John Clarke, Alex I. Braginski (Eds.)

The SQUID Handbook

Vol. II Applications of SQUIDsand SQUID Systems

InnodataFile Attachment9783527609505.jpg

J. Clarke, A. I. Braginski (Eds.)

The SQUID Handbook

Vol. II

Related Titles

Buckel, W., Kleiner, R.

SuperconductivityFundamentals and Applications

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2004

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John Clarke, Alex I. Braginski (Eds.)

The SQUID Handbook

Vol. II Applications of SQUIDsand SQUID Systems

The Editors

Prof. John ClarkeDepartment of Physics366 LeConte HallUniversity of CaliforniaBerkeley, CA 94720-7300USAandMaterials Science DivisionLawrence Berkeley National LaboratoryOne Cyclotron RoadBerkeley, CA 94720jclarke@berkeley.edu

Prof. Dr. Alex I. BraginskiResearch Center J�lichIBN-2D-52425 J�lichGermanya.braginski@fz-juelich.de

& All books published by Wiley-VCH arecarefully produced. Nevertheless, authors,editors, and publisher do not warrant theinformation contained in these books,including this book, to be free of errors.Readers are advised to keep in mind thatstatements, data, illustrations, proceduraldetails or other items may inadvertentlybe inaccurate.

Library of Congress Card No.: applied for

British Library Cataloguing-in-Publication DataA catalogue record for this book is availablefrom the British Library.

Bibliographic information published byDie Deutsche BibliothekDie Deutsche Bibliothek lists this publicationin the Deutsche Nationalbibliografie; detailedbibliographic data is available in the Internet at.

� 2006 WILEY-VCH Verlag GmbH & Co. KGaA,Weinheim

All rights reserved (including those oftranslation into other languages).No part of this book may be reproducedin any form – nor transmitted or translatedinto machine language without writtenpermission from the publishers. Registerednames, trademarks, etc. used in this book,even when not specifically marked as such,are not to be considered unprotected by law.

Typesetting K�hn & Weyh, Satz und Medien,FreiburgPrinting Strauss GmbH, M�rlenbachBookbinding Litges & Dopf Buchbinderei GmbH,Heppenheim

Printed in the Federal Republic of Germany.Printed on acid-free paper.

ISBN-13: 978-3-527-40408-7ISBN-10: 3-527-40408-2

This Handbook is dedicated to the memory of

Robin P. Giffard, Christoph Heiden andJames E. Zimmerman.

VII

Volume I

Preface XI

1 Introduction 11.1 The Beginning 21.2 Subsequent Developments 51.3 The dc SQUID: A First Look 71.4 The rf SQUID: A First Look 121.5 Cryogenics and Systems 161.6 Instruments: Amplifiers, Magnetometers and Gradiometers 171.7 Applications 211.8 Challenges and Perspectives 241.9 Acknowledgment 26

2 SQUID Theory 292.1 Josephson Junctions 302.2 Theory of the dc SQUID 432.3 Theory of the rf SQUID 70

3 SQUID Fabrication Technology 933.1 Junction Electrode Materials and Tunnel Barriers 943.2 Low-temperature SQUID Devices 963.3 High-temperature SQUID Devices 1073.4 Future Trends 118

4 SQUID Electronics 1274.1 General 1284.2 Basic Principle of a Flux-locked Loop 1284.3 The dc SQUID Readout 1374.4 The rf SQUID Readout 1554.5 Trends in SQUID Electronics 165

Contents

VIII

5 Practical DC SQUIDS: Configuration and Performance 1715.1 Introduction 1725.2 Basic dc SQUID Design 1755.3 Magnetometers 1865.4 Gradiometers 1935.5 1/f Noise and Operation in Ambient Field 2005.6 Other Performance Degrading Effects 208

6 Practical RF SQUIDs: Configuration and Performance 2196.1 Introduction 2206.2 Rf SQUID Magnetometers 2206.3 Rf SQUID Gradiometers 2366.4 Low-Frequency Excess Noise in rf SQUIDs 2376.5 Response of rf SQUIDs to High-frequency Electromagnetic

Interference 2396.6 Characterization and Adjustment of rf SQUIDs 2416.7 The rf SQUID versus the dc SQUID 2446.8 Concluding Remarks and Outlook 246

7 SQUID System Issues 2517.1 Introduction 2547.2 Cryogenics 2557.3 Cabling and Electronics 2727.4 Data Acquisition and Rudimentary Signal Processing 2897.5 Characterization, Calibration and Testing 2927.6 Conditions Imposed on SQUID Systems by the Environment and

Applications 3097.7 Noise Suppression 3157.8 Signal and Noise Implications for the SQUID System Design 3357.9 Concluding Remarks and System Trends 344

Appendix 1 357Basic Properties of Superconductivity

Appendix 2 367Abbreviations, Constants and Symbols

Index 383

Contents

IX

Volume II

Preface XI

List of Contributors XV

8 SQUID Voltmeters and Amplifiers 1J. Clarke, A. T. Lee, M. M�ck and P. L. Richards

8.1 Introduction 38.2 Voltmeters 48.3 The SQUID as a Radiofrequency Amplifier 58.4 Microstrip SQUID Amplifier 208.5 SQUID Readout of Thermal Detectors 328.6 Nuclear Magnetic and Quadrupole Resonance and Magnetic

