SQUID Readout Electronics and Magnetometric Systems for Practical Applications 3527344888, 9783527344888

Table of contents :
Cover
SQUID Readout Electronics and Magnetometric Systems for Practical
Applications
Copyright
Contents
Preface
Acknowledgments
1 Introduction
2 Josephson Junctions
3 dc SQUID’s I–V Characteristics and Its Bias Modes
4 Functions of the SQUID’s Readout Electronics
5 Direct Readout Scheme (DRS)
6 SQUID Magnetometric System and SQUID Parameters
7 Flux Modulation Scheme (FMS)
8 Flux Feedback Concepts and Parallel Feedback Circuit
9 Analyses of the “Series Feedback Coil (Circuit)” (SFC)
10 Weakly Damped SQUID
11 Two-Stage and Double Relaxation Oscillation Readout
Schemes
12 Radio-Frequency (rf) SQUID
Index

Citation preview

SQUID Readout Electronics and Magnetometric Systems for Practical Applications

SQUID Readout Electronics and Magnetometric Systems for Practical Applications Yi Zhang Hui Dong Hans-Joachim Krause Guofeng Zhang Xiaoming Xie

Authors

Forschungszentrum Jüelich (Retired) Institute of Biological Information Processing Wilhelm-Johnen-Straße 52428 Jüelich Germany

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Prof. Dr. Hui Dong

Library of Congress Card No.: applied for

Prof. Dr. Yi Zhang

Shanghai Institute of Microsystem and Information Technology 865 Changning Road 200050 Shanghai China Prof. Dr. Hans-Joachim Krause

Forschungszentrum Jüelich Institute of Biological Information Processing Wilhelm-Johnen-Straße 52428 Jüelich Germany Dr. Guofeng Zhang

Shanghai Institute of Microsystem and Information Technology 865 Changning Road 200050 Shanghai China Prof. Dr. Xiaoming Xie

Shanghai Institute of Microsystem and Information Technology ShanghaiTech University University of Chinese Academy of Sciences 865 Changning Road 200050 Shanghai China

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at . © 2020 Wiley-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Print ISBN: 978-3-527-34488-8 ePDF ISBN: 978-3-527-81650-7 ePub ISBN: 978-3-527-81652-1 oBook ISBN: 978-3-527-81651-4

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v

Contents Preface ix Acknowledgments xi 1 1.1 1.2

Introduction 1

Motivation 1 Contents of the Chapters 3 References 8

2

Josephson Junctions 9

2.1 2.2

Josephson Equations 9 RCSJ Model 9 References 13

3

dc SQUID’s I–V Characteristics and Its Bias Modes 15

3.1 3.2 3.3

SQUID’s I–V Characteristics 15 An Ideal Current Source 19 A Practical Voltage Source 19 References 21

4

Functions of the SQUID’s Readout Electronics 23

4.1 4.2 4.2.1 4.2.2

Selection of the SQUID’s Bias Mode 23 Flux Locked Loop (FLL) 23 Principle of the FLL 24 Electronic Circuit of the FLL and the Selection of the Working Point 25 “Locked” and “Unlocked” Cases in the FLL 28 Slew Rate of the SQUID System 29 Suppressing the Noise Contribution from the Preamplifier 29 Two Models of a dc SQUID 29 References 31

4.2.3 4.2.4 4.3 4.4

5

Direct Readout Scheme (DRS) 33

5.1 5.2

Introduction 33 Readout Electronics Noise in DRS

33

vi

Contents

5.2.1 5.2.2 5.3 5.3.1 5.3.2 5.4

Noise Characteristics of Two Types of Preamplifiers 34 Noise Contribution of a Preamplifier with Different Source Resistors 37 Chain Rule and Flux Noise Contribution of a Preamplifier 39 Test Circuit Using the Same Preamplifier in Both Bias Modes 40 Noise Measurements in Both Bias Modes 42 Summary of the DRS 43 References 43

6

SQUID Magnetometric System and SQUID Parameters 45

6.1 6.2

Field-to-Flux Transformer Circuit (Converter) 45 Three Dimensionless Characteristic Parameters, 𝛽 c , Γ, and 𝛽 L , in SQUID Operation 48 SQUID’s Nominal Stewart-McCumber Characteristic Parameter 𝛽 c 49 SQUID’s Nominal Thermal Noise Parameter Γ 52 SQUID’s Screening Parameter 𝛽 L 54 Discussion on the Three Characteristic Parameters 55 References 56

6.2.1 6.2.2 6.2.3 6.2.4

7

Flux Modulation Scheme (FMS) 61

7.1 7.2 7.2.1 7.2.2 7.2.3 7.2.4 7.3 7.3.1 7.3.2 7.3.3 7.3.4 7.3.5 7.4

Mixed Bias Modes 61 Conventional Explanation for the FMS 63 Schematic Diagram of the FMS 63 Time Domain and Flux Domain 65 Flux Modulation 66 Five Additional Notes 71 FMS Revisited 73 Bias Mode in FMS 74 Basic Consideration of Synchronous Measurements of I s and V s Experimental Synchronous Measurements of Δi and VR s 75 Transfer Characteristics of the Step-Up Transformer 78 V (Φ) Comparison Obtained by DRS and FMS 80 Conclusion 81 References 82

8

Flux Feedback Concepts and Parallel Feedback Circuit 85

8.1 8.2 8.3 8.3.1 8.3.2 8.3.3 8.4

Flux Feedback Concepts and History 85 SQUID’s Apparent Parameters 87 Parallel Feedback Circuit (PFC) 89 Working Principle of the PFC in Current Bias Mode 89 Working Principle of PFC in Voltage Bias Mode 94 Brief Summary of Qualitative Analyses of PFC 97 Quantitative Analyses and Experimental Verification of the PFC in Voltage Bias Mode 99 The Equivalent Circuit with the PFC in Voltage Bias Mode 99 Introduction of Two Dimensionless Parameters r and Δ 101

8.4.1 8.4.2

74

Contents

8.4.3 8.4.4 8.4.5 8.4.6 8.5 8.6

Numerical Calculations 103 Experimental Results 108 Noise Comparison and Interpretation 111 Two Practical Designs for PFC 114 Main Achievements of PFC Quantitative Analysis 116 Comparison with the Noise Behaviors of Two Preamplifiers 117 References 119

9

Analyses of the “Series Feedback Coil (Circuit)” (SFC)

9.1 9.1.1 9.1.2 9.2 9.3 9.4 9.4.1 9.4.2 9.4.3

121 SFC in Current Bias Mode 121 Working Principle of the SFC in Current Bias Mode 121 Noise Measurements of a Weakly Damped SQUID (Magnetometer) System with the SFC 123 The SFC in Voltage Bias Mode 125 Summary of the PFC and SFC 127 Combination of the PFC and SFC (PSFC) 129 PSFC Analysis Under Independence Conditions 129 PSFC Experiments and Results 132 Conclusion of the PSFC 136 References 137

10

Weakly Damped SQUID 139

10.1 10.2 10.3 10.4 10.4.1 10.4.2 10.4.3

Basic Consideration of Weakly Damped SQUID 139 SQUID System Noise Measurements with Different 𝛽 c Values 140 Statistics of SQUID Properties 143 Single Chip Readout Electronics (SCRE) 147 Principle of SCRE and Its Performance 148 Equivalent Circuit of SCRE 149 Differences Between the Conventional Version of Readout Electronics with an Integrator and SCRE 152 Two Applications of SCRE 153 Suggestions for the DRS 154 References 155

10.4.4 10.5

11

Two-Stage and Double Relaxation Oscillation Readout Schemes 157

11.1 11.2 11.3

Two-Stage Scheme 158 ROS and D-ROS 164 Some Comments on D-ROS and Two-Stage Scheme 168 References 169

12

Radio-Frequency (rf) SQUID 171

12.1 12.2 12.2.1

Fundamentals of an rf SQUID 171 Conventional rf SQUID System 176 Block Diagram of rf SQUID Readout Electronics (the 30 MHz Version) 176

vii

viii

Contents

12.2.2 12.3 12.3.1

rf SQUID System Noise in the 30 MHz Version 178 Introduction to Modern rf SQUID Systems 180 Magnetometric Thin-Film rf SQUID and a Conventional Tank Circuit with a Capacitor Tap 181 12.3.2 Improved rf SQUID Readout Electronics 184 12.3.3 Tank Circuit Operating Up to 1 GHz with Inductive Coupling 188 12.3.4 Modern rf SQUID System 190 12.3.4.1 Microstrip Resonator 190 12.3.4.2 Coplanar Resonator 192 12.3.4.3 Instability of rf Bias Current 194 12.3.5 Substrate Resonator 196 12.3.6 Regarding the rf SQUID’s Thermal Noise Limit 200 12.4 Further Developments of the rf SQUID Magnetometer System 201 12.4.1 Achievement of a Very Large 𝜕V rf /𝜕Φ in a Low-Impedance System 201 12.4.2 Multiturn Input Coil for a Thin-Film rf SQUID Magnetometer with a Planar Labyrinth Resonator 204 12.4.3 Modern rf SQUID Electronics 208 12.5 Multichannel High-T c rf SQUID Gradiometer 211 12.6 Comparison of rf SQUID Readout with dc SQUID Readout 214 12.7 Summary and Outlook 215 References 218 Index 225

ix

Preface Time flies! Thirteen years ago, as a research professor at Shanghai Institute of Microsystem and Information Technology (SIMIT), Chinese Academy of Sciences (known as Shanghai Institute of Metallurgy by that time), I was charged with a challenging mission, to start a team on superconducting electronics research. From the institute, it was a quite straightforward decision, as the whole institute had been gradually shifting from materials science research toward electronics and systems. And for myself, it was not so easy to start something new at the age over 40, with a strong background on superconducting materials, some basic knowledge on electronics but little on superconducting electronics. Just when I was wondering how to do that, Prof. P.H. Wu, a member of Chinese Academy of Sciences, a famous professor in the field of superconducting electronics in China, who had worked at Research Center Julich (FZJ) Germany, recommended me Dr. Yi Zhang, a reputable German scientist at FZJ, born in Shanghai, acknowledged globally for his excellent research on the development of high T c radio-frequency superconducting quantum interference devices (high temperature superconducting [HTS] rf SQUIDs), their readout electronics and systems. I contacted FZJ without hesitation, inviting Yi to act as a consultant to our first project on SQUID-based Magnetocardiography (MCG) system. This request letter opened the door of cooperation between SIMIT and FZJ. To date, our cooperation has developed from a project collaboration between two professors to the establishment of two joint research laboratories and further to a virtual joint research institute. The cooperation also has been extended from superconductivity to topological insulators and quantum computing. After some formal procedures, I got the approval of my request letter from Prof. Dr. Joachim Treusch, the former chairman of the board of directors of FZJ, Prof. Dr. Sebastian Schmidt, a current member of the board of directors of FZJ, and Prof. Dr. Andreas Offenhäusser, director of IBN2 (Institute of Bio and Nanoscience, now Institute of Biological Information Processing), which Yi belonged to. Besides the support from the top management, the involvement of Prof. Dr. Hans-Joachim Krause, team leader of magnetic sensors in IBN2, was another important step for our successful cooperation. Our joint research on dc SQUID started from the development of asymmetrical SQUID characteristics, in an attempt to simply SQUID readout and system

x

Preface

design. The adventure was full of excitement and frustration. Early in the morning, we sat together, planning the work of the day, late in the evening, we summarized our results from the notes we made during the day, sometimes exciting progress, sometimes frustrating results, and sometimes confusing results which we could not describe easily. I still remember how excited we were when we first observed the asymmetrical flux-current characteristics of a SQUID on the oscilloscope, and I remembered as well how much we were frustrated when we learnt that the desired asymmetrical characteristics did not lead to the lower noise we had sought for so long. The notes piled up day after day, getting thicker and thicker, we called them “Rabe’s Diary.” After numerous discussions back and forth, we succeeded in interpreting our results, which led to our first joint publication and our joint patent on the so-called “SQUID Bootstrap Circuit,” and to many other joint publications in the following 10 more years. The SQUID research was more difficult than we first thought because setting up SQUID systems for applications requires the involvement of people from several different disciplines. A complete understanding for SQUID systems needs comprehensive knowledge not only in quantum physics and low-temperature physics but also in material science and electronics engineering. In fact, electrical or electronics engineers are always needed for system development. Therefore, it is very important to establish a common language that is easily accessible for all people. That was how we got the idea to write this book. Yi Zhang contributed most to writing of this book, with his experience in SQUID research for 34 years, including more than 10 years of joint research with SIMIT. We have aimed to write this book in a way that is easily understandable for engineers and students, in order to overcome the formidable barrier of “quantum” physics. In this book, e.g. dc SQUIDs are simply treated as resistor-like elements, which are modulated by the magnetic flux. We hope that this book will be appreciated by all people interested in developing and working with SQUIDs and SQUID systems. By inviting engineers into the SQUID “family,” we will have a better chance to transform SQUID from a laboratory toy to an enabling technology that will eventually shape our life. This book is largely a documentation of the joint achievements accomplished in the cooperation between SIMIT and FZJ in the field of superconducting electronics. We believe that the ongoing collaboration between the two parties will continue to grow, and the cooperation will bring more achievements not only in the field of superconducting electronics but also in other fields in the future. November 2019.

Xiaoming Xie Shanghai, China

xi

Acknowledgments It is our pleasure to acknowledge the generous assistance that has been offered throughout the preparation of this book. Without such help, our task would not have been possible. We owe a special debt of gratitude to all the colleagues from China and Germany who have contributed to the works mentioned in this book. Special thanks to Dr. M. Mück for your constructive comments on Chapter 6. Very special thanks to Dr. H. Soltner for the English language reading. We finally express our heartfelt gratitude to Wiley.

1

1 Introduction 1.1 Motivation Superconducting QUantum Interference Devices (SQUIDs) are well known because they are the most sensitive sensors for measuring magnetic flux. In magnetometry, a SQUID with a field-to-flux transformer circuit (converter) √ construct is a magnetometer with high field sensitivity in the range of fT/ Hz (one millionth of the earth’s magnetic field). Therefore, the study of SQUID systems has never stopped. Many books and reviews have elaborated on the SQUID principle and SQUID magnetometric systems as well as SQUID applications, e.g. “Superconductor Applications: SQUIDs and Machines” edited by B. B. Schwartz and S. Foner [1], “Physics and Applications of the Josephson Effect” edited by A. Barone and G. Paterno [2], and the NATO proceedings “SQUID Sensors: Fundamentals, Fabrication and Applications” edited by H. Weinstock [3]. In particular, “The SQUID Handbook,” edited in 2004 by John Clarke and Alex I. Braginski comprehensively summarizes SQUID’s theory and practice since SQUIDs have been discovered [4]. Hence, this book has become the new “bible” for researchers in the field. Furthermore, the review of “SQUID Magnetometers for Low-Frequency Applications” by Tapani Ryhänen et al. presented a novel formulation for SQUID operation and SQUID magnetometers for low-frequency applications, taking into account the coupling circuits and electronics [5]. Structurally, a direct current (dc) SQUID is a superconducting ring interrupted with two Josephson junctions. Predicatively, SQUIDs have very rich physical meanings, e.g. the Aharonov–Bohm effect, flux quantization, Meissner effect, Bardeen–Cooper–Schrieffer (BCS) theory, and the Josephson tunnel effect. However, starting from the view of electronic circuits, our first question is on what a dc SQUID is. In magnetometry, a dc SQUID should be regarded as a resistor-like element where its dynamic resistance is modulated by the flux Φ threading the SQUID’s loop. In the readout technique, the dynamic resistance of the SQUID, Rd (Φ) = 𝜕V /𝜕I, i.e. the derivative of the voltage with respect to current, is the fundamental readout quantity, which is embodied in the current–voltage (I–V ) characteristics of the SQUID. Here, the changing I–V characteristics are limited by two curves at the integer (upper limit) and half-integer (lower limit) of the flux quantum Φ0 , which reflect the quantity SQUID Readout Electronics and Magnetometric Systems for Practical Applications, First Edition. Yi Zhang, Hui Dong, Hans-Joachim Krause, Guofeng Zhang, and Xiaoming Xie. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

2

1 Introduction

of magnetic flux in the SQUID loop. There is already abundant “know-how” to read out a resistor R. For example, one can measure a voltage V across R with a constant current flowing through R or measure a current I through R when a constant voltage V is connected to R in parallel. A dc SQUID can either be operated at constant current by measuring the voltage across it (called current bias mode) or at constant voltage by measuring the current through it (called voltage bias mode). In either bias mode, only the SQUID’s V (Φ) or I(Φ) characteristics emerge. Similar to the change in I–V characteristics with the flux, V (Φ) and I(Φ) are also modulated by Φ. In brief, the essence of all three SQUID characteristics is recording the SQUID’s dynamic resistance changes, Rd (Φ). Generally, a SQUID system consists of the SQUID sensor and its readout electronics. The small SQUID signal leads to difficulty in reading out the SQUID’s signal without additional noise contributions from the readout technique. Conventionally, one hopes to suppress such noise contribution below the intrinsic SQUID noise δΦs . In other words, the measured system noise almost reaches δΦs . The main noise source in readout electronics is the preamplifier, which possesses two independent noise sources: the voltage noise V n and the current noise I n . Both of these noise sources are innate to the amplifier chip and cannot be changed. In order to compare these two noise contributions in a SQUID system, both types √ of electronic noise should be translated into a flux noise, δΦe , in units of Φ0 / Hz with SQUID’s transfer coefficient of 𝜕V /𝜕Φ or 𝜕I/𝜕Φ. In fact, the original SQUID parameters including the transfer coefficients are also innate to the particular SQUID and cannot be changed. However, the SQUID’s apparent parameters at the input terminal of the preamplifier can be modified. Over the past half century, people have developed different readout schemes, where the electronic noise δΦe is suppressed by increasing the apparent transfer coefficients once a preamplifier is selected. Indeed, the modification of the apparent parameters is the main thread running through the book. Here, we will change the perspective to discuss the optimization of the SQUID system noise, i.e. how to match the SQUID parameters with the readout electronics. According to the type of superconducting material used, SQUIDs can be divided into two groups: the low-temperature superconducting (LTS) SQUID, also called low-T c SQUID, usually operated at 4.2 K (liquid helium temperature); and the high-temperature superconducting (HTS) SQUID, also called high-T c SQUID, usually operated at 77 K (the liquid nitrogen temperature). The LTS material is typically niobium and HTS material is yttrium barium copper oxide (YB2 Cu3 O7−x ). However, according to the working principles, the dc SQUID mentioned above is completely different from the radio frequency (rf) SQUID, which is a superconducting ring interrupted with only one junction. To read the signal from an rf SQUID, it is inductively coupled to an rf tank circuit, which connects to the readout electronics. In this book, LTS (low-T c ) dc SQUID and HTS (high-T c ) rf SQUID systems, which are often used in magnetometry, will be highlighted. We will share our experiences and lessons, mostly from our own works, with readers, college students, and graduates in physics and engineering who have an interest in SQUID techniques, e.g. how to set up a simple SQUID system for themselves.

1.2 Contents of the Chapters

1.2 Contents of the Chapters The book is organized into 12 chapters, where most of the content (from Chapters 2–11) is about the dc SQUIDs, and only the last chapter is related to rf SQUIDs. However, the dc SQUID bias reversal scheme [6], the 1/f noise study [7, 8], and the special readout scheme for the nano-SQUID [9, 10] are not included. Chapter 1: This chapter is devoted to our motivation above and the subsequent chapter contents – why did we write this book, and what is it about? Chapter 2: Because the Josephson junction (JJ) is the key element of SQUIDs, Josephson’s equations should be first introduced. Then, JJs are analyzed with the resistively and capacitively shunted junction (RCSJ) model, thus introducing two important parameters: the Stewart–McCumber parameter 𝛽 c and the thermal rounding parameter Γ. To observe the features of JJs, one often uses the I–V characteristics, where the hysteresis behavior depends on the values of both 𝛽 c and Γ. Actually, the I–V characteristics describe the changing dynamic resistances Rd of the JJ, i.e. Rd = 𝜕V /𝜕I. It was experimentally verified that the value of Rd depends not only on the junction shunt resistor RJ but also on the junction critical current I c . Generally, JJs without hysteresis are suitable for SQUID operation. In fact, one habitually transforms the parameters 𝛽 c and Γ of the JJ into SQUID operation. Chapter 3: For readout electronics, the dc SQUID is regarded as dynamic resistance Rd (Φ) modulated by the flux threading into the SQUID loop. The SQUID’s I–V characteristics can be divided into three regions, and the SQUID is operated in the flux-modulated region (II). In fact, the behavior of Rd (Φ) is embodied in a SQUID’s I–V characteristics. To measure a resistance Rd , one can impress a known current (current bias) into a SQUID and observe the voltage across the SQUID’s dynamic resistance Rd . Alternatively, one can apply a constant voltage to the SQUID (voltage bias) and measure the current passing through Rd . Owing to the small Rd ≈ 10 Ω of the SQUID, an ideal current bias mode for SQUID operation can easily be realized. In contrast, an ideal voltage bias mode can hardly be achieved, as will be shown in the course of the chapter. Chapter 4: Almost all SQUID readout electronics developed over the past half century have a common feature: they establish a so-called flux-locked loop (FLL) to realize linearization of the output voltage V out (Φ) of the readout electronics; i.e. V out is proportional to the flux change Φ. In this chapter, the principle and realization of the FLL are explained. It is a nulling method where a compensation flux always follows the measured flux, thus resulting in a total flux change of zero in the SQUID loop. In the FLL, the concept of the working point W comes up, and the “locked” and “unlocked” cases are discussed. In the FLL, a small flux change ΔΦ near the working point W appears transiently, and a counter flux −ΔΦ immediately compensates it so that the SQUID is continuously operated at a constant flux state. Therefore, the SQUID’s Rd (Φ) near W can be expressed as Rd (Φ) = Rd + ΔRd , where Rd is considered a fixed resistance and ΔRd is a minor change with flux. According to the SQUID’s bias modes, ΔRd is translated into the readout quantity ΔV (or ΔI). For example, in practice, a current-biased SQUID can be regarded as a voltage source, ΔV = ΔΦ × (𝜕V /𝜕Φ), connecting to the fixed Rd in series (which seems to be the internal resistance

3

4

1 Introduction

of the voltage source), where (𝜕V /𝜕Φ) is the SQUID’s flux-to-voltage transfer coefficient at the working point W. The description of the SQUID by means of a differential dynamic resistance is a new model concept. Chapter 5: In the case of a direct readout scheme (DRS) where the SQUID directly connects to a preamplifier, the electronics noise δΦe is usually much larger than the SQUID intrinsic noise δΦs . Two types of preamplifiers, commercial op-amps (e.g. AD797 from Analog Devices Inc. or LT1028 from Linear Technology Corp.) and parallel-connected bipolar pair transistors (PCBTs) (e.g. 3 × SSM2210 or 3 × SSM2220 from Analog Devices Inc.), are the most commonly used. Here, the noise characteristics, V n and I n , of these two types of preamplifiers are measured separately. Nevertheless, a DRS exhibits several advantages; e.g. the SQUID’s original parameters can be directly determined, and the noise contributions from both sides, δΦe and δΦs , can be separately analyzed. Especially, the SQUID’s transfer coefficient 𝜕V /𝜕Φ (𝜕I/𝜕Φ) at the working point W plays two important roles: (i) √ it bridges different kinds of noise sources, thus unifying all noise in units of Φ0 / Hz, as the SQUID is a flux sensor; and (ii) a large transfer coefficient is beneficial for reducing δΦe . In fact, it was experimentally confirmed that the noise contribution of δΦe does not depend on the SQUID’s bias modes. Furthermore, for strongly damped SQUIDs, δΦe in DRS dominates the system noise δΦsys . Chapter 6: In a SQUID magnetometric system, one strives for a high magnetic field sensitivity δBsys , which involves two aspects: a field-to-flux transformer circuit (converter) and an ordinary SQUID system with an FLL. The former converts a magnetic field signal B into a flux Φ threading the SQUID loop, while the latter reads out the picked-up Φ. In Section 6.1, the requirements of the converter are discussed. In Section 6.2, we show that the SQUID system is characterized by three dimensionless parameters, 𝛽 c , Γ, and 𝛽 L . Note that the definitions of 𝛽 c and Γ for only a single JJ are given in Chapter 2. During SQUID operation, both parameters must be given a new connotation. Four SQUIDs with different 𝛽 c values were characterized. Here, a reasonable interpretation of the observed absence of hysteresis in the SQUID’s I–V characteristics at high 𝛽 c is given. For SQUID operation, the dimensionless parameter 𝛽 L particularly describes the modulation depth of the SQUID. Importantly, 𝛽 L ≈ 1 imposes a design condition on the product Ls I c – namely, all electrically readable values of SQUID parameters increase with increase in the SQUID’s nominal 𝛽 c . Chapter 7: The flux modulation scheme (FMS) was first introduced to the SQUID readout in 1968 and quickly became the standard readout technique for current-biased SQUIDs. To date, FMS electronics have been the most extensively used. The basic idea of the FMS is to perform an up-conversion of the SQUID’s voltage swing at the input terminal of the preamplifier with a step-up transformer, thus reducing the noise contribution of δΦe . In contrast to a DRS, where a dc circuit (amplifier and integrator) is employed, the FMS is an ac circuit, e.g. operating in the 100 kHz frequency range, because the transformer can pass only ac signals. However, a SQUID is often used to detect magnetic flux signals Φ with slow changes, even quasi-static signals. To resolve this challenge, a high-frequency modulation of the SQUID signal is employed in order to transform the low-frequency magnetic flux signal into the high-frequency

1.2 Contents of the Chapters

regime. After up-conversion, demodulation is employed to convert the flux signal back to the low-frequency regime, thus realizing the transitions between ac and dc circuits. If a SQUID is shunted by an element with impedance Zs (e.g. the transformer), a change in the bias mode occurs. We introduce a dimensionless parameter 𝜒 = Zs /Rd to quantitatively characterize the bias modes. In Section 7.1, we first introduce the so-called “mixed bias mode” concept. In Section 7.2, the FMS is discussed along with a conventional explanation. In Section 7.3, we revisit the FMS by analyzing the bias mode and the transfer characteristics of a step-up transformer. Chapter 8: The DRS with flux feedback circuits in the “head stage” at the cryogenic temperature is highlighted in this chapter and in Chapter 9. The chapter starts with a comprehensive comparison of the different feedback schemes that have been employed in the recent decades. The techniques of additional positive feedback (APF), bias current feedback (BCF), and noise cancellation (NC) are categorized and discussed. Generally, there are two typical kinds of flux feedback circuits, the parallel feedback circuit (PFC) described in Chapter 8 and the series feedback circuit (SFC), which will follow in Chapter 9. Indeed, we often use the differential chain rule of Rd = (𝜕V /𝜕I) = (𝜕V /𝜕Φ)/ (𝜕I/𝜕Φ) to analyze the flux feedback circuits. With the PFC, (Rd )PFC and (𝜕V /𝜕Φ)PFC increase synchronously, while (𝜕I/𝜕Φ)PFC = (𝜕I/𝜕Φ) remains constant. However, using the SFC, (Rd )SFC decreases with the simultaneous increase in (𝜕I/𝜕Φ)SFC because (𝜕V /𝜕Φ)SFC = (𝜕V /𝜕Φ). In fact, the large (𝜕V /𝜕Φ)PFC has the benefit of suppressing the preamplifier’s V n . Separately, the large (𝜕I/𝜕Φ)SFC reduces the noise contribution from the preamplifier’s I n . Although the behaviors of the apparent V (Φ) or I(Φ) in the two bias modes with the flux feedbacks (PFC and SFC) are very different, the effects of δΦe suppression are the same. The PFC consists of a resistor Rp connected to a coil Lp in series that shunts to the SQUID, where Lp couples to the SQUID with a mutual inductance Mp . To simplify the analysis, we always take the flux feedback circuit in voltage bias mode, where two branches, the SQUID and PFC, are independent. Thus, the critical conditions of both flux feedbacks are easily obtained. In addition, we quantitatively analyze the PFC parameters and give their recommended regimes of operation. Indeed, it was experimentally proved that our analyses of both flux feedbacks agree well with the measured data. Chapter 9: The SFC consists of a coil Lse connected to the SQUID in series, where Lse couples to the SQUID with a mutual inductance Mse . Because of SFC, the SQUID’s apparent parameters (Rd )SFC at the input terminal of the preamplifier are reduced, thus reducing the preamplifier’s current noise contribution, δΦIn . A possible combination of the PFC and SFC is also discussed in Chapter 9. In practice, the two flux feedbacks via Mse and Mp are not independent, so adjusting Mse can also change Mp of the PFC, and vice versa. This leads to difficulties in reaching the designed mutual inductances. According to our experience, we do not recommend employing both flux feedbacks at the same time. For general SQUID applications, we suggest two practical concepts with flux feedback in a DRS: (i) an op-amp (preamplifier) with the PFC and (ii) a PCBT with the SFC.

