transistors. An

argument may be made in favour of using transconductance because it is common

to all three terminal amplifying devices. In the case of common base circuits

however, the use of α and β often produces a more elegant expression. The

equivalence of analysis using current gain and transconductance can be shown.

76

CHAPTER 4

For the transimpedance gain of 4.1, solutions are given using current gain and

transconductance. The transimpedance of the circuit in Fig. 4.1 is given by,

vciin

=β0RL

β0 + 1· 1

1 + s τT α0(4.1)

An equivalent expression without invoking α or β is,

vciin

=gm rbe RL

gm rbe + 1· 1

1 + s τT gm rbegm rbe+1

(4.2)

The impedance looking into the emitter is,

zin =rbe + rbβ0 + 1

·1 +

s τβ rbrbe+rb

1 + s τT α0(4.3)

Equation 4.3 is a first order pole – zero standard form. The zero occurs at a

lower frequency than the pole; consequently, a range of frequencies exists where

the phase shift increases from 0 at lower frequencies towards +90 before re-

turning to 0. It is instructive to observe graphically the real part of zin and the

phase of zin (Fig. 4.2) using some sensible values. The real part shows that at low

frequencies zin is given by rbe+rbβ0+1

and as frequency increases zin → rb. The phase

of zin illuminates the impedance transforming nature of the transistor. A range

of frequencies exists where the phase of zin increases from zero, reaches a maxi-

mum, and returns to zero at high frequencies. In this region the base spreading

resistance appears partially or wholly inductive when viewed from the emitter of

the transistor. It will be shown shortly that series inductance in the base network

is transformed – over a certain range of frequencies – into a negative resistance.

Similar arguments follow for capacitance transformation.

4.3 Addition of Parasitic Inductance in the Base

Network

It is assumed that the base lead possesses parasitic inductance, LP . The induc-

tance may arise intentionally by design or be formed as part of a ground plane.

77

CHAPTER 4

Figure 4.2 – Black, real part of zin, red, phase of zin, both from Fig. 4.1. Small signalparameters, rbe = 172.5 Ω, cbe = 139 pF, β0 = 100, rb = 39.66 Ω.

It may be found as a parasitic in another component such as a decoupling capac-

itor. The multiplicity of ways in which the parasitic inductance can be added to

the circuit can lead to difficulty in finding a circuit board layout and component

technology which minimises the inductance. The circuit of Fig. 4.3 differs only

slightly from that of Fig. 4.1, it may be shown that the transimpedance gain of

this amplifier is identical to that of the amplifier in Fig. 4.1. The base spreading

resistance and the base inductance form a series LR network with a time constant

τL. The impedance looking into the emitter is given by,

zin =rbe + rbβ0 + 1

·1 +

s τL rb+s τβ rbrbe+rb

+s2 τβ τL rbrbe+rb

1 + s τT α0(4.4)

iin

gmvbe

RL

cberbe

rb

LP

Figure 4.3 – Common base amplifier with base spreading resistance, frequency dependent andbase inductance.

78

CHAPTER 4

Inserting s = j ω and considering only the real parts,

ℜ(zin) =rbe + rbβ0 + 1

·

(

− τβ τL rbrbe+rb

+ rb τT α0 τLrbe+rb

+ τb rb τT α0

rbe+rb

)

ω2 + 1

1 + ω2 τT 2 α02

(4.5)

A frequency independent input resistance may be obtained if the frequency de-

pendent terms of the real part are arranged such that they cancel. The value of

τL required to fulfil this condition is,

τL =τT α0 (τT α0 rbe + rb τT α0 − τβ rb)

rb (−τβ + τT α0)(4.6)

While this condition is satisfied the input impedance takes the form,

zin = k · (1 + s τ) (4.7)

k =rbe + rbβ0 + 1

(4.8)

τ =τβ (τT α0 rbe + rb τT α0 − τβ rb)

(−τβ + τT α0) (rbe + rb)(4.9)

This circuit cannot be realised easily because there are unavoidable parasitic

elements, such as signal source capacitance, missing from Fig. 4.3, which modify

the circuit behaviour. Furthermore, the input resistance is quite sensitive to

changes in the value of τL; obtaining the precise value from a distributed set of

parasitic elements may be difficult.

A second value of τL which is of interest is the value which is just sufficient to

make the input resistance fall to zero as frequency tends to infinity. Setting (4.5)

to zero and solving in the limit ω → ∞ yields,

τL = − τβ τT α0

−τβ + τT α0

= τT (4.10)

In this circuit, where the transistor biasing current flows through the signal source

to ground, the condition for a range of frequencies over which the input resistance

is negative is τL > τT . If τL = τT , ℜ(zin) → 0 as f → ∞.

79

CHAPTER 4

iin RE

gmvbe

RL

cberbe

rb

LP

Figure 4.4 – Common base amplifier with base spreading resistance, frequency dependants,base inductance and emitter biasing.

4.4 The Addition of an Emitter Biasing Resistor

It is sometimes impractical or undesirable to have the transistor biasing current

flowing through the signal source. It is common to bias the transistor with other

components and employ level shifting, the simplest form thereof is a blocking

capacitor between the signal source and the emitter electrode. This is shown in

Fig. 4.4; the blocking capacitor is considered short circuit at all frequencies of

interest so in the small signal model RE is connected directly to the emitter of

the transistor.

The addition of RE significantly complicates the problem of finding zin, how-

ever it is still possible analytically. In the prior circuit of Fig. 4.3 if the base

network inductance is sufficiently large there will be a frequency at which the

input resistance falls below zero; at higher frequencies the input resistance re-

mains less than zero. In Fig. 4.4 a bounded range of values of τL exist that yield

a negative input resistance over a bounded range of frequencies. It is impossible

to produce a constant input resistance with this circuit. Analysis to show this

falls along similar lines to the prior circuit. The frequency dependent terms can

not be set such that they cancel and consequently no value of τL will provide

a frequency independent input resistance. The range of frequencies over which

a negative input resistance exists is of interest, as are the boundary conditions.

This analysis also falls along similar lines as that given for Fig. 4.3. An expression

is obtained for the real part of the impedance looking into the emitter (4.11); this

is set to zero and solved for τL (4.12).

80

CHAPTER 4

ℜ(zin) =(−τβ rb ω

2 τL + rbe + rb)RE (−τβ rb ω2 τL +RE + rbe + rb + β0RE)

(−τβ rb ω2 τL +RE + rbe + rb + β0RE)2 + (ωRE τβ + ω τβ rb + ω rb τL)

2

+(ω rb τL + ω τβ rb)RE (ωRE τβ + ω τβ rb + ω rb τL)

(−τβ rb ω2 τL +RE + rbe + rb + β0RE)2 + (ωRE τβ + ω τβ rb + ω rb τL)

2

(4.11)

τL =1

2 rb· τβ

(

2 rbe + β0RE − 2A± B

A

)

(4.12)

where,

A = ±((rbe + rb) (rb + rbe + β0RE +RE)

) 12 (4.13)

and,

B =((−4 (β0RE + 2 rbe)A

2

+A(β0

2RE2 + 4RE (2 rbe + rb)β0 + 4 rbe (2 rbe +RE + 2 rb)

)) 12 (4.14)

The denominator of (4.11) is a quartic polynomial in ω, it has four roots. Two

roots are complex and may be ignored, the other two are real. Setting the two

real roots equal to each other and solving the resulting expression for τL yields the

boundary conditions. The boundary conditions are found where the real part of

zin is a minimum at zero ohms. A set of values of τL is sought which satisfy both

zin = 0 and d(zin)/df = 0. This is mathematically analogous to setting the two

real roots at the same frequency. As the two roots are set at the same frequency

only one value is sought, but two values of τL are solutions because the original

expression for ℜ(zin) is quartic. For a certain set of small signal parameters,

there are two frequencies at which ℜ(zin) has only one root; a different value of

τL yields each of them. Values of τL between these boundaries will yield two

roots. In the case of two roots, the region of frequencies between the roots will

be where the real part of zin is negative. Two complex boundaries exist but they

are ignored. The boundaries are given by (4.12). Note that the sign acting on

the root in the expression for ‘A’ must be the same for both occurrences of ‘A’ in

81

CHAPTER 4

the main expression. Fig. 4.5 shows a plot of frequency versus ℜ(zin) for each of

the solution conditions using some example small signal parameters. The small

signal parameters used to produce the plot are given in the caption. The yellow

(dash dot) line represents LP ≈ 1.5 mH and is a real root. The blue (long dash)

line represents LP ≈ 11.44 nH and is the other real root. The red (dash) and

green (solid) lines represent complex roots. The value of τL that represents the

blue (long dash) line is the minimum τL that will result in instability. The yellow

(dash dot) line is the maximum τL which maintains instability. The value of τL

produces zero or more – a maximum of two – real roots of zin. If more than zero

real roots exist, the amplifier is potentially unstable. The value of the inductor,

LP , is not especially significant, but the value of the time constant of the base

spreading resistance and the inductor, τL, is important.

The problem may be approached from another perspective by finding a range

of values of gm for which the amplifier is stable or unstable for a given rb and LP .

gm may be related to the base network through β0 and rbe. A logical extension

is then to use gm = (e IC) / (k T ) to obtain a range of values of IC for which the

amplifier is stable with a given LP and rb. It is difficult to estimate values for

LP and to a lesser degree rb, several methods of measuring rb exist [68, 69]. It is

therefore preferable to estimate a worst case scenario and ensure that the circuit

design parameters that are easily controlled will not result in instability. The

transimpedance of Fig. 4.4 exhibits a second order low pass network which may

be found using standard circuit analysis.

4.5 Addition of Emitter Capacitance

The addition of capacitance at the emitter – the signal source capacitance –

further complicates the problem but it is still analytically tractable. By using

methods identical to those used in the prior section it may be shown that the

new time constant τE = CE RE has no effect on the range of frequencies over

which ℜ(zin) is negative given a certain value of τL and a particular set of small

signal parameters. The extension to the method used previously is to note that

τE is absent from the expression for the solution of the roots of the real part

of zin and so cannot influence the boundary conditions. A graphical example is

82

CHAPTER 4

Figure 4.5 – Plots of ℜ(zin) showing the values of τL that satisfy the exactly one real rootcondition. The red (dash) and green (solid) lines represent the values of τL that produce rootsof ℜ(zin) in jω. The yellow (dash dot) and blue (long dash) lines are the upper and lowerbounds of τL that produce roots of ℜ(zin) in ω. Note that both the blue (long dash) andyellow (dash dot) lines have only one root. The violet (dash space) line is an example valueof τL which lies between the blue (long dash) and yellow (dash dot) values. It has two rootsand the region of frequencies where negative input resistance exists is not limited to layingbetween the frequencies that satisfy the exactly one real root condition. To generate this figurerbe = 172.5 Ω, cbe = 139 pF, β0 = 100, RE = 680 Ω, rb = 39.66 Ω, Lp = 15 nH.

shown in Fig. 4.7, using similar small signal parameters as Fig. 4.5, which shows

that a wide range of τE has no effect on the boundary conditions. Note that all

of the lines fall through the axis at the same frequency; τE does not affect the

range of frequencies having negative input resistance. The value of CE plays a

part in tuning the frequency of oscillation in a poorly designed amplifier. The

transimpedance of the circuit in Fig. 4.6 is third order low pass and may be found

using standard circuit analysis.

4.6 Addition of Load Capacitance and Collector

- Base Junction Capacitance

Unfortunately the final example, shown in Fig. 4.8, is not analytically tractable.

Therefore Maple, a symbolic/numerical mathematics package, has been used. It

can be shown indirectly that the circuit of Fig. 4.8 has a range of frequencies over

83

CHAPTER 4

iin RE

gmvbe

RL

CE

cberbe

rb

LP

Figure 4.6 – Common base amplifier with base spreading resistance, frequency dependence,base inductance, emitter biasing and emitter capacitance.

which the real part of the input resistance is negative, and that the boundaries of

this region are still independent of τE . A numerical solution for the roots of the

input resistance can be found without requiring any information regarding the

value of τE ; therefore one may conclude that τE does not appear in the expressions

being evaluated. If τE did appear in the expressions it would also appear in the

solution, unevaluated. ccb and τC – the time constant of RL and CL – do affect

the range of frequencies over which ℜ(zin) is negative for a given τL.

A general rule for assessing the likelihood of oscillation is if τL > τT , oscillation

is likely but not certain. Observation of the real part of the impedance looking

into the emitter using a circuit simulator such as SPICE can give an indication

of the likelihood of oscillation, assuming the simulation models are accurate. The

Figure 4.7 – Plot of ℜ(zin) with frequency showing the independence of the roots of ℜ(zin)from τE . At high frequencies all lines tend to zero ohms. rbe = 172.5 Ω, cbe = 139 pF, β0 = 100,RE = 680 Ω, rb = 39.66 Ω, LP = 15 nH, CE = 0.01 pF to 100 nF in decade steps.

84

CHAPTER 4

iin RE

gmvbe

RL

CE

cberbe

rb

LP

ccb

CL

Figure 4.8 – Common base amplifier hybrid–π model neglecting Early effect.

transimpedance expression for Fig. 4.8 has five poles and four zeros.

4.7 Colpitts Oscillator

Negative resistance oscillators are found in electronic systems serving a multi-

tude of purposes. A popular negative resistance oscillator is the circuit after

Colpitts [67,70,71]. The common collector Colpitts oscillator bares strong resem-

blance to the amplifier circuits considered in this chapter. Fig. 4.9 is nearly a

standard form of the Colpitts oscillator. The base emitter diffusion capacitance

is lumped with an external capacitance into C1. C2 is the source capacitance,

RE is the emitter biasing resistance, R2 and R3 bias the transistor and LP is the

parasitic base inductance. RB is a ‘base stopper’ resistor. C3 grounds the base

for AC signals. The introduction of the collector load resistor to the oscillator

is not significant provided ccb is small. If ccb is sufficiently large that significant

coupling exists between the base and collector at the frequencies of interest then

increasing RL tends to increase the impedance looking into the emitter eventually

making it positive. For this reason a true common collector Colpitts oscillator has

no intentional elements in its collector circuit. In this chapter it is assumed that

RL is relatively low valued such that τC does not limit the high frequency per-

formance. A compatible assumption is that RL is insufficiently large to prevent

ℜ(zin) becoming negative.

85

CHAPTER 4

C3 R2

LP RB C1

C2 RE iin

RLR1

V s

Figure 4.9 – Common collector Colpitts oscillator with collector load resistance.

4.8 Analysis of the Common Base Amplifier from

an Oscillator’s Perspective

Up to this point it has been assumed that the desirable circuit function is ampli-

fication. If however we treat the common base amplifier as a common collector

Colpitts oscillator and allow the collector load resistor to remain, the circuit has

two modes of operation. The stability of the amplifier may be discovered by

considering its merits as an oscillator.

Ignoring ccb and CL, the circuit of Fig. 4.9 has the small signal representation

given in Fig. 4.6. The circuit may be analysed to find the frequency of oscillation

as a function of the circuit parameters. A further value of interest is the minimum

transconductance required for oscillation to begin.

4.8.1 Oscillation Frequency

The method of analysis can be found in a standard text such as Razavi [72]. The

principle is to provide an initial excitation and find the gain due to the excitation.

The excitation may take the form of a voltage or current source; the placement

of the source must be such that when it is removed the network is not modified.

For oscillation to begin a frequency must exist where both the real and imaginary

parts of the denominator of the transimpedance function fall to zero.

86

CHAPTER 4

The real part of the denominator of the transimpedance function of Fig. 4.6

is given by,

(−rb β0 cbe RE CE − Lpβ0 cbe − Lpgm RE CE )ω2

+ β0 + rb gm + RE gm + β0 gm RE (4.15)

The imaginary part is given by,

− ω3Lpβ0 cbe RE CE

+ (rb gm RE CE + β0 RE CE + rb β0 cbe + RE β0 cbe + Lpgm)ω (4.16)

Setting the imaginary part (4.16) to zero and solving yields,

ω2 =rb gm RE CE + β0 RE CE + rb β0 cbe + RE β0 cbe + Lpgm

Lpβ0 cbe RE CE

(4.17)

It is possible to find the value of a particular component which will yield the min-

imum transconductance required for oscillation. This is accomplished by substi-

tuting (4.17) into the real part (4.15) and differentiating the resulting expression

with respect to the component in question. For example, to find the value of CE

that requires the minimum transconductance for oscillation – all other parame-

ters being defined and held constant –substitute ω2 into the real part (4.15) and

differentiate with respect to CE. Setting the result to zero and solving for CE

yields,

C2E =

Lpβ0 cbe (rb β0 cbe + RE β0 cbe + Lpgm)

RE2(rb2β0 cbe gm + rb β0

2cbe + Lpgm2rb + Lpβ0 gm) (4.18)

For a particular set of small signal parameters the value of CE given by (4.18)

will yield oscillation with the minimum device transconductance. The frequency

of oscillation for this, minimum transconductance, value of CE is found by back

substitution into (4.17).

87

CHAPTER 4

Figure 4.10 – Plot showing the value of gm required for oscillation as a function of CE forseveral values of rb = 1 mΩ (red, dot), 0.1 Ω (green, dash), 1 Ω (violet, dot dash), 10 Ω (darkred, long dash), 39.66 Ω (blue, solid) and 100 Ω (brown, dash space). cbe is constant throughout,small signal parameters are given in Fig. 4.5, LP = 20 nH.

4.9 The Effectiveness of Increasing the

Base Resistance

A common method of stabilising an oscillating amplifier is the addition of a

resistance which increases the base spreading resistance, rb. In Fig. 4.9 this

extra base resistance is labelled RB. It is illuminating to plot the minimum

transconductance as a function of an example circuit parameter, CE , as rb –

which is now taken to be the sum of rb and RB – varies. Unfortunately, cbe

is linked to gm and the interaction between them clouds the picture, therefore

assume that the oscillator designer adds an extra capacitance in parallel with

cbe such that the combination of the two, labelled C1 in Fig. 4.9, does not vary

significantly with gm; this is common practice. The effect of increasing rb on the

necessary gm required for oscillation is shown in Fig. 4.10; small signal parameters

are given in the caption.

88

CHAPTER 4

4.10 Practical Methods of Assessing Oscillation

A spectrum analyser provides a productive method of assessing oscillation on the

test bench. A free object possessing capacitance to ground, for example a hand,

is also useful. The circuit under observation is connected to the analyser. The

background signals should be recorded prior to applying power. The circuit may

not oscillate immediately, particularly if it is cold. Oscillation in the RF region

will often result in a waveform consisting of several harmonics. As the active

device(s) increase in temperature from cold to the operating temperature a shift

in oscillation frequency may be observed. Once the oscillation frequency has sta-

bilised it is sometimes possible to lower the frequency of oscillation by bringing an

object close to the circuit. This principle underpins several musical instruments

including the Theramin [73]. The emitter node is particularly sensitive in the

common base amplifier configuration. The magnitude of the frequency variation

is dependent on the magnitude of the additional capacitance to ground formed

by the object with respect to the capacitances already present in the circuit.

It most circuit designs a common base amplifier will not drive a transmission

line directly. The output impedance of a common base amplifier is approximately

equal to RL so a buffer amplifier is often used. If the buffer has insufficient band-

width to reproduce the oscillation frequency, it will be difficult to observe it

directly using an analyser. Assuming that the buffer amplifiers have insufficient

bandwidth, their action on the oscillation signal is likely to be rectification. Con-

sequently, observing a baseband output that has a large change in DC offset when

an object possessing capacitance to ground is brought near to the RF common

base stage is suggestive of oscillation. In such situations near field probes may

be used with an analyser to observe the oscillation. Note that any wide band

active circuit’s DC offset will rise to some degree when an object is brought near

because the object acts as an antenna and background signals are capacitively

coupled into the amplifier. In the case of oscillation however the DC shift is

likely to be more pronounced as the amplitude of the oscillation will change as

the circuit capacitances increase. One may also find that quiescent DC voltages

in the RF amplifier are unusually sensitive to foreign objects and physical dis-

turbance. ‘Wiggling’ the DC power input connector on the circuit board a little

89

CHAPTER 4

may cause significant changes in quiescent conditions for a short period. If this

kind of behaviour is observed oscillations should be sought. It is unfortunately

more difficult to show that a circuit does not oscillate than to show the opposite.

4.11 High Frequency Techniques

Practical methods regarding the physical requirements of high frequency circuits

which function as designed are often neglected or summarised as ‘high frequency

techniques’. High speed transistor circuits, especially those run at higher quies-

cent currents and so larger transconductances, are more susceptible to oscillation

when inappropriate board layout and construction is employed. Below are the

points that have been found most useful in helping to achieve a stable amplifier.

• Minimise stray inductance and capacitance in the layout. Reduce the board

area taken up by RF stages. Increase track widths; this decreases self

inductance of the track. Increase distance between track and ground plane,

this decreases capacitance to ground. Only run long traces which are driven

by low impedances.

• Avoid ground planes under high speed transistor amplifiers and operational

amplifier stages. This decreases capacitance to ground.

• Ensure microwave transistor emitters are well connected; connect all the

emitter tags. This reduces the resistance and inductance in series with the

emitter.

• When decoupling nodes to ground use a range of capacitances. For exam-

ple 220 µF and 220 nF and 100 pF in parallel. Ensure that the smallest

value capacitance is closest to the node. Aluminium electrolytic capacitors

become quite inductive as frequency increases.

• Connect top and bottom ground planes together at regular intervals. Con-

nections should be spaced less than 110

th of the wavelength that corresponds

to the unity power gain frequency of the active devices.

90

CHAPTER 4

• Ground planes must not be permitted to appear as transmission lines be-

cause there will be no common ground reference for the circuits on the

board. The transistor in common base mode views the signal source with

respect to the base node, not with respect to wherever the signal source

ground enters the circuit board, or any other location in between the two.