Resonance Imaging 568.7 The Axion Detector 81

9 SQUIDs for Standards and Metrology 95J. Gallop and F. Piquemal

9.1 Introduction 969.2 SQUIDs in Voltage Metrology 979.3 Cryogenic Current Comparator (CCC) 1019.4 Other Current Metrological Applications of SQUIDs 1239.5 Future Trends and Conclusion 129

10 The Magnetic Inverse Problem 139E. A. Lima, A. Irimia and J. P. Wikswo

10.1 The Peculiarities of the Magnetic Inverse Problem 14110.2 The Magnetic Forward Problem 14510.3 The Magnetic Inverse Problem 16810.4 Conclusions 254

11 Biomagnetism 269J. Vrba, J. Nenonen and L. Trahms

11.1 Introduction 27111.2 Magnetoencephalography 27411.3 Magnetocardiography 32111.4 Quasistatic Field Magnetometry 34211.5 Magnetoneurography 34611.6 Liver Susceptometry 35111.7 Gastromagnetometry 35611.8 Magnetic Relaxation Immunoassays 360

Contents

12 Measurements of Magnetism and Magnetic Properties of Matter 391R. C. Black and F. C. Wellstood

12.1 Introduction 39212.2 The SQUID Magnetometer–Susceptometer 39212.3 Scanning SQUID Microscopy 409

13 Nondestructive Evaluation of Materials andStructures using SQUIDs 441H.-J. Krause and G. Donaldson

13.1 Introduction 44213.2 Detection of Magnetic Moments 44513.3 Magnetic Flux Leakage Technique 44813.4 Static Current Distribution Mapping 45213.5 Eddy Current Technique 45313.6 Alternative Excitation Techniques 46713.7 Conclusion and Prospects 472

14 SQUIDs for Geophysical Survey and Magnetic Anomaly Detection 481T. R. Clem, C. P. Foley, M. N. Keene

14.1 Introduction 48314.2 Magnetic Measurements in the Earth’s Field 48414.3 Operation of SQUIDs in Real World Environments 49414.4 Data Acquisition and Signal Processing 49914.5 Geophysical Applications of SQUIDs 50414.6 Magnetic Anomaly Detection Systems using SQUIDs 52714.7 Future Prospects 536

15 Gravity and Motion Sensors 545Ho J. Paik

15.1 Introduction 54615.2 The Superconducting Accelerometer 54715.3 Superconducting Transducer for Gravitational-Wave Detectors 54815.4 Superconducting Gravity Gradiometers (SGGs) 55415.5 Applications of the SGG Technology 56315.6 Outlook 575

Appendix 581Physical Constants, Abbreviations and Symbols

Index 617

ContentsX

XI

We hope that this two-volume Handbook will provide an in-depth, systematictreatment of Superconducting QUantum Interference Devices (SQUIDs) andtheir many applications. Our intent is to offer the reader a reasonably complete,balanced and up-to-date presentation of the entire field, with as few omissionsand duplications as possible. Although our publisher initially suggested that oneor two of us write the Handbook, we pointed out that the field had become solarge and diverse that this would be an almost impossible undertaking. Manyaspects of SQUIDs, especially applications, have become so specialized that nosingle person can realistically provide adequate coverage. Consequently, weinvited various colleagues collectively to write a comprehensive treatise. Fortu-nately, virtually everyone we asked graciously agreed to participate.The first volume of the Handbook, published in 2004, contained seven chapters

devoted to the fundamental science, fabrication and operation of low-Tc and high-Tc, dc and rf SQUIDs. After an introductory overview, subsequent chapters wereentitled SQUID Theory, SQUID Fabrication Technology, SQUID Electronics,Practical DC SQUIDs: Configuration and Performance, Practical RF SQUIDs:Configuration and Performance, and SQUID System Issues. Appendix 1 brieflydescribed the Basic Properties of Superconductivity and Appendix 2 listed theacronyms and symbols used in the Handbook.Volume II contains eight chapters concerned with applications using SQUIDs

as sensors and readout devices.In Chapter 8, Clarke, Lee, M�ck and Richards describe the theory and imple-

mentation of SQUID voltmeters and amplifiers. The first sections describe mea-surements of quasistatic voltages, the use of the dc SQUID as a radiofrequencyamplifier, and the extension of the frequency range into the microwave regime bymeans of a microstrip input circuit. Subsequently, the application of SQUIDs toread out thermal detectors and their multiplexing in the time- and frequency-domains are discussed. SQUID amplifiers for nuclear magnetic resonance andmagnetic resonance imaging are reviewed, and various examples are presented.The chapter concludes with a brief discussion of the implementation of a near-quantum-limited SQUID amplifier on a detector to search for the axion, a candi-date for cold dark matter.

