emily barrentine

The Development of a Transition-Edge Hot-Electron

Microbolometer for Observation of the Cosmic

Microwave Background

by

Emily Margaret Barrentine

A dissertation submitted in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

(Physics)

at the

University of Wisconsin – Madison

2011

c© Copyright by Emily Margaret Barrentine 2011

All Rights Reserved

i

Abstract

In this thesis the development of a Transition-Edge Hot-Electron Microbolometer

(THM) is presented. This detector will have the capacity to make sensitive and

broadband astrophysical observations when deployed in large detector arrays in fu-

ture ground- or space-based instruments, over frequencies ranging from 30-300 GHz

(10-1 mm). This thesis focuses on the development of the THM for observations of the

Cosmic Microwave Background (CMB), and specifically for observations of the CMB

polarization signal.

The THM is a micron-sized bolometer that is fabricated photolithographically.

It consists of a superconducting Molybdenum/Gold Transition-Edge Sensor (TES)

and a thin-film semi-metal Bismuth microwave absorber, both of which are deposited

directly on the substrate. The THM employs the decoupling between electrons and

phonons at low temperatures (∼100-300 mK) to provide thermal isolation for the

bolometer. The devices are read out with Superconducting Quantum Interference

Devices (SQUIDs).

In this thesis a summary of the thermal and electrical models for the THM

detector is presented. The physical processes within the detector, with particular

attention to electron-phonon decoupling, and the lateral proximity effect between the

superconducting leads and the TES, are also discussed. This understanding of the

ii

detector and these models are used to interpret measurements of thermal conductance,

noise, responsivity and the transition behaviour of a variety of THM test devices. The

optimization of the THM design, based on these models and measurements, is also

discussed, and the thesis concludes with a presentation of the recommended THM

design for CMB applications. In addition, a planar-microwave circuit design and

a quasi-optical scheme for coupling microwave radiation to the THM detector are

presented.

iii

Acknowledgements

An adequate acknowledgement is very difficult to write at this point. I am im-

mensely grateful and lucky for the many people who have contributed both directly

to the work described in this thesis or who have provided expertise, advice and assis-

tance. None of the work presented here would be possible if it were not for their input

or contributions. It also goes without saying that any deficiencies in this thesis are due

to my own modest abilities and do not reflect upon the quality of their contributions.

This research project and my graduate education have been completed in very

close collaboration with colleagues and advisers at NASA-Goddard Space Flight Cen-

ter. I must thank the Wisconsin Space Grant Consortium, the NASA-Goddard Grad-

uate Student Researcher’s Program, the NASA Co-operative Education Program and

all those at the Detector Systems Branch and Detector Development Laboratory at

NASA-Goddard Space Flight Center.

Specifically, I would like to thank my NASA adviser, Thomas Stevenson, who

initially agreed to sponsor me in the GSRP program and who provided excellent

guidance as well as much direct input into the THM design. I must thank as well, Wen-

Ting Hsieh, who contributed to and oversaw much of the THM fabrication. I am very

indebted to Kongpop U-Yen who contributed to much of the THM microwave design,

provided helpful advice on microwave simulations and who assisted and oversaw the

preliminary CPW probe station measurements. I must also thank Nga Cao who

contributed to the THM microwave termination design and who also advised on early

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microwave simulations. I must especially thank Ari Brown and Kevin Denis who

dedicated an extraordinary amount of time and patience to training me in the Detector

Development Laboratory and who also completed fabrication of the THM test devices.

I also wish to thank Fred Finkbeiner who completed additional measurements of one

of the S-N-S test devices.

I would like to acknowledge the very specific contributions of those at NASA-

Goddard DDL in the fabrication of these THM test devices. Fabrication of the

‘THM2003’ devices was completed by Tim Miller and Christine Jhabvala. Fabrication

of the ‘THM4/THM5’ devices was completed with the assistance of Kevin Denis, Ari

Brown and Travis Travers. Fabrication of the ‘us25/us23’ devices was completed by

Kevin Denis, Peter Nagler and Dorothy Talley. Fabrication of the ‘THMA4/THMA24’

test devices was completed with assistance from Ari Brown.

Thanks also go to the originators of this project, mainly Shafinaz Ali, who with

assistance from Kari Kripps, and advice from Thomas Stevenson, Jay Chervenak, Ed

Wollack, Harvey Moseley and Dominic Benford at NASA-Goddard, guided the design

of the first THM detectors. I would like to thank Kent Irwin of NIST who provided

a two-stage SQUID readout. I am grateful for the experience and explanations of

Dan McCammon and Mark Lindeman who made the theoretical aspects of TESs and

bolometers understandable and applicable and who provided advice for interpreting

many of the THM measurements. I wish to thank Jack Sadleir for his advice con-

cerning the lateral proximity effect. I also wish to thank Don Brandl for enjoyable

and illuminating discussions of superconductivity theory and TESs and who created

some of the initial numerical programs to make predictions about the S-N-S effect.

v

I wish to thank my fellow graduate students during my time in the Observational

Cosmology lab here at University of Wisconsin, Amanda Gault, Peter Hyland, and

Siddharth Malu, who provided assistance and advice in the laboratory and discussions

about cosmology and who made the lab a very supportive and fun place to work. I

am extraordinarily grateful for the work done by Sara Stanchfield, who completed the

cryogenic testing of the black body source described in this thesis, and who also com-

pleted some of the E-M simulations and microwave measurements for the design of

the THM optical coupling scheme. I also thank Zubair Abdulla, Eric Katzlenick and

Amy Lowitz for their important contributions to the THM optical coupling design.

As this is a culmination of a very specific dream I have had since I was about

10 years old, I also wish to express my gratitude at this moment to my teachers

throughout all levels of my schooling. I thank my elementary school teacher, Mr.

Steve Kaio-Maddox, who handed me the first book I ever read about physics, and put

up with my persistent questions. I also need to acknowledge the late Carl Sagan, who

gave meaning and purpose to my life though he had already passed away by the time

I read his first book in high school. I thank the dedicated and caring teachers I had

in all disciplines at Deer Park High School. I also thank the gifted teachers on the

faculty of the Bryn Mawr College Physics Department during my years there from

1999-2003, and especially my advisers there, Matthew Rice and Juan Burciaga. It is

through their support and the exceptional environment which they nurtured within

that department that I gained both the ability and the confidence to pursue this degree

at the graduate level.

Most importantly however, I need to express my gratitude to my adviser, Peter

vi

Timbie. I feel extraordinarily lucky to have been his student. His unequaled patience,

guidance, as well as enthusiasm and understanding, made both this project, and my

happiness as a graduate student during these last five years, possible. One could not

ask for a better mentor.

vii

Dedication

I wish to dedicate this thesis to my family. To my mother and father, Marianne

and Wayne Barrentine. To my grandmother, Delores Barrentine, and in memory of

my grandfather, Harold Barrentine. To my grandmother and grandfather, Janet and

John Salomone. This is dedicated to them because of their care, encouragement and

love, and their support of my education and the development of my curiosity and

seriousness towards all areas of life. It is also dedicated to them because they have

accepted my absence from their daily lives for much of the last twelve and a half years

in the pursuit of this degree. I also wish to dedicate this thesis to my uncle, Carl

Barrentine, for his encouragement of all of his nieces to pursue higher education and

to ask questions about the deeper meanings of life and the workings of the universe.

Above all, however, this thesis is dedicated to my sister, Erin Barrentine, without

whom I would not be able to even contemplate existence.

viii

Contents

Abstract i

1 Introduction 1

1.1 The History and Science of the CMB . . . . . . . . . . . . . . . . . . . 1

1.1.1 CMB Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Inflation & Gravitational Waves . . . . . . . . . . . . . . . . . . 6

1.1.3 CMB Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.1.4 Other Detector Science Applications . . . . . . . . . . . . . . . 15

1.2 CMB Detector Technology & THM Advantages . . . . . . . . . . . . . 16

1.2.1 History of CMB Detectors & Future Detector Needs . . . . . . . 18

1.2.2 THM Detector Advantages . . . . . . . . . . . . . . . . . . . . . 22

1.3 Thesis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 General Bolometer & TES Theory 24

2.1 Basic Bolometer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 The Superconducting Transition . . . . . . . . . . . . . . . . . . . . . . 27

2.3 The Transition Edge Sensor . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4 Basics of SQUIDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

ix

2.5 SQUID Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.6 The TES Bolometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.7 Electrothermal Feedback & Bias Conditions . . . . . . . . . . . . . . . 52

2.8 Bolometer Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3 Physical Effects in the THM Detector 56

3.1 The Hot-Electron Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.1 Hot-Electron Effect Theory . . . . . . . . . . . . . . . . . . . . 58

3.1.2 Measurements of the Hot-Electron Effect . . . . . . . . . . . . . 61

3.1.3 HEB Detector History . . . . . . . . . . . . . . . . . . . . . . . 62

3.2 Other Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2.1 Andreev Reflection . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2.2 Boundary Conductance . . . . . . . . . . . . . . . . . . . . . . . 65

3.2.3 Wiedemann-Franz Conductance . . . . . . . . . . . . . . . . . . 69

3.2.4 Radiation Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.3 The Superconducting Proximity Effect . . . . . . . . . . . . . . . . . . 71

3.3.1 Superconductivity Theory & the Proximity Effect . . . . . . . . 72

3.3.2 Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . . . 74

3.3.3 Usadel Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3.4 The Characteristic Lengths of the Proximity Effect . . . . . . . 78

3.3.5 S-N Bilayer Theory . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.3.6 S-N Bilayer Measurements . . . . . . . . . . . . . . . . . . . . . 83

3.3.7 S-S’-S Theory & Predictions . . . . . . . . . . . . . . . . . . . . 84

3.3.8 S-S’-S Measurements in the Literature . . . . . . . . . . . . . . 85

x

3.3.9 S-N-S Theory & Predictions . . . . . . . . . . . . . . . . . . . . 87

3.3.9.1 Likharev-Usadel Model . . . . . . . . . . . . . . . . . . 87

3.3.9.2 deGennes-GL Model . . . . . . . . . . . . . . . . . . . 87

3.3.9.3 Dubos-Usadel Model . . . . . . . . . . . . . . . . . . . 89

3.3.9.4 van Dover-GL Model . . . . . . . . . . . . . . . . . . . 91

3.3.9.5 Kuprianov-Lukichev-Usadel Model . . . . . . . . . . . 93

3.3.10 S-N-S Measurements in the Literature . . . . . . . . . . . . . . 97

3.3.11 Conclusions of Modeling the Lateral Proximity Effect . . . . . . 99

4 The THM Thermal Model & Detector Optimization 100

4.1 THM Thermal Model & Noise Sources . . . . . . . . . . . . . . . . . . 100

4.1.1 Ideal Model Theory & Noise . . . . . . . . . . . . . . . . . . . . 102

4.1.2 Non-Ideal Model Theory & Noise . . . . . . . . . . . . . . . . . 108

4.2 Thermal & Microwave Optimization . . . . . . . . . . . . . . . . . . . . 116

4.2.1 Photon Background Noise & Detector Loading Conditions . . . 117

4.2.2 Microwave Circuit Constraints . . . . . . . . . . . . . . . . . . . 119

4.2.3 Fabrication & Material Constraints . . . . . . . . . . . . . . . . 121

4.2.4 Stability Constraints & Optimal Bias Conditions . . . . . . . . 122

4.2.5 Electron-Phonon Versus Electron-Electron . . . . . . . . . . . . 123

4.2.6 Phonon-Phonon Versus Electron-Phonon . . . . . . . . . . . . . 130

4.2.7 SQUID & Johnson Noise Constraints . . . . . . . . . . . . . . . 132

4.2.8 The Optimal Design for the THM Detector . . . . . . . . . . . . 134

4.2.9 General Optimization for CMB Bolometric Detectors . . . . . . 143

xi

5 THM Test Devices & “Dark” Measurements 150

5.1 Test Devices & Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.2 “Dark” Cryogenic Test Setup . . . . . . . . . . . . . . . . . . . . . . . 157

5.2.1 Dewar & Cryogenic Setup . . . . . . . . . . . . . . . . . . . . . 157

5.2.2 SQUID Readout Setup . . . . . . . . . . . . . . . . . . . . . . . 160

5.3 Inquiry into a Non-Ideal THM Model . . . . . . . . . . . . . . . . . . . 162

5.3.1 I-V Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.3.2 Thermal Conductance Measurements . . . . . . . . . . . . . . . 168

5.3.3 Responsivity Measurements . . . . . . . . . . . . . . . . . . . . 172

5.3.4 Noise measurements . . . . . . . . . . . . . . . . . . . . . . . . 172

5.4 Inquiry into the Hot-Electron Effect . . . . . . . . . . . . . . . . . . . . 179

5.4.1 THM2003 Test Device . . . . . . . . . . . . . . . . . . . . . . . 179

5.4.2 us25 Test Devices . . . . . . . . . . . . . . . . . . . . . . . . . . 180

5.4.3 Conclusions of Inquiry into Electron-Phonon Effect . . . . . . . 188

5.5 Inquiry into the Lateral Proximity Effect . . . . . . . . . . . . . . . . . 189

5.5.1 S-S’-S Junction Measurements . . . . . . . . . . . . . . . . . . . 189

5.5.2 S-S’-S Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 196

5.5.3 S-N-S Junction Measurements . . . . . . . . . . . . . . . . . . . 196

5.5.4 S-N-S Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 204

5.6 Inquiry into Detector NEP . . . . . . . . . . . . . . . . . . . . . . . . . 205

6 THM Microwave Design & Simulations 207

6.1 THM Microwave Design . . . . . . . . . . . . . . . . . . . . . . . . . . 207

6.1.1 Microstrip Transmission Lines . . . . . . . . . . . . . . . . . . . 210

xii

6.1.2 Double Slot Antenna Design . . . . . . . . . . . . . . . . . . . . 215

6.1.3 Four-Fold Slot Antenna Design . . . . . . . . . . . . . . . . . . 217

6.1.4 Radial & Rectangular Stubs . . . . . . . . . . . . . . . . . . . . 223

6.1.5 Low-pass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

6.1.6 Couplers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

6.1.7 Termination Structure . . . . . . . . . . . . . . . . . . . . . . . 228

6.1.8 DC Chokes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

6.1.9 Impedance Transformers . . . . . . . . . . . . . . . . . . . . . . 237

6.2 THM Microwave Testing Schemes & Preliminary Measurements . . . . 239

6.2.1 CPW Probe Station Microwave Design & Measurements . . . . 239

6.2.2 Optical Coupling Schemes . . . . . . . . . . . . . . . . . . . . . 249

6.2.2.1 The Horn Antenna . . . . . . . . . . . . . . . . . . . . 250

6.2.2.2 The Extended Hemispherical Lens . . . . . . . . . . . 253

6.2.2.3 The Rexolite Lens . . . . . . . . . . . . . . . . . . . . 256

6.2.3 Black Body Source . . . . . . . . . . . . . . . . . . . . . . . . . 257

6.2.3.1 Nichrome Black Body Source . . . . . . . . . . . . . . 259

6.2.3.2 Finline Black Body Source . . . . . . . . . . . . . . . . 263

6.2.4 External RF Source Coupling . . . . . . . . . . . . . . . . . . . 268

7 Conclusion 269

7.1 Summary of Understanding of the THM Design . . . . . . . . . . . . . 269

7.2 Future Work Involving The Lateral Proximity Effect . . . . . . . . . . 271

7.3 Future Microwave Work . . . . . . . . . . . . . . . . . . . . . . . . . . 273

7.4 Future Work for Scaling to Large Arrays . . . . . . . . . . . . . . . . . 274

xiii

7.5 A Last Word: The Recommended THM Design . . . . . . . . . . . . . 274

xiv

List of Figures

1.1 Timeline of the universe, courtesy WMAP-Science team [114]. . . . . . 2

1.2 The history of CMB experiments. . . . . . . . . . . . . . . . . . . . . . 5

1.3 The Thompson scattering process through which a quadrupole moment

in the photon anisotropy from the perspective of the target electron

gives rise to a polarization in the CMB. Blue field lines on the incoming

photons indicate colder and thus lower energy photons and the red field

lines indicate hotter and thus higher energy photons. The net effect is

an outgoing electric field magnitude which is stronger in one direction.

Figure adapted from Hu & White [47]. . . . . . . . . . . . . . . . . . . 12

1.4 The two types of polarization patterns in the CMB. An example of an

E-mode polarization pattern which exhibits a divergence and a B-mode

polarization pattern which exhibits a curl. . . . . . . . . . . . . . . . . 13

1.5 The measured temperature angular power spectrum of the CMB. (Cour-

tesy WMAP science team [114]). The amplitude here is plotted in terms

of the variance of the fluctuations multiplied by the average CMB tem-

perature T = 2.73 K squared, such that the plot has units temperature

squared. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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1.6 The measured E-mode polarization (EE) and temperature-E-mode cross-

correlation (EB) power spectra, and measured upper limits on B-mode

polarization (BB) power spectrum of the CMB. (Courtesy BICEP Sci-

ence Team [15]). The predicted B-mode polarization angular power

spectrum in the CMB is also shown by the grey line, with the contri-

butions near l ∼ 100 coming from primordial gravitational waves. The

B-mode component from gravitational lensing of E-mode to B-mode

peaks at l ∼ 1000. The signal is plotted in terms of the variance of the

fluctuations multiplied by the average CMB temperature T = 2.73 K

squared, such that the plot has units of temperature squared. . . . . . . 14

1.7 The basic detection schemes for CMB observing using coherent receivers

or incoherent bolometric detectors. . . . . . . . . . . . . . . . . . . . . 17

1.8 The frequency range of CMB experiments by type as a function of time. 19

1.9 The number of detector/receivers for CMB experiments as a function

of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1 The components to a simple bolometer. . . . . . . . . . . . . . . . . . . 25

2.2 The energy gap interpretation of superconductivity. Quasiparticles oc-

cupy the free electron states in the superconductor and electrons bound

in Cooper pairs occupy the bound states. Note however, that unlike

bound electron states in the other cases, in a superconductor, electrons

bound in Cooper-pair states do carry an electrical current. . . . . . . . 30

xvi

2.3 The RTES vs. T curve of a superconducting TES from a THM test

device of variation THM5 (this and other THM test devices will be

described in Chapter 5). . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 A superconducting ring allows only quantized values of magnetic flux

through the center due to quantization of the superconducting wave-

function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5 A Josephson junction is two superconductors separated by a thin insu-

lating link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.6 A “two-fluid” circuit diagram for a Josephson junction. . . . . . . . . . 39

2.7 An example of the I-V characteristics of a Josephson junction with zero

capacitance. Figure courtesy Gallop [31]. . . . . . . . . . . . . . . . . . 40

2.8 A simple SQUID: a superconducting ring broken by two Josephson junc-

tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.9 An example of the I-V characteristics of a current-biased SQUID for

different magnetic flux values. . . . . . . . . . . . . . . . . . . . . . . . 42

2.10 The measured voltage as a function of magnetic flux for the current-

biased NIST-Series Array SQUID which was used to read out the THM

test devices. Further measurements and details of this SQUID readout

are presented in Chapter 5. . . . . . . . . . . . . . . . . . . . . . . . . 43

2.11 A SQUID as a magnetometer. . . . . . . . . . . . . . . . . . . . . . . . 45

2.12 Electronic feedback to linearise SQUID. . . . . . . . . . . . . . . . . . . 46

2.13 The linear output of the NIST series array SQUID used to read out

THM test devices with feedback. . . . . . . . . . . . . . . . . . . . . . 46

xvii

2.14 The electical circuit for a voltage-biased TES read out by a SQUID

amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.15 The thermal circuit for an ideal bolometer detector. . . . . . . . . . . . 49

3.1 (a) The bilayer superconducting proximity effect. (b) A lateral super-

conducting proximity effect. (c) The two different TES designs where

the lateral or bilayer proximity effect play a role: a S-S’-S junction (left)

and a S-N-S junction (right). . . . . . . . . . . . . . . . . . . . . . . . . 73

3.2 Superconductivity theories, their relationships, and applicability. . . . . 75

3.3 The normal metal coherence length for a dirty Nb-Au-Nb junction (fol-

lowing Equation 3.21) as a function of temperature for Au RRR = 1&2

is plotted for 30 nm and 300 nm Au thicknesses. Room temperature

Au resistivity is assumed to be 3 · 10−8 Ω · m and Nb Tc = 8.3 K is

assumed. For typical S-N-S THM devices one expects ξN = 0.1− 0.4 µm. 80

3.4 The normal metal coherence length for a Nb-Mo/Au-Nb junction as

a function of temperature for Au RRR = 1 is plotted for 300 nm Au

thickness. Room temperature Au resistivity of 3·10−8 Ω·m, Nb Tc = 8.3

K and Mo/Au Tc = 200 mK are assumed. For the Sadleir-G-L model

(which matches the low temperature Usadel Model), the normal metal

coherence length is also plotted assuming Mo resistivity of 5.3 · 10−8

Ω ·m, and also assuming Sadleir et al.’s measured fit to ξi = 738 nm.

For all these models for typical THM devices ξN ∼ 0.1 − 1.0 µm is

predicted, with the coherence lengths predicted by the fit values on the

higher side of this range. . . . . . . . . . . . . . . . . . . . . . . . . . . 82

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3.5 Critical current behaviour of a Nb-Mo/Au-Nb S-S’-S TES as predicted

by the Sadleir et al. model using Equation 3.29. The legends on each

plot indicate lead-to-lead lengths in meters. a) Ic calculated assuming

Au resistivity of ρ = 3.3 · 10−8 Ω·m, and Au thickness of 300 nm. b)

Ic assuming Mo resistivity of ρ = 5.3 · 10−8 Ω·m with bilayer thickness

dominated by Au thickness of 300 nm. c) Ic calculated assuming the

measured fit value by Sadleir et al. for ξi = 738 nm. For all predictions

λr = 79 nm (the measured value from Sadleir et al.), Nb Tc = 8.4 K,

and Mo/Au TcN = 170 mK have been assumed. The predictions for Ic

using the measured fit ξi value are ∼ 0.5−4 orders of magnitude higher

than predictions using ξi values calculated from resistivity. . . . . . . . 86

3.6 Critical current behaviour of a Nb-Au-Nb S-N-S TES predicted by the

Likharev-Usadel model using Equation 3.31. The legend on each of the

graphs indicates different lead-to-lead lengths in meters. a) Assuming

a Au resistivity of ρ = 3.3 · 10−8 Ω ·m and a Au thickness of 30 nm. b)

Assuming a Au resistivity of ρ = 3.3 · 10−8 Ω ·m and a Au thickness of

300 nm. In both cases a Nb transition temperature Tc = 8.4 K and a 3

µm wide device is assumed. This model predicts Ic ∼ 0.1 − 100 µA at

∼ 200 mK for a 3 µm long THM device. . . . . . . . . . . . . . . . . . 88

xix


deGennes-GL model using Equation 3.32. The legend on each of the



Assuming a Au resistivity of ρ = 3.3 · 10−8 Ω ·m and a Au thickness of

300 nm. In both cases a Nb transition temperature Tc = 8.4 K and a 3

µm wide device is assumed. This model predicts Ic ∼ 10 nA-10 µA at

∼ 200 mK for a 3 µm long THM device. . . . . . . . . . . . . . . . . . 90


Dubos-Usadel model using Equations 3.33 and 3.34. The legend on each

of the graphs indicates lead-to-lead lengths in meters. a) Assuming Au

resistivity of ρ = 3.3·10−8 Ω·m and Au thickness of 30 nm. b) Assuming

Au resistivity of ρ = 3.3 · 10−8 Ω ·m and Au thickness of 300 nm. In

both cases Nb Tc = 8.4 K and a 3 µm wide device is assumed. This

model predicts Ic ∼ 1 nA-0.1 µA at ∼ 200 mK for a 3 µm long TES.

The discontinuity in the curve for the shortest devices indicates the

crossover from the high to the low temperature limit. . . . . . . . . . . 92

xx


van Dover-GL model using Equation 3.35. The legend on each of the



Assuming a Au resistivity of ρ = 3.3 · 10−8 Ω ·m and a Au thickness

of 300 nm. In both cases a Nb transition temperature Tc = 8.4 K and

Aρnγρs

= 0.24 is assumed. This model predicts Ic ∼ 0.1− 10 µA at ∼ 200

mK for a 3 µm long THM device. . . . . . . . . . . . . . . . . . . . . . 94

3.10 Critical current predictions for a Nb-Au-Nb S-N-S TES for the Likharev-

Usadel, deGennes-GL, Dubos-Usadel and van Dover-GL models. Here

a Nb transition temperature Tc = 8.4 K, Au resistivity of ρ = 3.3 · 10−8

Ω ·m, and Au thickness of 180 nm is assumed for a 3 µm x 3 µm device.

Also plotted for comparison purposes is the critical current behaviour

for a 3 µ long Mo/Au S-S’-S TES, following the Sadleir et al. model

and using their measured fit values for ξi and λr, with TcN = 180 mK.

For the S-N-S case, with a bias current of ∼1 µA, an effective TES Tc

of 100-400 mK is expected across the junction. . . . . . . . . . . . . . . 95

xxi

3.11 Limiting Γc =Rc

Rjunctionpredicted from Kuprianov-Lukichev-Usadel the-

ory requiring there be less than 20% change in critical current due to

finite contact resistance between superconducting Nb leads and a nor-

mal Au TES. A Au resistivity of ρ = 3.3 ·10−8Ω·m, thickness of 30−300

nm, and a 3 µm wide device is assumed at 150 mK. For typical THM

test devices, for Γ . 10 − 15%, only a small shift in critical current is

expected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.1 The basic geometry of the THM components. . . . . . . . . . . . . . . 101

4.2 The complete thermal model for the THM. . . . . . . . . . . . . . . . . 103

4.3 The ideal thermal model for the THM detector. . . . . . . . . . . . . . 104

4.4 A thermal model for a non-ideal THM. . . . . . . . . . . . . . . . . . . 109

4.5 The non-ideal THM thermal model, simplified by the symmetry of the

device. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.6 Microstrip line geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.7 Stability limits on the SQUID readout input inductance, L. The r.h.s.

and l.h.s of Equation 2.21 are plotted and the maximum L for stability

is given by the point where the lines cross. (a) 20 µm2 TES, assuming

‘CMB-ground’ conditions, for Tbath = 240 mK.(b) 20 µm2 TES, assum-

ing ‘CMB-space’ conditions, for Tbath = 50 mK. (c) 3 µm2 TES, as-

suming ‘CMB-ground’ conditions, for Tbath = 240 mK.(d) 3 µm2 TES,

assuming ‘CMB-space’ conditions, for Tbath = 50 mK. ‘CMB-ground’

results are similar for Tbath = 50 mK. For these THM designs, stability

requires L ≤ 10 nH−1 µH. . . . . . . . . . . . . . . . . . . . . . . . . 124

xxii

4.8 The electron-phonon conductance of the combined TES and absorber

structure, Ge−p, the electron-electron conductance across the entire

TES and absorbing structure (absorber dominated in this case), Ge−e,

and the electron-phonon conductance of the absorber only, Ge−p,Abs

plotted as a function of absorber resistance, RAbs. Here we have as-

sumed a 10 µm x 10 µm x 300 nm Au TES and a 3 µm wide, 800 nm

thick, Bi absorber. For electron-phonon thermal conductance we have

assumed n = 5 and ‘literature values’ for Σs. (a) ‘CMB-ground’ condi-

tions and Tbath = 240 mK (b) ‘CMB-ground’ conditions and Tbath = 50

mK (c) ‘CMB-space’ conditions and Tbath = 50 mK. For these designs

RAbs . 80 − 100 Ω is required to avoid power bypassing the TES, and

RAbs . 3 − 80 Ω is required to avoid a temperature differential across

the absorber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.9 The estimated fraction of power which flows through the TES (PTES ∼Ge−e

Ge−e+Ge−p) as a function of absorber resistance, RAbs. Here we have

assumed a 10 µm x 10 µm x 300 nm Au TES and a 3 µm wide, 800 nm

thick, Bi absorber. For electron-phonon thermal conductance we have

assumed n = 5 and literature Σ values. (a) ‘CMB-ground’ conditions

and Tbath = 240 mK (b) ‘CMB-ground’ conditions and Tbath = 50 mK

(c) ‘CMB-space’ conditions and Tbath = 50 mK. For these designs Rabs .

40−100 Ω is necessary to obtain at least 80−100% power flow through

the TES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

xxiii

4.10 The TES electron-electron thermal conductance, Ge−e,TES plotted with

the total electron-phonon thermal conductance, Ge−p, as a function of

TES resistance, RTES. In this case we assume a Bi absorber geometry

of 3 µm x 6 µm x 800 nm. We also assume a Mo/Au TES of a fixed

width 10 µm and thickness of 300 nm (resulting in ρ ∼0.1 Ω). For

electron-phonon thermal conductance we have assumed n = 5 and lit-

erature Σ values. (a) ‘CMB-ground’ conditions and Tbath = 240 mK (b)

‘CMB-ground’ conditions and Tbath = 50 mK (c) ‘CMB-space’ condi-

tions and Tbath = 50 mK. Requiring Ge−p < Ge−e,TES, gives a constraint

of RTES < 0.5− 5 Ω depending on the application and bath temperature.129

4.11 The total electron-electron thermal conductance across the absorber

and TES, Ge−e, plotted with the total electron-phonon thermal conduc-

tance, Ge−p, and the absorber electron-phonon conductance, Ge−p,Abs,

as a function of TES width, wAu, for a square Mo/Au TES with Au

thickness 300 nm. Here we have assumed a 3 µm x 6 µm x 800 nm Bi

absorber for RAbs = 30 Ω. For electron-phonon thermal conductance we

have assumed n = 5 and literature Σ values. (a) ‘CMB-ground’ condi-

tions and Tbath = 240 mK (b) ‘CMB-ground’ conditions and Tbath = 50


wAu . 1− 20 µm is required to avoid a temperature differential across

the absorber (requiring Ge−e > Ge−p). There is no requirement to avoid

power bypassing the TES (requiring Ge−e > Ge−p,Abs). . . . . . . . . . . 131

xxiv

4.12 The electron-phonon thermal conductance, Ge−p, boundary thermal

conductance, Gboundary, and total thermal conductance Gtot, of a THM

detector as a function of TES width, wAu. Here we have assumed a

Bi absorber of size 3 µm x 6 µm x 800 nm and a square Mo/Au TES

with a Au thickness of 300 nm. We have assumed n = 5 and coupling

‘literature values’ for Σs, and a average literature value for the bound-

ary thermal conductance coefficient, Cb = 15.5 · 10−4 Km2

W[111]. (a)

‘CMB-ground’ conditions and Tbath = 240 mK (b) ‘CMB-ground’ con-

ditions and Tbath = 50 mK (c) ‘CMB-space’ conditions and Tbath = 50

mK. Gboundary is predicted to have some impact or dominate the total

thermal conductance over the full range of TES size scales for ground

and space observing conditions. . . . . . . . . . . . . . . . . . . . . . . 133

4.13 SQUID and Johnson noise as a function of TES resistance, RTES. We

assume a Bi absorber of size 3 µm x 6 µm x 800 nm and a 3 µ wide

Mo/Au TES with Au thickness of 300 nm (RTES = 0.1 Ω) and an

adjustable length and total resistance. We assume a shunt resistor with

Rs = 0.025 Ω and SQUID current noise of NEI = 4√Tbath · 10−12 A√

Hz.

(a) ‘CMB-ground’ conditions and Tbath = 240 mK (b) ‘CMB-ground’

conditions and Tbath = 50 mK (c) ‘CMB-space’ conditions and Tbath =

50 mK. A bath temperature of Tbath = 240 mK requires RTES < 1

Ω for ground-based observing to ensure that the Johnson and SQUID

noise are insignificant. At bath temperature of Tbath = 50 mK, RTES <

20− 100 Ω is required for ground- or space-based observing. . . . . . . 135

xxv

4.14 Thermal detector NEP as a function of TES width, wAu, for a square

Mo/Au TES with Au thickness 300 nm assuming a 3 µm x 6 µm

x 800 nm Bi absorber. Detector and background photon NEP for

‘CMB-ground’, ‘CMB-space’ and ‘FIR-spectral’ observing conditions

are shown. We have assumed n = 5 and ‘literature values’ for

Σs. (a) Tbath = 240 mK. (b) Tbath = 50 mK. At Tbath = 240 mK back-

ground limited NEP is obtainable for CMB ground observing only, for

wAu . 3 µm. At Tbath = 50 mK background limited NEP is obtainable

for both CMB ground and space observing for wAu = 3− 50 µm. . . . . 137


Mo/Au TES with Au thickness 300 nm, assuming a 3 µm x 6 µm


‘CMB-ground’, ‘CMB-space’ and ‘FIR-spectral’ observing conditions

are shown. We have assumed ‘measured n = 5’ values for Σs.

(a) Tbath = 240 mK. (b) Tbath = 50 mK. At Tbath = 240 mK background

limited NEP is obtainable for ground observing only, for wAu . 10 µm.

At Tbath = 50 mK background limited NEP is obtainable for both CMB

ground and space observing for wAu = 1− 100 µm. . . . . . . . . . . . 138

xxvi


Mo/Au TES with Au thickness 300 nm, assuming a 3 µm x 6 µm


‘CMB-ground’, ‘CMB-space’ and‘FIR-spectral’ conditions are shown.

We have assumed ‘measure n = 6’ values for Σ. (a) Tbath = 240

mK. (b) Tbath = 50 mK. At Tbath = 240 mK near background limited

NEP is obtainable for CMB ground and space observing, for wAu . 40

µm and wAu . 4 µm, respectively. At Tbath = 50 mK background

limited NEP is obtainable for CMB observing for wAu = 1− 1000 µm. . 139

4.17 The bolometer temperature, Tbolo, as a function of TES width, wAu, for

the THM design variations of Figure 4.14 for CMB observing conditions

(for FIR observing Tbolo = Tbath). (a) Tbath = 240 mK. (b) Tbath = 50 mK.140







xxvii

4.20 The thermal fluctuation NEP of a generic CMB observing bolometer

in comparison to background photon NEP. A temperature depen-

dence of the power flow, n = 2 is assumed. (a) ‘CMB-ground’

observing conditions with Tbath = 240 mK. (b) ‘CMB-space’ observing

conditions with Tbath = 240 mK.(c) ‘CMB-ground’ observing condi-

tions with Tbath = 50 mK. (d) ‘CMB-space’ observing conditions with

Tbath = 50 mK. For all of these cases, the thermal detector NEP in the

low power loading limit is also plotted. . . . . . . . . . . . . . . . . . . 145









xxviii

















xxix









5.1 In-process images of some of the key fabrication steps for the THM

test devices. (a) Optical image of bilayer TES on wafer THM5 after

completion of the Au ionmill and Mo RIE patterning. The Mo extends

out to make contact with the Nb leads which will be deposited next.

The Mo on the non-lead sides of the TES will be etched away during the

Nb patterning step. (b) Optical image of wafer THM4 after completion

of Nb (SF6 + 02) RI sloped sidewall etch. The sidewall can been seen

even in this optical image as Nb (blue) extends past the dark top edge. 153

xxx

5.2 In-process images of some of the key fabrication steps for the THM test

devices. (a) SEM image of THM4 showing good step coverage over a

Nb sloped sidewall. This view shows the Nb microstripline crossing the

gap in the Nb ground plane(see Chapter 6). The ground plane is seen

through the AlO dielectric layer. (b) SEM image of a 3 µm x 9 µm Bi

absorber test device (not incorporated into a THM) after liftoff. The

grainy structure of the evaporated Bi is observable. Thin Au contact

pads under the Bi help make contact to the Nb leads. . . . . . . . . . . 154

5.3 The resistivity of evaporated Bi resistivity measured for several test

wafers (with Bi thickness listed) as a function of temperature. An

increase in resistivity is seen as the temperature decreases. . . . . . . . 156

5.4 The dewar shielding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

5.5 The coldstage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5.6 The THM chip and 1st stage SQUID mount which fits inside a Nb

shielded cylindrical can. . . . . . . . . . . . . . . . . . . . . . . . . . . 160

5.7 Diagram of the 2-stage SQUID electrical connections for the read out

of the THM TES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.8 The 2nd stage SQUID bias line voltage as a function of the input cali-

bration signal to the 1st stage SQUID input coil. With feedback off the

second stage signal follows the input signal. With feedback on the bias

signal is zero no matter the input signal, as a nulling feedback signal,

which tracks the input signal, is being fed back to the 1st stage SQUID. 163

xxxi

5.9 The SQUID calibration curve. Plotted is the feedback signal to the

first stage SQUID as a function of the signal to the input coil. The 1st

stage feedback signal is what is read out. The gain factor is given by

the slope of this linear curve. . . . . . . . . . . . . . . . . . . . . . . . . 164

5.10 (a) Fabrication steps and cross-sectional view showing the layers for

each of the THM2003 test devices. Thicknesses are not drawn to scale,

but are listed on the labels for each layer. (b) Optical image of a test

device of variation THM2003. . . . . . . . . . . . . . . . . . . . . . . . 166

5.11 I-V curves for a device from wafer THM2003 at different bath tempera-

tures (labelled in the figure). The solid black line indicates RTES =0.32

Ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

5.12 P vs. Tbath measurement for devices from wafer THM2003. . . . . . . . 169

5.13 Noise measurements of the THM2003 test devices at 1 µV bias, 311 mK

bath temperature. The best fit is given by the non-ideal model. . . . . 174

5.14 Noise measurements of the THM2003 test devices at 0.5 µV bias, 311

mK bath temperature. The best fit is given by the non-ideal model. . . 175

5.15 Noise measurements of the THM2003 test devices at 0.2 µV bias, 311

mK bath temperature. The best fit is given by the non-ideal model. . . 176

5.16 Predictions of detector NEP for the ideal electron-phonon model and

the non-ideal model at 1 µV bias, showing contributions from thermal,

Johnson, and SQUID noise sources. The non-ideal model matches the

measured noise (Figure 5.13) best. . . . . . . . . . . . . . . . . . . . . . 178

xxxii


each of the us23/us25 test devices. Thicknesses are not drawn to scale,

but are listed on the labels for each layer. (b) SEM image of a test

device of variation us23/us25. . . . . . . . . . . . . . . . . . . . . . . . 181

5.18 The layer overlap and active thermal area of the TES and absorber for

the two micron-sized test devices of variation us25. . . . . . . . . . . . 183

5.19 Joule power applied to the absorber versus hot-electron temperature,

Tbolo, while the bath was held constant at 190 mK for the THM test

device of variation us25 with Au TES volume of 0.27 µm3. Fits to the

predicted forms of the power law dependence are shown. . . . . . . . . 185

5.20 Joule power applied to the absorber versus hot-electron temperature,

Tbolo, while the bath was held constant at 190 mK for the THM test

device of variation us25 with Au TES volume of 0.58 µm3. Fits to the

predicted forms of the power law dependence are shown. . . . . . . . . 186


each of the THM4/THM5 test devices. Thicknesses are not drawn to

scale, but are listed on the labels for each layer. (b) Optical image of a

test device of variation THM4/THM5. . . . . . . . . . . . . . . . . . . 190

xxxiii

5.22 (a) Schematic of contact between the Mo/Au TES and the Nb leads for

devices THM4/THM5. (b) SEM image of contact between the Nb leads

and TES for a 3 µm x 3 µm THM4/THM5 device with no absorber or

microwave circuit. The Mo layer which extends from the TES is visible

as a slight change in Nb thickness. The over-etching of the Nb leads

during the Nb etch on top of the Mo/Au TES is also clearly visible. . . 192

5.23 (a) Tc vs. lead-to-lead distance, L, for the 65 nm Mo/350 nm Au bilayer

devices of variation THM5. A curve fit is shown for the parameters TcN

= 240 mK, LN = 12 µm and m = 5. (b) Tc vs. lead-to-lead distance,

L, for the 55 nm Mo/350 nm Au bilayer devices of variation THM4.

A curve fit is shown for the parameters TcN = 170 mK, LN = 12 µm

and m = 5. Data points indicate nominal L values while the error bars

indicate uncertainties due to over-etching. Error bars for Tc indicate

uncertainty distinguishing the TES from the Nb lead transition. . . . . 193

5.24 Resistance vs. temperature curves for two of the THM test devices.

(a) Nominal 12 µm x 12 µm, 55 nm/350 nm thick, Mo/Au TES with

absorber of variation THM4. (b) Nominal 3 µm x 3 µm, 55 nm/350 nm

thick Mo/Au TES with absorber of variation THM4 with a transition

near the Nb lead transition. . . . . . . . . . . . . . . . . . . . . . . . . 195


each of the THMA4/THMA24 test devices. Thicknesses are not drawn

to scale, but are listed on the labels for each layer. (b) SEM image of

a test device of variation THMA4/THMA24. . . . . . . . . . . . . . . . 198

xxxiv

5.26 SEM and AFM images after ion-milling, cleaning of the Au surface, and

Nb deposition. A rough surface of the Au metal is seen, but no organic

contamination which would be indicated by dark or stringy residue can

be seen in these images. . . . . . . . . . . . . . . . . . . . . . . . . . . 200

5.27 Measured contact resistance, Rc (with error), for all devices tested.

These measurements are normalized to assume a 3 µm2 contact area

for each device. Error bars indicate error due to the measurement noise. 201

5.28 Chart summary of Table 5.8. The Kuprianov and Lukichev Rc limits

are also shown in the plot. . . . . . . . . . . . . . . . . . . . . . . . . . 202

6.1 Microwave design at 92 GHz to couple radiation to the THM detector

for the THM4 and THM5 test devices. Devices at 43 GHz were also

designed and fabricated. The signal from the double slot antenna is

transmitted via a low-pass filter to the RF-terminated THM detector.

Choked bias leads provide DC connections to both the TES (for bias

and SQUID read out) and the absorber (for DC Joule heating of the

absorber). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

6.2 Microwave design at 92 GHz to couple radiation to an array of 128

individual THM detectors for the us23/us25 test devices. The coupling

to such a large number of detectors was done in an attempt to find

the optimal lateral proximity effect S-N-S TES, with each of the 128

detectors of varying TES lead-to-lead length. In actual operation, with

an optimized THM device, one imagines only a single detector coupling

to the slot antenna without the branching network. . . . . . . . . . . . 212

xxxv

6.3 Optical image of one of the us23/us25 test device detector chips. Chip

size is 15 mm x 15 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

6.4 Two-detector coupled alternative slot antenna design for test devices

THM4/THM5. This device was designed specifically to test the perfor-

mance of the low-pass filter. A directional coupled line coupler trans-

mits the signal to two separate detectors, one line via a low-pass filter

and another line without the filter. By coupling a swept microwave sig-

nal into the slot antenna the filter response as a function of frequency

can be determined. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

6.5 The geometry of the THM4/THM5 double slot antenna design. . . . . 216

6.6 The simulated reflection (S11) from an input signal sent into a port

which feeds the single-line microstrip to the double slot antenna for

devices of variation THM4/THM5. The low level of reflection near 92

GHz indicates power radiated by the antenna. . . . . . . . . . . . . . . 218

6.7 An optical image of the 92 GHz double slot antenna on a test device of

variation THM5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

6.8 The geometry of the us23/us25 four-fold slot antenna design. . . . . . . 221

6.9 The simulated reflection (S11) at the port which feeds the microstrip to

the four-fold slot antenna for the test devices of variation us23/us25.