If the ground plane possesses significant parasitic inductance over the dis-

tance between the location of the signal source ground and the location of

the transistor base ground the circuit stability will be compromised.

• Decouple all power supplies at the terminals of all active devices and circuit

stages.

Sometimes ensuring the layout is well conceived is not sufficient to prevent

oscillation. In these cases extra components must be added to: 1) make the

impedances larger at high frequencies, 2) damp resonances or 3) increase the

transconductance necessary for oscillation. Many methods of achieving these

aims exist, but the additional general guidance below has proved useful.

• Use extra base resistance sparingly, increasing the base resistance reduces

gain, reduces bandwidth and carries a high noise penalty [74]. If frequencies

are sufficiently low, a ferrite bead may be used instead.

• Use several active devices in parallel to lower the individual transconduc-

tance, but maintain the ensemble transconductance. This carries a noise

benefit; multiple devices may be used in conjunction with an extra base

resistance such that the noise penalty of the extra resistance is partially

mitigated [74].

• In some cases a Zobel network (image filter) is a viable alternative to in-

creasing the base spreading resistance without suffering the same speed,

gain or noise penalty. For example a Zobel network connected from the

gate to drain of a power FET operating as a source follower can prevent

the impedance looking into the gate becoming negative at high frequencies,

due to a capacitive load on the source.

91

CHAPTER 4

4.12 Conclusion

Analytical results have been developed for a series of progressively more complex

common base small signal amplifier models which address the nature of, and

conditions necessary for, instability. The common base amplifier has been treated

as a Colpitts oscillator and the conditions necessary for oscillation established by

a second method. The common practice of using a ‘base stopper’ resistor has

been explored. A general design rule for avoiding oscillation (τL < τT ) has been

given. Practical approaches to observing oscillation and recovering an oscillating

amplifier have been discussed.

92

Chapter 5

1 MHz Opamp Based High

Capacitance Noise Measurement

System

This chapter reports a second versatile system for measuring excess noise and

multiplication in avalanche photo-diodes, using an operational amplifier based

transimpedance amplifier (TIA) front-end and operating by phase-sensitive de-

tection, which facilitates accurate measurement of multiplication and excess noise

in the presence of a high dark currents. The system can reliably measure the ex-

cess noise factor of devices with capacitance up to 5 nF at a centre frequency of

1 MHz. This system has recently been used to measure the thin, large area Si

pin avalanche photo-diodes which are reported in Chapter 6. The data obtained

from this measurement system is in good agreement with that obtained from the

10 MHz system described in Chapter 3.

5.1 Introduction

The problems of measuring the noise associated with large capacitance photo-

diodes, or photodiodes that for some reason must be connected to the TIA by

long cable lengths, have been discussed in detail at the start of Chapter 3. In

Chapter 3 a discrete TIA design was described with special attention being given

93

CHAPTER 5

Chopper Driver

Scitec Inst. 300C

Photo-currentLock-in-amplifier

Stanford SR830

Reference Signal

Excess NioseLock-in-amplifier

Stanford SR830

Laser

He-Ne

2.5mW, 633nm

Chopper

Bias Source

Keithley 236

TIAVoltage Amp

Minicircuits

ZFL-500LN+ HP 355D

f0 = 1 MHz

Friend, N = 8

Voltage Amp

ZFL-500LN+

Power Meter

VoltageAmplifier

Figure 5.1 – A system diagram of the 1 MHz operational amplifier based noise measurementsystem.

to the trade-off between, input impedance, noise, bandwidth and stability. In

this chapter a noise measurement system which has similar goals to the system

described in Chapter 3 but having differing specifications is described. The objec-

tive of this measurement system is to corroborate the data presented in Chapter 3

by excluding the possibility of measuring 1/f effects and RC limiting. By relin-

quishing an order of magnitude in frequency range the design problem becomes

much more tractable. Operation amplifiers are now viable front end devices. This

measurement system also benefits from better system noise performance then the

transistorised system under low capacitance conditions.

5.2 Noise Measurement System

The principle difference between this system and Li’s system [20] is the change

in the measurement bandwidth and centre frequency. The open loop gain of

the AD9631 at 10 MHz is approximately 12. It can be shown that the input

impedance is the feedback impedance reduced by this factor and is approximately

180 Ω. The input impedance and the APD capacitance form a filter which must

not limit the measurement bandwidth. In the 10 MHz case the maximum value

of APD capacitance due to RC limiting is approximately 70 pF. However in

the 1 MHz case the open loop gain is increased to approximately 112 and the

94

CHAPTER 5

input impedance falls to approximately 19.6 Ω yielding an RC limit of 8 nF. The

limitation on the APD capacitance due to stability problems that result from the

interaction of the APD capacitance and the open loop transfer function of the

opamp will be discussed shortly.

The structure of the measurement system is shown in Fig. 5.1. The laser

is mechanically chopped by a rotating, slotted, disk at 180 Hz and is incident

on the diode via a set of optics. The transimpedance amplifier (TIA) is shown

schematically in Fig. 5.2. It converts the diode current into a voltage. The voltage

is amplified using a commercial low noise, wide band communications amplifier

with a terminated gain of 25 dB. A precision stepped attenuator is used to select

the system gain permitting measurement of devices under injection conditions

which elicit both high and low excess noise. The noise signal is separated from the

low frequency component of the photo-current by a cascade of single tuned Friend

bandpass filter sections [75]. The filter sets the measurement bandwidth of the

system. The circuit diagram is shown in Fig. 5.10. The filtered signal resembles

an amplitude modulated noise waveform. Periods of device illumination produce

greater noise amplitude than periods of darkness. Voltage gain is provided by a

second commercial amplifier module, which is identical to the first, and follows

the filter. Further voltage gain is rendered by operational amplifiers (shown in

Fig. 5.13) prior to a wide band (250 MHz) squaring and averaging circuit. The

squared, averaged signal is amplified a further sixteen times. After squaring

and averaging the signal is approximately square with a fundamental frequency

of 180 Hz. The signal amplitude is a function of the noise power contained

within the pass band of the filter. The squaring circuit makes use of an Analogue

Devices AD835 analogue multiplier and is shown schematically in Fig. 5.14. The

averaging circuit is a first order RC network with a time constant of approximately

100 µs. The output from the squaring and averaging circuit is measured using

a lock-in-amplifier. The photo-current signal is taken from an additional output

of the TIA where the amplitude of the 180 Hz square wave is proportional to

the photo-current. The photo-current signal is measured using a second lock-in-

amplifier.

95

CHAPTER 5

R1

100 ΩAPD Bias Voltage

C1

1 µF

−

+A1

AD9631

C2

70 pFR3

33 Ω

R2

2.2 k

R4

22 Ω

+5V

C3

100 pFC4

220 nF

C5

100 pFC6

220 nF

R5

22 Ω

−5V

−

+

A2

TLE2141

+5VC8

220 nF

-5V C9

220 nF

R7

50 Ω

Photocurrent Output

−

+

A2

TLE2141

+5VC10

220 nF

-5V C11

220 nF

R8

50 Ω

Noise Output

Figure 5.2 – Opamp based transimpedance amplifier showing all components.

5.3 Transimpedance Front End

A schematic diagram of the TIA is shown in Fig. 5.2. In this figure A1 is the

transimpedance amplifier. The opamp is the same model reported by Li [20,27].

R2 sets the transimpedance gain at all frequencies of interest. C2 is computed

using the Analog Devices data sheet [30]. Together C2 and R3 add a zero to

the compensation network around 70 MHz and so provide leading phase shift

enhancing the phase margin of the feedback system and maintaining stability.

R4 and R5 are data sheet recommendations. C3−6 decouple the opamp power

supplies to ground. C1 grounds the biased terminal of the APD for AC signals

and filters any HF noise that would be injected into the APD terminal from the

SMU. R1 is used to isolate the output of the biasing source (Kiethley 236) from

C1 by raising the biasing source output resistance sufficiently to prevent it from

oscillating when presented with the capacitive load.

96

CHAPTER 5

5.3.1 Front-End Bandwidth

Considering the schematic in Fig. 5.2 and employing the same diode small sig-

nal model as in Chapter 3 (Fig. 3.4) and assuming the opamp is perfect, the

transimpedance in the mid-band is given by,

Vo = −(iin R2) (5.1)

Unfortunately the junction capacitance of the APD and the open loop transfer

function interact in such a way as to promote instability. A more realistic approx-

imation is that A1 is a first order low pass system having a frequency independent

gain A0 of 560 and time constant τ0 of 1.6 µs [30]. The transimpedance gain stage

formed by A1 and the surrounding components forms a system with three poles

and one zero.

A0R2 · (sR3C2 + 1)

(1 + s τ0)(

1 + A0

1+s τ0

)(

1 +cj R2+R2 C2+

A0 R2 C21+s τ0

+A0 R3 C21+s τ0

+R3 C2

1+A0

1+s τ0

s+cj R2 R3 C2

1+A0

1+s τ0

s2)

(5.2)

Using parameters from Fig. 5.2 and datasheet values for A0 and τ0, it can be

shown numerically that the form changes from three real poles when cj less than

approximately 1.5 nF to a conjugate pair and one real pole when cj is greater than

approximately 1.5 nF. Eventually when cj becomes very large the conjugate poles

tend towards the real axis. This is shown graphically in the pole zero diagram of

Fig. 5.3.

Example plots of the circuit response near to the measurement bandwidth

are shown in Fig. 5.4. In comparison an example uncompensated response is

shown in Fig. 5.5. A practical implementation of this circuit may be expected

to oscillate. Unlike the Li system in which R3 = 0 Ω [20], A0 divides C2 in all

terms of the denominator, consequently C2 must be considerably larger to have

the desired effect in this design. Furthermore the bandwidth of the circuit is a

much weaker function of cj than in Li’s design.

97

CHAPTER 5

Figure 5.3 – Pole-Zero plot showing the variation of pole location with cj = 0, 10, 100 and500 pF, 1, 1.5, 2, 3, 4, 5, 10, 20, 50, 200 nF, 1 µF. The colour order is a repeated pattern ofblue, red, green, purple, orange, brown and black. rd = 1 kΩ, R3 = 33 Ω, R2 = 2.2 kΩ, C2 =70 pF. A0 = 562, τ0 = 1.5915 µs.

Figure 5.4 – Approximation of the 1 MHz operational amplifier based noise measurementsystem front end response. A0 = 562. τ0 = 1.5915 µs. rd = 1 kΩ, C2 = 70 pF, R2 = 2.2 kΩ,R3 = 33Ω and cj = 1 pF, 100 pF and 1 to 5 nF in 1 nF steps.

98

CHAPTER 5

Figure 5.5 – Approximation of the 1 MHz operational amplifier based noise measurementsystem front end response assuming the TIA opamp is first order. A0 = 562. τ0 = 1.5915 µs.rd = 1 kΩ, C2 = 0 pF, R2 = 2.2 kΩ, R3 = 0 Ω and cj = 1 pF, 100 pF and 1 – 5 nF in 1 nFsteps.

iin cj rd I2n

V 2n

−

+ A1

AD9631

R2

InR22

C2R3

EnR23

R8Vout

R9

Figure 5.6 – Schematic showing the noise model of the TIA showing all of the noise generatorsand the photo-current. R8 and R9 are 50 Ω, and are not included in the noise analysis, it canbe shown that the contribution of these resistors is insignificant.

99

CHAPTER 5

5.3.2 TIA Noise Performance

The noise performance of the transimpedance amplifier can be assessed using the

standard opamp noise model which makes use of equivalent noise voltage and

noise current generators, V 2n and I2n. The equivalent circuit of the TIA from a

noise perspective is shown in Fig. 5.6. The approach is similar to that used in

Chapter 3. The total output noise voltage is the superposition of the value of each

source multiplied by the noise gain from the source to the output. The resulting

values are combined as the rooted sum of the squared values. The pertinent

results are shown in Fig. 5.7 and will be discussed shortly.

The output noise voltage resulting from the opamp input current noise gener-

ator, In, and the thermal noise of R2, InR2 are both identical to the photo-current

generator expression (5.2). The output noise voltage due to the opamp voltage

noise generator (Vn) is,

−A0 (rd +R2)

(

1 +R2 R3 C2 rd cj s

2

rd+R2+

(rd R3 C2+rd R2 C2+R2 rd cj R2 R3 C2) s

rd+R2

)

(1 + s τ0)(

rd +A0 rd1+s τ0

+R2

)(

1 +R2 R3 C2 rd cj s2

rd+A0 rd1+s τ0

+R2+ Qs

rd+A0 rd1+s τ0

+R2

) (5.3)

where,

Q =

((

rd +A0 rd1 + sτ0

+ R2

)

R3 +

(

rd +A0 rd1 + sτ0

)

R2

)

C2 + R2 rd cj (5.4)

The noise voltage at the output resulting from the thermal noise of R3, EnR3

is,

A0 rdR2C2 s

(1 + s τ0)(

rd +A0 rd1+s τ0

+R2

)(

1 +R2 R3 C2 rd cj s2

rd+A0 rd1+s τ0

+R2+ Qs

rd+A0 rd1+s τ0

+R2

) (5.5)

5.3.3 Characterisation of the TIA

This transimpedance amplifier is characterised by similar means to the 10 MHz

TIA reported in Chapter 3. The details of the circuit used to approximate a

current source are described in Section 3.11. The DC characteristic of the TIA

100

CHAPTER 5

Figure 5.7 – Top left: graph showing the equivalence of the noise gain between the analyticalmodel (symbols) and a SPICE small signal (.AC) simulation (lines) for each source in theequivalent circuit. EnR3 - black, EnR2 - red, Vn - green, In - blue and the transfer functionis shown in yellow. Top right: Analytical model output noise voltage versus frequency foreach source with cj as a parameter. cj = 1 pF (solid lines), 5 nF (dashed lines). Black linesrepresent shot noise from 1 µA, EnR3 - red. EnR2 - green. In - blue. Vn - yellow. Thetotal is shown in pink. Middle left: Measured circuit noise c.f analytical model. Solid linesrepresent measured data from 0 nF to 5 nF in 1 nF steps. Middle right: Measured circuit noisec.f SPICE noise simulation (.NOISE) using an Analog Devices AD9631 model. Solid linesrepresent measurements, 0 nF to 5 nF in 1 nF steps. The bottom graph shows the analyticalmodel noise signal to noise ratio (NSNR) against frequency with cj as a parameter. cj = 0, 10,100 and 500 pF then 1 nF – 5 nF in 1 nF steps for a 1 µA signal current.

101

CHAPTER 5

Figure 5.8 – DC response of the TIA showing the onset of non-linearity around 2 mA inputcurrent.

is shown in Fig. 5.8 and is linear until the opamp saturates against the power

supply rail. This occurs around at around 4 Volts and is partially due to the use

of R4 and R5, but dominantly due to the voltage drop across circuits internal to

the opamp.

Figure 5.9 shows the measured frequency response of the TIA front end as a

function of increasing APD capacitance. The agreement with the model which

assumes the opamp is first order is very reasonable, with the resonance effect

being slightly more pronounced in the model. The inset of Fig. 5.9 shows the

difference in roll off due to increasing cj more clearly.

Graphs depicting the measured and simulated noise characteristics of the TIA

are shown in Fig. 5.7. From the top right graph in this figure it is evident that the

voltage noise of the opamp is by between one half and two orders of magnitude

greater than any other source, depending on the junction capacitance. The noise

gain of this source is shown in the the top left graph, and is 15.2 dB (5.75 lin) at

1 MHz when cj is 1 nF and rd is 1 kΩ. As rd increases k, the frequency independent

gain for the Vn source, decreases reaching approximately 1.2 (lin) when rd is 10 kΩ.

Ultimately, however, the voltage noise of the opamp is the determining factor in

the noise of this TIA, and it can only be adjusted by selecting a different opamp.

The near equivalence of the analytical model and the SPICE model is shown

102

CHAPTER 5

Figure 5.9 – Frequency response of TIA. Input capacitance increases from 0 nF to 5.5 nF in0.5 nF steps.

by the middle graphs of Fig. 5.7. The early roll off of the analytical model could

be corrected by adding poles and zeros to the transfer function of the opamp to

force it to follow the open loop response published by Analog Devices more closely.

However this added complexity would cloud an already complicated picture and

add little overall.

The bottom graph in Fig. 5.7 shows the NSNR as a function of frequency for

several values of cj . Up to approximately 100 pF, the NSNR of this front end

is nearly equal to Li’s system [20]. Thereafter NSNR deteriorates due to the Vn

peaking, at 5 nF the NSNR is approximately -52 dB. By comparison the 10 MHz

system described in Chapter 3 also has a low frequency NSNR of just over -25 dB,

falling to approximately -55 dB for cj = 5 nF.

5.4 Bandwidth Setting Filter and ENBW

The effective noise power bandwidth (ENBW) of the measurement system is

defined by a cascade of single tuned Friend filter sections [75]. A simplified

schematic is shown in Fig. 5.10. The ENBW is in approximately the same

relative proportion to the centre frequency as in the system after Li [27] and as in

103

CHAPTER 5

R1

50 Ω

Input

−

+

A1

AD829

R2

2.7 k

R3

249 Ω

C1

100 pF

C2

100 pFR4

10 k

−

+ A2

AD829

x7

R5

2.7 k

R6

249 Ω

C3

100 pF

C4

100 pFR7

10 k

−

+ A3

LM7171

C5

220 nFR8

50 ΩOutput

Figure 5.10 – 1 MHz Single tuned 16th order bandpass filter.

the system described in Chapter 3. The frequency response of the combination

of the filter and front end is shown in Fig. 5.11 as a function of APD capacitance.

The effective noise power bandwidth of the system is computed by normalized

numerical integration of Fig. 5.11. The ENBW of the system as a function of

device junction capacitance is shown in Fig. 5.12 and is given by

Beff(cj) =1

G20

∫ f2

f1

|G(f, cj)|2 df. (5.6)

where Beff is the ENBW, G0 is the gain at the centre frequency, G(f, cj) is the

frequency and APD capacitance dependent TIA and filter gain. f1 and f2 are

placed sufficiently far from the centre frequency, f0, that practically all of the

area under the curve in Fig. 5.11 is integrated.

5.5 Voltage Amplifiers

The voltage amplifier schematic required to replicate the measurement system is

shown in Fig. 5.13. The objective of this amplification (and that provided by the

Minicircuits Amplifiers) is to scale the noise signal to a suitable value to generate

an output from the noise power meter with minimal ambiguity. The order and

gain of each amplifier is shown in the block diagram of Fig. 5.1.

Similar arguments apply to the nature of the noise signal and the design of

crest factor in this measurement system as for the 10 MHz system described in

Chapter 3.

104

CHAPTER 5

Figure 5.11 – Frequency response of TIA and ENBW setting filter with input capacitance.Input capacitance increases from 0 nF to 5.5 nF in 0.5 nF steps. Solid line 0 nF, medium dash5.5 nF.

Figure 5.12 – The effective noise power bandwidth as a function of APD capacitance. Circlesand triangles represent independent data sets.

105

CHAPTER 5

C1

220 nF

R1

50 Ω

From ZFL-500LN+

−

+

A1

LM7171

R2

470 ΩR3

220 Ω

R4

50 ΩC2

220 nF

To Power Meter

3x

Figure 5.13 – Opamp voltage amplifier.

Calibration measurements on the 0 dB attenuation, using a BPX65 (cj ∼ 4 pF)

photo-diode show output noise voltage increases with the root of photo-current

with excellent agreement (R2 > 0.99) over a range of photo-currents from 0 to

200 µA indicating that full shot noise has been measured. Further evidence for

the acceptability of a low crest factor is the agreement of the data gathered using

this system and that gathered using the 10 MHz system reported in Chapter 3.

This data is presented in Chapter 6. The likelihood of agreement between the two

systems while both are producing erroneous data is extremely small. While both

systems operate under the same assumptions slight differences in implementation

(for example the difference in background noise floor and the difference in total

gain) would have to conspire to produce a consistent and repeatable error in

both measurement systems. Moreover the consistency of data between attenuator

settings on individual systems is excellent.

The linearity of the system prior to the power meter can be examined by

similar methods to those described in Chapter 3 and by connecting a sinusoidal

signal generator to the TIA (Fig. 5.2) via the resistor chain (Fig. 3.16) and ex-

amining the output from the final amplifier prior to the power meter using a

spectrum analyser. This method allows the measurement of the majority of the

system gain. Measurement of the power meter transfer function is obtained sep-

arately. Signal distortion is manifested as extra peaks on the spectrum which are

not present in the generator output. Clipping of the system noise on the 0 dB

attenuation setting is not present in the reported design.

106

CHAPTER 5

R1

50 Ω

Input

−

+

−

+

Σ

+

++1

R2

330 Ω

R3

100 ΩAD835

R4

1 k

C1

0.1 µF

−

+

A1

TL071

R5

15 kR5

1 k

Output

Figure 5.14 – AD835 based power meter.

5.6 Noise Power Meter

The power meter design shown here was developed by B. K. Ng, and is based

on the Analog Devices AD835 wide band analogue multiplier. A representative

schematic is shown in Fig. 5.14, where the AD835 is shown in a block diagram

form. The inputs of the multiplier are connected together in order that the output

voltage is dominantly proportional to the square of the input. A small linear term

and a DC offset remain. The DC offset is lessened by trimming the offset null of

the TLO71 and finally removed by AC coupling the power meter output to the

lock-in-amplifier. The linear term is partially resultant from imperfection in the

squaring process, but it is likely that the majority of the linear term is a result

of regressing manually collected data possessing some uncertainty. The squared

noise voltage signal is applied to a first order low pass filter, R4 and C1, with a

time constant of approximately 100 µs. The filter is required to pass the 180 Hz

chopping signal such that the LIA can differentiate between light and dark noise,

but must filter the noise signal (around 1 MHz) to produce a signal representing

the average value of the squared noise voltage.