Preface

XII

In Chapter 9, Gallop and Piquemal describe the role of SQUIDs in standardsand metrology. After a brief discussion of highly accurate voltage measurement,the authors focus on the principles and accuracy limits of the cryogenic currentcomparator (CCC). Among its applications are measurements of resistance ratios,very low currents from superconducting electron transistors, and currents inbeams of charged particles. Other metrology applications include secondary ther-mometers based on magnetic susceptibility and resistance, and a primary thermo-meter based on Nyquist noise.In Chapter 10, Lima, Irimia and Wikswo tackle the magnetic inverse problem

that is central to interpreting measurements in biomagnetism, geophysics andnondestructive evaluation. They first describe the forward problem – the determi-nation of magnetic fields produced by distributions of magnetization and currentand by multipoles. They begin their discussion of the inverse problem with thelaw of Biot and Savart, and go on to discuss the imaging of distributions of mag-netization. An important aspect of the inverse problem is “silent sources” – forexample, source configurations that produce either an electric or a magnetic fieldbut not both. They conclude with a treatment of the three-dimensional inverseproblem – which, in general, has no unique solution – that highlights some of themost widely used algorithms.In Chapter 11, Vrba, Nenonen and Trahms address biomagnetism, unquestion-

ably the largest single consumer of SQUIDs. They begin with magnetoencephalo-graphy (MEG) – magnetic signals from the brain – and describe whole cortex sys-tems, types of sensors, fetal MEG, and data analysis with clinical examples. Theycontinue with magnetocardiography, describing the kinds of instrumentation,types of sensors, and clinical applications. There follows a miscellany of topics inbiomagnetism, including the measurement of static fields from the body, detect-ing signals propagating along nerves, the susceptibility of the liver as a diagnostictool, gastro-magnetometery, and immunoassay using magnetic labeling of cells.In Chapter 12, Black and Wellstood describe measurements of magnetism and

magnetic properties of matter. The first part describes the history, developmentand operation of the most widely used SQUID system, namely a commerciallyavailable magnetometer and susceptometer. Issues of accuracy and sensitivity arediscussed. The second part of the chapter is concerned with the scanning SQUIDmicroscope. The authors outline the special requirements for the SQUIDs andcryogenics, describe the techniques for scanning and image processing, and dis-cuss issues of spatial resolution. They conclude with a review of current andpotential applications.In Chapter 13, Krause and Donaldson give an overview of methods for nondes-

tructive evaluation. These include the detection of static magnetic moments, themagnetic flux leakage technique, static current distribution mapping, and theeddy current technique. A number of examples is presented. The chapter con-cludes with a brief discussion of alternative ways of exciting a magnetic response.In Chapter 14, Clem, Foley and Keene describe the application of SQUIDs to

geophysical survey and magnetic anomaly detection. They begin with issues ofmagnetic measurements in the presence of the Earth’s field and operating

Preface

XIII

SQUIDs in harsh environments, and continue with data acquisition and signalprocessing. A major portion of the chapter is concerned with geophysical applica-tions, ranging from rock magnetometry to a variety of prospecting and surveyingmethods. They conclude with an overview of the detection of magnetic anomalies,for example, buried ordnance.Finally, in Chapter 15, Paik addresses gravity and motion sensors. He describes

in turn a superconducting accelerometer, a superconducting transducer for gravi-tational-wave detectors, and the superconducting gravity gradiometer (SGG). Ap-plications of the SGG include precision tests of the laws of gravity, searching fornew weak forces, gravity mapping and mass detection, and inertial navigation andsurvey.In the Appendix, we duplicate Appendix 2 of Volume I and provide a list of addi-

tional acronyms and symbols for each chapter of Volume II.This very brief survey illustrates the remarkable diversity of the SQUID, which

finds applications to physics, astrophysics, cosmology, chemistry, materialsscience, standards, biology and medicine. We would like to believe that the Hand-book will be of use not only to practitioners of the art of SQUIDs but also to stu-dents and professionals working in these fields.In conclusion, we express our heartfelt thanks to the authors of both volumes of

the Handbook for their hard work, their attention to quality and accuracy and notleast for their patience and perseverance during our editing of their manuscripts.One of us (JC) expresses his grateful thanks to his assistant, Barbara Salisbury, forher unflagging help with all the manuscripts for both volumes. We owe an enor-mous debt of gratitude to the staff at Wiley-VCH, particularly to Dr. Michael B�r,who first asked us to co-write the Handbook, and to Mrs. Vera Palmer and Mrs.Ulrike Werner without whose expert guidance and extraordinary patience theHandbook would never have seen the light of day. Finally, we thank our wivesMaria Teresa and Grethe for their patience and understanding during our editingof both volumes of the Handbook, which took much of our time away from them.

Alex Braginski and John Clarke

Preface

XV

Volume I

Alex I. Braginski(Chapters 1 and 6)Research Centre J�lich, ISG-2, D-52425J�lich, Germany, (retired), andPhysics Department, University ofWuppertal, 42097 Wuppertal, GermanyA.Braginski@fz-juelich.de

Robin Cantor(Chapters 3 and 5)STAR Cryoelectronics, 25-A BisbeeCourt, NM 87508 Santa Fe, USArcantor@starcryo.com

Boris Chesca(Chapter 2)Institute of Physics, University ofT�bingen, Auf der Morgenstelle 14,72076 T�bingen, Germanyboris.chesca@uni-tuebingen.de

John Clarke(Chapter 1)Department of Physics, 366 LeConteHall, University of California, BerkeleyCA 94720-7300, USA, andMaterials Sciences Division, LawrenceBerkeley National Laboratory,1 Cyclotron Road, Berkeley CA 94720,USAjclarke@berkeley.edu

Dietmar Drung(Chapter 4, Appendix 2)Physikalisch-Technische Bundesanstalt,Abbestrasse 2–12, 10587 Berlin,GermanyDietmar.Drung@ptb.de