5

6

1 Introduction

Chapter 10: In many applications, the objective of a SQUID system is not to achieve utmost sensitivity but to rather have a SQUID system with simplicity, user-friendliness, robustness, with a high resistance against disturbances, good stability, and acceptable system noise δΦsys . In this way, we should abandon the traditional ideas to pursue a low readout electronics noise δΦe that is lower than the intrinsic SQUID noise δΦs . In contrast, tolerating a relatively large δΦs of a weakly damped SQUID with a large 𝛽 c to achieve a suitable δΦsys is a practical approach. Indeed, our novel paradigm for SQUID readout is to strive for equally high SQUID and electronics noise, δΦs ≈ δΦe , as a basis to set up a simple and reliable SQUID system. The drawback of always striving for lowest SQUID system noise is the vulnerability of the system to fitting the exact amount of feedback in PFC or SFC schemes, thus leading to complexity and instability of the SQUID readout circuitry. Our concept of a weakly damped SQUID system does not yield the very best system noise δΦsys but rather a δΦsys , which is suitable for applications. Most importantly, this concept tolerates deviations of the SQUID parameters in a large range, as we have shown by performing a statistical analysis of 101 SQUID magnetometers. Thereby, we proved the applicability of weakly damped SQUIDs with DRS to be employed in a multichannel SQUID system. For this purpose, “single-chip readout electronics” (SCREs) consisting of only one op-amp was developed. The equivalent circuit of the SCRE is used as the cover of this book. We characterized this system and demonstrated its applicability to magnetocardiography (MCG) and the transient electromagnetic (TEM) method in geophysical measurements. Chapter 11: Two special dc SQUID readout schemes, the two-stage scheme and the double relaxation oscillation (D-ROS) scheme, are introduced. Both of them are suitable for observation of the SQUID’s intrinsic noise δΦs , i.e. δΦe < δΦs . In fact, the δΦs values in the two readout schemes are quite different. The two-stage readout scheme possesses a very small δΦe , which can be lower than the δΦs of a SQUID with 𝛽 c < 1. In contrast, the un-shunted SQUID in the D-ROS scheme presents a large δΦs and a large 𝜕V /𝜕Φ, thus leading to δΦs < δΦe in the system. Actually, the two-stage readout scheme consists of a voltage-biased sensing SQUID and a sensitive SQUID-ammeter (the reading SQUID). The real trick of the two-stage scheme is the “flux amplifier,” where the reading SQUID measures only the “amplified flux.” In the closed voltage biased circuit of the sensing SQUID, a ring current ΔI = ΔΦ × (𝜕I/𝜕Φ)Sensing is modulated by the measured flux ΔΦ. Here, ΔI flows through a coil La which is inductively coupled to the reading SQUID via the mutual inductance Ma , thus generating a further flux in the reading SQUID. Thus, the flux of ΔI × Ma for the reading SQUID is amplified by a factor, GF = [(𝜕I/𝜕Φ)Sensing × Ma ], where GF > 1. Therefore, the system noise of the reading SQUID can be regarded as the readout noise δΦe for the sensing SQUID. In other words, the two-stage scheme realizes a readout electronics noise δΦe below the product of GF × (δΦs )Sensing . In brief, for intrinsic SQUID noise studies, the δΦs of most SQUIDs is calibrated with the two-stage readout scheme. The key elements of the D-ROS readout scheme are a hysteretic SQUID and a shunted circuit, the latter of which consists of a coil Lro and a resistor Rro in series. Here, the Lro is not coupled to the SQUID. When a constant current I b above the SQUID’s I c flows through the parallel circuit, the D-ROS becomes active

1.2 Contents of the Chapters

to oscillate. In fact, the initial motivation of the D-ROS readout scheme was to achieve a high flux-to-voltage transfer coefficient 𝜕V /𝜕Φ, e.g. in the 10 mV/Φ0 region, thus simplifying the readout electronics and improving the slew rate. As an important consequence, the δΦs and the value of 𝜕V /𝜕Φ in D-ROS scheme are high due to the un-shunted SQUID parameter 𝛽 c → ∞. However, the system noise δΦsys of D-ROS is still acceptable for recording signals of, e.g. human biomagnetism; moreover, the large δΦs (≈δΦsys ) improves the system robustness. Therefore, some commercial multichannel SQUID systems for MCG and magnetoencephalography (MEG) are equipped with the D-ROS scheme. Chapter 12: An rf SQUID is inductively coupled to a tank circuit that connects to the readout electronics. According to the parameter 𝛽 e of rf SQUIDs, there are two working modes: the dissipative mode and the dispersive mode. In the dissipative mode, the rf SQUID acts as a damping resistance for the tank circuit; i.e. the quality factor of the tank circuit, Q, is changed with changing flux ΔΦ. In the dispersive mode, the rf SQUID is regarded as an additional inductance LSQ inserted into the tank circuit, where the value of LSQ is modulated with varying ΔΦ. However, for both working modes, the readout electronics is the same. The system noise δΦsys of the rf SQUID consists of three independent parts: the intrinsic SQUID noise δΦs , the readout electronics noise δΦe , and the thermal noise of the tank circuit, δΦT , thus resulting in δΦ2sys = δΦ2s + δΦ2e + δΦ2T . Conventionally, the pumping (resonance) frequency f 0 of the tank circuit is limited to approximately 30 MHz due to the distributed inductance and capacitance of the connection wires between the tank circuit at, e.g. 4.2 K, and the readout electronics at room temperature (RT). Taking an (capacitor-) inductor-tap on the tank circuit, the f 0 can rise up to the gigahertz range; thus the impedance across the tank circuit becomes very high. However, the impedance at the tap point remains low. Therefore, a standard 50 Ω transmission line is employed to connect the tap point of the tank circuit with the readout electronics at RT, where a bipolar transistor acts as a low-noise preamplifier. We define a dimensionless ratio 𝜅 to describe the position of the tap, where 𝜅 = Zrf,input /Zrf,T , in which the impedance Zrf,input at the tap point should approximately be 50 Ω and the high impedance Zrf,T = 2𝜋f 0 LT Q appears across the LT C T tank circuit. Two main achievements have been attained in HTS rf SQUID research: (i) Instead of the conventional LT C T tank circuit, superconducting planar resonators or substrate resonators were developed for rf SQUID operation. This new kind of resonator possesses a high resonance frequency f 0 and a large quality factor Q0 , thus leading to a large Zrf,T across the tank circuit. To match the 50 Ω impedance, the ratio of 𝜅 ≪ 1 can reduce the effective temperature of the tank circuit, thus decreasing its thermal noise, δΦT . (ii) With such resonators, some high harmonic components of the V rf (Φ) characteristics can be observed, thus changing the shapes of V rf (Φ), where its slopes become steeper. Consequently, a large transfer coefficient (𝜕V rf /𝜕Φ) appears at the working point W, so the readout noise δΦe is suppressed. Ultimately, some HTS rf SQUIDs in resonator version demonstrated that their δΦsys was close to the SQUID’s thermal noise limit. Furthermore, using a planar HTS field-to-flux transfer coil system with a pick-up area of 10 × 10 mm2 in a three-layer structure, the HTS rf SQUID magnetometer

7

8

1 Introduction

consisting of a thin-film rf SQUID and this transfer coil system √ in flip-chip configuration reached a field sensitivity of approximately 10 fT/ Hz at 77 K.

References 1 Schwartz, B.B. and Foner, S. (1976). Superconductor Applications: SQUIDs

and Machines. New York and London: Plenum Press. 2 Barone, A. and Paterno, G. (1982). Physics and Applications of the Josephson

Effect. New York: Wiley. 3 Weinstock, H. (1996). SQUID Sensors: Fundamentals, Fabrication and Appli-

cations. Dordrecht: Kluwer Academic Publishers. 4 Clarke, J. and Braginski, A.I. (2004). The SQUID Handbook. Weinheim:

Wiley-VCH. 5 Ryhänen, T., Seppä, H., Ilmoniemi, R., and Knuutila, J. (1989). SQUID magne-

6 7 8 9 10

tometers for low-frequency applications. Journal of Low Temperature Physics 76 (5–6): 287–386. Simmonds, M.B. and Giffard, R.P. (1983). Apparatus for reducing low frequency noise in dc biased SQUIDs. US Patent 4, 389, 612. Dutta, P. and Horn, P.M. (1981). Low-frequency fluctuations in solids – 1-F noise. Reviews of Modern Physics 53 (3): 497–516. Weissman, M.B. (1988). 1/F noise and other slow, nonexponential kinetics in condensed matter. Reviews of Modern Physics 60 (2): 537–571. Lam, S.K.H. (2006). Noise properties of SQUIDs made from nanobridges. Superconductor Science & Technology 19 (9): 963–967. Cleuziou, J.P., Wernsdorfer, W., Bouchiat, V. et al. (2006). Carbon nanotube superconducting quantum interference device. Nature Nanotechnology 1 (1): 53–59.

9

2 Josephson Junctions 2.1 Josephson Equations In 1962, Brian Josephson predicted a macroscopic quantum phenomenon, the Josephson effect [1], which became the basis of the superconducting quantum interference device (SQUID). A Josephson junction (JJ) is defined as two superconducting electrodes with weak coupling between them. One of the most widely adopted JJs is the superconductor–insulator–superconductor (SIS) tunnel contact, where the electrodes are separated by a thin insulating layer in thin-film techniques. The dc and ac properties of a JJ are described in the two Josephson equations (Eqs. (2.1) and (2.2)), thereby initiating the discipline of “superconducting electronics” [1, 2]. In the dc Josephson equation, a current I flowing through a JJ is given by I = Ic sin δ

(2.1)

where δ is the phase difference in the macroscopic wave functions of the two superconducting electrodes. For each JJ, a critical current I c exists. When I < I c , the supercurrent I leads to a change in δ, while the voltage across the JJ remains zero (U = 0). Here, we assume that its current density is homogeneous in this junction area. When U ≠ 0 (e.g. I > I c ), the supercurrent I exhibits ac behavior, where the phase difference δ changes with time. The ac Josephson equation describes the relation between δ changes and the voltage U as follows: 2𝜋 2e U δ̇ = U = ℏ Φ0

(2.2)

where e is the charge of an electron, ℏ is Planck’s constant, and Φ0 ≈ 2.07 × 10−15 Wb is called the magnetic flux quantum.

2.2 RCSJ Model Generally, an SIS-type tunnel junction can be described by the model of a resistively and capacitively shunted junction, the so-called RCSJ model. In this model, the Josephson element, represented by its critical current I c , is connected in parallel with the junction capacitance C and a shunt resistance RJ SQUID Readout Electronics and Magnetometric Systems for Practical Applications, First Edition. Yi Zhang, Hui Dong, Hans-Joachim Krause, Guofeng Zhang, and Xiaoming Xie. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

10

2 Josephson Junctions

(III)

(II)

Figure 2.1 A Josephson element J, a capacitance C, and a resistance RJ are connected in parallel to form the RCSJ model.

(I) V (t)

J

RJ

Idc

C

as shown in Figure 2.1. When I dc > I c , an ac voltage across the parallel circuit appears, i.e. V (t) ≠ 0. In this case, there are three different currents flowing through the junction: (I) C[dV (t)/dt], the capacitive displacement current; (II) V (t)/R, the resistance current; and (III) I c sin δ(t), the ac supercurrent through the Josephson element [3, 4]. For instance, to understand the resistively and capacitively shunted junction (RCSJ) model, Sullivan and Zimmerman used the average angular velocity of a pendulum as a function of the applied torque to represent the analog of the current-to-voltage (I–V ) characteristics of the junction, thus introducing damping concept into JJs [5]. The kinetic energy term of the pendulum can be the analog of the term CV 2 /2, where C is the capacitance in branch (I) of Figure 2.1. Similarly, branches (II) and (III) have analogs in the pendulum system. To study the junction’s features, one often employs its I–V characteristics, which are obtained by using a quasi-dc electrical measuring method. There, one frequently varies the current I dc injected into the RCSJ junction while synchronously recording the voltage V (t) across the junction, or vice versa. How one can obtain the I–V characteristics will be described in detail in Chapter 3. In fact, the I–V characteristics include information on the resistive and hysteretic behavior, although the rich physical meaning of the Josephson effects in Eqs. (2.1) and (2.2) cannot be fully reflected. Usually, two parameters, 𝛽 c and Γ, are introduced to characterize the features of a JJ. In the RCSJ model, the Stewart-McCumber parameter 𝛽 c is denoted as 𝛽c = (2𝜋∕Φ0 )Ic CR2J

(2.3)

which describes the junction’s hysteresis behavior, or, the damping classes. Figure 2.2 sketches the I–V characteristics of junctions for different 𝛽 c values. Generally, there are two extreme cases: (i) for 𝛽 c ≪ 1, the junction is strongly damped with a small shunt resistance RJ . Consequently, the capacitive displacement current in Figure 2.1 does not play a role, so the I–V characteristics are single valued, i.e. nonhysteretic (see Figure 2.2a). In this case, the RCSJ model can be simplified to the so-called resistively shunted junction (RSJ) model [6]; (ii) when the resistance RJ is removed from the RCSJ model, the I–V characteristics become hysteretic, i.e. 𝛽 c → ∞ (see Figure 2.2b) [7]. Here, the so-called capacitively shunted junction (CSJ) model is typically realized by an un-shunted tunnel contact with an SIS three-layer construction. In fact, 𝛽 c = 1 is the boundary between the nonhysteretic and hysteretic regimes in the junction’s I–V characteristics. For 𝛽 c = 2, for example, hysteresis including a local loop clearly appears in the I–V characteristics (see Figure 2.2c).

2.2 RCSJ Model

l

l/lc 2

βc ≫ 1

βc ≪ 1 lc

1

0 0 (a)

1

2

2Δg/e

V/lcRJ

V

(b)

l/lc 2

βc = 2

1

0 0 (c)

1

2

V/lcRJ

Figure 2.2 I–V characteristics for different 𝛽 c values: (a) 𝛽 c ≪ 1 (RSJ model); (b) 𝛽 c ≫ 1, (CSJ model); (c) 𝛽 c = 2 (RCSJ model). Note that the coordinates are not normalized in (b), because Δg denotes the energy gap.

Another important feature of the junction is its thermal noise parameter Γ, which leads to significant rounding of the I–V characteristics [8, 9]. It is defined as the ratio of the thermal energy to the Josephson coupling energy, i.e. Γ = 2𝜋kB T∕(Ic Φ0 )

(2.4)

where k B is Boltzmann’s constant and T is the absolute temperature. With the two typical sketches of the I–V characteristics shown in Figure 2.3, one can gain an impression of the influence of Γ on the characteristics: (a) For 𝛽 c ≪ 1 (RSJ model), the I–V characteristics are displayed for the cases Γ = 0, Γ = 0.018, Γ = 0.33, and Γ = 1 (see Figure 2.3a). For Γ = 0, the resistive transition is obvious at (I/I c ) ≥ 1. An RSJ with I c = 10 μA yields Γ = 0.018 at T = 4.2 K (e.g. a niobium junction) or Γ = 0.33 at T = 77 K (e.g. a yttrium barium copper oxide [YBCO] junction). For a classic niobium junction, the rounding effect is not very prominent. However, the I–V characteristics of the YBCO junction are seriously rounded, so the value of I c of the JJ cannot be clearly identified at Γ = 0.33. Here, when a voltage is barely appearing over the junction, the large Γ blurs its two states, the superconducting state and the normal conducting state of JJ. Namely, when (I/I c ) >1, the larger Γ leads

11

12

2 Josephson Junctions

l/lc

l/lc 2

2

βc < 1 Γ=0

1

βc = 2

1

Γ=0

0.33 0.018

Γ = 0.1

1 0

0 0

(a)

1

2

0 V/IcRJ

(b)

1

2 V/IcRJ

Figure 2.3 Schematic illustrations of the I–V characteristics with different Γ values at 𝛽 c ≪ 1 (RSJ model) (a) and at 𝛽 c = 2 with Γ = 0 and 0.1 (b).

to a smaller dynamic resistance Rd . At Γ = 1, the Josephson effect is already suppressed, so a straight line appears in the I–V characteristics. (b) For 𝛽 c = 2, because of the large thermal noise at Γ = 0.1, the hysteretic I–V characteristics (gray lines, at Γ = 0) become a single-valued function, i.e. the hysteresis is removed. Indeed, the slope of the I–V characteristics becomes larger; i.e. the corresponding resistance decreases due to the rounding effect. In fact, the appearance of hysteresis in the I–V characteristics depends on both parameters, 𝛽 c and Γ. A dc SQUID is a superconductive ring interrupted with two JJs. The physical basis of a SQUID originates from the Josephson equations (Eqs. (2.1) and (2.2)). In principle, dc SQUIDs can be operated with junctions having arbitrary 𝛽 c and Γ values. For example, a conventional dc SQUID is operated at 𝛽 c < 1 in the RSJ model because single-valued I–V characteristics are required. However, the readout of a relaxation oscillation SQUID (ROS) is based on the hysteretic I–V characteristics of un-shunted SQUIDs, i.e. the CSJ model (see Chapter 11). Historically, two types of JJs have been employed for the production of conventional dc SQUIDs: (i) a point contact with two bulk superconducting materials [10] or a microbridge that is often used in early thin-film techniques [11] and described by the RSJ model; (ii) a standard tunnel junction with three layers (SIS) and a shunt RJ , i.e. a junction according to the RCSJ model. Figure 2.4 shows the I–V characteristics of two SIS junctions in the RCSJ model measured at 4.2 K. These two junctions are shunted by a resistor RJ = 1 Ω, thus leading to 𝛽 c ≪ 1, although the values of I c vary from approximately 6.4 to 34 μA. Note that one can clearly read the value of I c = 6.4 μA at Γ = 0.028 because the rounding effect actually is not strong. The dashed line describes the shunted junction’s normal resistance RN ≈ RJ . In the nonlinear regime, i.e. at low voltages (V < 25 μV), each point on the I–V curves can be characterized by Rd = 𝜕V /𝜕I, i.e. the dynamic resistance Rd , which increases with increasing I c , e.g. Rd ≈ 1.2 Ω (at point A) and 25 Ω (at point B), shown in Figure 2.4. Here, the value of 𝛽 c is proportional to I c at a constant product of CR2J . In addition to I c , the value of Rd also depends on RJ (not shown here).

References

50 B 25 Current (μA)

A Rd = ∂V/∂I

0

–25

–50 –50

–25

0 Voltage (μV)

25

50

Figure 2.4 I–V characteristics of two SIS junctions in the RCSJ model with different Ic values for 𝛽 c ≪ 1. Source: Adapted from Liu et al. 2012 [12].

Recently, the fabrication of nanoscale JJs in 25-nm-thick YBCO thin films with wire widths as narrow as 50 nm was reported [13]. This fabrication process utilized a finely focused gas field ion source from a helium ion microscope to directly modify the material on the nanometer scale in order to convert the irradiated regions of the film from superconductors into insulators. With this approach high-quality junctions were prepared. In a sense, this type of junction can be regarded as a nanoscale “varying-thickness bridge” (V.T.B.) [14]. Excitingly, the product of I c and RN for each junction in the RSJ model remains approximately constant at approximately 400 μV at 4.2 K, and this product is independent of the value of I c . As an example of this type of junction with I c = 5.6 μA, which is suited well for SQUID operations, a large RN value of 70 Ω is obtained. Nevertheless, junction preparation is not the main subject of this book, although such nanoscale junctions are very important for the low-noise operation of SQUIDs. To ensure the integrity of this book about the SQUID magnetometric system and its readout electronics, only some fundamental junction knowledge is introduced here. For the reader’s reference, there are many books and review articles that have described JJs in detail [15–17].

References 1 Josephson, B.D. (1962). Possible new effects in superconductive tunnelling.

Physics Letters 1 (7): 251–253. 2 Josephson, B.D. (1965). Supercurrents through barriers. Advances in Physics

14 (56): 419–451. 3 Stewart, W.C. (1968). Current–voltage characteristics of Josephson junctions.

Applied Physics Letters 12 (8): 277.

13

14

2 Josephson Junctions

4 McCumber, D.E. (1968). Effect of Ac impedance on Dc voltage–current char-

5 6 7 8 9 10 11 12

13

14

15 16 17

acteristics of superconductor weak-link junctions. Journal of Applied Physics 39 (7): 3113. Sullivan, D.B. and Zimmerman, J.E. (1971). Mechanical analogs of time dependent Josephson phenomena. American Journal of Physics 39 (12): 1504. Likharev, K.K. (1979). Superconducting weak links. Reviews of Modern Physics 51 (1): 101–159. Anderson, P.W. and Rowell, J.M. (1963). Probable observation of Josephson superconducting tunneling effect. Physical Review Letters 10 (6): 230. Ivanchenko, Y.M. and Zilberma, L.A. (1968). Destruction of Josephson current by fluctuations. JETP Letters-Ussr 8 (4): 113–115. Ambegaokar, V. and Halperin, B.I. (1969). Voltage due to thermal noise in Dc Josephson effect. Physical Review Letters 22 (25): 1364–1366. Zimmerman, J.E. and Silver, A.H. (1966). Macroscopic quantum interference effects through superconducting point contacts. Physical Review 141 (1): 367. Anderson, P.W. and Dayem, A.H. (1964). Radio-frequency effects in superconducting thin film bridges. Physical Review Letters 13 (6): 195. Liu, C., Zhang, Y., Mück, M. et al. (2012). An insight into voltage-biased superconducting quantum interference devices. Applied Physics Letters 101 (22): 222602. Cho, E.Y., Zhou, Y.W., Cho, J.Y., and Cybart, S.A. (2018). Superconducting nano Josephson junctions patterned with a focused helium ion beam. Applied Physics Letters 113 (2): 022604. Sandell, R.D., Dolan, G.J., and Lukens, J.E. (1976). Preparation of variable thickness microbridge using electron beam lithography and ion etching. IC-SQUID 76: 93–100. Schwartz, B.B. and Foner, S. (1976). Superconductor Applications: SQUIDs and Machines. New York and London: Plenum Press. Chesca, B., Kleiner, R., and Koelle, D. (2004). SQUID Theory. In: The SQUID Handbook (eds. J. Clarke and A.I. Braginski). Weinheim: Wiley-VCH. Barone, A. and Paterno, G. (1982). Physics and Applications of the Josephson Effect. New York: Wiley.

15

3 dc SQUID’s I–V Characteristics and Its Bias Modes The physics of a dc SQUID involves two phenomena: flux quantization and Josephson tunneling [1–3]. Flux quantization shows that the flux threading a closed superconducting loop (without the junctions) is quantized in units of the flux quantum Φ0 ; Josephson tunneling was already introduced in Chapter 2. The symbol for a dc SQUID is sketched as a circle representing the superconducting ring and two crosses for the JJs on the circle. For readout electronics, the dc SQUID acts as a varying dynamic resistance Rd (Φ) that changes its resistance with changing flux Φ. In order to illustrate and emphasize the flux-dependent dynamic resistance, in this book the SQUID symbol is often connected in series with a resistor Rd (see the dotted box in Figure 3.1). Similar to JJs, the SQUID’s features are characterized by its I–V curves. Generally, conventional dc SQUIDs are operated at 𝛽 c < 1, where their I–V characteristics are single valued, i.e. there is no hysteresis. Because the SQUID is a flux sensing element, its I–V curves describe the changes in the dynamic resistance Rd (Φ) = 𝜕V /𝜕I with the magnetic flux threading the SQUID loop. Essentially, reading the function Rd (Φ) of a dc SQUID is the mission for its readout electronics. We describe (i) how to measure and understand a SQUID’s I–V characteristics in Section 3.1; (ii) how to realize an ideal current source for biasing a SQUID in Section 3.2; and (iii) how to set up a practical voltage source to operate a SQUID in voltage bias mode in Section 3.3.

3.1 SQUID’s I–V Characteristics To study a dc SQUID’s features, the first step is to obtain its I–V characteristics. Generally, to measure an ordinary resistor R, one needs an ammeter for observing the current through R and a voltmeter for recording the voltage across R. In this way, one often records the SQUID’s I–V curves by applying an injected current I and recording the resultant voltage V across the SQUID. To avoid the influence of all connecting resistances, the SQUID’s I–V characteristics should be acquired by using a four-pole (terminal) measuring method (see Figure 3.1). A current source is connected to the SQUID at points I 1-probe and I 2-probe , while a voltmeter is connected in parallel to points V 1-probe and V 2-probe . In practice, a voltage source can SQUID Readout Electronics and Magnetometric Systems for Practical Applications, First Edition. Yi Zhang, Hui Dong, Hans-Joachim Krause, Guofeng Zhang, and Xiaoming Xie. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

3 dc SQUID’s I–V Characteristics and Its Bias Modes

Y

X

I1-probe Ib

V1-probe

Oscilloscope

Isource

Rd Vsource

16

A SQUID

ΔΦ V/I converter

V2-probe I2-probe

Figure 3.1 Schematic diagram of a four-pole (terminal) measuring method. Here, the I–V characteristics of a dc SQUID are displayed on an oscilloscope operated in X–Y mode. The symbol in the dotted box represents the SQUID. The voltage-to-current (V/I) converter is schematically sketched in the dashed box, where the voltage output is proportional to the current Ib flowing through a SQUID (a junction, or a resistor). The SQUID’s I–V characteristics are modulated by the flux change ΔΦ threading the SQUID loop. Figure 3.2 I–V characteristics of a resistor (a) and a diode (b), where V BR and V F denote the breakdown voltage and the forward voltage of a diode, respectively.

I (a)

(b)

VBR VF

V

be converted into a current source. Here, the current source is converted by a triangular wave voltage generator V source , schematically shown as V /I converter in Figure 3.1. For example, the 1 V pp signal of V source generates 10 μApp of current I b flowing through the SQUID. The voltage-to-current translation will be discussed in Section 3.2. The voltage difference between points V 1-probe and V 2-probe is amplified by an amplifier “A” with a gain of, e.g. 1000. The oscilloscope is operated in X–Y mode to quantitatively display and record the I–V characteristics in real time. The triangular wave voltage generator with a low frequency of several tens of hertz connects to the oscilloscope’s input “Y” representing the current I, while the output of the voltage amplifier connects to the oscilloscope’s input “X” recording the voltage V . Of course, the oscilloscope’s inputs “Y” and “X” can be exchanged, depending on the user’s usual practice.

3.1 SQUID’s I–V Characteristics

Figure 3.3 The SQUID’s I–V characteristics are divided into three regions. The SQUID is operated in the flux-modulated region (region II). Here, Ic2 is the critical current of the SQUID’s I–V characteristics for Φ = (2n + 1)Φ0 /2, and Ic1 is the critical current for Φ = nΦ0 , where n = 0, 1, 2, . . . . Note that the horizontal dashed line (gray) represents the constant bias current in current bias mode, while the perpendicular dotted line (gray) marks the constant bias voltage in voltage bias mode.

I RJ/2

III

Φ = nΦ0 Ic1 Ic2

II Φ = [(2n + 1)/2]Φ0

I V

Before we try to understand the I–V characteristics of a dc SQUID, let us review the I–V curves of a normal resistor and of a diode, as shown in Figure 3.2. Curve (a) of a resistor (dashed line) appears as a straight line, and its slope 𝜕V /𝜕I exhibits the resistance value in units of ohms (Ω). Curve (b) of a diode (solid line) can be roughly divided into three regions: (i) In the linear region (V > V F ), the resistance (𝜕V /𝜕I) remains constant, so the current flowing through the diode I D increases linearly with increasing voltage V across the diode, and vice versa. (ii) In the cut-off region between V F and V BR , I D approaches zero and is independent of V . Therefore, the resistance 𝜕V /𝜕I in this region approaches infinity. (iii) In the reverse breakdown region (V < V BR ), V remains equal to V BR , while I D increases rapidly, thus leading to 𝜕V /𝜕I → 0. In contrast to the I–V characteristics of the resistor or the diode, those of the SQUID are modulated by the applied flux Φ but are limited within two flux states of integer and half-integer Φ0 . In Figure 3.3, the I–V characteristics represent the different resistive behaviors at these two flux limits, where different critical currents are denoted as I c1 at nΦ0 and I c2 at (2n + 1)Φ0 /2. So, the SQUID possesses strongly nonlinear characteristics and exhibits its resistive performance, i.e. Rd (Φ). In general, the dc SQUID’s I–V characteristics can be divided into three regions: (I) Superconducting region (I < I c2 ): the SQUID remains in the superconductive state, i.e. V = 0 and 𝜕V /𝜕I = 0 as I changes. (II) Flux-modulated region (I c1 < I < I c2 ): the nonlinear I–V characteristics are periodically modulated by an applied flux Φ threading the SQUID loop. Additionally, all I–V characteristics are limited by the two extreme flux states, integer flux Φ = nΦ0 and half-integer flux Φ = (2n + 1)Φ0 /2, where n = 0, 1, 2, . . . . Here, all flux states represent the operating region of a dc SQUID. In this region, the function Rd (Φ) exhibits the nonlinearity at each point of the I–V characteristics during the flux changes. (III) Resistor region (I ≫ I c1 ): the I–V characteristics are linear and independent of Φ. In this region, the SQUID is regarded as a normal resistor that should be equal to RJ /2, the value of two parallel junction shunt resistors RJ , as described in Chapter 2.

17

3 dc SQUID’s I–V Characteristics and Its Bias Modes

In region II, the I–V characteristics not only include the performance of macroscopic physical phenomena, e.g. flux quantization and the Josephson tunneling effect, but also provide information on the SQUID’s three dimensionless parameters: the Stewart-McCumber parameter 𝛽 c , the thermal noise parameter Γ, and the screening parameter 𝛽 L . These three important parameters of the SQUID will be discussed in detail in Section 6.2. In SQUID readout electronics, the SQUID’s dynamic resistance Rd (Φ) is mostly measured by only one instrument, either by a voltmeter or an ammeter. In other words, there are two typical methods for measuring Rd (Φ): (i) recording the voltage across the SQUID with a known (constant) bias current flowing through the SQUID (the dashed horizontal line in Figure 3.3) or (ii) reading out the current flowing through the SQUID biased with a known (constant) voltage (the dotted perpendicular line in Figure 3.3). Both bias lines are also sometimes called the “load line.” In SQUID terminology, this Rd (Φ) measurement using a voltmeter is called “current bias mode,” and its equivalent circuit is shown in Figure 3.4a, whereas the measurement using an ammeter is called “voltage bias mode” (Figure 3.4b). To avoid any influences of the measuring instruments, the internal resistance Rinter of the voltmeter should approach infinity for the circuit shown in Figure 3.4a, whereas that of the ammeter should be close to zero for the circuit shown in Figure 3.4b. In contrast, the internal resistance Rinter of the ideal current source approaches infinity, and that of the voltage source should be close to zero. The reader should keep these concepts about Rinter in mind. Note that the dc SQUID is a low-resistance element. The values of Rd and the maximal resistance change ΔRd (Φ) normally range from several ohms to a few tens of ohms. Thus, the so-called “ideal source” is valid for only small values of Rd . The discussion on bias sources in Sections 3.2 and 3.3 is based on this fact.

A Ammeter

Ib

(a)

V

Δi

Vsource

Rd Voltmeter

Rd Isource

18

(b)

Figure 3.4 Principle circuits of current bias mode (a) and voltage bias mode (b). Here, the dc SQUID is regarded as a dynamic resistance Rd (Φ), which changes with the applied flux.

3.3 A Practical Voltage Source

RI Rd

Voltmeter

V

Rinter

Ib V/Ib converter

Figure 3.5 Practical circuit of the current bias mode for measuring SQUID dynamic resistance Rd (Φ): a current source is formed by a battery with Rinter and a large resistor RI connected in series, where Rinter ≪ Rd ≪ RI ≈ 100 kΩ.

V

3.2 An Ideal Current Source An ideal current source is characterized by an internal resistance (Rinter )current approaching infinity, compared to the SQUID’s dynamic resistance Rd . The ideal current source means that the current flowing out from the current source remains constant; i.e. this current does not change with a load change, e.g. ΔRd (Φ) during SQUID operation. In Figure 3.5, an ideal current source can easily be realized with a voltage source connected to a SQUID via a large series resistor RI . So, the converted current source presents a high equivalent internal resistance, (Rinter )current ≈ RI ≈ 100 kΩ. If the voltage and RI are already known, a (known) constant bias current I b flowing through the SQUID is realized due to the high ratio of (Rinter )current /(Rd ) ≈ 104 , where the change of Rd is approximately tens of ohms, as mentioned in Section 3.1. Such V /I b converters were also used for obtaining the I–V characteristics of SQUIDs and JJs, as shown in Figure 3.1. In this case, the SQUID is operated in current bias mode. Here, one employs only a voltmeter V to read out the SQUID signal, i.e. to directly obtain the flux-to-voltage characteristics, V (Φ), which is given by the product of Rd (Φ) × I b .