The transfer function for the multiplier circuit after Ng is,

Vo = 3.53 V 2i − 0.03 Vi + 0.8850 (5.7)

where Vo is the mean squared output voltage and Vi is the RMS input voltage

of a deterministic test signal.

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CHAPTER 5

Figure 5.15 – Measured power meter transfer characteristic.

5.7 Conclusion

In this chapter a new transimpedance amplifier based measurement system for use

in measurements of excess noise in avalanche photodiodes has been documented.

This amplifier has found use in situations where the device being measured has

high junction capacitance, up to 5 nF, and where the device has lower than usual

dynamic resistance, rd. The measurement of a much wider range of devices is now

possible than was previously. Noise measurements on devices held in cryogenic

chambers or high temperature ovens is, in principle, possible by connection of the

device and TIA by coaxial cable. This measurement system has been used to con-

firm measurements made using the 10 MHz transistor based noise measurement

system described in Chapter 3.

An area of particular interest at present is the use of narrow band-gap materi-

als including InAs and HgCdTe in X-ray detection applications. X-ray detectors

are often relatively large area devices, and when manufactured from narrow band-

gap materials may exhibit significant tunnelling currents. These devices would be

unmeasurable on prior systems due to bandwidth constraints or system instabil-

ity. The present work and the noise measurement system reported in Chapter 3

overcome these obstacle, opening new avenues of investigation.

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Chapter 6

Excess Avalanche Noise Thin

Silicon APDs

This chapter reports multiplication and excess noise factor measurements, and

non-local modelling results, in a set of Silicon pin diodes over a range of optical

injection wavelengths. A local model ionisation coefficient extraction is performed

to highlight the effect of the dead-space on the ionisation coefficients when com-

pared to the bulk coefficients. The data has been gathered using the two new

measurement systems, described in Chapters 3 and 5, specifically designed to

measure excess noise and multiplication in avalanche photo-diodes possessing

high junction capacitance and large tunnelling current.

6.1 Review of Impact Ionisation in Silicon

Silicon was one of the first semiconductor materials to have its bulk ionisation

properties investigated systematically. A portion of this systematic investigation

was undertaken by Bell Systems, where several of the early workers were based.

The first report of ionisation coefficients in Silicon was by McKay and McAffee [7]

both authors were with Bell. Impact ionisation was treated with the pre-existing

theory of gas discharge as a framework, which is also called ‘avalanche theory’ but

is also referred to as ‘Townsend’s beta theory’. They assumed α and β were equal

and used pn junctions, so the ionisation coefficient was reported as a function of

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CHAPTER 6

the peak electric field. The authors fitted the data to,

α0 =(1−M−1)

w(6.1)

where α0 is the ionisation coefficient, and has the units cm−1, M is the photo-

multiplication and w is the effective width of the device assuming a uniform

field. Microcomputers were not available to assist the early workers with taxing

integrals, and that the uniform field approximation produces closed form solutions

which can be derived, with some effort, by hand.

McKay published a separate work on the breakdown of Silicon diodes in 1954

[6] still assuming that the coefficients were equal. This work is significant, it is

the first work in which impact ionisation is considered in a similar way to the

present methods. It is also insightful, predicting the non-local nature of avalanche

multiplication – its dependence on carrier history, most evident in thin devices.

McKay gives a solution to a multiplication integral, which is derived from first

principles, for several junction geometries. The usefulness of the experimental

data appears to be limited by the device material quality.

P. A Wolff, also with Bell, and a collaborator of McKay and McAffe solved

the Boltzmann transport equation in a later publication of 1954 [8]. This was

done by separating the problem into three regions, firstly carriers with energies

appreciably less than the pair creation energy were solved, then carriers with

energies appreciably more than the pair creation energy, finally the third region

in which the carriers have energies approximately equal to the pair creation en-

ergy was resolved by extrapolating the solutions in the other two regions until a

smooth function resulted. This paper ends with a strict definition of ‘high field’.

The field is considered ‘high’ when a carrier can move from zero energy to the

pair creation energy in less distance than the average distance between phonon

scattering events. It is in circumstances where the electric field strength is suf-

ficiently high that the effect of phonon scattering on the ionisation process may

be neglected.

The first work that showed α > β was by Miller in 1957 [76]. Miller was also

working for Bell. His 1957 paper is the first in which note has been made of the

importance of the injection purity, this is not surprising when it is considered that

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CHAPTER 6

the prior workers assumed α and β to be equal so the initiating carrier type was

irrelevant. This work is the first Silicon paper in which the following expression

appeared,

1− 1

M= −

∫ w

0

αi (E) exp

[

−∫ x

0

[αi (E)− βi (E)] dx′.

]

dx. (6.2)

Miller stated his empirical formula for fitting multiplication data (8.2) in a

1955 Germanium paper [5]. The diodes reported by Miller [76] were pn step

junctions, although Miller preferred transistors for providing carriers of one type

(i.e. pure injection).

Chynoweth and McKay published two papers in 1957 [10, 77] expanding the

volume of available data. Chynoweth published again in 1958 [11] this publication

contains more data, and the first example of α and β plotted as functions of inverse

electric field. The empirical expression which bares his name is also published.

However it is clear that the expression is borrowed from gas discharge theory and

was proposed as early as 1936 [78].

α (β) = A exp

(B

ξ

)C

(6.3)

In the 1958 paper the power term does not appear.

Shockley entered the debate with a publication in 1961 [12] which argued

that the multiplication effect was produced by those carriers lucky enough not to

undergo collisions which randomise the motion of the carriers. Baraff provided

a theoretical paper which applied boundaries to both Shockley’s and Wolffe’s

solutions of the Boltzmann transport equation [9]. It was shown that Shockley’s

solution was more appropriate at low electric fields and that Wolffe’s was more

appropriate at high fields.

Throughout the 1960s and 1970s several authors published more data regard-

ing the values ionisation coefficients [79–90]. The coefficients of Van Overstraeten

and De Man [85] and Grant [87] are preferred due to the large number of samples

that their study undertook.

Other than the work of Robbins [91, 92] which showed the isotropic nature

of impact ionisation coefficients in Silicon, no significant work was carried out

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CHAPTER 6

thereafter until Pauchard [93] and Tan [94] both in 2000. These papers deal with

sub-micron width devices. Pauchard found no evidence of dead space effects in

a 0.8 µm structure. Tan et al. showed that excess noise is reduced in devices as

thin as 0.1 µm and that the choice of carrier injection becomes less significant as

depletion width decreases. It is however still preferable to select the more readily

ionising carrier to produce a low noise device.

Recent work has been reported by Massey et al. [36]. This data is in good

agreement with that of Van Overstraten and DeMan [85]. Massey concludes that

the dead space effect is less pronounced in Silicon than in GaAs and as a result

the local model produces a reasonable approximation of impact ionisation for low

values of multiplication even when the dead space is significant. Agrawal [95]

showed data from sub-micron structures in agreement with that of Massey.

6.2 Application: On-Chip Photonic Data

Transfer

The principle benefits of optical interconnects between cells of a VLSI design

were reported in a seminal paper by Goodman et al. [96] and further expanded

and updated by Miller [97]. This topic has received attention in the professional

electronics press also, for example [98, 99].

The principle of operation has been widely reported and may be found de-

scribed in many of the references herein. Briefly, two or more cells in a digital

chip are to communicate at a high data rate. An alternate scenario may be that a

clock signal having very low delay around the chip is to be transmitted to several

cells simultaneously. In any case the transmitting cell either generates a series of

NIR optical pulses or modulates an external NIR optical source [100, 101] which

is passed into the chip by some means, for example [102]. The modulated light

is coupled into a Silicon dioxide wave guide structure [103] which is formed by

standard lithographic processes on top of standard metallised interconnects. The

wave guide provides an equivalent function to a fibre-optic cable. The modu-

lated light arrives at a detector fabricated from a material suited to its detection,

InGaAs and Ge have been reported, [104–107] This detector is either grown epi-

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CHAPTER 6

taxially onto the Silicon structure or is produced separately and is mechanically

bonded both methods have been reported [108,109]. MSM and pin detectors have

been reported [110]. The state of the art appears to lie with a wave-guide pin

device which combines high -3 dB bandwidth of 47 GHz with good responsivity

0.8 A/W, low dark current, 0.072 nA at -1 V and a CMOS compatible operating

voltage of -3 V [111].

The principle advantages of intra-chip optical communications are,

• Unlike electrical interconnects, the maximum bandwidth of an optical in-

terconnect is unrelated to the number of nodes to which it is connected.

That is to say there is no optical effect which is equivalent to capacitive

loading of an electrical interconnect.

• Optical interconnects run in close proximity are incapable of interacting

with each other by capacitive coupling.

• Wave-guides are not confined to being planar structures. Unlike electrical

interconnects two wave-guides can pass through each other without the

information each is carrying being distributed into both.

• The possibility of adapting the route of an optical interconnect – on the fly

– based on the data being sent has been demonstrated [112].

• Latency in clock distribution can be avoided by passing the clock into several

areas of the chip simultaneously from a common optical source.

• The data rate that can be achieved in optical communications is signifi-

cantly greater than in electrical interconnects because the carrier frequency

is higher.

Ge on Si pn, unity gain pin and APD detectors have been reported in recent

times including [4, 106, 113–115]. These attempts to produce a NIR photodiode

on a Silicon wafer can be broadly categorised into three groups,

• Bonding structures composed of narrower band-gap materials including Ge

and InGaAs to a Si wafer [107, 109].

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CHAPTER 6

• Hetero-epitaxial growth for example Ge on Si [106, 116, 117]

• Homo-epitaxial growth of Si to produce a visible light interconnect [118].

The use of a thin high speed, low noise, Silicon multiplication region APD

is briefly considered by Chen et al. [118]. However, APDs are yet to receive as

much attention in this application as pin detectors.

APDs considered for this application must be very high speed, data rates of

100 Gbits−1 are desirable. The APD should also operate at CMOS voltages. Both

speed and voltage demands require a thin device which is likely to exhibit quan-

tum mechanical tunnelling. The device area will be determined by the necessity

of obtaining a high coupling efficiency between the wave guide and the device

surface. Operating voltage (or electric field), gain, area, speed and tunnelling

are contended against each other to find an optimum device design. Tunnelling

current scales with device area, the thinnest device in this chapter can provide a

useful estimate of the tunnelling, gain and voltage trade off. For example, based

on the data in this chapter, a 31 nm thick 5 µm x 10 µm device may be expected

to exhibit 5 nA tunnelling current at 4 V reverse bias and provide a multiplication

of ∼3.5 for 850 - 1064 nm light.

6.3 Application: Medical/Biological Imaging

Silicon is the principle material of choice when optical detection in the visible

region is required. A great variation in devices exists including CCD arrays

CMOS arrays, linear detector arrays for for photographic and video imaging and

position sensing or machine vision; as well as the pin / nip diodes which are of

greatest interest here. In medical fluorescence imaging (MFI), the detection of

the florescence wavelength is often important but in other cases more information

is available by sensing the fluorescence decay [119–121]. Capturing a spatially

resolved image from very little incident light is often required and high frame rate

is desirable. The case of medical fluorescence imaging is particularly interesting.

Early systems used vidicon tube detectors and gas lasers. They were somewhat

unwieldy and consequently fluorescence was observed in vitro [122]. The advent

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CHAPTER 6

of reliable semiconductor lasers that produce near UV lines, and the use of fibre

optics allows in vivo studies of tissue [123, 124].

The present clinical uses of MFI are,

• Early cancer diagnostics

• Identification of tumour boundaries

• Assessment of blood vessels

• Visualisation of lymph vessels

• Treatment response assessments

• Interactive dosimetry [125, 126]

Presently photon counting directly in CMOS [127–129] is generating consid-

erable interest as a suitable technology for florescence imaging and positron emis-

sion tomography [130] and chemical sensing [131] and as a replacement for LIDAR

systems such as [132, 133].

6.4 Chapter Objective

The objective of this chapter is to provide a systematic investigation of the extent

of the dead-space effect in thin Silicon avalanche photo-diodes by measurement of

excess noise as well as photo-multiplication. This work also provides experimental

confirmation that the RPL model and ionisation coefficients can effectively predict

the effects of impact ionisation in Si devices possessing dimensions approximately

equal to present FET gate lengths.

Excess noise measurements on five APD structures having intrinsic widths

over the range of interest for investigating dead-space are reported. Photo-

multiplication is measured at five wavelengths, providing a range of injection

conditions including, for the first time, pure injection in the thinnest device re-

ported to date. Excess noise measurements are obtained at 633 nm. The mea-

surement systems used to collect this data are described in Chapters 3 and 5. A

comparison of bulk coefficients and extracted coefficients (ignoring dead-space)

will be provided in order to assess the possible benefit provided by the dead-space

in thin structures.

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CHAPTER 6

6.5 Layer Details

Five p+in+ layers were grown epitaxially on As doped n+ substrates using chemi-

cal vapour deposition. Intrinsic layers with nominal region thickness w = 0.031,

0.079, 0.132, 0.223 and 0.347 µm were grown and capped with 0.2 µm thick p+

cladding layers. Square mesas of areas from 0.08 mm2 to 1.65 mm2 were fab-

ricated by reactive ion etching and their edges were chemically passivated with

a tetraethoxysilane layer. A top electrical contact was formed on the p+ layer

via sputtered AlSi and a small hole was etched into this contact to allow optical

access. SIMS measurements were performed, confirming the nominal growth val-

ues of w and the top p+ cladding layer and showing steep doping profiles at the

interfaces in each of the structures.

Table 6.1 – Layer details for thin Silicon layers.

Layer w [µm] Vbd [V]

8 0.031 ∼511 0.079 6.512 0.132 8.213 0.223 11.214 0.347 14.7


Current voltage (IV) measurements were performed without room lighting. For-

ward and reverse IV measurements were taken using an HP 4140B Picoammeter.

Forward and reverse bias measurements were undertaken on one device of each

size on each wafer, however for clarity only the 1 mm2 data are shown in Fig. 6.2.

Fitting of each characteristic to (2.1) is also shown in this figure. The emission

coefficient for all wafers except #12 is 1.15. In wafer #12 n = 1.1. Saturation

currents are between 25 fA and 50 fA for all wafers. The emission coefficient close

to unity indicates the dominant transport mechanism is diffusion. There is no

sign of series resistance in these layers.

In reverse, all of the layers except #8 show bulk breakdown and low leakage

currents. Layer #8 exhibits considerable tunnelling current, and does not break-

down prior to current limiting. However the measurement techniques used allow

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CHAPTER 6

the photo-generated current to be extracted from the tunnelling current even un-

der conditions where the tunnelling current is up to 60 dB greater in magnitude.

An example is shown in the Fig. 6.3 where 325 nm photo-current measurements

are shown overlaid on the dark current plots.


Junction capacitance as a function of reverse bias voltage is shown Fig. 6.1.

These data were gathered using an HP4275A multi-frequency LCR meter. A

1-D Poisson solver is used to fit the doping densities and intrinsic region width

to the measured data. The fitted intrinsic region widths yield the data shown

in Table 6.1. The Poisson solver is described in Appendix A. The diffusion

voltage is obtained from extrapolation of the forward bias capacitance voltage

characteristics towards the bias axis. The crossing point represents the diffusion

voltage, and is 1 V.

Figure 6.1 – Capacitance versus reverse bias voltage characteristics for 1 mm2 devices. #8,red circles #11, green circles, #12, blue circles, #13, pink circles, #14, grey circles. Black linesrepresent fitting to a 1-D Poisson solver.

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CHAPTER 6

Figure 6.2 – Forward bias current versus voltage characteristics (open symbols). Dashed linesrepresent a numerical, trial and correction, fitting to (2.1).

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CHAPTER 6

Figure 6.3 – Reverse bias dark current versus voltage characteristics (solid lines) and photo-current versus bias voltage (open circles) taken at 325 nm.

119

CHAPTER 6

Table 6.2 – Absorption coefficients at selected wavelengths in Silicon.

Wavelength[nm]

α [cm−1]Light reaching i region

[%]

325 1.30× 106 50× 10−9

442 27.7× 103 57532 9.00× 103 84633 4.00× 103 921064 26.86 99.95

6.8 Photo-multiplication

Photo-multiplication was measured at 325, 442, 532, 633 and 1024 nm. Excess

noise was measured as a function of reverse bias at 633 nm and will be discussed

shortly. 325 and 442 nm light was provided by a He-Cd laser, a diode pumped

solid state laser provided 1064 and 532 nm. 633 nm was obtained from a HeNe

laser making it suitable for noise measurement. The laser light is focused to a

spot on the device p+ layer through the optical window. The window dimensions

are considerably greater than the spot diameter. The absorption coefficient of Si

at these wavelengths is given in Table 6.2 [134].

The photo-multiplication versus reverse bias voltage data is shown in Fig. 6.4

and M-1 against reverse bias voltage is shown in Fig. 6.5 which exposes the lower

multiplication values. A non-local model using the parameters in Table 6.3 is used

to generate the lines in these figures. The modelling results are obtained using an

extended version of the RPL model [17], the particular implementation of which

is due to B. K. Ng [18]. This model accounts for photo-generation throughout

the structure, diffusion of photo-generated carriers and surface recombination.

The parameters associated with the generation of the modelling data are given in

Table 6.3. The ionisation coefficients used are due to Rang [90]. The ionisation

threshold energies are due to Tan et al. [94]. The hole diffusion length is after

Tyagi and Van Overstraeten [135]. The electron diffusion length is taken from del

Alamo and Swanson [136]. The model makes use of a flat electric field approx-

imation, which requires slight adjustment of the intrinsic region width to yield

the correct breakdown voltage. Not withstanding this adjustment, the modelling

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CHAPTER 6

Table 6.3 – RPL modelling parameters.

Parameter Value

Electron Ionisation coefficient (α)1.286×

106(

−1.4×106

ξ

)

cm−1

Hole Ionisation coefficient (β)1.438×

106(

−2.02×106

ξ

)

cm−1

Electron threshold energy (Ethe) 1.7 eV

Hole threshold energy (Ethh) 2.4 eV

Diffusion Voltage (Vd) 1 V

Top Cladding

Minority carrier diffusion length (Le) 1.8 µm

Mobility (µe) 800 cm2

V s

p+ cladding thickness 0.2 µmRecombination Velocity (S) 1200000 cm

s

Back Cladding

Minority carrier diffusion length (Lh) 1 µm

Mobility (µh) 100 cm2

V s

n+ cladding thickness 99.5 µmRecombination Velocity (S) 1200000 cm

s

data fits well in all layers except layer #8 when the photo-multiplication is less

than two, where the RPL model overestimates the photo-multiplication slightly.

The likely cause of this error is that the very small intrinsic region distance in

layer #8 requires a proportionally larger adjustment to fit the correct break-

down voltage in the flat field approximation. Despite the very high doping in the

cladding layers, the electric field must extend a (relatively) large distance into the

cladding layer compared to the intrinsic width. This yields a more trapezoidal

electric field profile than in the other layers which also have very high doping

in the cladding layers, but also have a much thicker intrinsic width, making the

error proportionally smaller.

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CHAPTER 6

Figure 6.4 – Photo-multiplication, M as a function of reverse bias voltage, V, for each wafer;obtained from the 10 MHz measurement system. Open symbols represent independent mea-surement datasets where the symbol colour represents 325 (blue), 442 (cyan), 532 (green), 633(red), 1064 nm (dark grey). Black lines represent non-local modelling data for 325 (solid),442 (long dashed), 532 (medium dashed), 633 (short dashed), 1064 nm (dotted) and pure holeinjection (dash – dot) using doping and intrinsic region widths obtained from 1-D Poisson fit-ted capacitance voltage characteristics then adjusted to provide the correct breakdown voltageusing a flat electric field approximation.

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CHAPTER 6

Figure 6.5 – Photo-multiplication, M-1 as a function of reverse bias voltage, V, for eachwafer; obtained from the 10 MHz measurement system. Open symbols represent independentmeasurement datasets where the symbol colour represents 325 (blue), 442 (cyan), 532 (green),633 (red), 1064 nm (dark grey). Black lines represent non-local modelling data for 325 (solid),442 (long dashed), 532 (medium dashed), 633 (short dashed), 1064 nm (dotted) and pure holeinjection (dash – dot) using doping and intrinsic region widths obtained from 1-D Poisson fittedcapacitance voltage characteristics then adjusted to provide the correct breakdown voltage usinga flat electric field approximation.

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CHAPTER 6

6.9 Excess Noise Measurement

For the 633 nm case, 92% of the light reaches the depletion region through the 0.2

µm p+ cap layer. The excess noise results obtained from the 10 MHz measurement

system, described in Chapter 3, are shown in Fig. 6.6. The same flat electric

field model is used to generate the coloured lines in this figure as for the photo-

multiplication data.

With the exception of #8, the agreement is good at lower multiplications in all

cases. The curvature of the noise data which is particularly apparent in #11 is due

to the falling dynamic resistance of the diode as voltage and, especially, current

increases. For the 10 MHz system the limit of measurement is approximately

rd = 200 Ω. In devices having low dynamic resistance and high series resistance

the limit is considerably increased as the device series resistance is effectively

added to the measurement system input impedance. For the red triangles in #11

at 6.3 V reverse bias rd = 236 Ω. All data after this voltage could, legitimately,

be discarded. To avoid this effect a lower primary photo-current may be used,

however this worsens the NSNR increasing the measurement uncertainty.

In #8 the data is in plausible agreement, if a little high. Measurement of

this layer is particularly taxing, as several factors that deteriorate the measure-

ment accuracy are present. The device has particularly high capacitance. This

increases the noise floor of the measurement system worsening the (already neg-

ative) NSNR. To improve the NSNR, a larger primary photo-current is desirable.