Catherine P. Foley(Chapter 7)CSIRO Industrial Physics, P.O. Box218, Lindfield, NSW 2070 AustraliaCathy.Foley@csiro.au

Mark N. Keene(Chapter 7)QinetiQ Ltd., St. Andrews Road,Malvern, Worcestershire WR14 3PS,UKmnkeene@qinetiq.com

Reinhold Kleiner(Chapter 2, Appendix 1)Institute of Physics, University ofT�bingen, Auf der Morgenstelle 14,72076 T�bingen, Germanykleiner@uni-tuebingen.de

List of Contributors

XVI List of Contributors

Dieter Koelle(Chapter 2 and 5 andAppendices 1 and 2)Institute of Physics, University ofT�bingen, Auf der Morgenstelle 14,72076 T�bingen, Germanykoelle@uni-tuebingen.de

Frank Ludwig(Chapter 3)Institute of Electrical Metrology andElectrical Engineering, TechnicalUniversity of Braunschweig, 38092Braunschweig, Germanyf.ludwig@tu-bs.de

Michael M�ck(Chapter 4)Institute of Applied Physics, Universityof Giessen, Heinrich-Buff-Ring 16,35392 Giessen, GermanyMichael.Mueck@ap.physik.uni-giessen.de

H. J. M. ter Brake(Chapter 7)Department of Applied Physics, TwenteUniversity of Technology, P.O. Box 217,7500AE Enschede, The NetherlandsH.J.M.terBrake@tn.utwente.nl

Jiri Vrba(Chapter 7)VSM MedTech Ltd, 9 Burbidge Street,Coquitlam, B.C., Canadajvrba@vsmmedtech.com

Yi Zhang(Chapter 6)Research Centre J�lich, ISG-2, 52425J�lich, Germanyy.zhang@fz-juelich.de

Volume II

Randall C. Black(Chapter 12)Quantum Design, Inc., 6325 Lusk Blvd.,San Diego CA 92121, USArandy@blacksdesign.com

John Clarke(Chapter 8)Department of Physics, 366 LeConteHall, University of California, BerkeleyCA 94720-7300, USA, andMaterials Sciences Division, LawrenceBerkeley National Laboratory,1 Cyclotron Road, Berkeley CA 94720,USAjclarke@berkeley.edu

Ted R. Clem(Chapter 14)Naval Surface Warfare Center PanamaCity, 110 Vernon Avenue, Panama CityFL 32407-7001, USAted.clem@navy.mil

Gordon B. Donaldson(Chapter 13)Department of Physics, University ofStrathclyde, Glasgow G4 0NG, UKg.b.donaldson@strath.ac.uk

Catherine P. Foley(Chapter 14)CSIRO Industrial Physics, P.O. Box218, Lindfield, NSW 2070 AustraliaCathy.Foley@csiro.au

XVII

John Gallop(Chapter 9)National Physical Laboratory, HamptonRd., Teddington TW11 0LW, UKJohn.Gallop@npl.co.uk

Andrei Irimia(Chapter 10)Department of Physics and Astronomy,Vanderbilt University, VU Station B351807, Nashville TN 37235, USAandrei.irimia@vanderbilt.edu

Mark N. Keene(Chapter 14)QinetiQ Ltd., St. Andrews Road,Malvern, Worcestershire WR14 3PS,UKmnkeene@qinetiq.com

Hans-Joachim Krause(Chapter 13)Institute of Thin Films and Interfaces,Research Center J�lich, 52425 J�lich,Germanyh.-j.krause@fz-juelich.de

Adrian T. Lee(Chapter 8)Department of Physics, University ofCalifornia, 363 LeConte Hall, BerkeleyCA 94720-7300, USAatl@physics7.berkeley.edu

Eduardo Andrade Lima(Chapter 10)Department of Biomedical Engineering,Vanderbilt University, VU Station B351807, Nashville TN 37235, USAeduardo.a.lima@vanderbilt.edu

Michael M�ck(Chapter 8)Institute of Applied Physics, Universityof Giessen, Heinrich-Buff-Ring 16,35392 Giessen, GermanyMichael.Mueck@ap.physik.uni-giessen.de

Jukka Nenonen(Chapter 11)Laboratory of Biomedical Engineering,Helsinki University of Technology,Espoo, FinlandJukka.Nenonen@neuromag.fi

Ho Jung Paik(Chapter 15)Department of Physics, University ofMaryland, College Park MD 20742,USAhpaik@physics.umd.edu

Fran�ois Piquemal(Chapter 9)Bureau National de M�trologie, LNE:Laboratoire National de M�trologie etd’Essais, Avenue Roger Hennequin 29,78197 Trappes cedex, Francefrancois.piquemal@lne.fr

Paul L. Richards(Chapter 8)Department of Physics, University ofCalifornia, 363 LeConte Hall, BerkeleyCA 94720-7300, USArichards@physics.berkeley.edu

Lutz Trahms(Chapter 11)Department of Bioelectricity andBiomagnetism, Physikalisch-Technische Bundesanstalt,Abbestr. 2–12, 10587 Berlin, Germanylutz.trahms@ptb.de

List of Contributors

Jiri Vrba(Chapter 11)VSM MedTech Ltd., 9 Burbidge Street,Coquitlam, B.C., Canadajvrba@vsmmedtech.com