The low level of reflection indicates power radiated by the antenna. . . 222

6.10 The predicted beam pattern at 92 GHz for the four-fold slot antenna

design for the test devices of variation us23/us25. As expected, the gain

into the Si substrate is greater than into vacuum. . . . . . . . . . . . . 223

xxxvi

6.11 The geometry of the radial stub design for devices of variation THM4/THM5.224

6.12 The geometry of the rectangular stub design for devices of variation

us23/us25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

6.13 Input impedance for the THM4/THM5 radial stub design. A very broad

bandwidth at 92 GHz indicating a short to ground (with impedance

Z ≪ 20 Ω) is predicted. . . . . . . . . . . . . . . . . . . . . . . . . . . 226

6.14 The geometry of the low-pass filter, consisting of three sub-filters, indi-

cated as “Part 1”, “Part 2”, and “Part 3”. “Part 1” and “Part 3” are

“stub” type filters and “Part 2” is a “stepped impedance” type filter. . 227

6.15 Optical image of the low-pass filter from test device of variation THM5. 228

6.16 Simulated transmission (S12) through the low-pass filter parts. Part “2”

defines the low frequency cut-off. Part “1” blocks the higher frequency

leaks from part “2”. Part “3” blocks the higher frequency leaks from

part “1”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

6.17 The simulated transmission (S12) through the low-pass filter for the

us23/us25 design. Note that “in-band” (∼ 75 − 110 GHz) the loss is

< −3 dB while the out of band rejection at frequencies > 180 GHz is

excellent (< −20 dB transmission). . . . . . . . . . . . . . . . . . . . . 230

6.18 The Wilkinson coupler design [91]. The design requirements are shown

in terms of the charateristic impedances of the input and output mi-

crostripline, Z0, and the microwave wavelength in the dielectric, λ. . . . 231

6.19 An optical image of the Wilkinson couplers in part of the 128-channel

splitter network on a test device of variation us23. . . . . . . . . . . . . 232

xxxvii

6.20 S-parameters showing transmission from port 1 (output of coupler) to

ports 2 and 3 (inputs to coupler) for the us23/us25 Wilkinson coupler

design. A 3 dB (equal) coupling/splitting is expected near 92 GHz. . . 233

6.21 The THMmicrostrip termination structure geometry. The large rectan-

gular microstrip structures provide capacitive coupling to ground. For

the us23/us25 design optimal dimensions at 92 GHz were W1 = 41 µm,

W2 = 21 µm, L1 = 72 µm, and L2 = 29 µm. For the THM4/THM5

design the optimal dimensions at 92 GHz were W1 = 54 µm, W2 = 10

µm, L1 = 90 µm and L2 = 30 µm. . . . . . . . . . . . . . . . . . . . . 235

6.22 Simulated reflection (S11) for an input signal into the microstrip termi-

nation structure with matched THM absorber (Z0 = RAbs = 31 Ω) for

the us23/us25 design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

6.23 The geometry and placement of the DC chokes on the us23/us25 devices

in order to provide TES bias and absorber access. A close up of the RF

termination on the THM absorber is shown in Figure 6.21. . . . . . . . 237

6.24 Simulated microwave transmission (S12) and reflection (S11) through a

microstripline (above) which is intersected by the DC choke design used

in test devices us23/us25. In the desired RF band of the detector the

presence of the DC connection has little effect on the RF performance. 238

6.25 The geometry of one of the stepped impedance transformers from the

us23/us25 test devices which transitions from a 19 Ω (6 µm wide) to a

30 Ω (3 µm wide) microstrip line. Lengths/widths are not drawn to scale.239

xxxviii

6.26 The simulated transmission (S12) and reflection (S11) through the mi-

crostrip impedance transformer shown in Figure 6.25. . . . . . . . . . . 240

6.27 One of the THM 43 GHz CPW test chip designs. Used mainly for cal-

ibration purposes, it includes microstrip throughlines of various length

as well as two microstrip line “opens”. Contact to each lines is made

by 3 CPW probe tip pads at the edges of the chip. . . . . . . . . . . . 242

6.28 Another of the THM CPW designs, with test lines to test the low-pass

filter, termination structure, and impedance transformer performances. 243

6.29 The predicted transmission (S11) and reflection (S12) through the coplaner

waveguide to microstrip transformer for the 43 GHz THM4/THM5

CPW test chip design. . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

6.30 The CPW cryogenic probe station measurement setup. . . . . . . . . . 245

6.31 Measured loss per millimeter of 3 µm wide microstrip throughline on

a test device of variation THM5. This measured loss is within the

measurement noise (∼ ±0.01 dB). . . . . . . . . . . . . . . . . . . . . . 246

6.32 Measured relative dielectric constant of Al2O3 dielectric found via cali-

bration measurement of the microstrip throughlines and opens on a test

device of variation THM5. Expected ǫr ≈ 10. . . . . . . . . . . . . . . . 247

6.33 Measured characteristic impedance of a 3 µm wide microstrip line,

found via calibration measurements of microstrip throughlines and open

lines on a test device of variation THM5. The measured impedance is

close to the 20 Ω design impedance. . . . . . . . . . . . . . . . . . . . . 248

xxxix

6.34 The two optical coupling schemes for testing the microwave compo-

nents and microwave response of the THM in the laboratory cryostat.

Components are not drawn to scale. . . . . . . . . . . . . . . . . . . . . 251

6.35 Dimensions of the optical components, separation distances and beam

waists. Figures are not drawn to scale. . . . . . . . . . . . . . . . . . . 252

6.36 The horn antenna geometry and dimensions. Here SE and SH are the

pyramidal horn dimensions projected into the E-field and H-field planes. 254

6.37 The simulated (above) and measured (below) beam pattern from the

pyramidal horn antenna at 92 GHz. Courtesy Sara Stanchfield. . . . . . 255

6.38 The extended hemispherical lens geometry. R is the spherical radius,

L is the cylindrical extension length. . . . . . . . . . . . . . . . . . . . 256

6.39 The simulated and measured beam pattern at 92 GHz from the horn

antenna after focusing by the Rexolite lens. Courtesy Sara Stanchfield.

Measurements are consistent with the simulated beam pattern. . . . . . 258

6.40 The nichrome blackbody source. . . . . . . . . . . . . . . . . . . . . . . 260

6.41 The measured emissivity of the nichrome black body source. . . . . . . 262

6.42 The geometry and layout of the finline black body source. . . . . . . . 263

6.43 The finline black body source. . . . . . . . . . . . . . . . . . . . . . . . 264

6.44 The measured emissivity of the finline black body source. Courtesy

Sara Stanchfield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

6.45 The simulated emissivity of the finline black body source with a 0.22

inch long extended resistor with resistivity of 250 Ωand total terminat-

ing resistance of 28.5 Ω. Courtesy Sara Stanchfield. . . . . . . . . . . . 267

xl

7.1 The basic microwave circuit for the recommended THM design. A

close-up of the THM detector is shown in Figure 7.2. . . . . . . . . . . 278

7.2 The geometry of the recommended THM design for CMB applications. 279

1

Chapter 1

Introduction

1.1 The History and Science of the CMB

1.1.1 CMB Cosmology

The early universe was a very dense and hot place. When it was only a few

minutes old it contained baryonic matter (protons, neutrons and electrons), neutri-

nos, ‘cold’ dark matter, radiation (photons), and presumably dark energy. Since the

beginning of the universe around 13.7 billion years ago, after an event called the Big

Bang, the universe has been expanding, becoming less dense and cooling. As the uni-

verse has expanded and cooled, the constituents of the universe have been interacting

and evolving. In the very early universe most of the energy of the universe was in

the form of radiation, but not very long after (70,000 years after the Big Bang) the

universe became matter dominated. Just recently we appear to be entering an era of

dark energy domination.

As the universe has cooled both dark matter and baryonic matter have condensed

through gravity from small inhomogeneities at early times, eventually forming the

2

Figure 1.1: Timeline of the universe, courtesy WMAP-Science team [114].

clusters, galaxies, stars and planets that we observe today. This standard cosmological

account (depicted in Figure 1.1) is based on observational evidence of the recession

of distant galaxies (and the supernovae embedded in them), observation of large scale

structure traced by galaxy surveys, observation of element abundances leftover from

primordial nucleosynthesis, and perhaps most importantly, from observations of the

Cosmic Microwave Background (CMB). It is these precision measurements of the CMB

which will be the focus and motivation of the detector technology developed in this

thesis, a Transition-Edge Hot-Electron Microboloter (or THM).

In the early universe, the temperature and density were high enough that photons

were continuously colliding via Thomson scattering with electrons, ionizing atoms that

3

tried to form. However, about 380,000 years after the Big Bang the universe had

cooled enough that the photons no longer had enough energy to ionize these atoms,

and the electrons and protons recombined, leaving the photons free to travel, for the

most part undeflected. These photons are observed today as the Cosmic Microwave

Background (CMB), a surprisingly uniform background radiation which arrives at our

telescopes from all directions in the sky. The CMB provides us with a snapshot of

the universe at the time of this recombination. Indeed, it is the earliest radiation, and

thus information, that we receive from the early universe.

The CMB is near perfect blackbody radiation with a temperature of 2.725 K

today (at the time of recombination it would have had a temperature of 3000 K).

Today the peak of the CMB blackbody spectrum is near 100 GHz, which corresponds

to a peak wavelength near 3 mm. The CMB was first observed by Penzias and Wilson

in 1964, a discovery for which they later earned the Nobel Prize in physics. An

incomplete history of past CMB experiments and their science goals (and detector

type which will be discussed in the following section) is listed in Figure 1.2. Early

observations of the CMB traced out this near perfect blackbody frequency spectrum

of the CMB. This was most accurately done by the FIRAS instrument on the COBE

satellite in 1994 for which another Nobel Prize was earned in 2006 [82]. Work in the

last two decades has gone into detailed mapping of the low-level temperature and ‘E-

mode’ polarization fluctuations in the CMB. These measurements are expressed as an

angular power spectrum where the amplitude of fluctuations in the CMB is plotted

as a function of the angular size scale of these fluctuations on the sky. This power

spectrum is a spatial frequency Fourier transform of the CMB as function of position

4

on the sky.

Before recombination the proton-electron-photon fluid in the early universe was

undergoing acoustic oscillations on all scales. These initial perturbations were seeded

by inflation. Over-dense regions in the fluid attracted more mass to them through grav-

ity. As these regions grew, radiation pressure built up, eventually forcing the fluid to

expand again. This interaction of gravity and radiation pressure led to over-dense and

under-dense oscillations in the proton-electron-photon plasma. Like a 3-dimensional

version of waves on a bound string, and like sound waves travelling through the air

in a closed pipe, these acoustic oscillations in the early universe plasma at the time

of last scattering were occurring at harmonic mode frequencies. Each harmonic mode

corresponded to a spatial scale of the universe and today corresponds to an angular

scale on the CMB sky. The temperature power-spectrum of the CMB is a measure

of the amplitude of these fluctuations at the time of last scattering, as a function of

angular size scale, or spherical harmonic mode number, l. Here large l corresponds to

a small angular distance on the sky.

The location and precise shape of acoustic peaks in the angular power spectrum

is determined by the density of dark matter and dark energy in the universe, the

expansion rate history of the universe and is also related to the timeline for structure

and star formation in the universe. The CMB has provided us with the most precise

method to pinpoint such cosmological parameters. The most recent constraints on

such cosmological parameters have been put in place by the WMAP satellite 7-year

results [75, 69].

Hope for extracting further cosmological information from the CMB focuses on

5

Figure 1.2: The history of CMB experiments.

6

observing the ‘B-Mode’ polarization in the CMB sky. A measurement of this signal will

allow us to challenge theories about events in the early universe at times significantly

earlier than recombination. The CMB polarization signal provides a unique probe

of inflation theory, an era in the universe’s history which occurred during the first

10−32 − 10−34 seconds.

1.1.2 Inflation & Gravitational Waves

Fundamentally there are three cosmological observations of the universe which

lead to puzzling questions about the origins of the universe. Together, the first two,

called the Cosmological Principal, are: 1) On large scales the universe is homogeneous;

2) On large scales the universe is isotropic. This principle states that on large scales

the universe has a uniform matter and photon density and that from any place in

the universe on average an observer sees the same thing. For instance, the Cosmic

Microwave Background is extremely smooth, so much so that regions on the sky that

are too far apart to ever have been in causal contact given our present expansion rate

have the same temperature. The last observation is: 3) The total density of the uni-

verse (including baryonic matter, neutrinos, radiation, dark matter and dark energy)

is very nearly identical to the critical density (ρ = 1.88h210−29 g·cm−3 at the present

time). According to Einstein’s general relativity, which relates the gravitational effect

of energy density to the curvature of space-time, a critical density means that the

universe is flat (curvature parameter, k = 0). In a general relativistic account of the

universe, the acceleration and curvature of the universe are defined by Friedmann’s

acceleration equation:

da2

dt2= −8πG

3a2ρ− kc2. (1.1)

7

Applying energy conservation this equation takes the form of:

da2

dt2= −4

3πGa(ρ+

3P

c2). (1.2)

Here a is the scale factor of the universe, and is defined to have current value of 1; at

earlier times it had a value < 1. The physical distance between any two points scales

with this factor. Thus da2

dt2is the acceleration of the expansion of the universe, which

is often written in terms of the Hubble constant, H = dadt

1a. P and ρ are the pressure

and density of the universe which can be also be written as a function of a and the

current density. Terms must be included in these equations to describe each type of

energy in the universe (matter, radiation, nuetrinos, dark energy etc.). For any given

type of energy, the density scales as a particular function of a where ρ ∼ 1an. n = 3 for

baryonic matter and dark matter, and n = 4 for radiation and relativistic neutrinos.

For dark energy in the form of a cosmological constant n = 0. In addition, for a given

type of energy, the density and pressure are related by an ‘equation of state’ (which

determines the value of n).

Inflation was initially proposed in 1981 by Guth [45] (with subsequent versions

in 1982 by Linde [77] and Albrecht & Steinhardt [2]) as a solution to the questions

posed by the three fundamental cosmological observations and which are assumed by

this standard model of cosmology. Inflation says that there was an era of very rapid

expansion for a very short time, very early on in the universe (the first 10−32 − 10−34

seconds). Due to this rapid expansion what were previously very small regions of space

expanded to a size much greater than the horizon-size of the early universe. This

meant that a universe that may have had curvature or inhomogeneities in the past

would now be approximately flat and smooth. In addition to providing an explanation

8

for the three cosmological observations, inflation also provides a way to set the initial

conditions of the universe and to seed the small scale perturbations that we observe in

the Cosmic Microwave Background. Inflation explains that these initial perturbations

began as quantum fluctuations that were blown up by this rapid early expansion.

Inflation also explains why there is no substantial number of magnetic monopoles

in our universe and provides the initial ‘setting into motion’ of cosmic or Hubble

expansion which we now observe at a much slower rate.

In inflation, the mechanism for this rapid expansion is a scalar ‘inflaton’ field. As

the scalar field value of the universe slowly ‘rolled’ down the slope of this inflationary

potential an exponential expansion occurred. During this short inflation era this scalar

field dominated the energy density of the universe. The inflationary field energy

density is described by an equation of state which produces a negative pressure. Thus

the r.h.s of equation 1.2 was dominated by a negative pressure term and a rapid

acceleration occurred, rather than a deceleration as occurs for the positive pressure

characteristic of matter and radiation. Interestingly, this negative pressure term is of

the same form that is needed to explain the dark energy contribution to the universe

which dominates our present day universe and is causing the current acceleration.

However, there is as of yet no clear connection between dark energy and inflation.

Inflation, although providing an explanation for the cosmological observations

and the origin of inhomogeneities which we already observe, has not yet provided a

unique prediction that has been adequately tested. Because of the large energy scales

involved in the inflation process, the theory predicts the production of gravitational

waves in the early universe [22]. The effect of these gravitational waves would be

9

observable in the CMB ‘B-mode’ polarization. The amplitude of the gravitational

waves produced during this time and their imprint left in the CMB would be directly

proportional to the height of the inflaton field potential and the energy scale of in-

flation (thought to be near 1015 GeV). The observation of B-mode polarization from

gravitational waves would be the first instance where a direct and unique prediction

of inflation could be confirmed. Gaussianity of the primordial acoustic oscillations

and a small spectral tilt in these oscillations are other predictions of inflation theory

which have been observed in the CMB, but these predictions do not provide such a

direct connection to the details of the inflation theory and process like the B-mode

polarization signal.

Although inflation is the most popular theory of the early universe due to its

great explanatory power, there are other theories about the earliest moments of the

universe. For example, a ‘cyclical’ model of the early universe may be able to explain

these cosmological observations as well [116]. This lower-energy model would not

create gravitational waves and thus would leave no imprint in the CMB in the form

of B-modes. Recently, it has been suggested that due to issues arising from the un-

predictive, eternal nature of inflation and fine-tuning that is required, that a cyclical

universe theory is a sounder and a more likely theory than inflation [116, 110]. Thus

a measurement of the B-mode polarization is a very important test. If B-modes are

not observed it may push theorists towards a new direction in our understanding of

the earliest times of our universe.

10

1.1.3 CMB Polarization

Polarization is defined as the direction of the E-field of an electromagnetic wave.

Unpolarized light is the superposition of waves with random E-field orientations. Po-

larized light has an E-field with a greater amplitude in one particular direction. The

polarization in the CMB is a result of Thomson scattering at the time of recombina-

tion [22, 47]. In an isotropic medium the net result from this scattering is unpolarized

light. However from observations of the CMB we observe that the medium was not

completely isotropic at the time of last scattering. These anisotropies can be modelled

as spherical harmonics made up of hot and cold spots from the perspective of the scat-

tering target electron at the center of the incoming CMB photons as shown in Figure

1.3. In the case of a quadrupole moment of the spherical harmonic, the incoming

hot/cold/hot/cold photons will sum to give a net polarization. In all other moments

the scattering photons sum to give a net zero polarization. Thus a quadrupole moment

is the sole cause of polarization in the CMB.

There are three possible types of inflation-seeded perturbations which give rise to

a quadrupole moment in the electron-proton-photon fluid at the time of last scattering

and thus to polarization in the CMB. These are: 1) scalar perturbations, which lead

to over- and under-densities in the fluid; 2) vector perturbations, which lead to vortex

flows in the fluid; 3) tensor perturbations, which lead to gravitational waves which

stretch space as they travel through the fluid [22, 47]. Scalar perturbations give rise

to an ‘E-mode’ type of polarization. They are also the source of the CMB temperature

fluctuations. Thus there is an expected correlation between the E-mode polarization

and the temperature signal in the CMB. The vector perturbations are expected to be so

11

small that no effects are observable in the CMB. The tensor fluctuations are expected

to give rise to both E-mode and B-mode polarization. The terms E-mode and B-

mode polarization indicate different types of patterns which the CMB polarization

can form on the sky. In Figure 1.4 these two patterns are shown. Although it can be

difficult to distinguish by eye, mathematically an E-mode polarization pattern exhibits

a divergence, while a B-mode pattern exhibits a curl.

In Figures 1.5 and 1.6 recent measurements and predictions of the CMB power

spectra are shown courtesy of the WMAP Science team [114] and the BICEP Science

team [15]. The well-measured temperature spectrum is shown in Figure 1.5, which has

now been mapped on an extensive range of angular scales. The first measurements

of the E-mode spectrum are also shown, and a confirmed correlation between this

type of polarization and the temperature power spectrum are shown in Figure 1.6.

The measured upper limit and predicted B-mode signal from an optimistic inflation

theory prediction is also shown in Figure 1.6. This predicted primordial B-mode signal

occurs at low l value and is an order of magnitude below the E-mode polarization

signal. At high l values a B-mode signal from the transferring of the primordial E-

mode polarization to a B-mode polarization due to gravitational lensing is expected. A

foreground B-mode polarization signal due to diffuse galactic sources from synchrotron,

thermal dust and spinning dust emission is also expected. In addition to improving

our ability to distinguish and remove these foreground signals, further improvements

in detector sensitivity are required to reach this predicted B-mode signal which occurs

at the 0.1 µK level.

In addition to a measurement of the primordial ‘B-mode’ polarization signal in

12

Get polarized light!

Figure 1.3: The Thompson scattering process through which a quadrupole

moment in the photon anisotropy from the perspective of the target elec-

tron gives rise to a polarization in the CMB. Blue field lines on the incom-

ing photons indicate colder and thus lower energy photons and the red

field lines indicate hotter and thus higher energy photons. The net effect

is an outgoing electric field magnitude which is stronger in one direction.

Figure adapted from Hu & White [47].

13

Figure 1.4: The two types of polarization patterns in the CMB. An exam-

ple of an E-mode polarization pattern which exhibits a divergence and a

B-mode polarization pattern which exhibits a curl.

Figure 1.5: The measured temperature angular power spectrum of the

CMB. (Courtesy WMAP science team [114]). The amplitude here is plot-

ted in terms of the variance of the fluctuations multiplied by the average

CMB temperature T = 2.73 K squared, such that the plot has units tem-

perature squared.

14

Figure 1.6: The measured E-mode polarization (EE) and temperature-E-

mode cross-correlation (EB) power spectra, and measured upper limits

on B-mode polarization (BB) power spectrum of the CMB. (Courtesy

BICEP Science Team [15]). The predicted B-mode polarization angular

power spectrum in the CMB is also shown by the grey line, with the

contributions near l ∼ 100 coming from primordial gravitational waves.

The B-mode component from gravitational lensing of E-mode to B-mode

peaks at l ∼ 1000. The signal is plotted in terms of the variance of

the fluctuations multiplied by the average CMB temperature T = 2.73 K

squared, such that the plot has units of temperature squared.

15

the CMB, there are other scientific rewards which can come from further and more

sensitive observations of the CMB polarization. These include measurements of gravi-

tational lensing in the ‘B-mode’ spectrum and observations of the post-recombination

re-scattering of CMB photons in the ‘E-mode’ spectrum to gain knowledge about the

re-ionization history of the universe. Measurements of the CMB temperature at small

angular scales to observe the Sunyaev-Zeldovich effect give cosmological information

about the history of the matter distribution in the universe.

1.1.4 Other Detector Science Applications

Although the THM detector as it is developed in this thesis is designed to op-

timally match a CMB observing scheme the THM detector also has applications to

far-infrared and sub-millimeter astronomy where bolometric detectors are also used. In

particular, the very low background noise requirements for these types of applications

make the use of a hot-electron rather than a membrane-isolated detector advanta-

geous, due to the ability of this detector to reach low noise levels without requiring

difficult fabrication of fragile membrane structures or legs [59].

Astrophysical sources in the sub-millimeter and far-infrared (300 GHz-400 THz,

wavelengths of 1 mm-700 nm) include faint optical and ultra-violet continuum emission

from galactic, AGN and star-forming region sources from the early universe which

have now red-shifted down to lower energies. Observations of such sources also probe

cosmology and the growth of structure in the universe. Other astrophysical sources

in the sub-millimeter and far-infrared are line emission sources in the local universe

from interstellar gas clouds, and the hot gas in the atmosphere of planets. In the sub-

millimeter and infrared applications high resolution spectroscopic capability is often

16

desired. Future science missions where a THM detector may have application include

SPICA, SAFARI and Millimetron [112, 10, 125]. Due to the cross-over applications

of the THM detector to this regime, in Chapters 4 & 5 the THM performance under

these conditions will also be discussed.

1.2 CMB Detector Technology & THM Advantages

For CMB applications there are two main schemes which have been developed:

1) coherent receivers and 2) incoherent bolometers. A basic ‘black box’ diagram of

the coherent receiver scheme and the incoherent bolometer scheme is shown in Figure

1.7. In both schemes one imagines microwave radiation from the sky coupling to an

antenna (which in the past has usually been a feedhorn antenna as is shown in the

diagram) with some filtering done to define the band and bandwidth of the signal. In

the coherent receiver the signal is first amplified by a low-noise amplifier, usually a

HEMT amplifier, and then detected by a square-law detector. In addition, mixers are

sometimes used to down-convert the signal to a lower frequency before amplification,

mainly due to increased amplifier noise and cost at higher frequencies. In this coherent

receiver scheme both the amplitude and phase of the incoming electromagnetic wave

are retained in the detection process. In the bolometer scheme the signal is absorbed

in the bolometer and the electromagnetic wave is converted to thermal power (this

process and the readout of this power signal is described in greater detail in Chapter

2). In this detection process the phase of the signal is not retained and the output is

only proportional to the power of the incoming signal.

There is a huge variety of detectors available for detecting the full spectrum of

radiation from astrophysical and non-astrophysical sources, from gamma-rays to radio

17

Basic Coherent Detector:Basic Coherent Detector:

Filter Amp Detector

OutputFeedhorn

Filter

Feedhorn/Antenna

Bolometer

Basic Bolometer (Incoherent) Detector:Basic Bolometer (Incoherent) Detector:

Output

Figure 1.7: The basic detection schemes for CMB observing using coherent

receivers or incoherent bolometric detectors.

waves. What is meant by a “detector” here is the device which actually detects the

electromagnetic radiation and converts it into a electronic signal one can measure,

such as a voltage or current. This output voltage or current is proportional to the

incoming radiation’s power/energy density/electric field. In addition to a detector

other components such as antennas, optics, transmission lines, filters, mixers, phase-

shifters, and polarization sensitive components are often used to help pick out and

channel the radiation signal as it travels from the source to the detector.

When choosing a specific type of detector one must appropriately match the

detector to the desired application. Considerations when choosing a detector for a

specific application include, but are not limited to: 1) the compatibility of the detector

to coupling to the sky with a particular type of antenna, optics, or transmission line

design; 2) the intrinsic detector noise, the noise of the detector readout, and noise from

18

optical components; 3) the incoming radiation frequency and bandwidth; 4) the time

response and speed of the detector; 5) the readout power dissipation and the readout

scheme required; 6) whether and/or what type of cryogenic system is necessary; 7) the

detector efficiency (which may include optical efficiency, quantum efficiency or thermal

efficiency); 8) the power or energy saturation level of the detector.

In general there are two types of radiation detectors: those which produce charge

and light in response to radiation, and those which convert the incoming radiation

into thermal power. The THM is a thermal-type detector: a bolometer. In these next

sections, the detector schemes for previous and future CMB-observing instruments will

be reviewed. The advantages that the THM detector has for fulfilling the requirements

of future CMB-observing instruments will also be highlighted.

1.2.1 History of CMB Detectors & Future Detector Needs

In general, even the low-noise amplifier of choice, the HEMT amplifier, has large

noise at high frequencies. In fact, the sensitivity of coherent receivers is limited by

quantum mechanics, with noise temperature TN ∼ hνkB. Thus for frequencies >100

GHz receiver schemes are usually not used and bolometers are preferred. In the past,

bolometer size has also scaled with the radiation wavelength, as a structure with size

on order of the radiation wavelength is needed to absorb this radiation. Due to this

constraint, at frequencies <100 GHz bolometers were typically not used and receivers

were preferable. This has now changed somewhat, due to the introduction of planar

microwave technology, where bolometer absorbers can be made that are significantly

smaller then the wavelength of the transmitted wave. An example of such a planar

microwave circuit adapted to the THM detector is discussed in Chapter 6. In Figure

19

Figure 1.8: The frequency range of CMB experiments by type as a function

of time.

1.8 the frequency range of the current and past CMB experiments which are listed in

Figure 1.2 is shown as a function of year and detector type.

To reach the predicted B-mode level, an order of magnitude improvement in

sensitivity from current instruments is required. The current single pixel noise of CMB

instruments is now near the background photon noise level for CMB observations from

space and the ground, for both receiver and bolometer technologies. (As a side note,

this is not the case for sub-millimeter and far-infrared detectors where single pixel

noise improvements are still possible.) For CMB observations, however, there are two

ways that sensitivity can be improved. One is to take a longer measurement and in

doing so take a longer time average of the noise signal. The other option is to take more

20

Figure 1.9: The number of detector/receivers for CMB experiments as a

function of time.

measurements at the same time by using an array of detectors. As we have already

reached the limit of reasonable observing times, lasting 3-7 years, the motivation for

current CMB detector development is not to reduce the noise of a single detector or

single pixel, but to develop detectors which can be easily scaled to 1,000-10,0000 pixel

size arrays. To show the trend towards large arrays in CMB instruments, in Figure 1.9

CMB observing instruments are plotted, for both coherent receiver- and bolometer-

based instruments, to show the total number of bolometers or receivers as a function

of time.

There is a variety of difficulties to scaling to large detector arrays. These dif-

21

ficulties are usually compounded when considering the constraints for a space-based

satellite mission, presumably a requirement for detecting primordial ‘B-mode’ polar-

ization. These main difficulties include: 1) the added power dissipation to read out a

larger number of detectors since both receiver and bolometer schemes require cryogen-

ics with limited cooling power; 2) the added space in the focal plane and instrument

itself to fit all these pixels and their readout wiring and optics; 3) the need to keep all

the devices in the array within certain uniformity bounds and with a high fabrication

yield across an array. The first of these issues can be addressed by adding a multi-

plexing scheme to the detector read out. The second challenge can be addressed by

moving from larger microwave components like feedhorns and waveguides to planar

microwave technology. The third challenge is a fabrication challenge and may provide

the ultimate fundamental limit to the ability to use larger and larger detector arrays.

Due to advantages in addressing all of the issues listed above, bolometers will

likely be preferable to HEMT-based receivers for larger CMB arrays. TES thermome-

ters when used in a bolometeric detector can be read out by multiplexed low-noise and

low-power dissipating Superconducting Quantum Interference Device (SQUID) ampli-

fiers. This multiplexing capability is unique for TES thermometers and is not the case

for semiconducting thermistors (a popular thermometer type for CMB bolometers pre-

viously). Bolometers and planar microwave components can also be fabricated pho-

tolithographically with great uniformity. Typical TES bolometers use micro-machined

structures or membranes to thermally isolate the detector. TES bolometers are con-

sidered one of the most promising candidates for large CMB arrays [11] and are already

in use in arrays of sizes 100-1000 [1, 12].

22

A relatively new technology, the Microwave Kinetic Inductance Detector (MKID)

has presented another option for CMB detector arrays [11]. Like TES bolometers,

MKIDs can easily couple to planar microwave transmission lines. Unlike most TES

bolometers, however, they do not require micro-machined structures, simplifying fab-

rication. They can also be frequency multiplexed in a simpler manner than that of

a TES bolometer SQUID multiplexing scheme. This multiplexing advantage is due

to the intrinsic resonance property of these detectors. Currently, MKID detectors

have not reached single-pixel background noise levels and the technology is much less

well understood than that for TES bolometers and the SQUID multiplexing associated

with these detectors. However, MKID detectors have the potential to be a competitive

alternative to TES bolometers in the future.

1.2.2 THM Detector Advantages

The THM is a type of TES bolometer which has several advantages for future

applications to large array CMB instruments. It shares some of these advantages

with the competitor detectors mentioned above, but some advantages are unique to

the THM detector design. As with other TES bolometers, it can be read out by a

multiplexed SQUID system, a technology which has now reached advanced stages.

Additionally the fabrication of these detectors is much simpler and more robust than

that for a typical TES bolometer as no fragile thermal isolation membranes are neces-

sary. This simplification could lead to higher fabrication yields for large arrays. The

lack of membranes is also advantageous as the size of the detector is very small, thus a

denser packing of the focal plane is possible. The absorbing part of the detector itself

is also much smaller than a typical TES bolometer, providing a smaller cross-section

23

to cosmic rays. The small size also leads to a very short thermal time constant in the

device (.1-100 µs) which will allow for rapid scanning of the sky. Like other TES

bolometers, background-limited noise performance is obtainable. Although the THM

is in the development stages, as we will present in this thesis, it is very near to becom-

ing an integrated and functioning CMB-type detector, and a competitive alternative

to membrane-based TES bolometers designs.

1.3 Thesis Summary

The structure of this thesis, which presents work done to develop a Transition-

Edge Hot-Electron Microbolometer (THM), is broken into three parts. The first part

(Chapters 2 & 3) gives a summary of the background knowledge necessary to under-

stand this type of detector, including general theory of how this type of TES bolometric

detector works and the physical processes involved within the detector. This section

is mainly a summary of previous work done by others. The second part is an overview

of the THM detector thermal model and design and the optimization of this THM

design (Chapter 4). The third and final part (Chapters 5, 6) presents “dark” DC,

and RF measurements of THM test devices, analysis of these measurements, as well

an introduction to the microwave design for radiation coupling to the THM detector.

The thesis concludes (Chapter 7) with a discussion of future work and prospects for

the THM detector and a presentation of the recommended THM design for CMB

applications.

24

Chapter 2

General Bolometer & TES Theory

2.1 Basic Bolometer Model

A bolometer consists of three basic parts: an absorber, thermometer, and a

thermal link to a cold reservoir. Sometimes these parts exist as separate physical

structures within the detector and sometimes these parts exist within the same physical

structure. An illustration of this simple bolometer model is shown in Figure 2.1. A

bolometer detects electromagnetic radiation via the thermalization of the incoming

electromagnetic waves by an appropriate absorbing structure. Thermalization occurs

when the power carried in the electric and magnetic fields of the observed radiation

is transferred to thermal energy in the absorbing material. Usually this energy is

transferred first to the kinetic energy of the electrons in the absorbing structure.

After the initial capture of electromagnetic power and thermalization within the

detector, energy flows out of the bolometer to the cold reservoir via the thermal link.

This cold reservoir may be a different part of the detector, or a detector substrate

which is heat-sunk to a cold stage. For most astrophysical bolometer applications this

cold stage is cooled to sub-kelvin temperatures to reduce noise. The mechanism for

25

Figure 2.1: The components to a simple bolometer.

heat flow along this thermal link involves scattering between energy carriers (usually

phonons or electrons). One of the unique properties of the Transition-Edge Hot-

Electron Bolometer (THM) is the “hot-electron” property of this thermal link which

will be discussed in more detail in Chapters 3 and 4. The properties of the thermal

link of a bolometer have important repercussions for the sensitivity and response of

the bolometer.

Unlike coherent detectors, no phase information of the incoming electromagnetic

wave is retained in the bolometer detection process. Rather, the thermal response in

the time domain of the detector is determined by the intrinsic thermal time constants of

the bolometer system. These internal thermal time constants, as well as the electrical

time constants of the thermometer readout, occur on time scales much longer than

the detected radiation frequency. For bolometers observing the Cosmic Microwave

Background this means that the internal time constants occur on scales much longer

than 10s of picoseconds.

In a bolometer, the incoming power that one wishes to measure is directly pro-

26

portional to the temperature of the absorbing detector. This temperature is measured

by a thermometer in good thermal contact with the absorber and is eventually read

out electrically as a voltage or current. There are many types of thermometers which

can be used in a bolometer, and indeed, in principle all that is necessary is a structure

which exhibits a property change with temperature. Some examples of these tem-

perature dependent properties are paramagnetization, which is utilized by magnetic

calorimeter detectors, kinetic inductance, utilized by Microwave Kinetic Inductance

Detectors (MKIDS), current flow, seen in Superconducting Tunneling Junction (STJ)

detectors, or resistance, as in semiconductor thermistors.

The THM uses one of the most sensitive resistive thermometers for astrophysical

bolometer detector applications called a Transition-Edge Sensor (TES). A TES is a

thermometer which makes use of the steep change in the resistance versus temperature

curve of a superconductor at its transition. TESs are currently being introduced into

astrophysical bolometer applications (see Figure 1.2), replacing the semiconductor

thermistors of the previous generation of bolometer detector arrays.

A bolometer detector is quite similar to a calorimeter detector, one of the pre-

ferred detectors in high energy astrophysics and x-ray astronomy. The thermal model

of the bolometer and the calorimeter are the same, however a bolometer measures

the continuous influx of power, while a calorimeter measures the energy of individual

photons. Thus, although there are no differences in the physical models nor the theory

that describes these two types of detectors, different figures of merit and detector qual-

ities are important. For example, for a calorimeter the important sensitivity parameter

is energy resolution, whereas for a bolometer the important sensitivity requirement is

27

the noise equivalent power (NEP). These differences, however, are not fundamental.

The theory which is reviewed in this and following chapters is applicable to both types

of detectors, and in fact one may imagine some applications where the THM could be

used in the photon-counting, calorimetric regime.

In the following sections a brief review of the basics of superconductivity theory

and the superconducting transition which are necessary to fully understand a TES

and the TES readout for the THM detector are given. This is followed by a review of

some of the details of bolometer theory and bolometer behaviour when it incorporates

the TES.

2.2 The Superconducting Transition

In a normal metal, a portion of the electrons, called free or conduction electrons,

are loosely bound to the atomic lattice. Thus a current is easily induced by applying

a potential difference across the metal. As these electrons travel through the metal

they scatter from the crystal structure due to deviations in the perfect periodicity

of the lattice caused by impurities in the metal, deviations in the lattice symmetry,

and thermal vibrations of the lattice structure. Quantized vibrations of the lattice

structure are called “phonons” in an analogy to the photon, the quantization of elec-

tromagnetic waves. Scattering of free electrons from impurities and with phonons

create a resistance to electron flow.

As the temperature of a metal decreases the thermal vibrations of the lattice

also decrease and the electrical “resistivity” of the material decreases. This is why a

measurement of the residual resistance ratio (RRR) of a material, the ratio of resistance

at room temperature to resistance at temperatures near absolute zero, is an important

28

indication of the purity of the material. Even at absolute zero one expects an electrical

resistance due to residual impurities in the metal. Surprisingly, in certain materials

this is not observed to be the case. Instead, as the temperature is lowered, the material

suddenly undergoes a phase transition to a new “condensed” state characterized by

zero resistivity. This is the phenomenon of superconductivity.

Below a critical temperature, Tc, critical current, Ic, and a critical applied mag-

netic field, Hc, a superconducting material will exhibit zero resistance. It is important

to note that each of these critical values is a function of the temperature, current

and applied field. The values of this critical temperature (for zero applied magnetic

field, and zero applied current) for elemental superconductors ranges from 10 mK to

10 K [115]. Critical temperatures as high as 130 K have more recently been observed

for “high-temperature” types of superconducting materials [115]. In addition to the

property of zero resistivity, below these critical values a superconductor also exhibits

the Meissner effect [86]. The Meissner effect occurs when superconducting currents,

induced on the surface of the metal, null any penetrating magnetic flux through the

superconducting surface.

Superconductors have been classified into two different types, Type I and Type

II, according to their magnetic transition behaviour. A Type I superconductor under-

goes a homogeneous magnetic transition. The transition of a Type II superconductor

is not so uniform. For a Type II superconductor there are two critical points to the

applied magnetic field as the superconductor transitions to a normal state. When

the applied field surpasses the first critical value, small normal regions which allow

magnetic field penetration, called vortices or fluxons, occur in the superconductor. As

29

the temperature and/or magnetic field increases these regions grow and spread across

the entire superconductor until a second critical magnetic field point when the entire

superconductor becomes normal. Though magnetic flux is allowed in these fluxon re-

gions, superconducting currents still persist across the superconductor until the second

critical point is reached, and thus there are not two separate transitions in the resis-

tance versus temperature transition curve of the superconductor. We expect the thin

film TES (a Mo/Au bilayer) for our THM bolometer to be a Type I superconductor.

A microscopic theory, called Bardeen-Cooper-Schrieffer (BCS) theory, explains

superconductivity (although it is still unclear whether this theory adequately explains

the microscopic behaviour of high-temperature superconductors) by introducing an

attractive force between electrons in the metal. This attractive force occurs via an

interaction called a Frohlich interaction with a phonon. Below a critical temperature

and current in certain materials this attractive force is stronger than the Coulomb

repulsion between the two negatively charged free electrons and the two electrons

become bound together into a condensed state called a “Cooper pair”. A property

of the coupling in this Cooper pair state is zero total momentum, with each of the

paired electrons contributing equal and opposite momenta to the Cooper pair total

momentum. In the Cooper pair state scattering which leads to resistance to current

flow is not allowed as such a collision would require a change in the total momentum

of the Cooper pair and a breaking of the Cooper pair. Thus electron pairs in the

superconducting state travel through the material with zero resistance.

One way to interpret the superconducting phase transition is in terms of an

energy gap, just as one might see in an insulator or semiconductor. This analogy is

30

Normal Metal Insulator Semiconductor Superconductor Normal Metal Insulator Semiconductor Superconductor

Free Electrons Energy Gap Bound Free Electrons Energy Gap Bound

E

Figure 2.2: The energy gap interpretation of superconductivity. Quasipar-

ticles occupy the free electron states in the superconductor and electrons

bound in Cooper pairs occupy the bound states. Note however, that unlike

bound electron states in the other cases, in a superconductor, electrons

bound in Cooper-pair states do carry an electrical current.

illustrated in Figure 2.2. In a superconductor an energy gap exists between the energy

of an electron bound together in a Cooper pair and the energy of a free electron in

the normal metal state. In a superconductor at absolute zero this energy gap is at a

maximum (ǫgap = 3.528kBTc) and all the electrons are in the superconducting state.

As the temperature rises the energy gap shrinks, and a small portion of the electrons

enter an unpaired, free electron state in the superconductor, called a “quasi-particle

state”. At the critical transition point, the energy gap rapidly shrinks to zero and all

the electrons in the metal enter the normal metal state.

One of the other striking properties of a superconductor is the extremely long

31

range coherence of the Cooper pairs in the material. This is also the result of the

inability of the Cooper pairs to scatter in the metal. Unlike a normal metal where

the electrons constantly scatter and change phase, the coupling of electrons in Cooper

pairs means that the Cooper pairs retain their phase over long distances. In addition

to the phase coherence of individual Cooper pairs, all of the Cooper pairs in the same

superconductor are described by the same quantum wave function and phase.

This coherence property is directly responsible for many other technological ap-

plications of superconductivity beyond the TES. One of the most notable examples of

this is the development and use of Josephson junctions. A Josephson junction is a weak

insulating link between two separate superconductors. These Josephson junctions are

useful devices which exploit the quantum coherence property of the superconductor

and the interference effects that result. Josephson junctions have had an important

impact on the development of the TES bolometer since they were integral to the de-

velopment of Superconducting Quantum Interference Devices (SQUIDS). SQUIDs are

sensitive meters of magnetic field and current which are uniquely qualified to read out

the TES, and which will be described in more detail in the following sections of this

chapter.