3.3 A Practical Voltage Source In Figure 3.4b, an ideal bias voltage source V source has an internal resistance (Rinter )voltage approaching zero, so the closed circular (ring) current Δi depends only on the load Rd (Φ) in SQUID operation. In this case, using an ammeter A, one obtains the SQUID’s flux-to-current characteristics I(Φ) in voltage bias mode. However, an ideal voltage source is difficult to realize because its internal resistance (Rinter )voltage cannot be set small enough, compared to the small dynamic resistance Rd of the SQUID. In practice, reaching the ratio of (Rinter )voltage /Rd ≈ 10−3 is already not easy. There are only two realized concepts for the SQUID operation in voltage bias mode, as illustrated in Figure 3.6. In concept (a), one applies a constant current I b flowing through a SQUID shunted by a resistor Rinter of ≤1 Ω, where Rinter < Rd of the SQUID. In this case, most of the I b flows through Rinter , thus generating a

19

3 dc SQUID’s I–V Characteristics and Its Bias Modes

A Ib

Vb

i Isource

20

Rg

Rd

Vout

– Rd

Rinter

ig

A +

Vb

(a)

(b)

Figure 3.6 Two possible concepts for realizing voltage bias mode. (a) A current source Isource translates into a voltage source V b = Isource × Rinter , where Rinter ≪ Rd . (b) An op-amp acts as a current-to-voltage converter. The voltage of ig × Rd at the inverting terminal is pulled to V b applied at the noninverting terminal. Note that the function V out (Φ) represents the SQUID’s flux-to-current characteristics I(Φ) in voltage bias mode.

quasi-constant bias voltage V b across the parallel circuit. Furthermore, Rinter can be regarded as the internal resistance of V b . A flux change results in a change in the SQUID’s Rd (Φ), thereby changing the ring current i flowing through the SQUID, as recorded by an ammeter A in voltage bias mode. In fact, the voltage source V b may not be stable enough for SQUID operation because the ratio of Rinter /Rd ≈ 0.1 is not sufficiently small (e.g. Rinter = 1 Ω and Rd in the 10 Ω range). On the other hand, the power dissipation W R of Rinter is expressed as WR = Vb2 ∕Rinter Of course, one can take, e.g. a Rinter of 0.01 Ω to stabilize V b , but the W R increases 100 times to maintain the same V b at 4.2 K. Indeed, this concept, shown in Figure 3.6a, is not often used in applications, because a large W R caused by a small Rinter leads to more boiling of the liquid helium. The other concept is shown in Figure 3.6b. An operational amplifier (op-amp) acting as a current-to-voltage converter is often used to realize the voltage bias mode. By applying a bias voltage V b at the op-amp’s noninverting terminal, the voltage at the op-amp’s inverting terminal is pulled to V b , where ig × Rd = V b must be clamped. In fact, the current ig flows from the op-amp’s output to the ground via Rg and Rd , thus causing the voltage V out = ig × (Rg + Rd ). Generally, Rg ≫ Rd , so we obtain V out ≈ ig × Rg = (V b /Rd ) × Rg ; i.e. V out is inversely proportional to Rd . Consequently, in this concept, V out (Φ) at the output of the op-amp (current-to-voltage converter) represents the SQUID’s flux-to-current characteristics I(Φ). In short, the key feature in concept (b) is that the voltages at the two input terminals of the op-amp must be the same, they have to be balanced. In principle, Rinter of the voltage source V b is the resistance of the connecting wires between the SQUID at cryogenic temperature (e.g. at 4.2 K, liquid helium temperature) and the preamplifier (op-amp) at room temperature (300 K). The influence of this resistance Rinter on the SQUID system’s behavior is discussed in Ref. [4]. In brief, the dc SQUID I–V characteristics directly describe the behavior of Rd (Φ). For dc SQUID operation in current bias mode, an ideal current source

References

is easily realized. However, strictly speaking, the SQUID can be operated only in nominal voltage bias mode due to the relatively large ratio (Rinter /Rd ) in Figure 3.6. As a matter of fact, many people are unfamiliar with voltage bias mode.

References 1 Deaver, B.S. and Fairbank, W.M. (1961). Experimental evidence for quantized

flux in superconducting cylinders. Physical Review Letters 7 (2): 43–46. 2 Doll, R. and Nabauer, M. (1961). Experimental proof of magnetic flux quanti-

zation in a superconducting ring. Physical Review Letters 7 (2): 51–52. 3 Josephson, B.D. (1962). Possible new effects in superconductive tunnelling.

Physics Letters 1 (7): 251–253. 4 Dong, H., Zhang, G.F., Wang, Y.L. et al. (2012). Effect of voltage source inter-

nal resistance on the SQUID bootstrap circuit. Superconductor Science and Technology 25 (1): 015012.

21

23

4 Functions of the SQUID’s Readout Electronics Different SQUID readout schemes have been developed in the past half century. All SQUID readout electronics should have three common functions: (i) structuring the bias circuit; (ii) establishing the flux locked loop (FLL) to linearize the output voltage, i.e. V out is proportional to the flux change Φ; and (iii) suppressing the noise contribution from the preamplifier. For readout electronics, the dc SQUID is equivalent to a dynamic resistance Rd (Φ) modulated by the applied flux, as mentioned in Chapter 3.

4.1 Selection of the SQUID’s Bias Mode To construct readout electronics for a dc SQUID, one should select the SQUID’s bias mode first, i.e. by employing either a current source or a voltage source to bias the SQUID, thus determining the readout quantity for SQUID operation. In addition, the bias current I b or the bias voltage V b should be adjustable in the readout electronics because each SQUID needs its own bias condition. In current bias mode, one chooses a proper bias current I b flowing through the SQUID to obtain the optimal V (Φ) characteristics. In voltage bias mode, a suitable bias voltage V b across the SQUID is selected to yield the best I(Φ) characteristics. Both characteristics are periodically modulated by the flux Φ with a period of Φ0 . Our readout techniques are based on the three important characteristics (curves) shown in Figure 4.1. The I–V characteristics exhibit the SQUID’s resistive performance, while V (Φ) and I(Φ) are its only two projections upon the load line. Either projection can be used to read out the SQUID’s Rd (Φ).

4.2 Flux Locked Loop (FLL) From Figure 4.1, we learn that V (Φ) and I(Φ) are periodical functions. For a practical SQUID magnetometric system, one hopes to have a linear relation between the output voltage V out of the readout electronics and the measured flux Φ. Therefore, the linearization of the V out (Φ) is the second function of the SQUID’s readout electronics. Note that there is no difference between the bias modes with respect to the linearization requirements. Here, we focus only on SQUID Readout Electronics and Magnetometric Systems for Practical Applications, First Edition. Yi Zhang, Hui Dong, Hans-Joachim Krause, Guofeng Zhang, and Xiaoming Xie. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

24

4 Functions of the SQUID’s Readout Electronics

I

I Rd = ∂V/∂I Ib

W Φ = [(2n + 1)/4]Φ0

Φ

Vb V

Φ0

V Φ0 Φ

Figure 4.1 The I–V characteristics are the SQUID’s essence, while V(Φ) (lower right) and I(Φ) (upper left) are two projections of it. The working point W is usually set to (2n + 1)Φ0 /4 for the I–V characteristics, and the tangent line at W (gray dashed line) denotes the SQUID’s dynamic resistance Rd = 𝜕V/𝜕I at W (upper right). The points on V(Φ) and I(Φ) mark the possible working points W.

SQUID operation in current bias mode because most people are accustomed to reading out the SQUID’s voltage signal V (Φ). Assuming that the measured Φ(t) is a linear function with time t, V out (t) should be proportional to this flux change Φ(t) (see Figure 4.2a). However, the linear Φ(t) threading the SQUID loop becomes the voltage change with multiple periods in the flux domain at the input of the readout electronics due to the SQUID’s V (Φ) characteristics (Figure 4.2b). To realize a linear relationship between the readout electronics’ output voltage V out (t) and the measured flux Φ(t), the so-called “flux locked loop” (FLL) is developed [1, 2]. To understand the principle of the FLL, one should learn the relations between the time domain (Φ(t) and V out (t) in Figure 4.2a) and the flux domain (V (Φ) in Figure 4.2b), because people are accustomed to observing the SQUID’s signal in the time domain, but also to analyzing this signal in the flux domain. For simplicity, we first introduce the principle of the FLL in the flux domain. 4.2.1

Principle of the FLL

For FLL operation, we should first introduce the concept of the working point W. When converting the periodic characteristics at the input of the readout electronics into a linear function of V out (Φ) at the output, only one point is chosen on the V (Φ) characteristics. This point is called the working point W, which is usually set to (2n + 1)Φ0 /4 for the I–V characteristics (n = 1,2,3,…), where the transfer coefficient 𝜕V/𝜕Φ usually reaches its maximum, as marked in Figure 4.1.

4.2 Flux Locked Loop (FLL)

Figure 4.2 (a) A linearly changing flux Φ(t) with time and the expected output voltage V out (t) of the SQUID readout electronics, if the originally periodic V(Φ) characteristics of a current-biased SQUID in the flux domain (b) has already been concealed by means of a flux locked loop (FLL).

Φ, Vout Φ (t)

Vout(t) t (a) Vs

Φ0

Φ

(b)

The basic idea of the FLL is that as an applied (measured) flux change ΔΦ causes a deviation from the flux state at W, an opposite flux change −ΔΦ instantaneously occurs to compensate for ΔΦ, leading to zero flux change in the SQUID loop. Thus, the working point W from V (Φ) remains seemingly unchanged. The SQUID is effectively operating as a null detector. For this flux feedback (−ΔΦ) process, a metaphor is very appropriate: two teams in a tug-of-war competition are working hard, but the position marker in the middle does not move. Contrary to the conventional tug-of-war competition, in FLL operation, one team representing ΔΦ is offensive, but the other team corresponding to (−ΔΦ) is just defensive. Here, the feedback flux (−ΔΦ) is provided by V out of the readout electronics. Thus, both fluxes are always evenly matched, so the periodically modulated (nonlinear) characteristics of V (Φ) shown in Figure 4.2b are hidden there. Consequently, the V out of the readout electronics is proportional to the flux change ΔΦ with the help of the FLL. Such compensation methods are frequently used in control engineering with nonlinear sensors. 4.2.2 Electronic Circuit of the FLL and the Selection of the Working Point The FLL is always employed in a SQUID magnetometric system, where the “head stage” consists of two elements at cryogenic temperature, i.e. a coil Lf coupled to the SQUID with a mutual inductance Mf , as illustrated in Figure 4.3a. A current I f flowing through Lf generates a flux Φf = I f × Mf , which acts as a feedback flux, −ΔΦ. Sometimes, this “head stage” can be regarded as a combined element with four terminals. For the readout electronics, the two terminals on the right represent the SQUID signal input, while the two terminals on the left are for feedback −ΔΦ, thus providing a possibility to set up a closed FLL system, i.e. to linearly

25

26

4 Functions of the SQUID’s Readout Electronics

Mf

S1

Lf1

Lf

S1 RJ

Lf

Lloop

JJ

Lf2

S2 Washer

If

S2 Lf2

Lf1 (a)

Substrate

(b)

Figure 4.3 In dc SQUID magnetometry, the “head stage” consists of at least two elements, a coil Lf and a dc SQUID, where Lf is always coupled to the SQUID with a mutual inductance Mf . Together, Lf and the SQUID can be regarded as a combined element with four terminals (a). A practical layout is fabricated by scanning electron microscopy (SEM) (b). Here, RJ is the shunt resistor of the Josephson junction (JJ), while the spiral multiturn Lf is integrated on the SQUID’s washer.

R2

R1

P

Ib Mf

C

R

–

+ ΔΦ

Vp(out)

Vout

–

+

I

Vw

Lf If

–ΔΦ

Rf

Figure 4.4 A fundamental circuit of a SQUID’s readout electronics with an FLL in current bias mode. Here, “P” denotes a proportional linear amplifier, and “I,” an integrator. The output of the P amplifier is monitored at V p (out). The head stage of the SQUID and Lf in the dashed box can be regarded as a four-terminal element. Here, ΔV out remains proportional to ΔΦ.

read out the minute flux changes ΔΦ threading the SQUID loop. Figure 4.3b shows a practical SQUID “head stage” with a feedback coil Lf in a planar structure with multilayer thin films, where Lf is integrated on the SQUID-washer and two JJs of SIS type are shunted by the RJ . Here, the two contact pads of the coil Lf are denoted as Lf1 and Lf2 , while the other two pads, S1 and S2 , are for the SQUID. Next, we discuss how the feedback flux −ΔΦ = I f × Mf , i.e. the feedback current I f , can be generated by the electronic circuit. Figure 4.4 illustrates a fundamental circuit of a SQUID’s readout electronics in current bias mode for an FLL. In terms of control engineering, the circuit is a typical proportional–integral (PI) controller with a closed feedback loop. Here, P denotes a proportional amplifier, and I, an integrator. Owing to the PI controller, a flux change ΔΦ acting as an input signal of the readout electronics leads to ΔV out at the output of the integrator, which generates a compensation flux −ΔΦ via Rf and Mf , i.e. −ΔΦ = If Mf =

Mf ΔVout Rf

(4.1)

4.2 Flux Locked Loop (FLL)

Namely, −ΔΦ is generated by the ΔV out with the translation factor of Mf /Rf . As ΔΦ = −ΔΦ, the voltage and its corresponding flux state at the working point W are locked. The amplification factor of the P amplifier is given by R2 /R1 . The I integrator usually works within the cut-off frequency f c determined by the time constant RC and exhibits a low-pass filter effect. In the f < f c regime, the integrator acts as an open loop op-amp with infinite gain. There are two accomplishments of the integrator: (1) in the balanced case, i.e. when the voltages at the two input terminals of the op-amp (integrator) are equal, V out ≠ 0 is valid due to the existence of a memory element C; (2) when the voltages at the two input terminals of the I integrator are not equal, the integrator immediately becomes saturated due to its infinite gain. If the negative feedback circuit (FLL) is closed, performance (1), the voltage balance between two input terminals of the op-amp (integrator), always remains. However, performance (2) describes the unlocked case in FLL operation. To introduce the concept of the working point W, let us first assume that the FLL is open (e.g. the feedback resistor Rf is temporally removed). The amplified SQUID’s V (Φ) with a gain of R2 /R1 is present at the inverting terminal of the I integrator (monitoring point of V p (out)), where the SQUID’s V (Φ) characteristics exhibits the periodic characteristics shown in Figure 4.2b. The so-called “selecting the working point W” means adjusting a voltage V w at the noninverting terminal of the integrator. Usually, the working point W is selected by manually adjusting the voltage V w , which is schematically illustrated in Figure 4.5. Here, V p (out) is the periodic function V (Φ), and the voltage V w should aim for a target voltage Vw′ at V p (out) for selecting the working point W. In principle, any point on the SQUID’s V (Φ) curve with 𝜕V /𝜕Φ ≠ 0 can be chosen as the working point W for the FLL operation. However, in practice, the working point W is usually located on V (Φ) at Φw ≈ (2n + 1)Φ0 /4, where the flux-to-voltage transfer coefficient 𝜕V /𝜕Φ reaches a maximum. Only in the balanced case of Vw = Vw′ is the integrator not saturated despite its “infinite” gain. In one period of Φ0 , two points from the SQUID’s V (Φ) characteristics (the black one and the gray one) have the same voltage value of Vw′ with opposite slope polarities (see Figure 4.5). Owing to the slope polarity, only one point (e.g. the black point with a positive slope) can be selected as the working point W to build a negative feedback loop for FLL operation. For the other point (gray), because of the positive feedback loop, the equilibrium condition is not reached, and the working point will jump to the nearest point that yields a stable negative feedback. In fact, the working point W is automatically set at a suitable slope for a stable operation in the FLL, as V w is manually determined. Vp(out)

Vw′

Vw

Adjustable

Figure 4.5 Schematic diagram of working point selection from V(Φ) in a period of Φ0 . Here, an adjustable voltage V w at the noninverting input terminal of the integrator aims for the value of Vw′ at the working point W from V(Φ) at V p (out), where the flux-to-voltage transfer coefficient 𝜕V/𝜕Φ reaches a maximum.

Φ

27

28

4 Functions of the SQUID’s Readout Electronics

4.2.3

“Locked” and “Unlocked” Cases in the FLL

In FLL operation, there is a causal relationship, i.e. ΔΦ is an action and −ΔΦ is its reaction. Actually, this relationship is a dynamic balance process, where the working point W is transiently displaced and then rapidly returns. Any ΔΦ can be regarded as the integration of flux microchanges 𝜎Φ. Next, we analyze two cases shown in Figure 4.6. Assuming that the working point W at Φ0 /4 is selected (see Figure 4.6) and the FLL is closed, i.e. in the “locked” case, any 𝜎Φ will lead to transient displacement from the selected working point W, thus causing a voltage microchange 𝜎V at the input. This 𝜎V is amplified by the “P” amplifier and applied at the inverting input terminal of the integrator “I,” so the two terminals of the integrator lose balance, thus generating a compensation voltage at V out that produces −𝜎Φ into the SQUID loop via Rf and Mf to reset W immediately. Thus, the working point W returns to its original state, and a new balance between the two terminals of the integrator “I" is established again; however, 𝜎V out ≠ 0 appears at the integrator output and is saved in the integrator capacitance C. In the next moment, 𝜎Φ′ makes a new transient displacement, thus leading to a new −𝜎Φ′ for resetting the working point W while accumulating the new 𝜎V out at V out . Because of this process of “displacement” and “return” described in Figure 4.6a, ΔV out at the output of integration is proportional to ΔΦ even though ΔΦ ≫ Φ0 . Thus, the periodically modulated (nonlinear) characteristics of V (Φ) are hidden, as mentioned above. In the “unlocked” case shown in Figure 4.6b, once the transient displacement 𝜎Φ from the working point W selected at Φ ≈ (2n + 1)Φ0 /4 is larger than Φ0 /4, the slope polarity of this working point on the SQUID’s V (Φ) characteristics will be reversed. The flux feedback then becomes positive, thus leading to instability in the FLL. In this case, the SQUID system is temporarily in an unstable state. Positive feedback will drive the system to the next stable point at the next period of the V (Φ) characteristics one flux quantum Φ0 higher. These flux jumps are often seen as precursor states before a SQUID completely unlocks (i.e. the output V

V

∂V > 0 ∂Φ

∂V >0 ∂Φ –σV

σV

W

W

Φ (a)

∂V 106 Φ0 /s is a serious mission for a SQUID’s readout development [3–7]. In fact, the slew rate depends on the readout schemes below, e.g. in Chapter 7 (flux modulation scheme [FMS]) and Chapter 10 (weakly damped SQUID).

4.3 Suppressing the Noise Contribution from the Preamplifier Because a SQUID is a flux sensor with very low intrinsic noise δΦs , it is challenging to develop readout electronics with an electronic noise contribution δΦe below the intrinsic SQUID noise δΦs . The third function of the readout electronics is to suppress the preamplifier noise contribution by improving the SQUID input circuit “head stage.” For example, the first SQUID readout electronics with a FMS were developed in 1967 [1], as will be described in Chapter 7. In fact, the optimization of the SQUID input circuit to suppress the noise contribution δΦe from the readout electronics is the main subject throughout this book. Namely, the main point of this book is to introduce and discuss different readout schemes developed in the past half century.

4.4 Two Models of a dc SQUID Before we discuss low-noise SQUID magnetometric systems in the forthcoming Chapters 5–11, we should summarize which key values of a SQUID can be read out by readout electronics. In fact, the SQUID readout principle is based on the I–V characteristics, in terms of the dynamic resistance Rd (Φ). Here, the two functions of V (Φ) and I(Φ) are two projections of the I–V characteristics of SQUID when fixing the bias current and the bias voltage, respectively.

29

30

4 Functions of the SQUID’s Readout Electronics

Rd (fixed)

Rd (fixed)

Rd(Φ)

ΔVs = ΔΦ × (∂V/∂Φ)

ΔIs = ΔΦ × (∂I/∂Φ)

ΔΦ (a)

(b)

(c)

Figure 4.7 For readout electronics, a dc SQUID can be described with either the resistance model (a) or the differential model, assuming that the SQUID operates in current bias mode (b) and in voltage bias mode (c).

In fact, the Rd (Φ) of a dc SQUID can be generally divided into two models shown in Figure 4.7: a resistance model (a) and two differential models, which depend on the bias modes (b, c). In the former case, Rd (Φ) can be expressed as Rd (Φ) = Rd + ΔRd (Φ) where Rd is considered as a fixed resistance and ΔRd (Φ) is a dynamic resistance periodically modulated by Φ. According to SQUID bias modes, the ΔRd (Φ) part is translated into the readout quantity V (Φ) (or I(Φ)) displayed in Figure 4.1, which is often characterized by the voltage swing V swing (or the current swing I swing ) of the period Φ0 /2. In FLL operation, the working point W is transiently displaced and reset due to the microchange 𝜎Φ, so that the SQUID is operated in the differential model. For example, the V (Φ) in current bias mode is hidden, and only the value of Rd and the flux-to-voltage transfer coefficient 𝜕V /𝜕Φ at the working point W become visible. Equivalently, under voltage bias I(Φ) is hidden and only the Rd and the flux-to-current transfer coefficient 𝜕I/𝜕Φ can be observed. Therefore, the SQUID can be regarded as a flux-dependent voltage (or current) source connected in series to the fixed Rd , which acts as the source’s internal resistance. However, the differential model is sometimes borrowed to analyze V (Φ) (or I(Φ)) characteristics, where Rd and 𝜕V /𝜕Φ (or 𝜕I/𝜕Φ) are assumed approximately constant at their slopes of V (Φ) (or I(Φ)). According to the SQUID differential model in FLL, only three values of a SQUID at the working point W are electronically readable: (1) The first is the value of the fixed dynamic resistance Rd, i.e. 𝜕V /𝜕I at W, from the I–V characteristics. (2) 𝜕Rd /𝜕Φ translates to the transfer coefficient 𝜕V /𝜕Φ in current bias mode or to 𝜕I/𝜕Φ in voltage bias mode. One of the two is indeed effective in readout electronics.

References

(3) The SQUID’s intrinsic noise δΦs shows up as a voltage noise δV s or as a current noise δI s according to the SQUID’s bias mode. Here, the transfer coefficient bridges the intrinsic flux noise δΦs and the observed δV s (or δI s ), namely, δV s = δΦs × 𝜕V /𝜕Φ (or δI s = δΦs × 𝜕I/𝜕Φ). Actually, these three readable values of a SQUID at W determine the SQUID system noise. In practice, all readable values depend on the SQUID parameters, 𝛽 c , Γ, and 𝛽 L , which will be discussed in Chapter 6.

References 1 Forgacs, R.L. and Warnick, A. (1967). Digital-analog magnetometer utilizing

superconducting sensor. Review of Scientific Instruments 38 (2): 214–220. 2 Drung, D. (1996). Advanced SQUID read-out electronics. In: SQUID Sensors:

3 4

5

6

7

Fundamentals, Fabrication and Applications (ed. H. Weinstock), 63–116. Dordrecht/Boston/London: Kluwer Academic Publishers. Wellstood, F., Heiden, C., and Clarke, J. (1984). Integrated dc SQUID magnetometer with a high slew rate. Review of Scientific Instruments 55 (6): 952–957. Bick, M., Panaitov, G., Wolters, N. et al. (1999). A HTS rf SQUID vector magnetometer for geophysical exploration. IEEE Transactions on Applied Superconductivity 9 (2): 3780–3785. Chwala, A., Stolz, R., Ramos, J. et al. (1999). An HTS dc SQUID system for geomagnetic prospection. Superconductor Science and Technology 12 (11): 1036–1038. Leslie, K.E., Binks, R.A., Foley, C.P. et al. (2003). Operation of a geophysical HTS SQUID system in sub-Arctic environments. IEEE Transactions on Applied Superconductivity 13 (2): 759–762. Humphrey, K.P., Horton, T.J., and Keene, M.N. (2005). Detection of mobile targets from a moving platform using an actively shielded, adaptively balanced SQUID gradiometer. IEEE Transactions on Applied Superconductivity 15 (2): 753–756.

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33

5 Direct Readout Scheme (DRS) 5.1 Introduction In a SQUID magnetometric system, a SQUID that is operated at cryogenic temperature, e.g. at 4.2 K, has very low intrinsic noise, but the signals it measures are also very small and are read out by semiconductor electronics operated at room temperature (RT). The SQUID intrinsic noise δΦs is given as a flux noise in units √ of Φ / Hz , the electronics noise is characterized by its voltage noise in units of √0 √ V/ Hz, and its current noise in units of A/ Hz. The link between the flux noise and the electronics noise is the flux-to-voltage transfer coefficient 𝜕V /𝜕Φ of the SQUID in current bias mode or the flux-to-current transfer coefficient 𝜕I/𝜕Φ in voltage bias mode. To unify√ the noise units, the noise from readout electronics is usually expressed as Φ0 / Hz and denoted by δΦsys . In most applications, researchers wish to have a SQUID system with low system noise δΦsys . In the case of a direct readout scheme (DRS) when the SQUID directly connects to a preamplifier, e.g. an op-amp at RT (see Figure 5.1), unfortunately, the noise δΦe is larger than the SQUID intrinsic noise δΦs . In other words, δΦe in DRS dominates the SQUID system noise, i.e. δΦsys = [δΦ2e + δΦ2s ]1∕2 ≈ δΦe . Traditionally, people strive toward suppressing the noise δΦe below the noise δΦs [1, 2]. In this chapter, we only focus on the analysis of the noise component δΦe in DRS, while the fundamental mechanism and the performance of the SQUID’s intrinsic noise δΦs can be found in [3, 4].

5.2 Readout Electronics Noise in DRS In DRS, the readout electronics noise δΦe mainly originates from the preamplifier noise δΦpreamp ; therefore, we apply only δΦe = δΦpreamp in the following. In this case, δΦpreamp dominates δΦsys . Here, two noise sources contribute to δΦe : the √ preamplifier’s voltage noise V n originally characterized in units of nV/ Hz and √ the current noise I n with units of pA/ Hz. Here, V n and I n are always assumed to be uncorrelated. To understand the serious consequences of δΦe > δΦs , one often takes the√following example: when a SQUID has an intrinsic flux noise of δΦs = 1 μΦ0 / Hz SQUID Readout Electronics and Magnetometric Systems for Practical Applications, First Edition. Yi Zhang, Hui Dong, Hans-Joachim Krause, Guofeng Zhang, and Xiaoming Xie. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

34

5 Direct Readout Scheme (DRS)

Rg

1Ω – In Rs

A + Vn

AD797 or LT1028

Vout

Figure 5.1 The test circuit of an op-amp functioning as a voltage amplifier contains two noise sources, V n and In , where a source resistor Rs is grounded. The product of In × Rs is an independent voltage noise V n . Both V n and In × Rs contribute to the total voltage noise at the noninverting input of the op-amp.

(in frequency spectrum), the equivalent preamplifier noise δΦe should√be below √ 1 μΦ0 / Hz, which can be translated to a voltage noise V n < 0.1 nV/ Hz with the SQUID’s typical flux-to-voltage transfer coefficient 𝜕V /𝜕Φ = 100 μV/Φ0 in current bias mode. Unfortunately, such a low-noise preamplifier is nonexistent. In fact, the types of commercial preamplifiers suited for SQUID operation are very limited. Two different kinds of preamplifiers, op-amps and discrete elements (transistors), can be utilized for the SQUID readout. Here, a preamplifier acts either as a voltage amplifier in current bias mode (see Figure 4.4) or as a current-to-voltage converter in voltage bias mode (see Figure 3.6b). When ignoring the current the typical voltage noise of an assumed low-noise op-amp noise (I n ) contribution,√ √ (or transistor) of 1 nV/ Hz corresponds to δΦe = V n /(𝜕V /𝜕Φ) = 10 μΦ0 / Hz, which is already one order of magnitude higher than the SQUID intrinsic noise δΦs . Consequently, DRS is usually not considered for SQUID operation due to the large noise contribution from the electronics, i.e. δΦe ≫ δΦs . 5.2.1

Noise Characteristics of Two Types of Preamplifiers

The discussions in this subsection are based on Refs. [5, 6]. Two types of preamplifiers, either commercial op-amps (like AD797 or LT1028) or the parallelconnected bipolar pair transistor (PCBT) (e.g. 3×SSM2210 or 3×SSM2220), are most commonly used to read out a SQUID in DRS. First, let us understand the noise behavior of the two types of preamplifiers by analyzing the noise characteristics, V n and I n . The test circuit shown in Figure 5.1 is a voltage amplifier with a gain of approximately Rg /(1 Ω) for current-biased SQUIDs when Rg ≫ 1 Ω. At the noninverting input of the op-amp, a source resistor Rs is grounded, which may represent the fixed Rd in Figure 4.7b. Each amplifier, e.g. op-amp, possesses two independent noise sources, V n and I n , that contribute to the voltage noise. Actually, V n and I n can act as an ideal voltage source and an ideal current source, respectively, as defined in Section 3.2. Both behaviors are independent of Rs . The total noise voltage power δVe2 at the input terminal of the op-amp is expressed as δVe2 = Vn2 + (Rs × In )2 + 4kB TRs

(5.1)

where k B is the Boltzmann constant and T is the absolute temperature. Here, the contribution of V n does not depend √ on Rs , but I n does. In the test circuit, the thermal voltage noise of 0.13 nV/ Hz of the 1 Ω resistor at the inverting input in RT and the thermal noise of Rs at 4.2 K can be neglected in most cases.

5.2 Readout Electronics Noise in DRS

In

102

101

101 Vn

Vn (nV/√Hz)

103

In (pA/√Hz)

Figure 5.2 The measured noise spectra of In (left vertical axis) and V n (right vertical axis) for a commercial AD797 (op-amp).