Since there is little multiplication available in this layer, a large primary photo-

current is generally desirable but, the increased photo-current reduces the device

dynamic resistance. The dynamic resistance is further reduced by the very large

tunnelling current which flows as ‘breakdown’ is approached. In spite of the

taxing measurement the #8 data is in plausible agreement with the modelling.

The excess noise results obtained from the 1 MHz measurement system, de-

scribed in Chapter 5, are shown in Fig. 6.7. These data are in good agreement

with the 10 MHz excess noise data. Confirmation of the data with two separate

measurement systems and their general agreement with the non-local modelling

forms a robust argument in favour of the accuracy of the data. The agreement

between the 10 MHz and 1 MHz data shows no bandwidth limiting or 1/f effects

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CHAPTER 6

in the devices.

Figure 6.6 – Excess noise factor versus photo-multiplication at 10 MHz. Closed symbolsrepresent independent datasets at 633 nm. Coloured lines represent non-local modelling datafor 325 (blue, solid), 442 (cyan, long dashed), 532 (green, medium dashed), 633 (red, shortdashed), 1064 nm (dark grey, dotted) and pure hole injection (black, dash – dot) using dopingand intrinsic region widths obtained from 1-D Poisson fitted capacitance voltage characteristicsthen adjusted to provide the correct breakdown voltage using a flat electric field approximation.Short dashed grey lines represent McIntyre’s lines of constant. k = 0 to 1 in 0.1 steps.

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CHAPTER 6

Figure 6.7 – Excess noise factor versus photo-multiplication at 1 MHz. #8 red closed symbolsrepresent the lower measurement bound, dark red closed symbols represent the upper measure-ment bound. In all other graphs closed symbols represent independent datasets. Red closedsymbols represent 633 nm measurements, green closed symbols represent 543 nm measurements.Coloured lines represent non-local modelling data for 325 (blue, solid), 442 (cyan, long dashed),532 (green, medium dashed), 633 (red, short dashed), 1064 nm (dark grey, dotted) and pure holeinjection (black, dash – dot). Short dashed grey lines represent McIntyre’s lines of constant. k= 0 to 1 in 0.1 steps.

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CHAPTER 6

Figure 6.8 – Extracted α (left) and β (right) using local model, flat field approximation.

6.10 Local Ionisation Coefficient Comparison

To assess the effect of the dead-space in this set of devices, the method after

Stillman and Wolfe is used [137]. Ionisation coefficients are extracted from these

devices assuming that the dead-space is negligible and the local model applies.

A flat electric field profile is also assumed so that (1.9) and (1.16) can be used in

conjunction with,

ξ =Vt + Vd

w(6.4)

where Vt is the terminal voltage, Vd is the diffusion voltage and w is the intrinsic

region width obtained from CV fitting.

Figure 6.8 shows the extracted α and β respectively for all layers. The bulk

coefficients of Van Overstraeten and De Man [85] are shown to agree well with

the extracted coefficients as intrinsic width increases.

6.11 Conclusion

A systematic empirical investigation of the effect of dead-space in Silicon has

been reported in this chapter. Measurements of photo-multiplication and excess

noise factor on a series of five Silicon pin photo-diode structures including, the

thinnest intrinsic width, widest area, APD structure reported to date. These

results have been confirmed using the two new measurement systems reported in

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CHAPTER 6

Chapters 3 and 5. It has been shown that the RPL model is sufficiently accurate

to predict the effective reduction in ionisation coefficients over a wide range of

nominal intrinsic region width. From this work it is clear that the dead-space

starts to affect the device characteristics when the region width is approximately

0.4 µm.

The intrinsic width of the thinnest structure in this work is similar in di-

mension to present FET gate lengths in VLSI designs and microprocessors. In

circumstances where impact ionisation is a possibility, the use of one or more thin

regions, if they can be accommodated by the design, allows much higher electric

fields to be developed in those regions while the ionisation coefficients remain

considerably below the bulk value. This effectively increases device speed and re-

duces power dissipation (assuming no significant tunnelling or carrier feedback)

when compared to a thicker region in an otherwise similar device. For example in

a 71 nm long MOS transistor channel, sustaining an electric field of 0.25 MV/cm,

α is reduced by a factor of 45 over the bulk value that a 400 nm channel length

transistor experiences. β is negligible, so carrier feedback is unlikely to have a

significant effect.

An area of present interest is the use of optical on-chip interconnects as a

replacement for electrical interconnects. Used in this way, optical interconnects

offer several reported advantages over electrical interconnects [96, 112]; the most

important arguably being higher bandwidth and lack of crosstalk. Several ap-

proaches for detection of light in on-chip optical interconnects exist, including

the use of hetro-structures of Ge grown on Si, and separately fabricated detectors

bonded to Si wafers [104–106, 108–111, 113–115]. The use of Si as an absorption

and multiplication region [118] has also been evaluated. It is expected that thin

(high speed, low noise) small area Si avalanche photo-diodes may be integrated

into VLSI designs at the transistor level, where Silicon will form either the en-

tire device, or the multiplication region of a device possessing a separate NIR

absorber, Ge or InGaAS for example.

128

Chapter 7

Excess Noise Measurements and

Impact Ionisation Coefficients in

4H-SiC

This chapter reports multiplication and excess noise factor measurements and

local modelling results in a thick (2.7 µm) Silicon Carbide nip APD and a thick

(0.57 µm) SAM-APD. Multiplication and excess noise data has been gathered for

optical injection at 244 and 325 nm, where 244 nm provides pure hole injection

in the thicker device. Ionisation coefficient extraction is performed using a local

model to verify experimentally the ionisation coefficient at lower electric fields.

The data have been gathered using the measurement system first reported by Li

and described in Chapter 2.

7.1 Introduction

4H Silicon Carbide is a wide, indirect band gap (3.23 eV) [138] visible blind,

radiation tolerant [139] semiconductor. It is chemically quite inert, pressure tol-

erant, has a large bulk breakdown field, and is tolerant of high power densities. It

also has a high carrier saturation velocity ∼ 8.0 x 106 cms−1 for 8 off the c-axis

towards <1120> [140]. The military and civilian potential of Silicon Carbide

detectors has been noted [141,142] in the following areas: Missile/rocket exhaust

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CHAPTER 7

plume detection, tracking and chemical composition analysis [143], chemical and

biological battlefield reagent detectors, medical fluorescence imaging, fire/smoke

detection, furnace control, water purification, internal combustion engine moni-

toring, UV radiation dose monitoring, pollution monitoring, short range non line

of sight communications, autocorrelation of pulse lasers and covert space-to-space

communications. SiC is presently a multi-billion dollar industry in the combined

areas of: High power microwave transistors, power semiconductor rectifiers and

switches, high temperature linear devices, super bright blue and green LEDs,

blue and UV laser diodes (as a substrate material) and synthetic gemstones. SiC

growth and fabrication is considerably more mature than GaN. SiC power de-

vices have been demonstrated at temperatures exceeding 600C. NASA Glenn

Research Center has published several papers regarding characterisation of SiC

high voltage, high power, devices at 500C for up to 2000 hours [144, 145].

7.1.1 Bulk Growth

For Silicon Carbide semiconductor manufacturing to flourish, a high level of inte-

gration with pre-existing Silicon device production technology is required. This

implies the growth of large area wafers. A modification to the Lely process was

developed and reported by Tairov and Tsvetkov [146,147]. A seed crystal is held

in a cavity, polycrystalline SiC is vaporised at 2400C and then is allowed to

condense onto the seed crystal. The majority of SiC growth is done on or up to

8 off the c-axis <0001>. Other growth directions have been attempted. The

bulk growth rate is in the order of a few millimetres per hour.

7.1.2 Epitaxy

Chemical Vapour Deposition (CVD) is a favourable process for SiC epitaxy. It

requires little adaptation from Si to SiC processes. Growth rates of several mi-

crons per hour are reported for a temperature of 2000C [141,148]. The addition

of HCL to the process, and the use of ‘warm wall’ CVD reactors can increase

the growth rate to a few hundred microns per hour [149]. This makes the epi-

taxial growth of substrates financially viable; which is attractive because defects

in the substrate can cause defects in the epitaxial layers, thereby affecting device

performance.

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Micropipes were, until recently, a significant problem. A micropipe is thought

to develop when several screw dislocations occur in close proximity, if the Burgers

vector is sufficiently high, the volume enclosed by the dislocations will be empty,

hence a pipe [150]. The diameter is in the region of a micrometer. The epitaxial

layers are of considerably higher quality, in terms of defects/cm2, than substrates.

Electronic devices are manufactured by standard methods from the epitaxial lay-

ers i.e. photo-lithography, dry etching and evaporation of metal contacts [151].

Wet etching is troublesome because SiC is chemically quite inert.

7.1.3 Doping

Doping of semiconductors is often achieved by diffusion of dopants. SiC dopants

tend not to diffuse easily below 1800C [142], which is favourable for high tem-

perature operation but a hindrance for device production. Wide bandgap semi-

conductors can be difficult to dope. This is due to the large ionisation energies,

and compensation effects of most substitutional impurities. The main n and p-

type dopants in SiC are nitrogen and aluminium respectively. Nitrogen creates

a relatively shallow donor level in the band-gap (50 ± 20 meV) [152], and is an

excellent donor atom. However aluminium creates a deeper acceptor level (220

± 20 meV) [152]. Aluminium doping provides acceptable p type layers. A wide

doping range is possible for both n and p type layers. Doping of a whole epi-

layer is achieved by adding dopants during CVD. Commercial layers produced

by CVD are available with doping in the range 9 x 1014 to 1 x 1019 cm−3. It is

not possible to selectively dope parts of the substrate during growth. Therefore,

ion implantation is sometimes used, which causes damage to the crystal struc-

ture. The damage is partially repaired by high temperature annealing. When the

temperature used for the implantation is high the lattice can self heal to some

degree. Ion implantation is only possible to a depth of ∼1 µm. Selective doping

by ion implantation presently produces inferior quality layers.

7.1.4 Commercial Wafers

In 1989 Cree Inc. announced the commercialisation of 2.5” SiC substrates.

150 mm SiC wafers are presently available with epitaxial layers laid down to

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the purchasers’ specification. In May 2007 Cree announced the commercial pro-

duction of 4” wafers with zero micropipes [153]. Other manufacturers include

Dow Corning (formally Sterling Semiconductor), II-VI, SiCrystal and Nippon

Steel, SemiSouth and United Silicon Carbide.

7.1.5 Application: Covert NLOS Communications

Non line of sight communication schemes have been considered for at least forty

years [154]. Some advantages are,

• Immunity to RF propagation effects which hinder or preclude the use of

short range RF links in built up areas.

• A very large spectrum is available, without licensing requirements.

• The possibility of considerably higher channel capacity than presently used

RF systems.

• Undetectable at ‘stand off’ ranges, due to atmospheric absorption.

• Negligible solar background in the UV-C band (200 nm – 280 nm).

• Low susceptibility to ‘jamming’.

Figure 7.1 shows a schematic of an example channel geometry. The principle of

operation is as follows: a transmitter, sighted at distance from a receiver, emits

UV radiation into the atmosphere. Some of this radiation interacts with the

molecules and aerosols in the atmosphere, by Rayleigh scattering and absorption

respectively. A portion of the scattered radiation falls onto the receiving device

and is converted into an electrical signal. A ‘single scatter model’, such as the

one proposed by Luettgen [155], limits a photon to one scattering event, during

its transit from transmitter to receiver.

The number of scattering events encountered by a photon is not limited.

However the simplifying assumption, that each photon is only scattered once, is

valid when the product of the transmitter receiver distance, and the scattering

coefficient, ks, is less than or equal to 0.1 [156]. The scattering coefficient is a

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X

ZF1

ReceiverF2

Transmitter

βRθR βT

θR

ξmin

ξmaxScatteringVolume

Figure 7.1 – A schematic of an NLOS transmission channel, redrawn from [155].

weak function of wavelength. Taking 274 nm as an example ks = 5 x 10−4 m−1,

the valid range of the model is 200 meters.

Although several NLOS ‘test beds’ have been reported including [157, 158],

Silicon Carbide has been largely ignored as a detector technology. No practical

work has been reported on NLOS systems using a Silicon Carbide detector; such

work would be of considerable benefit.

7.1.6 Application: Water Purification

Water purification systems based on UV have been in use for at least one hundred

years [159]. Methods of purification may take several forms in addition to UV

irradiation, including application of gaseous chlorine or hypochlorite to the water

supply. UV purification systems based on vacuum device technology are com-

mercially available and are in widespread use [160]. Recently, several researchers

have published test-bed systems in which the UV lamp is replaced by an array of

UV LEDs [161–164]. UV LEDs have the usual advantages that semiconductors

have over vacuum devices, most notably,

• Reduced power consumption / increased efficiency

• Mechanical ruggedness

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• Longer operating life

• Simpler electronics required to drive the device

The ruggedness combined with the long operating life make the LED a par-

ticularly suitable choice for field operation, in developing countries, for example.

It has been estimated that 1.6 million deaths were caused throughout the world

by water-borne diseases in 1999 alone [165]. The application of Silicon Carbide

detectors to the design of these systems is in an equivalent role to the photo-diode

in light output stabilisation of a laser. The photo-diode monitors the UV radi-

ance in the treatment chamber and controls the emitter characteristics – current

is an obvious choice – to produce the desired dose (energy per unit surface area).

It is also possible to prevent water leaving the device untreated in the event of

emitter failure. In standard UV lamp systems a photo-diode may be used to

similar effect, however because the lamp life is generally short – they are replaced

yearly – there may be no monitoring of the lamp output at all.

7.1.7 Application: SiC Power Electronic Devices

Silicon Carbide has been lauded for some time as a replacement for Silicon in

power semiconductor applications [166]. SemiSouth has significant interests in the

power SiC market and has obtained at least eleven patents (for example [167]) for

device designs and processing techniques during the last decade. These have been

awarded in areas related to the production of SiC SBDs and JEFTs. Recently -

February 2011 [168,169] - SemiSouth began shipping normally-off power JFETs,

which are capable of greater power dissipation and higher operating temperatures

than a similarly sized Silicon MOSFET. However the potential of SiC is yet to be

fully exploited. Close inspection of the published characteristics [170] shows that

the material is imperfect, the drain - source leakage currents, especially at elevated

temperatures is significant ∼ 200 µA max at 1.7 kV. Furthermore the trench

JFET design appears to make heat removal from the device problematic. In the

high blocking voltage design TJC = 2.6C/W. The 1.2 kV blocking voltage design

is similarly specified to a Si Power MOSFET. The device package – TO247 – is

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not a limiting factor of heat removal in this design at present1, lending weight to

the argument that there is still room for improvement in the commercial devices.

Cree are noted as field leaders in the production of SiC wafers, optoelectronic

emitters based on SiC substrates and power and RF devices. Cree have pro-

duced zero micropipe 4” wafers since 2007 [153], SemiSouth presently market

a ‘premium’ grade 3” wafer with 15 micropipes / cm2 [171]. Cree also has in-

terests in RF and Power SiC products. They presently produce more than 20

SBDs for power electronics applications. A recent - January 2011 - addition to

the Cree product line is an enhancement mode SiC MOSFET [172]. Some of

the early power devices made in SiC have been Trench JFETs [173] however the

appearance of a commercial MOS transistor not surprising when it is realised

that Cree patented a method of establishing an oxide layer on Silicon Carbide

in 2005 [174] and several methods for forming a MOS device in SiC, the most

relevant of which appears to be [175]. The Cree device is generally of higher

specification than the SemiSouth devices. Transconductance is similar to Silicon

MOSFETs of lower specifications so large (>20V) – by Si standards – gate drive

voltages are required, the gate charge is small by Si standards however. The

transconductance is approximately two orders of magnitude less than a Si BJT

at a collector / channel current of 20 A. Several reports of SiC BJTs exist in the

academic literature [176, 177].

7.2 Review of Impact Ionisation in Silicon Car-

bide

Work on impact ionisation in Silicon Carbide was initially hampered by material

problems [178–180]. New device fabrication techniques have been developed to

permit the manufacture of devices which are capable of bulk breakdown for exam-

ple, positive edge termination [181], the use of a guard ring [182] and multi-step

junction edge termination [183]. It has been demonstrated that micropipes and

dislocations can adversely affect the electrical properties of devices [184, 185].

The earliest measurements on 4H-SiC, conducted on devices exhibiting uni-

1A sample of package vendors give the minimum TJC for the T0247 package as 0.4C/W

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form breakdown characteristics was reported by Konstantinov [186, 187]. Mul-

tiplication characteristics of three thick n−p+ mesa diodes with doping varying

from 5.2 x 1016 cm−3 to 1.2 x 1018 cm−3 were reported. DC photo multiplication

was measured with the device excited by 325 nm light. The ionisation coefficients

were obtained in the form proposed by Thornber [188] by fitting a local model

to each of the multiplication characteristics. Konstantiov et al. showed that the

impact ionisation in holes is considerably greater than for electrons.

A pulsed electron beam induced current and phase sensitive detection was

used by Raghunathan et al. [189, 189] to obtain a value for beta over a range

of temperatures from Shottky barrier diodes. Raghunathan et al. assumed α is

negligible. In the range of electric fields close to breakdown carrier feedback is an

essential process, this is evidenced by the super–linear increase in multiplication

with bias close to breakdown.

Yan et al. [190] reported noise measurements in which a p+pnn+ reach through

photo-diode was excited with 325 nm yielding k = 0.1. Ng et al. [191] reported

short (230 nm) and long (365 nm) wavelength multiplication and excess noise

results on two thin devices (w of 0.105 µm and 0.285 µm). The light was injected

from the p side of both samples. The ionisation coefficients were obtained by fit-

ting the data to a non-local RPL model [18]. Non uniform optical generation and

non-uniform electric fields were accommodated in this model. Agreement with

the experimental data was found by using electron and hole ionisation thresh-

old energies of 12 eV and 8 eV respectively. Multiplication increased and excess

noise decreased as the wavelength increased and carrier injection became more

hole dominated. The lowest excess noise of k = 0.1 and k = 0.15 for device

widths of 0.105 µm and 0.285 µm respectively was observed at 365 nm.

Hatakeyama et al. [192] reported the anisotropy of the impact ionisation co-

efficients on the <0001> and <1120 > faces. They used CW optical excitation

at 350 nm and so measured DC photo-multiplication. The impact ionisation co-

efficients were established by fitting the multiplication verses bias voltage, and

breakdown voltage versus doping density.

Guo et al. [193] measured a n+n−n−p+ SAM-APD at 351/363 nm and 270 nm

achieving a k of 0.2 and 0.3 respectively. The device thickness in this case was

120 nm. Guo et al [194] also reported a p+ppn+ SAM-APD having an intrinsic

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region of 180 nm, which was measured using 363 nm light. The authors report k

of 0.12 for this device.

Loh et al. [195] reported photo-multiplication measurements and impact ion-

isation coefficients based on three structures including one sample which is used

in this work. The measurements were carried out using 244 nm and 325 nm light.

A local model was used to extract the ionisation coefficients from multiplication

versus bias data. By accounting for the mixed injection of the 325 nm light in

several nip structures The authors were able to estimate the aggregate value of

α as approximately twenty five times less than β, k = 0.04.

The temperature dependence of the impact ionisation mechanism in 4H-SiC

has been investigated by Konstantinov et al. [187]. They show that α increases

whilst β decreases with temperature. A similar effect is seen in the 6H polytype.

The movement of the breakdown voltage as a function of temperature in 4H-

SiC is determined by holes, not electrons as in 6H-SiC. This is a result of the

greater enhancement of β with temperature than α. The hole impact ionisation

coefficients in 4H-SiC and 6H-SiC SBDs were obtained by Raghunathan et al.

[196] for temperatures over the range 300 to 450 K using their P-EBIC technique.

The positive temperature coefficient of the breakdown voltage in 4H-SiC has been

confirmed by the work of Neudeck et al. [197] Lee et al. [198], Ozpineci et al. [199]

and Vassilevski [140]. The breakdown voltage of 4H-SiC increases linearly at a

rate of 16 mV/C up to 257C in Yan et al.’s work [200] and at 10 mV/C up to

200C in Guo et al.’s work [193]. This suggests that 4H-SiC has thermally stable

avalanche breakdown behaviour.

Despite the wide variation in the published impact ionisation coefficients in

4H-SiC as shown in Fig. 7.21, all the authors have demonstrated that β is much

greater than α especially at low fields. Konstantinov et al. [186, 187] have sug-

gested that the asymmetry in the impact ionisation coefficients, is a property

of devices in which the electric field is parallel to the c-axis direction (E‖c).Hatakeyama et al. [192] measured breakdown voltage in a <1120> face p+n diode

that was 60% of that on a <0001> face p+n diode confirming the large anisotropy

of the extracted impact ionisation coefficients. Anomalous breakdown fields were

also demonstrated by Nakamura et al. [201]. The breakdown fields along the

<1120> and <0338> plane were about 75% of that measured in the <0001>

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plane for 4H-SiC. The breakdown field exhibits a positive temperature coefficient

of 200 V/cmK, 600 V/cmK and 400 V/cmK for <0001>, <1120> and <0338>

plane diodes, respectively. Tanaka et al. [202] have observed larger leakage cur-

rents and lower breakdown voltages along the <1120> direction compared with

<0001> direction.

The anisotropy impact ionisation in 4H-SiC has been ascribed to the con-

duction band splitting into a number of mini bands. This occurs only along the

<0001> direction (c-axis) [203]. A second process thought to be responsible is

the high energy loss of hot electrons due to the Bragg reflections from the zone

edges inherent to non-cubic SiC. The splitting between the first and second mini-

conduction bands is sufficiently large to assume very few electrons will occupy the

higher mini-band. The electrons in the first mini-band can only initiate impact

ionisation by means of scattering by phonons or crystal imperfections [186, 187].