In addition to providing a method to read out the TES thermometer, the the-

ory of Josephson junction behaviour and theory of weak link (and specifically the

superconducting proximity effect) has an unexpected overlap with the theory of TES

behaviour. This overlap is especially applicable in small TES devices on the micron

size scale. Understanding these small size TES devices is particularly important to the

design of the THM due to the important role size plays in the sensitivity and thermal

32

properties of these detectors.

The review of superconductivity given above will be sufficient for understanding

some of the basics of the TES detector operation. In Chapter 3 some of the semi-

microscopic superconductivity theories will be reviewed in more detail, in particular

the theories that lead to understanding of the superconducting proximity effect, in

order to more fully understand the physics and optimal design properties of the THM

and to put measurements in context.

2.3 The Transition Edge Sensor

A Transition-Edge Sensor is a superconductor used as a resistive thermometer.

The superconducting transition makes a very sensitive thermometer since the slope

of the resistance versus temperature curve can be very steep (the transition width in

a superconductor with no applied current is only limited by thermal and magnetic

field fluctuations). Thus a very small change in the TES temperature can result in

a very large change in resistance. An example of a sharp transition resistance versus

temperature (RTES vs. T ) curve of a superconducting TES from one of the THM

test devices is shown in Figure 2.3. There are three main challenges to using the TES

as a thermometer, all of which have been addressed by the low temperature detector

community within the last two decades. Although these challenges have been solved in

both practice and theory there are still application-specific solutions to be considered,

and the issues are important when considering the optimal THM design.

One of these challenges is to design a TES that has the precise Tc necessary to

fulfil application-specific noise requirements. One way to do this is to simply select an

appropriate bulk superconducting material which exhibits a transition temperature

33

Figure 2.3: The RTES vs. T curve of a superconducting TES from a THM

test device of variation THM5 (this and other THM test devices will be

described in Chapter 5).

34

at the required Tc for the detector. However, when bulk superconductors are not

adequate, and more precision in Tc is required, a modification of the bulk Tc is possible

by modifying the bulk superconducting material. One of these methods is by magnetic

doping of the superconductor. The most widely used process, however, is to employ

the superconducting bilayer proximity effect. Here a normal metal layer is sandwiched

together with a superconducting layer such that the proximity of the normal metal

layer shifts the superconducting Tc [80].

Another challenge is to find a way to keep the TES in the sensitive resistive

transition region during operation. This problem of stability is partially solved by

the concept of electrothermal feedback. A feedback loop exists between the electrical

power dissipated in the detector when it is electrically biased and the thermal power

dissipated in the detector by incoming radiation. This feedback will be negative when

the TES is voltage-biased to read out the TES resistance [49]. The electrical power

dissipation from the bias circuit follows the usual Joule power dependence (P0 =

V 20

RTES) where V0 is the voltage applied across the TES and RTES is the TES resistance.

To illustrate this feedback concept, one imagines a resistive TES detector where

the voltage across the TES resistor is kept constant. When incoming power dissipates

in the detector the temperature of the TES rises by a small amount. This increase

in temperature results in an increase in TES resistance due to the positive slope of

the superconducting resistance versus temperature curve. The increase in resistance

reduces the Joule power since the voltage is constant. The drop in Joule power leads

to a decrease in temperature, stabilizing the detector back in the transition region.

One of the final obstacles to developing TES bolometric detectors for astrophysi-

35

cal applications was to develop a low-noise method to read out the TES resistance. As

TES detectors are typically low impedance devices (RTES . 1 Ω), the read out for the

device must also have a low input impedance in order to read out the detector signal

with low noise. One method of doing so is to use the most sensitive low-impedance

ammeters, a SQUID. Since SQUID performance is integral to TES operation and ap-

plications, in the next two sections a brief overview of SQUID theory and operation

is given.

2.4 Basics of SQUIDs

A simple design for a SQUID is a superconducting ring intersected by two Joseph-

son junctions (two weak insulating layers). It is helpful to first examine what happens

in a superconducting ring, and then what happens in a single Josephson junction to

gain a more intuitive understanding of SQUID behaviour.

A superconducting current travelling through a single superconducting ring sur-

rounding a normal region, like the one shown in Figure 2.4, can be describe by a

single BCS wavefunction. The geometry of the ring requires the important boundary

condition that the phase and amplitude of the wavefunction match as the wavefunc-

tion completes each integral number of loops around the ring. Specifically, the phase

difference must be equal to 2πn, where n is an integer. Thus the wavelength of the su-

perconducting wavefunction is quantized, just like a Bohr electron orbiting the nucleus

of a hydrogen atom. The wavelength of the superconducting wavefunction is related

to the magnetic flux, Φ, through the center of the ring produced by the circulating

superconducting current due to Ampere’s law. Thus the allowed magnetic flux is also

36

i s

Φ

Figure 2.4: A superconducting ring allows only quantized values of mag-

netic flux through the center due to quantization of the superconducting

wavefunction.

quantized by the relation [95]:

Φ = nh

2e= nΦ0 (2.1)

Here h is Planck’s constant, e is the electron charge and Φ0 is the flux quantum called

a fluxon which has a value of 2.07 · 10−15 T ·m2.

A single Josephson junction consists of a single superconductor broken by a weak

insulating link as is shown in Figure 2.5. As the insulating region is so thin a supercur-

rent may tunnel across this weak link. Because the two superconducting sections are

separate, the superconducting current in each can no longer be described by the same

wavefunction with the same phase, but rather each is described by separate wave-

functions. In the weak link region the two separate supercurrents interfere and this

interference manifests itself in the amplitude of the superconducting current, which

has a sinusoidal dependence on the phase difference across the link. The maximum

37

Insulator

Superconductor 1 Superconductor 2

is

Insulator


is

Insulator


is

Superconductor 1 Superconductor 2Superconductor 1 Superconductor 2Superconductor 1 Superconductor 2Superconductor 1 Superconductor 2

is

Figure 2.5: A Josephson junction is two superconductors separated by a

thin insulating link.

superconducting current, is, is set by the critical current of the weak link, ic [95]:

is = ic sin∆φ. (2.2)

Here ∆φ is the phase difference across the junction. This dependence is called the

DC Josephson effect. In the absence of any applied voltage a superconducting current

will tunnel across the junction with a sinusoidal dependence on the phase difference

across the link.

When a voltage is applied across the link another effect also occurs, called the

AC Josephson effect. This effect is due to the fact that an AC current cannot travel

through a superconductor without some impedance, creating a voltage across the

superconductor. To understand this one must think of a superconductor with changing

current in terms of a ‘two-fluid’ model, where for a changing current, some of the

current is carried by a small number of quasi-particles (this is not the case for a DC

supercurrent). A changing current creates an electric field in the superconductor which

will accelerate the quasi-particles and create resistance. Thus if there is a changing

current (or phase difference across the Josephson junction, following the DC Josephson

38

effect) then a voltage, V , will develop as well. This dependence is given by [54]:

V =ℏ

2e

d∆φ

dt. (2.3)

With these two relations one can examine the current-voltage (I-V) properties of a

Josephson junction. The Josephson junction can be represented electronically as the

circuit shown in Figure 2.6 and an equation describing the total current flow through

a Josephson junction, I, as:

I = CdV

dt+V

R+ ic sin∆φ. (2.4)

Relating the voltage to the phase difference,

I =Cℏ

2e

d2φ

dt2+

ℏ

2eR

d∆φ

dt+ ic sin∆φ. (2.5)

Now one can imagine that a DC current is sent through the Josephson junction.

If the current is less than the critical current of the junction no voltage appears and the

current is entirely superconducting. If the current is greater than the critical current

of the junction then a voltage will appear across the junction, and in addition, an AC

superconducting current will tunnel back and forth across the junction. From these

equations one can plot the I-V curve of the Josephson junction. An analytical solution

is possible to the equations for the case where there is no capacitance. In the more

realistic case of a capacitive junction the solution can only be solved numerically [31].

A plot of the shape for the low-capacitive case is shown in Figure 2.7.

In addition to an insulating weak link between the separate superconductors, a

similar weak link can occur when two superconductors are intersected by a thin normal

layer, or a thin superconductor with a lower transition temperature than the super-

conducting leads [76]. In these cases instead of a tunnelling current a superconducting

39

C

R

isI I

in,C

in,R

Figure 2.6: A “two-fluid” circuit diagram for a Josephson junction.

40

Figure 2.7: An example of the I-V characteristics of a Josephson junction

with zero capacitance. Figure courtesy Gallop [31].

41

proximity effect can effectively allow a superconducting current to travel across the

normal junction. The criterion for such a weak link is that the critical current of the

junction be less than the critical current of the superconducting leads. These types

of weak links will produce behaviour similar to that of a Josephson junction. This

theory will be returned to in Chapter 3 when the lateral proximity effect in the TES

is discussed.

One can now begin to understand what might occur in a simple SQUID device

where a superconducting ring is intersected by two weak links as shown in Figure 2.8.

First, unlike the superconducting ring without junctions, the magnetic flux through

the ring is neither quantized nor frozen in. Instead magnetic flux is allowed to pene-

trate via the weak link portions of the ring while the remaining ring remains super-

conducting. Although the flux is no longer quantized, the amplitude of the current

able to flow through the link and the magnetic flux through the center of the ring

is dependent on the phase difference across the weak link regions. In the case of a

current-biased SQUID where a constant current is sent through the device, the I-V

characteristics of the SQUID obtained by numerical solution of the Josephson junction

equations, Kirchoff’s laws and energy analysis [31] are reproduced in Figure 2.9. Here

one sees that at a particular current bias the voltage across the SQUID has a periodic

dependence on the magnetic flux through the SQUID. In Figure 2.10 a measurement

of this dependence is shown for a current-biased NIST series array SQUID. One can

see that the SQUID is sensitive to magnetic flux at a scale far below the fluxon level.

Although the details of SQUID theory are complex, the basic physical process

is at heart the interference pattern seen in the voltage and current characteristics of

42

Josephson junctionscurrent bias

direction

Figure 2.8: A simple SQUID: a superconducting ring broken by two

Josephson junctions.

Figure 2.9: An example of the I-V characteristics of a current-biased

SQUID for different magnetic flux values.

43

0

0.50

1..00

1.50 0Φ

SQ

UID

Vo

ltag

e

(m

V)

Magnetic Flux

-0.50

1..00

1.50-

-

Figure 2.10: The measured voltage as a function of magnetic flux for the

current-biased NIST-Series Array SQUID which was used to read out the

THM test devices. Further measurements and details of this SQUID read-

out are presented in Chapter 5.

44

a SQUID as a result of the coherence properties of a superconductor and a phase

difference across the Josephson junction weak links.

2.5 SQUID Operation

How can the SQUID work in actual operation as an ammeter or magnetometer?

Perhaps the most obvious answer to this is as a magnetometer. The set-up of the

SQUID as a magnetometer is shown in Figure 2.11. Here the SQUID is current-biased

and the signal which is read out is the voltage across the SQUID. The SQUID voltage

is directly proportional to the magnetic flux through the SQUID. In a more flexible

magnetometer design an input coil is used and magnetic flux through the input coil

is coupled to the SQUID loop. The SQUID voltage is directly proportional to the

magnetic flux through the input coil and the mutual-inductance of the input coil and

SQUID loop. In most cases an additional pick-up coil is also added to the input coil

of the SQUID to provide further amplification and flexibility.

From this set up it is fairly straightforward to see how the SQUID functions

as a ammeter. The current one wishes to measure can be sent through the input

coil that couples to the SQUID loop. Again, the voltage across the SQUID is directly

proportional to the mutual inductance of the input coil and SQUID loop and the input

coil current. In the case of the TES bolometer readout, the TES is voltage-biased

and the TES current is passed through the SQUID input coil. This TES current is

proportional to the TES resistance and thus the bolometer temperature and incoming

bolometer power.

In actual operation the SQUID is usually operated with electronic feedback as

shown in Figure 2.12. A feedback coil is placed near the SQUID loop and electronic

45

Pick-up Coil

SQUID

Iin

X X

Ibias

Iin

X X

Ibias

Iin

X X

Ibias

Iin

X X

Ibias

Iin

X X

Ibias

Iinput

X X

Ibias

X X

Ibias

X X

Ibias

X X

Ibias

X X

Ibias

X X

Ibias

X XX X

Ibias

V

Input Coil

SQUID

Figure 2.11: A SQUID as a magnetometer.

feedback is set up by monitoring the voltage output of the SQUID and sending a nulling

current through the feedback coil to keep the magnetic flux through the SQUID loop

constant. This linearises the output of the SQUID as is shown in Figure 2.13. SQUIDs

can also be operated with multiple SQUIDs in series in order to increase amplification.

One of the main advantages to using SQUIDs to read out TES detector arrays

is the ability to easily multiplex the SQUID readout. Many different multiplexing

schemes are possible and are being developed for the readout of TES arrays using

both frequency-division, time-division and code-division methods [52, 50, 66]. This

multiplexing provides a low-power dissipation method to read out the large TES de-

tector arrays, which are necessary for the next generation of precision astrophysical

instruments.

46

Feedback

box

Feedback

coil

Input

coil

Feedback

box

Feedback

box

Feedback

box

Feedback

box

Feedback

coil

Input

coil

SQUID

Figure 2.12: Electronic feedback to linearise SQUID.

Fee dback Lock October 23, 2005

-0.0003

-0.00025

-0.0002

-0.00015

-0.0001

-0.00005

0

0.00005

0.0001

-0.00003 -0.00002 -0.00001 0 0.00001 0.00002 0.00003

Input Coil (A)

Fe

ed

ba

ck

Cu

rre

nt

(A)

Figure 2.13: The linear output of the NIST series array SQUID used to

read out THM test devices with feedback.

47

2.6 The TES Bolometer

Some of the details of the thermal and electrical behaviour of the TES in op-

eration will now be reviewed. To do so the basic equations governing the behaviour

of a voltage-biased TES bolometer will be presented, following the derivations and

procedure found in Chapter 1 and 3 of Cryogenic Particle Detection [26] and in the

PhD Thesis of M. Lindeman [78]. In Chapter 4 explicit expressions for the noise and

response of the THM detector will be derived from these equations. In Figure 2.14

and Figure 2.15 two schematics of the TES electrical and thermal circuits are shown.

The voltage-bias of the TES is accomplished by current-biasing a shunt resistor, Rs,

in parallel with the TES, where Rs ≪ RTES. The TES resistor is in series with the

SQUID input coil, with inductance L. Here any parasitic resistances or inductances in

the circuit are neglected. The thermal circuit shown for the TES is an ideal thermal

model for a bolometer where the detector, an absorber/TES at temperature Tbolo and

with heat capacity C, is in thermal contact with a cold bath at temperature Tbath via

a thermal link with thermal conductance, G.

One can write down a differential equation describing both the thermal and

electrical behaviour. The two equations are coupled by the introduction of Joule

power into the device with the voltage-bias of the TES:

CdTbolodt

= Psignal + P0 − Pbath (2.6)

LdITESdt

= V0 − ITESRs − ITESRTES(ITES, Tbolo). (2.7)

Here P0 =V 2

0

RTES= I2TESRTES is the Joule power dissipated in the TES. RTES is the

resistance of the TES, which is a function of both the TES temperature, Tbolo, and

TES current, ITES. V0 is the voltage bias. Psignal is the incident radiation on the TES

48

VSQUID

X X

ISQUID Bias

Rt

RTES

t

IBias

SQUID

VSQUID

ISQUID Bias

R

RTES

IBias

SQUID

VSQUID

X X

ISQUID Bias

Rs

R

IBias

SQUID

Input

Coil

L

EST

V0

ITES

Figure 2.14: The electical circuit for a voltage-biased TES read out by a

SQUID amplifier.

49

G

Tbolo

TES/

Absorber

T bath

P + P signal 0

C

Pbath

Figure 2.15: The thermal circuit for an ideal bolometer detector.

50

absorber and Pbath is the power that flows from the detector to the cold bath. This

power flow from the detector to the cold bath (as will be detailed further in Chapters

3 and 4) usually follows a power law dependence on temperature as below [26]:

Pbath = κ(T nbolo − T nbath). (2.8)

Here κ and n depend on the method of energy flow across the thermal link and are

directly related to the thermal conductance of the link G(T ) by

G(T ) =dPbathdT

= nκT n−1. (2.9)

To learn about the detector response during operation, these equations are usu-

ally examined in the small signal/linear limit where the change in current and temper-

ature of the device are small compared to the current and temperature with only the

TES bias [26]. In this regime, the current, and temperature variables are expanded in

terms of ∆T and ∆I:

∆I = ITES − I0 (2.10)

∆T = Tbolo − T0 (2.11)

Here T0 and I0 are the equilibrium values of the TES when voltage-biased with no in-

coming signal power. Inserting a Taylor expansion of these definitions in the electrical

and thermal equations results in the following:

CdTbolodt

≈ ∂

∂Tbolo[I2TESRTES(ITES, Tbolo)− κ(T nbolo − T nbath)]0∆T

+∂

∂ITES[I2TESRTES(ITES, Tbolo)]0∆I + Psignal (2.12)

LdITESdt

≈ ∂

∂Tbolo[−ITESRTES(ITES, Tbolo)]0∆I

+∂

∂ITES[−ITESRs − ITESRTES(ITES, Tbolo)]0∆I (2.13)

51

which becomes, after applying the partial derivative,

Cd∆T

dt= I20

∂RTES(ITES, Tbolo)

∂Tbolo|0 ∆T − nκT n−1

0 ∆T + 2I0R0∆I

+I20∂RTES(ITES, Tbolo)

∂ITES|0 ∆I + Psignal (2.14)

Ld∆I

dt= −I0

∂RTES(I0, Tbolo)

∂Tbolo|0 ∆T + [−Rs −RTES(I0, T0)

−I0∂RTES(ITES, Tbolo)

∂ITES|0]∆I (2.15)

Using the definitions for thermal conductance, G, given by Equation 2.9 and the

current and temperature dependence of the TES resistance given by the definitions:

α ≡ T0R0

∂RTES(Tbolo, ITES)

∂Tbolo|I0 (2.16)

β ≡ I0R0

∂RTES(Tbolo, ITES)

∂ITES|T0 (2.17)

the two equations can be rewritten as:

Cd∆T

dt= [

P0α

T0−G(T0)]∆T + (2 + β)V0∆I + Psignal (2.18)

Ld∆I

dt= −V0α

T0∆T + (−Rs − (1 + β)R0)∆I. (2.19)

Here R0 ≡ RTES(I0, T0).

Several methods have been used to solve these two coupled differential equations

in both the time and frequency domain for both simple and more complex thermal

models and assuming different functions for the signal power with time [78], [83], [81],

[30]. Perhaps the most straightforward method to solve these equations and to learn

about the detector response is to arrange them into matrix form following the method

developed by Lindeman [78]. When this is done the two coupled differential equations

52

are solvable in either the time or frequency domain. From these solutions fall out the

response of the detector, the time response and the noise performance. In Chapter 4

this will be done step-by-step for two specific THM thermal models in order to predict

the responsivity and noise performance of the THM detector. However, at the moment

only a few of the consequences of the general TES bolometer model in the effects of

electrothermal feedback, stability and noise in the TES detector will be discussed. For

the derivation and details of these effects and the bolometer/calorimeter electrical-

thermal model from the above equations the reader is again referred to Chapters 1

and 3 of Cryogenic Particle Detection [26], and the PhD thesis of M. Lindeman [78].

2.7 Electrothermal Feedback & Bias Conditions

The application of a voltage bias to the TES creates a negative feedback between

the electrical and thermal systems. This greatly improves the range of operating

conditions under which the TES is stable. Under negative feedback the detector can

still become unstable however due to phase shifts of electrothermal oscillations in the

detector caused by the inductance in the TES electrical circuit. The stability of the

detector under a constant voltage-bias depends on both the electrical bias circuit and

the TES thermal response and is guaranteed when the conditions for strong negative

feedback are met [26]:

Rs ≪ RTES (2.20)

C

G> (

P0α

G(T0)T0− 1)

L

Rs +R0(1 + β)(2.21)

The constraints this puts on the bias conditions of the THM detector will be examined

in Chapter 4.

53

The natural thermal response time of the detector under un-biased conditions is

determined by the ratio of the heat capacity and thermal conductance, τ = CG. The

influence of negative feedback in the detector due to voltage-biasing decreases this

thermal time constant. In the case of the strong negative feedback the thermal time

constant can be significantly faster than the natural thermal time constant.

2.8 Bolometer Noise

Noise in the detection process comes from three sources: photon background

noise (which is ‘external’ to the detector), ‘internal’ detector noise, and detector read-

out noise. These noise sources are important to the THM design optimization, which

will be discussed later in Chapter 4. All these sources of noise contribute to the total

NEP (Noise Equivalent Power) of a detector.

Photon background noise is an external, unavoidable noise source, due to statis-

tical fluctuations in the arrival rate of photons at the detector. This fluctuation noise

is given by:

NEPphoton = hν

√

2∆ν

ννηn0(1 + ηn0) (2.22)

for single mode observing [128]. Here ν is the observing frequency, ∆νν

is the fractional

bandwidth, and η is the optical efficiency. n0 is the photon occupation number, and is

a function of the observing frequency and in the case of the CMB, the CMB blackbody

temperature and emissivity. To achieve the best possible detector performance for a

specific observing condition requires that NEPdetector < NEPphoton where NEPdetector

is the noise due to internal detector noise and readout noise. Thus in this thesis the

photon noise, specifically photon noise for CMB observing conditions, will be used as

54

a sensitivity benchmark for the THM design.

The internal detector noise and read out noise comes from thermal fluctuations

between the hot detector and the cold reservoir, from Johnson noise fluctuations in

the resistive TES, as well as Johnson voltage/current noise from the bias circuit, and

current noise from the SQUID amplifier read out. Each of these “detector” noise

sources will be described briefly below.

The Johnson noise sources in the detector are the resistive TES itself, the shunt

resistor and any other parasitic resistance in the TES electronic circuit. The Johnson

noise from the TES resistance in terms of a voltage fluctuation across the TES is given

by:

NEVJ,TES =√

4kBTboloRTES. (2.23)

Similarly, the Johnson noise in terms of an equivalent voltage fluctuation noise across

the TES due to the shunt resistor is given by:

NEVJ,s =√

4kBTbathRs. (2.24)

The SQUID amplifier noise in the form of current noise in the readout circuit is given

by [51]:

NEISQUID =

√

2ξh

L. (2.25)

Here h is Planck’s constant, L is the input inductance of the SQUID, which is de-

termined by the number of turns in the input coil, and ξ is a multiplicative constant

which typically ranges from 10-1000. In the next chapter the manner in which these

voltage and current noise sources contribute to the total noise in terms of an equivalent

power fluctuation noise at the input of the detector will be derived.

55

Thermal power fluctuation noise across the bolometer link is given by [26]:

NEPthermal =√

4kBT 2bathG(Tbath) (2.26)

in the case that Tbolo ∼ Tbath. When the TES temperature is considerably higher than

the bath temperature (Tbolo ≫ Tbath) an additional factor is added to this expression.

This factor describes the thermal conductance along the temperature differential of

the weak link. In the case of radiative energy transfer, which holds for cases where the

mean free path of the energy carriers is large compared to the length of the thermal

link the expression is given by [26]:

NEPthermal =

√

4kBT 2bathG(Tbath)

( TboloTbath

)n+1 + 1

2. (2.27)

In the case of the THM where the thermal link occurs between the electrons and

phonons of the detector directly on the substrate this radiative limit holds. In the case

of diffusive energy transfer, where the mean free path of the energy carriers is small

compared to the length of the link, which may apply to typical bolometers where the

thermal link is controlled by phonon scattering along membranes or suspended legs,

the expression is given by [26]:

NEPthermal =

√

√

√

√4kBT 2bathG(Tbath)

n

2n+ 1

( TboloTbath

)2n+1 − 1

( TboloTbath

)n − 1. (2.28)

Numerically, the increase to the NEP by these added factors is similar for both the

radiative and diffusive cases. This additional factor becomes important to the THM

detector performance. It imposes a fundamental limit to the NEP possible for the

THM detector under CMB-observing conditions, and indeed CMB bolometric detec-

tors in general, as will be shown in Chapter 4.

56

Chapter 3

Physical Effects in the THM Detector

Although the THM detector appears to be a relatively simple device, consisting of a

thin metal film absorber, and a thin bilayer or monolayer superconducting TES, the

physical processes occurring within the THM detector are many and complex. Due

to the low operating temperature of these detectors, as well as their small size, the

physical processes which dominate the the THM behaviour are also ones not usually

observed in larger or warmer detectors and devices. In the first part of this chapter

some of the dominant thermal processes within the THM detector, and the present

theoretical understanding of these processes in the literature are reviewed. In Chapter

4 these thermal processes will be used to construct a thermal model for the THM

bolometer. In the second part of this chapter the theory of the superconducting

proximity effect will be reviewed. In these sections some of the applicable S-N-S and

S-S’-S proximity effect theoretical models will be used to make predictions for the

THM detector. In Chapter 5 these predictions will be compared to measurements of

THM test devices.

57

3.1 The Hot-Electron Effect

As was described in Section 2.2 when the phenomenon of superconductivity was

discussed, vibrational states called phonons exist in the crystal lattice. These vi-

brational states can be modelled as quantum harmonic oscillators with a quantized

phonon energy and frequency in a manner analogous to the quantization of electro-

magnetic waves as photons [65]. These quantized waves can occupy three dimensions

in a bulk solid, and can exist as either transverse or longitudinal modes. The energy

distribution of phonons in a solid is dependent on temperature. In Debye theory the

maximum phonon frequency is given by the Debye frequency, ωD = (6π2)1

3v, or the

Debye Temperature, TD = ℏ(6π2n)1

3 v, where v is the sound velocity of the solid and

n is the ion density. The phonon population peaks at phonon energies of kT , and at

phonon wavenumber q = kT~v

[65].

These vibrational states shift the positions of the atoms from their static states

and change the periodic potential that the conduction electrons in a metal see. This

effect is noticed in both the electron effective mass, the property of electrical resistivity

and in the appearance of superconductivity, where electron pairs couple together via

a phonon interaction. However, even in a non-superconducting metal, conduction

electrons scatter via the emission or absorption of a phonon. The rate at which

this scattering occurs is dependent on the energy spectrum and the density of states

of phonons and electrons in the metal. This scattering rate also depends on the

dimensionality of the metal, the dominant phonon modes (transverse or longitudinal),

and the level of disorder. During this scattering process a net thermal energy can be

exchanged between the electron and phonon systems. At low temperatures and within

58

small volumes electron-phonon scattering becomes infrequent and the electron-phonon

thermal conductance becomes very small. It is this small electron-phonon conductance

which will provide the dominant thermal isolation mechanism for the THM detector.

In this section the theories that describe this electron-phonon scattering in metals

under various regimes are reviewed. Measurements from the literature which confirm

or do not confirm parts of these theories for various materials are also presented, with a

focus on gold and bismuth, the materials chosen for the THM detector. A brief history

is then given of the application of this effect as a basis of electromagnetic radiation

detectors, and specifically, the development of Hot-Electron Bolometers (HEBs) and

Hot Electron Direct Detectors (HEDDs).

3.1.1 Hot-Electron Effect Theory

Theoretical models of the electron-phonon scattering time in a metal, τe−p, at

least over a limited temperature range, follow the form of [25],[123]:

1

τe−p= α∗T n−2

e . (3.1)

The power-law parameter, n, and the constant α∗ depend on the temperature regime

and properties of the material with predicted values ranging from n = 3.5 − 6. This

scattering rate can be converted to a net heat flow between the two systems which

follows the form of:

Pe−p = ΣV (T ne − T np ) (3.2)

Ge−p = nΣV T n−1e . (3.3)

Here Pe−p is the power flowing from the electron system to the phonon system, and

Ge−p is the thermal conductance that describes that link. These thermal models

59

assume a ‘2-Temperature’ model in which other methods of thermal exchange are

ignored and both the electron and phonon system are assumed to be in equilibrium

with themselves [55]. This assumption will be examined in the context of the THM

design in Chapter 4.

Σ is a temperature independent, material dependent constant, and V is the

volume. The temperature independent constant, Σ, is related to the scattering rate

constant, α∗ and in various models can be derived theoretically. In one of the most

general electron-phonon model it is given by [123]:

Σ = 0.524α∗γ (3.4)

Here γ is the electronic heat capacity constant of the metal. However, agreement

between measured and theoretical values of Σ vary. Instead of using theoretical pre-

dictions for Σ values in the design of the THM previously measured values are used,

reviewed in the following section.

The most general models of electron-phonon scattering take place in a pure 3-

dimensional metal. Most of these theories argue that the electrons scatter only with

longitudinal phonon modes [127, 32, 4, 123], however, some work has considered a

model where scattering with transverse phonon modes is also possible [93]. In this

pure 3-dimensional model the electrons interact via emission or absorption of a single

phonon but do not scatter with impurities in the metal. For temperatures below the

Debye temperature, these models all predict n = 5 with differing Σ prefactors.

Electron-phonon scattering has also been examined for the case where the metal

exhibits a large amount of disorder. In a disordered film the electron-phonon interac-

tion is complicated by interference effects due to electrons scattering with impurities.

60

A film’s disorder is characterized by the quantity ql where q is the dominant phonon

mode wave-vector and l is the electron mean free path. It should be noted that this

quantity is temperature and mode dependent. In the case that ql > 1 then the pure

metal model holds. However, when ql < 1 the metal is considered disordered and scat-

tering theories predict n = 6, assuming three-dimensionality and that the scattering

is dominated by longitudinal phonon modes [100]. A disordered theory, allowing for

the possibility of coupling to transverse phonon modes to dominate the scattering pre-

dicts n = 4 [104]. In disordered metals, a transition region between longitudinal and

transverse mode domination (qT l ∼ 1 where qT is the transverse phonon wavevector)

is also predicted. In this transition region n = 5 may be expected [104]. It should be

noted that in addition to n, Σ may depend on the level of disorder as well.

There have also been attempts to model 2-dimensional thin metal films where

the phonons are confined to a thin layer. These models predict an amplification of the

electron-phonon coupling with n = 3−3.5 in the pure limit [40] and n = 4.5−5 in the

disordered limit [25], [9]. It is important to note that measured values of n = 4, while

conforming to a possible 2D model of electron-phonon scattering (and a disordered 3D

model with transverse dominated scattering) also conform to heat flow determined by

a boundary thermal conductance between the phonons in the film and phonons in the

detector substrate. This boundary thermal conductance and how it may complicate

the simple hot-electron model of the detector will be discussed in Sections 3.2.2 and

4.2.6.

One can justify the extension of the theory of the electron-phonon effect in

a metal to a superconductor like a TES by considering the two-fluid model of the

61

superconductor as was described in Section 2.2. In this model there are normal-

acting electrons, called quasiparticles, excited out of the bound Cooper pair state,

and also superconducting electrons bound in Cooper pairs. In the transition region,

as is the case for a TES, the energy gap is rapidly approaching zero. In this operating

region, the superconductor is so close to the normal state that the vast majority of

the electrons are in the normal quasi-particle state, with very few Cooper pairs. Thus

a superconducting hot-electron bolometer in the transition can be treated as a normal

metal from the perspective of the hot-electron effect [103], [37].

3.1.2 Measurements of the Hot-Electron Effect

There have been many measurements of the hot-electron effect in a variety of

metals including both noble metals and alloys (such as AuCu, Cu, Au, NiCu, Nb)

which support a pure 3D model for electron phonon scattering, and demonstrate n = 5

temperature dependence [123], [32], [35], [102]. Disordered n = 6 behaviour has been

observed in some semi-metals (Ti, Hf, Bi) [38], [70] and alloys. There have also been

measurements of disordered n = 4 behaviour in Au films [105]. Presently there have

been no measurements of disordered noble-metal films which demonstrate n = 6 be-

haviour, and even in such metals which would be considered disordered, n = 5 has been

measured [25],[34], [96]. These measurements of n = 5 behaviour in disordered metals

could possibly be explained as occurring in the transition region between longitudinal

and transverse phonon domination [104]. There are a few measurements indicating the

possibility of 2D effects with measured values of n = 4− 4.5 in both thin membrane-

suspended films and thin films on a bulk substrate indicating a strengthening of the

electron-phonon effect [21], [73], [60].

62

The THM detector is a two-component bolometer with a Au or Mo/Au TES

and a Bi absorber. Previous hot-electron measurements of Au films have indicated

n = 5 [25], [60] and there have also been indications of n = 4 in disordered Au films

[105]. Bismuth, a semi-metal, has shown instances of n = 6 behaviour [70] as well as

n = 5 behaviour [23]. These previous measurements of the hot-electron effect in both

Bi and Au were instrumental in the design and optimization of the THM detector.

For this purpose we chose to use an average literature value of ΣAu = 4.2 · 109 Wm3K5

for Au and ΣBi = 2.4 · 108 Wm3K5 for Bi (assuming n = 5) [89], [23] as a guide to design

and for measurement comparison. These values will be referred to in later sections as

‘literature values’. In Chapter 5 measurements of the thermal conductance of THM

test devices are presented, and κ, Σ and n values for these Au-Bi composite THM test

devices are extracted.

3.1.3 HEB Detector History

The idea of employing the hot electron effect in a bolometer detector scheme was

first realized in semi-conductors [94], [64]. The possibility of using a superconduct-

ing thin film to detect the hot-electron effect when heated by a bias or optical signal

was then explored [106], [88] [103]. The details and possible optimization of a super-

conducting hot-electron bolometer was first formally proposed by Gershenzon et al.

[36] and the first measurements of a millimeter-size superconducting Nb hot-electron

bolometer by the same group [33], [36].

In parallel to the development of a hot-electron bolometer for direct detection,

hot-electron bolometers were also developed for use in coherent receivers as RF mixers

[130]. In fact, HEB technology in this application is more mature than that for

63

HEDDs. Superconducting Nb HEBs currently perform at the lowest sensitivity levels

in coherent mixer technology, and are in use in astrophysical applications [61]. The

THM and other HEDDs are employed in a different operating scheme, however, and

thus have very different requirements than the HEB mixers.

Since the first HEDD by Gershenzon et al., the development of a direct super-

conducting bolometer detector for astrophysical applications has made major leaps

forward in sensitivity. At the forefront of this push towards greater sensitivity has

been the development of a bulk Ti bolometer on nanometer size-scales by a contin-

uation of the Gershenzon et al. group. In their most recent work, reported by Wei

et al. they report minimum thermal conductance measurements of G ∼ 1 · 10−16 WK

[122], [58]. This HEDD design is optimized for terahertz frequencies and currently

holds the record for the lowest thermal conductance of a bolometer, with predicted

NEP below the background limit for spectroscopy from space. This group reports a

measured optical NEP of 3·10−19 W√Hz

[59].

Solar “boundary” microcalorimeters for use in high-count rate solar x-ray obser-

vations have also employed micron-sized Mo/Au bilayer TESs with larger and thicker

Au absorbers [5]. The size and temperature range of their operation, makes it likely

that an electron-phonon effect may play some role in the calorimeter thermal bot-

tleneck. Another group is also developing a Mo/Au bilayer TES with a separate Bi

absorber for far-infrared spectral observing and report G = 1 · 10−10 WK, with further

improvement still necessary to reach background limited levels [53].

The THM described in this thesis is a Mo/Au bilayer or Au TES with a Bi

absorber. The separate absorber allows for separate matching to a microwave trans-

64

mission line and antenna and the SQUID readout, a flexibility not possible for single-

element TES bolometers. The bilayer TES also allows for greater flexibility and tuning

of the TES Tc, an important detector requirement, especially when trying to obtain

great uniformity in large detector arrays (this is cited as one of the main limits to

present array application for the infrared single-element Ti hot-electron nanobolome-

ters [59]).

In addition, the THM is optimized for CMB observing conditions in the millime-

ter wavelength regime. As will be discussed in Section 4.2.8 and 4.2.9 a low thermal

conductance, although optimal for far-infrared applications, is not necessarily opti-

mal for CMB observing. With modifications to the coupling scheme, however, the

THM might also have applications to this sub-millimeter regime. We present our

measurements of the performance of test THM devices in detail in Chapter 5. Based

on thermal conductance measurements in our hot-electron devices NEP near or below

the photon background noise for CMB and far-infrared spectral observing is predicted,

making the THM a competitive detector choice for future astrophysical applications,

in particular in large detector arrays.

3.2 Other Effects

3.2.1 Andreev Reflection

Another effect in the THM device concerns the thermal isolation of the detector

from heat flow into the superconducting leads. Superconducting leads are necessary

to couple the incoming microwave power to the absorber as well as to provide the

DC voltage bias across the TES and to make a connection to the SQUID readout

65

circuit. When electrons attempts to flow from the detector across the boundary to the

leads, there are no quasi-particle states available to the majority of the electrons due

to the energy gap in the superconducting leads and most of the electrons are reflected

back into the detector. A Cooper pair can flow across the interface and current flow

is possible, however to conserve the number of particles, a ‘hole’ is reflected back at

the interface, and no thermal energy is carried across. This process is called Andreev

reflection [3]. The thermal conductance from the lower Tc/normal TES into the higher

Tc leads is described by [3]:

GAndreev =π

4f0(

mevfπ

)2ǫgap

√

ǫgap2kBT

e−ǫgap2kBT . (3.5)

Here me is the electron mass and vf is the Fermi velocity of the electrons in the TES.

ǫgap is the energy gap between the Cooper pair state of the superconductor and the

normal metal or the superconductor with a lower Tc. f0 ∼ 1 is a constant. In the case

of a THM TES with Tc = 200 mK and with Nb leads of Tc = 8 K, this results in a

predicted GAndreev ∼ 10−100 WK, significantly lower then the expected electron-phonon

thermal conductance to the bolometer cold bath. It is possible for multiple Andreev

reflections to occur, which may increase this thermal conductance [67].

3.2.2 Boundary Conductance

As the THM detector is a thin film deposited directly on the substrate the ther-

mal conductance between the film and substrate at the interface must also be consid-

ered. A thermal boundary ‘Kapitza’ resistance at the interface between two materials

was first discovered at the solid/liquid He II boundary by Kapitza in 1961 [56]. Fol-

lowing this discovery further theoretical and experimental work was undertaken to

66

try to understand this thermal boundary resistance, not only between a solid/liquid

barrier, but also between a liquid/liquid interface and a solid/solid interface such as

occurs between the detector and substrate in the THM [79].

The theoretical work broadly falls into two models which describe the phonon

transmission across a mismatched impedance boundary: the acoustic mismatch model

(AMM) and the diffuse mismatch model (DMM). In both of these models the thermal

conductance calculation depends on the number of incident (and counter-incident)

phonons, and the probability of their transmission across the boundary. The acoustic

mismatch model uses the acoustic analog of optical laws to describe the transmission

of photons at an interface between two boundaries [84], [63], [79]. In this model the

transmission of phonons across a boundary is determined by the angle of incidence of

the phonons, the phonon modes (transverse or longitudinal) and the respective sound

velocities on either side of the interface.

For solid/solid interfaces where the surface is not clean, and at higher temper-

atures where the majority of phonons have frequencies greater than 100 GHz, many

of the phonons scatter from the rough surface. In this case the thermal boundary

conductance is better described by the diffuse mismatch model first introduced by

Swartz [111]. The diffuse mismatch model assumes that all of the phonons scatter

back into the material from the rough surface and lose all correlation or “memory”.

In this model the phonon transmission is determined only by the density of states on

either side of the boundary. Thus if two solids have the same acoustic properties then

half of the incident phonons will be transmitted, and half reflected. The effect of the

diffuse model is to decrease the thermal conductance slightly from what is predicted

67

by the acoustic model for dissimilar solids.

The two mismatch models have been shown to be in relatively close agreement

with solid/solid boundary thermal conductances measured for temperatures below 50

K [111]. More recent models of thermal boundary resistance to describe experimental

results which differ from the simple analytical AMM and DMM models have focused

on atomic level computational simulations of the crystal interfaces [14].

Mathematically, the power flow across the boundary for both the acoustic and

diffuse model is given by the expression:

Pboundary =A

4Cb(T 4

1 − T 42 ) (3.6)

and the thermal conductance is given by:

Gboundary =A

CbT 3 (3.7)

Here A is the surface contact area at the boundary and Cb is a material dependent

constant which depends on the material properties of both sides of the interface and

whether the transmission follows an acoustic or diffuse model. T1 is the temperature

of the hotter material, and T2 is the temperature of the colder material. For Au on

Si or AlO, which will be applicable to our THM detectors, theoretical predictions for

Cb range from 12− 19 · 10−4 Km2

Wfor both the acoustic or diffuse models [111]. In the

upcoming chapters of this thesis, an average value of Cb = 15.5 ·10−4 Km2

Wwill be used

for design considerations and measurement comparison purposes.

What is the effect on the boundary resistance when a very thin film is deposited

on the substrate? The AMM and DMM models discussed so far occur in a bulk regime

where the typical phonon wavelength is much smaller than the film thickness. In thin

68

films at room temperature, the dominant phonon wavelength is often on the same

order as the the film thickness and interference wave-effects between phonons become

important, altering the boundary conductance theory [14]. There has been some

work examining thin films at Kelvin and sub-Kelvin temperatures where the phonon

wavelength can be much longer than the film thickness. The problem of adequately

defining a phonon temperature at this local scale also complicates the interpretation

of the boundary conductance of a thin film [14], [48].

Wellstood et al. [123], while discussing the hot-electron effect in low temperature

thin films, make the argument that when the phonon wavelength is much larger than

the film thickness, it is impossible to talk about phonons within the film itself, but

rather the phonons in the substrate and the thin film may be considered the same

system. Huberman et al. [48] apply this same argument to interpret anomalous room

temperature thermal conductance measurements in Pb on diamond, by assuming that

the phonons within a mean free path of the interface are strongly coupled together

with the substrate phonons and may be considered the same system.

As far as the author is aware there is still disagreement about this interpreta-

tion, and others argue that a boundary conductance due to the impedance mismatch

between thin film and substrate will still exist, even when long wavelength phonons

dominate, although this mismatch model might need to be refined [21], [87] [111] [29].

Some early work into the boundary conductance in thin low temperature films ar-

gues that the transmitted phonon rate and spectrum is modified from bulk behaviour

[29]. According to this model, and confirmed by measurements, one expects to see an

increase in the phonon transmission rate across the boundary up to an order of mag-

69

nitude depending on power dissipation level for regimes where the dominant phonon

wavelength is larger than 2x the film thickness [29]. In semiconductor multilayers

there are also theoretical models taking into account wave-interference effect which

predict that an increase in thermal boundary conductance will occur when the film

layer is much smaller than the phonon mean free path [107].