Frederick C. Wellstood(Chapter 12)Center for Superconductivity Research,Department of Physics, University ofMaryland, College Park MD 20742-4111, USAwell@squid.umd.edu

John P. Wikswo(Chapter 10)Departments of BiomedicalEngineering, Physics and Astronomy,Molecular Physiology and Biophysics,Vanderbilt University, VU Station B351807, Nashville TN 37235, USAjohn.wikswo@vanderbilt.edu

List of ContributorsXVIII

1

8SQUID Voltmeters and AmplifiersJohn Clarke, Adrian T. Lee, Michael M�ck and Paul L. Richards

8.1 Introduction 38.2 Voltmeters 48.3 The SQUID as a Radiofrequency Amplifier 58.3.1 Introduction 58.3.2 Mutual Interaction of SQUID and Input Circuit 68.3.3 Tuned Amplifier: Theory 108.3.4 Untuned Amplifier: Theory 128.3.5 Tuned and Untuned Amplifiers: Experiment 138.3.6 To Tune or Not to Tune? 168.3.7 SQUID Series Array Amplifier 178.3.8 The Quantum Limit 188.3.9 Future Outlook 198.4 Microstrip SQUID Amplifier 208.4.1 Introduction 208.4.2 The Microstrip 218.4.3 The Microstrip SQUID Amplifier: Gain 218.4.4 The Microstrip SQUID Amplifier: Noise Temperature 268.4.5 High-Tc Microstrip SQUID Amplifier 318.4.6 Future Outlook 318.5 SQUID Readout of Thermal Detectors 328.5.1 Introduction 328.5.2 Transition-Edge Sensors 338.5.3 SQUID Multiplexers 358.5.3.1 Time-Domain Multiplexing 358.5.3.2 Frequency-Domain Multiplexing 398.5.4 TES Bolometers 458.5.4.1 TES Bolometer Designs 468.5.4.2 Bolometer Performance 498.5.5 TES Calorimeters and Nonequilibrium Detectors 508.5.5.1 Calorimeter Designs 518.5.5.2 Calorimeter Noise Performance 52

8.5.6 SQUID Readout of Non-TES Detectors 538.5.6.1 Magnetic Calorimeter 538.5.6.2 SIS Tunnel Junction 548.5.6.3 NIS Junctions 558.5.6.4 Kinetic-Inductance Thermometer 558.5.7 Future Outlook 568.6 Nuclear Magnetic and Quadrupole Resonance and

Magnetic Resonance Imaging 568.6.1 Introduction 568.6.2 Principles of NMR and NQR 578.6.3 SQUID-Detected NMR and NQR 618.6.3.1 NQR of 14N 618.6.3.2 Spin Noise 648.6.3.3 NMR of Hyperpolarized 129Xe 678.6.3.4 Liquid-State Proton NMR and MRI 698.6.4 Future Outlook 808.7 The Axion Detector 81

8 SQUID Voltmeters and Amplifiers2

8.1Introduction

Volume I of this handbook is concerned with the theory, fabrication and perfor-mance of dc and rf SQUIDs, and with the implementation of SQUIDs as magnet-ometers and gradiometers using appropriate superconducting input circuits.Most of these devices are used at frequencies ranging from zero to a few kilohertz,for example for quasistatic measurements of susceptibility, for geophysical appli-cations and for biomagnetism. In this chapter we are concerned with the use ofSQUIDs as voltmeters and amplifiers. Since rf SQUIDs are almost never used forsuch purposes, we confine ourselves to dc SQUIDs.Broadly speaking, we can divide these applications into three frequency ranges.

The first is the measurement of quasistatic voltages – for example, thermoelectricvoltage and the voltage generated by quasiparticle charge imbalance in a super-conductor. These voltmeters are described briefly in Section 8.2. The second fre-quency range extends from a few tens or hundreds of hertz to perhaps 100 MHz,and is discussed in Section 8.3. Major applications include readout schemes forbolometers and calorimeters for particle detectors, discussed in Section 8.5, andnuclear magnetic resonance (NMR) , nuclear quadrupole resonance (NQR) andmagnetic resonance imaging (MRI), discussed in Section 8.6. In the frequencyrange up to a few megahertz, the SQUID is generally operated in a flux–lockedloop, while at higher frequencies it is operated open–loop, with an applied fluxnear (2n + 1)U0/4 (U0 = h/2e » 2.07 � 10–15 Wb is the flux quantum and n is aninteger) chosen to maximize the flux–to–voltage transfer coefficient (¶V/¶U)I ” VU.At frequencies up to, say, 100 MHz, the conventional square–washer SQUIDdesign described in Chapter 5 is entirely adequate. In the third range of frequen-cies, a few hundred megahertz to a few gigahertz, however, the parasitic capaci-tance between the input coil and the SQUID washer can substantially reduce thegain of the conventional design. An alternative option is the so–called microstripSQUID amplifier, in which the input coil is used as a resonant microstrip. Thisdevice is described in Section 8.4. Applications of the microstrip amplifier includethe axion detector described in Section 8.7 and a postamplifier for the radiofre-quency single–electron transistor (RFSET).