The impact ionisation initiated by electrons along the <0001> direction is greatly

suppressed, leading to a much larger breakdown field along <0001> in 4H-SiC.

A theoretical study of β in 4H-SiC was performed by Bellotti et al. [204]

in which a full band Monte Carlo simulation was reported. Their simulation

showed a strong anisotropy in hole ionisation. The range of values of β in the

direction perpendicular to c-axis (E⊥c) were found to be about 5 to 10 times

higher than those parallel to c-axis (E‖c) for particular field. This is a consequenceof the anisotropy of the Brillouin zone [204]. These differences in β introduce

considerable differences in the transport characteristics in different directions.

Agreement with Raghunathan et al.’s experimental results [196] was achieved

when inter-band tunnelling effects near the crossing and mixing points in the

valence bands was included in the model to account for the transport parallel to

the c-axis. Bertilsson et al. [205] have reported a two-dimensional drift-diffusion

model which was used in conjunction with the anisotropy in the hole ionisation

coefficients reported by Bellotti et al. [204] to explain the discrepancy in the

experimental results of Konstantinov et al. [187] and Raghunathan et al. [196].

The authors also suggest that the anisotropy of β is greater at lower fields and

reduces with increasing electric field strength.

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7.3 Chapter Objective

In this work confirmation of the published ionisation coefficients is sought by

measuring photo multiplication and excess noise on two thick 4H-SiC structures.

There is considerable disparity between the published ionisation coefficients of 4H-

SiC [186, 189, 191, 192, 195, 204].The variation is most pronounced at low electric

fields such as those more typically found in power electronic devices. The diodes

used in this work were diodes were produced primarily for other applications but

were generously provided by The Semiconductor Technology Laboratory which

is part of General Electric Global Research. All of the wafers and epitaxial

layers were grown 8 off the c-axis <0001> using metal-organic chemical vapour

deposition (MoCVD), the layers were purchased from Cree Research (Durham,

NC, USA).

7.4 Structure Details

Two samples are used in this work the schematic cross sectional view of sample

B, a 4H-SiC non-reach-through SAM-APD, is shown in Fig. 7.2. The structure

consists of a 0.2 µm n+ cap layer (ND = 4 x 1018 cm−3), a 1.35 µm n− absorption

layer (ND = 7.5 x 1015 cm−3), a 0.45 nm n+ charge layer (ND = 3 x 1017 cm−3),

a 2.69 µm n− multiplication layer (ND = 7.5 x 1015 cm−3), and a 2 µm p+ layer

(NA = 2 x 1018 cm−3) grown on an n-type substrate. These devices are effectively

0.66 mm x 0.66 mm square mesa diodes, and have a positively bevelled edge. The

diodes were etched using an inductively coupled plasma (ICP) etching process and

Figure 7.2 – Sample B layer details.

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Figure 7.3 – Sample A layer details.

passivated using a thermal SiO2 film with an additional plasma enhanced chemical

vapour deposition SiO2 film on top. Optical access windows were formed in the

anode contact. Cathode contacts were fabricated by sputtering Ni, whilst the

anode contacts were formed by depositing a layered stack of Ti/Al/Ti/Ni. An

earlier version of the fabrication process is described in more detail by Yan et

al. [206]. Similar methods were employed in the production of sample A. A

schematic cross-sectional view of sample A is shown in Fig. 7.3. The structure

is composed of a 0.19 µm n+ cap layer (ND = 3.5 x 1018 cm−3), a 0.66 µm n−

absorption layer (ND = 2 x 1016 cm−3), a 0.197 µm n+ charge layer (ND = 6.7

x 1017 cm−3), an n− 0.582 µm multiplication layer (ND = 5 x 1015 cm−3), a

1.87 µm p+ layer (NA = 2 x 1018 cm−3), a 0.457 µm n+ buffer layer (ND =

1 x 1018 cm−3) grown on an n+ substrate. These devices are fabricated as a

square array of 16 circular mesa diodes. The effective diameter of the junction is

106.6 µm.


Capacitance voltage characteristics were measured for sample A using a HP4275A

LCR meter coupled to a Keithly 237 SMU. Sample B was measured using an

Agilent B1500A semiconductor analyser. The layer widths and doping densities of

each layer were established by fitting capacitance voltage measurements to a one

dimensional Poisson model which is described in Appendix A. The capacitance

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Figure 7.4 – Capacitance voltage fitting of Sample A. Black circles represent measured datawhile the red line is developed from a one dimensional Poisson fitting algorithm.

Figure 7.5 – Capacitance voltage fitting of Sample B. Black circles represent measured datawhile the green line is developed from a one dimensional Poisson fitting algorithm.

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voltage fitting of samples A and B are shown in Figs. 7.4 and 7.5 respectively.

Sample B does not punch through its charge layer from 0 to 600 V bias. Be-

tween 600 V and breakdown at approximately 625 V the collection efficiency of

244 nm does not change by a measurable amount. Sample B is modelled using a

three region approximation. The fitted doping densities and region widths sug-

gest that the electric field will not punch through the charge layer before 772 V,

considerably above breakdown. The n− region is fully depleted by 50 V. An ac-

ceptable fit to the CV curve of sample B is given by assuming an n+n−p+ structure

0.45 nm n+ charge layer (ND = 3 x 1017 cm−3), a 2.6935 µm n− multiplication

layer (ND = 7.5 x 1015 cm−3), and a 2 µm p+ layer (NA = 2 x 1018 cm−3).

In sample A at least four, perhaps five, regions are partially depleted prior

to breakdown at approximately 193 V. Punch through of the multiplication layer

occurs at a few Volts. Depletion of the absorption layer begins around 160 Volts

and appears to be complete by 190 V however fitting doping densities and re-

gion widths under this interpretation leads to an anomalous breakdown voltage

value when using the coefficients of Loh et al. [195] and Ng [191]. Fitting accu-

rately in this high voltage region also leads to doping densities and/or depletion

widths which are incompatible with SIMS data. The high series resistance of this

layer coupled with the small depletion capacitance (∼ 0.5 pF) provides sufficient

uncertainty to overlook the lack of perfect agreement in the CV fitting close to

breakdown. It will be shown shortly that the series resistance in this layer is

approximately 10 kΩ. The overall error of the last few measured data points is

a maximum of 43 fF greater than the modelled result. Errors of this order can

be ascribed to the movement of the probe towards the copper plate which holds

the device after calibration of the CV test setup and similar usually negligible

systematic errors.

The electric field associated with samples A and B at 192.3V and 620V re-

spectively are shown in Figs. 7.6 and 7.7. In these figures the optical generation

and ionisation coefficient profiles are also shown, these are discussed throughout

the chapter.

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Figure 7.6 – Several parameters of Sample A as a function of distance at 192.3 V reverse bias.Electric field (black, solid), electron ionisation coefficient α (red, dash–dot), hole ionisationcoefficient β (violet dash–dot–dot), absorption of 244 nm light shone onto the surface of the n+

capping layer (blue, long dash) and absorption of similarly injected 325 nm light (dark green,medium dash). Light enters the n+ layer from the right of the figure.

Figure 7.7 – Several parameters of Sample B as a function of distance at 620 V reverse bias.Electric field (black), electron ionisation coefficient α (red, dash–dot), hole ionisation coefficientβ (violet dash–dot–dot), absorption of 244 nm light shone onto the surface of the n+ cappinglayer (blue, long dash) and absorption of similarly injected 325 nm light (dark green, mediumdash).

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Figure 7.8 – Sample A forward bias for the sixteen samples in the array (solid lines). Longdashed line shows an ideal diode with ideality factor of two. The medium dashed line shows asimilar diode with Rs = 10 kΩ.


Current voltage (IV) measurements were performed without room lighting (flo-

rescent tubes), however no difference could be found between IV curves taken

under room lighting and those obtained in darkness. Forward IV measurements

were taken using an HP 4140B Picoammeter. Reverse IV measurements on sam-

ple A were taken using an Agilent B1500A. Sample B reverse characteristics were

obtained with a Keithly 237.

Forward bias measurements were undertaken on all devices in both sample

sets; sixteen devices of sample A and twelve devices of sample B. Data relating

to samples A and B are shown in Figs. 7.8 and 7.9. The measurement range

is limited to a maximum current of 100 µA. Currents greater than this value

are known to deleteriously reduce the reliability and lifetime of these devices.1

Because the series resistance of sample B is modest, one device was sacrificed in

order to measure higher currents and approximate the series resistance. Series

resistance in sample A fits approximately 10 kΩ, in sample B, 130 Ω is the ap-

propriate value, see Figs. 7.8, 7.9 and 7.10. The analytic method for modelling

1P. M. Sandvik, private communication 2009. Also see US Patent No. 7,002,156.

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Figure 7.9 – Sample B forward Bias for all samples. Other than the two devices whichwere dead on arrival (pink and dark yellow) the devices show highly consistent forward biascharacteristics. The black line represents an ideality factor of two.

a forward biased diode and series resistance yields a solution which requires a

numerical analysis because it contains a Lambert W function. The two stage

fitting procedure described in Chapter 2 was used to obtain the saturation cur-

rent, Is, emission coefficient, n, and series resistance, Rs. The fitted emission

coefficient of two indicates that majority carrier recombination within the de-

pletion region dominates the transport [179, 207]. It is impossible to establish

the relative distribution of current between bulk and surface recombination in

these layers because all devices are of the same area. The reverse bias current

voltage characteristics of sample A are shown in Fig. 7.12, and are measurement

system limited up to breakdown for twelve devices. All devices show generally

abrupt breakdown characteristics. Current rises from the near the measurement

limit by at least two orders of magnitude over a few volts reverse bias. The

inconsistency in breakdown observed in sample A may be ascribed to the large

and quite variable series resistance of each diode. Diodes with negligible series

resistance fabricated on a localised area of a wafer may be expected to exhibit

identical breakdown behaviour under ideal conditions. Several diodes exhibit real

dark current levels above the measurement floor prior to breakdown (for example

the grey line in Fig. 7.11), this may be ascribed to higher levels of trap assisted

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CHAPTER 7

Figure 7.10 – Sample B high current forward bias characteristic. Black circles represent data,red line represents an ideal diode with 130 Ω series resistance.

thermal generation in these devices than in those surrounding them.

The reverse bias current voltage characteristic of sample B is shown in Fig. 7.12,

in which two 244 nm photo-current plots are also presented. The dark current

is measurement limited up to breakdown. The true dark current is much lower

than this measurement suggests. The curve shown does not appear to exhibit

impact ionisation, unless the impact ionisation is purely electron initiated, in

which case it would not be apparent until very close to breakdown. However it

has been shown by several workers including Konstantinov [186,187], Ng [18,191]

and Loh [195] that β > α in 4H-SiC hence a generally hole dominated dark current

multiplication curve may be expected. The two pure hole injection photo-current

plots show the likely shape of the real dark current prior to breakdown. Measure-

ments taken with a HP 4140B Picoammeter up to 50 Volts reverse bias which is

the maximum for this measurement equipment, show dark currents near to the

measurement limit of 1 pA for most devices. It is possible that this too is an

overestimation of the dark current. Generally extremely low dark currents are

expected in SiC due to the very low intrinsic carrier density (∼ 10−9 cm−3 [152])

which is a function of band gap. The ideal generation current would be negligible.

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CHAPTER 7

Figure 7.11 – Sample A reverse current voltage characteristics for all 16 devices.

Figure 7.12 – Sample B reverse current voltage characteristics for one device (black, solid).Photo-current due to 244 nm injection is shown for two optical intensities (red, long dashand blue, medium dash). The optical intensity of the blue line is approximately an order ofmagnitude greater than the red line.

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CHAPTER 7

Figure 7.13 – Light absorption as a function of wavelength [208,209] for 4H-SiC (black circles),6H-SiC (blue circles) and the extrapolation due to Ng [18].

7.7 Photo-Multiplication Characteristics

The multiplication and excess noise characteristics were measured simultaneously

using the measurement system after Li et al. [27] which is described in Chapter 2.

The multiplication is measured by an AC (LIA) measurement of photo-current

passed through a transimpedance amplifier. The uniformity of several diodes

were checked prior to multiplication and noise measurements. This is accom-

plished by scanning the devices with 325 nm light with a low reverse bias (-5V)

applied. The spot size is focused such that it is significantly smaller than the

device area. Under these conditions a uniform device presents a ‘flat top’ re-

sponse. The method is similar to that described by Guo et al. [193]. Uniformity

between devices is confirmed by the grouping of the excess noise measurements.

Excess noise is approximately proportional to M2 or M3 when β = α [28]. Mul-

tiplication differences due to non-uniformity may be expected to exhibit a strong

spread in excess noise results. However, the spread in excess noise is small for

several devices for multiplication values up to approximately two hundred and

fifty. A typical reverse bias photo-current and dark current curve of sample B

is shown in Fig. 7.12. The photo-current is several orders of magnitude greater

than the measured dark current, the multiplication due to the two levels of light

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Figure 7.14 – Relative light absorption in 4H-SiC at several wavelengths. Black – 230 nm,red – 244 nm, blue – 280 nm, pink – 325 nm, green – 360 nm.

intensity shown are similar for a prescribed reverse bias voltage. This implies that

heating and space charge effects have not influenced the results. The multiplica-

tion characteristics of samples A and B are shown in Figs. 7.16 and 7.18. The

multiplication factor increases substantially as breakdown is approached. The

breakdown voltage is fitted using Miller’s empirical expression [5],

M =1

1−(

VVbd

)n (7.1)

where M is the multiplication at a reverse bias voltage V . Vbd is the breakdown

voltage and n is a fitting parameter which is proportional to the ‘sharpness’ of

the breakdown. The breakdown voltage of Sample A is 193.78 V and sample B is

624.60 V. The measured breakdown voltage of sample A is hampered by the large

series resistance, however agreement is adequate. The agreement for sample B is

good. In this work the 4H-SiC absorption coefficients of Sridhara et al. [208,209]

are used and the extrapolation due to Ng [18] is also employed. Both are shown

with 6H-SiC data in Fig. 7.13. In sample A approximately 97% of 244 nm light

generates e-h pairs between the window and the high field region contributing to

pure hole injection. Practically all of the remaining 3% generates carriers inside

the depletion region contributing a mixed injection component, the proportion of

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CHAPTER 7

Figure 7.15 – Sample A multiplication characteristics as a function of reverse bias for 244 nm(circles) and 325 nm (squares). The black solid line represents pure hole multiplication, blue– 244 nm mixed injection, green – 325 nm mixed injection and red represents pure electroninjection.

244 nm generating carriers in the p+ layer is negligible. At 325 nm the proportion

of light generating carriers prior to the electric field is approximately 12%, a

further 9% generates carriers inside the depletion region. At the end of the p+

layer nearly 38% of all the incident light will have been absorbed. Under 325 nm

injection the primary carrier conditions are split quite evenly between generation

in the high field region and diffusion/drift of electrons and holes into the high field

regions from the p+ layer and n layers respectively. In performing this calculation

it has been assumed that carriers generated within the n+ capping layer, the

absorption region and half of the charge layer contribute to pure hole injection.

This is plausible because the electrons which are produced in the generation of

carriers in this region have practically no chance of ionising the lattice prior to

leaving the electric field region because they will not experience the high field

of the multiplication layer. Holes generated in the n+ cap, absorber and half of

the charge layer will travel towards the multiplication region and so contribute to

pure hole injection. Despite the relatively even injection conditions in this device

is clear from the measured data that holes dominate the ionisation process in this

device at 244 nm and at 325 nm. In sample B at 244 nm >99.9% of the light is

absorbed in the cladding layers outside the electric field. The remaining <0.1%

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CHAPTER 7

Figure 7.16 – Sample A multiplication characteristics as a function of reverse bias for 244 nm(circles) and 325 nm (squares). The black solid line represents pure hole multiplication, blue– 244 nm mixed injection, green – 325 nm mixed injection and red represents pure electroninjection.

is absorbed within the depletion region. This device may be assumed to exhibit

pure hole injection at 244 nm. At 325 nm carriers are generated throughout the

structure. In the n+ cladding layers approximately 23% of the light is absorbed,

in the depletion region approximately 23% is absorbed. By the end of the p+

layer around 60% of the 325 nm light is absorbed.

Multiplication data at 244 nm and 325 nm for samples A and B versus reverse

bias voltage are shown in Figs. 7.18 and 7.16 where M − 1 is plotted to expose

the low multiplication values. Figs. 7.17 and 7.15 show the multiplication versus

reverse bias in the region near to breakdown.

While mixed injection prevails, the difference in multiplication characteristics

under short and long wavelength injection conditions is not very great. This is

due to the overwhelming propensity of holes to ionise the lattice compared to

electrons. Even a small quantity of holes ionising the lattice can produce long

ionisation chains which cause electron dominated optical generation conditions

to produce hole dominated multiplication characteristics. This is demonstrated

by the positioning of the 325 nm curve close to the pure injection 244 nm curve

compared to the predicted pure electron curve in both devices.

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CHAPTER 7

Figure 7.17 – Sample B multiplication characteristics as a function of reverse bias for 244 nm(circles) and 325 nm (squares). The black solid line represents pure hole multiplication, blue– 244 nm mixed injection, green – 325 nm mixed injection and red represents pure electroninjection.

Figure 7.18 – Sample B multiplication characteristics as a function of reverse bias for 244 nm(circles) and 325 nm (squares). The black solid line represents pure hole multiplication, blue– 244 nm mixed injection, green – 325 nm mixed injection and red represents pure electroninjection.

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CHAPTER 7

7.8 Excess Noise Characteristics

The excess noise characteristics are of considerable interest primarily because

pure injection data is available for the first time, secondly because sample B

is relatively thick compared to the majority of devices for which excess noise

results have been published. The thickest device published thus far is by Ng et

al. [18, 191], the intrinsic width is 285 nm; sample B is nearly ten times thicker.

Figs. 7.19 and 7.20 show excess noise characteristics produced by injection of

244 nm and 325 nm light into samples A and B respectively. The excess noise

data corresponds to the multiplication data shown in Figs. 7.15, 7.16, 7.17 and

7.18. Due to the large series resistance of sample A excess noise cannot be ac-

curately measured while the multiplied photo-current causes the series resistance

to significantly affect the device multiplication characteristics. Good agreement

can be seen between the data taken for both wavelengths especially when the

low quantity of noise is considered. In sample A 244 nm injection corresponds

to McIntyre’s effective k of 0.1, and 325 nm injection corresponds to 0.45. In

sample B 244 nm injection corresponds to McIntyre’s effective k of 0.007 while

325 nm injection corresponds to 0.045. The 244 nm injection excess noise data

for sample B falls below F = 2 in some cases. This is especially noted in the

black circles and green circles where higher starting photo-current was used to

ensure an accurate measurement of noise. It is possible that there is a slight cal-

ibration error, F axis offset is a function of the disparity between the calibration

shot noise and the APD noise either due to uncertainty in the calibration noise

or APD noise measurement or to uncertainty in the bandwidth calibration of the

measurement system for the calibration device capacitance or the APD capaci-

tance. To mitigate this as far as possible sample B was tested at unity gain with

varying optical power in order to produce a shot noise calibration. This calibra-

tion was compared and found to be very nearly identical to a UV enhanced Silicon

reference diode (Perkin Elmer UV–BQ–40) when adjusted for capacitance effects.

Consequently sample B shot noise data was used as the calibration for Figs. 7.19

and 7.20, and as a result no bandwidth corrections were necessary. Despite this

careful checking, a slight error remains. It may be suggested that, because α is

small, there must be many hole ionisations in each transit of the device, if the

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CHAPTER 7

Figure 7.19 – Sample A excess noise characteristics as a function of multiplication. Circlesrepresent 244 nm data, squares – 325 nm. Black solid line – pure hole injection, blue solid line– 244 nm mixed injection, green medium dashed line – 325 nm mixed injection and the red longdashed line represents pure electron injection. Grey lines represent McIntyre’s lines of constanteffective k from 0 to 0.5 in 0.05 steps.

Figure 7.20 – Sample B excess noise characteristics as a function of multiplication. Circlesrepresent 244 nm data, squares – 325 nm. Black solid line – pure hole injected, blue solid line– 244 nm mixed injection, green medium dashed line – 325 nm mixed injection and the redlong dashed line represents pure electron injection. The grey lines represent McIntyre’s lines ofconstant effective k from 0 to 0.1 in 0.01 steps.

154

CHAPTER 7

number of ionisations required per transit forces each carrier to ionise directly

after the expiration of the dead space some grouping of carriers would result and

F < 2 is possible. This effect is unlikely to be seen at higher multiplication be-

cause α and β tend to converge as electric field increases. The increasing carrier

feedback would permit long hole ionisation chains to begin at random positions

in the lattice due to ionisation of the lattice caused by relatively few electrons,

thus breaking up the grouping of carriers. It is just as likely, however, looking

at the red circles, that the true curve falls on F = 2 at multiplications less than

forty and that the slightly low result is a systematic error.

7.9 Ionisation Coefficient Extraction

Ionisation coefficients have been extracted from the multiplication and excess

noise data gathered from samples A and B using the local framework proposed

by McIntyre [3]. This extraction was performed partially by extraction from mul-

tiplication and excess noise data and thereafter by trial and correction approach

the main steps of the extraction method are outlined below.

• Using the ionisation coefficients of Ng and Loh, fit the measured breakdown

voltage of a nip / pin (sample B) to that of a device possessing a flat electric

field profile by adjusting the intrinsic region width, w, of the device.

• Using this fitted intrinsic width, extract ionisation coefficients from the data

by use of the expressions given in Section 1.6.

• By trial and correction adjust the coefficients such that they fit the multi-

plication and excess noise data of Sample B using the CV fitted (piecewise

linear) region widths and doping densities.

• Use the extracted coefficients to model Sample A.