In Chapter 4 the question of hot-electron versus boundary conductance domi-

nation in the THM detector design will be explored using the predictions from the

bulk AMM and DMM models, while acknowledging that a boundary conductance

significantly higher then these values might be expected due to the thin film effect.

Ultimately, however, since theoretical models are still in disagreement, measurements

of the thermal conductance of THM test devices, which will be presented in Chap-

ter 5, will be allowed to make the argument that the hot-electron effect provides the

dominant thermal isolation mechanism for the THM detector.

3.2.3 Wiedemann-Franz Conductance

Another important thermal exchange mechanism in the THM detector involves

electron-electron scattering within the absorber and TES and between the absorber

and TES. This thermal exchange provides for thermal equilibrium within the ‘hot’

electron system of the detector. The electron-electron thermal conductance in a metal

is linearly related to the temperature and is given by the free electron Fermi gas model

as [65]:

Ge−e =πn2

ek2BTτe

3me. (3.8)

Here ne is the electron density and τe is the electron scattering time. Because the

electrical resistivity in a metal is also related to the electron density and scattering

70

rate ρ = mnee2τe

[65], the electron-electron thermal conductance can be related to the

resistance. In this way the electron thermal conductance can be predicted from a

measurement of the electrical resistivity of a metal without knowing the electron-

scattering rate or the electron concentration. This expression for the electron-electron

thermal conductance is given by the Wiedemann-Franz law:

Ge−e =L0T

R(3.9)

Here R is the resistance across the length of the device and L0 is the Lorenz number

which is given by L0 =π2k2

B

2e. The Wiedemann-Franz law is usually applicable at

low temperatures where the temperature is much less than the Debye temperature

(T < TD10) [90].

3.2.4 Radiation Loss

The hot-electron gas in the THM detector is also able to transfer heat to the

external environment via blackbody radiation into the impedance-matched transmis-

sion line. This radiation loss is determined by a thermal conductance modelled by

one-dimensional blackbody radiation given by [101]:

Gγ =rπ2k2bTe

3h(3.10)

Here r is the transmission factor, where r = 1 for a matched load and r < 1 for

an unmatched load, and where Te is the electron temperature. For a hot-electron

bolometer at 50 mK and r = 1, Gγ = 4.7 · 10−14 WK. Here the thermal conductance is

evaluated for the full photon frequency range of the 1-D blackbody radiation, which

is dominated by photons with frequencies between 0 − 5kBTeh

. This radiation thermal

conductance is usually much lower than the electron-phonon conductance, and we

71

expect that even for our smallest THM sizes (volumes on order 1 µm3, Ge−p ∼ 10−13

WK) it can be ignored. However, in smaller, nanometer hot-electron bolometers it may

become necessary to account for this heat loss [101].

3.3 The Superconducting Proximity Effect

The superconducting proximity effect occurs a short distance from the interface

between a normal metal and a superconductor. The presence of a normal metal mod-

ifies the superconductor’s properties and the presence of the superconductor modifies

the normal metal’s properties a short distance from the interface. In the superconduc-

tor, the BCS energy gap, ǫgap, is minimized near the interface. In the normal metal,

superconducting Cooper pairs from the superconductor are able to travel a distance

into the metal while retaining their coherence and the normal metal may exhibit su-

perconducting characteristics. The proximity effect can also occur at the interface of

superconductors of differing energy gaps.

The design of the TES in the THM detector includes two mechanisms by which a

superconducting proximity effect can occur. The first mechanism is a proximity effect

in the bilayer Mo/Au TES between the superconducting Mo film and the normal Au

film. This first effect is exploited intentionally in the THM design to modify the TES

transition temperature to the precise temperature necessary for the application-specific

noise requirements. The second mechanism is a lateral proximity effect between the

Nb superconducting leads of the TES and the superconducting bilayer Mo/Au TES.

Because of this lateral proximity effect the TES can effectively become an S-S’-S

junction in certain designs. This second effect was discovered as the TES lead-to-lead

length was decreased to the micron scale, and the TES transition temperature shifted

72

closer to the superconducting lead transition temperature (these results are detailed

in Section 5.5).

Strong shifts in Tc make it difficult to fabricate micron-sized Mo/Au TESs with

the low Tcs (50-300 mK) necessary for the THM application-specific noise require-

ments. With this motivation in mind, we have also explored the possibility of using

a micron-sized Nb-Au-Nb, S-N-S junction as a TES. The advantages in this concept

is a simplification of the system, freedom from thickness constraints due to bilayer

thickness ratios, and a lowering of the unmodified junction transition temperature,

TcN , since the normal Au metal has an unmodified transition temperature of TcN = 0

K, compared to TcN = 50− 300 mK for a bilayer S-S’-S junction.

In Figure 3.1 two different THM designs with these two different geometries are

shown. In the following sections the superconductivity theories which describes the

superconducting proximity effect in each of these designs will be reviewed in more

detail. Some predictions of the effect of the lateral proximitization on TES transition

temperatures for various S-N-S and S-S’-S THM test devices will also be presented. In

Section 5.5 these predictions will be compared to measurements of S-N-S and S-S’-S

THM test devices.

3.3.1 Superconductivity Theory & the Proximity Effect

The first experimental investigations into superconductivity took place at a

macroscopic level. London theory and theories which describe the electrodynamics of

a superconductor were developed and applied at this level to describe properties such

as penetration depth and the Meissner effect without a fundamental understanding

of the microscopic origin of superconductivity. Later on, a fundamental microscopic

73

normal metal

superconductor

superconductor normal metal

normal metal

SS’

NS

Lead LeadTES

S N S

Lead LeadTES

(a)

(b)

(c)

Figure 3.1: (a) The bilayer superconducting proximity effect. (b) A lateral

superconducting proximity effect. (c) The two different TES designs where

the lateral or bilayer proximity effect play a role: a S-S’-S junction (left)

and a S-N-S junction (right).

74

theory, BCS theory, was able to describe the macroscopic properties of superconduc-

tors by proposing that the electrons paired together into Cooper pairs via a phonon

interaction.

Although BCS theory works well to describe some properties of bulk super-

conductors such as the energy gap in a superconductor as a function of temperature,

infinite electrical conductivity and phase coherence, it is too unwieldy to describe more

detailed systems where there is a great deal of variation as a function of position or if

there are high magnetic fields. In these situations, as is the case when describing the

lateral proximity effect, more general phenomenological or ‘semi-macroscopic’ theo-

ries are usually applied instead. There are two branches of superconductivity theories

which will be focused on in this thesis in order to understand and make predictions

for the proximity effect in the THM detector: Ginzburg-Landau Theory and Usadel

Theory. A flow chart summarizing these theories, their range of applicability, and a

list of the specific applied theories which will be discussed in this thesis, is shown in

Figure 3.2.

3.3.2 Ginzburg-Landau Theory

Ginzburg-Landau theory introduces a quantity, called an order-parameter, ψ, to

describe the superconducting state, where ψ can be a function of both position and

temperature. This quantity is analogous to the superconducting wavefunction given

by BCS Theory. Ginzburg-Landau theory in fact pre-dates BCS theory and does not

rely on fundamental assumptions about the microscopic properties of a superconductor

[39]. However, the Ginzburg-Landau psuedowavefunction can be related physically to

75

BCS Theory:

No variation with position

No high fields

Ginzburg-Landau Theory:

T~Tc, or ~small,

vary slowly with position

No energy dependence

Usadel Theory:

Full T range

& A vary slowly with position

Energy dependence

Microscopic

Semi-microscopic

S-N-S Theories:deGennes

van Dover et al.

S-N-S Theories:Likharev

Dubos et al.

Kuprianov & Lukichev

S-S’-S Theory:Sadleir et al.

S-S’-S Theory:Kozorezov et al.

S-N Bilayer Theory:Martinis et al.

Θ

or Phenomemological

Figure 3.2: Superconductivity theories, their relationships, and applicabil-

ity.

the density of superconducting electrons, ns, by [115]:

|ψ|2 = ns. (3.11)

Near the superconducting transition Ginzburg-Landau theory can be derived from

BCS theory and the psuedo-wavefunction can also be related to the BCS energy gap

by [42]:

ψ = √

7ζ(3)ne4πTc

. (3.12)

where ζ is Riemann’s function, ne is the number density of electrons in the normal

metal and = ǫgap2.

Ginzburg-Landau theory as it is applied to superconductors is an application of a

more general theory, Landau’s homogeneous theory of second order phase transitions.

This theory applies to any physical process which undergoes a second order phase

transition which can be defined by a complex order parameter [74]. Landau’s theory

76

defines the free energy of a system which undergoes a phase transition in terms of this

complex energy parameter and applies a minimization principle which requires the

thermodynamic system find the state which minimizes this free energy with respect

to the order parameter and the magnetic field distribution. For a superconductor

this free energy minimization requirement is described by the two Ginzburg-Landau

differential equations [98, 17, 115], the first describing the order parameter:

αψ + β|ψ|2ψ +1

2me(~

i∇− 2e

c~A)2ψ = 0; (3.13)

and the second defining the superconducting current density, ~j, due to the diamag-

netism of the superconductor:

~j =e~

ime

(ψ∗∇ψ − ψ∇ψ∗)− 4e2

mecψ∗ψ~A. (3.14)

Here ~A is the vector potential responsible for the magnetic field in the superconductor.

The coefficients α and β determine the solution for ψ and ~j.

The solution to these two differential equations in terms of both the order param-

eter ψ, or the current density ~j, can be found by applying the appropriate boundary

conditions. In certain applications analytical solutions are possible, usually when sim-

plifying assumptions are made. If analytical solutions are not possible then numerical

solutions can be found. Ginzburg-Landau theory has been shown to work well to

describe situations where the superconductor is near the transition, and the order pa-

rameter and current vary slowly with position. For situations beyond this description

a more general semi-microscopic theory, Usadel theory, can be applied.

77

3.3.3 Usadel Theory

Usadel theory is a phenomenological microscopic approach based on a “non-

equilibrium” theory of superconductivity in which the correlations corresponding to

normal or superconducting electrons in a metal are both described by a Green’s func-

tion that depends on position, x, and electron energy, E [43, 118]. Usadel theory

describes superconductivity by a complex order parameter, θ(x, E), which relates to

the density of superconducting or normal electrons. In a normal reservoir θ(x, E) = 0.

In an infinite superconductor, the order parameter θ(x, E) is related to the BCS energy

gap by [44]:

tan θ(x, E) = iE

(3.15)

and the order parameter is related to the density of states, n, by:

n(x, E) = n0Re[cos θ(x, E)] (3.16)

There are 2, 1-D Usadel equations [44],[118]:

~D

2

∂2θ(x, E)

∂x2+ [iE − (

~

τsf+

~D

2(∂ϕ(x, E)

∂x+

2e

~Ax)

2) cos θ(x, E)] sin θ(x, E)

+∆(x) cos θ(x, E) = 0 (3.17)

∂

∂x[(∂ϕ(x, E)

∂x+

2e

~Ax)

2)(sin θ(x, E))2] = 0. (3.18)

Here ϕ is the real superconducting phase, τsf is the spin-flip scattering rate, Ax is the

component of the potential along the x direction andD is the diffusivity in the normal-

state metal where D = vF l and l is the electron mean free path. The Usadel differential

equations are solved by applying the appropriate boundary conditions. With further

simplifications analytical solutions are often possible, in other cases numerical solutions

are necessary.

78

The author will not proceed to discuss or derive any more of the details of

the Usadel or Ginzburg-Landau theories in this thesis, but in the following sections

solutions will be reproduced which have already been derived for THM-applicable

situations. These solutions will be used in order to make specific predictions for the

THM test devices. For the details of the derivations the reader is referred the references

given.

3.3.4 The Characteristic Lengths of the Proximity Effect

Before presenting specific solutions, the characteristic length scale of the su-

perconducting proximity effect will first be introduced. The normal metal coherence

length, ξN , is the distance into which electron-pairs retain their coherence, and thus

superconducting behaviour, as they travel from a superconductor into a normal metal.

Specifically, the critical current through a S-N-S junction, as will be shown explicitly

in the following sections, scales as Ic ∼ e−L/ξN , where L is the lead-to-lead distance

(the length of the normal metal). In a S-S’-S junction, this critical current dependence

does not necessarily follow this form, however, the normal metal coherence length of

the junction region still plays an important role.

The mean free path of an electron in a normal metal is given by [4]:

l =r2n9.2 · 10−17m

ρ. (3.19)

Here rn is the dimensionless radius parameter for the metal and ρ is the electrical

resistivity. The effective mean free path of a thin film, taking into account thickness

dependence, is [68]:

leff = (1

l+

1

d)−1 (3.20)

79

where d is the metal film thickness. This expression requires that the electron mean

free path does not significantly exceed the film thickness.

For an S-N-S junction (where TcN = 0) following both microscopic theory [16]

and semi-macroscopic Usadel theory [76],[68] the temperature dependent normal metal

coherence length in the dirty limit is given by:

ξN,d =

√

~vF leff6πkbT

. (3.21)

This dirty limit is characterized by electron mean free path, l < ξ0 where ξ0 is the

Cooper pair coherence length (the size of an individual Cooper pair) given by ξ0 =

~vF2πkBT

[68],[76]. In the clean limit, l > ξ0, the normal metal coherence length is given

by [68]:

ξN,c = (2πT

~vF+

1

leff)−1. (3.22)

THM devices (with normal Au film of thicknesses, d ∼ 30 − 300 nm) have

effective mean free paths of l ∼ 10− 40 nm, which is much less than the Cooper pair

coherence length (ξ0 ∼ 200 nm). Thus, although the clean limit expressions will be

included in this section, the dirty limit will be assumed for the S-N-S and S-S’-S THM

predictions in the following sections. In Figure 3.3 the normal metal coherence length

in the dirty limit for a Nb-Au-Nb junction as a function of temperature is plotted for

a typical THM devices.

In the case of an S-S’-S junction (TcN > 0) there are two temperature regimes

that are considered when deriving the normal metal coherence length. In the regime

where T ∼ TcN , in the clean limit, Usadel theory and G-L theory give [68]:

ξN,c =~vF

2πkbTcN

√

0.701TTcN

− 1. (3.23)

80

0 1 2 3 4 5 6 7 8

4.0x10-8

8.0x10-8

1.2x10-7

1.6x10-7

2.0x10-7

2.4x10-7

2.8x10-7

3.2x10-7

3.6x10-7

N

(m

)

Temperature (K)

30 nm, RRR=1

30 nm, RRR=2

300 nn, RRR=1

300 nm, RRR=2

ξ

Figure 3.3: The normal metal coherence length for a dirty Nb-Au-Nb

junction (following Equation 3.21) as a function of temperature for Au

RRR = 1&2 is plotted for 30 nm and 300 nm Au thicknesses. Room

temperature Au resistivity is assumed to be 3 ·10−8 Ω ·m and Nb Tc = 8.3

K is assumed. For typical S-N-S THM devices one expects ξN = 0.1− 0.4

µm.

81

And in the regime where T ≫ TcN , in the clean limit, Usadel theory gives [68]:

ξN,c =~vF

2πkbT(1− 0.28TcN

T)−1. (3.24)

In the dirty limit where T & TcN , Usadel and G-L theory give [68]:

ξN,d =

√

π~vF leff24(T − TcN)

. (3.25)

In the dirty limit where T ≫ TcN , Usadel theory gives [68]:

ξN,d =

√

~vF leff6πkBT

(1− 2

ln 4TTcN

)−1. (3.26)

Sadleir et al. have also developed a theory for S-S’-S bilayer devices following

Ginzburg-Landau theory where the temperature dependent normal metal coherence

length is given by [97]:

ξN =ξi

√

TTcN

− 1(3.27)

where ξi is the normal metal coherence length at absolute zero, and a parameter which

they fit to their measurements. Here T > TcN is assumed. They assume for prediction

purposes that ξi =√

π~vF leff24kBTcN

such that Equation 3.27 matches Equation 3.25. Sadleir

et al. found that theoretical predictions using this expression (calculated based on

the superconducting layer of the bilayer only) matched their results to an order of

magnitude. However, they stress that no theoretical expression exists to calculate this

parameter for the bilayer system. In Figure 3.4 the normal metal coherence length

in the dirty limit for a Nb-Mo/Au-Nb junction as a function of temperature for Au

or Mo resisitivity is plotted for a typical bilayer thickness following the expressions

from Equations 3.25, 3.26 and 3.27. The normal metal coherence length is also plotted

assuming Sadleir et al.’s measured fit values for ξi. For both S-N-S and S-S’-S models

for a typical THM detector, ξN ∼ 0.1− 1.0 µm.

82

2 4 6 8

0.0

2.0x10-7

4.0x10-7

6.0x10-7

8.0x10-7

1.0x10-6

1.2x10-6

1.4x10-6

1.6x10-6

1.8x10-6

2.0x10-6

N (

m)

Temperature (K)

Usadel 300 nm Au RRR=1

Sadleir-G-L, 300 nm Au RRR=1

Sadleir-G-L, 300 nm Mo RRR=1

Sadleir-G-L, Sadleir et al. !t

ξ

Figure 3.4: The normal metal coherence length for a Nb-Mo/Au-Nb junc-

tion as a function of temperature for Au RRR = 1 is plotted for 300

nm Au thickness. Room temperature Au resistivity of 3 · 10−8 Ω ·m, Nb

Tc = 8.3 K and Mo/Au Tc = 200 mK are assumed. For the Sadleir-G-L

model (which matches the low temperature Usadel Model), the normal

metal coherence length is also plotted assuming Mo resistivity of 5.3 ·10−8

Ω ·m, and also assuming Sadleir et al.’s measured fit to ξi = 738 nm. For

all these models for typical THM devices ξN ∼ 0.1− 1.0 µm is predicted,

with the coherence lengths predicted by the fit values on the higher side

of this range.

83

3.3.5 S-N Bilayer Theory

Applying Usadel theory Martinis et al. derive an expression for the transition

temperature of a normal-superconducting bilayer film as [80]:

Tc = TcN [2ds

πkBTc0λ2Fns

1

1.13(1 + dsns

dnnn)

1

t]dsnsdnnn . (3.28)

Here TcN is the unmodified superconducting film transition temperature, ds is the

superconducting film thickness, dn is the normal metal thickness, and ns and nn are

the electron density of states in the superconducting and normal metal films respec-

tively. λF is the Fermi wavelength for the normal metal, and t is the transmission

coefficient, which characterizes the transparency of the interface between the normal

and superconducting sandwich and is directly related to the electrical resistivity of

this interface. t = 1 indicates a perfectly transparent interface.

3.3.6 S-N Bilayer Measurements

In practice the Martinis et al. equation must be fit to experimental results to

develop a consistent recipe for bilayer Tc in a specific deposition system. In bilayer

deposition, the deposition is best done in a dedicated and conditioned system, without

breaking vacuum in order to increase the transparency between the layers and the

reproducibility of the transition temperature.

Many of the TESs developed and in use as astrophysical detectors are bilayer

TESs due to the flexibility the bilayers allows in the choice of the transition temper-

ature of the TES. The fabrication techniques for Mo/Au and Mo/Cu bilayer TESs

have been widely perfected [28, 108] and such devices are now in use in astrophysical

applications [109]. We haven chosen to use the Mo/Au bilayer system in the THM due

84

to this maturity of the Mo/Au TES design as it is in use in membrane-isolated TES

detectors for microcalorimeter and bolometer detector arrays and due to the fact that

transition temperatures of bilayer films can be produced to conform to the desired Tc

for CMB applications (50-300 mK). In the next sections of this chapter we discuss

the details of the ‘lateral’ proximity effect, an addition to this bilayer proximity effect,

which can complicate the obtainable Tcs for the THM detector.

3.3.7 S-S’-S Theory & Predictions

Sadleir et al. [97] have developed a model of the lateral proximity effect for

Nb-Mo/Au-Nb S-S’-S junctions based on Ginzburg-Landau theory. Here the solution

for the critical current at the center of the junction is given by:

Ic(T, L) =hf 2

rLd

6πeµ0λ2rξN(T )exp

−LξN(T )

. (3.29)

Here fr ∼ 1, λr is the local penetration depth in the leads and ξN is the temperature-

dependent normal metal coherence length given by Equation 3.27 and d is the Mo/Au

film thickness. The transition temperature, Tc, is defined as the temperature at which

voltage appears across the junction, and thus marks the beginning of the TES transi-

tion. It is shifted from the bulk bilayer transition temperture, TcN , due to this effect.

The Tc is given by the relation [97]:

Tc − TcNTcN

= (ξiLln

hf 2r λrLd

6πeµ0λ2rξN(Tc))2. (3.30)

This solution applies when T > TcN and L ≫ ξN . Here the normal metal coherence

length is given by equation 3.27. In Figure 3.5 predictions for Ic are plotted as a func-

tion of temperature following Equation 3.29 assuming a typical THM bilayer device.

This model predicts a transition temperature which is shifted from TcN = 170 mK

85

to Tc = 600 mK−1 K for a 3 µm long device with bias current of ∼ 1 µA. For the

lead-to-lead lengths we have considered here, the predictions for Ic using measured

fit ξi values from Sadleir et al. are ∼ 1 − 4 orders of magnitude higher than those

predicted using electrical resistivity.

Kozorezov et al. [71] have also developed a model based on the more general Us-

adel equations to describe the behaviour of a bilayer TES as a S-S’-S weak link. They

derive an analytical solution for the case of a large contact resistance at the interface

and more general numerical solutions in the case of a transparent interface. They find

that unlike the Sadleir et al. S-S’-S model and the models for S-N-S junctions (which

will be examined in the next section) the critical current does not fall exponentially as

Ic ∼ exp−L/ξN (here they assume ξN follows the form given in Equation 3.21 rather

than the S-S’-S forms) and that even for T ≫ TcN there is a finite energy gap in the

junction region.

3.3.8 S-S’-S Measurements in the Literature

Recent measurements of micron-sized Mo/Au bilayer TESs, similar to THM

devices with Nb leads, have indicated strong shifts in the TES Tc due to this lateral

effect [97]. For example, for devices with lead-to-lead lengths of 8 µm (the shortest they

tested), Sadleir et al. saw Ic = 1 mA at 200 mK (for TcN = 180 mK) and measured

an effective Tc = 400 mK with a bias current of 1 µA. Although these measurements

fit to the model developed by Sadleir et al. using ξi as a fitting parameter, they do

not fit as well to predictions of this model when ξi is calculated from resistivity, as

is shown in Figure 3.5. Measurements of our own similar THM devices are presented

in Chapter 5 where an even stronger proximity effect is observed in the smallest 3

86

c)

b)a)

Figure 3.5: Critical current behaviour of a Nb-Mo/Au-Nb S-S’-S TES as

predicted by the Sadleir et al. model using Equation 3.29. The legends on

each plot indicate lead-to-lead lengths in meters. a) Ic calculated assuming

Au resistivity of ρ = 3.3 · 10−8 Ω·m, and Au thickness of 300 nm. b)

Ic assuming Mo resistivity of ρ = 5.3 · 10−8 Ω·m with bilayer thickness

dominated by Au thickness of 300 nm. c) Ic calculated assuming the

measured fit value by Sadleir et al. for ξi = 738 nm. For all predictions

λr = 79 nm (the measured value from Sadleir et al.), Nb Tc = 8.4 K, and

Mo/Au TcN = 170 mK have been assumed. The predictions for Ic using

the measured fit ξi value are ∼ 0.5 − 4 orders of magnitude higher than

predictions using ξi values calculated from resistivity.

87

µm long devices. Interestingly, a strong lateral proximity effect has not been seen in

hot-electron Ti nano-bolometers [122], likely because of the high resistivity of these

devices.

3.3.9 S-N-S Theory & Predictions

We now summarize models which predict the critical current behaviour of a S-

N-S junction where the normal metal has a critical temperature of TcN = 0. For all

of these models the form of the normal metal coherence length given in Equation 3.21

has been assumed.

3.3.9.1 Likharev-Usadel Model

The first model comes from the application of Usadel theory in the dirty limit.

The solution to this ‘Likharev-Usadel’ model, in terms of the critical current through

the junction region as a function of the lead to lead length, L, is given by [76]:

Ic(T, L) =π∆2

4ekBTΣ∞i=1

8√2i+ 1L

π2(2i+ 1)2ξN(T ) sinh(√2i+1LξN (T )

). (3.31)

In Figure 3.6 the critical current given by this expression is plotted as a function of

temperature for a variety of L values for a typical THM device. This model predicts

Ic ∼ 0.1− 100 µA at ∼ 200 mK for a 3 µm long device.

3.3.9.2 deGennes-GL Model

The second model was originally derived by deGennes [16] applying Ginzburg-

Landau theory in the dirty limit. In this ‘deGennes-GL’ model the critical current as

a function of lead-to-lead length and temperature is given by [46]:

Ic(T, L) =π∆2

2ekBTcRn

L

ξN(T )exp

−LξN(T )

. (3.32)

88

a)

b)

Figure 3.6: Critical current behaviour of a Nb-Au-Nb S-N-S TES predicted

by the Likharev-Usadel model using Equation 3.31. The legend on each of

the graphs indicates different lead-to-lead lengths in meters. a) Assuming


Assuming a Au resistivity of ρ = 3.3 ·10−8 Ω ·m and a Au thickness of 300

nm. In both cases a Nb transition temperature Tc = 8.4 K and a 3 µm

wide device is assumed. This model predicts Ic ∼ 0.1 − 100 µA at ∼ 200

mK for a 3 µm long THM device.

89

Here Rn is the total normal resistance of the junction and Tc is the transition temper-

ature of the superconducting leads, not the junction region. In Figure 3.7 the critical

current given by this expression is plotted as a function of temperature for a variety

of L values for a typical THM device. This model predicts Ic ∼ 10 nA-10 µA at ∼ 200

mK for a 3 µm long device.

3.3.9.3 Dubos-Usadel Model

The third model is a Usadel-based model which extends the Usadel model to

a broader temperature range in the long junction device limit regime. This model,

derived by Dubos et al. [24] examines two different temperature regimes, characterized

by the quantity defined as the Thouless energy of the junction, ǫc, when ǫc(T, L) =

2kBTξN (T )2

L2 . The Dubos high temperature regime is defined when ǫc ≪ kBT and the low

temperature regime where ǫc ≫ kBT . In addition to these two temperature regimes,

the junction may be considered a long junction if ǫc ≪ ∆, where ∆ is the energy

gap in the superconducting leads. In most of the device size and temperature range

for the THM the long device limit holds, with a maximum ǫc ∼ 10−24J, compared

to ∆ ∼ 10−22 J in the Nb leads. In this high temperature, long device regime, the

solution is given by [24]:

Ic(T, L) =32

eRn(3 + 2√2)ǫc(T, L)

L3

ξN(T )3exp

−LξN(T )

. (3.33)

In the low temperature, long junction regime, the solution is given by [24]:

Ic(T, L) =10.82ǫc(T, L)

eRn

(1− 1.30 exp−10.82ǫc(T, L)

3.2kBT). (3.34)

In Figure 3.8 the predictions following both Equation 3.33 and Equation 3.34 are

shown for a variety of possible THM devices. This model predicts Ic ∼ 1 nA-0.1 µA

90

a)

b)


by the deGennes-GL model using Equation 3.32. The legend on each of




nm. In both cases a Nb transition temperature Tc = 8.4 K and a 3 µm

wide device is assumed. This model predicts Ic ∼ 10 nA-10 µA at ∼ 200

mK for a 3 µm long THM device.

91

at ∼ 200 mK for a 3 µm long device.

3.3.9.4 van Dover-GL Model

Another correction to the Likarev-Usadel theory is given by van Dover et al.

[119]. Here they apply Ginzburg-Landau theory and allow for the suppression of the

superconducting energy gap at the boundary due to the inverse proximity effect from

the normal metal, mimicking a SN-N-NS junction model. In this ‘van Dover-GL’

model the critical current of the junction is given by [119]:

Ic(T, L) =π∆2

2ekBTf 2(T )

Aρnγρs

L

ξN(T )exp

−LξN(T )

(3.35)

Here A = ψN (0)ψS(0)

is the ratio of Ginzburg-Landau order parameter on either side of the

boundary, ρn and ρs are the respective normal state electrical resistivities, γ = mn

msis

the ratio of the effective electron mass in the normal state for the superconducting

leads and the normal metal and f(T ) ∼ 1 for T ≪ Tc, where Tc is the transition

temperature of the Nb leads.

van Dover et al. note that the parameter A, in addition to modelling the sup-

pression of the order parameter at the boundary due to the inverse proximity effect,

can also model the effect of a finite contact resistance between the superconducting

and normal contacts. Experimentally, they found the relation Aρnγρs

= 0.24 in their

Nb-Au-Nb devices, though they argue that this parameter cannot be known a priori

and should be fit to measurements. However, this value will be used as a guide for

predictions for THM devices. In Figure 3.9 the predictions for the critical current fol-

lowing Equation 3.35 are shown, assuming this experimental relation holds, for typical

THM devices. This model predicts Ic ∼ 0.1− 10 µA for a 3 µm long device at ∼ 200

92

a)

b)


by the Dubos-Usadel model using Equations 3.33 and 3.34. The legend on

each of the graphs indicates lead-to-lead lengths in meters. a) Assuming

Au resistivity of ρ = 3.3 · 10−8 Ω · m and Au thickness of 30 nm. b)

Assuming Au resistivity of ρ = 3.3 · 10−8 Ω ·m and Au thickness of 300

nm. In both cases Nb Tc = 8.4 K and a 3 µm wide device is assumed. This

model predicts Ic ∼ 1 nA-0.1 µA at ∼ 200 mK for a 3 µm long TES. The

discontinuity in the curve for the shortest devices indicates the crossover

from the high to the low temperature limit.

93

mK.

Figure 3.10 shows a comparison of all the S-N-S theories (the Likharev-Usadel,

deGennes-GL, Dubos-Usadel and van Dover-GL models) that have been introduced

thus far for a 3 µm x 3 µm TES device. This plot also includes the Sadleir et al. fit

model for a similar S-S’-S device. This plot provides a relevant theoretical comparison

for the results of transition measurements for some of the shortest 3 µm long S-N-S

and S-S’-S THM test devices, which will be presented in Chapter 5. For the S-N-S

case, with a bias current of 1 µA, an effective Tc of 100− 400 mK is expected across

the junction.

3.3.9.5 Kuprianov-Lukichev-Usadel Model

The final S-N-S model is one which explicitly takes into account a finite contact

resistance or non-unity transmission factor between the boundaries by Kuprianov and

Lukichev [72] applying Usadel theory. They define a contact resistance Rc at the

interface between the superconducting leads and normal junction, which modifies the

critical current behaviour of the junction. This contact resistance is parametrized in

terms of the ratio Γc =Rc

Rjunction, where the normal resistance across the junction region

is Rjunction and where the total resistance across the junction is Rn = (1+2Γc)Rjunction.

The general solutions to the theory are numerical. However, in the case of a S-N-S

junction where Γc . ξNL

their numerical solutions predict a decrease in the critical

current of less than 20%. In Figure 3.11 we plot this requirement for Γc as a function

of device length, for a typical THM device. As long as the contact resistance is less

than 10− 15% of the normal metal resistances of the junction then one expects only

small shifts in the critical current.

94

a)

b)


by the van Dover-GL model using Equation 3.35. The legend on each of




nm. In both cases a Nb transition temperature Tc = 8.4 K and Aρnγρs

= 0.24

is assumed. This model predicts Ic ∼ 0.1 − 10 µA at ∼ 200 mK for a 3

µm long THM device.

95

Figure 3.10: Critical current predictions for a Nb-Au-Nb S-N-S TES for the

Likharev-Usadel, deGennes-GL, Dubos-Usadel and van Dover-GL models.

Here a Nb transition temperature Tc = 8.4 K, Au resistivity of ρ = 3.3 ·

10−8 Ω·m, and Au thickness of 180 nm is assumed for a 3 µm x 3 µm device.

Also plotted for comparison purposes is the critical current behaviour for

a 3 µ long Mo/Au S-S’-S TES, following the Sadleir et al. model and

using their measured fit values for ξi and λr, with TcN = 180 mK. For the

S-N-S case, with a bias current of ∼1 µA, an effective TES Tc of 100-400

mK is expected across the junction.

96

2.0x10-6

4.0x10-6

6.0x10-6

8.0x10-6

1.0x10-5

0.0

0.1

0.2

0.3

0.4

L (m)

30 nm

300 nm

Γ

Figure 3.11: Limiting Γc =Rc

Rjunctionpredicted from Kuprianov-Lukichev-

Usadel theory requiring there be less than 20% change in critical current

due to finite contact resistance between superconducting Nb leads and a

normal Au TES. A Au resistivity of ρ = 3.3·10−8Ω·m, thickness of 30−300

nm, and a 3 µm wide device is assumed at 150 mK. For typical THM test

devices, for Γ . 10−15%, only a small shift in critical current is expected.

97

3.3.10 S-N-S Measurements in the Literature

Sub-micron size S-N-S junctions at Kelvin temperatures, called “microbridges”,

made with Nb superconducting leads and normal metal film junctions have been de-

veloped previously for Josephson junction applications. We now examine this previous

work and the agreement and disagreement to the theories considered here in order to

provide context for measurements of THM S-N-S devices which will be presented in

Chapter 5. This previous work is summarized in Table 3.1.

Warlaumont and Buhrman [121] developed Nb-Cu-Nb microbridges where the

normal metal junctions ranged from 0.2−2 µm in length and the Cu bridge region was

60 nm thick. They claim consistency in magnitude and shape to the Likharev-Usadel

model for these Nb-Cu-Nb devices, although they only explicitly present results for

similarly sized Pb-Cu-Pb devices, where they measure critical currents of 100 µA for

a 0.2 µm device at sub-Kelvin temperatures.

van Dover et al. [120],[119] and de Lozanne et al [18] (both from the same

group) present measurements of Nb-Au-Nb and Nb-Cu-Nb microbridges where the

normal metal junction lengths range from 0.2−1.6 µm and thicknesses from 120−240

nm. They measured a critical current of 0.2 mA at 3.7 K for their longest 1.6 µm

Cu device and a critical current of 1 − 10 mA at sub-Kelvin temperatures for their

0.2 − 0.5 µm long Au devices. To explain measurements where they observed lower

critical currents then expected from the Likharev-Usadel model, they developed the

van Dover-GL model described in the previous section.

van Dover et al. also fit Ic(T ) measurements to the normal metal coherence

98

length parameter, ξN . In this case the normal metal coherence length measured agreed

within a factor of 2 with the predicted normal metal coherence length assuming the

dirty model definition and the measured normal metal resistivity. In all their devices

they measure Aρnγρs

= 0.1 − 0.5, except in one of their devices where they measure

Aρnγρs

= 0.00016, they believe due to poor interface conditions between the Nb and

normal metal. Thus, they find that a poor interface can have striking effects on the

predicted Ic, lowering it by several orders of magnitude in an unpredictable manner.

Sauvageau et al. [99] also fabricated Nb-Cu-Nb junctions, with lengths ranging

from 0.25− 0.8 µm and thicknesses of 50− 100 nm. They also followed a fabrication

procedure where the Cu and Nb depositions where completed in quick succession

without breaking vacuum to avoid contamination at the interface. They measured

critical currents of 70 µA for their longest 0.8 µm device at 1.6 K. They saw Ic

behaviour as a function of temperature which fit the form predicted by all of the S-

N-S models which feature the Ic ∼ exp −Lξn

dependence. Fitting these measurements

to the normal metal coherence length given in the dirty limit, their measurement of

the electron mean free path matched that predicted from the normal metal resistivity

and thickness within a factor of 2. In some devices they measure what appears to be

a residual contact resistance which may be due to contamination at the interface.

Dubos et al. [24] made Nb-Cu-Nb devices with length ranging from 0.7− 1 µm

and thicknesses from 370-670 nm. They measured a critical current of 55 µA near 1 K

for their longest 1 µm device, and critical current of 1 mA near 400 mK for their 0.8

µm long device. The behaviour of the Ic vs. T curve matches the Dubos-GL model

very closely where they fit their measurements to the Thouless energy parameter.

99

Table 3.1: A summary of S-N-S ‘microbridge’ devices which have been

developed previously and which are similar to the Nb-Au-Nb THM design.Group Type Length Typical Ic Theory match

Warlaumont & Buhrman Pb-Au-PB

Nb-Cu-Nb

0.2-2.0 µm 100 µA Likharev-Usadel

van Dover et al. & de

Lozanne et al.

Nb-Cu-Nb

Nb-Au-Nb

0.2-1.6 µm 1-10 mA van Dover-GL

Sauvageau et al. Nb-Cu-Nb 0.25-0.8 µm 70 µA all, ∼ exp −LξN

Dubos et al. Nb-Cu-Nb 0.7-1.0 µm 50 µA-1 mA Dubos-GL

The measured Thouless energy for their model matches that predicted from resistivity

measurements and the thickness by a factor of 1− 3.

3.3.11 Conclusions of Modeling the Lateral Proximity Effect

Measurements of S-N-S and S-S’-S devices have been shown to match behaviour

predicted from semi-microscopic theory. However, there is usually some disagreement

(if small in some cases) between critical current behaviour and measured fit values

for the normal metal coherence length and similar parameters when calculated from

resistivity alone. There also exists uncertainty concerning the role of contact resistance

in measurements of test devices when compared to models which assume a transparent

interface. This concern will be returned to in Chapter 5 when measurements of similar

THM devices are presented. Nevertheless, there is reason to believe that these S-

N-S and S-S’-S models and predictions should give a good indication of the lateral

proximity effect in THM test devices.

100

Chapter 4

The THM Thermal Model & Detector

Optimization

4.1 THM Thermal Model & Noise Sources

The basic design of the THM detector consists of a thin-film semi-metal Bi

microwave absorber/RF termination in contact with a micron-sized superconducting

bilayer Mo/Au TES deposited directly on the detector chip substrate. Nb supercon-

ducting leads provide electrical contact to the TES in order to voltage bias the TES

and also provide for microwave termination. A Nb microstrip transmission line trans-

mits microwave radiation to the detector and terminates in the absorber. This basic

structure is shown in Figure 4.1. Unlike typical bolometers where the thermal link

is controlled by carefully designed and fabricated membrane structures, the thermal

isolation between “hot” electrons in the detector and cooler phonons in the detector

(and detector substrate) is controlled by the weak electron-phonon coupling at low

temperatures within the small THM volume.

There are many different physical processes involved in the THM detector that

101

Nb µstrip

Nb lead

Bi

Absorber

Mo/Au TES

Nb lead

RF power

Dielectric substrate

DC bias leads and

µstrip termination structure

Figure 4.1: The basic geometry of the THM components.

102

affect the heat flow within the detector and the heat flow from the detector to the

colder reservoir. The physical processes for these heat flow mechanisms were described

in Chapter 3. In the complete THM thermal model shown in Figure 4.2 all of these

physical processes are denoted as thermal conductances connecting thermal and/or

physical regions of the detector together. Incoming microwave power is deposited first

in the electrons in the absorber. These electrons are in contact with electrons in the

TES and absorber via Weidemann-Franz scattering, characterized by thermal conduc-

tance, Ge−e. The electrons in both the absorber and TES are in thermal contact with

phonons via electron-phonon scattering, characterized by thermal conductance, Ge−p.

Phonon-phonon scattering occurs at the interface of the absorber and TES with the

detector substrate, and is characterized by thermal conductance, Gboundary. Andreev

reflection due to the energy gap between the superconducting TES with a lower Tc and

the higher Tc leads, and between the normal absorber and superconducting microstrip

line, inhibits heat loss through the leads and transmission line connections, and is

characterized by thermal conductance GAndreev,.

4.1.1 Ideal Model Theory & Noise

In this section, expressions for the noise and responsivity of the THM detector

will now be derived for an ideal thermal model. For this ideal model of the THM

detector the hot-electron thermal conductance, Ge−p, controls the heat-flow from the

bolometer to the substrate and the other paths of thermal heat flow are neglected.

This assumption holds for Ge−p < Ge−e, Gboundary and GAndreev < Ge−p. A schematic

for this ideal thermal model is shown in Figure 4.3. The derivation below follows the

matrix method developed by M. Lindeman [78].

103

Absorber

Electrons

TES

Electrons

RF Power

µstrip

Ge-p

GAndreev Ge-e

Ge-p

TES

Phonons

Absorber

Phonons

Substrate Phonons

Gboundary

GAndreev

Leads

Figure 4.2: The complete thermal model for the THM.

104

G e-p

Tbolo

TES/

Absorber

T bath

P + P signal 0

Pbath

Figure 4.3: The ideal thermal model for the THM detector.

105

The derivation begins with the two electrical and thermal differential equations

in the small signal limit given by Equations 2.18 and 2.19. This time, however, noise

source terms (described in Chapter 2) are also taken into account. Power fluctuations

in the detector due to thermal fluctuations, NEPthermal, and the power fluctuations

inside the detector due to Johnson noise voltage fluctuations of the resistive TES

(NEPJ,TES = I0NEVJ,TES where I0 is the bias current) are added as source terms

in the thermal equation. This TES voltage fluctuation noise, as well as the voltage

fluctuations across the shunt resistor, NEVJ,s, are also added to the source terms in

the electrical equation. It is important to note that only voltage fluctuations inside

the detector and not those which occur outside the detector in the readout circuit

contribute to the power fluctuation source terms. Equations 2.18 and 2.19 become

respectively:

Cd∆T

dt= [

P0α

T0−Ge−p(T0]∆T+(2+β)V0∆I+Psignal+NEPthermal+NEPJ,TES (4.1)

Ld∆I

dt= −V0α

T0∆T + (−Rs − (1 + β)R0)∆I +NEVJ,s −NEVJ,TES. (4.2)

It happens that for bolometer performance one is usually interested in solving

these equations in the frequency domain. This is useful since a bolometric instrument

is usually operated in a chopping mode between a signal and calibrated source at a

particular frequency. Thus at this point a Fourier transform of the equations from

time to frequency space is completed as below:

∆T (t) −→ ∆T (ω) (4.3)

d∆T (t)

dt−→ ıω∆T (ω) (4.4)

∆I(t) −→ ∆I(ω) (4.5)

106

d∆I(t)

dt−→ ıω∆I(ω). (4.6)

The thermal and electrical equations can be written in frequency space as:

ıωC∆T = [P0α

T0−Ge−p(T0)]∆T+(2+β)V0∆I+Psignal+NEPthermal+NEPJ,TES. (4.7)

ıωL∆I = −V0αT0

∆T + (−Rs − (1 + β)R0)∆I +NEVJ,s −NEVJ,TES (4.8)

Dividing both sides of each of the equations above by C and L respectively it becomes

apparent that the two equations can be written in matrix form:

ıω

(

∆T

∆I

)

=

(

P0αT0C

− Ge−p(T0)C

(2+β)V0C

−V0αLT0

−Rs−(1+β)V0L

)

(

∆T

∆I

)

+

(

Psignal+NEPthermal+NEPJ,TES

CNEVJ,s−NEVJ,TES

L

)

(4.9)

Combining the like terms, the expression simplifies to:

M

(

∆T

∆I

)

=

(


CNEVJ,s−NEVJ,TES

L

)

(4.10)

where the matrix M is given by:

M =

(

m1 m2

m3 m4

)

≡(

ıω +Ge−p(T0)

C− P0α

T0C−1(2+β)V0

CV0αLT0

ıω + Rs+(1+β)R0

L

)

. (4.11)

To solve for the response of the THM, the inverse of the matrix M is taken:

(

∆T

∆I

)

=M−1

(


CNEVJ,s−NEVJ,TES

L

)

(4.12)

where M−1 is given by

M−1 =

(

m−11 m−1

2

m−13 m−1

4

)

≡(

m4

m1m4−m2m3

−m2

m1m4−m2m3

−m3

m1m4−m2m3

m1

m1m4−m2m3

)

. (4.13)

The response of the detector to the incoming signal power can now be read off by

completing the matrix multiplication in Equation 4.12 and ignoring the noise terms.