100

100 10–1

100

101 102 Frequency (Hz)

103

104

Experimentally, these two independent noise sources of the op-amp can be calibrated separately. In the case of Rs ≈ 0, i.e. the input terminal directly connected to the ground, we obtain δV e = V n (see Eq. (5.1)), so V out in Figure 5.1 represents V n amplified by a gain of Rg /(1 Ω), e.g. G = 1000. In this case, V n of the op-amp √ (AD797) is calibrated, as plotted in Figure 5.2. For AD797, V n ≈ 1 nV/ Hz in the white noise range and a corner frequency f c ≈ 3 Hz for the crossover between 1/f noise and the white noise were measured (see the right axis of Figure 5.2). To accurately measure I n , one often plunges a large Rs of, e.g. 10 kΩ, into liquid helium (4.2 K). In this case, the three noise parts of δV e in Eq. (5.1), namely, items, the meaV n , (Rs × I n ), and 4k B TRs , have their own contributions. Two√ sured V n and the calculated thermal noise (4k B TRs )1/2 = 2.7 nV/ Hz, are already known. We then separate the item (Rs × I n )2 from the acquired δVe2 and plot I n in Figure 5.2 (see the left axis). In contrast to the behaviors of V n , I n presents a large low-frequency noise, i.e. the √ 1/f noise contribution to I n is quite strong. Here, a current noise of I n ≈ 1 pA/ √ Hz can be obtained at frequencies above 1 kHz, √ while I n increases up to 40 pA/ Hz at 1 Hz. For I n = √ 1 pA/ Hz and Rs = 10 kΩ, the current noise contribution of (Rs × I n ) = 10 nV/ Hz should dominate the total observed voltage noise power δVe2 in Eq. (5.1), because the noise power of Vn2 is less than 1% of δVe2 and the thermal noise of 4k B TRs is approximately 10%. Indeed, a large Rs highlights the contribution of I n in δV e . Therefore, the expression I n ≈ δV e /Rs can be approximated for the case of Rs > 10 kΩ. Although all elements in the PCBT circuit depicted in Figure 5.3, e.g. the low-noise pair transistors (SSM2220), are commercially available, complete PCBT electronics have not yet been commercialized. In fact, the function of the circuit is similar to that of a voltage amplifier consisting of an op-amp with two input terminals (V b+ and V b− ) and one output terminal, where the gain factor is determined by the ratio R2 /R1 . The PCBT preamplifier can be divided into three parts: (i) a symmetric amplifier set up by three transistor pairs (3 × 2); (ii) a current source providing the collector current I col determined by the common voltage of the emitters V e , while the bases of the transistors are considered grounded; and (iii) an op-amp acting as a voltage amplifier, where

35

36

5 Direct Readout Scheme (DRS)

V–

Rcol Rcol – Icol /2

Icol /2 3 × SSM2220

Vb+ Rd

Rs

T1

T2

T3

Ve

T3′

Vb– T2′

Icol

Vout

+

Vc2

Vc1

Op-amp

T1′

R2 1 kΩ R1 1Ω

Current source V+

Figure 5.3 Test circuit of a PCBT as a voltage amplifier with G = 1000 for calibrating two noise sources, V n and In . In principle, a PCBT consists of three parts: a symmetric amplifier including three paired transistors, a current source, and an op-amp. Here, a current-biased SQUID is connected at the noninverting input V b+ of the PCBT.

two collector voltages V c1 and V c2 are applied at its input terminals and its output connects to a resistive-feedback network consisting of R1 and R2 . When a voltage change ΔV occurs at the input terminal V b+ (e.g. the SQUID signal), ΔV must be mirrored at V b− to balance the op-amp, and two identical resistors (Rcol ) with high precision should be required at the √ inputs of op-amp. The PCBT s preamplifier exhibits a low voltage noise, V = V ∕ 6, but a large current noise, n n √ s s s 6, where Vn and In denote the noise of one single transistor. Here, In = In × √ the factor 6 represents the fact that six transistors are connected in parallel. In principle, one can use six discrete transistors for this circuit, but three paired transistors (e.g. 3×SSM2220) effectively improve the symmetry and reduce the temperature drift. Using the same measuring method as described in the noise calibration of the op-amp, we obtained the noise spectra of the PCBT as shown in Figure 5.4. The measured voltage noise spectrum V√ n of the PCBT in the white noise regime yields a white noise floor of 0.35 nV/ Hz. This √ value is determined by the Pythagorean sum of the thermal noise of 0.17 nV/ Hz (at 300 K) of a 1 Ω resistor (R1 ) at the inverting √ terminal (the input V b+ is grounded) and the claimed voltage noise of 0.32 nV/ Hz as specified for the SSM2220 in the datasheet (https:// www.alldatasheet.com/datasheet-pdf/pdf/49084/AD/SSM2220.html). In other words, the thermal noise of R1 is not negligible in the measured V n of the PCBT. Therefore, we suggest replacing the 1 Ω resistor (R1 ) with a smaller resistor, e.g. approximately 0.3–0.4 Ω, for the PCBT. When the noninverting terminal of the PCBT, V b+ , is open (this case often happens during a SQUID test), the preamplifier becomes saturated. Thus, the V out of the PCBT is close to the power

5.2 Readout Electronics Noise in DRS

Figure 5.4 The measured noise spectra of In (left vertical axis) and V n (right vertical axis) of the PCBT.

103

10

101

10–1

100

Vn

100

101 102 Frequency (Hz)

103

Vn (nV/√Hz)

In (pA/√Hz)

In 2

10–1 104

supply voltage, so the output current V out /R2 in Figure 5.3 should not exceed the allowed maximal output current of the transistors. With the power supply voltage of 12 V, R2 ≥ 500 Ω should be designed to limit the maximal output current of the PCBT to 24 mA. Similar to Figure 5.2, the measured V n of the PCBT shows a corner frequency f c of a few hertz. In other words, the V n spectrum of the PCBT shown in√Figure 5.4 mostly exhibits white noise. Compared to the current noise the measured current noise I n of the of 1 pA/ Hz of the AD797 (op-amp), √ PCBT clearly increases to 5–6 pA/ Hz in√the white noise regime, and at 1 Hz, this noise reaches approximately 150 pA/ Hz, as shown in the current noise spectrum depicted in Figure 5.4. Sometimes, a PCBT is also called an “ultralow-noise preamplifier.” Actually, it may exhibit an ultralow voltage noise V n but at the expense of an increased current noise I n . Similar to a PCBT consisting of discrete elements, a preamplifier consisting of two low-noise√junction field effect transistors (Toshiba SK 146) in √ parallel, where I n ≈ 0.1 pA/ Hz at 100 kHz and V n ≈ 0.4 nV/ Hz, was reported by J. Knuutila et al. However, they did not show the noise behavior of this preamplifier at low frequency, e.g. lower than 100 Hz [7]. 5.2.2 Noise Contribution of a Preamplifier with Different Source Resistors Thus far, we have focused on the electrical noise of preamplifiers. In this subsection, we discuss the equivalent flux noise δΦe of the readout electronics contributed by the preamplifier’s V n and I n during SQUID operation, where the SQUID’s dynamic resistance Rd is regarded as a source resistor Rs in Figures 5.1 and 5.3. In the following discussion, we assume that a SQUID is still operated in current bias mode; therefore, V (Φ) is read out, and the preamplifier acts as a voltage amplifier. During SQUID operation, the preamplifier’s V n is independent of Rd ; i.e. it continues to contribute to the total noise, but the voltage noise of V In = I n × Rd caused by I n increases with increasing Rd , which varies from several ohms to tens of ohms at 4.2 K. Actually, compared to the contributions of

37

5 Direct Readout Scheme (DRS)

101

I and III

100 δVe = [V 2n + (In × Rs)2]1/2

38

10–1

II = In × 10 Ω

10–2 101 III

I 10

0

II = In × 50 Ω

10–1 10–2 10–1

100

101 102 Frequency (Hz)

103

104

Figure 5.5 The voltage noise δV e of an AD797 acting as a voltage preamplifier. Here, the upper graph represents the noise contributions when Rs = 10 Ω, and the lower graph represents the noise contributions when Rs = 50 Ω. Each plot contains three curves: (I) V n ; (II) V In = In × Rs (Rs = 10 or 50 Ω); and (III) total noise δV e . In the upper plot for Rs = 10 Ω, curves I and III are actually overlapping. In the lower plot for Rs = 50 Ω, δV e is dominated by V In below 1 Hz.

√ V In and V n , the thermal voltage noise of Rd can be neglected, e.g. 0.07 nV/ Hz at Rd = 20 Ω. Therefore, the total voltage noise δV e at the preamplifier input 2 1∕2 ] = [Vn2 + (In × Rd )2 ]1∕2 . terminals consists of only two terms: δVe = [Vn2 + VIn First, let us use two different resistors, Rs , of 10 and 50 Ω, which tentatively replace the SQUID dynamic resistance Rd , to simulate the noise contributions from V n and V In during SQUID operation. The corresponding V n and V In values of the op-amp (AD797) are indicated in Figure 5.5. Three curves are presented for the cases Rs = 10 Ω (upper) and 50 Ω (lower): (I) V n ; (II) V In = I n × Rs ; and (III) δV e , where the data of the preamplifier’s V n and I n are taken from Figure 5.2. Because of the large V n of the AD797, V In is negligible at small Rs = 10 Ω. In this case, δV e is dominated by only V n , so curves I and III are overlapping (upper curve). If Rs increases to 50 Ω, V In of curve II becomes apparent only in curve III at f < 1 Hz (lower plot). It is clear that for a certain Rs (SQUID’s Rd ), the δV e of an op-amp (e.g. AD797, LT1028) is dominated by the noise source V n . In contrast, the contribution of current noise source I n can be ignored. In a SQUID system, the flux noise is expressed as δΦe = δV e /(𝜕V /𝜕Φ) = V n /(𝜕V /𝜕Φ). Therefore, a large transfer coefficient 𝜕V /𝜕Φ at the SQUID’s working point W is beneficial for reducing the flux noise contribution from the preamplifier. Figure 5.6 elucidates the √ relation between δV e and Rs for the PCBT. Owing to the low V n = 0.35 nV/ Hz, the contribution from V In is highlighted when Rs = 10 Ω, and V In begins to dominate δV e below 30 Hz. However, for Rs = 50 Ω, V In dominates δV e over the whole frequency range. In this case, curves II and III are overlapping, and curve I (V n ) can be ignored.

5.3 Chain Rule and Flux Noise Contribution of a Preamplifier

101 III

100 δVe = [V 2n + (In × Rs)2]1/2

I 10–1 II = In × 10 Ω 10–2 101 100

III I

10–1 10

II = In × 50 Ω

–2

10–1

100

101 102 Frequency (Hz)

103

104

Figure 5.6 The total voltage noise δV e of a PCBT acting as a voltage preamplifier. Here, the upper graph represents noise contributions when Rs = 10 Ω and the lower graph represents noise contributions when Rs = 50 Ω. Each plot contains three curves: (I) V n ; (II) V In = In × Rs (Rs = 10 or 50 Ω); and (III) δV e . In the upper plot, the white noise of δV e is still determined by V n ; nevertheless, V In dominates below 30 Hz. In the lower plot, δV e is dominated by V In over the whole frequency range.

In brief, both types of preamplifiers, the op-amp (AD797, LT1028) and PCBT (3×SSM2210 or √ 3×SSM2220), exhibit shortcomings, i.e. √either a high voltage I n = 6 pA/ Hz in the white noise noise V n = 1 nV/ Hz or a high current noise√ range. Here, the voltage noise V n = 0.35 nV/ Hz of the PCBT is already very low and is well suited for SQUID operation. To give the √ readers an impression about low V n , we take the analogy that V n = 0.35 nV/ Hz is equivalent to the thermal noise of a 7.4 Ω resistor at 300 K. In contrast, for the PCBT, the 1/f noise of I n is much greater. In brief, these noise characteristics of the two types of preamplifiers lead us to the goal of suppressing the dominant noise source contribution in SQUID readout electronics according to the Rd values of different SQUIDs.

5.3 Chain Rule and Flux Noise Contribution of a Preamplifier At every point in the SQUID’s nonlinear I–V characteristics, Rd obeys the relation Rd = 𝜕V /𝜕I; i.e. Rd is the reciprocal of the derivative of the I–V characteristics. Next, we introduce a very important relation, the differential chain rule. As the three elements V , I, and R in Ohm’s law are functions of the same variable Φ, one may get the following relation: Rd = 𝜕V ∕𝜕I = [(𝜕V ∕𝜕Φ)∕(𝜕I∕𝜕Φ)]

(5.2)

39

40

5 Direct Readout Scheme (DRS)

where (𝜕V /𝜕Φ) and (𝜕I/𝜕Φ) are the SQUID’s transfer coefficients at the working point W for the flux-locked loop (FLL) in current bias mode and voltage bias mode, respectively. Using the SQUID’s transfer coefficients, one can translate the electrical noise into flux noise, e.g. δV /(𝜕V /𝜕Φ) = δΦ. For more detailed knowledge about the general mathematics of chain rule, the reader can refer to [8]. In fact, this differential chain rule (Eq. (5.2)) is very important for SQUID operations and has two main functions: (1) We can easily understand that a preamplifier with V n and I n yields the same contribution to the equivalent flux noise δΦe of the readout electronics regardless of whether the SQUID is operated in current bias mode or in voltage bias mode. (2) In fact, SQUID operation always exhibits only one transfer coefficient, which depends on the bias mode. If two parameters at W, the SQUID’s transfer coefficient, e.g. (𝜕V /𝜕Φ) in current bias mode, and Rd are known, we can use the chain rule to obtain the other transfer coefficient of the SQUID in voltage bias mode, i.e. (𝜕I/𝜕Φ) = (𝜕V /𝜕Φ)/Rd , although it is hidden. For the function (1), the flux noise contribution δΦe from the preamplifier can be expressed as follows: (a) In current bias mode, δΦe = δV e /(𝜕V /𝜕Φ), where 2 1∕2 ] = [Vn2 + (In × Rd )]1∕2 δVe = [Vn2 + VIn

(5.3)

(b) In voltage bias mode, δΦe = δI e /(𝜕I/𝜕Φ), where δIe = [IV2 n + In2 ]1∕2 = [(Vn ∕Rd )2 + In2 ]1∕2

(5.4)

However, δΦe does not change in the two bias modes because δV e = δ (I e × Rd ) and (𝜕I × Rd )/𝜕Φ = (𝜕V /𝜕Φ), thus leading to δΦe = δI e /(𝜕I/𝜕Φ) (in voltage bias mode) = δV e /(𝜕V /𝜕Φ) (in current bias mode). Function (2) will often be employed as a constraint condition subsequently, e.g. in Chapter 8 (parallel feedback circuit [PFC]) and Chapter 9 (series feedback circuit [SFC]). 5.3.1

Test Circuit Using the Same Preamplifier in Both Bias Modes

In principle, the basic circuits for both bias modes are introduced in Chapter 4. To compare the noise performance in the two typical bias modes, i.e. current bias and voltage bias, we built a test circuit with one preamplifier, where the SQUID can be operated in either bias mode selected by a double-pole double-throw (DPDT) switch [9]. Indeed, the experiment has two purposes: (i) we experimentally verify that the flux contribution δΦe from the preamplifier is independent of bias modes; (ii) it is proved that DRS is not suited for reading out strongly damped SQUID. The test circuit with one preamplifier and switchable bias mode contains two op-amps, as schematically shown in Figure 5.7a. The first op-amp (OP-1) is an ordinary forward voltage amplifier with a gain G1 = R2 /R1 (R2 ≫ R1 ), which is connected to the SQUID for both bias modes. When K 1 and K 2 of the DPDT switch are set at position 1, the SQUID is biased by the constant current V b /Rb when Rb ≫ Rd , as discussed in Section 3.2. In this case, the second op-amp (OP-2) plays the role of a second-stage reverse voltage amplifier with a gain G2 = RI2 /RI1 ,

5.3 Chain Rule and Flux Noise Contribution of a Preamplifier

1 Rb

R2

R1

2 Vb

– OP-1 +

Rd

2

RI1

RV2

RI2

1 Vm

– OP-2 +

RV1 (a) R2

R1

Rd

RI2

RI1

– OP-1 +

RV1

Rb

Vm

– OP-2 +

Vb (b) Rd

Amplifier OP-1

(c)

RV 2 Vb

Rb – OP-2 +

Vm

RV1

Figure 5.7 (a) Schematic diagram showing the test circuit, where the same preamplifier OP-1 is employed for two bias modes. Switching of the bias modes is realized by synchronous switches K 1 and K 2 . At position 1, the SQUID is operated in current bias mode, while at position 2, it is operated in voltage bias mode. (b) At position 1, the test circuit in (b) is a voltage amplifier with two stages, where OP-1 is the preamplifier. (c) In voltage bias mode (at position 2), the circuit of the current-to-voltage converter is modified. Unlike in Figure 3.6b, the SQUID voltage signal in this configuration is first amplified by OP-1, while the bias voltage is applied at the noninverting input terminal of OP-2 via a voltage divider consisting of RV1 and RV2 . Note that Rb plays the role of Rg in Figure 3.6b.

thus resulting in a total voltage gain of (R2 RI2 )/(R1 RI1 ) when combined with OP-1. Here, the noninverting input terminal of OP-2 is grounded via a small RV 1 , which has no influence on the gain. Also, at position 1 of the DPDT, the test circuit in Figure 5.7b is a voltage amplifier with two stages, where OP-1 serves as the preamplifier. When K 1 and K 2 are switched to position 2, the SQUID is operated in voltage bias mode, and its equivalent circuit is shown in Figure 5.7c. A combination of OP-1 with OP-2 acts as a current-to-voltage converter. Here, the role of OP-1 does not change, i.e. it continues to work as a voltage amplifier. However, the bias voltage V bias is not applied to OP-1 directly, as described by the

41

5 Direct Readout Scheme (DRS)

current-to-voltage converter in Figure 3.6b, but rather to the noninverting input terminal of OP-2 via a voltage divider consisting of RV 1 and RV 2 , i.e. V bias = (V b RV 1 /RV 2 ) when RV 2 ≫ RV 1 . This voltage is equivalent to a bias voltage ′ = Vbias ∕G1 = (Vb RV 1 ∕RV 2 )∕(R2 ∕R1 ) applied to the SQUID, where (R2 /R1 ) is Vbias the gain of OP-1. In this circuit, the voltages at the two input terminals of OP-2 should maintain balance. The output voltage V m of OP-2 generates a current I g flowing through Rb and the SQUID’s dynamic resistance Rd . The voltage across ′ = Ig × Rd . In this case, V m is proportional the SQUID should be clamped to Vbias to I g (Rb ≫ Rd ), thus establishing a current-to-voltage conversion. The total gain of the test circuit shown in Figure 5.7b is given by Rb /Rd . Note that OP-1 shares a partial contribution to the gain when Rb /Rd > R2 /R1 . In brief, the test circuit ensures that the same preamplifier (OP-1) is employed for both bias modes. 5.3.2

Noise Measurements in Both Bias Modes

In DRS, the measured SQUID system noise δΦsys is the sum of δΦs and δΦe ; i.e. δΦsys = [δΦ2s + δΦ2e ]1∕2 . Here, a niobium (low-T c ) SQUID with 𝛽 c ≈ 0.3 (Ls = 350 pH, 2I c ≈ 8 μA, RJ ≈ 7.5 Ω) was measured by using the test circuit with one preamplifier shown in Figure 5.7. In current bias mode, the recorded SQUID’s V (Φ) curve (see inset (a) in Figure 5.8) is symmetrically quasi-sinusoidal. 𝜕V /𝜕Φ ≈ 85 μV/Φ0 and Rd ≈ 8 Ω were measured at the working point W. In voltage bias mode, the I(Φ) curves (inset (b) in Figure 5.8) resulted in 𝜕I/𝜕Φ ≈ 10 μA/Φ0 at W when Rd ≈ 8 Ω. This result means that the differential chain rule 𝜕V /𝜕Φ = (𝜕I/𝜕Φ) × Rd ≈ 80 μV/Φ0 is maintained when the bias mode is changed from current bias to voltage bias. Figure 5.8 shows two SQUID system noise δΦsys spectra plotted in curves I (current bias) and II (voltage bias). The two measured δΦsys in the FLL are nearly the same and are dominated by the preamplifier noise contribution, i.e. δΦsys ≈ √ √ δΦe = V n /(𝜕V /𝜕Φ) = 1 nV/ Hz/(85 μV/Φ0 ) ≈ 11 μΦ0 / Hz. The divergence 103 Flux noise (μΦ0 /√Hz)

42

(a) V 2

10

(b) W I

Φ

W Φ

I 101 5 × 100 100

II 101

102

103

Frequency (Hz)

Figure 5.8 Noise measurements of a strongly damped SQUID with 𝛽 c ≈ 0.3 in two bias modes. Curves I and II represent the SQUID system noise δΦsys in current bias mode and in voltage bias mode, respectively. The two insets show V(Φ) and I(Φ) denoted by (a) and (b), respectively. The working points W are marked. Here, a commercial op-amp (OP-1, AD797) was employed as a preamplifier in the circuit depicted in Figure 5.7.

References

between curves I and II below 10 Hz may be caused by an unexpected low-frequency vibration of the probe during one measurement or by the different working points selected in the two bias modes. In brief, the noise measurements shown in Figure 5.8 demonstrate that (i) the electronics noise δΦe does not depend on the SQUID’s bias mode; i.e. the differential chain rule (Eq. (5.2)) is valid, and (ii) the intrinsic noise δΦs of a SQUID with strongly damped junctions cannot be observed in DRS because δΦe of the op-amp (e.g. AD797) dominates δΦsys .

5.4 Summary of the DRS In this chapter, we discussed two electrical noise sources, voltage noise V n and current noise I n , of two types of preamplifiers, an op-amp (AD797) and a PCBT (3×SSM2220). The influence of Rs (Rd ) on δΦe was also demonstrated. Furthermore, it was experimentally confirmed that the noise contribution of δΦe does not depend on the SQUID’s bias modes. In most cases of DRS, e.g. a SQUID with strongly damped junctions, the system noise δΦsys is dominated by the readout electronics noise δΦe , i.e. δΦsys ≈ δΦe . Nevertheless, DRS exhibits several advantages: (i) The principle of the DRS is very easy to understand; (ii) the noise contributions from both sides, δΦe and δΦs , can be separately analyzed; (iii) the readout electronics need only two adjustments, i.e. determination of the bias current (or voltage) and selection of the working point W for FLL locking, so the DRS is very user-friendly; (iv) it provides the possibility to determine the three original readable SQUID parameters (δΦs , Rd , and 𝜕V /𝜕Φ) mentioned in Chapter 4. Furthermore, the SQUID’s transfer coefficient 𝜕V /𝜕Φ (𝜕I/𝜕Φ) at the working point W in FLL plays two important roles: (i) it bridges different kinds of noise sources, thus unifying all noise in √ units of Φ0 / Hz as the SQUID is a flux sensor; (ii) a large transfer coefficient is beneficial for reducing δΦe for observation of the SQUID’s intrinsic noise δΦs . Consequently, DRS is still attractive for some SQUID applications, although δΦe is larger than δΦs . In fact, different readout schemes have been developed in this way in the last 50 years to increase the apparent transfer coefficient at the input terminal of the preamplifier.

References 1 Forgacs, R.L. and Warnick, A. (1967). Digital-analog magnetometer utilizing

superconducting sensor. Review of Scientific Instruments 38 (2): 214–220. 2 Drung, D. and Mück, M. (2004). SQUID electronics. In: The SQUID Handbook

(eds. J. Clarke and A.I. Braginski), 128–165. Weinheim: Wiley-VCH. 3 Kurkijärvi, J. (1972). Thermal fluctuation noise in a superconducting flux

detector. Applied superconductivity conference, Annapolis, Maryland, USA (1 May 1972). New York: Institution of Electrical and Electronics Engineers, Inc: 581–587.

43

44

5 Direct Readout Scheme (DRS)

4 Weinstock, H. (1996). SQUID Sensors: Fundamentals, Fabrication and Applica-

tions. Dordrecht: Kluwer Academic Publishers. 5 Zhao, J., Zhang, Y., Lee, Y.H., and Krause, H.J. (2014). Investigation and

6

7

8 9

optimization of low-frequency noise performance in readout electronics of dc superconducting quantum interference device. Review of Scientific Instruments 85 (5): 054707. Zhang, G.F., Zhang, Y., Hong, T. et al. (2015). Practical dc SQUID system: devices and electronics. Physica C: Superconductivity and Its Applications 518: 73–76. Knuutila, J., Kajola, M., Seppä, H. et al. (1988). Design, optimization, and construction of a dc SQUID with complete flux transformer circuits. Journal of Low Temperature Physics 71 (5–6): 369–392. Osler, T.J. (1973). Chain rule for derivatives of arbitrary order. 1. General formulas. Notices of the American Mathematical Society 20 (1): A138–A139. Zeng, J., Zhang, Y., Qiu, Y. et al. (2014). Superconducting quantum interference devices with different damped junctions operated in directly coupled currentand voltage-bias modes. Chinese Physics B 23 (11): 118501. https://doi.org/10 .1088/1674-1056/23/11/118501.

45

6 SQUID Magnetometric System and SQUID Parameters Before we discuss the main topic of the low-noise readout technique in Chapters 7–11, the concept of a SQUID magnetometric system should first be introduced. For a SQUID magnetometer, one strives for low magnetic field noise δBsys (i.e. high magnetic field sensitivity), which involves two aspects: a field-to-flux transformer circuit (converter) and an ordinary SQUID system with an FLL. The former converts a magnetic field signal B into a flux Φ threading the SQUID loop, characterized by the field-to-flux transfer coefficient (𝜕B/𝜕Φ) in units of T/Φ0 , while the latter reads out ΔΦ, which is proportionally translated to the output voltage, V out , of the readout electronics. Here, δBsys is the product of (𝜕B/𝜕Φ) and δΦsys . To achieve a low δBsys (or, a high sensitivity), both quantities should be small. In Section 6.1, the requirements of the converter are discussed. In Section 6.2, we show that the SQUID system is characterized by three dimensionless characteristic parameters, 𝛽 c , Γ, and 𝛽 L , which determine the SQUID system noise power δΦ2sys , i.e. not only the SQUID intrinsic noise δΦ2s but also the readout electronics noise δΦ2e , in units of Φ20 ∕Hz.

6.1 Field-to-Flux Transformer Circuit (Converter) Compared with the ordinary SQUID with feedback coil Lf for FLL operation shown in Figure 4.3, in a magnetometric system, a dc SQUID is also coupled to an additional coil Lin with mutual inductance Min , as shown in Figure 6.1a. Here, Lin can connect to different pickup antennas Lpickup to construct closed field-to-flux transformer circuits (converter). Both Lin and Lpickup must be superconducting. The pickup antennas could be constructed in different configurations, e.g. wire-wound axial pickup antennas (see Figure 6.1b) [1–3]. There are other possibilities to construct a planar (LTS or HTS) magnetometer integrated with the field-to-flux transformer on a SQUID chip (not shown here) [4–6].

SQUID Readout Electronics and Magnetometric Systems for Practical Applications, First Edition. Yi Zhang, Hui Dong, Hans-Joachim Krause, Guofeng Zhang, and Xiaoming Xie. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

6 SQUID Magnetometric System and SQUID Parameters

Lf

Rf

Mf SQUID

46

P1

Lin

Min

(i) P 1 P2

FLL Vout

(ii) Readout electronics

P1 P2

(iii) l

B P1 P2 l ∂B/∂l

P2 (a)

Lpickup

l ∂ 2B/∂l 2

(b)

Figure 6.1 (a) Schematic “head stage” of a SQUID magnetometric system, consisting of an additional coil Lin and an ordinary SQUID system with readout electronics, feedback resistor Rf , and coil Lf for FLL operation. Usually, a multiturn spiral input coil Lin is integrated on top of the SQUID washer; (b) three possible wire-wound pickup antennas forming a magnetometer (i), a first-order gradiometer (ii), and a second-order gradiometer (iii) to connect with Lin to form a closed field-to-flux converter. Here, only the flux threading the SQUID loop is measured.

A large flux capture area of a separate pickup loop Lpickup can be imposed on the small loop of the SQUID washer via a multiturn spiral input coil Lin . To optimize the field-to-flux converter system, three issues should be considered at the same time: (1) A measured magnetic field B in units of Tesla (T) induces a current flowing in the two connected superconducting coils of the closed field-to-flux converter. To achieve a minimum 𝜕B/𝜕Φ, the inductance of the detection antenna, Lpickup , should match with Lin [1, 7]. As shown in Figure 6.1a, P1 and P2 denote the connection points of Lin and Lpickup . The former is mostly constructed in planar structure, while the latter can be set up in planar structure or wire-wound configurations. Some references describe inductance matching [7–9] and antenna structures for magnetometers and for gradiometers of different orders [2, 3, 7, 10, 11]. Planar multiloop SQUIDs can also act as sensitive magnetometers [12, 13]. For some insensitive magnetometers with single layer structures used in special applications, the SQUID washer can be simply used as a pickup loop Lwasher mismatched with the SQUID loop inductance Ls [14, 15]. (2) The SQUID loop inductance Ls plays an important role in the magnetometer system. Usually, a large Ls is provided by a large geometrical SQUID loop and leads to a small coefficient 𝜕B/𝜕Φ [16–18] but also to a large SQUID intrinsic flux noise δΦs [19]. In brief, one should find a suitable Ls as a good compromise to achieve a minimal δBsys . In SQUID magnetometric systems, most Ls values are designed in the range of 200–500 pH for liquid helium-cooled SQUIDs [17, 18, 20] and in the range from 50 to 100 pH for YBCO dc SQUIDs at 77 K [4, 21–24]. With Ls in such ranges, δBsys = (𝜕B/𝜕Φ) × δΦsys may achieve low levels. According to the geometrical size of the SQUID loop, Ls (without Lin ) can be calculated as given in Refs. [25–27]. Actually, the screening effect of the closed superconducting field-to-flux converter reduces the effective SQUID inductance Ls,eff via the multiturn spiral input coil Lin integrated on the washer [28].