• Iteratively adjust the coefficients until both samples A and B fit the multi-

plication and excess noise data within permissible tolerances.

This process is quite slow and the fitting shown in this chapter represents ap-

proximately forty hours work completed over three days. The fitted coefficients

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CHAPTER 7

are shown plotted in Fig. 7.21 with the coefficients of Ng [191] Loh et al. [195]

and Konstantinov et al. [186] Multiplication and excess noise data produced us-

ing these coefficients is shown as lines in Figs. 7.15, 7.16, 7.19, 7.17, 7.18 and

7.20. The ionisation coefficients can be expressed using Chynoweth’s expression.

The ionisation coefficient for electrons, α is separated into a low field and high

field regions to improve the fitting of Chynoweth’s expression to the extracted

coefficients. The low field and high field α expressions overlap across a relatively

broad range of electric filed values around 2.5 MV/cm. The expressions for alpha

and beta are given in (7.2), (7.3) and (7.4).

αl = 1.933× 104 · exp[

−(2.888× 106

ξ

)]4.828

(7.2)

αh = 1.878× 106 · exp[

−(9.134× 106

ξ

)]1.459

(7.3)

β = 6.000× 106 · exp[

−(1.387× 106

ξ

)]0.9600

(7.4)

Figure 7.21 – Ionisation coefficients of 4H-SiC versus inverse electric field published by variousworkers including those extracted in this work. Bellotti et al. [204] (black squares), Konstantinovet al. [186] (black, short dashed lines), Ng et al. [191] (β – dark red triangles, α – dark bluetriangles), Loh et al. [195] (green, long dashed lines) and this work (β – red circles, α – bluecircles).

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CHAPTER 7

7.10 Comparison of GaN and SiC: GaN and Al-

GaN Detectors

Wide band gap semiconductors promise mechanical ruggedness, high breakdown

fields and solar blindness. The main contenders in wide band gap detectors are

summarised in [210]. Gallium Nitride is a wide, direct band gap semiconductor

(3.4 eV), with high breakdown field and high electron mobility. This offers the

possibility of very wide bandwidth devices, with high responsivity (due to the

direct band gap). GaN has excellent mechanical properties and is chemically quite

inert. It is sometimes reported that GaN is intrinsically solar blind. However,

there is significant background radiation at sea level, around 362 nm, which is

the cut off wavelength of GaN. The ternary, AlGaN, can be made solar blind,

however, operating from 200 nm to 362 nm. The cut off wavelength can be

tuned by adjusting the fraction of the ternary alloy. A review of wavelength of

detection with composition is given in [139]. The following table shows a selection

of published work on AlGaN and GaN photo-diodes. The dark current density is

a figure of merit related to the quality of the material.

Dark Current Density Authors

2.33 to 4.65 mA/cm−2 at -150 V K. A. McIntosh et al., 1999 [211]

3.54 µA/cm−2 at -90 V B. Yang et al., 2000 [181]

5.556 mA/cm−2 at -30 V Carrano et al.,2000 [212]

20 mA/cm−2 at -100 V S. Verghese et al., 2001 [213]

230 uA/cm−2 at -5 V R. McClintock et al., 2004 [214]

1 mA/cm−2 at -70 V R. McClintock et al., 2005 [215]

10 µA/cm−2 at -45 V S-C. Shen et al., 2007 [216]

10 µA/cm−2 at -100 V Y. Yoshizumi et al., 2007 [217]

400nA/cm−2 at -15 V H. Jaing et al., 2007 [218]

24 µA/cm−2 at -60 V T.Tut et al., 2007 [219]

35.4 µA/cm−2 at -80 V J.B. Limb et al., 2008 [220]

45 µA/cm−2 at -80 V J. L. Pau et al., 2008 [221]

6.37 µA/cm−2 at -120V T. Tut et al., 2008 [222]

7.87 mA/cm−2 at -100 V Bayram et al., 2008 [223]

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CHAPTER 7

Dark Current Density Authors

88 mA/cm−2 at -80 V Bayram et al., 2008 [224]

9.55 µA/cm−2 at 55 V L. Sun et al., 2010 [225]

133 µA/cm−2 at 100 V E. Cicek et al., 2010 [226]

In comparison 4H-SiC is mature and in high quality devices the dark current

is often hidden below the instrument limit [227], often less than 1 pA until very

close to breakdown. While GaN/AlGaN is presently well suited to production of

optoelectronic emitters it is unsuitable for production of devices which operate

under large electric fields such as APDs. High defect densities caused by the lack

of a native substrate limits the electric field to values below the bulk breakdown

level. At the present time usefulness of GaN/AlGaN as a detector remains limited

to the research environment.

7.11 Comparison of GaN and SiC: GaN Power

Electronics

It is predicted that GaN will eventually outperform Si and SiC power devices

[210, 228]. This is principally due to its larger critical electric field, low intrinsic

carrier concentration and high carrier mobility. However, the breakdown voltage

of physically realisable GaN devices has, thus far, been less than the predicted

value. The realisation of GaN’s bulk properties in working devices have been ham-

pered by the material problems and semiconductor processing [211,212,216,229]

The high defect density and number of dislocations found in the epitaxial layers

grown on lattice-mismatched Sapphire or SiC substrates leads to high leakage

current and premature microplasmic breakdown before the bulk breakdown field

is reached. Dmitriev et al. [229] have reported the breakdown of linearly graded

pn junctions in the electric field range of 1.5 to 3 MV/cm. Their devices have

reverse leakage current of 10 mA/cm−2. The pn junction breakdown exhibited a

positive temperature coefficient, but the breakdown was of a localised microplas-

mic nature due to defects. The progress of commercial GaN power switching

158

CHAPTER 7

devices compared to SiC can be loosely approximated by observing the patents

sought by Cree and their competitors as well as in the academic literature, for

example [230].

Irrespective of which material GaN or SiC wins the high power/high voltage

electronics ‘battle’, in which SiC presently has the upper hand, the two materials

are for the moment at least, bound together. SiC is a favourable substrate mate-

rial for growth of GaN epilayers. Cree, arguably the leader in SiC production, has

significant interests in the production of SiC wafers to manufacture GaN LED

products besides their SiC on SiC wafer/epi-layer production, power and RF de-

vice lines. If the outstanding GaN material problems can be solved it is possible

that the research effort expended on SiC growth will not have been wasted.

7.12 Conclusion

The avalanche multiplication and excess noise characteristics of two thick 4H-SiC

APDs with intrinsic region widths of 0.59 µm and 2.7 µm have been investigated

using 244 nm and 325 nm injection. For the first time pure hole injection multipli-

cation and excess noise data have been obtained in a SiC APD. The excess noise

characteristics of the 244 nm injection correspond to an effective k of 0.007 in the

case of sample B and 0.1 in the case of sample A. This unexpectedly low noise

in thicker devices has been used to extend the range of experimentally verified

ionisation coefficients into the low field region where these devices operate.

A local model has been used in conjunction with a principally trial and correc-

tion fitting approach to extract new ionisation coefficients which, while following

the work of Ng et al. [191] and Loh et al. [195] at high fields, properly reflect the

multiplication and very low noise obtained at lower electric fields.

159

CHAPTER 7

160

Chapter 8

An APD SPICE Model for

Circuit Simulation

This chapter presents a simple SPICE model for use in simulation of detector

circuitry. The model may be used to represent a real or fictitious device and

operates in large signal and small signal simulations. The source code is presented

in Appendix B.

8.1 Introduction

When simulating the electrical characteristics of new transimpedance amplifier

designs or other electro-optical systems it is often assumed that the photo-diode

detector is simply a perfect current generator. It is often desirable to ensure

that certain characteristics of real photo-diodes do not detrimentally influence

the circuit behaviour. In these cases the required parasitic components are often

added to the simulation, as deemed necessary, depending on the situation. A

more universal approach is desirable; to that end a - nearly - universal model of

a linear mode APD has been developed.

The main requirement of this model is to reproduce, in a SPICE simulator,

characteristics of a pre-existing semiconductor detector such that the predicted

performance of a proposed circuit may be evaluated when used with the given

photo-diode. The photo-diode being modelled will have been measured using

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CHAPTER 8

standard characterisation tests including,

• Reverse bias current voltage characteristics (IV)

• Reverse bias capacitance voltage characteristics (CV)

• Reverse bias photo-multiplication (Multiplication)

• Excess noise factor (Noise)

• Responsivity over a range of wavelengths (responsivity)

Not all of these tests are necessary; it is perfectly valid to develop a model of a

fictitious diode. For example the instrumentation designer might like to simulate

how a new measuring system will respond to use with a diode possessing very high

tunnelling currents. In which case, some tunnelling characteristics which expose

the new measuring systems weaknesses may be contrived without reference to

a real photo-diode. In this discussion it is assumed that a real device is being

modelled and that it is desirable to replicate this device’s characteristics with

some accuracy.

The model must be efficient with the simulators resources, as the majority

of the memory and processing power should be available for the simulation of

the circuit not the APD. Several circuit based models are described in the liter-

ature which attempt to represent accurately the behaviour of an APD at high

frequencies [231–238]. This is challenging when – as in SPICE – only lumped cir-

cuit models are available. The number of lumped elements required for accurate

device modelling is often quite large. These models are separate from the local

model and from the RPL, recurrence and Monte-Carlo models which attempt

to model the semiconductor statistically. A phenomenological approach, which

lends itself to the type of model required has been sought.

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CHAPTER 8

8.2 Characteristics of the Model

The model includes,

• Capacitance as a function of reverse bias

• Multiplication as a function of reverse bias

• Excess noise as a function of multiplication (local model)

• Series resistance

• Responsivity

• Bulk leakage current as a function of reverse bias

• Surface leakage current

• Tunnelling current as a function of reverse bias

• The gain bandwidth product of the APD

• The noise bandwidth of the APD (experimental)

All of these functions can have their variables parametrised by the calling

script. For example it is possible to supply a list of series resistances and observe

the effect these have on noise as a function of multiplication by running several

simulations (one at each resistor value) and collecting the data into one graph.

Similarly a table of responsivity, VBD, nm and k values could be provided to

model differing injection conditions which elicit different multiplication noise and

responsivity profiles. Examples of these methods are shown in section 8.4.

The model does not include,

• Geiger mode (SPAD) operation

• Transit time effects

• Stray inductance or capacitance and packaging parasitic components

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CHAPTER 8

Table 8.1 – Node connections for the APD model.

Node Function

LIrradiance falling on the device in Watts. Use for illuminating the

device in .AC .TRAN .DC .OP .NOISEA Anode terminalC Cathode terminal

MA voltage representing the large signal multiplication. Use forobtaining a voltage which represents the photo-multiplication in.OP .DC .TRAN. This node is meaningless in .AC and .NOISE

FA voltage representing the large signal excess noise factor. Use forobtaining a voltage which represents the excess noise factor in.OP .DC .TRAN. This node is meaningless in .AC and .NOISE

CapA voltage representing the large signal parallel plate capacitanceof the device. Use for obtaining the same in .OP, .DC, .TRAN.

Meaningless in .AC and .NOISE

This model is not appropriate for Geiger mode operation; convergence problems

abound. Furthermore ‘M’ and ‘F’ are ill-defined in Geiger mode. Transit time

effects are not included. At the frequencies where these effects begin to dominate,

SPICE is unlikely to be an appropriate choice of circuit simulator because it is

parameter based, and is incapable of properly representing distributed effects.

More appropriate simulators exist including Microwave Office, Ansys HFSS and

Agilent ADS, which better represent the distributed high frequency effects using

methods such as numerical electromagnetic solvers. Similar arguments apply to

stray inductance and capacitance and packaging parasitics. The package para-

sitics can be added easily if desired. A simple transit time model which could be

integrated into this model was proposed by Jou [232].

8.3 Parameters of the Model

The node connections for the APD model are given in Table 8.1, the parameters

of the model are given in Table 8.2. The parameters and methods by which they

may be obtained are described below. It is necessary to supply all parameters.

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CHAPTER 8

RL

L

−+

V1

B1C1 B2

D1 B5 B3 B4 B10

R2C

A

Q2

−+

V2

Q1

B6

−+

B7

Cap

−+

B8

M

−+

B9

F

100

Ci

200

201

Figure 8.1 – A schematic representation of the APD model. Node labels are internal to thesub-circuit and can be reused in a calling circuit. B1 represents the bulk dark current, B2

represents the primary photo-current and B5 represents the multiplied bulk dark and primaryphoto-current. B3 represents the tunnelling current, B4 provides the surface leakage currentand B10 provides the small signal noise. B7−9 supply the large signal and DC reverse biascapacitance, multiplication and excess noise factor. B6 and surrounding components producethe value for small signal noise. V1 is a zero volt DC source used to sum the currents from B1

and B2.

If, for example, tunnelling current does not apply to a particular device, set ITM

= 0 and ITC = -30. This will render the tunnelling current effectively switched

off for all practical currents.

8.3.1 Series Resistance

The device series resistance (RS) may be fitted to,

Ia = Is

(

expq (Va − IaRs)

n k T− 1

)

(8.1)

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CHAPTER 8

Figure 8.2 – APD model parameters. Parameter names are typeset in agreement with thesource code in Appendix B.

Parameter Function

RS

Device series resistance. Obtain this by performing forward IVs upto high currents (100 mA) on a few sacrificial devices, and then fitthe data for RS in series with a diode approximated by Shockley’sdiode equation.

RResponsivity Amps/Watt. This can be obtained by measuring thedevice being modelled and a reference device using a monochroma-tor and white light source.

VBDThe breakdown voltage of the device. Obtain this by regressingphoto-multiplication data to Miller’s empirical expression [5].

NMThe multiplication exponent of Miller’s empirical expression. Ob-tained simultaneously with VBD.

CJMDevice reverse bias capacitance. This is the linear coefficient of1/C2(V) when using the linear regression method.

CJC y-axis intercept of 1/C2(V) in the linear regression method.

C0y-axis intercept of C(V) in the inverse second order polynomialregression of C(V).

aReciprocal coefficient of C(V) in the inverse second order polyno-mial regression of C(V).

bReciprocal squared coefficient of C(V) in the inverse second orderpolynomial regression of C(V).

IBULKBulk dark current at unity gain. Bulk dark current experiencesmultiplication.

ISURFSurface leakage current. Surface leakage does not experience mul-tiplication.

ITMLinear coefficient of the linear regression of the common logarithmof the tunnelling current.

ITCY-axis intercept of linear regression of the common logarithm ofthe tunnelling current.

k McIntyre’s effective k [3].

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CHAPTER 8

Where Ia is the diode forward bias dark current, Is is the saturation current, q

is the electron charge, n is the emission coefficient, k is Boltzmann’s constant, T

is the absolute temperature of the junction, Rs is the series resistance and Va is

the voltage across the diode. The appearance of Ia on both sides of the equation

makes fitting analytically slightly complex, however it is possible by resorting to

Lambert’s W. An iterative, numerical, trial and correction method is often used.

8.3.2 Multiplication

Photo-multiplication is modelled by regressing measured multiplication data to

Millers empirical expression [5],

M =1

1−(

VVbd

)n (8.2)

where M is the photo-multiplication, V is the reverse bias voltage, Vbd is the

reverse breakdown voltage and n is a coefficient of curvature of the multiplication

profile. nm and Vbd are extracted from the regression.

8.3.3 Device Capacitance

Two methods are provided for modelling the reverse bias device capacitance as a

function of reverse bias voltage the first requires a linear regression of the reverse

bias capacitance voltage characteristic to,

C(V ) = cjm · V + cjc (8.3)

Where cjm and cjc are model parameters, V is the reverse bias voltage and C(V )

is the device capacitance as a function of reverse bias voltage. This regression is

often sufficient as the regressed capacitance need only fit the device capacitance

at higher reverse bias voltages.

The second method is a regression of C(V ) to a second order inverse polyno-

mial,

C(V ) =a

V 2+

b

V+ C0 (8.4)

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CHAPTER 8

where C0, b and a are the parameters being extracted, and C(V ) is given above.

This regression requires an additional parameter but often fits a wider range of

the reverse bias capacitance – voltage characteristic.

8.3.4 Tunnelling Current

Band to band tunnelling current, which is the only tunnelling current modelled,

is implemented by regressing the common logarithm of the measured tunnelling

current to a linear expression,

IT = ITm · V + ITc (8.5)

where ITm and ITc are the model parameters being found, V is the reverse bias

voltage and IT is the measured tunnelling current. Tunnelling current does not

experience avalanche multiplication.

8.3.5 Bulk and Surface Leakage Currents

The bulk, Ibulk, and surface, Isurf , leakage currents can be estimated by observa-

tion of dark reverse current voltage characteristics. The bulk and surface currents

can only be separated if devices of differing area are available. Masks available

to the EPSRC National Centre for III-V Technologies often have 25, 50, 100 and

200 µm radius mesa diodes fabricated in close proximity to each other. For de-

vices of differing area, leakage current which is dominated by bulk processes is a

function of normalised device area. Therefore if the device dark current density

normalised to device area is constant the leakage current is bulk dominated. Sim-

ilarly surface leakage current dominates if the dark current density is constant

when normalised to device perimeter. Generally both processes are found in real

devices and these two terms may be fitted iteratively. In this model bulk leak-

age is bias dependant and experiences avalanche multiplication; surface leakage

current does not experience avalanche multiplication.

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CHAPTER 8

8.4 Examples of the Model

In this section examples of the model are shown in LTSpice. The simulations

reproduce the characterisation experiments that were used to gather the data

necessary to produce the model. This is not representative of how the model is

intended to be used but shows example characteristics of a modelled device. The

model is principally that of a thin silicon pin diode, however to show tunnelling

in the model the tunnelling is taken from a similar thickness AlInAs device, as is

the device capacitance. The responsivity is not representative of any particular

device.

169

CHAPTER 8

8.4.1 DC Characteristics

Example DC characteristics are shown in Fig. 8.3. In this simulation, L, M, F,

Cap, the anode and cathode voltages, currents and device power dissipation can

be meaningfully interrogated.

Figure 8.3 – Top left: LTSpice schematic used to produce the DC characteristics. The APDis connected in a photo-current experiment where V2 is an SMU and V1 represents the opticalpower (Watts). V1 is parametrised between 0 and 100 V by the .step command. V2 is sweptfrom 0 to 13.5 V by the .dc command. Top right: Total current as a function of reverse biasvoltage with optical intensity as a parameter. It is the result of the DC sweep shown on theleft. Multiplication can be observed above 7 V. Tunnelling can be observed in the lower fourtraces. Bottom: Capacitance as a function of reverse bias voltage, obtained by plotting the‘voltage’ on the Cap node.

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CHAPTER 8

8.4.2 Optical Characteristics

Disparate injection conditions can be simulated by the use of tables of parameters.

This is shown in Fig. 8.4

Figure 8.4 – Top left: An LTSpice schematic showing the use of tables of parameters to sim-ulate sets of several parameters. This permits simulation of several wavelengths illuminatingthe device. Reponsivity (R), the shape of the multiplication curve (nm) and the effective k(k) are parametrised and selected from three tables by the .step command. The APD param-eters are shown in blue text. Top Right: M-1 versus reverse bias voltage for several injectionconditions. Bottom Left: Excess noise factor, F as a function of multiplication, M for severalinjection wavelengths. Bottom Right: The diode current may be used to observe the change inresponsivity with wavelength – incident optical power is constant for this simulation.

171

CHAPTER 8

8.4.3 Small Signal Characteristics

An example of small signal characteristics can be obtained by .ac analysis and

a schematic similar to Fig. 8.5. The excess noise characteristics are available in

small signal simulation by the use of the .NOISE command. It can be shown that,

in the mid-band, the small signal noise of the APD and the DC or large signal

(.TRAN) excess noise factor (available on the ‘F’ node) are equivalent. Relatively

little information is available regarding the frequency dependence of excess noise.

In RC limited devices it seems likely that the excess noise will roll off by the same

means as a periodic signal. In transit time limited devices however, and at optical

modulation frequencies where the transit time of the device and the period of the

modulating signal are similar, it is unclear if the full excess noise will be available

at the device terminals. An experiment that may begin to provide some data is

discussed in Chapter 9.

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CHAPTER 8

Figure 8.5 – Left: An LTSpice schematic showing the analysis of small signal response (.AC).Right: The resulting graph of cathode current as a function of frequency with reverse biasvoltage as a parameter. Note the inherent gain bandwidth product. This model is RC limitedunder all circumstances. The lowest, blue, line does not follow the others because the devicecapacitance changes markedly over the reverse bias range.

Figure 8.6 – Left: In this simulation excess noise is treated with a small signal model whereeach (successively larger) DC bias represent a change in ‘operating point’. This is useful whenthe SNR of a receiver system is required under certain conditions, for example simulatingthe SNR over a range of multiplications to evaluate the optimum combination of TIA gainand device multiplication as a function of the SNR at the TIA output. Right: Small signalresult (.NOISE) of excess noise factor of an APD. the roll-off of this characteristic may notbe physically representative as very little information exists regarding noise gain bandwidthproduct. However the low frequency values follow McIntyre’s model [3]. It can be shown thatthe small signal noise shown here leads to the large signal noise shown in Fig. 8.4.

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8.4.4 Large Signal Characteristics

Large signal characteristics are available via the anode and cathode voltages and

currents as well as the M, F and Cap nodes. An example of large signal APD

parameters in a 1 MHz operational amplifier based TIA is shown in Fig. 8.7. In

this figure one aspect of APD operation is exposed which can not be observed

easily in a laboratory. Namely that a large enough photo-current coupled with

a large enough multiplication and sufficient series resistance may cause the dark

multiplication and the light multiplication to differ in value. This effect applies

to excess noise also and represents a region of operation where the F versus M

characteristics may be expected to ‘bend over’ – become more nearly parallel

to the bias axis. An example of this type of bending is seen in the middle left

graph of Fig. 6.6, in the case of this data it is the low dynamic resistance of the

diode which causes some of the noise power to pass to ground without entering

the TIA. The shape of the graphs for these two measurement shortcomings is

indistinguishable.