107

The responsivity of the device, S(ω), is defined as the ratio of change in TES current

to a change in input power:

S(ω) ≡ ∆I

Psignal=m−1

3

C(4.14)

where the value ofm−13 in terms of the bolometer properties can be read off from theM

and M−1 definitions in Equations 4.11 & 4.13. The current noise (NEI ≡ 〈∆I ·∆I〉)

also falls out from the matrix equations by completing the matric multiplication and

ignoring the input signal term:

∆I =m−1

3

C(NEPthermal + I0NEVJ,TES) +

m−14

L(NEVJ,s −NEVJ,TES) (4.15)

NEI(ω) = [(m−1

3

C)2(NEP 2

thermal + I20NEV2J,TES)

+(m−1

4

L)2(NEV 2

J,s +NEV 2J,TES)

−m−13 m−1

4 I0CL

NEV 2J,TES +NEI2SQUID]

1

2 . (4.16)

Here only the correlated terms have survived and the uncorrelated current noise from

the SQUID readout has been added directly into the current noise expression at this

point. To convert to an equivalent power fluctuation in the detector one divides by

the responsivity:

NEP (ω) =NEI(ω)

S(ω)(4.17)

The results for responsivity, NEI and NEP will be left in this form, however

the matrix elements can be read off from Equations 4.11 and 4.13. The expressions

for the noise terms, NEVJ,TES, NEVJ,s, NEPthermal are given by Equation 2.23, 2.24

and 2.27. For the case of a THM with a Au or Mo/Au TES with Au volume VTES

108

and a Bi absorbing structure with volume VAbs, Ge−p(T ) = ne−pκe−pTne−p−1 where

κe−p = ΣAuVTES + ΣBiVAbs.

4.1.2 Non-Ideal Model Theory & Noise

In this section, the noise and response of the THM detector will now be derived

for a thermal model of a device which is not “ideal”. Specifically, we will examine the

situation where the thermal conductance due to electron-electron scattering within

the detector is on the same order as the electron-phonon thermal conductance within

the detector (Ge−p & Ge−e). The other thermal conductances will continue to be

neglected. In this situation the THM detector is no longer in thermal equilibrium and

the absorber and TES must be divided into multiple thermal sections, each with its

own temperature.

An example of such a thermal model for a non-ideal, thermally disconnected

THM is shown in Figure 4.4 where a large absorber is divided into three parts. The

specific manner of thermal disconnection in this model matches that of THM test de-

vices of variation ‘THM2003’ which are described and presented in Chapter 5. How-

ever, other types of thermal disconnection or other geometries will lead to similar

effects on the THM response and noise, and the method of calculating these proper-

ties would follow the method described here. Two effects result from this thermally

disconnected model: additional noise terms arise from internal thermal fluctuations,

and the detector responsivity is reduced because some of the incident power to the

device bypasses the TES.

In this particular non-ideal THM the incoming signal power is dissipated evenly

along the absorber structure except in the central thermal region which overlaps the

109

Absorber End

T abs

½ Psignal ½ Psignal P 0

G e - p G e - p

T c

TES/ Central

Absorber

G e - e

G e - p

T bath

T abs Absorber

End

G e - e

Figure 4.4: A thermal model for a non-ideal THM.

TES. Due to the symmetry of the power dissipation in this device the thermal model

can be simplified to that shown in Figure 4.5. Here G1(T ) = ne−pκe−p,1Tne−p−1,

G2(T ) = ne−pκe−p,2Tn−1 and G3(T ) = ne−eκe−eT

ne−e−1. κe−p,1 = ΣBi23VAbs, κe−p,2 =

ΣBi13VAbs + ΣAuVTES and κe−e = 8L0

3RAbsfor this particular ‘THM2003’ device, where

RAbs is the total resistance across the absorber (see Chapter 5 for a description of this

device geometry) and ne−e = 2. Each section has a composite heat capacity of C1 and

C2.

To derive the noise and responsivity for this non-ideal THM one must take a

step back from the electrical and thermal equations simplified and applied in the low

signal limit that are given by Equations 2.18 and 2.19, as now there is more than one

temperature variable and more than one region of the detector to account for. For

section ‘1’ of the detector the thermal differential equation is:

C1dT1dt

= Psignal − Pbath,1 − P12. (4.18)

Here Pbath,1 is the net power flow out of part ‘1’ to the cold bath, and P12 is the net

110

P0 Psignal

G 1

T1

TES/ Central

Absorber

G2

T bath

Absorber Ends

G3

C1 T2 C2

Pbath,1 Pbath,2

P12

Figure 4.5: The non-ideal THM thermal model, simplified by the symme-

try of the device.

power flow out of part ‘1’ to part ‘2’. All the signal power is dissipated in the absorber

ends. This equation can be written more explicitly as:

C1dT1dt

= Psignal − κe−p,1(Tne−p

1 − Tne−p

bath )− κe−e(Tne−e

1 − Tne−e

2 ) (4.19)

A similar thermal differential equation for section ‘2’ of the detector can be written

as:

C2dT2dt

= P0 − Pbath,2 + P12 (4.20)

C2dT2dt

= I2TESRTES(ITES, T2)− κe−p,2(Tne−p

2 − Tne−p

bath ) + κe−e(Tne−e

2 − Tne−e

1 ). (4.21)

Here it is assumed that the signal power dissipation, Psignal, bypasses the central part

of the absorber because for this THM variation the absorber is short-circuited by

the low-resistance TES; there is no signal power source term and only Joule power

111

dissipation, P0. The electrical equation describing the TES behaviour remains the

same as in the ideal case:

LdITESdt

= V − ITESRs − ITESRTES(ITES, T2). (4.22)

A Taylor expansion of these three equations is completed as before in order to examine

the low-signal/linear limit, in terms of ∆T1 , ∆T2 and ∆I:

∆I = ITES − I0 (4.23)

∆T1 = T1 − T0,1 (4.24)

∆T2 = T2 − T0,2. (4.25)

Here I0, T0,1 and T0,2 are the equilibrium values with only bias power dissipation in

the device due to a voltage bias, V0. Completing a Taylor expansion of Equations

4.19, 4.21 and 4.22 results in:

C1dT1dt

≈ ∂

∂T1[−κe−p,1(T ne−p

1 − Tne−p


1 − Tne−e

2 )]0∆T1

+∂


1 − Tne−p


1 − Tne−e

2 )]0∆T2

+Psignal (4.26)

C2dT2dt

≈ ∂


2 − Tne−p


2 − Tne−e

1 )]0∆T1

+∂


2 − Tne−p


2 − Tne−e

1 )]0∆T2

+∂

∂ITES[I2TESRTES(ITES, T2)]0∆I

+∂

∂T2[I2TESRTES(ITES, T2)]0∆T2 (4.27)

LdITESdt

≈ ∂

∂T2[−ITESRTES(ITES, T2)]0∆T2

+∂

∂ITES[−ITESRs − ITESRTES(ITES, T2)]0∆I. (4.28)

112

After partial differentiation, these equations become:

C1d∆T1dt

= −ne−pκe−p,1T ne−p−11,0 ∆T1 − ne−eκe−eT

ne−e−11,0 ∆T1 + ne−eκe−eT

ne−e−12,0 ∆T2

+Psignal (4.29)

C2d∆T2dt

= −ne−pκe−p,2T ne−p−12,0 ∆T1 − ne−eκe−eT

ne,e−12,0 ∆T2

+ne−eκe−eTne−e−11,0 ∆T1 + I20

∂RTES(ITES, T2)

∂T2|0 ∆T2

+2I0RTES(I0, T2,0)∆I + I20∂RTES(ITES, T2)

∂ITES|0 ∆I (4.30)

Ld∆I

dt= −I0

∂RTES(ITES, T2)

∂T2|0 ∆T2

+[−Rs −RTES(I0, T2,0)− I0∂RTES(ITES, T2,0)

∂ITES|0]∆I. (4.31)

When these equations are written in terms of the thermal conductances G1, G2 and

G3, and α and β, they become:

C1d∆T1dt

= [−G1(T1,0)−G3(T1,0)]∆T1 +G3(T2,0)∆T2 + Psignal (4.32)

C2d∆T2dt

= G3(T1,0)∆T1 + [−G2(T2,0)−G3(T2,0) +P0α

T2,0]∆T2 + (2 + β)V0∆I (4.33)

Ld∆I

dt= −V0α

T2,0∆T2 + (−Rs − (1 + β)R0)∆I. (4.34)

Here, R0 ≡ RTES(I0, T2,0).

The noise sources for this thermally disconnected model will now be considered.

There are now thermal power fluctuations across three thermal paths: NEPthermal,1,

NEPthermal,2, NEPthermal,3. Here NEPthermal,1 is the thermal fluctuation noise be-

tween region ‘1’ at temperature T1 and the bath at temperature Tbath. NEPthermal,2

is the thermal fluctuation noise between region ‘2’ at temperature T2 and the bath at

113

temperature Tbath. NEPthermal,3 is the thermal fluctuation noise between region ‘1’ at

temperature T1 and region ‘2’ at temperature T2. Adding these source noise terms,

along with the Johnson noise terms, to the three differential equations they become:

C1d∆T1dt

= [−G1(T1,0)−G3(T1,0)]∆T1 +G3(T2,0)∆T2 + Psignal

+NEPthermal,1 +NEPthermal,3 (4.35)

C2d∆T2dt

= G3(T1,0)∆T1 + [−G2(T2,0)−G3(T2,0) +P0α

T2,0]∆T2

+(2 + β)V0∆I +NEPthermal,2 −NEPthermal,3

+I0NEVJ,TES (4.36)

Ld∆I

dt= −V0α

T2,0∆T2 + (−Rs − (1 + β)R0)∆I +NEVJ,s −NEVJ,TES (4.37)

A Fourier transform to frequency space is again completed:

∆T1(t) −→ ∆T1(ω) (4.38)

d∆T1(t)

dt−→ ıω∆T1(ω) (4.39)

∆T2(t) −→ ∆T2(ω) (4.40)

d∆T2(t)

dt−→ ıω∆T2(ω) (4.41)

∆I(t) −→ ∆I(ω) (4.42)

d∆I(t)

dt−→ ıω∆I(ω) (4.43)

In frequency space, the three differential equations become:

ıωC1∆T1 = [−G1(T1,0)−G3(T1,0)]∆T1 +G3(T2,0)∆T2 + Psignal

+NEPthermal,1 +NEPthermal,3 (4.44)

114

ıωC2∆T2 = [−G2(T2,0)−G3(T2,0) +P0α

T2,0]∆T2 +G3(T1,0)∆T1

+(2 + β)V0∆I +NEPthermal,2 −NEPthermal,3

+I0NEVJ,TES (4.45)

ıωL∆I = −V0αT2,0

∆T2 + (−Rs − (1 + β)R0)∆I +NEVJ,s −NEVJ,TES. (4.46)

After some rearrangement this becomes:

ıω∆T1 =−G1(T1,0)−G3(T1,0)

C1∆T1 +

G3(T2,0)

C1∆T2

+Psignal +NEPthermal,1 +NEPthermal,3

C1(4.47)

ıω∆T2 =G3(T1,0)

C2∆T1 +

−G2(T2,0)−G3(T2,0) +P0αT2,0

C2∆T2 +

(2 + β)V0C2

∆I

+NEPthermal,2 −NEPthermal,3 + I0NEVJ,TES

C2(4.48)

ıωL∆I = − V0α

LT2,0∆T2 +

−Rs − (1 + β)R0

L∆I +

NEVJ,s −NEVJ,TESL

. (4.49)

Writing these equations into matrix equation form:

ıω

∆T1∆T2∆I

=

G1(T1,0)−G3(T1,0)

C1

G3(T2,0)

C10

G3(T1,0)

C2

−G2(T2,0)−G3(T2,0)+PαT2,0

C2

(2+β)V0C2

0 −V0αLT2,0

−Rs−(1+β)R0

L

∆T1∆T2∆I

+

Psignal+NEPthermal,1+NEPthermal,3

C1

NEPthermal,2−NEPthermal,3+I0NEVJ,TES

C2

NEVJ,s−NEVJ,TES

L

(4.50)

After rearranging, this matrix equation becomes:

M

∆T1∆T2∆I

=


C1


C2

NEVJ,s−NEVJ,TES

L

(4.51)

115

where the matrix M is defined as:

M =

m1 m2 m3

m4 m5 m6

m7 m8 m9

≡

ıω +G1(T1,0)+G3(T1,0)

C1G3(T2,0) 0

−G3(T1,0)

C2ıω +

G2(T2,0)+G3(T2,0)

C2− P0α

T2,0C2− (2+β)V0

C2

0 V0αLT2,0

ıω + Rs+(1+β)R0

L

. (4.52)

After inverting the matrix, this equation can be solved for the bolometer response in

the non-ideal case:

∆T1∆T2∆I

=M−1


C1


C2

NEVJ,s−NEVJ,TES

L

(4.53)

where the inverse matrix, M−1 is defined as

M−1 =

m−11 m−1

2 m−13

m−14 m−1

5 m−16

m−17 m−1

8 m−19

. (4.54)

Following the same procedure as before in the ideal case, completing the matrix mul-

tiplication, ignoring the noise terms, the relation for responsivity can be read off:

S(ω) ≡ ∆I

Psignal=m−1

7

C1. (4.55)

The detector current noise, NEI, can also be read off from the equation, ignoring the

signal power and inserting the uncorrelated SQUID current noise directly into the

solution at this point:

NEI(ω) = [(m−1

7

C1

)2(NEP 2thermal,1 +NEP 2

thermal,3)

+(m−1

8

C2

)2(NEP 2thermal,2 +NEP 2

thermal,3 + I20NEV2J,TES)

+(m−1

9

L2)2(NEV 2

J,s +NEV 2J,TES)−

m−17 m−1

8

C1C2

NEP 2thermal,3

−m−18 m−1

9

C2LNEV 2

J,TES +NEISQUID]1

2 . (4.56)

116

Finally, the detector NEP in given by:

NEP (ω) =NEI(ω)

S(ω). (4.57)

A conclusion of this derivation is that the noise performance of the THM suffers

in the thermally disconnected detector model. Additional noise terms arise due to

thermal power fluctuations between the thermally disconnected regions in the detector.

In addition, the responsivity of the device decreases due to some of the power bypassing

the TES. These effects will be confirmed in measurements of a THM test device in

Chapter 5. An important part of optimizing the THM design will be to require

that the electron-phonon thermal conductance of the THM be less than the internal

electron-electron thermal conductance across the detector. As the electron-electron

Wiedemann-Franz conductance scales inversely with the resistance, this translates

into keeping the resistance of the TES and absorber low. This constraint will be

parametrized in more detail in Section 4.2.5.

4.2 Thermal & Microwave Optimization

In the second part of this chapter the specific optimization concerns for the

THM detector will now be detailed. One of these concerns is detector optimization to

obtain a thermally ‘ideal’ device, however, there are also additional concerns related

to obtaining background limited noise performance. Other concerns which must be

considered in the THM design include more practical constraints on the detector

design in order to simplify fabrication and the microwave coupling design. Before this

detailed optimization of the THM design is presented, however, the specific observing

and operating conditions which will be assumed, and which are specific to optimizing

117

the THM detector for CMB observing, are presented.

4.2.1 Photon Background Noise & Detector Loading Conditions

To calculate the loading power on a THM detector observing the CMB from

space one starts by calculating the photon occupation number for the CMB blackbody

source which follows Bose-Einstein statistics [92]:

n0 =ε

ehν

kTbb − 1. (4.58)

Here ε is the emissivity of the blackbody, ν is the observation frequency, and Tbb is

the blackbody temperature. The sky power that is absorbed by the detector when

observing this blackbody source is given by:

Psky = hνηn0∆ν

νν (4.59)

for the single-mode case, where ∆νν

is the fractional bandwidth and η is the optical effi-

ciency. The photon noise under these blackbody loading conditions can be calculated

using the expression given by Equation 2.22.

We now consider two different background conditions for observing the CMB:

from ground and space. Observing the CMB blackbody with Tbb = 2.7 K, at ν =92

GHz from space, assuming a 50% optical efficiency and a broad bandwidth of 20%,

the background photon noise is NEPphoton = 4.3 · 10−18 W√Hz

and the power loading

from the sky is Psky = 1.4 · 10−13 W. Observing the CMB blackbody at 92 GHz from

the ground (with additional sky background which mimics an effective blackbody of

temperature Tbb = 25 K) and assuming a 50% optical efficiency and a broad bandwidth

of 20%, the background photon noise is NEPphoton = 3.6 · 10−17 W√Hz

and the power

loading from the ground is Psky = 2.9 · 10−12 W. These two observing conditions will

118

be designated ‘CMB-ground’ and ‘CMB-Space’ in this chapter and in later chapters

of this thesis.

In addition to these two CMB observing conditions, for comparison purposes

in Sections 4.2.8 and 4.2.9 of this chapter, and later when measurements of actual

test devices are presented in Chapter 5, the THM’s performance under the CMB

observing conditions described above will be compared to its performance when ob-

serving the far-infrared sky from space. As was discussed in Section 1.1.4, observing in

the far-infrared is another application of great scientific interest for hot-electron type

bolometers. For observations of spectral lines in the far-infrared, the background power

loading is much lower than the CMB power loading, and the detector performance and

optimization criteria are very different. For the ‘FIR-spectral’ observing case, we as-

sume an observed Psky = 6.6 · 10−20 W from space at 1 THz, assuming 25% optical

efficiency and a narrow bandwidth of 0.1% corresponding to a resolving power of ν∆ν

.

The background photon noise for these observing conditions is NEPphoton = 1 · 10−20

W√Hz

.

For optimization purposes and later for measurement comparison, throughout

this thesis an electrical bias power of twice the sky background power, P0 = 2Psky, will

be assumed for ‘CMB-ground’, ‘CMB-space’ and ‘FIR-spectral’ observing. Through-

out this thesis we will also consider two bath temperatures: Tbath = 240 mK, cor-

responding to a bath temperature easily obtainable using a 3He sorption fridge, and

Tbath = 50 mK corresponding to a lower bath temperature obtainable using an Adi-

abatic Demagnetization Refrigerator (ADR), for example. For optimization purposes

we will also assume specific values for the electron-phonon material coefficient Σ, cor-

119

responding to average ‘literature values’ (ΣAu = 4.2 · 109 Wm3K5 and ΣBi = 2.4 · 108

Wm3K5 , see Section 3.1.2) and measured values corresponding to THM test device re-

sults for ‘measured n = 5’ (ΣAu = 1.0 · 109 Wm3K5 and ΣBi = 3.0 · 108 W

m3K5 , see test

device ‘THM2003’ in Section 5.3) and ‘measured n = 6’ (ΣAu = 1.5 · 108 Wm3K6 and

ΣBi = 8.8 · 106 Wm3K6 , see test device ‘us25’ in Section 5.4).

4.2.2 Microwave Circuit Constraints

To couple incoming radiation from free space to the small volume of the THM

detector an optical coupling scheme is necessary as coupling from free space directly

into the small bismuth absorber is inefficient. This optical coupling method needs to

provide a directed beam pattern on the sky. One would also like to create filters to

define the the detection bandwidth and polarizers to provide polarization sensitivity.

One way to accomplish this is to optically couple the incoming radiation to a planar

antenna which is much larger than the absorber and to transmit the signal to the

THM detector via planar transmission lines. Optical coupling via a horn antenna and

waveguide transmission line is another possible method. For present purposes we will

consider a microwave coupling scheme which only incorporates a planar antenna which

couples to a planar microstrip transmission line. In Chapter 6, however, an optical

scheme which incorporates additional optical components in order to provide efficient

optical coupling to a microwave source will be presented.

A planar transmission line is a compact way to transmit microwaves. Unlike

transmission in a waveguide, the electromagnetic fields are transmitted on a scale

much smaller than the radiation wavelength. A superconducting transmission line

also provides a low loss method to transmit the signal from one location to another.

120

In addition, it is possible to design microwave structures such as filters, couplers, and

terminations on a chip, allowing for the possibility to build a polarimeter (with a

polarization sensitive antenna and filter) on a single chip.

In the microwave design for the THM we have chosen to transmit the signal via a

superconducting microstrip transmission line which terminates on the THM absorber

as shown in Figure 4.1. More details of this microwave design and the coupling from a

planar antenna to the THM detector are given in Chapter 6. A microstrip termination

structure, which will also be described in Chapter 6, allows for the abrupt termination

of microwave power in the THM absorber.

A microstrip transmission line consists of a narrow conducting microstrip line

over a dielectrically insulated conducting ground plane as shown in Figure 4.6. The

characteristic impedance of a microstripline follows the form of [91]:

Zmicrostrip =120π

√ǫe[

Wd+ 1.393 + 0.667 ln W

d+ 1.444]

. (4.60)

Here ǫe is the effective dielectric constant of the dielectric between the ground plane

and microstrip line and is given by:

ǫe =ǫr + 1

2+ǫr − 1

2

1√

1 + 12dW

(4.61)

where ǫr is the relative dielectric constant, W is the microstrip width, and d is the

dielectric thickness.

The microstrip termination structure and the constraints this places on the ab-

sorber geometry and resistivity of the THM must be considered. Impedance matching

between the terminating microstrip characteristic impedance and the absorber resis-

tance is required. The width of the absorber must also match the microstrip width.

121

µstrip line

dielectricd

ground plane

W

Figure 4.6: Microstrip line geometry.

Thus there is a double constraint on the dimensions and resistivity of the absorber.

As will be detailed in the following sections, we would generally like to make the ab-

sorber as small as possible in order to minimize the electron-phonon conductance of

the bolometer for minimal thermal noise. A small microstrip/absorber width (higher

impedance) is compatible with this small absorber. An optimal size is a square,

micron-sized absorber coupling to micron-sized Nb microstrip line. A square absorber

of evaporated bismuth with resistance ∼ 20− 30 Ω (with thickness 0.8-1 µm) couples

well to a 3 µm wide Nb microstripline with characteristic impedance ∼ 20− 30 Ω on

a silicon or alumina dielectric of thickness ∼ 0.75− 1.5 µm.

4.2.3 Fabrication & Material Constraints

In the exploration of the THM design in this thesis we have only considered

device geometries obtainable by standard photolithography and fabrication techniques.

These fabrication processes are discussed in more detail in Chapter 5 when all of the

THM test devices are described. The main constraint these fabrication processes

122

impose is on the minimum square-area of the THM detector. Structures less than

2− 3 µm in size are not easily obtained with standard photolithography, and smaller

sizes are only possible by making use of e-beam lithography. As will be demonstrated

in this chapter, although a small size is usually desirable to minimize electron-phonon

coupling, other effects such as heating, and a strong lateral proximity effect from the

Nb leads will conspire to make the optimal absorber and TES size occur at, or above,

this photolithography size limit for CMB observing.

The material choice for the THM is a Mo/Au or Au superconducting TES and a

semi-metal Bi absorber. Bismuth was chosen both for its high resistivity at cryogenic

temperatures for impedance matching to the microstrip transmission line, and its low

electron-phonon coupling coefficient, Σ. A Mo/Au bilayer was chosen for the TES

due to the reliability of the Mo/Au bilayer system for obtaining precise transition

temperatures for the TES, as discussed in Sections 3.3.6 and 5.1.

4.2.4 Stability Constraints & Optimal Bias Conditions

Even with negative feedback a TES can become unstable if electrothermal fluc-

tuations are phase shifted in the feedback process. The requirements for stability are

given by Equations 2.20 and 2.21 and are determined by the time response of the

thermal and electrical circuits of the THM detector. We now investigate what con-

straints if any this places on the design of the THM bias circuit. In particular, we are

concerned with the inductance, L, the TES resistance, RTES, and the bias power, P0.

The second of these stability equations (Equation 2.21) indicates that for any

design or application, as the inductance in the readout circuit increases, the device

becomes less stable. As the load resistance and TES resistance increase (as long as

123

Rs ≪ RTES) the device becomes more stable (for a constant voltage bias, or constant

Joule power dissipation). For a THM device with a minimum volume absorber (3

µm x 6 µm x 800 nm), and a square TES (assuming RTES = 0.1 Ω) for both ground

and space-based observing conditions, stability requirements are met when L ≤ 10

nH−1 µH. This can be seen in Figure 4.7, where the left and right sides of Equation

2.21 are plotted for a 20 µm2 and 3 µm2 TES for ‘CMB-ground’ and ‘CMB-space’

loading conditions, respectively. As long as the inductance of the circuit is kept below

these limits the geometry and the operating conditions (such as the bath temperature

and bias power) of the THM detector are unconstrained. Typical single dc SQUID

amplifier readouts regularly obtain input inductances of L ∼ 1 nH-100 pH. However,

it should also be noted that SQUID current noise (see equation 2.25) also increases

with a low input inductance, and a balance between stability and current noise may

be necessary to consider when designing the SQUID readout.

The stability does not place any constraints on the bias power in principle.

However, the TES voltage bias should be chosen so that the TES is in the small

signal/linear regime. The responsivity is dependent on the bias conditions, but the

detector NEP, if thermally dominated, is independent of responsivity and bias power.

If the bias power is too high, however, heating occurs in the detector and the thermal

noise rises due to its temperature dependence. The degrading effect of this heating on

the thermal noise of the detector will be shown in more detail in Section 4.2.8.

4.2.5 Electron-Phonon Versus Electron-Electron

As argued in Section 4.1.2 an important component of the THM device opti-

mization is to require an ideal thermal design, where Gee > Gep, to avoid thermal

124

(a)

r.h.s

l.h.s

(b)

r.h.s

l.h.s

r.h.s

l.h.s

(c)

r.h.s

l.h.s

(d)

(H) (H)

(H)(H)

tim

e (

s)

tim

e (

s)

tim

e (

s)

tim

e (

s)

Figure 4.7: Stability limits on the SQUID readout input inductance, L.

The r.h.s. and l.h.s of Equation 2.21 are plotted and the maximum L for

stability is given by the point where the lines cross. (a) 20 µm2 TES,

assuming ‘CMB-ground’ conditions, for Tbath = 240 mK.(b) 20 µm2 TES,

assuming ‘CMB-space’ conditions, for Tbath = 50 mK. (c) 3 µm2 TES,

assuming ‘CMB-ground’ conditions, for Tbath = 240 mK.(d) 3 µm2 TES,

assuming ‘CMB-space’ conditions, for Tbath = 50 mK. ‘CMB-ground’ re-

sults are similar for Tbath = 50 mK. For these THM designs, stability

requires L ≤ 10 nH−1 µH.

125

disconnection within the detector and reductions to responsivity and sensitivity. To

illustrate this optimization we now consider a generic THM with a Mo/Au TES and

Bi absorber and examine what happens to the thermal conductances when the size

and/or resistance of the TES and absorber are varied. There are of course many

design variations possible which will meet this optimization requirement. The exer-

cises in this section are only to demonstrate the procedure and thought process for

THM optimization. As was noted in Section 4.2.2 both the absorber resistance and

the geometry of the absorber must be optimized to match the microstrip impedance.

However for the time being we will separate these two issues and only examine the

thermal properties due to the absorber resistance.

First we examine what happens when the absorber resistance is varied by ad-

justing the absorber length. In this case we assume a fixed Mo/Au TES geometry

with 300 nm thick Au layer, and a Au area of 10 µm x 10 µm. We also assume a Bi

absorber of a fixed width of 3 µm and thickness of 800 nm (resulting in ρ ∼ 30 Ω)

with a variable length which changes the absorber resistance, RAbs. The requirement

for an ideal model, such that there be no thermal disconnection across the absorber,

is that the total electron-phonon conductance of the detector, Ge−p ≪ Ge−e, where

Ge−e is the electron-electron conductance across the absorber. The requirement for

no power bypassing the TES is less stringent: Ge−p,Abs ≪ Ge−e, where Ge−p,Abs is the

electron-phonon thermal conductance of the absorber only. In Figure 4.8 these three

thermal conductances are plotted for a few typical THM designs. For these designs

RAbs . 80− 100 Ω is required to avoid power bypassing the TES, and RAbs . 3− 80

Ω is required to avoid a temperature differential across the absorber.

126

s

Ge-p

Ge-p, Abs

Ge-e

G (

W/

K)

(Ω)

(c)

G (

W/

K)

Ge-p

Ge-e

Ge-p, Abs

(a) (b)

G (

W/

K)

Ge-p, Abs

Ge-e

Ge-p

ss (Ω) (Ω)

Figure 4.8: The electron-phonon conductance of the combined TES and

absorber structure, Ge−p, the electron-electron conductance across the en-

tire TES and absorbing structure (absorber dominated in this case), Ge−e,

and the electron-phonon conductance of the absorber only, Ge−p,Abs plot-

ted as a function of absorber resistance, RAbs. Here we have assumed a 10

µm x 10 µm x 300 nm Au TES and a 3 µm wide, 800 nm thick, Bi absorber.

For electron-phonon thermal conductance we have assumed n = 5 and ‘lit-

erature values’ for Σs. (a) ‘CMB-ground’ conditions and Tbath = 240 mK

(b) ‘CMB-ground’ conditions and Tbath = 50 mK (c) ‘CMB-space’ condi-

tions and Tbath = 50 mK. For these designs RAbs . 80− 100 Ω is required

to avoid power bypassing the TES, and RAbs . 3 − 80 Ω is required to

avoid a temperature differential across the absorber.

127

This power separation in the detector as a function of this absorber resistance

can also be estimated. Assuming a similar and/or small temperature difference be-

tween the bath and absorber and absorber and TES, the fraction of incoming power

dissipated in the absorber which then flows through the TES, PTES, can be approx-

imated by PTES ∼ Ge−e

Ge−e+Ge−p,Abs. Plots of this power fraction are shown in Figure

4.9 for the same device assumptions used in Figure 4.8. For 80 − 100% power flow

through the TES, RAbs . 40− 100 Ω is required. Note that although this calculation

gives a useful indication of the power-loss effect, the true power ratio must be more

accurately calculated by doing a full responsivity calculation, assuming an appropriate

disconnected thermal model.

The dependence of these thermal conductances on the TES resistance can also

be examined. In Figure 4.10 the value of the electron-phonon conductance of the

entire detector, Ge−p is plotted against the electron-electron conductance across the

TES, Ge−e,TES, as a function of TES resistance, RTES. In this case we assume a fixed

Bi absorber geometry of 3 µm x 6 µm x 800 nm. We also assume a Mo/Au TES of a

fixed width of 10 µm and thickness of 300 nm (resulting in ρ ∼ 0.1 Ω) with a variable

length which changes with RTES. Requiring Ge−p < Ge−e,TES gives a constraint of

RTES < 0.5− 5 Ω for a typical THM device.

In addition to exploring this effect in terms of a constraint on the absorber or

TES resistance, it can also be explored in terms of its constraint on the geometry of the

THM. In Figure 4.11 the thermal conductances are plotted for a square TES device

(RTES = 0.1 Ω) as a function of TES width, wAu, assuming a fixed absorber geometry

with an absorber resistance that dominates the total device resistance (RAbs = 30 Ω).

128

(c)

(Ω)s

PT

ES

PT

ES

(Ω)s

(b)(Ω)s

(a)

PT

ES

Figure 4.9: The estimated fraction of power which flows through the TES

(PTES ∼ Ge−e

Ge−e+Ge−p) as a function of absorber resistance, RAbs. Here we

have assumed a 10 µm x 10 µm x 300 nm Au TES and a 3 µm wide, 800

nm thick, Bi absorber. For electron-phonon thermal conductance we have

assumed n = 5 and literature Σ values. (a) ‘CMB-ground’ conditions

and Tbath = 240 mK (b) ‘CMB-ground’ conditions and Tbath = 50 mK

(c) ‘CMB-space’ conditions and Tbath = 50 mK. For these designs Rabs .

40 − 100 Ω is necessary to obtain at least 80 − 100% power flow through

the TES.

129

(c)(Ω)

(Ω)

(b)(Ω)

(a)

G (

W/

K)

Ge-p

Ge-e,TES

Ge-e,TES

Ge-p

Ge-p

Ge-e,TES

G (

W/

K)

G (

W/

K)

Figure 4.10: The TES electron-electron thermal conductance, Ge−e,TES

plotted with the total electron-phonon thermal conductance, Ge−p, as a

function of TES resistance, RTES. In this case we assume a Bi absorber

geometry of 3 µm x 6 µm x 800 nm. We also assume a Mo/Au TES of

a fixed width 10 µm and thickness of 300 nm (resulting in ρ ∼0.1 Ω).

For electron-phonon thermal conductance we have assumed n = 5 and

literature Σ values. (a) ‘CMB-ground’ conditions and Tbath = 240 mK (b)

‘CMB-ground’ conditions and Tbath = 50 mK (c) ‘CMB-space’ conditions

and Tbath = 50 mK. Requiring Ge−p < Ge−e,TES, gives a constraint of

RTES < 0.5− 5 Ω depending on the application and bath temperature.

130

For these THM designs wAu . 1−20 µm is required to avoid a temperature differential

across the absorber (requiring Ge−e > Ge−p), but there is no requirement for wAu to

avoid power bypassing the TES (requiring Ge−e > Ge−p,Abs).

It should be noted that the usually cited requirement against self-heating and

a phase differential between superconducting and normal regions in a TES detector

due to this same effect of electron-electron scattering vs. electron-phonon scattering

(RTES ≤ π2L0TcG

nα) [51] gives a similar resistance requirement as is calculated here

for the more naive Ge−e > Ge−p requirement. Following the same THM design as-

sumptions as in Figure 4.10 this self-heating stability requirement gives optimal TES

resistance: RTES ≤ 0.4 Ω for Tbath = 240 mK and RTES ≤ 3 Ω for Tbath = 50 mK,

assuming n = 5 and α = 100.

4.2.6 Phonon-Phonon Versus Electron-Phonon

Thus far the phonon-phonon boundary thermal resistance between phonons in

the detector and the detector substrate/cold-stage has been neglected in our opti-

mization concerns. To do so we have assumed that the electron-phonon conductance

between the electrons to the phonons in the detector, Ge−p, is much smaller than the

phonon-phonon conductance between the detector phonons and substrate phonons,

Gboundary. This assumption will now be examined by comparing predictions for Ge−p

and Gboundary for the THM detector. Here we assume that the boundary thermal con-

ductance is given by Equation 3.7. Again, we assume a minimal Bi absorber of size 3

µm x 6 µm x 800 nm and a square Mo/Au TES with a Au thickness of 300 nm and

examine the thermal conductances dependence on TES width, wAu. In Figure 4.12

the relevant thermal conductances are compared for various bath temperatures and

131

(c)

Ge-p

Ge-e

Ge-p, Abs

G (

W/

K)

(m)

(a)

G (

W/

K)

Ge-p

Ge-e

Ge-p, Abs

G (

W/

K)

(b)

Ge-p

Ge-e

Ge-p, Abs

(m) (m)

Figure 4.11: The total electron-electron thermal conductance across the

absorber and TES, Ge−e, plotted with the total electron-phonon ther-

mal conductance, Ge−p, and the absorber electron-phonon conductance,

Ge−p,Abs, as a function of TES width, wAu, for a square Mo/Au TES with

Au thickness 300 nm. Here we have assumed a 3 µm x 6 µm x 800 nm

Bi absorber for RAbs = 30 Ω. For electron-phonon thermal conductance

we have assumed n = 5 and literature Σ values. (a) ‘CMB-ground’ con-

ditions and Tbath = 240 mK (b) ‘CMB-ground’ conditions and Tbath = 50


wAu . 1 − 20 µm is required to avoid a temperature differential across

the absorber (requiring Ge−e > Ge−p). There is no requirement to avoid

power bypassing the TES (requiring Ge−e > Ge−p,Abs).

132

loading conditions. The total thermal conductance of the THM, Gtot (summing Ge−p

and Gboundary in series), is also plotted.

From Figure 4.12 one can see that the boundary conductance is predicted to have

some impact or even dominate the total thermal conductance (Ge−p ≥ Gboundary) over

the full range of TES size scales for ground and space observing conditions. As was

discussed in Chapter 3, however, there is some dispute to whether this prediction for

Gboundary is correct for thin films like the THM detector and there are arguments that

the actual boundary thermal conductance may be higher for thin films or that there

will be no effective boundary resistance at all since the phonons in the detector may

be considered as the same system as the substrate phonons. Thus we will allow for the

possibility that phonon-phonon boundary scattering plays some role in the thermal

bottleneck of the THM bolometer and look for confirmation in the temperature power-

law dependence of the thermal conductance measurements of actual THM devices

(n = 4 for boundary conductance, n = 4 − 6 for electron-phonon conductance) in

order to disprove or prove a hot-electron dominated assumption for the THM.

4.2.7 SQUID & Johnson Noise Constraints

Although for the remainder of the chapter we will consider device performance

where the detector noise is dominated by thermal fluctuation noise we now briefly

discuss the situations when this is not the case and the limits this places on the TES

resistance. In Figure 4.13 the NEP contributions to detector noise from the SQUID

readout and Johnson noise from the TES resistor and a shunt resistor in the bias

circuit are plotted as a function of TES resistance assuming a THM with a minimal Bi

absorber of size 3 µm x 6 µm x 800 nm and a 3 µm wide Mo/Au TES with Au thickness

133

(a)(m)

G (

W/

K)

Ge-p

Gboundary

Gtot

(m)

(b)

Gboundary

Ge-p

Gtot

G (

W/

K)

(c)(m)

G (

W/

K)

Gboundary

Ge-p

Gtot

Figure 4.12: The electron-phonon thermal conductance, Ge−p, boundary

thermal conductance, Gboundary, and total thermal conductance Gtot, of a

THM detector as a function of TES width, wAu. Here we have assumed

a Bi absorber of size 3 µm x 6 µm x 800 nm and a square Mo/Au TES

with a Au thickness of 300 nm. We have assumed n = 5 and coupling

‘literature values’ for Σs, and a average literature value for the boundary

thermal conductance coefficient, Cb = 15.5 · 10−4 Km2

W[111]. (a) ‘CMB-

ground’ conditions and Tbath = 240 mK (b) ‘CMB-ground’ conditions and

Tbath = 50 mK (c) ‘CMB-space’ conditions and Tbath = 50 mK. Gboundary is

predicted to have some impact or dominate the total thermal conductance

over the full range of TES size scales for ground and space observing

conditions.

134

of 300 nm (ρ = 0.1 Ω) and an adjustable length and total resistance. We assume a

shunt resistor with Rs = 0.025 Ω. The background photon NEP for CMB observing

is also plotted. For this estimate the measured current noise of the NIST Series Array

currently installed in our THM testing setup is assumed (NEI = 4√Tbath · 10−12

A√Hz

). A bath temperature of Tbath = 240 mK requires RTES < 1 Ω for ground-

based observing to ensure that the Johnson and SQUID noise are insignificant. At

bath temperature of Tbath = 50 mK, RTES < 20 − 100 Ω is required for ground- or

space-based observing.

4.2.8 The Optimal Design for the THM Detector

Now we examine the optimal geometry of the THM detector in terms of noise

performance given the constraints discussed in the preceding sections. If the detector

occupies the ideal thermal regime (the internal electron-electron conductance is high

compared to the electron-phonon conductance between the electrons and the cold

bath) further optimization is necessary to obtain a detector NEP at or below the

background photon NEP for the desired application.

In this section the detector NEP for a THM device is presented as a function of

TES area, for different observing conditions and bath temperatures. In this case we

have minimized the Bi absorber size as much as possible and also impedance matched

it to a 30 Ohm microstrip line. The absorber termination area is kept square to

maximize electron-electron thermal conductance and to keep the THM in the ideal

detector regime. The only variable is the TES volume, which is parametrized as the

square Mo/Au TES width, wAu, assuming a typical bilayer Au thickness of 300 nm.

‘Literature values’ for n = 5, ‘measured n = 5’, and ‘measured n = 6’ values are

135

(c)

(Ω)

NE

P (

W/wH

z)

Johnson

SQUID

Photon

(b)(Ω)

Photon

SQUID

Johnson

(a)(Ω)

NE

P (

W/wH

z)

NE

P (

W/wH

z)

Photon

SQUID

Johnson

Figure 4.13: SQUID and Johnson noise as a function of TES resistance,

RTES. We assume a Bi absorber of size 3 µm x 6 µm x 800 nm and a 3

µ wide Mo/Au TES with Au thickness of 300 nm (RTES = 0.1 Ω) and

an adjustable length and total resistance. We assume a shunt resistor

with Rs = 0.025 Ω and SQUID current noise of NEI = 4√Tbath · 10−12

A√Hz

. (a) ‘CMB-ground’ conditions and Tbath = 240 mK (b) ‘CMB-ground’

conditions and Tbath = 50 mK (c) ‘CMB-space’ conditions and Tbath = 50

mK. A bath temperature of Tbath = 240 mK requires RTES < 1 Ω for

ground-based observing to ensure that the Johnson and SQUID noise are

insignificant. At bath temperature of Tbath = 50 mK, RTES < 20− 100 Ω

is required for ground- or space-based observing.

136

investigated for the electron-phonon Σ. For CMB loading conditions (‘CMB-ground’

and ‘CMB-space’) the radiative expression for the thermal NEP (Equation 2.27) is

used. In addition, the THM noise performance for CMB observing is compared to

the THM noise performance under ‘FIR-spectral’ observing conditions and to the

far-infrared photon background noise. In all cases the thermal noise of the detector

is assumed to dominate the detector noise. These variations are shown in Figures

4.14-4.16.