6.1 Field-to-Flux Transformer Circuit (Converter)

Table 6.1 Parameters of LTS SQUID magnetometers with Ap of 5 × 5 mm2 . Ls

(pH)

350

350

350

480

480

620

Turns of Lin

—

3.5 × 2

4.5 × 2

5.5 × 2

2.5 × 2

5.5 × 2

2.5 × 2

620 4.5 × 2

Ls,eff

(pH)

235

180

140

365

190

420

310

𝜕B/𝜕Φ

(nT/Φ0 )

1.47

1.50

1.63

1.24

1.53

1.09

1.28

Table 6.2 Parameters of LTS SQUID magnetometers with Ap of 10 × 10 mm2 . Ls

(pH)

350

480

Turns of Lin

—

7.5 × 2

6.5 × 2

620 5.5 × 2

Ls,eff

(pH)

120

192

265

𝜕B/𝜕Φ

(nT/Φ0 )

0.55

0.47

0.4

In planar magnetometer structures with two pickup areas Ap , Zeng et al. investigated the 𝜕B/𝜕Φ related to turns of input coil Lin and SQUID inductance Ls , as listed in Tables 6.1 and 6.2 [9]. Further optimization of the field-to-flux transfer converter can be realized by an intermediary transformer [7]. (3) In theory, the value of 𝜕B/𝜕Φ can be reduced almost indefinitely by an enlargement in the pickup area, if the matching (Lpickup ≈ Lin ) remains. That is, δBsys may be continuously improved by increasing the pickup area. Recently, using an antenna gradiometer√with a pickup area 45 mm in diameter, an equivalent δBsys of 0.15 fT/ Hz was obtained for ultra-low field magnetic resonance imaging measurements in a magnetically shielded room [29]. Note that the obtainable δBsys strongly depends on the operating environment of the SQUID magnetometer. Nowadays, more and more high-frequency disturbances make the SQUID’s characteristics worse, thus increasing δΦsys and limiting δBsys . In some environments, SQUID systems cannot work at all. Here, two main issues, the pickup area (a) and Josephson resonances (b), should be taken into account: (a) To date, ambient disturbances have become a serious problem, especially when operating a SQUID magnetometer system in an unshielded or a lightly shielded environment. The SQUID operation is disturbed by an increasing number of communication signals that occupy all frequency bands, so the SQUID system flux noise δΦsys increases when enlarging the pickup antenna. For a required δBsys , the SQUID pickup area designed should be as small as possible because reducing the pickup area of the magnetometer (i.e. both Lin and Lpickup ) may effectively protect the SQUID against high-frequency interference [30]. In fact, the measured δΦsys in a superconducting shield, where most disturbances from outside are suppressed, is often much lower than the noise measured in a real environment, even in the white noise regime. A small pickup area could narrow down this difference with or without the

47

48

6 SQUID Magnetometric System and SQUID Parameters

superconducting shield. Certainly, the obtainable δBsys (or δΦsys ) depends on the measurement environment, which should determine a suitable pickup area. As a good compromise between the pickup area and δBsys in noisy environments, a SQUID √ magnetometer with a pickup area 2.5 mm in diameter and δBsys ≈ 20 fT/ Hz was developed, where a multiloop structure was employed [31]. All in all, high-frequency protection plays an important role in SQUID magnetometer operation [30]. (b) The second interference comes from the SQUID itself. On the one hand, an rf voltage V (t) appears across the junctions due to the ac Josephson effect mentioned in Eq. (2.2), thus inevitably leading to rf radiation during dc SQUID operation. On the other hand, numerous resonances occur, caused by the parasitic elements on the SQUID chip, and their natural quality factors may be very large in superconducting structures [32]. For example, a superconducting transmission line resonates at half the wavelength, 𝜆/2, where the SQUID washer can act as a ground plane [33]. In a modern SQUID magnetometer system, at least two coils, Lf and Lin , are integrated on the SQUID chip using planar thin-film techniques (see Figure 6.1). Furthermore, the resonant frequencies and their harmonics may be close to the Josephson frequency, thus resulting in resonances that lead to a distortion of SQUID’s characteristics, thus increasing δΦs . This negative effect from resonances becomes stronger with increasing inductances of Lin and Lpickup , when both act as receiving antenna system. Thus, one should consider methods for eliminating rf resonances during layout design, e.g. using resistive microstrips to shunt superconducting transmission lines [34]. In fact, this adverse effect can only be reduced by damping the resonances, but it cannot be completely avoided [33]. Because this book is focused on the readout system noise δΦsys , we only want to tell the reader that the noise caused by resonances cannot be ignored in a modern SQUID magnetometer system and that this noise contributes to the total system noise δΦsys . In brief, how one can achieve an optimal field-to-flux transfer coefficient (𝜕B/𝜕Φ) with a field-to-flux transformer circuit is not the major topic of this book. Here, we examine only briefly three caveats in this closed transformer coil system, i.e. inductance matching between Lpickup and Lin , a proper SQUID inductance Ls for an optimal δBsys , and a possible small pickup area for a required δBsys . For low-noise SQUID magnetometer operation, we also need to protect the system against high-frequency interference and to damp the resonances from the structures on the SQUID chip.

6.2 Three Dimensionless Characteristic Parameters, 𝜷 c , 𝚪, and 𝜷 L , in SQUID Operation If a closed field-to-flux transformer circuit already exists, the SQUID magnetometric system is governed by the properties of the SQUID system. The noise of an ordinary SQUID system is characterized by δΦsys , which is the sum of the

6.2 Three Dimensionless Characteristic Parameters, 𝛽 c , Γ, and 𝛽 L , in SQUID Operation

readout electronics noise and intrinsic SQUID noise, i.e. δΦ2sys = δΦ2e + δΦ2s . The main topic of this book is how to achieve a minimal δΦsys . From the perspective of physics, SQUID operation is based on two effects: flux quantization and the Josephson effect. However, regarding readout electronics, a dc SQUID can be regarded as a flux-dependent dynamic resistor Rd (Φ), or, its differential model, as shown in Figure 4.7. In our discussion, we assume that a dc SQUID always has two identical JJs; i.e. their parameters I c , C, and RJ are identical. In order to achieve a minimal δΦsys , we should expose the interrelations among Rd , (𝜕V /𝜕Φ), (𝜕I/𝜕Φ), δΦs , and δΦe , at the working point W in a SQUID’s differential model. In the interrelations, the three fundamental characteristic parameters of the SQUID, i.e. 𝛽 c , Γ, and 𝛽 L , play a major role. However, 𝛽 c and Γ originate from characterizing the JJs, and their important behaviors for RCSJ-type JJs have been discussed in Chapter 2. Only 𝛽 L pertains to the SQUID itself; that is, it is unique to the SQUID. In fact, the three dimensionless characteristic parameters are determined by four basic quantities of SQUID: the inductance of the SQUID loop Ls , the junction critical current I c , the shunt resistance per junction RJ , and the parasitic junction capacitance C. If both character parameters 𝛽 c and Γ are directly taken from a junction to SQUID operation, they are indeed nominal. However, one often confuses the 𝛽 c and Γ behaviors of JJs with those of SQUIDs. Importantly, the three dimensionless characteristic parameters, 𝛽 c , Γ, and 𝛽 L , further codetermine the three electrically readable SQUID values mentioned in Section 4.4, where, e.g. δV is the manifestation of δΦs in current bias mode, and the fixed resistance Rd and the transfer coefficient 𝜕V /𝜕Φ are responsible for δΦe . Next, we separately discuss the features of SQUID I–V characteristics and SQUID V (Φ) (or I(Φ)) with different characteristic parameters 𝛽 c , Γ, and 𝛽 L . 6.2.1 SQUID’s Nominal Stewart-McCumber Characteristic Parameter 𝜷 c According to the interpretation of parameter 𝛽 c in the junction case, the SQUID’s I–V characteristics should exhibit hysteresis when 𝛽 c > 1, and should be single valued for 𝛽 c < 1. When discussed in conjunction with a SQUID, the SQUID’s nominal Stewart-McCumber parameter 𝛽c = (2𝜋∕Φ0 ) × Ic R2J C is also used to describe damping intensity in the regime of 𝛽 c < 1. The SQUIDs can be further divided into three regions: strongly, intermediately, and weakly damped [19]. To avoid hysteresis, conventional SQUIDs were mostly designed to operate in the strongly damped regime (𝛽 c < 1), where RJ was selected to be only a few ohms. Some groups noted that (i) a SQUID can be operated without hysteresis even if 𝛽 c ≥ 1 [35], and (ii) δΦsys always achieves its minimum in the range of 1 < 𝛽 c < 2 [17, 36]. Interestingly, it was observed that the signal swing of dc SQUID increases with increasing nominal 𝛽 c in 2012 [37]. A systematic study of the relationship between a nominal SQUID’s 𝛽 c and the three electrically readable values was performed in 2013 [38]. We used niobium SQUIDs in which we changed only RJ while keeping the other three basic quantities, Ls , I c , and C, unchanged, thus varying 𝛽 c from 0.1 to 17. Surprisingly, the expected hysteresis was not visible

49

6 SQUID Magnetometric System and SQUID Parameters

16 8 0 –8 –16

βc ≈ 0.1 (#1) Sine wave

0.0 (a)

15

Square wave

0 –15 –30

Working point

1.0 1.5 0.5 Applied flux (Φ0)

βc ≈ 3.5 (#3)

30

βc ≈ 0.4 (#2)

Voltage (μV)

Voltage (μV)

50

2.0

βc ≈ 17 (#4)

0.0 (b)

Working point

0.5 1.0 1.5 Applied flux (Φ0)

2.0

Figure 6.2 Measured V(Φ) curves of current-biased SQUIDs with Ls ≈ 350 pH: (a) in the case of low nominal 𝛽 c values of 0.1 (SQUID #1, RJ = 5 Ω) and 0.4 (#2, RJ = 10 Ω) and (b) in the case of high 𝛽 c values of 3.5 (#3, RJ = 30 Ω) and 17 (#4, RJ = 65 Ω). The flux scale origin is arbitrary. The measured 2Ic values of all SQUIDs were approximately 8 μA. The shape of the V(Φ) curves is quasi-sinusoidal for 𝛽 c < 1 but quasi-rectangular for 𝛽 c > 3.5. No hysteresis is observed. Sine and square waves are traced with dotted lines to guide the eye.

even at 𝛽 c = 17. The V (Φ) curves of four SQUIDs with typical nominal 𝛽 c values are shown in Figure 6.2. In Figure 6.2a, the shapes of V (Φ) curves shown for 𝛽 c < 1 (SQUIDs #1 and #2) are quasi-sinusoidal. With a nominal 𝛽 c value increasing from 0.1 to 0.4, the SQUID voltage swing V swing , the flux-to-voltage transfer coefficient 𝜕V /𝜕Φ, and Rd at the working point W (obtained from the I–V characteristics, not shown here) increased from 8 to 28 μV (V swing ), 30 to 110 μV/Φ0 (𝜕V /𝜕Φ), and 3.5 to 10 Ω (Rd ), respectively. The shapes of V (Φ) curves in the cases of the two SQUIDs with nominal 𝛽 c > 1 (SQUIDs #3 and #4), as shown in Figure 6.2b, are nearly rectangular without any hysteresis visible. In this range of 𝛽 c , V swing increased slightly from V swing = 50 μV (𝛽 c = 3.5) to 54 μV (𝛽 c = 17), while both 𝜕V /𝜕Φ and Rd increased further from 350 to 600 μV/Φ0 and from 25 to 50 Ω, respectively. All the data are listed in Table 6.3. When 𝛽 c was increased from 0.1 to 17, the 𝜕V /𝜕Φ of the SQUID increased from 30 to 600 μV/Φ0 , thus reducing δΦe ≈ V n /(𝜕V/𝜕Φ) by 20 times, as analyzed in Chapter 5. The δΦsys values listed were experimentally √ measured by DRS with a commercial op-amp as a preamplifier (V n ≈ 1 nV/ Hz) in FLL mode. Based on Figure 5.5, δΦe is√simplified to V n /(𝜕V /𝜕Φ) and decreases from approximately 33 to 1.7 μΦ0 / Hz as 𝛽 c increases from 0.1 up to 17. Here, the value of (𝜕V /𝜕Φ) is acquired at the working point W at the V (Φ) characteristics. Furthermore, the δΦs values are estimated using the relationship δΦ2s = δΦ2sys − δΦ2e . Two limiting cases occur: δΦe dominates δΦsys because of a small 𝜕V /𝜕Φ for SQUID #1, while δΦs dominates δΦsys due to a large 𝜕V /𝜕Φ for SQUID #4 with 𝛽 c ≈ 17. The minimal δΦsys of √ 4.2 μΦ0 / Hz appears at 𝛽 c ≈ 3.5 for SQUID #3. In this case, δΦs and δΦe equally contribute to δΦsys , i.e. δΦs ≈ δΦe . The values listed in Table 6.3 show two useful pieces of information: (i) the three electrically readable values of current-biased SQUIDs, Rd , 𝜕V /𝜕Φ, and δV (δΦs ), increase with increasing nominal 𝛽 c and (ii) for a large nominal 𝛽 c , a large δΦs

6.2 Three Dimensionless Characteristic Parameters, 𝛽 c , Γ, and 𝛽 L , in SQUID Operation

Table 6.3 SQUID parameters measured by the DRS. SQUID no.

#1

#2

#3

#4

𝛽 c a)

—

0.1

0.4

3.5

17

RJ a)

Ω

5

10

30

65

Rd b)

Ω

3.5

10

25

50

V swing c)

μV

8

28

50

54

𝜕V/𝜕Φc)

μV/Φ0 √ μΦ0 / Hz √ μΦ0 / Hz √ μΦ0 / Hz

30

110

350

600

33

9.1

2.9

1.7

33

9.3

4.2

7.2

*

1.9

3.0

7

δΦe δΦsys d) δΦs

a) The nominal value. b) Derived from the I–V characteristics. c) Obtained from traces in Figure 6.2. d) Measured SQUID system noise in FLL mode. *not determined. Here, Ls = 350 pH for all SQUIDs.

is present, while a low δΦe benefits from a large 𝜕V /𝜕Φ, and vice versa. In the measurements with different nominal 𝛽 c values, the weight of δΦe on δΦsys was different although the preamplifier was the same. Suppressing δΦe below δΦs is the traditional goal for developing a SQUID readout technique. Of course, this goal is hardly realized with a DRS for strongly damped SQUIDs, as demonstrated in Chapter 5. Taking SQUID #2 as an example, if the condition of, e.g. δΦe = 0.7 × δΦs (i.e. δΦe < δΦs ) is fulfilled, a preamplifier √ with an equivalent voltage noise of Vn∗ = 0.7 × δΦs × (𝜕V ∕𝜕Φ) ≈ 0.15 nV∕ Hz is required, where δΦs and 𝜕V /𝜕Φ data are taken from Table 6.3. Regrettably, such a low-noise preamplifier at RT does not exist. However, the goal of δΦe < δΦs is easily realized with a large nominal 𝛽 c , even using a commercial op-amp as a preamplifier, e.g. in the case of SQUID #4. In the case of JJs, the single-valued I–V characteristics are independent of the applied flux Φ (under a normal condition). However, during SQUID operation, the I–V characteristics depend on Φ. It is believed that each SQUID’s I–V characteristics with different I c modulated by Φ correspond to a value of 𝛽 c when considering the SQUID as a junction. In other words, a flux-dependent function 𝛽 c (Φ) exists during SQUID operation. The values of the SQUID’s nominal 𝛽 c increase as Φ increases from Φ0 /2 to Φ0 . Namely, the SQUID’s nominal 𝛽 c at half-integer Φ0 reaches its minimum, where the SQUID’s I c achieves its minimum. Note that the values of 𝛽 c denoted in Figure 6.2 and in Table 6.3 represent the SQUID’s nominal 𝛽 c at the working point W, i.e. a quarter of an integer Φ0 . In brief, the Stewart-McCumber parameter 𝛽 c originates from the JJ, and the value of 𝛽 c traditionally is responsible for the onset of hysteresis for the JJ. In this subsection, we have introduced the new concept of a SQUID’s nominal 𝛽 c (Φ) to distinguish it from the conventional 𝛽 c of a junction. In fact, its particularity in SQUID operation has been found; i.e. the SQUID’s nominal 𝛽 c plays a major role in δΦsys , when using DRS.

51

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6 SQUID Magnetometric System and SQUID Parameters

6.2.2

SQUID’s Nominal Thermal Noise Parameter 𝚪

The dimensionless thermal noise parameter is expressed as Γ = 2𝜋k B T/I c Φ0 . Generally, Γ describes the rounding effect of the I–V characteristics of a JJ and is a quantity that reflects the apparent reduction in the critical current I c of the junction in the presence of thermal (Johnson) noise [39]. Here, a large Γ leads to a small apparent I c and a small apparent Rd and thus more rounded I–V characteristics of JJs, as schematically shown in Figure 2.3. For SQUID operation, two important statements regarding Γ should be noted: (i) traditionally, the value of Γ is not followed with interest in low T c SQUID operation, as long as the condition Γ ≪ 1 is fulfilled. For an example of Γ < 0.05 at 4.2 K, the influence of the thermal noise rounding effect on the I–V characteristics is typically neglected [40] (see Figure 2.3a); (ii) using Γ, one can approximate the maximal SQUID inductance, Ls(max) , at different operating temperatures, e.g. a practical Ls(max) ≈ 1 nH at 4.2 K [41]. However, based on thermal noise considerations, J.E. Zimmerman gave the following expression Ls(max) < Φ20 ∕kB T [42], which is much larger than 1 nH. Our interest is to determine what a SQUID’s nominal Γ is, because we need a clear answer to the following questions: Why does hysteresis not occur at 𝛽 c ≈ 17 for SQUID #4 in Figure 6.2b? Furthermore, what role does a SQUID’s nominal Γ play in SQUID operation? The estimated SQUID intrinsic noise δΦs in Table 6.3 is much higher than the pure SQUID thermal noise at 4.2 K caused by the junction’s resistance [39, 41]. Therefore, the real SQUID intrinsic noise δΦs must be dominated by other noise sources. Next, we attempt to provide a possible explanation for this phenomenon. In fact, in addition to the thermal noise of junctions, Voss noted that an excess power of voltage noise Sv across a tunnel junction is produced by random switching between the superconducting and nonzero voltage states in the case of 𝛽√ c >0 [43]. Along this idea, we may interpret the increase in δΦs up to 7 μΦ0 / Hz (SQUID #4) as resulting from an increase in Sv of the two SQUID’s junctions for 𝛽 c ≈ 17. Indeed, both the thermal noise and the random switching noise contribute to δΦs , i.e. δΦ2s = 2kB T[L2s ∕RJ + 2Rd ∕(𝜕V ∕𝜕Φ)2 ] + SV ∕(𝜕V ∕𝜕Φ)2

(6.1) 2kB TL2s ∕RJ

Here, the thermal noise contains two parts [43]: a static term and a dynamic term 4k B TRd /(𝜕V /𝜕Φ)2 . In brief, Sv dominates δΦs at 𝛽 c > 1. A white noise change can be equivalent to a temperature change ΔT. To describe the power of voltage noise Sv caused by random switching, we introduce a noise temperature T*, which is usually employed to characterize amplifier noise in rf techniques. We then rewrite the expression in Eq. (6.1) as δΦ2s = 2kB (T + T ∗ )[L2s ∕RJ + 2Rd ∕(𝜕V ∕𝜕Φ)2 ] 2

(6.2)

where the item Sv /(𝜕V /𝜕Φ) is converted to be an additional part of the thermal noise with T*. All the parameters in Eq. (6.2) can be determined experimentally, yielding the √ value of (T + T*) ≈ 30 K for δΦs ≈ 7 μΦ0 / Hz (SQUID #4). From the definition of Γ, (T + T*) ≈ 30 K leads to the SQUID’s nominal thermal rounding parameter

6.2 Three Dimensionless Characteristic Parameters, 𝛽 c , Γ, and 𝛽 L , in SQUID Operation

1.0

I/(2Ic)

Figure 6.3 Three simulated I–V characteristics of a symmetric SQUID with Γs = 0.05, 0.35, and 0.5 and 𝛽 c = 17, 𝛽 L = 1 at applied integer flux Φ = nΦ0 . The inset shows the measured I–V characteristics of SQUID #4, and these characteristics agree best with the simulation at Γs = 0.35. Symbols V and I denote the time-averaged dc voltage and bias current, respectively.

Γ = 0.05 Γ = 0.35 Γ = 0.50

0.5 1.0

SQUID #4

0.5

0.0

0.0

0.0

0.0

0.5 V/(IcRJ)

0.5

1.0

1.0

Γ* ≈ 0.35, which is much larger than the typical value of Γ = 0.05 (T ≈ 4.2 K, I c = 4 μA) used in many traditional simulations. This high Γ* value results in strong rounding of the I–V characteristics, thus causing the hysteresis to disappear at 𝛽 c ≈ 17. Figure 6.3 shows three simulated I–V characteristics with different simulated Γs values based on the model of a symmetric SQUID with 𝛽 c = 17 and 𝛽 L = 2Ls I c /Φ0 ≈ 1. For Γs = 0.05, the I–V characteristics are obviously hysteretic. However, the hysteresis disappears at Γs = Γ* = 0.35 (T = 30 K), which agrees with the measured I–V curve of SQUID #4 shown in the inset. At Γs = 0.5, the rounding effect is even stronger. According to Eq. (6.1), when neglecting the power of random switching voltage noise Sv , i.e. T* = 0, the noise limit of a dc SQUID is the sum of the static and dynamic noise as follows: δΦ2s = 2kB T[L2s ∕RJ + 2Rd ∕(𝜕V ∕𝜕Φ)2 ]

(6.3)

The so-called minimal observable noise (δΦs )min considers √ only static noise generated by RJ [19], thus resulting in (δΦs )min ≈ 1.4 μΦ0 / Hz for RJ ≈ 5 Ω and Ls ≈ 350 pH (SQUID #1 in Table 6.3). In most cases, the practically measured δΦ2s is roughly four times higher than the thermal (Nyquist) noise due to down mixing effects in the SQUID [40]. In brief, Zeng et al. [38] regarded SQUID with two parallel junctions as one junction and the SQUID’s I–V characteristics under different Φ as the junction’s I–V characteristics with different Γ* values. The SQUID’s nominal Γ* is no longer a constant but rather a function of the applied flux, i.e. Γ*(Φ). In practice, SQUID #4 shown in Table 6.3 can be considered to operate at (T + T*) = 30 K with Γ* = 0.35. Similar to the JJ, a hysteresis in SQUID operation appears that depends on both nominal parameters, 𝛽 c (Φ) and Γ*(Φ). At Φ ≈ nΦ0 , 𝛽 c (Φ) reaches its maximum and Γ*(Φ) yields its minimum, so the hysteresis is most likely to occur. The phenomena were experimentally observed that some SQUIDs can still work at W (quarter of integer Φ0 ), where the SQUID characteristics are hysteresis-free, although a hysteresis already appears near Φ ≈ nΦ0 . In fact, the random switching noise, Sv , increases with increasing SQUID’s nominal 𝛽 c , thus increasing both the SQUID intrinsic noise δΦs and the SQUID’s

53

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6 SQUID Magnetometric System and SQUID Parameters

nominal thermal rounding parameter Γ* marked with T*. Consequently, the absence of hysteresis in the SQUID I–V characteristics is maintained up to the SQUID’s nominal 𝛽 c ≫ 1, e.g. 𝛽 c ≈ 17. In contrast, the strongly damped SQUID #1 shown in Table 6.3 with a small SQUID’s nominal 𝛽 c exhibits small δΦs and 𝜕V /𝜕Φ values, which makes it difficult for readout electronics to observe δΦs . 6.2.3

SQUID’s Screening Parameter 𝜷 L

The SQUID’s screening (shielding) parameter 𝛽 L is also dimensionless and is defined by 𝛽L = 2Ls Ic ∕Φ0

(6.4)

The 𝛽 L is a pure SQUID characteristic parameter that normalizes the product of Ls I c , which is important for the design of a SQUID [19]. The 𝛽 L describes the modulation of the dc SQUID’s critical current change ΔI c between integer and half-integer flux, as the I–V characteristics shown in Figure 6.4. In other words, the characteristic parameter 𝛽 L determines the SQUID signal swing. Geometrically, the value of 𝛽 L is given as (I c(max) − ΔI c )/ΔI c , where I c(max) is the maximum critical current of the SQUID, appearing at Φ = nΦ0 (n = 0,1,2,…). For intuitive understanding, we provide geometrical images of three typical values of 𝛽 L : (1) at 𝛽 L ≈ 1, ΔI c is modulated by 50% of I c(max) , as shown by the black I–V characteristics; (2) for 𝛽 L → 0, ΔI c approaches I c(max) , as shown in the lower gray curves; (3) for 𝛽 L → ∞, ΔI c reduces to zero (not shown here). Sometimes, the modulation depth of ΔI c /I c(max) is also used to evaluate 𝛽 L , i.e. ΔI c /I c(max) = 50% in case (1), 100% in (2), and 0% in (3). For 𝛽 L < 0.3, although ΔI c /I c(max) ≈ 100% remains almost constant, the absolute value of ΔI c decreases with decreasing I c(max) , thus leading to a small swing in the SQUID signal. When 𝛽 L > 5, ΔI c /I c(max) decreases as the function of 1/𝛽 L , thus also resulting in a small ΔI c [44]. Overall, 𝛽 L ≈ 1 is a suitable value for SQUID operation when the absolute ΔI c might reach the maximum. In fact, the characteristic parameter 𝛽 L = 1 plays a very important role in SQUID design. The SQUID inductance Ls is often selected for a purpose first, e.g. Ls = 200–400 pH for low-T c SQUID magnetometric systems, and the target value of 2Ls I c = Φ0 is used to determine the design value of I c , which is ensured by the fabrication process. Figure 6.4 Geometrical images of three typical 𝛽 L values, 𝛽 L ≈ 0 (lower gray line), 1 (black), and 5 (upper gray), sketched in the SQUID’s I–V characteristics. For 𝛽 L ≈ 1, (Ic(max) − ΔIc ) ≈ ΔIc , where ΔIc reaches the maximum.

I RJ/2 βL ≈ 5 Ic(max)

βL ≈ 1

ΔIc Ic(max) – ΔIc

ΔIc(max) ≈ 6 μA βL ≈ 0

Ls ≈ 350 pH V

6.2 Three Dimensionless Characteristic Parameters, 𝛽 c , Γ, and 𝛽 L , in SQUID Operation

In SQUID operation, the physical limit of the maximal critical current change, ΔI c(max) , without any noise contributions (Γ → 0) can be expressed as ΔIc(max) ≈ Φ0 ∕Ls

(6.5)

As an example, for the SQUIDs with Ls = 350 pH, ΔI c(max) should be approximately 6 μA. If the voltage is taken as a readout quantity, e.g. if the SQUID is operated in current bias mode, the physical limit of ΔV (max) across the SQUID increases with increasing shunt resistor RJ , i.e. ΔVmax ≈ (RJ ∕2) × ΔIc(max) = (RJ ∕2) × (Φ0 ∕Ls )

(6.6)

For RJ = 5 Ω (or 30 Ω), ΔV max should be approximately 15 μV (or 80 μV). Practically, the measured V swing was 8 μV at RJ = 5 Ω and 50 μV at RJ = 30 Ω. The reduction of V swing may be caused by the thermal noise parameter Γ*. Indeed, the tendency of the measured V swing listed in Table 6.3 fits well with the increase in RJ . 6.2.4

Discussion on the Three Characteristic Parameters

Thus far, we have discussed the functions of the three characteristic parameters in SQUID operation. In contrast to the conventional parameters 𝛽 c and Γ of JJs, the SQUID’s nominal 𝛽 c (Φ) and Γ*(Φ) are modulated by the applied flux. Here, a reasonable interpretation of the absence of hysteresis in the SQUID’s I–V characteristics at 𝛽 c ≈ 17 is based on the hypothesis that the SQUID is operated at an equivalent temperature of T* ≈ 30 K, i.e. Γ* ≈ 0.35. For SQUID operation, the dimensionless characteristic parameter 𝛽 L particularly describes the modulation depth of the SQUID’s ΔI c with the flux, e.g. ΔI c yields its maximum at 𝛽 L ≈ 1, where ΔI c reaches 50% of I c(max) . Importantly, 𝛽 L ≈ 1 imposes a design condition on the product Ls I c . Three dimensionless characteristic parameters, 𝛽 c (Φ), Γ(Φ), and 𝛽 L , codetermine the three electrically readable SQUID values, thus further determining δΦs and δΦe . We experimentally verified that the three readable values increase with increasing SQUID’s nominal 𝛽 c , as shown in Figure 6.5, where the curves are fitted with the data of the four SQUIDs shown in Table 6.3. Clearly, the curves have no statistical meaning due to the very small number of samples; instead, they show only their tendencies. The data extracted from these curves, e.g. at 𝛽 c = 0.5, 1, and 3, as listed in the inset of Figure 6.5, may be referenceable because the deviation from the measured data is not large in our latter experiments. Furthermore, one can speculate that as 𝛽 c approaches infinity, the three electrically readable SQUID values approach their asymptotic limits. Using a two-stage readout scheme, Wang et al. measured the SQUID intrinsic noise δΦs with varied 𝛽 c from 0.3 up to 13.5 [45]. The results verified the rationality of the curve δΦs in Figure 6.5. During SQUID operation, the SQUID’s nominal 𝛽 c (Φ) determines the occurrence of hysteresis, which should depend on the SQUID bias modes. In our early study, it was observed that some hysteretic V (Φ) in current bias mode becomes nonhysteretic I(Φ) in voltage bias mode [35]. Especially, the seriously hysteretic V (Φ) in current bias mode become a nonhysteretic I(Φ) in voltage bias mode, as a small resistor Rs is directly shunted to the SQUID, i.e. Rs ≪ RJ . Here, the SQUID’s

55

6 SQUID Magnetometric System and SQUID Parameters

*

800

12 50

30 6

Rd

40

20

600 ∂V/∂Φ

*

400

From the fitting curve: βc ∂V/∂Φ δΦs Rd 11 143 1.7 0.5 1 15 200 2.3 3 24 310 3.3

3 10 βc = 1

0

δΦs

0 0

5

10

200

∂V/∂Φ (μV/Φ0)

9

Rd (Ω)

δΦs (μΦ0 /√Hz)

56

0

15

βc

Figure 6.5 The tendencies of the three electrically readable SQUID values with increasing SQUID’s nominal 𝛽 c .

nominal 𝛽 c may have a reduction [46]. Nevertheless, the rf external shunt effect is difficult to evaluate due to the distribution of high-frequency energy. For a practical SQUID system, we suggest that the nominal 𝛽 c should not be larger than 3. To design a SQUID magenetometric system, we recommend the following two steps: (1) The SQUID’s characteristic parameter 𝛽 L ≈ 1 should be fulfilled first, as most analyses and designs are based upon this value. Namely, the requirement of 𝛽 L ≈ 1 is essential to ascertain the product of Ls I c . Once Ls from 200 to 400 pH is selected for low-T c SQUIDs, e.g. for a sensitive SQUID magnetometer (see Section 6.1), then I c can be determined, i.e. 2Ls I c = Φ0 . (2) The next step is to use the junction shunt resistance RJ to further determine the value of the SQUID’s nominal 𝛽 c , if the junction capacitance C remains unchanged. Furthermore, the so-called optimal nominal 𝛽 c depends on the readout scheme. For example, to obtain minimal SQUID system noise δΦsys in a DRS, the selection of the nominal 𝛽 c depends on the employed preamplifier to balance both δΦs and δΦe . However, from a conventional standpoint, δΦe must be suppressed below δΦs , e.g. when a flux modulation readout scheme (FMS) is employed for a strongly damped SQUID. Here, Table 6.3 gives only an overview of the interrelations among Rd , (𝜕V /𝜕Φ), (𝜕I/𝜕Φ), δΦs , and δΦe , in current-biased SQUIDs when using an op-amp as preamplifier. A more detailed analysis will be performed in the chapter on the weakly damped SQUID (Chapter 10).

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7 Flux Modulation Scheme (FMS) We have experimentally demonstrated in Chapter 5 that the choice of bias mode does not affect the noise properties of the SQUID system. Specifically, neither the SQUID intrinsic noise δΦs nor the noise contribution δΦe from the readout electronics depends on whether the readout is performed in current or voltage bias mode. In direct readout scheme (DRS), the readout electronics noise δΦe is much higher than the SQUID intrinsic noise δΦs , thus dominating the system noise δΦsys for a strongly damped SQUID with 𝛽 c < 1 (see Table 6.1). In fact, to suppress δΦe the matching between the SQUID and readout electronics has always been a serious challenge in the development of SQUID systems. During the last half a century, five solutions have been employed to improve this matching: (1) the flux modulation scheme (FMS) [1]; (2) additional positive feedback (APF), i.e. a parallel feedback circuit shunting to the SQUID [2]; (3) the two-stage scheme, which uses a secondary SQUID to measure the output of the primary SQUID [3, 4]; (4) a weakly damped SQUID in a DRS [5]; and (5) an un-shunted SQUID with a reference junction (or a reference SQUID) to form the double relaxation oscillation SQUID (D-ROS) readout scheme [6]. Before we analyze the FMS (solution (1) above), let us first introduce the concept of the so-called “mixed bias mode” in Section 7.1. In Section 7.2, the FMS is discussed along with a conventional explanation. In Section 7.3, we revisit the FMS by analyzing the bias mode and the transfer characteristics of a step-up transformer.