8.1 Introduction 3

8.2Voltmeters

One of the earliest applications of the dc SQUID was as a voltmeter. The sensorwas in fact a SLUG (superconducting low-inductance undulatory galvanometer)[1] described briefly in Chapter 5. In essence, the SLUG consists of a bead ofPbSn solder frozen around a length of Nb wire. The critical current measured be-tween the two superconductors is periodic (often multiply periodic) in the super-current passed along the Nb wire. In the early days of this device, it was possibleto measure changes in this current of about 1 lA Hz–1/2. The fact that the inputcircuit had a low inductance – a few nanohenries – enabled one to measure volt-ages developed by much smaller resistances than had been previously possiblesince the time constant of the measurement could be kept to below one second.Figure 8.1 shows the original voltmeter circuit used with a SLUG. The voltage

source Vs was connected in series with a standard resistor rs and the Nb wire ofthe SLUG. The SLUG was operated in a flux-locked loop (Section 4.2) that fed acurrent is into rs to maintain a null current in the Nb wire: evidently the value ofVs is given by isrs. With a SLUG current resolution of 1 lA Hz–1/2 determined bythe readout electronics, the voltage resolution for rs = 10–8 X was 10–14 VHz–1/2.This represented a five orders of magnitude improvement over the resolution ofsemiconductor amplifiers. Since the Nyquist voltage noise across a 10–8 X resistanceat 4.2 K is about 1.5 fV Hz–1/2, these early measurements were not Nyquist noiselimited. Nonetheless, the SLUG voltmeter was used successfully to make mea-surements of the characteristics of superconductor–normal metal–superconduc-tor (SNS) Josephson junctions [2] and of thermoelectric voltages [3]. Subsequently,it was used in studies of the resistance of the SN interface [4] and to make the firstmeasurements of quasiparticle charge imbalance in superconductors [5].

4 8 SQUID Voltmeters and Amplifiers

5 mm

Copper wire

Niobium wire

Solder

Vs

rs

V

I

I

is is

From output of

flux-locked loop

Fig. 8.1 The SLUG. The configuration of a voltmeter measuringa voltage source Vs has been superimposed on a photograph.

8.3 The SQUID as a Radiofrequency Amplifier

The development of much lower noise SQUIDs with multiturn input coils,notably the Ketchen Jaycox square-washer design [6], has greatly reduced theequivalent current noise. For example, for a low-Tc SQUID with a flux noise of2 � 10–6U0 Hz–1/2 at frequencies above the l/f knee (f is frequency) of typically1 Hz, coupled to an input coil with a mutual inductance of 5 nH, the currentnoise S1=2I (f ) » 1 pA Hz–1/2. At 4.2 K, this resolution enables one to make Nyquist-noise-limited measurements in resistors

r & 4kBT/SI(f ) » 200 X . (8.1)

In making this estimate, we have neglected the effects of current noise in theSQUID loop which induces noise voltages into the input circuit. This subject isdiscussed at length in Section 8.3. These devices are generally used with currentfeedback to the standard resistor to obtain a null balancing voltmeter [7].Voltmeters have also been based on high-Tc SQUIDs operating at 77 K [8–10].

The unavailability of flexible, bondable wire made from a high-Tc superconductormeans that normal wire must be used to connect the components in the inputcircuit. Contact resistance between this wire and the YBa2Cu3O7–x (YBCO) inputcoil adds to the total resistance. As a result, the voltage resolution is limited toroughly 1 pV Hz–1/2.SQUID packages suitable for use as voltmeters are available commercially from

several companies. The SQUID is enclosed in a niobium can to shield it fromambient magnetic noise. The two ends of the input coil are connected to niobiumpads to which external niobium wires can be clamped with screws to producesuperconducting contacts. Thus, the user can readily couple any desired externalcircuit to the SQUID.

8.3The SQUID as a Radiofrequency Amplifier

8.3.1Introduction

This section is concerned with the use of the dc SQUID as a radiofrequency (rf)amplifier. We confine our attention to the situation in which the SQUID is oper-ated open loop, biased near (2n + 1)U0/4 to maximize VU. A thorough discussionof such amplifiers is quite complicated. Although these issues are often ignoredin the design of SQUID input circuits, the coupling of a circuit to a SQUID maymodify its properties significantly, while at the same time the SQUID reflectsboth a nonlinear impedance and a voltage noise source into the input circuit. Themodification of the SQUID by a coupled inductance was pointed out by Zimmer-man [11], and studied extensively in a series of papers by Clarke and coworkers[12–14]. The fact that the SQUID loop contains a noise current that is partiallycorrelated with the voltage noise [15] across the SQUID was computed by Tesche

5

8 SQUID Voltmeters and Amplifiers

and Clarke [16], and subsequently used by various authors to calculate the noisetemperature of amplifiers [13, 14, 17–20]. A complete treatment of these issueswould make this chapter unwieldy, and we limit ourselves to summarizing thekey theoretical results and to describing some experimental amplifiers.