8.5 Conclusion

This chapter has presented a simple SPICE model of an APD intended for the use

in simulating the interaction between real APDs and front-end circuits. Examples

of the model characteristics have been provided. The source code is presented in

Appendix B.

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CHAPTER 8

Figure 8.7 – Top Left: This figure shows an opamp TIA with bandwidth of approximately1 MHz driven by the APD model, model data is shown in the blue text. Transient analysis(.TRAN) is preformed to examine the behaviour of the TIA and APD combination in the timedomain. The diode reverse bias voltage is a parameter. Top Right: Transient response of thetransimpedance amplifier under several APD bias voltages. Bottom Left: This figure is noteasily produced in a laboratory. It shows the time dependence of the photo-multiplication. Athigher gains the photo-current causes a non-negligible IR drop across the device series resistancechanging the junction bias sufficiently to influence the multiplication (and noise). Bottom Right:This is a plot of TIA output voltage on with a logarithmic y-axis. It allows the effect of darkcurrent (in this case tunnelling current) mechanisms on the time averaged output voltage to beseen more easily.

175

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176

Chapter 9

Future Work

One aspect of this thesis is the development of measurement systems for charac-

terisation of electro-optical devices. Many thesis have discussed the future work

regarding specific material systems for a range of purposes but often when III-V

materials are concerned telecommunications is the key market. In this section

some ideas are presented regarding improvements to the measurement methods in

use. The ideas in this chapter are not necessarily original, but most of them were

developed independently (i.e. without prior knowledge of their pre-existence).

These ideas have been collecting over the course of this work, and several have

already been implemented at Sheffield in some form with varying degrees of suc-

cess.

9.1 Microwave Small Signal Bandwidth Mea-

surements

The basis for this idea is that a measurement tool with comparable functionality

to a ‘lightwave component analyser’ – such as the Agilent N4373C – is desirable.

However the LCA is expensive and beyond the means of many research groups.

Consequently a vector network analyser (VNA) may be substituted with the

addition of some external components. Not only is the VNA less expensive, but

its usefulness is not limited to electro-optical problems. The cost may be shared

by several research groups, as is the case at Sheffield.

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CHAPTER 9

CW Laser

1330/1550nm

EAM

CIP 40G-SR-EAM-1550CIP 40G-IR-EAM-1300

Bias TeePicosecond 5544

SMU

Bias TeePicosecond 5544

SMU

VNA

Figure 9.1 – Block diagram of VNA system.

9.1.1 Measurement System

A block diagram of one implementation is shown in Fig. 9.1. The CW laser is

modulated by an electro-absorption modulator (EAM), which is biased by an

SMU via a bias tee. The VNA output signal is also applied to the EAM via the

bias tee. A second tee is used to apply DC bias to the device from a second SMU.

The modulated current signal drops a voltage across the tee, and this voltage is

applied to the input of the VNA.

This implementation has several flaws relating to the impedance matching of

the various components, but most importantly the EAM and the APD. These

flaws yield results with unexplainable low frequency attenuations, various reso-

nances, and oddly shaped high frequency roll-offs. Calibration of the VNA by

the use of a calibration substrate can not help at all as the impedances when

the calibration substrate is in use are different from when the APD is in use.

The principle difficulty arises because the APD (essentially an un-matched high

impedance source) is connected to the surrounding equipment by a 50 Ω a coaxial

cable which is electrically very long. “Never put a photo-detector on the end of

an [electrically] long cable” [40].

This does not mean that transimpedance amplifiers, which are unmatched by

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CHAPTER 9

Iph cj rd

RS

Transmisson LineZ0 = 50 Ω

VNAZin = 50 Ω

Figure 9.2 – Diagram of a series matched APD – transmission line – VNA system.

design, are useless at microwave frequencies! Practically all optical communica-

tions systems use IC transimpedance amplifiers. Reviewing some datasheets it

is easy to see that while the input impedance is not always low, 100 Ω is not

uncommon, the electrical distance between the unmatched APD and the TIA is

often specified to an exacting degree. It is common to see a graph of frequency

response as a function of the length and thickness of the bond-wire that connects

the APD to the TIA. The selection is often made based on the capacitance of

the detector. It is possible to make use of a TIA approach in ultra wide-band

measurements too, and some ideas on this topic will be discussed shortly.

9.1.2 Matching

The difficulty arising in ‘ultra wide-band’ measurements (for example 10 MHz –

50 GHz) is that all wide band impedance matching techniques have some asso-

ciated difficulty or trade-off. For the style of measurement proposed impedance

matching is required, unless the TIA approach, which will be addressed shortly, is

being followed. There are many ways of approaching matching an APD to a line,

and many ways of practically implementing that matching. Three possibilities

will be discussed which are representative of the major obstacles.

A diagram showing a series matching arrangement is shown in Fig. 9.2. In

this figure several key assumptions are made including that the series resistance

of the device is less than 50 Ω and that the series resistance can be supplemented

with an external resistor RS to force a series match to the line. This matching

is difficult to implement practically. Ideally, a set of APDs would be fabricated

and tested for series resistance. Having obtained the value of series resistance

internal to the diode the correct value may be added to another production run

by deposition and laser trimming of a thin film resistor on the substrate, close to

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CHAPTER 9

ZT

C1

T2

Z0 = 50 Ω RS

iph cj rd

L1

−

+VB

T1

Z0 = 50 Ω

C2

VNAZin = 50 Ω

Figure 9.3 – A variation of the parallel matching arrangement after Xie [23] adapted forbandwidth measurements.

the diode. Not only is this process expensive, it depends on the devices having

similar series resistance across production runs and the availability of the various

technologies. A second (inferior) possibility is the use of a very small (0201)

microwave resistor epoxied onto the header or ceramic plate carrying the wafer

and connected to the device by usual wire bonding methods.

This method can only work when the transit time is large or the capacitance

is extremely small (∼ 10s fF). The necessity of a small capacitance is due to the

low pass filter formed by RS and cj . It is possible to terminate the VNA end of

the transmission line with an open circuit, no voltage division is required there

to provide a wideband match. This is because the reflected signal will be com-

pletely absorbed when it returns to the diode end of the transmission line, hence

resonances are impossible as standing waves can not build up. In practice the

transmission line will almost always be terminated however. A further assump-

tion is that the dynamic resistance of the diode is always very large compared to

50 Ω.

In the next solution it is assumed that, since the bandwidth is being mea-

sured, the quantum efficiency is a secondary concern (it can be measured by

other methods if desired) and it is permissible to introduce a frequency indepen-

dent attenuation in the measurement. This method borrows heavily from the

noise measurement after Xie [23]. The system diagram is shown in Fig. 9.3. The

APD is connected to a line, T2, which is electrically short (< λ/10 at the max-

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CHAPTER 9

iph cj rd

RS

R1 C1

R2

L1

Figure 9.4 – Passive components are used to form an all–pass filter with a frequency indepen-dent impedance of 50 Ω.

imum frequency of interest e.g. 50 GHz). The line is then terminated by ZT

which is 50 Ω. C1 is necessary to prevent the biasing voltage VB driving a DC

current through ZT . Similar arguments apply for the use of C2 and the VNA

input impedance. The purpose of T2, C1 and ZT is to provide a termination at

the APD end of T1. The other end of T1 is terminated by the input impedance of

the VNA. Provided RS is much less than 50 Ω and rd much larger than 50 Ω the

system will be matched at all nodes. There is still the RC limit of the APD and

the characteristic impedance. If the transit time limit occurs after the RC limit

the system will measure an artificially low bandwidth. ZT , C1 and T2 would, ide-

ally, be fabricated with the APD on the semiconductor substrate. However this

may be impractical. It should be possible however to epoxy a wafer to a ceramic

header or disk and epoxy several sets of resistors and capacitors (assuming 0201

or 0402 size) around the wafer and then use standard wire bonding techniques to

make the connections.

The prior examples have chosen to neglect the capacitive nature of the APD

and assume that at all frequencies of interest the APD is dominantly resistive

(because cj is very small ∼200 fF or less) while this is not an unreasonable as-

sumption in some cases it can not be guaranteed, and even 200 fF yields an

impedance of 16 Ω at 50 GHz. As frequency increases the above methods, while

better than nothing, will eventually fail to operate as designed. The final method

attempts to make the APD look resistive at all frequencies. The idea for this

method, while is is probably not original, came partly from a particularly taxing

undergraduate tutorial sheet in which the proof of a conjecture about the rela-

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CHAPTER 9

tionship between the circuit components is sought, and partly from audio power

amplifiers where some image filters are also called Zobel networks.

The circuit diagram is shown in Fig. 9.4. In this diagram the APD is shown as

cj , rd, iph and RS. And it is assumed that RS is less than 50 Ω. It is also assumed

that across the range of bias voltages which the bandwidth will be measured the

change in cj is negligible. High speed detectors tend to have small depletion

capacitances. R1 is ideally a laser trimmed semiconductor resistor, but it could

be a 0201 microwave component, or otherwise fabricated into a MMIC or flip-

chip arrangement with the detector substrate. Its purpose is to increase the total

value of R1 +RS to 50Ω. C1 is ideally an IC capacitor. The purpose of which is

to block the biasing voltage from passing a large current through L1 and R2. A

suitable value for a 10 MHz -3 dB point in a 50 Ω system is 1 nF. The capacitor

must be able to sustain the biasing voltage, so it can not be arbitrarily thin. This

value will require a considerable area of semiconductor and it may be necessary to

relinquish the 10 MHz – 100 MHz frequency range and use 100 pF instead. The

value of R2 should be equal to R1+RS, this causes the network to be resonant at

all frequencies. It can be shown that when R2 = R1 +RS the value of L required

to provide a frequency independent response is L = C R2. Ideally L is a spiral

inductor fabricated near to the APD on the semiconductor substrate by normal

metallisation methods and a small air bridge. For an APD with cj = 1 pF the

required value of L is 2.5 nF. It is important to note that the objective is to

provide an impedance looking into the network of 50 Ω at all frequencies, it does

not remove the bandwidth limit of the APD formed by cj and the surrounding

resistors.

9.2 Microwave Noise Measurements

In this section the measurement of noise above 10 MHz will be discussed. This

area is interesting both from a measurement design perspective and from a device

perspective. The device ideas presented in the following discussion are due to

Marshall1. Consider an APD with a modulated optical source incident on the

window. The period of the modulation is less than the transit time of the device.

1Andrew R. J. Marshall private communication 2010.

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CHAPTER 9

For example in InGaAs the saturated electron drift velocity is ∼ 1 × 107 cm/s.

A 2.56 µm device would be required for a 40 GHz signal to have the same period

as the transit time. It may be hypothesised that the optically generated carriers

would be generated at regular intervals of the wavelength throughout the device

and that assuming impact ionisation was not taking place, or that one of the

ionisation coefficients was zero. The carriers would travel together, this represents

a reduction in the variance of the position of the carrier population which may

lead to a sub shot noise result. The effect, if it exists, should be detectable prior

to the frequency where an integer number of wavelengths fits into a device i.e.

as the wave period approaches the transit time. To approach this work, a thick

(∼ 5 µm) three region structure, manufactured from one of the III-V materials

that is commonly used in telecommunications is preferable. If the objective is

to show that the effect exists is would be prudent to choose the material with

the slowest saturation velocity as this should yield any observable effect at the

lowest frequency. This work would be interesting even if the noise is shown to

follow a shot process. For many years excess avalanche noise has been reported in

telecommunications material systems at 10 – 30 MHz. No work has been reported

showing the noise of a telecommunications APD at the bandwidths it is intended

to operate. While BER measurements are routinely reported, these deal directly

with the quality of the detector within a communications systems perspective.

The underlying physical processes require direct investigation.

9.2.1 Measurement Systems

The measurement systems discussed below are original and have been published

[29]. The systems are an improvement over the Xie system [23]. While the

systems appear similar to that of Xie, the measurement method is somewhat

different, and the results potentially more stable and repeatable.

There are two possible improvements to the design proposed by Xie et al. [23].

Both are essentially improvements to the method by which the instrumentation

is calibrated. The introduction of the calibrated noise source (HP346B) permits

the use of direct noise figure measurement as opposed to hot/cold measurements,

which is a considerable improvement. The noise figure meter (N8973A, or an older

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CHAPTER 9

model such as the N8970) is designed such that the noise source is connected

to the device (for example an LNB) under test. Of course if the device is an

electro-optical transducer this is impossible as there is no place to attach the

noise source. This leads to the use of a pre-test calibration followed by hot/cold

measurements. It would be preferable to use the noise figure analyser according

to its design principle i.e. with the noise source in the measurement. The NFA

is provided with prior calibration – by the manufacturer – of the noise sources

contribution to the system. The system gain is also computable by measuring

the effect on the noise output when the noise source is switched on and off – it

is pulsed by the noise figure meter. The time average of the change in noise level

can provide the gain from the noise input port to the NFA input port. The prior

knowledge of the known noise input from the calibrated source (HP346B) allows

the NFA to compute the gain and noise figure nearly instantly, a considerable

improvement in measurement speed, accuracy, precision and repeatability. The

use of real time signal averaging is also possible, effectively increasing the system

signal to noise ratio. The noise source can not be directly applied to the detector.

However, a secondary port can be created which permits the connection of an

APD and the noise source to the NFA simultaneously. Two example designs

are provided, the first uses a 50 Ω matched topology similar to that of Xie et

al. [23] The second describes a similar overall structure but using a commercial

transimpedance amplifier.

The APD multiplication, excess noise factor and noise power bandwidth can

be established simultaneously in one measurement. The limitation of the sys-

tem bandwidth can be alleviated by two methods. Firstly a higher maximum

frequency noise figure meter can be obtained. Agilent Technologies presently

manufactures noise figure meters/analysers capable of directly measuring up to

26 GHz. The use of heterodyne techniques could extend this considerably. How-

ever a relatively inexpensive alternative is to use a lower bandwidth noise figure

meter but begin measuring bandwidth once the APD has been biased to achieve

a high gain. The high frequency roll off due to a finite gain bandwidth product

can be observed at lower frequencies; the unity noise gain bandwidth product can

then be inferred. A potential problem with this method is that carrier feedback

will randomise the ionisation position for any given carrier so noise reduction, if it

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CHAPTER 9

exists, will only be observable at relatively low gains. The importance of correct

impedance matching cannot be overemphasized.

9.2.1.1 50 Ω System

The system diagram shown in Fig. 9.5 shows the structure of the measurement

setup. An SMU drives a bias tee composed of L1 and C1. An example of a

suitable tee is PicoSecond Model 5541A. The APD is connected to a microwave

DC block, C1, and this is in turn connected to a termination (50 Ω). The DC block

and the termination must be electrically close to the APD even at the highest

measurement frequency. From the point of view of the first amplifier the APD is

a Norton source coupled to the end of a properly terminated transmission line.

Approximately half of the noise power will escape to ground via R1, the rest will

enter the measurement system. It is possible to calibrate the measurement system

either manually (i.e. use a 50 Ω signal generator to list a table of adjustments

for each frequency and post process the measured device data based on these

reading) or automatically by using the HP 346B Noise source hooked to the first

amplifier input instead of the APD. The attenuator setting must be noted down

when the calibration is carried out. The first amplifier in the chain must be of

the lowest possible noise. Examples include Minicircuits ZFL-1000LN+, ZX60-

33LN+ and Pasternack PE1513. The ZFL-1000 has low noise and a reasonably

flat gain versus frequency profile from 100 kHz to 1 GHz however bandwidth is

limited to 1 GHz. The ZX60-33LN+ has exceptionally low noise, and reasonable

gain versus frequency characteristics from 50 MHz to 3 GHz. The PE1513 has

relatively poor noise especially as frequency increases, the gain versus frequency

profile is not ideal either; however it is the only device which covers the whole

frequency range of the NFA. Unless APDs possessing bandwidths below 50 MHz

are to be routinely measured the author’s preferred choice would be the ZX60-

33LN.

The specifications of the second and third amplifiers are considerably less

critical than the first. Any microwave device with reasonable noise and gain versus

frequency characteristics will be acceptable. The stepped attenuator should be of

the precision type for example the Trilithic RSA35-100 (0 dB to 100 dB in 10 dB

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CHAPTER 9

SMU

L1

C1

R1 APD

C2

50Ω Amplifier

10MHz to 3GHz

50Ω Amplifier

10MHz to 3GHz Stepped Attenuator0 – 100dB

50Ω Amplifier

10MHz to 3GHz

PowerCombiner

HP 346BNoise Source

Agilent N8973A

Figure 9.5 – 50 Ω excess noise measurement system.

steps) would be ideal. The power combiner may be of any type which covers

the required bandwidth. A suitable resistive splitter/combiner is the Minicircuits

ZX10E-14-S+.

The maximum device capacitance is approximately 2 pF to obtain a 3 dB point

of approximately 3 GHz. R1 must be electrically close to the APD, consequently

it is unlikely that the noise contribution of this resistor could be minimised by

cooling as was reported by Xie et al. [23]. If the APD was measured at low tem-

perature however it would be plausible to place R1 and C1 in the cryostat cham-

ber with the APD, hence obtaining a noise advantage at lower temperatures. A

laser is often used to excite electro-optical transducers in characterisation experi-

ments. In this case the laser should be a gas laser possessing a single longitudinal

mode, preferably frequency and amplitude stabilised. Noise characterisation ex-

periments using semiconductor lasers have always proved impossible, the laser

relative intensity noise (RIN) is too great to permit measurement of the detec-

tor noise. Multi-longditudinal mode lasers often have baseband mode beating

effects [40]1, and are generally unsuitable in this experiment.

1Samual Goldwasser private communication 2011

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CHAPTER 9

SMU

L1

C1

R1 APD

C2

50Ω Amplifier

10MHz to 3GHz

50Ω Amplifier

10MHz to 3GHz Stepped Attenuator0 – 100dB

50Ω Amplifier

10MHz to 3GHz

PowerCombiner

HP 346BNoise Source

Agilent N8973A

Figure 9.6 – Transimpedance amplifier excess noise measurement system.

9.2.1.2 TIA CW Noise Measurement System

The structure of this measurement system is nearly identical to the 50 Ω system

previously described. The principle difference is the use of a transimpedance

amplifier front end instead of a 50 Ω system. Fig. 9.6 shows the system diagram.

C1 provides an AC ground for the APD such that the very great majority

of the noise current flows into the TIA. Example TIAs are given in the figure.

Commercial TIAs often have input impedance which is not a good approximation

to a virtual earth. As a result the maximum permissible device capacitance is

often lower than in the 50 Ω system case and is dependent on the particular

TIA in use. The MAX3910 provides ∼9 GHz small signal bandwidth and nearly

linear output voltage to input current relationship for photo-currents in the range

0 to 900 µApk−pk. The small signal gain is approximately 1.6 kV/A in the linear

region.

Unlike the 50 Ω system it is not possible to connect the noise source to the TIA

input for calibration purposes. This is a major limitation of the TIA measurement

compared with the 50 Ω measurement. Calibration of the TIA signal path with

the noise source is only possible at the TIA output. A plausible method of

calibration is to use a unity gain wide band (much wider than the device that is

being measured) pin diode which is known to exhibit shot noise. Any deviation

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CHAPTER 9

Iph cj rd

−

+

A1AD829

C3

68 pF

−5V

+5V

R5

2.2 kΩ

C1

1.5 pF

R2

1 kΩ

R1

1 kΩ

R3

50Ω

Agilent/HP 346B −

+

A2AD829

C4

22 pF

−5V

+5V

R4

1 kΩ

R6

50Ω

C2

220 nF

Output

Figure 9.7 – Implemented 10 – 20 MHz transimpedance based excess noise measurment frontend with noise source port.

from shot noise can be calibrated out.

Fig. 9.7 shows a simple circuit which accomplishes the summing operation

with a TIA. The bandwidth is limited to 20 MHz, and the noise figure of the

circuit prior to any photo-generated noise is +23 dB (measured) consequently a

large (> 10 µA) photo-current is required to affect a significant increase in the

noise figure. The circuit does however possess a stable (if high) noise floor within

the operating bandwidth and the measured noise is repeatable over a period of

several hours and is nearly identical across the available measurment bandwidth.

These three features are lacking from the hot/cold measurements presently in use.

This particular implementation is of little practical value beyond being a proof

of concept however. A much lower noise, wider band circuit preferably avoiding

opamps is required to perform meaningful research.

9.3 Laser Noise “Cancelling” Front End

The work in this section involves the application of Hobbs’ noise cancelling circuit

to the PSD style measurment systems that are in use at Sheffield. The objective

is to make any laser usable in excess noise measurements. Presently only gas

lasers and often only He-Ne lasers have noise characteristics that are suitable for

excess noise measurements.

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I1

+VS

Q1Q2

−+

-15V+15V R12

R4

−+

C2-15V+15V

R2

Q5 Q4

R3

Q6

Q7

I2

−VS

Q8

Q3

−+

-15V+15V R1

−

+ V1

Output

Figure 9.8 – Noise “cancelling” front end modified after Hobbs [40]. This version has the DCservo detached from the output node. I1 is composed of the AC photo-current, DC photo-current + laser RIN. I2 is composed of the DC photo-current + laser RIN all multiplied by ∼1.01 – 2. The exact ratio is not important.

This work follows on from that of Hobbs [40] pp. 721 – 731. The principle

of operation relies on a circuit which allows division of two currents in an ar-

bitrary ratio, and that an analogue of this operation is the combination of two

currents. If these currents share a common noise component, which is not ran-

dom (for example laser RIN) it may be made to cancel such that a much smaller

signal component that was previously heavily corrupted is unmasked. The cir-

cuits Hobbs presents perform this function, unfortunately, do not permit the use

of PSD because the photo-current must have a DC component.