7.1 Mixed Bias Modes The five readout schemes introduced above share the common idea of modifying the SQUID input circuit to enhance the SQUID’s apparent transfer coefficient at the input terminals of the preamplifier. Note that none of the readout schemes can change the SQUID’s intrinsic (original) properties, especially its intrinsic noise δΦs . In fact, most of the five readout schemes are operated in the so-called “mixed bias mode,” which is unfamiliar to many people, and often employ a modified input circuit at the “head stage” at cryogenic temperature, as sketched in Figure 7.1. Here, the SQUID is shunted by an element with impedance Zs . As a bias current I b flows through the parallel circuit, the SQUID can be biased in different modes. We introduce a dimensionless parameter 𝜒 = Zs /Rd SQUID Readout Electronics and Magnetometric Systems for Practical Applications, First Edition. Yi Zhang, Hui Dong, Hans-Joachim Krause, Guofeng Zhang, and Xiaoming Xie. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

62

7 Flux Modulation Scheme (FMS)

Ib Is

Iz

A

Rd Zs

Figure 7.1 The input circuit (head stage) is a SQUID shunted by an element Z s . According to the value of 𝜒 = Z s /Rd , the SQUID can be biased in either current or voltage bias mode. Here, Ib = Iz + Is is always valid. The flux feedback coil Lf depicted in Figure 4.3 is omitted here.

for analyzing the input circuits shown in Figure 7.1. There are two extreme cases: (1) Zs is absent (𝜒 → ∞), and (2) Zs is shorted, i.e. 𝜒 → 0. Case (1) for 𝜒 → ∞ can be attained by connecting the SQUID directly to a high-impedance op-amp, e.g. point “A” in the DRS described in Chapter 5. In this case, I z = 0, and the current I s flowing through the SQUID is equal to the bias current, I s = I b , thus determining the SQUID’s operation in current bias mode. In case (2), for 𝜒 → 0, I b flows predominantly through Zs (e.g. a very small resistor R), i.e. I z ≈ I b , thus producing a constant voltage V b across Zs and the SQUID. Here, V b remains constant and independent of Φ. Therefore, the I(Φ) characteristics are obtained using an ammeter connected to the SQUID in series, as shown in Figure 3.4b, i.e. the ammeter is connected to the SQUID in series. In fact, the SQUID is operated in voltage bias mode. However, case (2) for 𝜒 → 0 is impractical, as discussed in Chapter 4. In brief, the maximal transfer coefficient 𝜕V /𝜕Φ at 𝜒 → ∞ or 𝜕I/𝜕Φ at 𝜒 → 0 is reached in the two extreme cases, where the SQUID’s measured I–V , V (Φ), and I(Φ) characteristics can be denoted as the “original” ones. If the impedance Zs is finite, namely, 0 < 𝜒 < ∞, the bias mode becomes mixed, where the current I s flowing through the SQUID, or the voltage V b across the SQUID, cannot remain constant due to the varying Rd (Φ). In other words, the unstable I s deviates from the ideal current bias mode due to 𝜒 < ∞, while the voltage V b across the SQUID is also unstable because 𝜒 > 0. When a constant bias current I b flows through the circuit shown in Figure 7.1, according to Kirchhoff ’s law, this current divides into two currents, I s and I z , but both are modulated by Φ. Thus, the SQUID’s V (Φ) and I(Φ) characteristics appear side by side, but the swings in both V (Φ) and I(Φ) cannot reach their maximum. The appearance means that both characteristics are not “original.” Because the readout electronics can read only one quantity, either voltage or current, one does not notice the existence of the other quantity. When 𝜒 = 1, compared to that in the ideal current bias case, the apparent swing in V (Φ) across the SQUID is reduced by 50% from the original value at the input terminal of the preamplifier, i.e. point “A” in Figure 7.1, due to the voltage divider formed by Rd and Zs . Similarly, the apparent swing in I(Φ) through the SQUID is also suppressed by 50%. Therefore, 𝜒 = 1 is the boundary between the two bias modes. Except for the two extreme cases, 𝜒 → ∞ and 𝜒 → 0, we can divide the bias mode further into two regimes: (i) in a nominal current bias mode with 𝜒 > 1, V (Φ) across the SQUID is highlighted, and we therefore recommend reading out voltage signals across the SQUID, i.e. V (Φ); (ii) in a nominal voltage bias mode

7.2 Conventional Explanation for the FMS

with 𝜒 < 1, the current changes flowing through the SQUID, i.e. I(Φ), should be recorded. However, the swings in both V (Φ) and I(Φ) are reduced in mixed bias mode. Actually, the parameter 𝜒 indicates the preferable readout quantity, voltage or current, for the readout electronics. Now, the SQUID bias mode can be quantifiable using the dimensionless parameter 𝜒, with which we evaluate the bias modes of the above five readout schemes. In fact, two of the five, FMS and APF, are operated in a mixed bias mode of 1 < 𝜒 < 10, where the readout quantity is the voltage: in the FMS, the SQUID is connected to the primary winding (PW) of a step-up transformer, while in APF, the SQUID is shunted by a branch consisting of a coil and an in-series-connected resistor. In these two readout schemes, the apparent transfer coefficients 𝜕V /𝜕Φ at the input of the preamplifier increase, although the SQUID’s V swing does not reach the original maximum in the ideal current bias mode of 𝜒 → ∞. In the two-stage scheme, a small resistor R ≤ 1 Ω acts as Zs in Figure 7.1 so that the strongly damped sensing SQUID is nominally operated in voltage bias mode with 𝜒 ≤ 0.5. Thus, the readout quantity is the current, which is read out by a SQUID ammeter (the reading SQUID). In D-ROS, the shunt element (circuit) is necessary to operate the SQUID. The two-stage scheme and D-ROS are discussed in detail in Chapter 11. Only one of the five schemes, the DRS with a weakly damped SQUID, can operate in ideal current bias mode with 𝜒 → ∞. Using this scheme, one obtains the SQUID’s original I–V and V (Φ) characteristics.

7.2 Conventional Explanation for the FMS The FMS was first introduced to SQUID readout in 1967 and quickly became the standard readout technique for current-biased SQUIDs [1]. To date, FMS electronics has been the most extensively used. The basic idea of the FMS is to step up the SQUID’s signal swing at the input terminal of the preamplifier with a transformer, thus reducing the noise contribution of δΦe . In contrast to a DRS, where a dc circuit (amplifier and integrator) is employed, the FMS is an ac circuit, e.g. operating in the 100 kHz frequency range, because the transformer can pass only ac signals. However, a SQUID is often used to detect magnetic flux signals Φ with slow changes, even quasi-static signals. To resolve this contradiction, a modulation and demodulation technique is employed to realize the transitions between dc and ac circuits. 7.2.1

Schematic Diagram of the FMS

In the FMS, the key element is a step-up transformer. The SQUID is matched to a preamplifier via a four-terminal step-up transformer. Here, the SQUID is connected to the transformer’s PW, while a large capacitor C iso in series with this winding acts as dc isolation, forcing the dc bias current I b to flow through only the SQUID. Therefore, it is commonly believed that the FMS is typically operated in current bias mode, where the readout quantity is the voltage. The secondary winding (SW) connects to the preamplifier and can be considered open because

63

64

7 Flux Modulation Scheme (FMS)

Ib

Ciso

Mf

M1 1:n

Ch A

Lf

PW SW iMO

RM

if

Rf

C

M2

Rh

(1)

R

M

(2) Ch

Vout

– +

I

Rh

Δφ VM

Figure 7.2 A schematic diagram of the FMS. The SQUID is biased by a constant current Ib . Two high-pass filters with a time constant Rh C h are located between A and M and between Δ𝜑 and M.

of the very high input impedance of the amplifier (op-amp). The original motivation of the FMS was that the SQUID voltage signal V s would be stepped up at the preamplifier’s input terminal by a factor of n, which is the ratio of SW/PW, thus resulting in a large apparent signal swing for a reduction in δΦe . A schematic diagram of the FMS is shown in Figure 7.2, consisting of five parts: (1) the input circuit consists of a SQUID, a transformer, and a flux feedback coil Lf coupled to the SQUID with mutual inductance Mf . The SQUID and the coil Lf form a four-terminal element (head stage), as discussed in Section 4.2 (see Figure 4.3). This “head stage” of the FMS is essentially an ac circuit and is calibrated by the amplitude- and phase-to-frequency transfer characteristics of the transformer. Usually, the transformer is set either at room temperature or at cryogenic temperature. Some works used two transformers, one of which was operated at room temperature and the other was positioned near the SQUID at cryogenic temperature to reduce its thermal noise [7]. (2) The amplifiers marked as “A” may include a low-noise preamplifier and a main amplifier. (3) The multiplier “M” with two input terminals and one output terminal acts as a demodulator to restore the original SQUID’s V (Φ). (4) The square-wave voltage generator V M produces a modulation flux ΦMO = iMO × Mf = Φ0 /2 via the resistor RM , and this V M also acts as a reference voltage signal for the multiplier M. Here, a phase shifter Δ𝜑, which is inserted between V M and the input voltage (2) of M, can adjust the phase between the two input terminals, (1) and (2) of M, to make them in-phase. (5) The integrator “I” with the time constant RC is employed for flux-locked loop (FLL) operation, as described in Section 4.2, where the voltage at the output, V out , is translated into a current if , thus generating the compensation flux −ΔΦ = if × Mf = V out × Mf /Rf . Consequently, ΔV out is proportional to the ΔΦ threading the SQUID loop. In Figure 7.2, there are two monitoring points, M1 at the input of A for the SQUID’s modulated signal and M2 at the output of M for the demodulated (restored) signal. In the following discussion, we assume that a bias current I b is already selected and that the swing in the SQUID’s V (Φ) almost reaches its maximal value.

7.2 Conventional Explanation for the FMS

Table 7.1 Comparison of schematic diagrams of a DRS and the FMS.a) DRS

FMS

(1) Input circuit

SQUID and coil Lf (head stage)

Head stage + step-up transformer and C iso

(2) Amplifierb)

dc amplifier

ac amplifier

(3) Multiplier

None

Present

(4) Square wave voltage generator (5) Integrator

None

Present

Voltage (V w ) at noninverting terminal determines the working point W

The ground is defined as the working point voltagec)

a) The numbering in the table matches that in the text above. b) In principle, the dc/ac amplifiers are the same in the frequency range from dc to 1 MHz. c) The positive and negative parts of the SQUID’s V (Φ) are symmetric at M2 , so the transfer coefficient 𝜕V /𝜕Φ reaches its maximum at V = 0, where Φ = [(2n + 1)Φ0 ]/4. To select the working point, the noninverting terminal of the integrator should be grounded, as shown in Figure 7.2.

Table 7.1 lists a comparison of the schematic diagrams of a DRS and the FMS. In the FMS, the SQUID’s V (Φ) characteristics at M1 are modulated into ac signals with a certain frequency and are then amplified and demodulated. The shape of the restored SQUID signal at M2 , i.e. at the input terminal of the integrator I, becomes the same as that in a DRS except for the dc offset. Compared to a DRS, the FMS adds the circuits (elements) of “M,” “V M ,” and “Δ𝜑” shown in Figure 7.2 for modulation and demodulation. This approach is the main point explained in Sections 7.2.2 and 7.2.3. Indeed, one expects a low δΦe in FMS for strongly damped SQUIDs, although the FMS is much more complicated than a DRS. 7.2.2

Time Domain and Flux Domain

In fact, FMS is not easy to understand. Before we explain and discuss the flux modulation technique in much detail in Section 7.2.3, let us look at two special examples in Figure 7.3 to understand the relation between the flux domain and the time domain: (a) a linearly changing flux Φ(t) results in a periodic SQUID’s V s (Φ) with a period of Φ0 and a periodic SQUID’s V s (t) with a period of Δt Φ0 . Both periodic functions look similar, but the variables are different: V s (Φ) appears in the flux domain, while V s (t) is recorded in the time domain; (b) a changing flux Φ(t) with a triangular waveform contains upward and downward sections in the time domain. However, in this example, V s (Φ) presents a periodic function of V (Φ) within the interval of (5/4)Φ0 in the flux domain, where V (Φ) is reversible. In the upward and downward sections, the function V s (t) at t re exhibits two mirrored parts, which reflect the SQUID characteristics in the time domain.

65

66

7 Flux Modulation Scheme (FMS)

Φ

Φ Φ(t)

Φ0 O

Φ0 ΔtΦ0

t

O

Vs

ΔtΦ0

tre

t

Vs

(a)

(b)

Figure 7.3 (a) A linearly changing flux Φ(t) results in a periodic SQUID’s V s (Φ) with a period of Φ0 and a periodic SQUID’s V s (t) with a period of ΔtΦ0 . (b) In the time domain, a triangular waveform flux Φ(t) contains upward and downward sections, which are indicated by two arrows. Here, the maxima of ΔΦ are 3.5Φ0 in (a) and (5/4)Φ0 in (b).

Within the time interval of Δt Φ0 , the flux changes by one flux quantum Φ0 , so the external flux change ΔΦ should fulfill the slowly changing flux condition of Δt Φ0 ≫ T M = 1/f M (f M is the modulation frequency) in the FMS operation. In the following discussion, it can be assumed that the flux does not change during the modulation period T M . Indeed, Figure 7.3 schematically illustrates the relations among Φ(t), V s (Φ), and V s (t). In the following, we often change between the flux domain and the time domain. V s (t) is often taken to describe the signal transformation in the readout electronics shown in Figure 7.2, while V s (Φ) is used to analyze the principle of the FMS. 7.2.3

Flux Modulation

In order to pass the slowly changing magnetic field signal picked up by the SQUID through the transformer, FMS is introduced. There are different modulation wave shapes with a modulation frequency f M of typically 100 kHz (i.e. a period T M of 10 μs) [8]. Generally, the square-wave modulation method is employed. Here, the modulation flux of ΦMO = Φ0 /2 is generated by the oscillator V M via a resistor RM and the mutual inductance Mf (see Figure 7.2). Unlike in the DRS, the readout voltage signal is not the V s (t) across the SQUID directly, but the modulated and stepped up V M1 (t) at the SW of the transformer in FMS. Firstly, let us view the difference between V s (t) and V M1 (t). Here, V s (t) contains dc and ac components shown in Figure 7.3. Owing to the ac input circuit of the “head stage,” only modulated ac components can pass through the step-up transformer, but its dc component is already filtered there. The ac components of V s (t) are stepped up at V M1 (t), so the ac V M1 (t) (monitored at M1 in Figure 7.2) should be symmetric to the ground of the electronics, where V ground = 0. Namely, the time-averaged value of V M1 (t) remains zero in a modulation period Φ0 , when Φ(t) is linearly changed. Figure 7.4 shows the relation of ΦMO (t), V s (Φ), and V M1 (t) in a modulation period T M of 10 μs. Here, ΦMO (t) is a square-wave flux with an amplitude of

7.2 Conventional Explanation for the FMS

Φ0

Φ WM′ Φ0 /2

WM′

WM O

5 μs

10 μs

t

WM Vs

0′ VM1

Figure 7.4 Φ(t), V s (Φ), and V M1 (t) in a modulation period for the case of a modulation frequency of 100 kHz. Here, a square wave with the amplitude of Φ0 /2 is present in Φ(t) in a period T M ; the two modulation points sketched as dots, WM and W′M , with an interval of Φ0 /2, are set on the hidden V(Φ) at Φ = (2n + 1)Φ0 /4 (n = 0, 1, 2, …), in V s (Φ) domain. Returning into time domain, V M1 (t) becomes a straight line. Note that the voltage coordinate axes represent V s (Φ) and V M1 (t), respectively. The difference between voltage origins, O and 0′ , represents the dc component (dc offset) of V(Φ), while only the ac component of V(Φ) is stepped up at V M1 (t), where 0′ is V ground = 0.

Φ0 /2 and a duty cycle of 1 : 1, which corresponds to two modulation points ′ (marked as two dots), WM and WM , set on the SQUID’s V (Φ). Let us assume ′ that WM and WM are just located at (1/4)Φ0 and (3/4)Φ0 of a hidden V (Φ) (the dotted curve). Both points appear alternately every 5 μs in the time domain. Interestingly, the square-wave modulation flux in Φ(t) becomes a voltage straight line at V M1 (t) = V M1 (0′ ) = 0, where 0′ is the V ground , i.e. the ground (origin) of V M1 (t). The voltage difference between O and 0′ indicates the dc component ′ (off-set) of V s (Φ). At WM and WM , the two maximum transfer coefficients (𝜕V /𝜕Φ) are reached, but with opposite polarities. Any flux change ΔΦ(t) even in a quasi-static process in time domain can be regarded as a hidden V (Φ) shift ′ of right or left in the flux domain, thus shifting the positions of WM and WM on V (Φ). Consequently, a square-wave V M1 (t) symmetric to ground 0′ appears. ′ In other words, the voltages V M1 (t) of WM and WM in Figure 7.4 do not remain at V ground = 0. ′ Note that the square wave modulation points, WM and WM , should not be confused with the working point W for the FLL operation mentioned in Section 4.2. The latter is necessary in a SQUID magnetometry system, but the former is just a transition process in FMS. In fact, any ΔΦ can be characterized by its size (amplitude) and direction, which should be contained in V M1 (t) after the flux modulation. Now, let us take three cases of flux changes, i.e. ΔΦ2 > ΔΦ1 > 0 and ΔΦ3 < 0, to analyze FMS. In the left column of Figure 7.5, the two gray modulation points set on V s (Φ) at ′ the original flux state described in Figure 7.4, and two black points, WM and WM ,

67

68

7 Flux Modulation Scheme (FMS)

Vs

VM1 ΔΦ1 WM′

t 0′

WM ΔΦ1 O (a) Vs

VM

Φ

VM1

ΔΦ2

WM′

VM t

0′ WM

ΔΦ2

O (b)

Φ VM1

Vs ΔΦ3 WM

O (c)

VM t

0′

WM′ ΔΦ3

Φ

Figure 7.5 The left column shows the positions of the modulation points, WM and W′M , on the hidden V s (Φ), while the right column displays the recordings of V M1 (t) during one modulation period, where V M (gray) is the modulation voltage originating from the FMS electronics depicted in Figure 7.2. Note that in order to fit the scale of V s (Φ) in the left column, the scales of V M1 used in the right column are already divided by the step-up factor n of the transformer. Furthermore, in the graphs in the left column, the solid arrows indicate the direction of ΔΦ.

indicate the actual flux state on V s (Φ), while the right column records the V M1 (t) within a modulation period of, e.g. 10 μs. ′ As the two black points of WM and WM are shifted by ΔΦ in flux domain, ′ they lead to V M1 (t) ≠ 0. Because the two voltage changes of WM and WM are opposite, they generate the square wave voltage (black) in the right column (time domain). Comparing Figure 7.5a with b, we infer that the voltage amplitude of the square wave at M1 is proportional to the ΔΦ amplitude. Furthermore, comparing Figure 7.5a with c, the direction of ΔΦ determines if V M1 (black) and the oscillator V M (gray) are in-phase or out-of-phase in the right column. Here, V M (t) will be applied at input (2) of the multiplier M in Figure 7.2. In short, the square wave signals V M1 (t) in Figure 7.5 record the amplitudes and the directions of ΔΦ in the three typical cases of FMS. In the case of the square-wave flux modulation, we indeed learned the transformation from the flux domain to the time domain. For general FMS readout electronics, the measured flux ΦA (t) becomes V M1 (t) via the flux domain, i.e. a nonlinear operation of SQUID and a Φ0 /2 flux

7.2 Conventional Explanation for the FMS

modulation (see Figure 7.2). Here, the SQUID’s main feature is the periodical V (Φ) in current bias mode. Now, let us observe the evolution process of ΦA (t) in FMS. Figure 7.6 gives the profiles of the interrelation of the four functions, ΦA (t), ΦMO (t), V s (Φ), and V M1 (t). Here, Φ(t) and V M1 (t) as well as ΦMO (t) are in time domain, while V s (Φ) is in flux domain. In order to understand the principle of FMS, we begin with a simple case. Firstly, we focus on the coordinate system of Φ(t) in Figure 7.6a. Here, the total flux consists of two components: (1) a linearly changing flux ΦA (t) (black line) with amplitude of Φ0 within a time interval Δt Φ0 ≫ T M = 10 μs (the modulation period). Note that ΦA (t) is just a hypothetical component in Φ(t); (2) the gray square-wave chain is the image sketch of Φ0 /2 modulation flux ΦMO with, e.g. only 12 modulation periods, where the sequence numbers are marked. Two consecutive numbers, odd and even, form a modulation period T M . In addition, the gray straight line is just an auxiliary line with Φ0 /2 offset to ΦA (t). The region between the black and gray straight lines shows the range of flux change of the sum of ΦA (t) and a square wave flux of Φ0 /2 modulation. The maximal flux change ΔΦ reaches (3/2)Φ0 , thus leading to 3/2 periods in V s (Φ). A flux change ΦA (t) in time domain can be regarded as a flux shift of V s (Φ) in flux domain, compared to ΦMO . In Figure 7.6b, one V s (Φ) curve with period of Φ0 at two states, 0 and Φ0 /2, of modulation flux ΦMO can be distinguished by V (Φ)0 (lower) and V (Φ)1/2 (upper), respectively. Note that the vertical voltage offset V offset between V (Φ)0 and V (Φ)1/2 is artificially added to facilitate visualization. Actually, the two curves are overlapping in the shaded area. The two ′ , introduced above appear on V (Φ)0 and V (Φ)1/2 , modulation points, WM and WM in accordance with odd and even numbers. Within a period time T M , the WM of ′ of even numbers appears odd numbers appears on the V (Φ)0 for 5 μs, while WM ′ always mark the actual on the V (Φ)1/2 for 5 μs, alternately. Here, WM and WM flux states, as ΦA (t) changes. So, V s (Φ) in Figure 7.6b is translated from ΦA (t) in Figure 7.6a in the case of FMS. Now, we return to time domain to observe the signal V M1 (t) at M1 of Figure 7.2, which evolves from a linearly changing flux ΦA (t), i.e. a straight line in the coordinate system of Φ(t) (in Figure 7.6a) via a square wave flux modulation of Φ0 /2 and a nonlinear operation of SQUID characteristics. The corresponding voltages ′ alternately appear at different points on V s (Φ) in the flux V s of WM and WM domain (in Figure 7.6b), thus resulting in a sequence of square wave voltages with different amplitudes in the time domain, as described in Figure 7.5. Now, these ac voltage signals can pass through the step-up transformer and are monitored at M1 , denoted as V M1 (t). In a slowly changing flux process, V M1 (t) can be considered as stepping curves, i.e. during each step time (T M /2 = 5 μs), the voltage value of V M1 (t) is approximately constant. At M1 , V M1 (t) becomes two positive and negative symmetrical curves with intermittence, where both envelopes reflect the periodical character of V (Φ). Immediately, the V M1 (t) is amplified by “A” and applied at the input (1) of the multiplier M denoted as V input1 , as shown in Figure 7.6c (lower half ). The other input (2) of M connects the reference square wave voltage V input2 generated by V M via a phase shift Δ𝜑, as shown in Figure 7.2. The input voltage V input2 with the sequence numbers is schematically shown in Figure 7.6c (upper half ). Here, we can draw vertical dotted lines to

69

70

7 Flux Modulation Scheme (FMS)

Φ

3Φ0 2

ΦA(t)

Φ0 Φ0 2

4 6 8 10 12 14 16 18 20 22 24 Φ MO

2

t

o

1

3

5

7

9 11 13 15 17 19 21 23

ΔtΦ0

0′

Vs

VM1

(a)

Vs 2 4

V(Φ)0

9

11 13 15

7

0′

5

3

1

8 17

19

10 21

WM

12 14 16

24

20

18

V(Φ)1/2

Voffset

23

Φ0 /2

(b)

22

WM′

6

Φ

Φ0

Vinput2 2

4

6

8

10 12

14 16

18 20

22

24

t

0 1

3

5

7

9

11 13

15

17 19

21

23

ΔtΦ0

Vinput1(VM1) WM′ t

0′ WM

(c) VM2 (Vout)

t

0′ W

ΔtΦ0

(d)

Figure 7.6 Working principle diagram of FMS demonstrated with Φ(t), V s (Φ), and V M1 (t) (a), where a linearly changing flux of Φ0 with time and the modulation flux ΦMO with amplitude Φ0 /2 in the time interval of ΔtΦ0 act as Φ(t); two inversely symmetric V(Φ) with Φ0 /2 offset due to flux modulation (b); two symmetric V M1 (t) appear cyclically (c); the restored V(Φ) after the demodulation monitored at M2 (d).

7.2 Conventional Explanation for the FMS

describe the one-to-one correspondence between V input2 (upper graph) and ′ marked in V input1 (lower). The output signal of M reverses the curve of WM Figure 7.6c, thus obtaining the continuous V M2 (t) at the monitor point M2 , as shown in Figure 7.6d. Here, V M2 (t) is a modification of ΦA (t) via the SQUID’s V (Φ). The V M2 (t) at the input terminal of the integrator I no longer contains the modulation square wave component and should have the same performance as that in DRS shown in Figure 7.3a, except the dc component in Figure 7.6b. In order to restore a linearly changing ΦA (t) in time domain, using FLL, one obtains the output voltage of the integrator I, V out (t), which should be a straight line (gray line), shown in Figure 7.6d. At last, V out (t) is proportional to ΦA (t). Indeed, everything we do in FMS is for the purpose of allowing a voltage change across the SQUID, e.g. caused by quasi-static flux change, to pass through the step-up transformer. With this example visualized in Figure 7.6, the principle of flux modulation and demodulation has been explained. All in all, FMS translates the SQUID’s voltage from dc to ac with flux modulation, and restores it from ac to dc with a multiplier “M.” The motivation of FMS is to gain the step-up factor n of the transformer, thus reducing δΦe from the preamplifier. 7.2.4

Five Additional Notes

Based on the principle of FMS, we emphasize five points for the readout electronics shown in Figure 7.2: (a) At the input terminal (1) of the multiplier M, a phase shift Δ𝜑 between the SQUID signal and the modulation square wave voltage generated by V M exists, due to, e.g. the phase-frequency characteristics of the step-up transformer, which is discussed in Section 7.3.4. The additional controllable phase shifter Δ𝜑 between the input terminal (2) of “M” and the voltage oscillator V M is employed to compensate for this unknown phase shift of the SQUID signal at the input terminal (1). The phase shifter Δ𝜑 makes both signals at the inputs of M in-phase, thus enhancing the demodulation efficiency and improving the authenticity of V M2 (t) restoration. (b) The selection of the working point V w at the input of the integrator I for FLL operation was introduced in DRS (Section 4.2). In FMS, the steepest slopes (𝜕V /𝜕Φ) of the restored V M2 (t) (see Figure 7.6d) appear at 0′ (the ground) due to the ac circuit, where Φ = (2n + 1)Φ0 /4 (n = 0, 1, 2, …). Therefore, the noninverting terminal of the integrator I for selecting V w in Figure 7.2 is grounded, so that V w = 0 for FLL is already selected in FMS. However, for FLL operation in DRS, the working point W requires to select V w ≠ 0. Recall that we should not confuse the working point W with the two modula′ tion points (WM and WM ); the former is for FLL operation, but the latter are intermediately utilized for flux modulation. Note that two modulation points ′ ) have already disappeared after demodulation, i.e. at M2 . (WM and WM (c) In FMS, the modulation frequency f M will determine the output bandwidth of the readout electronics at V out . In the case of Figure 7.6c, we have a linearly changing flux ΦA (t) with amplitude of Φ0 within a time interval Δt Φ0 = 12T M , namely, f ΔΦ0 = 1/Δt Φ0 ≈ 8.3 kHz. In this case, the restored V M2

71

72

7 Flux Modulation Scheme (FMS)

Vs

VM1 1

ΔΦ

VM

2 WM′

t 0′

WM 1 O

2

ΔΦ

Φ

Figure 7.7 If ΔΦ < Φ0 /4 (see point 1) or ΔΦ > Φ0 /4 (see point 2) in T M /2, the output voltages at V s and V M1 are the same, i.e. V(Φ) becomes multivalued.

looks like a stepping curve consisting of 24 steps, and does not authentically reflect its true shape. Therefore, as f ΔΦ0 further rises, V M2 becomes rougher, thus losing more and more resemblance with the input signal. Generally speaking, for f M = 100 kHz, the output bandwidth of the readout electronics at V out is limited to less than 10 kHz. (d) In FMS, not only the output bandwidth but also the slew rate is limited by f M . Figure 7.7 presents two cases: If ΔΦ < Φ0 /4 in T M /2 (case 1), a square wave voltage appears at V s and V M1 . If ΔΦ > Φ0 /4 in T M /2 (case 2), the voltages ′ have got the same voltages as in case 1. However, one value at WM and WM of V s can correspond to more than one flux state. This multivalued relation limits the maximum flux change to ΔΦmax = Φ0 /4 in T M /2 (5 μs), so that the slew rate of FMS with f M = 100 kHz should be limited to 5 × 104 Φ0 /s. (e) Up to now, FMS has been discussed in time domain and in flux domain, but some discussions can also be performed in frequency domain. Here, we take an example of 1/f noise of the preamplifier to highlight an advantage of FMS and the analysis in frequency domain. It is known that the 1/f noise is a serious problem of δΦe in DRS, especially for the current noise source, I n , of the PCBT preamplifier, as discussed in Chapter 5. Figure 7.8 schematically shows a frequency spectrum of V M1 (t), while the assumptive spectrum of ΔΦ(t) is shown in the inset. In FMS, the square wave V M1 (t) in time domain translates into some large peaks at f M and its odd harmonics in frequency domain, where the spectra of ΔΦ(t) are attached at both sides of the peaks. A high-pass filter consisting of Rh and C h is set between the amplifier A and the multiplier M (see Figure 7.2) to filter all components below its cut-off frequency f c = 1/(Rh C h ). In Figure 7.8, the dotted line presents the magnitude-frequency characteristics of the high-pass with e.g. f c ≈ 40 kHz indicated with a vertical dashed line. In this case, the 1/f noise from the amplifier is practically filtered out at input (1) of M, because 1/f noise is mostly present at frequencies f c ≈ 40 kHz. In FMS, the use of the high-pass filter is an advantage of the ac circuit. Furthermore, the high-pass suppresses the drift of the readout electronics.