8.3.2Mutual Interaction of SQUID and Input Circuit

Consider a SQUID with loop inductance L and two identical Josephson junctionseach with critical current I0, self capacitance C and shunt resistance R. For a typi-cal SQUID in the 4He temperature range, the noise parameter C ” 2pkBT/ I0U0 ~0.05. The noise energy e(f ) ” SU(f )/2L is optimized [15] when bL ” 2LI0/U0 = 1.The Stewart–McCumber parameter [21, 22] bC ” 2pI0R2C/U0 should be somewhatless than unity to avoid hysteresis in the current voltage (I–V) characteristic (seeChapters 1 and 2). Under these conditions, one finds the following results [15].The maximum flux-to-voltage transfer coefficient is

VU ” ¶Vj =¶UjIB » R=L (8.2)

where IB is the value of the bias current that maximizes VU, and the flux in theSQUID is near (2n + l)U0/4 (n is an integer). When VU is maximized, the spectraldensity of the voltage noise across the SQUID, which is assumed to arise fromNyquist noise in the shunt resistors, is [15]

SV ðf Þ » 16 kBTR . (8.3)

The current noise in the SQUID loop has a spectral density [16]

SJðf Þ » 11 kBT=R (8.4)

and is partially correlated with the voltage noise with the cross spectral density[16]

SVJðf Þ » 12 kBT . (8.5)

Figure 8.2(a) shows an input circuit consisting of a voltage source Vi in series witha resistance Ri, the inductance Lp of a pickup coil, a stray inductance Ls, a capaci-tor Ci and the input inductance Li of the SQUID. Depending on the application,some of the components may be omitted. The mutual inductance to the SQUIDis Mi = ki(LLi)1/2, where ki £ 1 is the coupling coefficient. The SQUID reflects acomplex impedance into the input circuit which is derived from the dynamicinput impedance Z of the SQUID; in turn, Z can be related to the flux-to-currenttransfer function JU ” (¶J/¶U)IB by the equation [18]

–JU = jx/Z = 1/L + jx/R . (8.6)

6

8.3 The SQUID as a Radiofrequency Amplifier

The parameters Z, L and R refer to currents flowing around the SQUID loop. Atx = 0, –JU reduces to the inverse of the dynamic input inductance L, while forx > 0 there are resistive losses, represented by the dynamic input resistance R.Figure 2(b) shows a schematic representation of L and R, which define theresponse of the SQUID to an applied flux U.Figure 8.3 shows the variation of L/L and R/R with applied flux [13] for four

values of bias current. Typically, SQUIDs are operated with IB » 2I0. We observethat both parameters depend strongly on U, with L/L becoming negative in someregions.We next discuss the effect of the input circuit on the SQUID parameters.

Throughout this discussion we assume that the SQUID is operated open-loop,with its current and flux biases adjusted to maximize VU. We also assume that theloading of the readout amplifier on the SQUID is negligible. To illustrate thepoint, consider a superconducting pickup inductance Lp in series with a strayinductance Ls connected across the input inductance Li, as in a magnetometer. Weassume that the SQUID is current-biased at a voltage corresponding to a Joseph-son angular frequency xJ. In the absence of parasitic capacitance, currents in the

7

o

Fig. 8.2 (a) Schematic of a generic tunedamplifier. The voltage source Vi is connectedin series with a pickup loop of inductance Lp,a stray inductance Ls, a capacitor Ci, a resis-tance Ri and the input inductance Li of theSQUID. (b) Dynamic input impedance of the

SQUID represented by an inductance L andresistanceR. In both figures, J is the currentinduced in the SQUID loop by signal andnoise sources in the input circuit. (Repro-duced with permission from ref. [13].)

Fig. 8.3 Simulated values of L/L and R/R. versusreduced flux U/U0 for a bare SQUID versus flux Ufor four values of bias current. SQUID parameterswere bL= 1.0, bC = 0.2 and C » 0.06 (Reproducedwith permission from ref. [13].)

8 SQUID Voltmeters and Amplifiers

SQUID loop at xJ and its harmonics will induce currents into the input circuit. Itis easy to show that the SQUID loop inductance will be reduced by the presence ofthe input circuit to a value

Lr ¼ ð1� k2ieÞL , (8.7)

where

kie ¼ ki½ðLi þ Lp þ LsÞ=Li��1=2 (8.8)

is the effective coupling coefficient between the SQUID and the total inductanceof the input circuit. Other parameters of the SQUID take the reduced values V rU,JrU, Zr and Rr corresponding to a SQUID with loop inductance Lr. In practice,things may be not so simple: parasitic capacitance between the coil and theSQUID washer modifies the coupling between them at the Josephson frequencyand its harmonics. In the limiting case where this parasitic capacitance preventsany high-frequency currents from flowing in the input circuit, the SQUID param-eters are unaffected by the input circuit [12]. In a real system, the result is likely tobe somewhere between the two extremes; it will also depend, for example, on thenumber of turns in the input coil which determines the parasitic capacitance. Bystudying a series of SQUIDs with 20-turn input coils, Hilbert and Clarke [13]found that VU was increased by roughly the expected amount (corresponding tothe reduced loop inductance) when the previously open coil was shorted.We are now in a position to consider the modification of the input circuit by the

SQUID impedance reflected into it. A productive way of writing the result is interms of the output voltage across the SQUID in the presence of a signal appliedto the input circuit shown in Figure 8.2(a). After some calculation, one finds [12]

VðxÞ ¼ V rNðxÞ þMiV rUViðxÞ þMi JrNðxÞðRi þ l=jxCiÞ=LTZTðxÞ � JrU M2i ðRi þ l=jxCiÞ=LT

� �. (8.9)

Here, V rN(x) and JrN(x) are the reduced voltage and current noises of the SQUID,

and Vi(x) is the input voltage applied to the resistance Ri and capacitance Ci inseries with the total inductance of the input circuit LT = Li + Lp + Ls. The totalimpedance of the (uncoupled) input circuit is