With reference to Fig. 9.8, the laser is tapped by a beam splitter, the ratio of

the two output beams is not significant, there is no requirement for them to be

of equal intensity but one must be equal to or greater than the other. One of the

output beams is chopped by some means which does not completely extinguish

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CHAPTER 9

the signal in the off period, the other is CW. I1 is the photo-diode under test. It

is illuminated by the chopped laser light which includes the RIN. I2 is a second

photo-diode which is not necessarily identical to the first (for example a BPX65).

It is illuminated by the CW output beam which also contains the RIN but not the

phase information associated with the chopping process. The servo ensures that

the DC conditions across the two differential pairs are balanced the cancellation

occurs due to the addition of the current in the collector of Q4 and the collector

of Q1. The collector of Q1 passes the PSD signal the (wanted) shot and excess

(if multiplication is present) noise of the APD represented by I1, the Laser RIN

and a DC component of photo-current. Q4’s collector passes the laser RIN and a

DC component of photo-current. It is critical that the ratio of the RIN derived

current in the two photo-diodes and the ratio of the DC components be identical.

This is the reason why there must always be a DC component of photo-current.

If the restriction on the ratio of the RIN currents and the DC currents did not

exist, a pair of current sources in parallel with I1 and I2 would be sufficient. It

may be possible to alleviate this necessity by measuring the DC component of

current in the two diodes and using a pair of controlled current sources (easily

manufactured from an opamp and a JFET) to maintain a DC current of the

correct ratio. This would also allow high dark current and photo-multiplication

to be appropriately handled. No method of achieving the measurement of the

DC component has been forthcoming which does not also disturb the balance

of the currents sufficiently to make their cancellation only partially effective. If

a solution can be found it would make noise measurements readily available at

many more wavelengths.

9.4 Improved Squaring Circuit

When using the two measurement systems reported in Chapters 3 and 5 the

system noise floor is strongly affected by the junction capacitance of the APD.

The output of the noise power meter has a DC component proportional to its

input noise. When using large capacitance APDs the output of the power meter

may saturate due to the DC component. The appropriate action is to reduce the

system gain by adjusting the attenuator, this may be necessary even if the power

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CHAPTER 9

P.G.50 Ω C1

L1

SMU

16 Ω

16 Ω

SPAD

220 nF

Ch1

16 Ω

220 nF

Ch2

CT

Figure 9.9 – An on-wafer SPAD experiment.

meter did not posses a DC offset proportional to its input noise. It would be

possible to detect this non-linearity by checking that two adjacent attenuation

settings do produce a factor of ten change in noise output voltage. However,

since the output is saturated it is impossible to continue without increasing the

attenuation. In situations where the output saturates due to the DC but the

noise power meter would otherwise operate correctly, system sensitivity is lost

due to the design of the power meter. A design of power meter having no DC

component in its output is desirable.

9.5 An On-Wafer SPAD Setup

This is an extremely simple system to produce a SPAD measurement system for

devices still in wafer form. The underlying principle is borrowed from Dimler

et al. [239]. It’s practical usefulness may be limited as it breaks Hobbs’ ‘rule’

of electro-optical design – diodes are connected by necessarily on long cables.

Nevertheless, a setup similar to the one shown in Fig. 9.9 was constructed and

initial results appeared promising, however its use was discontinued and it was

dismantled without a proper assessment of its suitability.

A preferred embodiment is shown in Fig. 9.10. The box is conducting (e.g

Aluminium). The impedance matching and balance capacitors are SMT compo-

nents mounted on an FR4 PCB as close together as physically possible, ground

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CHAPTER 9

P.G.

Seriestermination

50 Ω Coax

Wide BandBias Tee

SMU

Triax

50 Ω Coax 50 Ω Coax

SPAD

This coax and diode combinationrepresents a probe and probestation

16 Ω 16 Ω 220 nF

16 Ω

20 pF 190 pF

220 nF

50 Ω Coax

CRO CH150 Ω termination

50 Ω Coax

CRO CH250 Ω termination

Shield

Core

It may be necessary to addan LNA here if one is usedin CH1

In may be necessary to have an LNAhere in series with a continuouslyvariable attenuator

Figure 9.10 – An on-wafer SPAD experiment – a preferred implementation.

planes are essential. ∼16 Ω may be manufactured with two 33 Ω in parallel. It

is assumed that the diode series resistance is much lower than 50 Ω and that the

diode dynamic resistance never falls as low as 50 Ω. If these conditions are met

it is likely (although it has not been proved) that all nodes see 50 Ω in all direc-

tions. Balance is found when the sum of the trimmer capacitor and the 190 pF

capacitor (Fig. 9.10) equal the sum of the SPAD junction capacitance and the

cabling capacitance from the aluminium box to the probe-station including the

probe and all strays. The CRO is set up to display channel one subtracted from

channel two.

192

CHAPTER 9

9.6 NLOS UV Communications with SiC

In Chapter 7 an application of SiC detectors to NLOS communication was de-

scribed. Presently, there are no reported field trials of NLOS communication

systems using SiC detectors. Several ‘test beds’ exist using PMTs and GaN

photo-diodes for example [157,158,240–242]. While the idea of NLOS communi-

cations is not a new idea a comparison of SiC and GaN in a real world application,

a pair of personal radios for example, using a GaN/AlGaN transmitter would be

of interest. Comparing dark current densities tends not to yield much practically

useful information at the system level. BER should be one of the metrics by

which the systems are measured, as well as (accelerated) lifetime testing, and

maximum range.

193

CHAPTER 9

194

Chapter 10

Conclusion

The objectives of this work were threefold, firstly to develop new excess noise mea-

surement systems that increase the range of measurable junction capacitances of

an APD. Secondly to verify the impact ionisation coefficients in 4H-SiC, especially

at lower electric fields, where considerable disagreement exists in the literature.

The final objective was to systematically explore the dead-space effect in Silicon

by providing multiplication and excess noise measurements on a series of differing

thickness devices.

The design of two measurement systems capable of measuring APDs with

junction capacitance up to two orders of magnitude greater than the prior Sheffield

system, and greater than any other reported, has been described. The design and

optimisation of these systems and associated material has been presented in de-

tail. The systems use phase sensitive recovery techniques allowing the avalanche

noise power and photo-current to be measured unambiguously from the system

noise and leakage current of the test device. Both measurement systems systems

have been used to confirm the validity of the data in the high capacitance devices.

Avalanche noise measurements have been performed on a range of Silicon Car-

bide nip and separate absorption and multiplication avalanche photo-diodes with

i region widths, w, of 2.7 µm and 0.57 µm. with the objective of experimentally

verifying the value of α over a range of lower electric fields. This has been ac-

complished, and a new values of α and β have been published based on the data.

This work also represents the first reported (and lowest noise) measurement of

pure hole injection in 4H-SiC.

195

CHAPTER 10

Using the systems described, avalanche noise measurements have been per-

formed on a range of Silicon pin diodes. These devices have widths, w, ranging

from 0.35 µm to 0.031 µm. The Silicon results show that, for w < 0.4 µm, the

extracted and non-local modelled impact ionisation coefficients begin to deviate

from the bulk coefficients. This is the first systematic measurement of the effect

of the dead-space in Silicon and includes the thinnest reported device to date.

Using the experimental data and analysis herein it is possible to estimate, for

example, the length a transistor channel must be reduced to in order to support

a certain electric field while yielding a desired maximum value of current multi-

plication. Such information is of considerable use to IC designers. This work has

also verified that the RPL model accurately represents characteristics in devices

as thin as 71 nm, and provides acceptable analysis at 31 nm.

196

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Appendix A

A One Dimensional Poisson

Solver

This appendix lists the expressions required to model a 2, 3, 4 or 5 region semi-

conductor. This model is used to model the SiC devices in Chapter 7. Figure A.1

shows the progression of depletion through a five region structure as the magni-

tude of reverse bias voltage is increased. When fitting the CV profile of a diode

structure with several regions, there will be a set of voltages where the diode

behaves like a two region structure, and a further set of voltages where it be-

haves like a three region structure and so on. It is necessary to know what the

terminal voltage will be at each transition such that the correct set of equations

can be used to find the relevant depletion distances. The transition voltages

are obtained by considering the area under the electric field curve and the one

dimensional Poisson equation.

227

APPENDIX A

N1 N2 N3 N4 N5

P-Type

Capping Layer

N-Type

Multiplication

Layer

N-Type

Charge Layer

N-Type

Absorption Layer

N-Type

Capping Layer

ξ

x

ξ

x

ξ

x

ξ

x

E1

E1

E2

E1

E2

E3

E1

E2

E3

E4

A1A2

X1

X2

X1

X2

X3

X1

X2

X3

X4X1

X2

X3

X4

X5

Figure A.1 – Top: Schematic diagram of a reverse biased SAM-APD. Upper Middle: Twopartially depleted regions (p+–n). Middle: One fully depleted region and two partially depletedregions (p+–n−–n+). Lower Middle: Two fully depleted regions and two partially depletedregions (p+–n−–n+–n−). Bottom: Three fully depleted regions and two partially depletedregions (p+–n−–n+–n−–n+).

228

APPENDIX A

A.1 Two Regions

A1 =1

2X2E1 (A.1)

A2 =1

2X1E1 (A.2)

E1 = −qN2 X2

ε=

qN1 X1

ε(A.3)

by appropriate substitutions it may be shown that,

A2 =1

2

qN22X2

2

εN1

(A.4)

A1 = −1

2

X22 qN2

ε(A.5)

The terminal voltage is then,

Vt = −1

2

qN2X22

ε+

1

2

qN22X2

2

εN1

(A.6)

Setting the terminal voltage to zero and solving for X2,

X2 =

(−2N1

2Vt εN2 q + 2N1 Vt εN22 q

) 12

N2 q (−N1 + N2 )(A.7)

X1 is then,

X1 = −N2 X2

N1

(A.8)

229

APPENDIX A

A.2 Three Regions

By The same approach as for two regions, expressions for E1, E2, X1 and X3 may

be written. X2 is known (from for example SIMS or the growth order sheet)

because the multiplication region is fully depleted.

E2 = −qN3 X3

ε(A.9)

E1 = E2 − qN2 X2

ε(A.10)

E1 =qN1 X1

ε(A.11)

Solving for X1,

X1 = −N3 X3 + N2 X2

N1

(A.12)

The terminal voltage is,

Vt =1

2E1 X1 +

1

2E2 X3 +

1

2(E1 + E2 )X2 (A.13)

Setting (A.13) to zero and solving for X3 yields,

X3 =1

2

−b+√b2 − 4 a c

a(A.14)

a = −N3 +N3

2

N1

(A.15)

b =

(

−2X2 + 2N2 X2

N1

)

N3 (A.16)

c =N2

2X22

N1

− N2 X22 − 2

εVt

q(A.17)

230

APPENDIX A

A.3 Four Regions

Similarly for four regions, expressions for E1, E2, E3, X1 and X4 may be written.

X2 and X3 are known because the multiplication and charge regions are fully

depleted.

E1 =qN1 X1

ε(A.18)

E3 = −qN4 X4

ε(A.19)

E2 = E3 − qN3 X3

ε(A.20)

E1 = E2 − qN2 X2

ε(A.21)

(A.22)

Solving for X4,

X1 = −N4 X4 + N3 X3 + N2 X2

N1

(A.23)


Vt =1

2X1 E1 + X2

(1

2E1 +

1

2E2

)

+ X3

(1

2E2 +

1

2E3

)

+1

2X4 E3 (A.24)

Setting (A.24) to zero and solving for X4 yields,

X4 =1

2

−b+√b2 − 4 a c

a(A.25)

a = −N4 +N4

2

N1

(A.26)

b = 2N4 N3 X3

N1

+ 2N4 N2 X2

N1

− 2X3 N4 − 2X2 N4 (A.27)

231

APPENDIX A

and

c = 2N3 X3 N2 X2

N1

+N2

2X22

N1

− N3 X32 +

N32X3

2

N1

− N2 X22

− 2X2 N3 X3 − 2εVt

q(A.28)

A.4 Five Regions

Similarly for five regions, expressions for E1, E2, E3, E4, X1 and X4 may be writ-

ten. X2, X3 and X4 are known because the multiplication, charge and absorption

regions are fully depleted.

E4 = −qN5 X5

ε(A.29)

E3 = E4 − qN4 X4

ε(A.30)

E2 = E3 − qN3 X3

ε(A.31)

E1 = E2 − qN2 X2

ε(A.32)

E1 =qN1 X1

ε(A.33)

Solving for X5,

X1 = −N5 X5 + N4 X4 + N3 X3 + N2 X2

N1

(A.34)


Vt =1

2X1 E1 + X2

(1

2E1 +

1

2E2

)

+ X3

(1

2E2 +

1

2E3

)

+

X4

(1

2E3 +

1

2E4

)

+1

2X5 E4 (A.35)

232

APPENDIX A

By the same process as in the prior problems, setting to zero and solving for X5,

X4 =1

2

−b+√b2 − 4 a c

a(A.36)

a =N5 2

N1− N5 (A.37)

b = 2

(

−X2 − X3 − X4 +N4 X4 + N2 X2 + N3 X3

N1

)

N5 (A.38)

and,

c = −2X3 N4 X4−N3 X32−N2 X2

2−N4 X42−2X2 N4 X4−2X2 N3 X3−2

εVt

q+

N42X4

2 + 2N4 X4 N3 X3 + 2N4 X4 N2 X2 + N32X3

2 + 2N3 X3 N2 X2 + N22X2

2

N1

(A.39)

233

APPENDIX A

234

Appendix B

APD SPICE Model Source Code

This appendix lists the SPICE model for the APD discussed in Chapter 8.

.subckt APD L A C M Cap F

* Light entering the APD. V(L) is External Pin for Light entering the

device Units: Volts represents Watts

R L 0 1G

* Series Resistance

Rs C 100 Rs

* A sensing voltage source to properly execute multiplication on the

* bulk dark current and primary photocurrent

V1 100 Ci 0

* Primary Photocurrent

BPR Ci A i=R*V(L)

* Multiplication value computed using Millers Empirical Approximation

* having regressed some real multiplication data

BM M 0 v=1/(1-(V(Ci,A)/VBD)**nm)

* Capacitance due to physical dimensions. Capacitance is obtained by

* linear regression of 1/C(Bias)^2 data. Comment out this line or the

* next as appropriate.

* BC Cap 0 v=(1/(Cjm*V(Ci,A)+CJc))**(0.5)


* inverse second order polynomial regression of [C(Bias) C = C0 +

* a/bias + b/bias^2]

*BC Cap 0 v=(C0 + a/V(100,A) + b/(V(100,A)**2))

235

APPENDIX B


* regression of an exponential decay of the form

* y = y0 + a*exp(-b*bias)

BC Cap 0 v=(C0 + a*exp(-b*V(100,A)))

* A Non Linear Capacitor. ‘x’ represents the voltage across the

* non-linear capacitance.

CJ A Ci Q=V(Cap)*x

* Bulk Dark Current (is Multiplied).

Bgen Ci A i=Ibulk

* Surface Dark Current (is Not Multiplied).

Bsuf 100 A i=Isurf

* This line multiplies the Bulk dark current and the primary

* photocurrent.

BI 100 A i=I(V1)*(V(M)-1)

* Tunnelling current. The tunnelling current is modelled by performing

* the linear regression of the common log of some tunnelling current

* data ie log10(Itunnel) = m*Bias + c.

* This is generally a good enough fit for a first order model.

* Tunnelling current is not multiplied.

BT 100 A i=10**(itm*V(Ci,A) + itc)

* Excess Noise Factor Using the Local Expression for F(M) where pure

* carrier injection prevails.

* The modeling of mixed injection is possible by adjusting k and

* providing new coefficients for Millers approximation of M

BF F 0 v=k*V(M) + (2-(1/V(M)))*(1-k)

* two times the total current that will be multiplied by the shot

* formula. i.e. N^2 = 2*q*BTI

BTI 201 200 i=2*(I(BI)*V(M)*V(F)+Isurf+I(BT))

* Two Common Base transistors share the photocurrent and each generate

* 1/2 of the required noise

Q1 0 202 200 pnp

Q2 0 202 201 npn

.model npn npn (IS=0.01f BF=1 cje=1p)

.model pnp pnp (IS=0.01f BF=1 cje=1p)

* A voltage source senses the noise current flowing from Base to Ground

236

APPENDIX B

V2 202 0 0

* A current source in the main diode circuit develops the appropriate

* noise current

BN 100 A i=I(V2)

* A diode is used to model forward Bias

D1 A 100 diode

.model diode D(is=0.1f rs=1)

.ends

237

APPENDIX B

238

Appendix C

Phase-Sensitive Detection

Phase sensitive or “Lock-in” detection is a method increasing the signal to noise ratio

of an electronic system by decreasing the bandwidth. A complimentary viewpoint is

that the time taken to make an observation using the system is increased in order

to gather more information. The method requires the use of a linear phase detector,

an analogue circuit function which, more recently, has been implemented using DSP

techniques, followed by a filter. A simplified schematic of a linear phase detector and

RC filter is shown in Fig. C.1. The linear phase detector has two inputs labelled ‘input

signal’ and ‘reference signal’ the output of the linear phase detector is connected to R3

which forms part of an RC filter.

The ‘input signal’ is an appropriately scaled voltage taken from an experimental

setup. The signal is often heavily corrupted by noise. The reference signal is a periodic

signal which has its phase information imparted to the input signal. In the case of

the experimental work described in this thesis a slotted disk is placed in the path of

the light source and rotated at a known frequency (the frequency is measured by a

LED/photo-diode arrangement on the chopper assembly). The APD biasing voltage

could be modulated and a CW light source used instead. This would be advantageous

because mechanical choppers are affected by ‘jitter’ which is caused by mechanical

deficiencies.

The linear phase detector operates by inverting the gain of the amplifier on the

transitions of the reference signal. The first order filter represents any arbitrarily com-

plex form of signal averaging. The time constant of R3 and C1 must be much longer

than the period of the reference signal.

239

APPENDIX C

−

+

R1

R2

R3

C1

Phase DetectorOutput

ReferenceSignal

Input

Signal

Figure C.1 – Simplified Schematic of a linear phase detector.

For the purposes of simplified analysis assume the input signal is of the form,

A cos (ω t+ φ) (C.1)

and the reference signal is a square and changes state at 0, πω ,

2πω etc. The period of

the signal is T = 2πω . The output voltage is then,

〈A cos (ω t+ φ)〉|π/ω0 − 〈A cos (ω t+ φ)〉|2π/ωπ/ω (C.2)

Integrating to perform the averaging,

〈Vout〉 =ω

2π

∫ π/ω

0A cos (ω t+ φ) dt.− ω

2π

∫ 2π/ω

π/ωA cos (ω t+ φ) dt. (C.3)

which yields a DC signal proportional to φ and A,

Vout = −A sin (φ)

π− A sin (φ)

π(C.4)

Vout = −2A sin (φ)

π(C.5)

A corrupting signal component may be considered by adding another frequency to the

input signal which is not equal to the reference frequency,

cos (ω t+∆ω t) (C.6)

The phase shift of the original signal, φ, is now varying at the difference in frequency of

the corrupting signal and the input signal. The resulting output is a sinusoidal signal

240

APPENDIX C

Figure C.2 – Black: In-phase input signal, Red: Corrupted input signal, Blue: Output signalresulting from the corrupted input signal.

with a period equal to the difference in frequency,

Vout = −2A sin (∆ω t)

π(C.7)

An example is shown in Fig C.2 in which the nominal angular frequency is 2π rads−1,

the corrupting signal is 2.2π rads−1. The output signal is therefore 0.2 rads−1.

241

APPENDIX C

242

Appendix D

High Order Nature of the

Common Base Amplifier

This appendix gives a brief analysis which shows that a hybrid-π transistor model which

includes the frequency dependence of β and the APD junction capacitance (Fig. D.1)

is second order, and that the second order behaviour becomes apparent when rb ≈ rbe

Summing currents at the base node (b)

− vb

rb− vb − ve

rbe− (vb − vc) · s · ccb = 0 (D.1)

summing currents at the collector (c)

− vc

RL− gm · (vb − ve) + (vb − vc) · s · ccb = 0 (D.2)

rb

b

cbe rbe

iin

e

cj

gmvbe

c

RL

Figure D.1 – Common base amplifier hybrid-π small signal model

243

APPENDIX D

and summing currents at the emitter

− iin + gm · (vb − ve) +vb − ve

rbe− ve · s · cj = 0 (D.3)

To include the base emitter junction capacitance (cbe) let the rbe term in Eq. D.1

become

rbe =r′

be

1 + s · cbe · r′

be

(D.4)

The combination of the following stage input capacitance and the parasitic capacitance

associated with the load resistor can be included by letting the RL term in Eq. D.2

become

RL =R

′

L

1 + s · CL ·R′

L

(D.5)

where CL is the combined capacitance. To include the emitter biasing resistor RE , let

Eq. D.3 become

− iin + gm · (vb − ve) +vb − ve

rbe− ve · s · cj −

ve

RE= 0 (D.6)

Solving the node equations Eq. D.1–D.3 without CL, cbe or RE yields a second order

response of the formvc

iin= k · (1 + s · τ1)

( s2

ω20+ s

ω0·q+ 1)

(D.7)

where

k = −gm · rbe ·RL

gm · rbe + 1(D.8)

τ1 =ccb · (rb · gm · rbe + rb)

gm · rbe(D.9)

1

ω20

=((rb + gm · rb · RL +RL) · rbe +RL · rb) · cj · ccb

gm · rbe + 1(D.10)

1

ω0 · q=

(RL · gm · rbe +RL + gm · rb · rbe + rb) · ccbgm · rbe + 1

+(rbe + rb) · cjgm · rbe + 1

(D.11)

244

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