7.3 FMS Revisited

VM1 Amplitude

Normalized amplitude

1

0.7 0.5

O

ΔΦ(t)

2

4

6

f

8

Frequency (kHz)

f fc

1

5 3 Frequency (100 kHz)

7

Figure 7.8 The frequency spectrum of V M1 . The peaks present the Fourier components translated from the square wave modulation voltage V M with f M = 100 kHz, where the spectra of the measured signal are located on both sides of the peaks. The dotted line describes the characteristics of the high-pass filter consisting of Rh and C h with f c ≈ 40 kHz. The inset schematically zooms in the spectrum of the measured signal ΔΦ(t).

With FMS, many people successfully performed SQUID measurements for different applications. In order to extend the bandwidth and the slew rate of the readout electronics in FMS, the modulation frequency f M has been enhanced up to the MHz range [9–11]. FMS readout electronics has been already commercialized by some companies, e.g. “STAR Cryoelectronics” in the United States (https://starcryo.com/), “ez-SQUID” in Germany (http://ez-squid.de/Home .htm), and Photon technology Co., Ltd. in China (http://www.sconphoton.com/).

7.3 FMS Revisited There are two common misconceptions about FMS. Firstly, it is often believed that FMS is operated in ideal current bias mode because the constant bias current I b flows only through SQUID, due to the isolation capacitor C iso in Figure 7.2. Secondly, it is assumed that the SQUID intrinsic noise δΦs may be observed since the SQUID voltage signal V M1 at the input of the preamplifier has already been stepped up by the transformer, thus leading to a reduction of the preampliof the transformer’s fier noise contribution δΦe by n times, where n is the ratio √ (SW/PW). When taking a common op-amp with V n ≈ 1 nV/ Hz as an example, at n = 10, the equivalent the preamplifier in DRS is assumed to √ voltage noise of√ simply become (1 nV/ Hz)/10 = 0.1 nV/ Hz. In FMS, we raise two questions: (i) Does the current I s flowing through the SQUID shunted by PW of the transformer via C iso remain constant? (ii) Can the square wave voltage V s analyzed in Section 7.2 pass through the transformer with a limited bandwidth?

73

74

7 Flux Modulation Scheme (FMS)

Ib

Ciso

Rd Δi

Zs

PW Φ0 /2

1:n

A SW

Is

Vout

Figure 7.9 SQUID input circuit of FMS with a step-up transformer and preamplifier. The PW of the transformer acts as a shunt element with an impedance Z s . The ac voltage V s generated by the ac ΦMO of Φ0 /2 (f M ≈ 100 kHz) produces a ring current Δi through the closed circuit, consisting of the SQUID’s Rd , the isolation capacitor C iso , and the PW. The product of Δi and Z s is the ac voltage across the parallel circuit of SQUID and PW.

In this section, we will revisit FMS for the first time after this scheme has been developed for almost half a century, by experimentally examining the bias mode of FMS and analyzing the transfer function of the step-up transformer [12]. 7.3.1

Bias Mode in FMS

Practically, Figure 7.9 shows the zoomed “head stage” of the FMS, which belongs to the diagram of the readout electronics in Figure 7.2. Our revisited works begin with a discussion of the bias mode. Let us first consider two cases: (1) When the applied flux is quasi-statically changed, the branch consisting of C iso and PW can be considered as open and a constant bias current I b flowing through the SQUID is independent of a flux change ΔΦ. Only under the condition I s = I b = constant, the SQUID is operated in the ideal current bias mode with 𝜒 → ∞, where the SQUID’s original (intrinsic) I–V characteristics and V (Φ) curve can be obtained in DRS, as described in Section 7.1; (2) In FMS, the SQUID is shunted by the ac impedance Zs of the PW, where C iso only provides an ac path. In spite of constant I b , an ac-modulated V s appears across the SQUID due to ΦMO and generates an additional ac ring current Δi(Φ) flowing through the closed circuit consisting of SQUID, C iso , and PW of transformer, so that I s depends on the flux change ΔΦ, i.e. I s = I b − Δi(Φ), according to Kirchhoff’s current law. Thus, the SQUID is actually operated in the mixed bias mode. Our first work is to prove the existence of Δi(Φ) in Figure 7.9. 7.3.2

Basic Consideration of Synchronous Measurements of Is and V s

In practice, it is undisputed that the readout quantity of the readout electronics in all FMS is always the voltage. The key consequence of the ac shunt Zs in FMS is the appearance of Δi(Φ). When ΦMO changes by Φ0 /2, it can be recorded by synchronously measuring the change in the current I s passing through the SQUID and the ac voltage V s across the SQUID. In fact, the use of Δi and V s can yield the value of 𝜒 to determine the bias mode quantitatively. For simplicity in the following experiments, we temporally replaced the shunt impedance Zs (PW of transformer) by a variable resistor Rs , as depicted in Figure 7.10, to avoid phase shifts and a nonlinear frequency response. Furthermore, a sinusoidal ΦMO of amplitude Φ0 /2 with a low frequency below 1 kHz is taken, and the SQUID’s I–V characteristics are adjusted varying just between the two extreme flux states, nΦ0 and (n + 1/2)Φ0 . Here, both the ammeter A measuring I s and the voltmeter V recording V s across PW indicate I(Φ) and

7.3 FMS Revisited

Ib

Figure 7.10 The simplified input circuit by replacing Z s with a variable resistor Rs and using a sinusoidal ΦMO of Φ0 /2. Here, “A” and “V” represent an ammeter and a voltmeter, respectively.

Ciso

Rd Δi Rs

Φ0 /2

A

V

Is

V (Φ) synchronously. Note that the ac voltage V s is practically measured across the parallel circuit of SQUID and Rs . If both I(Φ) and V (Φ) are readable at the same time, it means that the SQUID operates in the mixed bias mode, as analyzed in Section 7.1. 7.3.3

Experimental Synchronous Measurements of 𝚫i and VR s

In order to determine I s , we only need to measure the ring current Δi, whereas I b is known. Note that Δi is the ac current swing as ΔΦ = Φ0 /2. In order to synchronously measure the change in Δi and its corresponding voltage V Rs across Rs (see Figure 7.10), the real test circuit shown in Figure 7.11 was designed. The measured SQUID (SQ1) was biased with a constant I b and modulated by an ac flux ΦMO of amplitude Φ0 /2, as commonly done in FMS. Here, a superconducting coil Li is connected in series with the measured SQUID (SQ1), but there is no mutual inductance between them. The ac ring current Δi is measured by the second SQUID (SQ2), where Δi flows through the closed circuit consisting of SQ1, the Li , the capacitor C iso , and shunting resistor Rs . To accomplish this, the planar superconducting coil Li is practically integrated on the washer of SQ2 and inductively coupled via the mutual inductance Mi to SQ2. Practically, Li and SQ2 constitute an ammeter A of Figure 7.10, while its readout electronics Ib Φ0 /2

ΔViout

Ciso

Vout A

Rd

E Rf

SQ1

Lf

Li

SQ2 Mf

Rs Δi Is

Mi

Figure 7.11 Test circuit for determining V out and Δi of the SQUID (SQ1) with sinusoidal flux modulation of amplitude Φ0 /2 and constant Ib at different shunts Rs . The current Δi is derived from the measured ΔViout of the SQUID (SQ2), read out by the readout electronics “E” operated in FLL as described in the text. The ac voltage V out is monitored at the output of the linear amplifier with a gain G = 1000. Here, SQ1 has a loop inductance Ls ≈ 150 pH and a junction shunt resistor with RJ = 9 Ω.

75

76

7 Flux Modulation Scheme (FMS)

“E” is operated in FLL via Rf and Mf . Thus, the voltage of ΔViout of SQ2 represents Δi flowing through SQ1. For calibration of the ammeter, we directly inject a known current through Li (without SQ1) to measure the change in ΔViout of SQ2, thus determining the current-to-voltage transfer coefficient of the SQ2 system, ΔViout ∕Δi = 530 mV∕μA at Rf = 5 kΩ, Mf = 1.5 nH. The voltmeter V in Figure 7.10 was realized in a simple manner. Here, an ac voltage across Rs , ΔV Rs = Δi × Rs , is synchronously generated and monitored at the output V out of a linear voltage amplifier (op-amp) with a gain of G = 1000 (see Figure 7.11). In the test circuit, an ideal ammeter with zero internal resistance and an ideal voltmeter with infinite internal resistance (i.e. very high input resistance of op-amp) were employed for the realization of the circuit depicted in Figure 7.10. Actually, ΔViout and ΔV out are the synchronous readings of I(Φ) and V (Φ) signals by the ammeter A and the voltmeter V. During the experiments, we used different Rs to determine Δi from the measured ΔViout and to record ΔV Rs across a given Rs at V out . All measured data are listed in Table 7.2. In experiments, the increase of Rs results in increasing ΔV Rs , but decreasing Δi. According to Ohm’s law, the right column of Δi × Rs should fit to the measured ΔV Rs . Here, the experimental data proves that V Rs ≈ Δi × Rs is established. From the data of Table 7.2, two important consequences are obtained: (i) with a shunt element (Rs ), the current I s cannot remain constant due to an ac ring current Δi in FMS, although the I b is constant. (ii) As Rs = 5 Ω, ΔV Rs drops to half of its maximum at Rs → ∞. In this case, Rd of SQ1 should also be 5 Ω, so that the boundary of two bias modes (𝜒 ≈ 1) is reached, as discussed in Section 7.1. When Rs < 5 Ω, the voltage bias mode takes over, so that the readout quantity should become the current Δi instead of the voltage ΔV Rs . In FMS, the PW impedance Zs of the step-up transformer is usually larger than Rd of SQ1. Therefore, FMS is operated in nominal current bias mode, i.e. to read out ΔV Rs . Now, we introduce Figure 7.12 to offer a simple graphic interpretation of our measured data, Δi and ΔV Rs , listed in Table 7.2, resulting from the complex nonlinear behavior at SQUID for different shunt resistances Rs . Practically, let us observe the SQUID’s behavior under a varying I s . Table 7.2 Measured Δi and V Rs . Rs (𝛀)

𝚫i (𝛍A)

𝚫V Rs (𝛍V)

𝚫i × Rs (𝛍V)

1.2

3.5

4

4.2

5.0

2.1

10

10.5

10.3

1.3

14

13.4

20.0

0.9

18

18

51.2

0.4

20

20.5

—

22a)

—

∞

a) In the absence of Rs .

7.3 FMS Revisited

15 Φ = nΦ0

Bias line Ib

10 I (μA)

Figure 7.12 Measured SQUID’s I–V characteristics at Φ = nΦ0 and Φ = (n + 0.5)Φ0 , when 𝜒 → ∞. The shunting effect of Rs leads to the reduction of Is = (Ib − Δi) and to a tilted bias (load) line, thus resulting in ′ Vswing < Vswing .

Φ = (n + 1/2)Φ0 Δi

5

0

′ Vswing

0

Vswing VRs

V0 10

Is

20 V (μV)

V0.5 30

40

When Rs is absent (𝜒 → ∞), one obtains the original I–V characteristics of the SQUID itself, e.g. at Φ = (n + 0.5)Φ0 (half-integer flux) and Φ = nΦ0 (integer flux), as shown in Figure 7.12. In this case, at a selected bias current I s = I b = 9 μA, the SQUID voltage swing V swing is denoted by the voltage difference, V swing = (V 0.5 − V 0 ), between two intersections of the I b horizontal line with the two limit I–V curves. Here, V 0 is the intersection voltage at Φ = nΦ0 and V 0.5 at Φ = (n + 0.5)Φ0 . If V 0 ≈ 0 (as assumed here), then V swing ≈ V 0.5 ≈ 22 μV, which is the maximal readable V swing of SQ1. In fact, this fits the measured data of the bottom row in Table 7.2 well, where ΔV Rs is the original V swing . When Rs is finite, the I–V curves of SQ1 in Figure 7.12 remain because the SQUID’s intrinsic characteristics will not be changed with external circuits, but I s varies, namely, the line of I s moves downward from I b to (I b − Δi), when Φ varies ′ ) is denoted as the voltage from nΦ0 to (n + 0.5)Φ0 . Here, the readout V Rs (Vswing difference of (V Rs − V 0 ) between two intersections of the I s (varying horizontal line) with the two limit I–V curves. Actually, the tilted bias (load) line leads to a reduction of ΔV Rs . It is easy to understand that when Δi is larger, ΔV Rs becomes smaller. For each row of Table 7.2, one could find its own bias (load) line, which fits to the Δi and ΔV Rs measured at different Rs (here, V 0 = 0). For the example of 𝜒 = 1 discussed above (see the second row in Table 7.2), the ′ is reduced by half to about 11 μV, which tilted bias (load) line indicates that Vswing can be obtained at V Rs , i.e. the intersection of the horizontal line I s = (I b − Δi) ′ ≈ 11 μV , Δi ≈ 2.1 μA and the I–V curve at Φ = (n + 0.5)Φ0 . In this case of Vswing can be obtained at I s = I b − Δi ≈ 7 μA from Figure 7.12, and Rs ≈ V Rs /Δi ≈ 5 Ω fits the given value of the second row in Table 7.2 well. Up to now, we have experimentally proved the existence of Δi in FMS and the use of the simple graphic interpretation fits to our measured data in Table 7.2. Both explain the reduction of the readout SQUID’s V swing due to the shunt effect of Zs . In our practical FMS operation, the impedance Zs of the transformer’s PW with inductance LPW ≈ 12 μH corresponds to 6.2 Ω with f M = 100 kHz for the fundamental frequency of the square wave, while Rd of the employed SQUID (SQ1) at the working point on Φ = (2n + 1)Φ0 /4 is less than 10 Ω, as seen in Figure 7.12. If 𝜒 ≈ 1, i.e. the SQUID indeed is operated at the boundary of two bias modes, as ′ of the SQUID signal may discussed in Section 7.1. Thus, the voltage swing Vswing

77

78

7 Flux Modulation Scheme (FMS)

be reduced by half at the PW of the step-up transformer, compared to that in the case 𝜒 → ∞. In other words, the shunt effect of PW leads to a loss of half of the gain from the PW/SW ratio of the step-up transformer. In brief, the assumption that FMS operates in ideal current bias mode has been proved wrong. In fact, the SQUID in FMS is practically in a mixed bias mode, thus reducing the SQUID’s voltage signal at the PW of the step-up transformer. This is the first imperfection of FMS. Therefore, FMS is usually well suited for strongly damped SQUID with small Rd , or, for high-T c SQUIDs. 7.3.4

Transfer Characteristics of the Step-Up Transformer

In all previous analyses of FMS, we assume that the SQUID’s signals at PW of the transformer can be stepped up at its SW (denoted as V M1 ) without any distortion. Our second revisited work is to check the transfer function of the step-up transformer, which contains the magnitude-frequency characteristic and phase-frequency characteristic between PW and SW. A home-made transformer with enameled wire winding on a ferrite ring was employed. The SW/PW ratio, n, was 20 and the inductances of LPW ≈ 12 μH and LSW ≈ 4.3 mH were measured with sinusoidal voltage at f M = 100 kHz. The coupling coefficient k between PW and SW was determined to be 0.98–0.99. However, the transfer characteristics of the SQUID input circuit depend not only on the transformer but also on the isolation capacitor C iso and on the dynamic resistance Rd of the SQUID. We utilized the test circuit shown in Figure 7.13 to obtain the complex transfer characteristics of the step-up transformer, as illustrated in Figure 7.14. The test circuit of Figure 7.13 simulates the SQUID input circuit of FMS. Here, the simulated SQUID’s voltage V s across a 1 Ω resistor is generated by the circumscribed V in via a voltage divider consisting of 1 kΩ and 1 Ω resistors. The resistor R stands for the SQUID’s Rd , while C iso is retained. Note that the capacitor C ′ shunting the SW is not a real one. In fact, it only represents the combined effect of the input capacitance of the preamplifier and the distributed capacitance of SW. In the calibration of this step-up transformer, we send V in with different frequencies and receive V M1 at SW. In Figure 7.14, the magnitude-frequency characteristics (a) and phasefrequency characteristics (b) describe the relationships between V in and V M1 . In measurements, the test circuit of Figure 7.13 employs C iso = 1 μF and different values of R, which plays a major role for the calibrations. Here, two resonances at f 0 ≈ 50 kHz and f 1 ≈ 210 kHz were observed in Figure 7.14a. One resonant circuit with f 0 is composed of LPW and C iso and the other resonant circuit with Vin 1 kΩ

1Ω

R

Vs

Ciso

PW

1:n

VM1 SW C′

Figure 7.13 The test circuit simulates the SQUID’s input circuit for measuring the magnitude-frequency and phase-frequency characteristics. The voltage across the 1 Ω resistor represents a SQUID voltage signal V s , while V M1 is the input signal of the preamplifier.

7.3 FMS Revisited

f1

20 10 0

135 90 45 0

–45

–10 (a)

180

Phase (°)

VSW/VSQ (dB)

30 26 dB f0

10

100 Frequency (kHz)

–90 10 1000 (b)

f0

f1

100 Frequency (kHz)

1000

Figure 7.14 Complex transfer characteristics measured by a frequency response analyzer FRA5087 (NF company), which was connected to the V in and V M1 terminals of the simulation circuit. The magnitude-frequency characteristics (a) and the phase-frequency characteristics (b) are plotted for different R with C iso = 1 μF. At f = 30 kHz, the curves from top to bottom correspond to R = 0, 1, 5, 10, 20, and 51 Ω, respectively. In (a), the dashed line at 26 dB corresponds to the stepping-up factor of n = 20.

f 1 consists of LSW and C ′ . The resistor R not only damps the LPW C iso circuit but also influences the LSW C ′ circuit. Here, R is connected in series to LPW C iso but parallel to LSW C′ via LPW and C iso . Increasing R will strongly dampen the LPW C iso circuit (f 0 ) but boost LSW C ′ (f 1 ) (see Figure 7.14a). The total bandwidth of the magnitude-frequency characteristics decreases with increasing R. At R = 5 Ω, f 0 practically disappears, while LSW C ′ (f 1 ) exhibits a suitable bandwidth. It means that this step-up transformer may match Rd = 5 Ω of a strongly damped SQUID well for FMS. Note that R = 0 cannot be considered in practice, although the bandwidth reaches its maximum. Increasing C iso shifts f 0 to the left, while f 1 remains constant (not shown here). The phase values are independent of R at f 0 and f 1 , but they rapidly change with f (see Figure 7.14b). At R = 5 Ω, the phase change with frequency f is relatively linear. Here, the phase shifts between PW and SW become 50∘ (degree) at f M , −10∘ at 3f M , −40∘ at 5f M , and so on. The dispersion effect destroys the recurrence of a square wave at SW. Actually, a large flat bandwidth is required for translating the square wave from PW to SW in the step-up transformer. For square wave ΦMO with f M = 100 kHz, our analysis focuses on the transfer characteristics in the frequency range between 100 and 700 kHz, i.e. from the fundamental to the seventh harmonics of the square wave. However, there are no flat transfer characteristics (magnitude and phase), especially as Rd ≥ 10 Ω (see Figure 7.14). It means that a distortion of the square wave during transformation is unavoidable. The square wave signal at V M1 (t) in time domain is constructed by the sum of the Fourier components with different amplitudes and phase shifts at f M , 3f M , 5f M , and 7f M , i.e. VM1 (t) = A1 cos(2𝜋 fM t + 𝜑1 ) + A3 cos(6𝜋fM t + 𝜑3 ) + A5 cos(10𝜋fM t + 𝜑5 ) + A7 cos(14𝜋fM t + 𝜑7 ) + · · ·

(7.1)

According to the transfer characteristics (magnitude and phase) at Rd = 10 Ω in Figure 7.14, we can obtain all parameters of Ai and 𝜑i of the Fourier contributions, thus combining the V M1 (t) at SW in time domain, when a square wave

79

7 Flux Modulation Scheme (FMS)

30

10 0

0 –10 –20 –30

–1 0

10

20

30 t (μs)

40

50

Input voltage (a.u.)

1

20 n = VSW/VPW

80

Figure 7.15 A square wave signal with dashed line (right axis) passing through the step-up transformer characterized in Figure 7.14 with R = 10 Ω is seriously distorted. The effective step-up factor ne is much less than the designed step-up factor n = SW/PW = 20.

60

voltage is applied at PW of the step-up transformer. The combined V M1 (t) is no longer a square wave voltage, but rather a complicated periodic function with higher harmonic components, as shown in Figure 7.15, which fits our experimental observation very well. The time-averaged value of V M1 (t) is approximately half of the expected square wave voltage. The second imperfection of SQUID operation in FMS is that a square wave voltage across PW with f M cannot pass through the step-up transformer without distortions during SQUID operation. In short, these two imperfections for SQUID operation in FMS, (i) the shunt effect of PW and (ii) the nonlinear transformation from PW to SW, reduce the benefit of the step-up transformer. In fact, the effective step-up factor ne is much less than the designed step-up factor n = SW/PW = 20. In our experiments, ne may be less than n/4. Therefore, one should never believe that in FMS, the noise contribution from δΦe is suppressed by the (step-up) factor n of the transformer. 7.3.5

V(𝚽) Comparison Obtained by DRS and FMS

The above discussions were performed both in time and frequency domains. Now, we focus on the SQUID’s V (Φ) characteristics in the flux domain. It has been emphasized many times that the SQUID’s original V (Φ) can be obtained only by DRS, where 𝜒 → ∞. In FMS, V (Φ) should reappear at M2 (after the demodulation) of Figure 7.2. Actually, the two imperfections discussed in Sections 7.3.3 and 7.3.4 lead to a distortion of the demodulated signal V (Φ) in FMS. To illustrate this distortion, we compare the SQUID signals, V (Φ) obtained by DRS (without flux modulation) and the FMS with Φ0 /2 square wave modulation. Usually, one utilizes a linearly varying Φ(t) with a low frequency f ≪ f M (quasi-static) to obtain V (Φ), which is a periodical function. In fact, the shape of the original V (Φ) may contain some odd harmonic components of V i (i = 3, 5, … (2n + 1)Φ0 ). In order to compare the V (Φ) characteristics obtained by DRS and FMS clearly, we employed a weakly damped SQUID with abundant odd harmonic components in V (Φ). In this case, V (Φ) is no longer quasi-sinusoidal. In the absence of the transformer (i.e. in DRS), although its V swing ≈ 50 μV, a transfer coefficient of 𝜕V /𝜕Φ ≈ 400 μV/Φ0 was reached, because the harmonic components enhanced its 𝜕V /𝜕Φ at the working point W on the original V (Φ) curve (see curve I in Figure 7.16). In FMS, V (Φ) will be firstly modulated by ΦMO

7.4 Conclusion

30

Voltage (μV)

Figure 7.16 The V(Φ) characteristics of a SQUID with a loop inductance Ls = 350 pH. The original V(Φ) (curve I) is measured with DRS, while the V(Φ)DM (curve II) after the demodulation is recorded by FMS. The working point W is marked, where 𝜕V/𝜕Φ reaches the maximum.

20

(I)

10

400 μV/Φ0

0 –10

W 110 μV/Φ0

(II)

–20 0.0

0.5

1.0 Φ (Φ0)

1.5

2.0

with f M and then reappear after the demodulation (see Figure 7.2). In principle, the demodulated V (Φ)DM at M2 should be identical to V (Φ) recorded by DRS. However, the V (Φ)DM (see curve II in Figure 7.16) became a quasi-sinusoidal function due to the transformation discussed above and √ illustrated in Figure 7.14. Here, the same SQUID only yields a 𝜕V /𝜕Φ ≈ 2 × 2 × 40 μV/Φ0 ≈ 110 μV/Φ0 at W of FLL. Thus, the SQUID’s 𝜕V /𝜕Φ was almost reduced by a factor of 4, as compared to curve I. Using DRS, one indeed observes the original (intrinsic) SQUID’s properties, e.g. I–V characteristics and V (Φ) in ideal current bias mode. In contrast, all intrinsic SQUID properties with FMS are masked by the transformer. In fact, when using a commercial readout electronics of FMS, it is very difficult to evaluate the real values of the SQUID’s V swing or 𝜕V /𝜕Φ or Rd at the working point, thus leading to the misconception that FMS is a very low-noise readout electronics on account of the ratio of SW/PW of the step-up transformer. In fact, the SQUID’s intrinsic noise δΦs cannot be measured easily with FMS.

7.4 Conclusion In conclusion, when the SQUID is operated in ideal current bias mode, the condition of 𝜒 = Zs /Rd → ∞ should be fulfilled, where I s = constant should remain valid. In practical FMS, the step-up transformer acts as an ac shunt, bringing the SQUID operation into the mixed bias mode, where 𝜒 usually lies between 0.5 and 3. Actually, almost all readout electronics in FMS detect the voltage signal, i.e. the SQUID is just operated in nominal current bias mode. The original motivation for FMS is that the transformer steps up the SQUID signal by a factor equal to the turn ratio SW/PW, thus providing both a good impedance matching and a suppression of δΦe . Nevertheless, using a transformer leads to three adverse effects: (i) The swing of the SQUID signal at the input stage V M1 is reduced (see Table 7.2). (ii) The nonlinear transformation from PW to SW reduces the benefit of the step-up transformer (see Figures 7.14 and 7.15). Furthermore, the transformer filters out higher harmonic components of the SQUID’s V (Φ), thus further reducing 𝜕V /𝜕Φ at the working

81

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7 Flux Modulation Scheme (FMS)

point (see Figure 7.16). (iii) Additional thermal noise from the transformer (not discussed here) is added [13]. Today, FMS has already been commercialized as the standard readout electronics for dc SQUIDs. One does not need to have much knowledge or experience on SQUID readout electronics to use it for applications. Generally, FMS is suitable for strongly damped SQUIDs with small Rd . In our experience, the virtual voltcan be equivalent to the V n of age noise Vn∗ of FMS, in the case of Rd < 10 Ω, √ an ultralow PCBT preamplifier with 0.3–0.4 nV/ Hz. Unfortunately, with FMS, one cannot observe the SQUID’s intrinsic noise δΦs in most cases. The data in Table 7.2 and in Figure 7.14, i.e. the two imperfections of FMS mentioned above, alert us that FMS is not suited as a readout technique for a SQUID with a large Rd > 10 Ω.

References 1 Forgacs, R.L. and Warnick, A. (1967). Digital-analog magnetometer utilizing

superconducting sensor. Review of Scientific Instruments 38 (2): 214–220. 2 Drung, D., Cantor, R., Peters, M. et al. (1990). Low-noise high-speed dc

3

4

5

6

7

8

9

10

superconducting quantum interference device magnetometer with simplified feedback electronics. Applied Physics Letters 57 (4): 406–408. Wellstood, F.C., Urbina, C., and Clarke, J. (1987). Low-frequency noise in dc superconducting quantum interference devices below 1 K. Applied Physics Letters 50 (12): 772–774. Foglietti, V., Giannini, M.E., and Petrocco, G. (1991). A double DC-SQUID device for flux locked loop operation. IEEE Transactions on Magnetics 27 (2): 2989–2992. Liu, C., Zhang, Y., Mück, M. et al. (2012). An insight into voltage-biased superconducting quantum interference devices. Applied Physics Letters 101 (22): 222602. Lee, Y.H., Kim, J.M., Kwon, H.C. et al. (1995). 3-Channel double relaxation oscillation SQUID magnetometer system with simple readout electronics. IEEE Transactions on Applied Superconductivity 5 (2): 2156–2159. Wellstood, F., Heiden, C., and Clarke, J. (1984). Integrated dc SQUID magnetometer with a high slew rate. Review of Scientific Instruments 55 (6): 952–957. Clarke, J., Goubau, W.M., and Ketchen, M.B. (1976). Tunnel junction dc SQUID: fabrication, operation, and performance. Journal of Low Temperature Physics 25 (1–2): 99–144. Koch, R.H., Rozen, J.R., Woltgens, P. et al. (1996). High performance superconducting quantum interference device feedback electronics. Review of Scientific Instruments 67 (8): 2968–2976. Penny, R.D., Lathrop, D.K., Thorson, B.D. et al. (1997). Wideband front end for high-frequency SQUID electronics. IEEE Transactions on Applied Superconductivity 7 (2): 2323–2326.

References

11 Matlashov, A., Espy, M., Kraus, R.H. et al. (2001). Electronic gradiometer

using HTc SQUIDs with fast feedback electronics. IEEE Transactions on Applied Superconductivity 11 (1): 876–879. 12 Hong, T., Wang, H., Zhang, Y. et al. (2016). Flux modulation scheme for direct current SQUID readout revisited. Applied Physics Letters 108 (6): 062601. 13 Carelli, P. and Foglietti, V. (1982). Behavior of a multiloop dc superconducting quantum interference device. Journal of Applied Physics 53 (11): 7592–7598.

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8 Flux Feedback Concepts and Parallel Feedback Circuit 8.1 Flux Feedback Concepts and History From Section 7.3, it is known that the flux modulation scheme (FMS) has difficulty in reading out the SQUID intrinsic noise, δΦs . The direct readout scheme (DRS) with feedback circuits in the “head stage” at cryogenic temperature is therefore highlighted in this chapter. Generally, there are two typical kinds of feedback circuits, i.e. the parallel feedback circuit (PFC) and series feedback coil (circuit) (SFC). Both feedback circuits are sketched in Figure 8.1. The PFC consists of a resistor Rp connected to a coil Lp in series that shunts to the SQUID, where Lp couples to the SQUID with a mutual inductance Mp (Figure 8.1a). The SFC consists of a coil Lse connected to the SQUID in series, where Lse couples to the SQUID with a mutual inductance Mse (Figure 8.1b). Here, the “four-terminal element” introduced in Chapter 4 for dc SQUID magnetometry remains, so the function of the feedback coil Lf and the principle of flux locked loop (FLL) operation will not be repeated here. In fact, both feedbacks are realized by a feedback flux ΦF via Mp or Mse . Namely, a current flowing through either the shunted PFC or the SFC (through the SQUID) generates a feedback flux ΦF , thus producing the total flux of Φ ± ΦF in the SQUID loop. Note that the measured Φ and ΦF must be coherent. In 1990, Drung developed “additional positive feedback” (APF) in current bias mode [1], as illustrated in Figure 8.2a. This work made a major breakthrough in the SQUID readout technique because it opened the door to incorporating a flux feedback circuit into the SQUID’s input circuit (head stage). In the APF system, the “head stage” is modified, but the readout electronics of the DRS described in Chapter 5 remains. Actually, the so-called APF is the SQUID shunted by the PFC and operated in nominal current bias mode, where the key aspect is enhancing the SQUID’s transfer coefficient (𝜕V /𝜕Φ) at the working point W. A large (𝜕V /𝜕Φ) reduces the noise contribution from the preamplifier’s voltage noise source, V n , which is an important component of δΦe from the readout electronics. In the FMS, one attempts to enlarge the signal swing V swing with the help of a step-up transformer, thereby increasing 𝜕V /𝜕Φ at W. However, in APF, the increase in 𝜕V /𝜕Φ is realized with asymmetric V (Φ) characteristics. From both works, we can learn that a large V swing leads to a large (𝜕V /𝜕Φ) at the working point but the opposite logical relation does not always hold. SQUID Readout Electronics and Magnetometric Systems for Practical Applications, First Edition. Yi Zhang, Hui Dong, Hans-Joachim Krause, Guofeng Zhang, and Xiaoming Xie. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

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8 Flux Feedback Concepts and Parallel Feedback Circuit

Mf

Mp

Mf

Rp

Lf

Lf

Lp

Mse

(a)

Lse

(b)

Figure 8.1 The SQUID’s “head stage” with the PFC and SFC. Here, the PFC is always shunted to the SQUID (a), while the SFC connects to the SQUID in series (b).