ZTðxÞ ¼ Ri þ jxðLi þ Lp þ LsÞ þ l=jxCi . (8.10)

The denominator of Eq. (8.9) contains the term �JrUM2i ðRi þ l=jxCiÞ=LT that rep-resents the impedance reflected into the input circuit from the SQUID. The terminvolving JrNðxÞ in the square brackets is the noise current generated in the inputcircuit by the SQUID. We can readily derive the voltage gain for a SQUID ampli-fier from Eq. (8.9):

Gv ¼MiV

rU

ZT � k2ieLJrUðRi þ l=jxCiÞ”MiV

rU

Z*T. (8.11)

8

8.3 The SQUID as a Radiofrequency Amplifier

Using Eq. (8.6), we see that the impedance can be written in the form

Z*T ¼ Ri 1þk2ieL

Lr� �

þ k2ieL

RrCiþ jx Li þ a2eL

RiRr �

1x2CiLr

� �þ Lp þ Ls

� �þ 1jxCi

:

(8.12)

We note that Rr and Lr contribute to both the real and imaginary parts of Z*T.Hilbert and Clarke [13] made extensive measurements of JrU as a function of U

by connecting a capacitor Ci across the input coil of both a 4-turn and a 20-turnSQUID. The resonant frequency f 00 and full width at half maximum (FWHM) ofthis tank circuit were measured directly with the SQUID biased with a large cur-rent (>> 2I0) where the SQUID has negligible inductive screening and a dynamicinput impedance of approximately 2Rr. The SQUID was then biased at its usualoperating point, and the Nyquist noise power P(f0) generated by the tank circuitwas measured at the output of the SQUID with a spectrum analyzer. The value ofV rU was determined from the height of the peak. The frequency f0 at which thenoise power peaked generally differed from f 00 , and yielded the inductance changeDLi reflected into the tank circuit. Similarly, the value of the FWHM, Df, yieldedthe resistance change DRi. From these values, it was straightforward to infer thevalues of L/Lr and R/Rr from Eq. (8.12) inserted into Eq. (8.11).The results for a 20-turn SQUID are shown in Figure 8.4 for three values of bias

current. The behavior of jV rUj is much as expected. However, the magnitudes ofthe curves suggest that the inductive screening of the SQUID is more effective atthe lowest bias current than at the highest bias current; this result is consistentwith a reduction in screening by the parasitic capacitance as the Josephson fre-quency increases. The measured values of L/Lr follow the trends in the simula-tions quite well. For example, at the lowest bias current there is a broad maximumat U = 0 and a negative region around U = –U0/2. The overall magnitude of L/Lr

9

Φ/Φ0Fig. 8.4 Measured values of L/ L r, R/R r and |V rU|L/R versus reduced flux U/U0for a SQUID with L » 400 pH, 2I0 » 6 – 1 lA, C » 0.5 pF and R » 8 X, correspondingto bL » 1 and bC » 0.2. The temperature was 4.2 K corresponding to C » 0.06. Biascurrent: (a) 4.0 lA, (b) 5.0 lA, (c) 6.0 lA. (Reproduced with permission from ref. [13].)

8 SQUID Voltmeters and Amplifiers

is generally in fair agreement with the simulations. On the other hand, the mea-sured values of R/Rr, which vary between +30 and –5, are in sharp disagreementwith the simulated values, which are always positive, with a maximum of about 2.Thus, DRi is evidently dominated by a mechanism other than the resistancereflected from the SQUID.Further investigation showed that the change in resistance in the tank circuit

was dominated by feedback from the output of the SQUID via the parasitic capac-itance between the washer and the input coil. Approximating the distributed ca-pacitance with a lumped capacitor, Hilbert and Clarke [13] were able to accountfor the observed change in resistance in the input circuit to within a factor of 2.The reader is referred to the original paper for details.This concludes our discussion of the input impedance of the dc SQUID and of

the mutual loading of the SQUID and input circuit. We next apply these ideas tothe design of SQUID amplifiers.

8.3.3Tuned Amplifier: Theory

To simplify our initial discussion we first neglect capacitive feedback, and laterreturn briefly to this issue. The circuit is shown schematically in Figure 8.2(a); weinterpret Ri as the impedance of the voltage source Vi. For Vi = 0, the noise voltageat the SQUID output can be written from Eqs. (8.8) and (8.9) as

VNðxÞ ¼ V rNðxÞ þ k2ieLV rUðRi þ l=jxCiÞJrNðxÞ=Z*TðxÞ (8.13)

where Z*TðxÞ is given by Eq. (8.12). We now assume that the amplifier is operatedat the resonant frequency f0 = x0/2p at which the imaginary terms in Z

*T tune to

zero:

x0 ¼ ½ðLi þ Ls þ k2ieLRi=RrÞCi=ð1þ k2ieL=LrÞ��1=2. (8.14)

Thus, at the resonant frequency, Eq. (8.13) reduces to

VNðx0Þ ¼ V rNðx0Þ þk2ieL Ri þ 1=jx0Cið ÞJrN x0ð ÞV rU

Ri þ DRi(8.15)

where

DRi ¼ k2ieLðRi=Lr þ 1=RrCiÞ . (8.16)

To simplify matters, we now assume that Q is high so that Ri