There is a large difference in most cases between the thermal fluctuation noise

in the CMB loading case and the thermal fluctuation noise of the same device in the

low-loading limit for the FIR-spectral observing case. This is due to the detector

temperature rising significantly above the bath temperature under the relatively high

CMB loading power, especially as the device volume and thus electron-phonon thermal

conductance becomes small. The magnitude of this heating is shown in Figures 4.17,

4.18 and 4.19 where the bolometer temperature, Tbolo (the electron temperature) is

plotted versus TES width, wAu, for each of the respective variations which are shown

in terms of NEP in Figures 4.14, 4.15 and 4.16.

The naive assumption that the lowest thermal conductance and smallest volume

THM is optimal to obtain the lowest detector noise is not the case for CMB appli-

cations. Instead the optimal device geometry is dependent on the specific loading

application and most sensitively on the bath operation temperature. For n = 5, CMB

ground observing optimal values occur at square TES sizes of 3−10 µm for Tbath = 240

mK. Lowering the bath temperature to Tbath = 50 mK allows the detector NEP to

drop below photon background NEP for both ground and space observing over a wide

137

(b)

NE

P (

W/wH

z)

Photon CMB-ground

Photon CMB-space

Photon FIR-spectral

Detector CMB-ground

Detector CMB-space

Detector FIR-spectral

(m)

(a)(m)

NE

P (

W/wH

z)

Photon CMB-ground

Photon CMB-space

Photon FIR-spectral

Detector CMB-ground

Detector CMB-spaceDetector FIR-spectral

Figure 4.14: Thermal detector NEP as a function of TES width, wAu, for

a square Mo/Au TES with Au thickness 300 nm assuming a 3 µm x 6

µm x 800 nm Bi absorber. Detector and background photon NEP for

‘CMB-ground’, ‘CMB-space’ and ‘FIR-spectral’ observing conditions are

shown. We have assumed n = 5 and ‘literature values’ for Σs.

(a) Tbath = 240 mK. (b) Tbath = 50 mK. At Tbath = 240 mK background

limited NEP is obtainable for CMB ground observing only, for wAu . 3

µm. At Tbath = 50 mK background limited NEP is obtainable for both

CMB ground and space observing for wAu = 3− 50 µm.

138

(b)

NE

P (

W/wH

z)

Photon CMB-ground

Photon CMB-space

Photon FIR-spectral

Detector CMB-ground

Detector CMB-space


(m)

(a)(m)

NE

P (

W/wH

z)

Photon CMB-ground

Photon CMB-space

Photon FIR-spectral

Detector CMB-ground

Detector CMB-spaceDetector FIR-spectral


a square Mo/Au TES with Au thickness 300 nm, assuming a 3 µm x 6 µm

x 800 nm Bi absorber. Detector and background photon NEP for ‘CMB-

ground’, ‘CMB-space’ and ‘FIR-spectral’ observing conditions are shown.

We have assumed ‘measured n = 5’ values for Σs. (a) Tbath = 240

mK. (b) Tbath = 50 mK. At Tbath = 240 mK background limited NEP is

obtainable for ground observing only, for wAu . 10 µm. At Tbath = 50 mK

background limited NEP is obtainable for both CMB ground and space

observing for wAu = 1− 100 µm.

139

(b)

NE

P (

W/wH

z)

Photon CMB-ground

Photon CMB-space

Photon FIR-spectral

Detector CMB-ground

Detector CMB-space


(m)

(a)(m)

NE

P (

W/wH

z)

Photon CMB-ground

Photon CMB-space

Photon FIR-spectral

Detector CMB-ground

Detector CMB-space



a square Mo/Au TES with Au thickness 300 nm, assuming a 3 µm x

6 µm x 800 nm Bi absorber. Detector and background photon NEP for

‘CMB-ground’, ‘CMB-space’ and‘FIR-spectral’ conditions are shown. We

have assumed ‘measure n = 6’ values for Σ. (a) Tbath = 240 mK.

(b) Tbath = 50 mK. At Tbath = 240 mK near background limited NEP is

obtainable for CMB ground and space observing, for wAu . 40 µm and

wAu . 4 µm, respectively. At Tbath = 50 mK background limited NEP is

obtainable for CMB observing for wAu = 1− 1000 µm.

140

(b)(m)

CMB-space

CMB-ground

FIR-spectral

Tb

olo

(K

)

(a)

Tb

olo

(K

)

CMB-ground

CMB-space

FIR-spectral

(m)

Figure 4.17: The bolometer temperature, Tbolo, as a function of TES width,

wAu, for the THM design variations of Figure 4.14 for CMB observing

conditions (for FIR observing Tbolo = Tbath). (a) Tbath = 240 mK. (b)

Tbath = 50 mK.

141

(b)(m)

CMB-space

CMB-ground

FIR-spectral

Tb

olo

(K

)

(a)

Tb

olo

(K

)CMB-ground

CMB-space

FIR-spectral

(m)




Tbath = 50 mK.

142

(b)(m)

CMB-space

CMB-ground

FIR-spectral

Tb

olo

(K

)

(a)

Tb

olo

(K

)

CMB-ground

CMB-space

FIR-spectral

(m)




Tbath = 50 mK.

143

spread of TES sizes, ranging from 1− 100 µm. For n = 6 and Tbath = 50 mK, optimal

TES sizes span an even wider range from 1− 1000 µm. In all cases, Tbath = 50 mK is

required for photon noise limited space-based CMB observing.

4.2.9 General Optimization for CMB Bolometric Detectors

One of the perhaps surprising results of the optimization of the THM detector

is that as the bath temperature is lowered the NEP dependence on size weakens. This

is the result of the competition between two effects in the detector. As the thermal

conductance is lowered (and the bath temperature and κ are lowered) the thermal

NEP tends to decrease (NEP ∼√

G(Tbath)). However, as the thermal conductance

decreases, the temperature of the detector, Tbolo, increases for a fixed power loading

to the device (Tbolo − Tbath ∼ P0

G) and this forces the thermal NEP to increase.

There is a fundamental limit to the improvement of detector thermal noise by

simply lowering the thermal conductance. Although we demonstrated that this was

the case for the THM detector design in the previous section, we can also demonstrate

that this is indeed the case for any bolometeric detector in a relatively high power

loading limit (like that for CMB observing) where T ≫ Tbath. This will be done in

this section for the case of a generic CMB detector with thermal conductance in the

form of Equation 2.9, and thermal NEP given in the radiative and the diffusive limits

by Equations 2.27 and 2.28. These radiative and diffusive limits hold for either a

detector directly on the cold substrate, as is the case for a boundary or hot-electron

type bolometer (radiative), or for the more common CMB-type bolometer where the

heat flow is controlled by phonon-phonon scattering along suspended thermal legs

(diffusive). Again it is assumed that the detector noise is dominated by the thermal

144

noise. And again the CMB observing performance is compared to that of a detector

in a low power loading limit (like the ‘FIR-spectral’ case) where the expression for

thermal NEP is given by Equation 2.26.

In Figures 4.20-4.24 the behaviour of a generic CMB-observing bolometer is

shown for n = 2− 6 (electron-electron, boundary or electron-phonon) controlled heat

flow as a function of the power flow coefficient, κ (P = κ(T nbolo−T nbath)). The difference

between the thermal noise in the radiative and diffusive regime is minimal. Again,

a large discrepancy exists in some regimes between the thermal noise in the CMB-

observing detector and the thermal noise in the low power loading limit. Although the

optimal κ and minimal thermal NEP do vary with the power law, n, and the chosen

bath temperature, Tbath, the behaviour across all devices is similar. Under the CMB

loading conditions the thermal noise minimum is limited to NEP ∼ 7 · 10−18 W√Hz

for ‘CMB-ground’ observing and NEP ∼ 2 · 10−18 W√Hz

for ‘CMB-space’ observing.

Significant improvements below this level, which might be naively assumed from the

low-power loading limit expression for thermal noise, are not possible. Although pho-

ton noise limited performance is certainly possible for certain optimal detector designs,

a CMB observing detector’s performance is fundamentally limited due to heating in

the bolometer.

145

NE

P (

W/wH

z)

(c)

Photon CMB-ground

Detector CMB-ground

Detector low power loading limit

(a)

Photon CMB-ground

Detector CMB-ground


NE

P (

W/wH

z)

(d)

Detector CMB-space


Photon CMB-space

(b)

NE

P (

W/wH

z)

Photon CMB-space

Detector CMB-space


NE

P (

W/wH

z)

Figure 4.20: The thermal fluctuation NEP of a generic CMB observing

bolometer in comparison to background photon NEP. A temperature

dependence of the power flow, n = 2 is assumed. (a) ‘CMB-ground’


conditions with Tbath = 240 mK.(c) ‘CMB-ground’ observing conditions

with Tbath = 50 mK. (d) ‘CMB-space’ observing conditions with Tbath = 50

mK. For all of these cases, the thermal detector NEP in the low power

loading limit is also plotted.

146

NE

P (

W/wH

z)

(c)

Photon CMB-ground

Detector CMB-ground


(a)

Photon CMB-ground

Detector CMB-ground


NE

P (

W/wH

z)

(d)

Detector CMB-space


Photon CMB-space

(b)

NE

P (

W/wH

z) Photon CMB-space

Detector CMB-space


NE

P (

W/wH

z)









147

NE

P (

W/wH

z)

(c)

Photon CMB-ground

Detector CMB-ground


(a)

Photon CMB-ground

Detector CMB-ground


NE

P (

W/wH

z)

(d)

Detector CMB-space


Photon CMB-space

(b)

NE

P (

W/wH

z)

Photon CMB-space

Detector CMB-space


NE

P (

W/wH

z)









148

NE

P (

W/wH

z)

(c)

Photon CMB-ground

Detector CMB-ground


(a)

Photon CMB-ground

Detector CMB-ground


NE

P (

W/wH

z)

(d)

Detector CMB-space


Photon CMB-space

(b)

NE

P (

W/wH

z)

Photon CMB-space

Detector CMB-space


NE

P (

W/wH

z)









149

NE

P (

W/wH

z)

(c)

Photon CMB-ground

Detector CMB-ground


(a)

Photon CMB-ground

Detector CMB-ground


NE

P (

W/wH

z)

(d)

Detector CMB-space


Photon CMB-space

(b)

NE

P (

W/wH

z)

Photon CMB-space

Detector CMB-space


NE

P (

W/wH

z)









150

Chapter 5

THM Test Devices & “Dark”

Measurements

5.1 Test Devices & Fabrication

The general plan for developing this novel type of TES bolometer was to start by

fabricating large (10−100 µm) THM devices and then to push towards smaller (3−10

µm size) THM devices with a more optimal design. Additionally, the first test devices

had only simple DC lead connections in order to focus the investigation on the TES

and DC characteristics of the absorber. Later, RF microwave circuits were coupled to

the THM devices to investigate the detector coupling to an RF source (these designs

and measurements are presented in Chapter 6). Along the way in these development

tests a physical process was encountered which had not previously been considered: a

strong lateral superconducting proximity effect in micron-size TES devices. Thus, a

side project was undertaken to investigate this effect and additional test devices were

fabricated for this purpose.

In Table 5.1 all of the THM test devices presented in this thesis are listed by wafer

151

Table 5.1: A summary of the THM test devices which are presented in

this thesis.Device Label Fabrication

Date

General Description TES size Absorber size Microwave

circuit?

Main Result

THM2003 2003 Mo/Au TES with Bi

absorber & Mo DC

leads

10-20 µm ∼100 µm No NEP, S & G measure-

ments confirm non-ideal e-p

model.

THM4 & THM5 2007-2008 Mo/Au TES with Bi

absorber & Nb µstrip

with AlO dielectric

3-24 µm 3-24 µm Yes (3 µm size

only)

Strong lateral proximity ef-

fect, µstrip transmission

and microwave absorption

observed.

us23 & us25 2009-2010 Au TES with Bi ab-

sorber & Nb µstrip

with Si dielectric

3-30 µm 3 µm Yes G measurements confirm

ideal e-p model, but no

transitions observed.

THMA4 2010 Au TES with Nb DC

leads

3-30 µm - No Change in fabrication

method, no transitions

observed

THMA24 2011 Au TES with Nb DC

leads

3-30 µm - No Additional cleaning, no

transitions observed

name and a summary of the design and the findings for each device type. A depiction

of the device geometry and layer order, as well as descriptions of the fabrication steps

for each of these devices, and optical or SEM images of test devices, can be found

later in this chapter when measurements from each of these test devices are presented

(Figures 5.10, 5.21, 5.17 & 5.25).

Fabrication of all of these test devices was carried out at the NASA-Goddard

Detector Development Laboratory (DDL) making use of their excellent facilities and

detector fabrication expertise. Much of the processing involved standard photolithog-

raphy with resolutions of ∼ 1 µm. However, fabrication of these devices included

many novel fabrication techniques which have been perfected at the DDL, and some

of these steps can be seen in the in-process images shown in Figures 5.1 and 5.2. Some

152

of these techniques which will be briefly described here have also been described in

Denis et al. [20].

The Mo/Au bilayer TESs were deposited in a load-locked vacuum chamber using

a dedicated bilayer sputtering deposition system. Transition temperatures for the

device wafers were targeted after conditioning of the system and characterization of

Mo and Mo/Au films on test wafers. A common process was an ion-milling of the

Au or Mo/Au TES structure to create the well defined TES area. This ion-milling

process also provided a moderately sloped sidewall to the Au for better step coverage

of the Bi and Nb over the Au TES.

Another of the common process steps for the THM test devices was a sloped

sidewall Reactive Ion Etch (RIE) Nb etch (SF2 & O2) for good step coverage over

the microstrip groundplane features and over the Nb leads. During this etch a thick

layer of photoresist is deposited over the Nb lead and microstrip pattern and the

photoresist is simultaneously etched with the Nb. Because of this, care must be taken

not to over-etch into the Nb features. In the THM design this was especially true on

the small area of Nb overlap at the contacts to the TES leeds where the photoresist

was thinner. This was a problem in the fabrication of test devices THM4/THM5 as

will be discussed later in this chapter.

Another concern that arose during the fabrication of these THM test devices

was proper stripping of the protective photoresist layer from the Au TES surface after

the ion-milling process, due to burning of the resist during the ion-milling step. This

cleaning was complicated by the fact that ultrasonic solvent baths could not be used

after the bilayer deposition process because they would cause a shift of the bilayer Tc.

153

MoAu

Nb

9 µm

Mo/Au TES

9 µm

(a)

(b)

Figure 5.1: In-process images of some of the key fabrication steps for the

THM test devices. (a) Optical image of bilayer TES on wafer THM5 after

completion of the Au ionmill and Mo RIE patterning. The Mo extends

out to make contact with the Nb leads which will be deposited next. The

Mo on the non-lead sides of the TES will be etched away during the Nb

patterning step. (b) Optical image of wafer THM4 after completion of Nb

(SF6 + 02) RI sloped sidewall etch. The sidewall can been seen even in

this optical image as Nb (blue) extends past the dark top edge.

154

(a)

(b)

Nb (ground plane)

Nb (µstrip)

2 µm

Nb

Bi

Au

5 µm

Figure 5.2: In-process images of some of the key fabrication steps for the

THM test devices. (a) SEM image of THM4 showing good step coverage

over a Nb sloped sidewall. This view shows the Nb microstripline crossing

the gap in the Nb ground plane(see Chapter 6). The ground plane is

seen through the AlO dielectric layer. (b) SEM image of a 3 µm x 9 µm

Bi absorber test device (not incorporated into a THM) after liftoff. The

grainy structure of the evaporated Bi is observable. Thin Au contact pads

under the Bi help make contact to the Nb leads.

155

Improvement in the wafer heatsinking to the chilled mount during ion-milling, as well

as the use of an UV exposure and ethanol rinse de-scumming procedure, following the

usual O2 ash and solvent clean, were able to solve this problem. This cleaning step

of the Au surface was of particular concern in the fabrication of some of the lateral

proximity test devices (THMA4,THMA24) and will be discussed in more detail in

Section 5.5.

Another concern was obtaining a high quality dielectric layer for the microwave

circuit of the THM test devices. For wafers us25 and us23 a BenzoCycloButene (BCB)

bonding method was used to bond a 1.45 µm thick Si single crystal wafer to a Si

backing wafer for two-sided patterning, such that the Si wafer could be used as the

microwave dielectric layer. This Si dielectric was adopted after pinhole problems were

found in some devices (THM4 & THM5) which were fabricated using an Al2O3 film as

the dielectric. For adhesion purposes to this Si dielectric, a thin non-superconducting

Mo or Ti layer was deposited immediately before the Au TES deposition.

By using evaporated bismuth, a high resistivity for the RF absorbing structure

was obtained to match to the high impedance transmission lines. The electrical resis-

tivity of evaporated bismuth exhibits an unusual temperature dependence. In Figure

5.3 the resistivity of evaporated Bi is shown as a function of temperature for sev-

eral test wafers. By depositing bismuth of thickness ∼1 µm resistances of 20-30 Ω/

at sub-Kelvin to Kelvin temperatures were obtained in order to properly impedance

match to the Nb microstripline. The bismuth lift-off required the use of non-aggressive

resist remover solvents (Resist Remover 5 or Acetone) as the bismuth material is very

sensitive to chemical etchants. After bismuth deposition the wafer could not be heated

156

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

2.50E-03

3.00E-03

3.50E-03

4.00E-03

4.50E-03

5.00E-03

3 53 103 153 203 253

Temperature (K)

Res

isti

vit

y (

Oh

ms

*cm

)

AB test wafer, 850 n m

THM D6 test wafer, 1200 n m



Figure 5.3: The resistivity of evaporated Bi resistivity measured for several

test wafers (with Bi thickness listed) as a function of temperature. An

increase in resistivity is seen as the temperature decreases.

above 70 C in order to prevent Au/Bi diffusion from occurring at the absorber/TES

interface.

More fabrication details and devices dimensions will be included in the following

sections as they become applicable to understanding and analysing these measure-

ments. Rather than presenting the measurements of these test devices in a strict

chronological order, the measurements and analysis presented in the following sec-

tions is organized by topic, although each device is discussed separately within each

157

of these sections. This is done in order to call attention to how the characteriza-

tion of each of these test devices has furthered our understanding of the THM de-

sign. These sections are: 1) Inquiry into a non-ideal THM model (THM2003) 2)

Inquiry into the hot-electron effect (THM2003,us25) 3) Inquiry into the lateral prox-

imity effect (THM4,THM5,us23,us25,THMA4,THMA24) 4) Inquiry into detector NEP

(THM2003,us25).

5.2 “Dark” Cryogenic Test Setup

5.2.1 Dewar & Cryogenic Setup

All devices were cooled to temperatures between 4 K and 100 mK during testing

using an RF shielded dewar with a magnetically shielded Adiabatic Demagnetization

Refrigerator (ADR). The outside of the dewar and thermal shields are shown in Figure

5.4. The dewar has three insulating compartments, starting with the outer room

temperature shield at 300 K, which is vacuum tight, and transitioning to a 77 K

shield heatsunk to a liquid N2 tank. A further inner shield at 4 K and a support

platform for the ADR are heatsunk to a 4 K liquid 4He tank. The 100-300 mK

coldstage where the test devices are mounted is thermally isolated from this 4 K stage

via suspension on Kevlar strings (shown in Figure 5.5). A ferric ammoniom alum

(FAA) salt pill is also thermally isolated from this 4 K stage via Kevlar suspension

and a mechanical heatswitch. The magnetic coil surrounding the salt pill provides

a 3 T field and is magnetically shielded from the surrounding dewar by a vanadium

permendur cylinder. The base temperature of the ADR is near 240 mK from a 4.2

K bath. A base temperature of 140 mK is possible when pumping on the 4He bath

158

77K ShieldVacuum Shield

4K Shield

Vacuum feedthroughs

Liquid N2 tank fillLiquid He tank fill

Figure 5.4: The dewar shielding.

to pressures . 10 Torr. The ADR and cryostat was previously used on the MSAM

II balloon flight and details of the cryogenic system have been described previously

[126].

Test devices were mounted on the coldstage by one of two methods: 1) Inside a

superconducting Nb magnetically shielded cylindrical can with heatsinking via metal

clamping of this can to the coldstage 2) Direct mounting to the coldstage via a sand-

wich of metal-metal contacts. In both of these cases, the THM chips were mounted via

Stycast-2850 epoxy or Apiezon-N thermal grease to thin copper or Invar plates which

were clamped to a copper mount. Short superconducting Al wirebonds were used to

159

Unshielded mount

Nb shielded mount

Heatsinking wires

Superconducting cable

Kevlar suspension

GRT

RuthOx Thermometer

Detector chip

Figure 5.5: The coldstage.

make electrical contact from the Au or Nb pads on the detector chip to Cu or Au

wirebonding pads on the mount. Twisted pairs of copper wires soldered to these pads

on the chip mount were heatsunk to the coldstage and plug into multi-pin connectors

on the coldstage. Superconducting cables carry these signals out of the dewar via RF

filtered vacuum feedthroughs. The superconducting cables are heatsunk to the 4 K

stage (via GE varnish and Au-plated Cu metal clamps) and also at the 77 K stage

(via the feedthrough connectors). Pictures of this coldstage setup are shown in Figure

5.5, and images of the two types of chip mounts in Figures 5.5 & 5.6.

160

SQUID ChipTHM Detector ChipShunt Resistor

Figure 5.6: The THM chip and 1st stage SQUID mount which fits inside

a Nb shielded cylindrical can.

In addition to the test devices themselves, several diagnostic thermometers are

mounted at various locations in the dewar. Two Ge resistive thermometers are located

on the 77 K and the 4 K stages. Two other thermometers are mounted directly to the

coldstage, one a Ge resistive thermometer covering a temperature range from 50 mK-5

K, and the other a Ruthenium Oxide resistive thermometer covering the temperature

range from 1 − 40 K. These two coldstage thermometers provided a temperature

measurement of the bath temperature for the bolometer detectors.

5.2.2 SQUID Readout Setup

For some of the measurements of these THM test devices the TES was read

out with a 2-stage SQUID system from NIST. The chip containing the 1st stage

SQUID was mounted on a superconducting Al plate (acting as a magnetic shield)

161

X X

X X

2nd stage feedback

1st stage feedback

2nd stage bias

1st stage bias

1st stage

input coil

2nd Stage

SQUID

1st Stage

SQUID

TES

V0Rs

Figure 5.7: Diagram of the 2-stage SQUID electrical connections for the

read out of the THM TES.

near the detector chip inside the Nb shielded can. Al superconducting wirebonds, a

few millimeters in length, make electrical connections from the TES to the input coil

of the SQUID in parallel with a 25 mΩ shunt resistor. Superconducting wires from

the output pads of this 1st stage SQUID were brought out of the canister and wired to

the 2nd stage SQUID array mounted on the 4 K stage inside a magnetically shielded

box made of Cryoperm. The electrical connections for this SQUID setup are shown

in Figure 5.7.

In this SQUID setup, the TES is voltage-biased to provide electrothermal feed-

162

back. The 1st and 2nd stage SQUIDs are both DC current biased. SQUID feedback

electronics monitor the 2nd stage voltage signal (via the 2nd stage bias line) and re-

spond to current through the 1st stage input coil by sending an output nulling current

to the feedback coil near the 1st stage SQUID. The 2nd stage feedback line is usually

not used except for diagnostic purposes. The 1st stage feedback nulling signal is what

is read out and gives a voltage that is linearly proportional to the TES current.

The SQUID response was calibrated and the appropriate bias and feedback set-

tings found by removing the connections to the TES and sending a known AC current

through the input coil from a function generator. The DC current biases for the 1st

and 2nd stage SQUIDs were adjusted while monitoring the 2nd stage bias signal to

find the DC bias settings which gave the maximum amplitude response to this input

signal. The feedback gain and offset settings were also adjusted to make sure the

feedback electronics adequately nulled the input signal. This calibration procedure is

shown in Figures 5.8 and 5.9.

5.3 Inquiry into a Non-Ideal THM Model

In this section measurements of one of the THM test devices of variation THM2003

are presented. These results have previously been reported in Barrentine et al. 2008

[8]. This device was the first device we tested and we determined it conformed to the

non-ideal thermal model which is described in detail in Chapter 4. This THM test

device consisted of a thin Bi/Au absorber and bilayer Mo/Au TES. An optical image

and dimensions for this test device as well as the layer order and fabrication details are

shown in Figure 5.10. The Bi absorber layer is 500 nm thick under a 190 nm thick Au

layer. The bilayer TES is 40 nm thick Mo under a 180 nm thick Au layer. Mo leads

163

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-0.0 -0.03 -0.01 0.01 0.03 0.05

Signal to input coil (

IN M

ON

(V

)

feedback off

feedback on

Input Coil

-6.0 -3.6 -1.2 1.2 3.6 6.0

(µA)

2n

d s

tag

e S

QU

ID B

ias

Figure 5.8: The 2nd stage SQUID bias line voltage as a function of the

input calibration signal to the 1st stage SQUID input coil. With feedback

off the second stage signal follows the input signal. With feedback on the

bias signal is zero no matter the input signal, as a nulling feedback signal,

which tracks the input signal, is being fed back to the 1st stage SQUID.

164

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

-0.05 -0.03 -0.01 0.01 0.03 0.05

Signal tInpu t Coil (V)

1st

Sta

ge F

B (

V)

Input Coil (µA)

-6.0 -3.6 -1.2 1.2 3.6 6.0

Figure 5.9: The SQUID calibration curve. Plotted is the feedback signal

to the first stage SQUID as a function of the signal to the input coil. The

1st stage feedback signal is what is read out. The gain factor is given by

the slope of this linear curve.

165

attached to the absorber allow for direct Joule heating of the absorber by applying a

DC current. The TES transitions at 310-317 mK and has a normal resistance of 0.32

Ω. The absorber has a resistance of 17 Ω across the absorber bias leads.

As described in the previous section, the TES was wired in parallel with a shunt

resistor of resistance, Rs = 25 mΩ, and was read out by the NIST 2-stage SQUID. The

test device and 1st stage SQUID chip were placed within the Nb cylindrical shield,

and cooled to 180− 300 mK.

5.3.1 I-V Measurements

The current-voltage (I-V) characteristics of the TES for this device were mea-

sured by applying a slow AC bias current to the TES bias circuit and observing the

SQUID response. In Figure 5.11 the I-V curves for the TES are shown at several

bath temperatures. With Tbath < Tc, at low TES bias-voltage, the TES remains in

the superconducting regime. As the TES voltage and current increase the TES transi-

tions to the normal linear regime. In the transition region the Joule power dissipation

(P0 = IV ) is constant since the TES temperature is approximately constant in the

transition and constant bias power is required to maintain the TES at this fixed tem-

perature above the cold stage. This accounts for the negative slope and non-ohmic

behaviour of the I-V curve in the transition. As the bath temperature approaches

Tc, the transition current decreases, and the I-V curve approaches the normal linear

regime with normal resistance, RTES =0.32 Ω. The I-V curves map out the TES bias

points in the transition. Possible bias points for this test device range from ∼ 0.2− 3

µV.

166

20um

Bi, 500 nm, evaporated deposition, liftoff

Au, 150nm, sputtering bilayer deposition, ionmill

superconducting Mo, 40nm, sputtering bilayer deposition, RIE

Au, 200 nm, evaporated deposition, liftoff

(a)

(b)

Figure 5.10: (a) Fabrication steps and cross-sectional view showing the

layers for each of the THM2003 test devices. Thicknesses are not drawn

to scale, but are listed on the labels for each layer. (b) Optical image of a

test device of variation THM2003.

167

Figure 5.11: I-V curves for a device from wafer THM2003 at different

bath temperatures (labelled in the figure). The solid black line indicates

RTES =0.32 Ω.

168

5.3.2 Thermal Conductance Measurements

To measure the thermal conductance which characterizes the bolometer thermal

link to the coldstage for this test device, measurements were made of the amount of

Joule power dissipation in the absorber, P , required to maintain the TES at Tc as

a function of the cold stage temperature. This was done by allowing the coldstage

temperature to gradually rise from 180 mK to 320 mK while applying a DC current

to the absorber via the separate DC leads. The I-V curve was monitored to see what

Joule power dissipation in the absorber was required to fully “normalize” the I-V

curve at each bath temperature (thus using the high temperature end of the RTES

vs. T superconducting transition curve for these measurements). A plot of these

measurements of P vs. Tbath is shown in Figure 5.12.

These P vs. Tbath measurements are fitted to the two thermal models introduced

in Section 4.1.1 and Section 4.1.2. The first is an ‘ideal model’, where electron-phonon

coupling is assumed to dominate and a n = 5 temperature dependence to the power

flow is assumed. Here the equation of fit is P = κe−p(T5c − T 5

bath). The other thermal

model is a ‘non-ideal model’, where the detector is broken into three geometrically and

thermally separate regions due to the high electrical resistance, and hence, low rate

of electron-electron scattering along the length of the absorber. In this model, both

electron-electron scattering and electron-phonon scattering are assumed to dominate

and n = 2 and n = 5 are assumed respectively. Here the equation of fit is:

P = (κe−p,TES +1

3κe−p,Abs)(T

5c − T 5

bath) +2

3κe−p,Abs(T

5Abs − T 5

bath) (5.1)

169

0.24 0.26 0.28 0.30 0.32

0.00E+000

1.00E-009

2.00E-009

3.00E-009

4.00E-009

5.00E-009 Non-Ideal Model Electron-Phonon

& Electron-Electron (n=5,2) Ideal Model Electron-Phonon (n=5) Ideal Model Phonon-Phonon (n=4)

Abs

orbe

r P

ower

(W

)

T bath

(K)

Figure 5.12: P vs. Tbath measurement for devices from wafer THM2003.

170

using the relation:

Tabs =

√

1

κe−e[(κe−p,TES + κe−p,Abs)(T 5

c − T 5bath) + κe−eT 2

c ]. (5.2)

Here TAbs is the temperature of the absorber ends and κe−e =8L0

3Rabsis assumed based

on the geometrical separation in this non-ideal thermal model and the measured re-

sistance of the absorber, RAbs = 17 Ω. In addition, an ideal model assuming bound-

ary (phonon-phonon) coupling is also fit to the data. Here the equation of fit is

P = κboundary(T4c − T 4

bath). It should kept in mind, that because of uncertainty in the

n value for electron-phonon coupling (see Chapter 3), n = 4 behaviour is not neces-

sarily exclusively indicative of phonon-phonon coupling, and n = 4 behaviour may be

indicative of electron-phonon coupling as well.

The fitted κ values for all three models are listed in Table 5.2. In the case of the

non-ideal model, Σ values are listed, since the sectioning inherent in the model allows

a fit to the Bi and Au portions of κe−p separately. These fit κ and Σ values are also

compared to κ values calculated from the device dimensions and ‘literature values’ for

electron-phonon scattering for Au and Bi or phonon-phonon boundary scattering (see

Section 4.2.1). Thermal conductances at 310 mK are calculated from each of these fit

values as well.

Due to the short temperature range of these measurements, all three of these fits

conform well to the measured P vs. Tbath curve. The non-ideal model fits to electron-

phonon Σ values only slightly lower than those predicted from the literature. The

ideal electron-phonon model also fits very closely to the predicted electron-phonon

κe−p values from literature. Although the ideal boundary model does give the best fit

to the measured curve, the fit κ value for this model is a factor of four below what

171

Table 5.2: Summary of P vs. Tbath fits for various models and thermal

conductances at 310 mK.Thermal Models Ideal e-p Ideal Boundary Non-Ideal Literature

κe−p,Abs+κe−p,TES

( WK5 )

1.73 · 10−6 - 4.38 · 10−7 1.23 · 10−6

κbounday =4ACb

( WK4 ) - 6.45 · 10−4 - 1.95 · 10−7

κe−e =L0

2RAbs( WK2 ) - - 2.87 · 10−9 -

ΣAu ( WK5m3 ) - - 1.0 · 109 4.5 · 109

ΣBi (W

K5m3 ) - - 3.0 · 108 2.4 · 108Ge−p,Abs (

WK) - - 5.84 · 10−9 1.58 · 10−8

Gep,tot (WK) 2.48 · 10−8 - 6.27 · 10−9 1.76 · 10−8

Gboundary (WK) - 7.69 · 10−8 - 2.32 · 10−8

Ge−e (WK) - - 4.5 · 10−10 -

might be expected from the literature. In all three cases the measured and calculated

electron-phonon or boundary conductance is significantly greater than the predicted

electron-electron conductance along the absorber, lending further support to a non-

ideal model for this test device. For the non-ideal model we measure electron-phonon

thermal conductance of Ge−p = 6.2 · 10−9 WK

at 310 mK, compared to an electron-

electron thermal conductance of Ge−e = 4.5 · 10−10 WK

along the absorber.

The P vs. Tbath measurements for this test device are consistent with any of these

three models, with preference going to a non-ideal based on the predicted electron-

electron thermal conductance from the absorber resistance measurement. In the fol-

lowing sections further measurements of these devices will provide more evidence for

this non-ideal model. These P vs. Tbath measurements also provide κ and Σ values

which will be used in the following sections to make predictions which can be compared

to measurements of responsivity and the noise performance for these devices.

172

5.3.3 Responsivity Measurements

The THM responsivity, S ( AW), was measured by passing a small AC current

through the absorber to provide direct Joule heating to the detector. The bolometer

response became non-linear for Joule power to the absorber, P > 0.35 nW. A lock-in

amplifier was used to read out the AC SQUID response, while the TES was DC voltage-

biased and the cold stage temperature was regulated. Table 5.3 lists the measured

responsivities of the THM test device at three TES voltage biases, (VBias = 0.2, 0.5, 1

µV), and two different cold stage temperatures, (Tbath = 309, 311 mK). These bias

points were chosen based on the I-V curves in Figure 5.11.

The measured responsivities are compared in Table 5.3 to the predicted respon-

sivities based on the ideal and non-ideal models using the fit values determined by the

P vs. Tbath measurements. The measured responsivities agree within a factor 1-2 to

the fit non-ideal thermal model predictions and to the fit electron-phonon ideal model.

However, the fit boundary ideal model overpredicts the responsivity by a factor of 1-4.

The extent to which the non-ideal model diverts power from the TES and reduces the

detector responsivity can be seen by comparing the predictions of the ideal models to

those for the non-ideal case.

5.3.4 Noise measurements

The measured NEP of this THM device at three TES bias points is shown in

Figures 5.13, 5.14 and 5.15. These noise measurements were done by measuring the

173

Table 5.3: Summary of responsivity measurements for the THM2003 test

device at three different voltage biases and two different temperatures.

Predictions for the expected responsivity are also shown using equations

for ideal and non-ideal models based on fit κ and Σ values from the P

vs. Tbath measurements (Table 5.2). The bias points corresponding to the

measured I-V curves in Figure 5.11 were also used for these predictions.Tbath (mK) V0 (µV) Measured S ( A

W) Non-ideal S ( A

W) Ideal e-p S ( A

W) Ideal boundary S

( AW)

309 0.2 2.7 · 104 8.1 · 103 1.1 · 104 3.1 · 104309 0.5 3.0 · 104 2.6 · 104 2.8 · 104 8.3 · 104309 1 3.3 · 104 5.2 · 104 5.6 · 104 1.7 · 105311 0.2 2.2 · 104 7.3 · 103 1.1 · 104 3.1 · 104311 0.5 2.6 · 104 2.0 · 104 2.8 · 104 7.98 · 104311 1 3.1 · 104 3.6 · 104 5.6 · 104 1.5 · 105

current fluctuation noise at the SQUID output (NEI), with zero power dissipation in

the absorber, and using the measured responsivity (Table 5.3) to convert to the equiv-

alent power fluctuation noise (NEP) of the detector. For these sensitive measurements,

the dewar (with only the SQUID feedback electronics powered by a battery source) was

placed inside a electromagnetically shielded room with the bias lines passing through

shielded connectors to the measurement electronics (a spectrum analyser and com-

puter readout). For the three bias points we measure detector NEP of 2 − 3 · 10−15

W√Hz

at 311 mK. For comparison, the predicted NEPs for this test device for both

thermal models (a non-ideal model or assuming a boundary or electron-phonon ideal

model) are also plotted.

For these predictions we use Equation 2.27 for the thermal NEP contributions.

The predicted NEPs also include contributions from SQUID noise. This SQUID

current noise was measured independently during the calibration setup, and was

174

101

102

103

104

1E-15

1E-14

NE

P (

W/H

z1

/2)

Frequency (Hz)

Non-ideal !t

Ideal e-p !t

Ideal boundary !t

Figure 5.13: Noise measurements of the THM2003 test devices at 1 µV

bias, 311 mK bath temperature. The best fit is given by the non-ideal

model.

175

101

102

103

104

1E-15

1E-14

1E-13

NE

P (

W/H

z1

/2)

Frequency (Hz)

Non-ideal !t

Ideal e-p !t

Ideal p-p !t

Figure 5.14: Noise measurements of the THM2003 test devices at 0.5 µV


model.

176

101

102

103

104

1E-15

1E-14

1E-13

NE

P (

W/H

z1

/2)

Frequency (Hz)

Non-ideal !t

Ideal e-p !t

Ideal p-p !t

Figure 5.15: Noise measurements of the THM2003 test devices at 0.2 µV


model.

177

2.3 · 10−11 A√Hz

at 4 K, with a predicted value at 300 mK of 5.57 · 10−12 A√Hz

, as

the SQUID noise for this device scales as NEI ∼ 1√T. The Johnson noise due to the

resistive TES and the shunt resistor in the TES bias circuit also contribute to the

detector NEP predictions.

In Figure 5.16 these individual contributions to the predicted noise are shown

for one of the bias points. In the case of the non-ideal model there is additional power

fluctuation noise due to thermal power fluctuations between the thermally separated

absorber ends and central absorber region and the TES, as well as a slightly increased

NEP from current noise sources due to the lower responsivity in this model. Thermal

noise dominates the detector noise for this device in both thermal models.

The peaks in the measured noise spectrum at low frequencies are believed to be

due to external electrical noise sources which can be minimized with further shielding

of the dewar. A high frequency roll off in the current noise (occurring at frequencies

> 30−70 kHz, and which is not shown in these plots) was observed due to the limited

bandwidth of the SQUID feedback loop. This roll off frequency shifted when the gain

settings of the feedback electronics were adjusted. This roll off in the noise spectrum

however is not related to the thermal time constants of the bolometer itself or the

TES electrical bias circuit. Based on the limits of the feedback electronics used in

our measurements this sets an upper limit for the measured thermal time constant of

the device with electrothermal feedback of τ < 10 µs. We expect, based on τ = CG, a

thermal time constant of τ < 0.1 µs at 300 mK for this test device.

The non-ideal model provides the best match to these noise measurements. The

ideal models do not adequately account for the noise. This result provides the final

178

101

102

103

104

1E-17

1E-16

1E-15

1E-14

NE

P (

W/H

z1

/2)

Frequency (Hz)

Non-ideal !t

Non-ideal !t Thermal

Non-ideal !t Johnson

Non-ideal !t SQUID

Ideal e-p !t

Ideal e-p !t Thermal

Ideal e-p !t Johnson

Ideal e-p !t SQUID

Figure 5.16: Predictions of detector NEP for the ideal electron-phonon

model and the non-ideal model at 1 µV bias, showing contributions

from thermal, Johnson, and SQUID noise sources. The non-ideal model

matches the measured noise (Figure 5.13) best.

179

evidence in support of a non-ideal model for this test device. As a side note, unlike

what is often observed in membrane-isolated bolometers and microcalorimeters we do

not observe any excess noise in this hot-electron device, once the non-ideal thermal na-

ture of the detector is accounted for. This may provide an indication that excess noise

may be less of a obstacle in substrate-type detectors and another possible advantage

of the THM design.

In the following sections we examine THM detector test devices with more

promising noise performances. These test devices are smaller devices, with a re-

designed absorber/TES coupling scheme for improved electron-electron vs. electron-

phonon optimization in order to fit to an ideal thermal THM model.

5.4 Inquiry into the Hot-Electron Effect

5.4.1 THM2003 Test Device

As was detailed in the preceding section thermal conductance measurements of

the THM2003 device alone were inconclusive in determining the hot-electron effect and

the power law dependence of power flow in the detector. However, the combination

of the DC measurements of this device (noise, responsivity and thermal conductance)

conform to a non-ideal model in which a hot-electron effect occurs with n = 5 be-

haviour. The best fits for the electron-phonon coupling coefficients in this model are

Σ = 1.0 · 109 WK5 for Au and Σ = 3.0 · 108 W

K5 for Bi, only slightly less than what

has been measured previously in the literature. No indications of a boundary thermal

resistance were seen, except for the slightly better fit of n = 4 to the P vs. T curve,

but some contribution to the thermal conductance from boundary resistance cannot

180

be ruled out by these measurements of this larger, non-optimal, THM test device.

5.4.2 us25 Test Devices

In this section thermal conductance measurements which provide more insight

into the hot-electron effect in a more optimally-sized THM detector are presented.

These results have previously been reported in Barrentine et al. 2011 [7]. These

are measurements of two micron-sized THM devices of variation us25 consisting of a

Ti/Au TES with niobium superconducting leads, and with an overlapping normal Bi

absorber. As will be discussed in section 5.5 these devices were fabricated to observe

the lateral proximity effect in a normal metal film. The Ti layer is for adhesion to

the Si substrate only and is non-superconducting. In Figure 5.17 the layers for test

devices of variation us23 and us25 are shown, as well as an optical image of one of the

larger us23/us25 test devices.

In this section measurements of two of the smallest THM test devices on wafer

us25 are presented. Device dimensions and resistances for these two test devices are

listed in Table 5.4. Here the TES active thermal volume (shown in Figure 5.18) is

considered to include only that volume between the Nb lead contacts and not the

volume which overlaps the contacts (due to the strong bilayer proximity effect with

the Nb lead). The absorber active thermal volume does include, however, the Bi

overlap onto the Nb microstrip transmission line.

Unfortunately, no superconducting transitions were observed in these us25 de-

vices. There was, however, a small linear drop in resistance from 4 K to 150 mK

181

(a)

(b)

3um Bi, 1.2 um, evaporated deposition, liftoff

Nb, 150nm, sputtering deposition, sloped-sidewall RIEnon-superconducting Ti, 5nm, sputtering deposition

Au (us25: 30nm, sputtering deposition, liftoff, presputter before Nb deposition) &

(us23: 60nm, sputtering deposition, wet-etch, presputter before Nb deposition)

Bi Absorber

Nb LeadsTES


layers for each of the us23/us25 test devices. Thicknesses are not drawn

to scale, but are listed on the labels for each layer. (b) SEM image of a

test device of variation us23/us25.

182

Table 5.4: Device dimensions and resistance measurements for the two

micron-sized test devices of variation us25.Au TES Volume (µm3) Bi Absorber Volume (µm3) TES Resistance at 190 mK R vs. T slope

3.2x6.0x.03= 0.58 3.0x12.0x0.8= 29 1.26 Ω .0127 ΩK

3.0x3.0x.03= 0.27 3.0x12.0x0.8= 29 2.52 Ω .0209 ΩK

(see Table 5.4). This shallow resistance versus temperature curve was sufficient to

provide thermometry for the measurement of the thermal conductance of these THM

devices. It also provided an advantage in that thermal conductance measurements

were possible over a wider temperature range than would normally be allowed for a

TES detector with a very sharp RTES vs. T curve. This wider temperature range al-

lowed for a more discriminatory measurement of the temperature dependence of power

flow in the detector and the corresponding n value.

These two test devices were cooled to 150 mK. 4-wire resistance versus tempera-

ture measurements were taken for each of the devices by passing a small AC excitation

current (1.22 µArms) through the TES and measuring the voltage with a lock-in am-

plifier while sweeping the bath temperature. To measure the thermal conductance,

the 4-wire resistance of each TES was measured at a fixed bath temperature while DC

power was applied directly to the absorber via separate bias leads to the absorber.

The TES temperature rise due to Joule heating in the absorber was inferred from the

zero power TES resistance vs. temperature curves.

In Figures 5.19 and 5.20 the applied power, P , is plotted as a function of hot-

electron temperature, Tbolo, up to temperatures of 6 K. Measurements below 300 mK

were not possible due to the shallow slope of the resistance vs. temperature curve.

The data was fit to the predicted ideal power law function for n values between 4 and

183

Figure 5.18: The layer overlap and active thermal area of the TES and

absorber for the two micron-sized test devices of variation us25.

184

Table 5.5: us25 thermal conductance measurements. Fits to κe−p values

for n = 5 and n = 6 are shown and compared to κe−p ‘literature values’.

Thermal conductances are predicted from these fits projecting down to

Tbath = 50 mK, assuming either ‘CMB-space’ or ‘FIR-spectral’ observing

conditions.TES volume 0.58 (µm3) 0.27 (µm3)

κe−p fit n = 5 ( WK5 ) 2.1 · 10−9 1.0 · 10−9

κe−p fit n = 6 ( WK6 ) 4.6 · 10−10 1.9 · 10−10

κe−p Literature ( WK5 ) 9.3 · 10−9 8.1 · 10−9

‘CMB-space’ Ge−p (WK) n = 5 1.1 · 10−11 9.7 · 10−12

‘CMB-space’ Ge−p (WK) n = 6 7.9 · 10−12 6.8 · 10−12

‘FIR-spectral’ Ge−p (WK) n = 5 6.6 · 10−14 3.1 · 10−14

‘FIR-spectral’ Ge−p (WK) n = 6 8.6 · 10−16 3.6 · 10−16

6. The best fit parameters and the resulting thermal conductances at 50 mK bath

temperature are listed in Table 5.5. Predicted thermal NEPs for the devices based on

these thermal conductances and also assuming a 50 mK bath temperature are listed

in Table 5.6. The values are calculated from the fit κe−p values. Here we assume that

thermal noise dominates and do not consider Johnson or SQUID noise contributions.

For the low loading far-infrared limit, NEP values are calculated using Equation 2.26.

For CMB observing the NEP values are calculated using Equation 2.27. For CMB

observing the detector temperature is determined assuming the ‘CMB-space’ loading

conditions described in Chapter 4. A TES bias power of twice the sky power is

assumed. For ‘CMB-space’ loading and n = 5, Tbolo ∼ 200 mK, for ‘CMB-space’

loading and n = 6, Tbolo ∼ 300 mK at a 50 mK bath temperature for the measured

thermal conductances.

185

Figure 5.19: Joule power applied to the absorber versus hot-electron tem-

perature, Tbolo, while the bath was held constant at 190 mK for the THM

test device of variation us25 with Au TES volume of 0.27 µm3. Fits to

the predicted forms of the power law dependence are shown.

186

Figure 5.20: Joule power applied to the absorber versus hot-electron tem-

perature, Tbolo, while the bath was held constant at 190 mK for the THM

test device of variation us25 with Au TES volume of 0.58 µm3. Fits to

the predicted forms of the power law dependence are shown.

187

Table 5.6: Predicted thermal NEP from thermal conductance measure-

ments of the two, micron-sized us25 test devices projecting down to

Tbath = 50 mK. These devices would make good CMB (or FIR) detec-

tors, with NEPdetector ≤ NEPphoton.

TES volume 0.58 (µm3) 0.27 (µm3)

‘CMB-space’ NEPthermal (W√Hz

) n = 5 3.2 · 10−18 3.4 · 10−18

‘CMB-space’ NEPthermal (W√Hz

) n = 6 4.6 · 10−18 4.9 · 10−18

‘FIR-spectral’ NEPthermal (W√Hz

) n = 5 9.5 · 10−20 6.6 · 10−20

‘FIR-spectral’ NEPthermal (W√Hz

) n = 6 1.1 · 10−20 7.0 · 10−21

‘CMB-space’ Photon NEP ( W√Hz

) 4 · 10−18 4 · 10−18

‘FIR-spectral’ Photon NEP ( W√Hz

) 1 · 10−20 1 · 10−20

In Table 5.5 κe−p ‘literature values’ are also calculated for these device dimen-

sions. Literature values for Au and Bi are characterized by an n = 5 dependence and

units for κe−p are thus in WK5 . As the detector is a hybrid TES/absorber structure,

Σ and n could not be determined for the Au and Bi structures independently. Based

on literature values we expect the Bi absorber structure to contribute more than 70%

of the total electron-phonon thermal conductance of these test devices. Interestingly,

however, the doubling in the measured best fit κe−p values, which corresponds to a

doubling of the gold TES volume in these two test devices, seems to indicate that the

gold structure may actually be dominating the thermal conductance.

The best fits for Tbolo =6 K-300 mK are consistent with n = 5 − 6, with a

better fit given by n = 6. A n = 4 fit, which might be expected from phonon-phonon

188

scattering at the boundary between the detector and substrate is not consistent with

our measurements. The best fit κe−p values however are a factor of 5-8 times lower

than what is predicted by literature κe−p values. The thermal conductance we measure

for these devices projecting down to a 50 mK bath temperature for CMB loading

conditions is Ge−p = 7− 8 · 10−12 W/K for n = 6. For the low power loading limit for

FIR observing conditions (Tbolo ∼ Tbath) this corresponds to Ge−p = 4 − 9 · 10−16 WK

for n = 6.

5.4.3 Conclusions of Inquiry into Electron-Phonon Effect

In conclusion, for both THM2003 and us25 test devices a power dissipation is

measured which is consistent with a hot-electron effect dominating the power flow in

the detector from the absorber and TES into the substrate cold bath. The measure-

ment of n ≥ 5 for the us25 test devices in particular confirmed this hot-electron effect.

The measured κe−p values are slightly or significantly lower than those measured for

Au and Bi in the literature, with a temperature dependence of the power flow following

a power law of n = 5 − 6. The observed n = 6 dependence in the us25 devices hints

that the previously unobserved disordered n = 6 behaviour in Au might be being ob-

served, but due to the composite nature of the detector in the ideal model, this n = 6

behaviour could also be due to the influence of the Bi layer. It is also interesting that

no indications of any electron-phonon 2-D effects which would suppress the n value

were seen in the thinner (30nm) Au TES devices. In the THM2003 non-ideal device

and in the us25 micron-size devices no conclusive evidence of a finite boundary resis-

tance, which would be characterized by n = 4 behaviour, is found. Electron-phonon

scattering appears to dominate the thermal resistance from the bolometer to the cold

189

bath for the THM detector.

5.5 Inquiry into the Lateral Proximity Effect

5.5.1 S-S’-S Junction Measurements

Measurements of the superconducting transition behaviour of THM test devices

on the micron-size scales are now presented. These measurements provided surprising

evidence for a dramatic lateral proximity effect on these size scales in the THM TES

devices. These results have previously been reported in Barrentine et al. 2009 [6].

These measurements corroborate measurements of similar TES devices of similar size

[97].

These antenna-coupled THM test devices from wafers THM4 & THM5 consist

of a 1.2 µm thick, 3 µm x 6 µm, 20 Ω evaporated Bi absorber which terminates

a 3 µm-wide Nb microstrip line as is shown in Figure 5.21. The absorber overlaps

a Mo/Au TES with superconducting Nb leads. The Mo pad extends out from the

Mo/Au TES bilayer, providing one of two avenues for contact to the Nb leads, as

depicted in Figure 5.21. The Nb leads and Nb microstrip were deposited over the

TES and then RI-etched to obtain a sloped Nb sidewall for good step coverage by the

absorber. The Bi absorber was deposited using the liftoff procedure and overlaps both

the Nb microstrip line and TES.

These test devices were fabricated with two different TES bilayer thicknesses, 55

nm Mo/350 nm Au (THM4) and 65 nm Mo/350 nm Au (THM5). The smallest TES

area is 3 µm x 3 µm square. TES devices were also fabricated with areas of 6 µm x

6 µm, 12 µm x 12 µm and 24 µm x 24 µm. As is visible in Figure 5.22, the actual

190

(a)

(b)

Au, 350nm, sputtering bilayer deposition, ionmill, reverse bias etch before Nb deposition

superconducting Mo (THM4: 55nm, sputtering bilayer deposition, ion mill)

(THM5: 65nm , sputtering bilayer deposition, ion mill)

3um Bi, 800nm, evaporated deposition, liftoff

Nb, 350nm, sputtering deposition, sloped-sidewall RIE


layers for each of the THM4/THM5 test devices. Thicknesses are not

drawn to scale, but are listed on the labels for each layer. (b) Optical

image of a test device of variation THM4/THM5.

191

lead-to-lead dimensions for the TES devices varied from device to device on the same

wafer due to a fabrication issue in which the Nb leads were over-etched at the point

of contact with the Au TES layer.

The test devices were cooled to 150 mK in the unshielded mount setup. 4-wire

resistance vs. temperature measurements were taken using a lock-in amplifier to send

a low amplitude (1.2 µArms) excitation current to the TES and to read out the TES

voltage via an isolation-amplifer. The TES transition temperature was discovered

to vary dramatically from device to device, depending on the distance between the

Nb lead contacts. In Figure 5.23 the measured TES Tc as a function of lead-to-lead

distance, L, for the two different bilayers is plotted. The large uncertainties in lead-

to-lead distance are due to the variation in the Nb over-etching at the leads for each

device. A dramatic shift towards higher Tc is seen with lead-to-lead distances L ≤ 12

µm. In fact, with some of the nominal 3 µm x 3 µm devices the transition temperature

is found to have shifted to the base of the Nb transition.

A curve fit is done for these measurements based on a relationship found for

resistance vs. temperature measurements of similar Mo/Au TES devices with Nb

leads [97] that follows the form:

Tc = TcN(1 + (LNL

)m). (5.3)

Here L is the lead-to-lead distance and LN is the characteristic distance scale at which

a shift in the Tc occurs. Our measurements follow this trend but with large scatter at

short distances, possibly due to the variation in actual lead-to-lead distance and/or

variation in the quality of the Nb contact with the TES. A best fit, however, is provided

by LN = 11-13 µm and m = 4− 6.

192

Figure 5.22: (a) Schematic of contact between the Mo/Au TES and the

Nb leads for devices THM4/THM5. (b) SEM image of contact between

the Nb leads and TES for a 3 µm x 3 µm THM4/THM5 device with no

absorber or microwave circuit. The Mo layer which extends from the TES

is visible as a slight change in Nb thickness. The over-etching of the Nb

leads during the Nb etch on top of the Mo/Au TES is also clearly visible.

193

Figure 5.23: (a) Tc vs. lead-to-lead distance, L, for the 65 nm Mo/350

nm Au bilayer devices of variation THM5. A curve fit is shown for the

parameters TcN = 240 mK, LN = 12 µm and m = 5. (b) Tc vs. lead-to-

lead distance, L, for the 55 nm Mo/350 nm Au bilayer devices of variation

THM4. A curve fit is shown for the parameters TcN = 170 mK, LN = 12

µm and m = 5. Data points indicate nominal L values while the error

bars indicate uncertainties due to over-etching. Error bars for Tc indicate

uncertainty distinguishing the TES from the Nb lead transition.

194

Table 5.7: Transition measurements for individual test devices from wafer

THM4/THM5.Lead to lead length, L (µm) Mo/Au Thickness Tc (K) ∆Tc (K) Normal Resistance (Ω) Absorber?

1977 65/350 0.243 0.007 13.82 no

24-30 65/350 0.263 0.010 0.033 yes

12-18 65/350 0.264 0.012 0.067 no

12-18 65/350 0.253 0.008 0.038 yes

6-12 65/350 ≥ 5.4 ≤ 3 - yes

3-9 65/350 ≥ 2.0 ≤ 6 - yes

3-9 65/350 7.3-8.5 ≤ 1.2 - yes

24-30 55/350 ≤ .240 ≤ 0.2 - yes

12-18 55/350 0.193 0.012 0.078 no

6-12 55/350 3.58 0.760 0.214 yes

3-9 55/350 7.6-8.1 ≤ 0.5 - yes

3-9 55/350 7.0-8.2 ≤ 1.2 - yes

The width of the transition was also found to increase with decreasing L. In

some of the smallest devices the transition is shifted so close to the Nb transition that

the true transition and transition width for the TES becomes difficult to determine.

The R vs. T curves at 1.2 µArms excitation current are shown in Figure 5.24 for a

nominal 12 µm long TES device (where the transition is hardly shifted at all from

the bulk transition temperature) and for a smaller nominally 3 µm long TES device

where the lateral proximity effect has shifted the TES transition up to the base of the

Nb lead transition at 8.1 K.

The transition temperature, normal resistance and the transition width for all of

the individual devices are listed in Table 5.7. The transition width, ∆Tc, was measured

as the width between 10% and 90% of the normal resistance. Tc was measured as the

midpoint between 10% and 90% normal resistance. For smaller devices the transition

temperature and breadth could not be determined in this manner and limits are given

instead. Most devices have an overlapping 3 µm wide Bi absorber.

195

Nb transition

Figure 5.24: Resistance vs. temperature curves for two of the THM test

devices. (a) Nominal 12 µm x 12 µm, 55 nm/350 nm thick, Mo/Au TES

with absorber of variation THM4. (b) Nominal 3 µm x 3 µm, 55 nm/350

nm thick Mo/Au TES with absorber of variation THM4 with a transition

near the Nb lead transition.

196

5.5.2 S-S’-S Conclusions

Based on the transition measurements of these micron-sized Mo/Au bilayer

TESs, the lateral proximity effect limits similar TES devices to lengths ≥12 µm in

order to retain a transition temperature within ∼200 mK of the Tc for long devices.

In addition to the shifting of the Tc and the increase in thermal noise due to this

higher bolometer temperature, the proximity effect will also increase the SQUID am-

plifier and Johnson noise contributions by broadening the transition, lowering α, and

decreasing the responsivity of the device.

There are still many unknowns about this bilayer S-S’-S system, such as the

role of contact resistance at the lead to TES interface and the nature of how the

bilayer aspect of the TES complicates the behaviour of the S-S’-S system. In Chapter

7 several possible steps are discussed which could be taken to minimize the lateral

proximity effect on micron-sized TES devices. The possibility of fabricating slightly

longer devices in order to work around this effect to create an optimal THM design is

also discussed. In addition, the possibility of using a single layer of Au for the TES,

and to make use of the Nb lateral proximity effect to obtain a reasonable transition

temperature as an S-N-S type junction has been suggested. The nascent exploration

of this idea, to create a novel type of TES device, is presented in the following section.

5.5.3 S-N-S Junction Measurements

We have explored the possibility of using a micron-sized Nb-Au-Nb, S-N-S junc-

tion as a TES. The advantages in this concept are a simplification of the system,

freedom from the thickness constraints due to bilayer thickness ratios, and a lowering

of the unmodified transition temperature, TcN , since the normal metal Au junction

197

has an unmodified transition temperature of TcN = 0 K. To explore this possibility

test devices were fabricated following procedures similar to those for Mo/Au bilayer

TES devices THM4 & THM5 where large shifts in Tc were observed as was presented

in the previous section. Figures 5.25 & 5.17 summarize the layer order and fabrication

method for the S-N-S devices that are presented here, and comparisons can be made

between these devices and the layer order and fabrication methods for the previous

bilayer devices THM4/THM5.

Four S-N-S variations were tested, which are designated: THMA4, THMA24,

us23 & us25. For devices designated us23 and us25, the Nb layer was deposited first

by sputtering, after which the vacuum was broken and the Nb layer patterned. Au was

then deposited via sputtering, preceded by a pre-sputter cleaning of the Nb surface.

The Au was patterned by liftoff in the case of us25, and wet-etch in the case of us23.

For the devices designated THMA4 and THMA24, the Au was deposited first via e-

beam deposition, then patterned via ion-milling, followed by a separate Nb deposition.

In the case of wafer THMA24, after the Au patterning an intensive cleaning of the

Au surface was performed, including a standard solvent clean, UV exposure clean and

in-situ reverse-bias etch immediately preceding the Nb deposition. This reverse bias

etch cleaning step etched ∼10 nm into the Au surface.

The Au surface of a test wafer which underwent the same processing as THMA24

was inspected with AFM and SEM imaging before and after the Nb deposition and

no organic residue was seen. Some of the images from this test wafer are shown in

Figure 5.26. For all of these S-N-S devices, an insulating, non-superconducting thin

layer of Mo (for THMA2 & THMA24) or Ti (for us23 & us25) was deposited beneath

198

(a)

(b)

Au (THMA4: 30nm, e-beam deposition & ionmill) &

(THMA24: 180nm, e-beam deposition & ion mill, reverse bias etch before Nb dep)

Non-superconducting Mo, 5nm, e-beam deposition, RIENb (THMA4: 150nm, sputtering deposition, sloped-sidewall RIE) &

(THMA24: 500nm, sputtering deposition, straight-sidewall RIE)

3um


layers for each of the THMA4/THMA24 test devices. Thicknesses are not

drawn to scale, but are listed on the labels for each layer. (b) SEM image

of a test device of variation THMA4/THMA24.

199

the Au for adhesion purposes to the Si substrate.

These S-N-S junctions, with lead-to-lead lengths ranging from 3-23 µm (and

widths ranging from 3-12 µm) were cooled in the unshielded mount setup. The tem-

perature was swept while monitoring the 4-wire resistance of each device with a lock-in

amplifier and a small AC excitation current. No transitions below the Nb transition

at 8.4 K were seen down to temperatures of 140 mK (and down to 50 mK for one

device of variation THMA24), using currents as low as 13-130 nArms. A magnetic coil

which provided a field up to 5-10 G in the vicinity of the TES device was also mounted

under the chip in order to test the device response to magnetic field. This was done

since a diffraction pattern due to a Josephson junction-like interference effect has been

observed in the critical current in response to magnetic flux in micron-sized bilayer

TES devices [97]. No effect was seen in the resistance measurements of THM test

devices when this field was ramped up and down in both directions perpendicular to

the TES plane.

In Table 5.8 a summary of these S-N-S measurements, indicating the limits

placed on the critical current for each of these variations, is shown. Limits on a

possible contact resistance, Rc, between the Nb and Au, which might impede the

lateral proximity effect are also listed. We measured non-zero Rc on a few devices on

us23, us25, and THMA4, however measurements of Rc are consistent with zero within

measurement error for devices of variation THMA24, which underwent more extensive

cleaning of the interface. These contact resistance measurements are shown in Figure

5.27.

The measured limit on contact resistance for THMA24 also falls near the min-

200

SEM image (left) and AFM image (right) of TES Au pad after ion-miling step and solvent, descum clean.

AFM image of TES Au after ion-milling, solvent, descum clean and reverse bias etch step..

SEM images of the TES Au pad (left) and the Au/Nb lead contacts (right) after Nb deposition.

5µm

µm

µm

2 µm 2 µm

Figure 5.26: SEM and AFM images after ion-milling, cleaning of the Au

surface, and Nb deposition. A rough surface of the Au metal is seen, but

no organic contamination which would be indicated by dark or stringy

residue can be seen in these images.

201

Figure 5.27: Measured contact resistance, Rc (with error), for all devices

tested. These measurements are normalized to assume a 3 µm2 contact

area for each device. Error bars indicate error due to the measurement

noise.

202

Ω

Figure 5.28: Chart summary of Table 5.8. The Kuprianov and Lukichev

Rc limits are also shown in the plot.

imum contact resistance limit given by Kuprianov and Lukichev [72] (discussed in

Chapter 3) for seeing noticeable shifts in Ic. In Table 5.8 the range for critical cur-

rents predicted by the S-N-S models discussed in Chapter 3 are also listed. For all

variations we rule out all or most of the critical current parameter space predicted by

these S-N-S models. The measurements and predictions in Table 5.8 are also summa-

rized in chart format in Figure 5.28.

203

Table 5.8: Summary of 4-wire resistance measurements of S-N-S test de-

vices. Ic limits with predictions for four variations of Nb-Au-Nb devices

are listed. Predictions were calculated using the Au thickness, d, measured

Au resistivity, ρ, measured Nb Tc = 8.4 K and the four S-N-S models de-

scribed in Chapter 3. Predictions for us23, us25, THMA2 & THMA24-1

were calculated for a 3µm × 3µm device at 140 mK, and measurements

were taken from 8 K-140 mK. The prediction for device THMA24-2 was

calculated for a 3µm × 12µm device at 50 mK and measurements were

taken from 4 K-50 mK. The resistivity, ρ, was measured using a larger

resistance, 960µm × 10µm Au test device on the same chip. Limits on

Rc were calculated by comparing the measured resistivity of each device

(with error due to measurement noise and dimension uncertainty) to ρ.

The contact resistance listed is for an assumed 3 µm2 contact area between

the Nb and Au. For the low contact resistances and low currents used in

the measurements we expected to see some evidence for superconducting

behaviour in these devices.Device L (µm) d (nm) ρ (Ω ·m) Ic predicted Ic Rc

us23 3− 23 60 2.4 · 10−8 190µA− 150nA ≤ 18nA ≤ 0.4Ω

us25 3− 23 30 4.0 · 10−8 3µA− 2nA ≤ 18nA ≤ 1Ω

THMA4 3− 15 30 3.1 · 10−8 8µA− 6nA ≤ 18nA ≤ 0.4Ω

THMA24-1 3− 12 180 3.2 · 10−8 490µA− 380nA ≤ 180nA ≤ 0.08Ω

THMA24-2 3.6 180 3.2 · 10−8 144µA− 5µA ≤ 425nA ≤ 0.05Ω

204

5.5.4 S-N-S Conclusions

Following typical TES fabrication procedures, in which the superconducting Nb

lead deposition and Au deposition are done separately, with robust cleaning of the

Au or Nb surface before deposition, we do not see any transitions in micron-sized

Nb-Au-Nb junctions at sub-Kelvin temperatures indicating that they could be used

as sensitive TESs in micro-bolometers. This result is in disagreement with theoretical

predictions for S-N-S type junctions, including those which allow for a small contact

resistance between the normal and superconducting layers.

It is possible that in contrast with the theoretical S-N-S models considered here,

and in contrast to what is observed in similar S-S’-S bilayer devices, even a small

contact resistance has a large weakening effect on S-N-S critical current behaviour. It

is also possible that magnetic impurities in the Au film which are also not considered in

these models are having a strong suppressive effect on the superconducting behaviour

of the films.

Previous results (Section 5.5.1) would appear to argue that the lateral proximity

effect in Nb-Mo/Au-Nb S-S’-S junctions is less sensitive to contact resistance between

the Nb and Au layer or impurity contaminations than for the Nb/Au/Nb S-N-S devices

described here. However, these null results for S-N-S junction test devices do indicate

that one possible method to weaken the lateral proximity effect and Tc shift in S-

S’-S devices would be to purposely create a dirty interface between the TES and

superconducting lead layers or to introduce sources of magnetic impurities into the

THM design. These options with regards to the lateral proximity effect in both S-S’-S

and S-N-S devices will be discussed in more length in Chapter 7.

205

5.6 Inquiry into Detector NEP

Under ‘CMB-space’ observing conditions the background photon noise level is

4 · 10−18 W√Hz

and for ‘FIR-spectral’ applications the background photon noise level

is 1 · 10−20 W√Hz

(see Section 4.2.1). The measured noise of the larger THM2003 test

device presented Section 5.3, which conformed to a non-ideal model, is thus still well

above CMB photon-noise levels at the device transition temperature near 300 mK.

Even projecting this device design down to temperatures of 50 mK and calculating

the NEP for this non-ideal model the noise is still slightly above background levels

(see Table 5.9).

In Table 5.6, and listed again in Table 5.9, the expected thermal fluctuation

noise is also calculated for each of the measured thermal conductances for the us25

test devices projected down to 50 mK. For these micron-sized us25 devices NEPs

near or below the background noise limit for CMB observing in space, and below the

background noise limit in the far-infrared are predicted. Unfortunately, at this device

size scale, the lateral proximity effect makes the low transition temperature necessary

to reach these low NEPs difficult to achieve. At this moment this is true for either a

3 µm long S-S’-S or a 3 µm long S-N-S device.

In the concluding Chapter 7 a new recommended mid-size design for the THM

will be presented. This design is necessary to make a functioning THM detector

which is not affected by a strong lateral proximity effect, and with the desired photon

noise limited noise performance for CMB observing, when it is operated at low (50

mK) bath temperatures. This design will not be optimal, however, for far-infrared

applications, where a smaller volume is still ultimately required to reach photon noise

206

Table 5.9: NEPthermal & Gep for the THM test device designs THM2003

and us25 predicted at 50 mK bath temperature assuming ‘CMB-space’

conditions. For THM2003 predictions the measured fit Σ values for this

device, n = 5 and a non-ideal model are assumed. For us25 predictions

the measured fit values κ value for the smallest device (with 3 µm long

TES), n = 6 and an ideal model are assumed. Predictions for the new

recommended design assume an ideal model and are based on these mea-

sured fit Σ and κ results where the exact Σ values assumed for ‘measured

n = 5’ and ‘measured n = 6’ are given in Section 4.2.1. For the THM

test devices thermal NEP is slightly above photon background noise. For

the new recommended design thermal noise below the photon background

noise is predicted.Device Label THM2003 us25 recommended THM-

‘measured n = 5’

recommended THM-

‘measured n = 6’

Au TES Volume 190nmx10µmx20µm 30nmx3µmx3µm 300nmx20µmx20µm 300nmx20µmx20µm

Bi Absorber Volume 500nmx20µmx100µm 800nmx3µmx12µm 800nmx3µmx12µm 800nmx3µmx12µm

κe−p (WKn ) 4.4 · 10−7 1.9·10−10 (n = 6) 1.3·10−7 (n = 5) 1.8·10−8 (n = 6)

Gep (WK) 3.8 · 10−11 6.8 · 10−12 (n = 6) 2.8 · 10−11 (n = 5) 3.0 · 10−12 (n = 6)

NEPthermal (W√Hz

) 9.0 · 10−18 5.0 ·10−18 2.3 ·10−18 3.3 ·10−18

NEPphoton ( W√Hz

) 4 · 10−18 4 · 10−18 4 · 10−18 4 · 10−18

limited levels. Further work and development following the suggestions which will be

discussed in Chapter 7 will be necessary to either make use of the lateral proximity

effect in a S-N-S type device, or avoid this lateral proximity effect in a S-S’-S type

device for a far-infrared THM detector. As a preview, the dimensions and performance

characteristics of the new recommended CMB THM design are also listed in Table 5.9.

207

Chapter 6

THM Microwave Design & Simulations

6.1 THM Microwave Design

In this chapter a microwave coupling scheme to the THM detector is presented.

This scheme makes use of planar microwave technology. The microwave designs pre-

sented here have been optimized for the 92 and 43 GHz frequency bands, however

via relatively simple scaling these designs are applicable over a wider range of CMB

frequencies (∼ 30 − 300 GHz). This planar microwave technology has a distinct ad-

vantage, and is probably necessary, for scaling to large detector arrays. In addition

to providing the THM with array-scalability, the planar microwave components of the

THM design also provide polarization sensitivity via the coupling of the detector to a

planar slot-antenna design. Other planar antenna designs are also possible (bow-tie,

spiral etc.) however the slot antenna approach is one which has been well-studied and

is most commonly used. This slot antenna is only sensitive to E-M radiation of a single

polarization direction. An array of detectors coupling to two slot antennas of orthog-

onal polarization directions would be capable of taking a polarization measurement of

the CMB.

208

In Figure 6.1 and Figure 6.2 the two microwave designs developed for the THM

detector test devices us23/25 and THM4/THM5 are shown and the main components

for each of these designs labelled. A chip-size optical image of a test device of variation

us23 is also shown in Figure 6.3. In addition to the two slot antenna chip designs de-

scribed in Figures 6.1 & 6.2, alternative variations to this slot antenna chip design were

also fabricated for test devices THM4/THM5. These alternative designs included a

two-detector arrangement for testing of the filter response shown in Figure 6.4. Addi-

tionally, 43 GHz coplaner waveguide (CPW) test chip designs were fabricated in order

to make cryogenic probe station measurements of individual microwave components.

These tests and the design of these CPW test devices are briefly described in Section

6.2.1.

The components to the THM microwave coupling design include: a slot antenna,

microstrip transmission line, a low-pass filter, various couplers, DC bias lead chokes,

and a microstrip termination structure. In the first part of this chapter, each of

the individual components to this design will be described and some of the E-M

simulations (using Sonnet, Ansoft Designer, Ansoft HFSS or CST Microwave Studio

E-M simulation software) which predict the performance of these components are also

presented.

These E-M simulation software programs all work by 2-D or 3-D modelling of the

microwave component under design. Common input parameters for these models are

the component geometry, and the microwave material properties, such as the relative

dielectric constant, the kinetic inductance and the resistivity. The simulations are

completed by dividing the model of the microwave component into small regions via

209

a meshing algorithm. An input microwave signal (usually at a particular frequency)

is modelled as an excitation via a microwave port on the design and the response of

the microwave component under design is simulated by solving Maxwell’s equations

within each of the mesh boundaries, solving for the E-field, H-field, current, voltage

and impedance properties at all locations in the microwave model geometry as a

function of the input microwave signal frequency.

A useful feature of many of these simulation software programs is the ability to

optimize the geometry of a design. For example, if a minimum transmission loss is de-

sired at a particular center band frequency an optimization routine can be run which

explores the geometry parameter space within a certain range and finds the optimal ge-

ometry which gives a maximum throughput at a certain frequency. This optimization

ability was used in the design of many of the THM microwave components.

These E-M simulation results are reported in terms of S-parameters. These S-

parameters assume that a microwave input signal is sent in via an input port in the

microwave design and the output signal at another port in the microwave design is

measured. The S-parameter, Sij , is a measure of the power transmission from one

port “j” to another port “i”. Likewise the S-parameter, Sii, is a measure of the power

reflection back into the input port “i”. S-parameters are reported in this chapter in

terms of decibels (dB). Here the relation between decibels to the magnitude of the

power ratio in and out of each of these ports is given by:

Sij = 10 logPi,outputPj,input

(6.1)

In the final part of this chapter, in Section 6.2, two off-chip optical coupling

schemes are introduced. These schemes have been developed to test the microwave

210

performance of these THM test devices in a laboratory cryostat. However, one could

imagine adapting a similar off-chip coupling scheme to couple the THM detector or

detector array to the CMB sky as part of a ground- or space-based telescope.

6.1.1 Microstrip Transmission Lines

The microwave circuit for the THM test devices consists of portions of supercon-

ducting Nb microstrip transmission line (previously discussed in Section 4.2.2). At its

narrowest, this microstrip transmission line is 3 µm wide. In the case of test devices

THM4/THM5 the transmission line consists of a 350 nm thick Nb microstrip line

over a 150 nm thick Nb ground plane separated by a 750 nm thick Al2O3 dielectric

(with expected relative dielectric constant of ǫR = 10). In the case of test devices

us23/us25 the microstrip is a 150 nm thick Nb microstrip line over a 250 nm thick

Nb ground plane separated by a 1.45 µm thick Si dielectric (with expected relative

dielectric constant of ǫR = 11.7).

The characteristic impedance of the narrowest 3 µm wide line is ∼20 Ω for

variations THM4/THM5 and ∼30 Ω for variations us23/us25. The small line width

of the 3 µm wide microstrip line makes geometrical and impedance matching possible

between the microstrip line and the optimal THM Bi absorber. This superconducting

Nb microstrip had a measured transition temperature between 8 and 8.5 K, with

measured critical currents > 10 mA. A surface kinetic inductance of Ls = 0.13 pH

was

assumed for the Nb microstrip lines during the E-M simulations of the microstripline

and other microstrip-type components based on surface impedance calculations [62].

The thin-film Nb, with a large energy gap of ǫgap ∼ 3.5kBTc at 200 mK, remains

superconducting at microwave frequencies below ∼ 600 GHz (fmax ∼ ǫgaph), allowing

211

Double Slot Antenna

Figure 6.1: Microwave design at 92 GHz to couple radiation to the THM

detector for the THM4 and THM5 test devices. Devices at 43 GHz were

also designed and fabricated. The signal from the double slot antenna

is transmitted via a low-pass filter to the RF-terminated THM detector.

Choked bias leads provide DC connections to both the TES (for bias and

SQUID read out) and the absorber (for DC Joule heating of the absorber).

212

Slot Antenna

Low-pass Filter

Coupler

THM and Microstrip Termination Structure

DC Chokes

Figure 6.2: Microwave design at 92 GHz to couple radiation to an array

of 128 individual THM detectors for the us23/us25 test devices. The

coupling to such a large number of detectors was done in an attempt to

find the optimal lateral proximity effect S-N-S TES, with each of the 128

detectors of varying TES lead-to-lead length. In actual operation, with

an optimized THM device, one imagines only a single detector coupling

to the slot antenna without the branching network.

213

Figure 6.3: Optical image of one of the us23/us25 test device detector

chips. Chip size is 15 mm x 15 mm.

214

THM Termination

Low Pass Filter

Coupled Line Coupler

Figure 6.4: Two-detector coupled alternative slot antenna design for test

devices THM4/THM5. This device was designed specifically to test the

performance of the low-pass filter. A directional coupled line coupler trans-

mits the signal to two separate detectors, one line via a low-pass filter and

another line without the filter. By coupling a swept microwave signal

into the slot antenna the filter response as a function of frequency can be

determined.

215

for near lossless transmission through the straight microstrip line portions of the THM

microwave circuit.

6.1.2 Double Slot Antenna Design

The slot antenna design for the test devices THM4 & THM5 (Figure 6.1) is a

double-slot design based on a similar design by Zmuidzinas & LeDuc [129]. In this de-

sign two slots are made in the ground plane of the superconducting microstrip circuit.

An incident E-M wave excites current in the superconductor surface surrounding these

slots. The slot antenna is sensitive to electric field modes which are perpendicular to

the direction of the slot length. A single slot gives an asymmetric beam pattern; the

double slot beam pattern is more symmetric. In this slot antenna design coupling

to the microstrip transmission lines is done by crossing the microstrip line over the

center of the ground plane slots and shorting the microstrip line to ground via a radial

stub described in Section 6.1.4. The two microstrip lines are combined into a single

microstrip line which then transmits the signal to the THM detector. The geometry

of the double slot antenna design is shown in Figure 6.5.

To find the correct design parameters, the impedance equations given in Zmuidz-

inas & LeDuc were solved, requiring an impedance matching of the antenna to the

microstrip line. Near 90 GHz, a match to the ∼20 Ω microstrip line was satisfied by a

23 Ω+0.156i Ω input impedance to the slot antenna, resulting in the following design

equations:

f0 = 100GHz · 1mmL

(6.2)

W

L= 0.04. (6.3)

216

L

S

W

narrowing radial stub

microstrip

Figure 6.5: The geometry of the THM4/THM5 double slot antenna design.

217

For a symmetric beam:

S

L= 0.5662. (6.4)

For a slightly antisymmetric beam, but lower sidelobes:

S

L= 0.5. (6.5)

Here the length of the slot antennas, L, is equal to the microwave wavelength in the

dielectric. In the Al2O3 dielectric the 92 GHz wavelength is ∼1 mm. The slot width

is W and the separation between the two slot antennas is S. To reduce the stray

input inductance for the microstrip to slot antenna coupling, the ground plane slot

is narrowed in the region where the microstrip overlaps. For this design the slot is

narrowed to a minimum width of 3 µm.

The two microstrip lines are coupled together via a Wilkinson power divider [91].

This design is discussed in Section 6.1.6. The signal from an E-M wave excitation

should excite the two antennas in phase. The Wilkinson coupler requires all out of

phase signals be absorbed in a resistive (Bi) film. An Ansoft HFSS simulation showing

the expected performance (S11) of this slot antenna design is shown in Figure 6.6. For

this design a ∼ 10 % bandwidth near 92 GHz is expected. Harmonic resonances > 150

GHz (not shown in Figure 6.6) are also predicted. These high frequency “leaks” are

eliminated by the introduction of the low-pass filter described in Section 6.1.5. Optical

images of the 92 GHz slot antenna from test device THM5 are shown in Figure 6.7.

6.1.3 Four-Fold Slot Antenna Design

The second slot antenna design, which was introduced for test devices us23 and

us25, is a four-fold slot antenna design by K. U-Yen (unpublished). This design, which

218

S11

Figure 6.6: The simulated reflection (S11) from an input signal sent into

a port which feeds the single-line microstrip to the double slot antenna

for devices of variation THM4/THM5. The low level of reflection near 92

GHz indicates power radiated by the antenna.

219

1 mm

Figure 6.7: An optical image of the 92 GHz double slot antenna on a test

device of variation THM5.

220

was optimized for a center frequency at 92 GHz for the THM test devices, consists of

four aperture slots in the ground plane. Slot line transmission lines couple to the end

of the slots and transmit the signal to a horizontal slot transmission line. This manner

of coupling transmission lines to the slot antenna provides a broad bandwidth for the

antenna. A microstrip transmission line (with a rectangular stub which provides a

short to ground) couples to the horizontal slot line in a similar manner as in the

double slot antenna design. The geometry of this four-fold antenna design is shown in

Figure 6.8. As was the case with the double slot antenna design, this antenna is also

only sensitive to incident E-M radiation which has electric field lines perpendicular to

the length of the slots.

Here the slot antenna length, L = 0.575 mm, is a half-wavelength of the 92 GHz

wavelength in the Si dielectric. The other slot antenna dimensions are optimized for

maximum absorption around the center frequency using Ansoft Designer, with optimal

dimensions for the 92 GHz design of W = 42 µm, S = 1 mm, WV = 10 µm, WH = 3

µm. Microwave simulations of this design included the effect of a 2 µm thick BCB

bonding material (ǫR = 2.6) directly under the ground plane layer on the opposite side

to the 1.45 µm thick Si dielectric. The simulated optimized four-fold 92 GHz antenna

performance is shown in Figure 6.9.

This four-fold antenna design has a broader bandwidth than the double slot

antenna design, with -10 dB bandwidth of ∼50% at 90 GHz. Harmonics (not shown

in Figure 6.9) are expected at frequencies > 150 GHz. A low-pass filter is again used

to limit transmission above these frequencies. The beam pattern for the slot antenna

(with no further optical coupling schemes, such as those which will be described in

221

wh

wv

L

W

S

Rectangular StubSlot lines in ground plane

Slot aperature in ground plane

Microstripline

Figure 6.8: The geometry of the us23/us25 four-fold slot antenna design.

222

1 0 3 0 5 0 7 0 9 0 1 1 0 1 3 0 1 5 0-4 0

-3 5

-3 0

-2 5

-2 0

-1 5

-1 0

-5

0

Frequency (GHz)

S1

1 (

dB

)

Figure 6.9: The simulated reflection (S11) at the port which feeds the mi-

crostrip to the four-fold slot antenna for the test devices of variation

us23/us25. The low level of reflection indicates power radiated by the

antenna.

223

Si wafer

Vacuum

Nb Groundplane

Vacuum

Figure 6.10: The predicted beam pattern at 92 GHz for the four-fold slot

antenna design for the test devices of variation us23/us25. As expected,

the gain into the Si substrate is greater than into vacuum.

Section 6.2.2) is also shown in Figure 6.10.

6.1.4 Radial & Rectangular Stubs

The radial stub design [113] and rectangular stub design, shown in Figures 6.11

& 6.12 respectively, provide for an effective microwave short circuit at the base of the

stub on the top microstrip layer to the ground plane. This provides coupling of the

slot antenna excitation currents in the ground plane to the microstrip transmission

line. The dimensions for the radial stub design at 92 GHz for devices THM4/THM5

are: α = 60 and R = 104 µm. The dimensions of the rectangular stub design at

92 GHz for devices us23/us25 are: L1 = 106 µm, L2 = 105 µm, W1 = 28 µm, and

224

R

α

Slot Antenna

3 µm microstrip feedline

Figure 6.11: The geometry of the radial stub design for devices of variation

THM4/THM5.

W2 = 5 µm. Sonnet simulations for the radial stub design’s input impedance when

terminating a 3 µm microstrip line for the THM4/THM5 design are shown in Figure

6.13 and show a broadband short to ground near 92 GHz.

6.1.5 Low-pass Filter

The low-pass filter design (shown in Figure 6.14) which blocks the high frequency

leaks from the slot antenna is based on a design by K. U-Yen [117]. The design

consists of three separate lumped element low-pass filters: two microstrip “stub” type

225

L1

L2

W1

W2Slot Line in ground plane

3 µm microstripline

Figure 6.12: The geometry of the rectangular stub design for devices of

variation us23/us25.

226

Figure 6.13: Input impedance for the THM4/THM5 radial stub design.

A very broad bandwidth at 92 GHz indicating a short to ground (with

impedance Z ≪ 20 Ω) is predicted.

227

Part 1 Part 2 Part 3

10 Ω

µstripline

10 Ω

µstripline

Figure 6.14: The geometry of the low-pass filter, consisting of three sub-

filters, indicated as “Part 1”, “Part 2”, and “Part 3”. “Part 1” and “Part

3” are “stub” type filters and “Part 2” is a “stepped impedance” type

filter.

filters and one microstrip “stepped impedance” type filter. These three filters combine

to block out frequencies from ∼ 150-600 GHz (at 600 GHz the microstrip becomes

non-superconducting and lossy). The design has a characteristic input and output

impedance of 10 Ω. An optical image of the low-pass filter on one of the test devices

of variation THM5 is shown in Figure 6.15. HFSS Ansoft Design simulations for the

us23/us25 design, showing the expected performance for the individual sub-filters and

the entire low-pass filter, are shown in Figures 6.16 & 6.17.

6.1.6 Couplers

The couplers in the microwave design for these test devices include either a direc-

tional coupled line coupler (in the case of the variation shown in Figure 6.4) or Wilkin-

son couplers [91]. The Wilkinson coupler geometry and design parameters are shown

in Figure 6.18. The Wilkinson coupler is used in the us23/us25 128 channel-splitting

network to isolate the detectors from each other; it rejects any out of phase signals

due to reflections in the circuit. The Wilkinson coupler is used in the THM4/THM5

microwave design to add the signals from the two slot antennas together. An optical

228

3 µm

Figure 6.15: Optical image of the low-pass filter from test device of varia-

tion THM5.

image of the Wilkinson coupler in a portion of the splitter network on a device of

variation us23 is shown in Figure 6.19. Simulations for the expected transmission be-

tween input and output ports for the Wilkinson Coupler design for the us23/us25 test

devices are shown in Figure 6.20. The directional coupler designs are not presented

here.

6.1.7 Termination Structure

The termination structure for the THM is perhaps the most important part

of the microwave circuit. It provides for the termination of microwave power in the

small THM absorber volume. The small volume of the absorber requires that the

incoming microwave radiation be absorbed over a short distance compared to the

effective wavelength of the radiation. The termination structure consists of two parallel

229

0 .0 0 1 0 0 .0 0 2 0 0 .0 0 3 0 0 .0 0 4 0 0 .0 0 5 0 0 .0 0 6 0 0 .0 0 7 0 0 .0 0F [GH z]

-8 7 .5 0

-7 7 .5 0

-6 7 .5 0

-5 7 .5 0

-4 7 .5 0

-3 7 .5 0

-2 7 .5 0

-1 7 .5 0

-7 .5 0

dB

(S(P

ort

1,P

ort

2))

Part “2”:

0 .0 0 1 0 0 .0 0 2 0 0 .0 0 3 0 0 .0 0 4 0 0 .0 0 5 0 0 .0 0 6 0 0 .0 0 7 0 0 .0 0F [GH z]

-7 5 .0 0

-6 5 .0 0

-5 5 .0 0

-4 5 .0 0

-3 5 .0 0

-2 5 .0 0

-1 5 .0 0

-5 .0 0

dB

(S(P

ort

1,P

ort

2))

Part “1”:

0 .0 0 1 0 0 .0 0 2 0 0 .0 0 3 0 0 .0 0 4 0 0 .0 0 5 0 0 .0 0 6 0 0 .0 0 7 0 0 .0 0F [GH z]

-7 0 .0 0

-6 0 .0 0

-5 0 .0 0

-4 0 .0 0

-3 0 .0 0

-2 0 .0 0

-1 0 .0 0

0 .0 0

dB

(S(P

ort

2,P

ort

1))

Part “3”:

Figure 6.16: Simulated transmission (S12) through the low-pass filter

parts. Part “2” defines the low frequency cut-off. Part “1” blocks the

higher frequency leaks from part “2”. Part “3” blocks the higher fre-

quency leaks from part “1”.

230

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0-1 2 0

-1 0 0

-8 0

-6 0

-4 0

-2 0

0

Frequency (GHz)

S12

(dB

)

Figure 6.17: The simulated transmission (S12) through the low-pass filter

for the us23/us25 design. Note that “in-band” (∼ 75− 110 GHz) the loss

is < −3 dB while the out of band rejection at frequencies > 180 GHz is

excellent (< −20 dB transmission).

231

2Z0

Z0

Z0

Z0

e2 Z0

e2 Z0

l/4

Lumped Resistor

Microstrip line

Figure 6.18: The Wilkinson coupler design [91]. The design requirements

are shown in terms of the charateristic impedances of the input and output

microstripline, Z0, and the microwave wavelength in the dielectric, λ.

232

210 µm

Figure 6.19: An optical image of the Wilkinson couplers in part of the

128-channel splitter network on a test device of variation us23.

233

5 0 .0 0 7 0 .0 0 9 0 .0 0 1 1 0 .0 0 1 3 0 .0 0 1 5 0 .0 0F [GH z]

-3 5 .0 0

-3 0 .0 0

-2 5 .0 0

-2 0 .0 0

-1 5 .0 0

-1 0 .0 0

-5 .0 0

0 .0 0

dB

Curv e Inf o

dB(S(Por t1,Por t1) )



Figure 6.20: S-parameters showing transmission from port 1 (output of

coupler) to ports 2 and 3 (inputs to coupler) for the us23/us25 Wilkinson

coupler design. A 3 dB (equal) coupling/splitting is expected near 92

GHz.

234

shorts to ground made of lumped capacitive and inductive microstrip elements as

shown in Figure 6.21.

This termination design provides absorption over a very wide band as is shown

by the Ansoft Designer simulations in Figure 6.22. Here the reflection S-parameter

(S11) is the reflected power when an input signal is sent into a port at the microstrip

line transmission side of the termination structure, before the absorber. The very low

reflection indicates absorption in the THM Bi absorber. This simulation assumes an

exact impedance match between the microstrip line and the Bi absorber. However,

even a large mismatch in absorber resistance results in minimal loss to reflection. Here

one can assume the fraction of reflected power, Γ, is given by the relation [91]:

Γ = (RAbs − Z0

RAbs + Z0)2 (6.6)

where RAbs is the absorber resistance and Z0 is the characteristic impedance of the 3

µm wide microstrip line. Even an absorber resistance difference of 10 Ohms will result

in . 10% loss of the incoming power to reflection. The termination structure design

is also modifiable over a range of THM geometries.

6.1.8 DC Chokes

DC chokes are necessary in order to attach bias leads to the THM TES. DC

chokes can also can provide DC access to the THM absorber without interrupting the

microwave circuit of the THM detector. The absorber access allows a DC bias to be

applied in order to dissipate Joule power in the absorber to measure thermal conduc-

tance and compare DC to RF response. DC chokes are inserted at the microstrip

termination structure as bias leads the TES, and at the microstrip transmission line

235

Nb microstripline

Nb

Nb

Bi absorber

TES

L1

L2

W1

W2

RF signal

Figure 6.21: The THM microstrip termination structure geometry. The

large rectangular microstrip structures provide capacitive coupling to

ground. For the us23/us25 design optimal dimensions at 92 GHz were

W1 = 41 µm, W2 = 21 µm, L1 = 72 µm, and L2 = 29 µm. For the

THM4/THM5 design the optimal dimensions at 92 GHz were W1 = 54

µm, W2 = 10 µm, L1 = 90 µm and L2 = 30 µm.

236

0 2 0 4 0 6 0 8 0 1 0 0 12 0 1 4 0 1 6 0 1 8 0 2 0 0-4 0

-3 5

-3 0

-2 5

-2 0

-1 5

-1 0

-5

Frequency (GHz)

S1

1 (

dB

)

Figure 6.22: Simulated reflection (S11) for an input signal into the mi-

crostrip termination structure with matched THM absorber (Z0 = RAbs =

31 Ω) for the us23/us25 design.

237

DC Chokes for TES bias leads

DC Chokes for absorber access

THM and RF termination

W1

L1

L2

W2

Microstrip Termination Structure

RF signal

Figure 6.23: The geometry and placement of the DC chokes on the

us23/us25 devices in order to provide TES bias and absorber access. A

close up of the RF termination on the THM absorber is shown in Figure

6.21.

before the termination at the absorber for absorber access, as is shown in Figure 6.23.

The inductive and capacitive lumped elements of this choke design create a microwave

“open” at the input to the chokes, but allow for the transport of DC currents. In

Figure 6.24 the simulated microwave transmission in a microstrip line intersected by

a DC Choke is shown for the us23/us25 choke design.

6.1.9 Impedance Transformers

At various places in the THM microwave circuits for test devices us23/25 and

THM4/THM5, microstrip impedance transformers are necessary to impedance match

between microstrip components of different characteristic impedances and whose impedance

is constrained by geometry or other design constraints. For example, this is done in

238

Port1 Port2

5 0 .0 0 7 0 .0 0 9 0 .0 0 1 1 0 .0 0 1 3 0 .0 0 1 5 0 .0 0F [GH z]

-3 5 .0 0

-3 0 .0 0

-2 5 .0 0

-2 0 .0 0

-1 5 .0 0

-1 0 .0 0

-5 .0 0

0 .0 0

dB

Curv e Inf o



Figure 6.24: Simulated microwave transmission (S12) and reflection (S11)

through a microstripline (above) which is intersected by the DC choke

design used in test devices us23/us25. In the desired RF band of the

detector the presence of the DC connection has little effect on the RF

performance.

239

400 µm

3 µm 6 µm

Figure 6.25: The geometry of one of the stepped impedance transformers

from the us23/us25 test devices which transitions from a 19 Ω (6 µm wide)

to a 30 Ω (3 µm wide) microstrip line. Lengths/widths are not drawn to

scale.

the us23/25 design after microwave coupling to the slot antenna to convert from 30

Ω (3 µm wide) microstrip to the 10 Ω (14 µm wide) microstrip input to the low-pass

filter. Another impedance transformer is used to convert from the 19 Ω (6 µm wide)

Wilkinson coupler output to the 30 Ω microstrip line which terminates on the 3 µm

wide absorber. These impedance transformers are all stepped-impedance transform-

ers, an example of the geometry of which is shown in Figure 6.25. The simulated

performance of one of these transformer designs is also shown in Figure 6.26.

6.2 THM Microwave Testing Schemes & Preliminary Mea-

surements

6.2.1 CPW Probe Station Microwave Design & Measurements

For devices THM4/THM5 additional microwave designs were fabricated at 43

GHz in order to test microwave components independently of the THM detector re-

sponse, in a 4 K cryogenic probe station. This probe station allowed for two-port

transmission measurements and one-port reflection measurements on a variety of test

lines which included variations of scaled versions of the 92 GHz microwave compo-

240

0 .0 0 2 5 .0 0 5 0 .0 0 7 5 .0 0 1 0 0 .0 0 1 2 5 .0 0 1 5 0 .0 0 1 7 5 .0 0 2 0 0 .0 0F [GH z]

-5 0 .0 0

-4 0 .0 0

-3 0 .0 0

-2 0 .0 0

-1 0 .0 0

0 .0 0

Y1

Curv e Inf o



dB

Figure 6.26: The simulated transmission (S12) and reflection (S11) through

the microstrip impedance transformer shown in Figure 6.25.

241

nents, including microstrip throughlines, terminations, filters, transformers and DC

bias chokes. The layouts of two of these 43 GHz probe station test chips are shown in

Figures 6.27 and 6.28. The co-planar waveguide (CPW) pads at the edge of this design

transition to 3 µm wide Nb microstrip lines using a CPW-to-microstrip transformer.

The Sonnet simulated performance for this transformer is shown in Figure 6.29.

The CPW probe station consists of a cryostat chamber, with two probe arms.

These probe arms are cooled to temperatures between 5− 7 K. The cold stage where

the chip is mounted is cooled to temperatures of 3−4 K with a pumped 4He bath. The

microwave signal is brought into the cryogenic vacuum chamger via coaxial cable and

connects to a spectrum network analyzer and microwave source which sweeps from 20

to 48 GHz. An IR filtered window with a camera allows for a view into the cryostat.

The probe tip positions on the end of the probe arms are adjustable after cooldown.

Pictures of this setup are shown in Figure 6.30. Calibration must be done at room

and cryogenic temperatures.

Preliminary measurements of test devices of variation THM4/THM5 were com-

pleted. The Nb microstrip transition was observed as the devices were cooled. The

Nb superconducting transition occurred at probe and chip temperatures of 8-8.5 K

where the throughline microwave performance was observed to transition from a lossy

to near lossless state. Calibrated measurements of the microstrip transmission loss at

4 K (removing the loss due to the CPW-to-microstrip transition) are shown in Fig-

ure 6.31. Measurements of the Al2O3 relative dielectric constant and characteristic

impedance of the 3 µm wide microstrip line are also shown in Figures 6.32 and 6.33.

Unfortunately, due to shorting issues on some of the test devices and high fre-

242

CPW ground contacts

CPW center conductor which transitions to microstrip

Figure 6.27: One of the THM 43 GHz CPW test chip designs. Used mainly

for calibration purposes, it includes microstrip throughlines of various

length as well as two microstrip line “opens”. Contact to each lines is

made by 3 CPW probe tip pads at the edges of the chip.

243

CPW ground contacts

CPW center conductor which transitions to microstrip

Figure 6.28: Another of the THM CPW designs, with test lines to test the

low-pass filter, termination structure, and impedance transformer perfor-

mances.

244

Figure 6.29: The predicted transmission (S11) and reflection (S12) through

the coplaner waveguide to microstrip transformer for the 43 GHz

THM4/THM5 CPW test chip design.

245

Probe Arms

Camera

Cable to Network Analyzer

and Source

Test Chamber

Window

Test chips

Probe tip

Figure 6.30: The CPW cryogenic probe station measurement setup.

246

Figure 6.31: Measured loss per millimeter of 3 µm wide microstrip

throughline on a test device of variation THM5. This measured loss is

within the measurement noise (∼ ±0.01 dB).

247

Figure 6.32: Measured relative dielectric constant of Al2O3 dielectric found

via calibration measurement of the microstrip throughlines and opens on

a test device of variation THM5. Expected ǫr ≈ 10.

248

Figure 6.33: Measured characteristic impedance of a 3 µm wide microstrip

line, found via calibration measurements of microstrip throughlines and

open lines on a test device of variation THM5. The measured impedance

is close to the 20 Ω design impedance.

249

quency calibration issues, we were unable to recover calibrated measurements of mi-

crowave absorption in the microstrip termination structure, or to measure other mi-

crowave components during these preliminary measurements of devices THM4/THM5.

Uncalibrated measurements of absorption were seen above the measurement noise

level at frequencies >35 GHz for a few of the CPW THM microstrip termination

structures. In the future, however, the probe station measurement setup may be a

very useful method for investigating the individual components necessary for coupling

microwaves to the THM detector.

6.2.2 Optical Coupling Schemes

To illuminate the devices with calibrated RF signals in the laboratory cryostat

we have devised two basic optical coupling schemes. These two schemes are depicted in

Figure 6.34 and are: 1) a black body microwave source placed inside the cryostat near

the THM detector chip; 2) an external variable frequency microwave source which is

brought into the cryostat via coaxial cable. In both of these cases the optical coupling

in the dewar is accomplished via the setup shown in Figure 6.34, in which the black

body source or the external source is coupled to a waveguide mount which feeds a

horn antenna mounted on the 4 K stage of the cryostat.

These optics are designed following quasi-optical Gaussian beam theory [41].

Contrary to how light behaves in geometrical optics, when the wavelength of the

light is much smaller than the optical components, with microwave light, where the

wavelength is macroscopic compared to the optical components, a beam never focuses

to a single point. Instead, the light is described as a Gaussian beam which focuses to

a beam of minimum spread, where this minimum beam size is defined as the Gaussian

250

beam “waist” radius, w0. The Gaussian beam waist is described in terms of a beam

width radius, w, and beam radius of curvature, R, at any location along the beam by

[41]:

w0 =w

√

1 + (πw2

λR)2

(6.7)

The location of this beam waist relative to the location of the beam at width, w, is

found using the relation [41]:

z =πw0

λ

√

w2 − w20 (6.8)

Optical coupling between two different Gaussian beams is done by matching the two

beam waist sizes and locations.

In the setup for coupling microwaves to the THM detector, the beam waist from

the horn antenna is matched via a Rexolite lens to the beam waist formed by an

extended hemispherical lens mounted to the slot antenna chip. The geometry of this

optical coupling method is shown in Figure 6.35. In the following sections the optimal

geometry for this optical coupling is derived. These optimal dimensions are listed in

Figure 6.35. In Section 6.2.3 and 6.2.4 the black body source design and the specifics

of routing the external source into the cold dewar are described.

6.2.2.1 The Horn Antenna

The horn antenna used is a pyramidal horn antenna with 15 dB gain attached

to a WR-10 waveguide mount. The horn antenna geometry is shown in Figure 6.36.

The beam waist and location for this pyramidal horn was determined by the following

procedure. First, the beam size and radius of curvature at the aperture is give by

251

200 mK

THM detector Chip

Slot antenna aperture

Extended hemispherical lens

4 K

Rexolite lens

Waveguide w/ horn antenna

Scheme 1: 4-20 K Blackbody source

embedded in waveguide

RF Source

300 K

Scheme 2:

External RF source

Figure 6.34: The two optical coupling schemes for testing the microwave

components and microwave response of the THM in the laboratory cryo-

stat. Components are not drawn to scale.

[124]:

wA =0.35wH + 0.5wE

2(6.9)

RA =SH + SE

2(6.10)

where wA is the average beam width at the aperture and RA is the average wavefront

radius of curvature at the aperture. These quantities can be calculated from the horn

geometry shown in Figure 6.36.

Using w = wA and R = RA the beam waist of this horn antenna, w0,in, is found

using Equation 6.7, assuming a 92 GHz center band frequency:

w0,in =wA

√

1 + (πw2

A

λRA)2

(6.11)

Using w = wa and Equation 6.8 one can solve for the beam waist location, zA, relative

252

Si Extended hemispherical Lens:

R=7 mm

L=2.8 mm

Rexolite Lens:

Radius of curvature= 12.0 mm

Diameter= 12.6 mm

Index of refraction=1.6

Horn Antenna:

Aperature width=6.3 mm

Aperature height=4.3 mm

w0,in= 2.0 mm w0,out= 5.1 mm

din=16.5 mm dout=8.8 mm

d0=1.7 mm

L

R

Figure 6.35: Dimensions of the optical components, separation distances

and beam waists. Figures are not drawn to scale.

253

to this aperture:

zA =πw0,in

λ

√

w2A − w2

0,in. (6.12)

Following this procedure for the horn antenna we calculate a beam waist radius of

w0,in = 2.0 mm, a distance of 1.65 mm behind the aperture plane of the horn an-

tenna, as shown in Figure 6.35. The beam pattern of the horn antenna was simulated

with CST Microwave Studio. The measurement of this beam pattern as well as the

simulation are shown in Figure 6.37.

6.2.2.2 The Extended Hemispherical Lens

The extended hemispherical lens geometry is shown in Figure 6.38. The optimal

dimensions for optical coupling to the THM slot antenna were chosen based on simula-

tions by Filopovic et al. [27] for an extended hemispherical lens coupling a similar slot

antenna design to free space, as our own simulations with HFSS and CST Microwave

studio of this coupling have been unsuccessful so far. We chose to use an extended

hemispherical lens characterized by the geometrical relation RL

= 0.39 for a Si lens

with ǫR = 11.7. This geometrical relation creates an approximate elliptical lens from

the extended hemispherical lens geometry. Filopovic et al. claim that this elliptical

lens design produces a beam waist at the curved surface of the hemispherical lens.

Another possible lens geometry for this design, which is also discussed in Filopovic et

al., is a hyper-hemispherical lens characterized by the geometrical relation RL= 0.27

for a Si lens. With the chosen elliptical lens we chose R = 7 mm to match to the

THM chip dimensions (15 mm x 15 mm) resulting in an extension length of L = 2.7

mm, with a predicted beam waist at the edge of the lens of w0,out =5.1 mm based on

Filipovic et al. simulations at 92 GHz. The elliptical lens is mounted directly to the

254

Horn Aperture

x=10.8 mm

Horn Aperture (face on)

wH= 6.3 mm

wE =

4.3

mm

Horn Throat

Horn Throat (face on)

lE =

1.3

mm

lH= 2.5 mm

SE = 10.9 mm

wE =

4.3

mm

Horn Aperture

x=10.8 mm

Horn Throat

SH = 11.0 mm

wH

= 6

.3 m

m

Cross-sectional view in E-plane

lE =

1.3

mm

lH =

2.5

mm

Cross-sectional view in H-plane

Figure 6.36: The horn antenna geometry and dimensions. Here SE and SH

are the pyramidal horn dimensions projected into the E-field and H-field

planes.

255

Figure 6.37: The simulated (above) and measured (below) beam pattern

from the pyramidal horn antenna at 92 GHz. Courtesy Sara Stanchfield.

256

L

R

Figure 6.38: The extended hemispherical lens geometry. R is the spherical

radius, L is the cylindrical extension length.

backside of the THM chip via a copper L-bracket mount and beryllium-copper spring

clamps to provide mechanical and thermal contact to the mount and coldstage.

6.2.2.3 The Rexolite Lens

The Rexolite lens dimensions and location (labelled as dout and din in Figure 6.35)

were determined by the available space in the dewar and requiring a match between

the horn antenna beam waist w0,in and the beam waist at the elliptical lens w0,out.

These parameters for the Rexolite lens can be calculated following the quasi-optical

thin lens approximation described here. First, dout and din are given by [41]:

din = f +M−1√

f 2 − f 20 (6.13)

dout = f +M√

f 2 − f 20 (6.14)

where M ≡ w0,out

w0,inand f0 =

πw0,outw0,in

λ. f ≡ the focal length of the Rexolite lens.

Assuming a thin lens approximation this focal length is given by [41]:

1

f=n2 − n1

n1

2

R(6.15)

257

where n2 = 1.59 is the index of refraction of Rexolite and n1 = 1.0 is the index of

refraction of air. R is the radius of curvature of both sides of the Rexolite lens. For

a focal length chosen to fit within the dewar space, the Rexolite curvature can be

determined, and the din and dout values found.

CST Microwave Studio simulations of the beam profile of the horn antenna Rex-

olite lens combination with optimal dimensions calculated from this method indicated

slight deviations of w0,out and dout from the values predicted by these calculations,

probably due to deviations of the Rexolite lens from the thin lens approximation.

Thus modifications to the Rexolite dimensions and placement were made via CST

studio simulations to find the optimal lens dimension and locations. These final di-

mensions are listed in Figure 6.35. Measurements and simulations of the horn antenna

beam pattern after refocusing by this Rexolite lens are shown in Figure 6.39.

6.2.3 Black Body Source

We have constructed and measured the characteristics of two cryogenic black

body microwave source designs. These black body sources are inserted into a waveg-

uide mount attached to the horn antenna on the 4 K stage of the dewar, and allow

testing of the response of the THM detector to a CMB-like source internal to the

cryostat. The black body source consists either of a distributed resistive element or a

lumped chip resistor which terminate a W-band waveguide. The termination can be

heated from 4 K to ∼ 20 K by Joule heating. At 4 K the black body source mimics

the 3 K black body loading from the CMB seen from a space-based instrument. At

20 K the black body source mimics the equivalent power loading from the CMB with

sky background for a ground-based instrument. A chip thermometer is mounted to

258

Figure 6.39: The simulated and measured beam pattern at 92 GHz from

the horn antenna after focusing by the Rexolite lens. Courtesy Sara

Stanchfield. Measurements are consistent with the simulated beam pat-

tern.

259

the black body source to monitor the black body temperature.

The black body source is thermally isolated from the waveguide and 4 K bath

via a thermal link which is characterised by a thermal conductance, Gbb. The black

body source also has a heat capacity, Cbb, consisting of the resistor heat capacity,

thermometer heat capacity, the heat capacity of any additional heater, as well as

the heat capacity of the adhesives used to mount these components and the insulating

substrate they are mounted on. The thermal time constant is τbb ≃ Cbb

Gbband determines

how fast the source power level can be chopped. The power, PJ , to heat the black

body is also determined by the thermal conductance where PJ ∼ Gbb(Tbb − Tbath).

A black body with near perfect emissivity (& 95%) is desired. In the following two

sections we describe and present thermal and microwave measurements of two black

body source designs, which fulfil these basic requirements.

6.2.3.1 Nichrome Black Body Source

The nichrome black body design is based on a similar black body source design

by McGrath et al. [85] for longer wavelengths. For our design a thin Nichrome film

(∼ 20 nm thick) with measured surface resistivity of ρ ∼ 200 Ω

is deposited on a

quartz wafer. A sliver of this quartz wafer is then inserted into the waveguide cavity

via a thin slot cut in the broad wall of the WR-10 waveguide. Outside of the waveguide

slot area, a 5 kΩ (∼ 8 kΩ at 4 K) resistor heater is mounted with silver epoxy. The

lead wires are Stycast to the quartz wafer. A ruthenium oxide chip thermometer is

mounted in a similar manner. This nichrome black body layout and the arrangement

of this black body source in the waveguide are shown in Figure 6.40.

The thermal link to the waveguide mount and 4K stage is provided by a copper

260

Waveguide Copper Block Mount

Heat Sink to 4K

Copper wire thermal linkThermometer

Resistor Heater

NichromePortion in waveguide

Figure 6.40: The nichrome blackbody source.

261

wire, the diameter and length of which can be adjusted. One end of this wire is glued

to the nichrome film with Ge varnish and the other end is soldered directly to a copper

plate which is clamped down to the copper waveguide mount. An additional thermal

link is provided by the thermal boundary conductance where the nichrome film and

the Quartz wafer come into contact with the copper waveguide mount as the wafer

is inserted into the waveguide. This boundary conductance was found to be quite

low, and did not dominate the thermal characteristics of the device. The thermal

conductance of the copper wire was adjusted to find the optimal Gbb which provided

a fast response, but still allowed for low power dissipation (< 100 mW) when heating

the source to 20 K. The heat capacity was also minimized as much as possible to

decrease the thermal response time.

The black body source design was cooled to 4 K in vacuum in an IR-shielded

laboratory cryostat and Joule power was applied to heat the device up to temperatures

of ∼ 20 K, while the bath temperature was held constant at 4 K. PJ vs. Tbb was

measured, as well as the thermal rise and fall time as the power was switched on

and off, allowing τbb, Gbb and Cbb to be determined. The maximum power needed

to raise the black body temperature to 20 K, PJ,max, was also determined. These

measurements of Gbb, Cbb, Pbb,max and τbb are listed in Table 6.1.

The microwave emissivity of the nichrome black body mounted inside the waveg-

uide was also measured via transmission and reflection measurements using a scalar

network analyzer with a microwave sweeper and external multipliers from 75 to 110

GHz. These measurements are shown in 6.41. We measure a thermal time constant

of 90 seconds, maximum power of 0.86 mW, and emissivity of 90 − 99%. Although

262

Table 6.1: The measured thermal characteristics of the nichrome black

body source.

Cbb (JK) Gbb (

WK) τbb (s) PJ,max (mW)

4.32 ·10−3 4.8 ·10−5 90 0.86

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

76 80 84 88 92

Frequency (GHz)

Em

issiv

ity

Figure 6.41: The measured emissivity of the nichrome black body source.

these thermal and microwave characteristics for this nichrome black body design are

acceptable, improvements are possible with the “finline” black body source design,

described in the following section.

263

Waveguide 1.5mm Kapton

Kapton film/circuit board

Figure 6.42: The geometry and layout of the finline black body source.

6.2.3.2 Finline Black Body Source

The second black body design is a modification of the nichrome black body

design and was developed to simplify the fabrication and shorten the time constant.

In this design a copper finline structure on a Kapton dielectric PC board (fabricated

by Brigitflex) is inserted in the E-plane of the WR-10 waveguide. The geometry of

this finline is shown in Figure 6.42 and a photo of the actual finline device is shown

in Figure 6.43. The finline structure provides a microwave transition from the 3-D

WR-10 waveguide to a 2-D slot line transmission line with characteristic impedance

of ∼100 Ω. A chip resistor of 100 Ω is soldered across the slot line to terminate the

microwaves and serve as the black body emitter/absorber. This resistor also serves

as the heating element for the black body as DC current can be passed through the

resistor.

264

Figure 6.43: The finline black body source.

265

Table 6.2: The measured thermal characteristics of the finline black body

source.

Cbb (JK) Gbb (

WK) τbb (s) PJ,max (mW)

9.7 ·10−3 2.9 ·10−3 3.3 34

As with the nichrome design a ruthenium oxide thermometer is soldered to the

finline structure to monitor the temperature of the source. One end of a copper wire

is soldiered to the copper layer of the PC board and the other end is clamped to the

copper waveguide mount which is heat sunk to 4 K. This copper wire controls the

thermal conductance between the finline black body source and the copper waveguide

mount in a similar manner to the nichrome black body.

Measurements of the thermal time constant, τbb, thermal conductance Gbb, heat

capacity Cbb, and Pmax,bb were taken while the device was cooled to 4 K and Joule

power was dissipated in the chip resistor. These measurements are shown in Table

6.2. Microwave emissivity measurements are shown in Figure 6.44. We measure a

thermal time constant of 3 seconds, maximum power of 34 mW, and emissivity of 60-

99%. These characteristics are acceptable, and the increase in the time constant is an

improvement over the nichrome design. A higher emissivity is possible (ǫ ∼ 90−98%)

for this design if an extended resistor is used in place of the chip resistor. The simulated

emissivity for an extended resistive film of length ∼ 0.22 inches, R = 250 Ωdeposited

along the finline for termination is shown in Figure 6.45.

266

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

75 80 85 90 95 100 105 110

Frequency (GHz)

Em

iss

ivit

y

Figure 6.44: The measured emissivity of the finline black body source.

Courtesy Sara Stanchfield.

267

Figure 6.45: The simulated emissivity of the finline black body source

with a 0.22 inch long extended resistor with resistivity of 250 Ωand total

terminating resistance of 28.5 Ω. Courtesy Sara Stanchfield.

268

6.2.4 External RF Source Coupling

The source in the external scheme is a HP model 83751A sweeper with external

active multipliers. It can be swept from 75−110 GHz. This signal is transmitted into

the dewar by a coaxial cable. This cable consists of an outer conductor of beryllium

copper (diameter of 1.19 mm) and inner conductor of silver plated beryllium copper

(diameter of 0.287 mm), separated by a PTFE dielectric (diameter of 0.94 mm). The

coaxial cable has characteristic impedance of 50 Ω. Room temperature loss in the

cable is ∼7 dBm

at 20 GHz. We use ∼ 0.5 m of cable to make the transition through

the dewar to the waveguide mount. This cable passes through a vacuum feed-through

at room temperature into the cryostat and is heat sunk as it travels through the 77

K and then 4 K stages. The coaxial cable connects to the copper waveguide mount

at the 4 K stage. Measurements of the performance of the us23/us25 test devices

are currently under way using the thermometry of the gradual transition from 4K

to 200 mK on these devices to make measurements of the total optical and on-chip

transmission efficiency. RF test on THMs with actual low temperature TES transitions

with Tc ∼ 200 mK will have to wait until we fabricate a working THM device in an

RF circuit. The design of such a device is described in the concluding Chapter 7.

269

Chapter 7

Conclusion

7.1 Summary of Understanding of the THM Design

In this thesis measurements of a variety of THM test devices have been presented.

A thermal model based on our understanding of TES and bolometer theory and the

theory of the physical processes in the THM detector (Chapters 2 and 3) for both an

ideal and a non-ideal THM detector has been developed and applied. These thermal

models make an understanding of thermal measurements and DC electrical measure-

ments of THM test devices possible. These measurements include I-V measurements,

thermal conductance measurements, responsivity, and dark noise measurements of

THM test devices (Chapter 5).

These models also allow the optimal THM design and performance (Chapter 4)

to be constrained while considering the very specific loading conditions for observing

the CMB from the ground or from space. Comparison to the THM performance un-

der FIR spectral loading conditions was also discussed in this thesis, as hot-electron

type bolometers have a unique advantage in FIR applications, but perform very dif-

ferently under these relatively low power loading conditions. In this discussion of the

270

THM optimization (Chapter 4), some time was also devoted to discussing the general

optimization of any bolometeric CMB detector, in order to further place the THM

performance in context.

Specifically, the investigations of the THM detector design presented in this

thesis have included measurements of the magnitude (κep and Σep) and temperature

dependence (n = 5− 6) of the hot-electron effect in a composite Au and Bi bolometer

(Section 5.4). These measurements confirm our understanding that the dominant

thermal process within the THM detector is due to scattering between electrons and

phonons. In addition, measurements of Mo/Au bilayer THM devices on the micron

size scale with Nb superconducting leads indicate the current geometrical limits on the

THM model and design due to the lateral proximity effect (Section 5.5). A relatively

simple procedure (described in Section 5.1) to fabricate optimal THM designs using

standard photolithography has also been demonstrated.

Concerning the coupling of microwave radiation to the small volume of the THM

detector a microwave design has been developed which provides polarization-selective

RF coupling to the THM detector via a planar slot antenna design, using planar

microstrip technology (Section 6.1). This development has included E-M simulations

of all components. Additionally, work has been done towards developing the necessary

off-chip optical components for coupling microwaves to the THM detector (Section

6.2.2), both for calibration in the laboratory cryostat, and in some modified form, for

actual observation of the CMB sky.

271

7.2 Future Work Involving The Lateral Proximity Effect

As was demonstrated by the measurements presented in Section 5.5, the obtain-

able NEP of THM devices with micron-size TES devices, in which the bilayer TES

forms a S-S’-S junction with the Nb superconducting leads, is limited due to shifting of

the effective Tc of these devices to higher temperatures. Based on these measurements

bilayer THM devices with lead-to-lead lengths of L ≥ 12 µm are necessary to avoid

large shifts in the transition temperature of the TES on the order of 500 mK-1 K for

a bulk TcN of ∼200 mK. These measurements confirm results by Sadleir et al. [97],

in which lead-to-lead lengths of L ≥ 16 µm are necessary to avoid large shifts in the

transition temperature of the TES on order of 100 mK for a bulk TcN of 170 mK. There

is still much irreproducibility and variation in this effect in bilayer devices, however,

possibly due to variations inherent in the fabrication process and due to the purity of

the deposited film and/or the contact resistance between the superconducting leads

and the TES.

To overcome the size constraints imposed by this lateral proximity effect, future

work on micron-sized THMs with bilayer TES devices might focus on developing

techniques to weaken this proximity effect. Possible techniques include inserting a thin

ferromagnetic layer (such as Ni, Fe, or Co) between the superconducting leads and the

superconducting bilayer TES. The phase coherence of a Cooper pair is very quickly

disrupted when travelling from a superconductor into a nearby ferromagnetic layer

and the coherence length in a ferromagnetic material is substantially shorter (by 1-3

orders of magnitude) than the coherence length in a normal metal due to interactions

with the opposing spins of each of the electrons in the Cooper pair [13] [19]. A similar

272

technique would be to purposely create or deposit an insulating contact resistance

layer between the superconducting lead and superconducting TES which would also

attenuate this lateral proximity effect.

Another avenue for developing a low Tc micron-sized TES device for the THM

was explored preliminarily in this thesis in Section 5.5. This idea is to create a new

type of TES made not of a bilayer, but of a normal monolayer, in which the lateral

proximity effect creates a S-N-S TES. In addition to the practical reasons of achieving a

micron-sized TES which can reach the optimal NEP for the THM detector, pursuing

the development of this type of TES is interesting simply from a novel technology

standpoint. Although this preliminary investigation resulted in no observation of a

superconducting transition, even in THM designs in which transitions are predicted

by simple S-N-S theories (see Section 3.3.9), we do not believe this method has been

ruled out. However, this technique may be more difficult to achieve than was originally

anticipated based on the strong lateral proximity effect observed in bilayer S-S’-S

devices.

Future exploration of this S-N-S TES idea might focus on a single deposition

procedure where the Au and Nb layers are deposited without breaking vacuum, as

was done by van Dover et al. [119], in order to take further precautions to avoid a

dirty interface. Another direction to explore in the development of an S-N-S TES

device would be the use of novel photolithography techniques or e-beam lithography

in order to reduce the lead-to-lead length of the devices below 3 µm.

All of these ideas for fabricating S-N-S or S-S’-S micron size THM devices are

especially worth pursuing for far-infrared hot-electron bolometers, where the advan-

273

tages provided by low thermal conductance and small volume are necessary to reach

background limited levels. However, for a CMB-observing THM detector this per-

formance is already achievable without pursuing these designs (as is presented in the

final section of this thesis, Section 7.5), and it will not be necessary, or arguably worth

the effort to follow these ideas further. This is because the optimal size for the THM

detector TES at low bath temperatures occurs at size scales above those which are

strongly affected by the lateral proximity effect (as shown in Chapter 4).

7.3 Future Microwave Work

The complete testing of the 92 GHz THM microwave design should include

measurements of transmission and reflection of the individual microwave components,

focusing on testing the microwave termination structure (Section 6.1.7), and low-pass

filter (Section 6.1.5) in order to test the frequency dependent microwave performance

of these components which have been predicted by E-M simulations. This type of

testing of individual components scaled to 43 GHz could be done in a cryogenic CPW

probe station measurement system as described in Section 6.2.1.

In addition, it is important to confirm the performance and efficiency of the

slot antenna design as it is coupled via the optical setup described in Section 6.2.2

in the laboratory cryostat. These tests would provide further confirmation of the

agreement between the E-M simulation and the actual performance via a measurement

of total optical efficiency from the RF source to the THM detector. More complete

simulations of the optical coupling efficiencies involved would also allow disentangling

of the optical efficiency of off-chip components and provide for on-chip microwave

efficiency measurements. These microwave measurements will also demonstrate and

274

provide a calibrated measurement of the THM RF responsivity and NEP under CMB-

loading conditions.

7.4 Future Work for Scaling to Large Arrays

Other future challenges that have not been addressed by this thesis occur with

the scaling from a single pixel THM detector to a large array of THM detectors. These

issues include uniformity concerns, and the design of optical coupling and SQUID

readout for an array of THMs. However, much work has been done and is in progress

in this area for membrane-isolated CMB bolometeric detectors, with current arrays of

1000s of detectors with multiplexed SQUID schemes obtainable (Section 1.2.1). As was

stated in Section 1.2, the THM detector has many advantages which may make array

scaling simpler, including the fact that no delicate thermal isolation legs or membranes

are required. One area of difference in the THM read out, however, is the high-speed

of these THM devices (τ . 1− 100 µs) compared to typical TES bolometers. Due to

the speed of these detectors, hot-electron type bolometers will likely need to be read

out with a microwave SQUID multiplexing scheme [57]. Although this topic of array

scalability is beyond the scope of this thesis, these issues will need to be considered

for the THM to prove applicable to the future science needs of CMB telescopes, and

specifically, the detection of B-mode polarization in the CMB.

7.5 A Last Word: The Recommended THM Design

As the last word of this thesis the recommended design for the THM detector

for CMB applications will now be presented. Although this design has not yet been

implemented in the form of a THM test device embedded in an RF circuit, the un-

275

derstanding of the THM thermal model and the physical effects in the model based

on the DC and “dark” measurements of similarly sized test devices presented in this

thesis give high confidence that this design will meet the requirements for a CMB

background-limited detector, and that in fact no, or very little further development is

necessary to provide a “working” CMB detector which might be inserted into a CMB

ground- or space-based telescope. In addition to this understanding of the THM phys-

ical processes, and the development of reliable and fairly straightforward fabrication

methods for the THM design, our investigation into the microwave design via E-M

simulations is to be complemented by impending microwave measurements, using the

test setup described in Section 6.2.2.

The recommended CMB THM design consists of a Mo/Au bilayer TES with a

Au thickness of ∼300 nm, and with a Mo/Au thermal area of ∼ 20 x 20 µm2 and with

a bilayer Tc = 80− 300 mK, depending on the precise bath temperature and whether

the detector is designed for ground- or space-based observing. This choice of TES

area provides for a lead-to-lead length significantly above the size scale where lateral

proximity effects from the superconducting Nb leads have been observed to have a

strong influence. The absorber for this recommended design consists of a ∼800 nm

thick Bi film, with total thermal area 3 µm x 12 µm, with a 3 µm x 3 µm terminating

resistive area and a 3 µm x 3 µm contact area overlapping the center of the Mo/Au

TES. This design is illustrated in Figures 7.1 and 7.2.

The recommended THM device would have the optimal thermal design following

the ideal thermal model. The predicted detector noise, as well as other important de-

tector characteristics, for this recommended THM detector design are listed in Tables

276

Table 7.1: Predicted thermal characteristics, including detector NEP for

the recommended CMB THM design for a ground-based instrument with

bath temperature of 240 mK. Mo/Au TES dimensions are 300 nm x 20

µm x 20 µm, with RTES ∼ 0.1 Ω. Bi absorber dimensions are 800 nm

x 3 µm x 12 µm with a terminating resistance of RAbs ∼30 Ω. Photon

background noise from the ground is NEPphoton = 3.6 · 10−17 W√Hz

. A

50% optical efficiency and bias power of twice the sky loading has been

assumed. The ‘measured n = 5’ and ‘measured n = 6’ design assumptions

are described in Section 4.2.1.Design Assumptions C ( J

K) G (W

K) τ (µs) Tc (mK) NEP ( W√

Hz)

measured n = 6 2.5·10−15 2.5·10−10 < 10 300 2.7·10−17

measured n = 5 2.1·10−15 2.3·10−9 < 1 244 8.5·10−17

7.1 and 7.2 for ground-based observing and in Table 7.3 for space-based observing. A

low bath temperature of 50 mK is necessary for space-based observing, and recom-

mended for ground-based observing as well. In both cases, photon-limited performance

is obtainable with this detector design.

277


the recommended CMB THM design for a ground-based instrument with

bath temperature of 50 mK. Mo/Au TES dimensions are 300 nm x 20 µm

x 20 µm, with RTES ∼ 0.1 Ω. Bi absorber dimensions are 800 nm x 3 µm x

12 µm with a terminating resistance of RAbs ∼30 Ω. Photon background

noise from the ground is NEPphoton = 3.6 · 10−17 W√Hz

. A 50% optical

efficiency and bias power of twice the sky loading has been assumed. The

‘measured n = 5’ and ‘measured n = 6’ design assumptions are described

in Section 4.2.1.Design Assumptions C ( J

K) G (W


Hz)

measured n = 6 2.4·10−15 1.9·10−10 < 13 280 2.0·10−17

measured n = 5 1.3·10−15 3.0·10−10 < 4 150 1.3·10−17


the recommended CMB THM design for a space-based instrument for bath

temperature of 50 mK. Mo/Au TES dimensions are 300 nm x 20 µm x 20

µm, with RTES ∼ 0.1 Ω . Bi absorber dimensions are 800 nm x 3 µm x

12 µm with a terminating resistance of RAbs ∼30 Ω. Photon background

noise from a space-based instrument is NEPphoton = 4.3 · 10−18 W√Hz

. A

50% optical efficiency and bias power of twice the sky loading has been

assumed. The ‘measured n = 5’ and ‘measured n = 6’ design assumptions

are described in Section 4.2.1.Design Assumptions C ( J

K) G (W


Hz)

measured n = 6 1.4·10−15 1.5·10−11 < 100 170 3.4·10−18

measured n = 5 7.0·10−16 2.8·10−11 < 30 80 2.3·10−18

278

Slot Antenna

Low-pass Filter

THM, Microstrip Termination and Bias Leads

Figure 7.1: The basic microwave circuit for the recommended THM design.

A close-up of the THM detector is shown in Figure 7.2.

279

Bi

AbsorberNb µstripMo/Au TES

Nb

Nb

20 µm

20 µm

RF Power

DC bias leads and

µstrip termination structure

3 µm

Au overlap on Nb for contact

Bi overlap on Nb for contact

Figure 7.2: The geometry of the recommended THM design for CMB ap-

plications.

280

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