Ib A Rd

Mp

Rd

Rp

Lp

Mp

V

Rp Vb

Lp

(a)

(c) Ib A Mse

Rd

Lse

(b)

Lp In

Rd

Rp

Mp

Mp

V

Rp

Lp Mse

Vb

Lse

(d)

Figure 8.2 Four schematic diagrams of a modified DRS at the “head stage” with the Lf coil omitted: (a) The APF circuit consists of Rp and Lp shunted to the SQUID operated in current bias mode. The voltage “V” across the parallel circuit is the readout quantity; (b) BCF is formed by a coil Lse coupled to the SQUID with Mse (SFC), where In (preamplifier’s current noise) flows through Lse and the APF circuit; (c) NC uses the same SQUID’s input circuit of the PFC as in (a) but is operated in voltage bias mode. Here, the current “A” through the parallel circuit is the readout quantity; (d) the SBC improves the NC scheme with an additional SFC, i.e. Lse . Generally, all coils in these figures are superconducting, and their impedances approach zero, i.e. 𝜔L → 0 at the low-frequency limit.

8.2 SQUID’s Apparent Parameters

Table 8.1 Feedback circuits in the two bias modes. PFC

SFC

PFC + SFC

Current bias modea)

APF

BCF

—

Voltage bias mode

NC

—

SBC

a) Nominal current bias mode.

In 1993, based on the APF circuit, Drung and Koch proposed a current feedback for suppression of the preamplifier’s current noise I n . This current feedback is named “bias current feedback (BCF),” which is an example of an SFC (see Figure 8.2b) [2, 3]. In 1995, Kiviranta and Seppä used the same SQUID’s input circuit of APF (i.e. PFC) but operated it in voltage bias mode, which is called “noise cancellation (NC),” and this work was published in [4] (see Figure 8.2c). The performances of APF and NC are quite different. The major difference is that NC does not enhance the SQUID’s transfer coefficient 𝜕I/𝜕Φ in voltage bias mode but rather directly suppresses the preamplifier’s noise contribution of V n with increasing dynamic resistance at W. In 2010, Xie et al. reported a modified combination of APF and BCF operated in voltage bias mode, which is called the “SQUID bootstrap circuit (SBC),” and is illustrated in Figure 8.2d [5]. Based on NC, the SBC additionally introduced an SFC to the SQUID. Since then, Xie’s group has studied the SBC in detail under different feedback conditions. To unify the technical terms, Table 8.1 summarizes the above configurations of the flux feedback circuits with PFC and SFC. In fact, the influence of the PFC has already been discussed as a parallel input circuit in Chapter 7 with regard to mixed bias mode and the dimensionless parameter 𝜒. Consequently, PFC destroys the SQUID’s ideal current bias mode. SFC, however, remains in the ideal current bias mode in the low-frequency limit (𝜔Lse → 0). In contrast, neither PFC nor SFC has any influence in the SQUID voltage bias mode. Apparently, PFC changes the transfer coefficient 𝜕V /𝜕Φ, and SFC changes 𝜕I/𝜕Φ. However, the nature of both feedbacks is to change the SQUID dynamic resistance Rd , as discussed in Section 4.4, at which the SQUID can be considered a resistive element for readout electronics. This concept of apparent Rd (Φ) will be repeatedly emphasized in the following discussion about the flux feedback.

8.2 SQUID’s Apparent Parameters Prior to detailed discussions about flux feedback circuits, we should introduce a new concept: apparent parameters. In a sense, the modified SQUID’s “head stage” with PFC (or SFC) can be regarded as a new device with apparent parameters. For example, a circuit consisting of a resistor R and two batteries with voltages V 1 and V 2 is illustrated in Figure 8.3a. There are three cases to be discussed. Note that we assume that (1) V 1 < V 2 and (2) the internal resistances of the batteries

87

8 Flux Feedback Concepts and Parallel Feedback Circuit

I

–

+

+ (a)

+

+ R′

Battery V2

or

I

–

–

Battery V2

R

Battery V1

88

– (b)

Figure 8.3 Schematic diagram of a circuit containing a resistor R and two batteries with voltages V 1 and V 2 (a); the ratio of V 2 /I is regarded as an apparent resistance R′ (b).

can be neglected. In case (a) of V 1 = 0 (short out), the ring current I is denoted by (I)a = V 2 /R; in case (b) of two batteries with equal polarities, the ring current (I)b = (V 2 − V 1 )/R; in case (c) of two batteries connected with opposite polarities, the ring current (I)c = (V 2 + V 1 )/R. It results in the relation (I)c > (I)a > (I)b . If the battery of V 1 and R would be concealed, only V 2 and I would be readable, thus leading to an apparent resistor R′ = V 2 /I (see Figure 8.3b). For the above three cases, we obtain the relation R′b > R′a > R′c . Indeed, only R′a records the real R, while R′b and R′c are the apparent resistance R′ . In the following, the apparent parameters will be marked below with an apostrophe or a subscript, e.g. the SQUID’s apparent dynamic resistance R′d or (Rd )PFC , where Rd presents the original SQUID’s dynamic resistance obtained from the I–V characteristics using the measurement circuit in Figure 3.1. In the PFC and SFC flux feedback circuits, the circuit model in Figure 8.3 and the concept of apparent parameters are often used. Additionally, instead of a pure SQUID, the SQUID with the PFC or SFC with new apparent parameters connects to the preamplifier at room temperature (RT) in the DRS. Here, the new design (the modified “head stage”) will determine the noise contributions from the readout electronics, δΦe . To realize minimal system flux noise δΦsys , any developments in the readout scheme are intended to find a new profitable balance among five parameters: the three readable SQUID parameters (values) near the working point W, i.e. the SQUID’s dynamic resistance Rd , the SQUID’s transfer coefficient 𝜕V /𝜕Φ (or 𝜕I/𝜕Φ), and the intrinsic flux noise δΦs exhibited as δV s (or δI s ), and two noise sources of the preamplifier (V n and I n ). Note that all five original parameters above are innate and can never be changed. However, the apparent parameters, R′d and (𝜕V /𝜕Φ)′ (or (𝜕I/𝜕Φ)′ ), of a modified “head stage” with PFC or SFC can be changed, thereby reducing the noise contribution from the preamplifier’s V n and I n , namely, δΦe in δΦsys . Conventionally, the ultimate goal for developing SQUID readout schemes is to observe the SQUID’s intrinsic flux noise δΦs , i.e. δΦsys ≈ δΦs . In brief, the PFC and SFC provide such possibilities. In this chapter, we focus on only the PFC, while the SFC will be separately discussed in Chapter 9.

8.3 Parallel Feedback Circuit (PFC)

8.3 Parallel Feedback Circuit (PFC) 8.3.1

Working Principle of the PFC in Current Bias Mode

As an entry point, we qualitatively analyze the PFC in current bias mode, namely, the APF scheme displayed in Figure 8.2a. In a normal forward amplifier, APF is connected to its noninverting input terminal. In the following, our analyses are always divided into two parts: In part (I), we discuss the apparent parameters of the SQUID with PFC, e.g. the change in the SQUID’s transfer coefficient (𝜕V /𝜕Φ)PFC and (Rd )PFC at the working point. In part (II), we focus on the flux noise contribution of the preamplifier, δΦe . In part (I), our discussion about the functions of PFC can be further divided into two steps, i.e. (i) without (shorted) and (ii) with the feedback coil Lp . In step (i), the SQUID is shunted by only the resistor Rp , whereby the SQUID acts as a varying Rd (Φ) and Lp is shorted out. In fact, this parallel circuit consists of a fixed resistor Rp and a varying Rd (Φ) (see Figure 8.4 left). The constant bias current I b is divided into two currents according to Kirchhoff ’s current law. One current I s flows through the SQUID, Is = Ib × {Rp ∕[Rp + Rd (Φ)]} and the other current I Rp through Rp , IRp = Ib × {Rd (Φ)∕[Rp + Rd (Φ)]} Here, the original SQUID’s Rd (Φ) is non-constant and nonlinear, so the current distributions of I s (Φ) and I Rp (Φ) depend on Φ, despite the constant I b = I s (Φ) + I Rp (Φ). Ib

Ib

Is

Is

IRp

IRp

Rd Rd(Φ)

ΔΦ

Rp

Δip

Rp

ΔVs = ΔΦ × (∂V/∂Φ)

Figure 8.4 In current bias mode, the equivalent circuit of a SQUID shunted by Rp is a parallel circuit of Rd (Φ) and Rp (left). At the working point in FLL, it evolves to a voltage source and two fixed resistors, Rd and Rp (right). Here, the SQUID’s differential model is employed, thus generating a ring current Δip .

89

8 Flux Feedback Concepts and Parallel Feedback Circuit

In FLL, because of the instantaneous offset and reset of the working point W, the SQUID indeed is operated in a differential model (see Section 4.4). Here, the SQUID is the equivalent of a battery with the voltage ΔV s = ΔΦ × (𝜕V /𝜕Φ) and a series-connected fixed Rd (as internal resistance of the battery) in current bias mode, as described in Figure 8.4 (right). So, a ring current Δip = ΔV s /(Rd + Rp ) appears and is equivalent to the change in the current distribution of I s (Φ) and I Rp (Φ) with Φ. Namely, Ib = Is (Φ) + IRp (Φ) = (Is − Δip ) + (IRp + Δip ) where I s and I Rp are invariant once the working point W is selected, while Δip exhibits the SQUID’s behavior. In fact, across the parallel circuit, the swing of V (Φ)Rp characteristics and the transfer coefficient (𝜕V /𝜕Φ)Rp are reduced due to the resistance load. The description of the SQUID in Figure 8.4 was also used in the analysis of the FMS in Section 7.1. However, in the FMS, the bias current I b flows only through the SQUID, and only the ac ring current Δi passes through the primary winding (PW) of the transformer (see Figure 7.2). Actually, DRS is a pure dc-circuit. With the shunting effect of the resistor Rp , according to Ohm’s law, the original SQUID’s I–V characteristics become increasingly steep, as shown in Figure 8.5, where the apparent I–V curves are sketched as dotted lines. In current bias mode, the horizontal line of I b intersects the two I–V curves at the flux limits, which correspond to integer- and half-integer-flux quantum, Φ0 . The voltage difference between the intersection points, a (a′ ) and b (b′ ), denotes the swing of the SQUID’s V (Φ) characteristics. Here, (Vb′ –Va′ ) < (Vb –Va ) means that the swing of V (Φ)Rp is reduced. The swing reduction was reported in Ref. [6]. In fact, three apparent parameters at W in the SQUID’s differential model are diminished due to the shunt resistor Rp : (1) (Rd )Rp < Rd because (Rd )Rp = Rd //Rp . (2) (𝜕V /𝜕Φ)Rp < 𝜕V /𝜕Φ because (𝜕V /𝜕Φ)Rp = [Rp /(Rd + Rp )] × (𝜕V /𝜕Φ). (3) (δV sn )Rp < δV sn , where δV sn is the exhibition of the SQUID intrinsic flux noise δΦs in current bias mode, i.e. δV sn = δΦs × (𝜕V /𝜕Φ) and (δV sn )Rp = δΦs × (𝜕V /𝜕Φ)Rp . However, δΦs remains. Figure 8.5 Two measured SQUID’s I–V characteristics: the original one with two solid lines representing two typical flux states (integerand half-integer-flux quantum) and the other one with Rp shunt plotted by two dotted lines.

35 30 25 I (μA)

90

20 15 10

a′

b′ a

b

5 0 0

20

40 V (μV)

60

80

100

8.3 Parallel Feedback Circuit (PFC)

V

I W1

II

W2 W3 S

G

III 0

1 ΔΦoff

2

Φ/Φ0

Figure 8.6 The SQUID’s V(Φ) characteristics without (step (i)) and with the PFC (step (ii)): curve I represents the SQUID’s original V(Φ) without the PFC. Compared to those of curve I, the swings of curve II, where the SQUID is shunted by Rp only (Lp is shorted), are reduced, and curve III with the complete PFC is asymmetrical. However, the apparent (𝜕V/𝜕Φ)PFC at the working point W3 is distinctly enhanced by the PFC and surpasses the (𝜕V/𝜕Φ) of curve I at W1 . Here, all curves should be obtained using DRS.

In other words, the original SQUID’s parameters are replaced by the apparent parameters due to the shunt Rp . Also, the changes of Rd and 𝜕V /𝜕Φ relate to the noise contribution of the preamplifier. A mode for displaying the SQUID’s I–V characteristics is not available in almost all commercial SQUID readout electronics; therefore, the most important SQUID readout becomes the original V (Φ) characteristics in current bias mode, as illustrated in Figure 8.6. There, the original swing and flux-to-voltage coefficient (𝜕V /𝜕Φ) at the working point W1 of curve I without Rp are larger than those of curve II with the shunt Rp at the working point W2 . In step (ii), when SQUID is shunted by the complete PFC (with Lp ) shown in Figure 8.2a, the swing of V (Φ) of curve III in Figure 8.6 remains the same as that of curve II in step (i). Now, we can use the SQUID’s differential model shown in Figure 8.4 to analyze PFC for the asymmetrical curve III. Because of the constant I b and Rd , in this model a constant current I PFC flowing through the PFC generates an invariant flux in the SQUID loop, which is denoted as a flux-offset, ΔΦoff , in curve III. Here, this ΔΦoff can be expressed as ΔΦoff = IPCF × Mp = Ib × [Rd ∕(Rp + Rd )] × Mp The other current flowing through the PFC is the ring current Δip (see Figure 8.4), which generates a feedback flux ΦPFC in the SQUID loop: ΦPFC = ΔiPFC × Mp = [ΔΦ × (𝜕V ∕𝜕Φ)∕(Rd + Rp )] × Mp Here, ΔΦ and ΦPFC are coherent, thus leading to the total flux Φtot = ΔΦ ± ΦPFC in the SQUID loop. The sign depends on the slope polarity of V (Φ) where the working point W is located. Indeed, the apparent V (Φ)PFC characteristics become very asymmetric in curve III of Figure 8.6. When Φtot = ΔΦ + ΦPFC , the corresponding slope of V (Φ)PFC becomes gradual, but at Φtot = ΔΦ − ΦPFC , it becomes steep. In the case of ΦPFC ≈ ΔΦ (the critical condition of APF), the apparent (𝜕V /𝜕Φ)PFC at the

91

8 Flux Feedback Concepts and Parallel Feedback Circuit

steep slope approaches infinity. The expression of the critical condition can be easily derived: Mp × (𝜕V ∕𝜕Φ) = Rd + Rp where the values of (𝜕V /𝜕Φ) and Rd are taken from original SQUID characteristics at the working point. In Figure 8.6, all working points W1 , W2 , and W3 for FLL operation are set at the maximum of (𝜕V /𝜕Φ), which is located at Φ = (2n + 1)Φ0 /4 on the V (Φ) curves. Especially, if the working point W3 is set at the steep slope of the asymmetric V (Φ) curve III with PFC, a very large (𝜕V /𝜕Φ)PFC is achieved. In Chapter 3, we explained that a dc SQUID is a resistive-like element for readout electronics and its resistance is modulated by the applied flux. Under a bias current I b , a large (𝜕V /𝜕Φ)PFC at W3 must be caused by a large (Rd )PFC change. In Figure 8.7, three I–V characteristics (two gray lines and one dashed line) located at Φ = (2n + 1)Φ0 /4 are plotted. The dashed line corresponds to W2 at V (Φ) of curve II (without Lp ) in Figure 8.6. In the case of the PFC, the gray solid lines represent two different I–V curves where n = odd number or even number due to the asymmetry of V (Φ)PFC . The flat (I–V )PFC characteristics present a large (Rd )PFC and correspond to the steep slope of V (Φ)PFC , e.g. at W3 on curve III in Figure 8.6. In contrast, the steep (I–V )PFC means a small (Rd )PFC at the gradual slope of V (Φ)PFC . In fact, Figure 8.7 explains the fact that an increase or a decrease in (𝜕V /𝜕Φ)PFC is the manifestation of a changing (Rd )PFC at W3 in current bias mode. Because 𝜕V /𝜕Φ = 0 at the two limiting flux states of nΦ0 and (2n + 1)Φ0 /2, the ring current ΔiPFC = 0, so the bounds of the SQUID’s I–V characteristics (black dashed lines) remain unchanged.

30

I (μA)

92

G

20 W3

S

Ib 10

0 0

20

40

60

80

100

V (μV)

Figure 8.7 The experimentally recorded data: the dashed lines represent the SQUID’s I–V characteristics in step (i) at the following three flux states (from top to bottom): Φ = nΦ0 , Φ = (2n + 1)Φ0 /4, and Φ = (2n + 1)Φ0 /2, respectively. The two gray solid lines plot the two I–V curves with the PFC (in step (ii)). Line G corresponds to the gradual slope of V(Φ)PFC , while line S corresponds to the steep slope. Here, we assume that the three I–V characteristics for Φ = (2n + 1)Φ0 /4 intersect at point W3 .

8.3 Parallel Feedback Circuit (PFC)

Vs III

I

IV II

0

1

2

Φ/Φ0

Figure 8.8 The V(Φ) characteristics with the PFC (APF) are schematically shown when [Mp × (𝜕V/𝜕Φ)]/(Rd + Rp ) ≈ 0 (dashed curve I), 0.5 (solid II), 0.65 (dashed III), and 0.8 (solid IV), respectively. Curve I is obtained at Mp = 0; i.e. Lp is short-circuited.

In practice, the PFC (APF) should work in the regime of [Mp × (𝜕V /𝜕Φ)]/ (Rd + Rp ) < 1, where V (Φ)PFC is asymmetrical and (𝜕V /𝜕Φ)PFC is increased. In Figure 8.8, four V (Φ)PFC characteristics with different PFC parameters are schematically shown, namely, when [Mp × (𝜕V /𝜕Φ)]/(Rd + Rp ) ≈ 0 (curve I), 0.5 (curve II), 0.65 (curve III), and 0.8 (curve IV). At the steep slope of V (Φ)PFC , (𝜕V /𝜕Φ)PFC increases with increasing values of [Mp × (𝜕V /𝜕Φ)]/(Rd + Rp ). In theory, (𝜕V /𝜕Φ)PFC on the steep slope of V (Φ)PFC at the working point approaches infinity at the critical condition [Mp × (𝜕V /𝜕Φ)]/(Rd + Rp ) ≈ 1. However, due to the voltage noise from the SQUID and the readout electronics, stable operation necessitates that (𝜕V /𝜕Φ)PFC be enhanced up to only a limited value, which is still away from the critical condition. In contrast, (𝜕V /𝜕Φ)PFC on the gradual slope is always smaller than the original 𝜕V /𝜕Φ. Although (𝜕V /𝜕Φ)PFC is reduced by half at the critical condition described above, the wide linear range at the gradual slope can be utilized for some special applications, as discussed below. When [Mp × (𝜕V /𝜕Φ)]/(Rd + Rv ) ≥ 1, hysteresis appears at the steep slope of the V (Φ)PFC characteristics; therefore, this case should be avoided. By putting together steps (i) and (ii) in our discussion about the apparent parameters of the SQUID with the PFC (part (I)), the two opposite effects, decreasing (𝜕V /𝜕Φ)Rp at W2 due to the Rp shunting effect (see curve II of Figure 8.6) and increasing (𝜕V /𝜕Φ)PFC at W3 on the steep slope with the PFC (curve III), appear simultaneously. However, the total effect is positive; i.e. the apparent (𝜕V /𝜕Φ)PFC at the working point W3 is indeed much larger than the original (𝜕V /𝜕Φ) at W1 in Figure 8.6. Now, we discuss part (II) of the preamplifier’s noise contribution, δΦe , with the PFC (APF). Here, we separate the two noise sources of the preamplifier, V n and I n , contributing to (δΦe )2 = δΦ2V n + δΦ2In . We can simply obtain the contribution from V n , denoted by δΦV n = V n /(𝜕V /𝜕Φ)PFC , which is clearly reduced, because (𝜕V /𝜕Φ)PFC is larger than (𝜕V /𝜕Φ) without the PFC. Note that (Rd )PFC > Rd at the working point leads to an increase in the voltage noise contribution caused by the preamplifier’s I n , i.e. V In = [I n × (Rd )PFC ]. However, the flux noise contribution from I n , δΦIn , remains constant, namely, with or without the PFC δΦIn = [In × (Rd )PFC ]∕(𝜕V ∕𝜕Φ)PFC = (In × Rd )∕(𝜕V ∕𝜕Φ)

93

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8 Flux Feedback Concepts and Parallel Feedback Circuit

because the values of (𝜕V /𝜕Φ)PFC increase in proportion to the (Rd )PFC values in current bias mode. It has been emphasized before that the SQUID intrinsic noise δΦs is not affected by the PFC. In fact, the PFC (APF) reduces only the contribution caused by δΦV n , thus highlighting δΦIn in δΦe . To suppress δΦIn , an SFC will be introduced in Chapter 9. In brief, the SQUID with the PFC can be regarded as a new flux detector with asymmetric I–V and V (Φ) curves, thus increasing (𝜕V /𝜕Φ)PFC at the working point on the steep slope of V (Φ)PFC , as [Mp × (𝜕V /𝜕Φ)]/(Rd + Rv ) < 1. In this (nominal) current bias mode (e.g. APF), the noise contribution from the preamplifier’s V n can be reduced because δΦV n = V n /(𝜕V /𝜕Φ)PFC . Here, the new flux detector is equivalent to a flux-to-voltage converter with an apparent transfer coefficient (𝜕V /𝜕Φ)PFC connected to an apparent (Rd )PFC in series, although all the SQUID’s original characteristics are innate and do not change. This inference will again be affirmed subsequently. 8.3.2

Working Principle of PFC in Voltage Bias Mode

Now, let us observe the behavior of the PFC in voltage bias mode, i.e. the NC scheme [4, 7]. Our analysis of NC is also divided into two parts, as referred to regarding APF analysis in Section 8.3.1. In part (I) concerning the SQUID’s signal changes, we first compare I(Φ) and the SQUID’s transfer coefficient (𝜕I/𝜕Φ) at the working point with and without the PFC. Part (II) discusses the flux noise contributions of the preamplifier’s V n and I n . The RT readout electronics for voltage-biased SQUIDs are shown in Figure 8.9a, while the FLL circuit is omitted. In Chapter 3, we learned that the preamplifier (e.g. an op-amp) acts as a current-to-voltage converter in voltage bias mode and its conversion gain depends on Rg /Rd . There is no difference between APF and NC in terms of PFC configuration. Unlike APF in current bias mode, the SQUID and the PFC acting as source resistors connect to the inverting input terminal of the preamplifier. The bias voltage source V b providing the clamp potential to the inverting terminal is applied at the noninverting input terminal of the preamplifier. The current I V b always flows out from V m via Rg to this parallel circuit of Rp and the SQUID dynamic resistance Rd (Φ), thus generating an equivalence voltage of V b at the inverting input terminal. The coil Lp of the PFC couples to the SQUID with a mutual inductance Mp . In voltage bias mode, the readout quantity is the current, and I V b is divided into two currents, I s and I PFC , through the SQUID and PFC, respectively. Figure 8.9b presents the equivalent circuit of both currents, where V b is regarded as an ideal voltage source with an internal resistance Rinter = 0. Therefore, the two currents are independent due to the parallel circuit, namely, Vb = Is × Rd = IPFC × Rp The I(Φ) characteristic of NC can be monitored from the preamplifier’s output V m ≈ I V b × Rg , where I V b = I s + I PFC . The current I s is modulated by Φ, i.e. the SQUID’s I(Φ) characteristics, while I PFC remains constant. In contrast to APF in current bias mode, both currents in voltage bias mode are independent because I s and I PFC are separately provided by V b ; therefore, no ring current ΔiPFC between

8.3 Parallel Feedback Circuit (PFC)

Rd (Is)Vb

Rg

Rp (IVFC)Vb Lp

–

Is

IVb

Rd

Vm

Rp Lp

+ Vb

Mp

(a)

Vb

Mp (b)

Vm (IVb)

0

IPFC

1 ΔΦ off

2

Φ/Φ0

(c)

Figure 8.9 (a) A SQUID shunted by a PFC (NC scheme) connects to the inverting input terminal of an op-amp, which acts as a current-to-voltage converter. The bias voltage, V b , at the noninverting input terminal provides a clamp potential to the inverting input terminal; (b) the equivalent input circuit of (a), where two independent closed current circuits, Is and IPFC , from V b flow through the SQUID branch marked by the dashed outline border and PFC branch marked by the dotted line (square), respectively; (c) a schematic of the I(Φ) characteristics of a voltage-biased SQUID observed at V m , which is the product of IVb and Rg .

the two branches exists. In fact, the SQUID is operating in ideal voltage bias mode, which is not influenced by the PFC. Thus the invariable I PFC generates a flux ΔΦoff = I PFC × Mp into the SQUID, which can be considered a dc flux offset for SQUID operation. In Figure 8.9c, there are two I(Φ) characteristics with Lp connected (black curve) or shorted (gray curve) in the PFC. Consequently, the shape of the I(Φ) characteristics with or without the PFC does not change; i.e. the swing of the SQUID’s I(Φ) and the symmetry of I(Φ) remain in the NC scheme. In other words, for a voltage-biased SQUID, the 𝜕I/𝜕Φ at the working point does not change with or without the PFC, i.e. (𝜕I/𝜕Φ) = (𝜕I/𝜕Φ)PFC . In part (II) concerning δΦe , there are still two independent noise currents from the preamplifier in the NC scheme: one is the current noise I n , and the other is the I V n caused by the V n of the preamplifier. According to the working principle of a current-to-voltage converter, the V b and V n at the noninverting input terminal are generated by two independent currents, I V b and I V n , which always flow outward from V m . The behaviors of I V b have just been described in part (I). In the following discussion, we assume that the dc voltage V b has already been adjusted to the optimal value; i.e. the I(Φ) characteristic reaches the maximum value of 𝜕I/𝜕Φ at the working point. The I n and V n characteristics of preamplifiers have already been discussed in Chapter 5. Indeed, I n always exists and does not depend on any source resistance at the input terminals of the op-amp. The other noise current, I V n , caused by

95

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8 Flux Feedback Concepts and Parallel Feedback Circuit

IVn Rd

Rg

Rp (Is)Vn

–

(IPFC)Vn

Vm

(Rd)PFC +

Lp

Vn

Mp (a)

IVn

(b)

Figure 8.10 The equivalent circuit of V n of an op-amp (preamplifier) in voltage bias mode. In the current-to-voltage converter (b), the parallel input circuit of the SQUID and PFC (a) can be regarded as the apparent dynamic resistance (Rd )PFC .

the preamplifier’s V n can be changed with the PFC, i.e. I V n = V n /(Rd )PFC , where (Rd )PFC is the apparent dynamic resistance of the parallel circuit consisting of the SQUID branch and PFC branch. The equivalent circuit of V n in voltage bias mode is shown in Figure 8.10, where the noise voltages of I V n × (Rd )PFC at the inverting input of the op-amp are clamped by V n at the noninverting input. Additionally, a large (Rd )PFC leads to a small I V n . In fact, I n and I V n always flow out from V m (the op-amp output) via Rg into (Rd )PFC. Here, two independent noise currents caused by the preamplifier are translated to the voltages, (V m )V n ≈ I V n × Rg and (V m )In ≈ I n × Rg , appearing at V m . In voltage bias mode with the PFC (e.g. the NC scheme), the flux noise caused by I n , δΦIn = I n /(𝜕I/𝜕Φ)PFC , remains, because the SQUID’s original (𝜕I/𝜕Φ) does not change, as discussed above in part (I). In contrast, the other component of δΦe , i.e. the flux noise δΦV n = I V n /(𝜕I/𝜕Φ) caused by V n , can be changed with the PFC. Now, let us study the behavior of (Rd )PFC with the PFC, while V n = I V n × (Rd )PFC must be clamped. According to the equivalent circuit shown in Figure 8.10, the noise current (I PFC )V n flowing through the PFC branch, denoted by (I PFC )V n = V n /Rp , is always constant. The (I PFC )V n generates a flux (ΦPFC )V n threading into the SQUID via the mutual inductance Mp , thus translating a voltage V s across the SQUID at the working point, Vs = (ΦPFC )V n × (𝜕V ∕𝜕Φ) = [(IPFC )V n × Mp ] × (𝜕V ∕𝜕Φ) It is very interesting that we have obtained the same equivalent circuit consisting of one resistor Rd and two voltage sources (batteries), V s and V n , as the one described in Figure 8.3. Here, V s and V n are coherent; therefore, the apparent resistance (Rd )SB of the SQUID branch depends on V s , i.e. (Rd )SB = Vn ∕[(Vn ± Vs )∕Rd ] = Rd × [Vn ∕(Vn ± Vs )] Here, the identity of the ± sign depends on the polarities of (𝜕V /𝜕Φ). According to the sign of V s , either (Rd )SB > Rd (negative sign), or (Rd )SB < Rd (positive sign).

8.3 Parallel Feedback Circuit (PFC)

Under the condition of (V n − V s ) = 0, (Rd )SB approaches infinity, so the current (I s )V n disappears. With this simple analysis, we obtain the critical condition of the NC scheme, namely, Mp × (𝜕V ∕𝜕Φ) = Rp or, Mp × (𝜕I∕𝜕Φ) = Rp ∕Rd However, in the original NC work, the critical condition was derived by a different method [4]. The current (I s )V n disappears under this condition, which may be the reason why the PFC in voltage bias mode was called the “noise cancellation” scheme. However, under the NC critical condition, I V n still flows through the PFC branch, namely, I V n = (I PFC )V n = V n /Rp , where Rp can be regarded as the apparent resistance (Rd )PFC illustrated in Figure 8.10. Compared to I V n without PFC, the (I PFC )V n with PFC is suppressed by a factor of Rp /Rd when Rp > Rd . In the other case of (V n + V s ) = 2V n (i.e. on the other slope of I(Φ)), at the NC critical condition, the current through the SQUID (I s )V n doubles, so the apparent (Rd )′SB is halved, i.e. (Is )V n = 2Vn ∕Rd = Vn ∕(Rd )′SB , where the value of V n /Rd is the noise current without the PFC. So, the total noise current I V n is the sum of (I s )V n and (I PFC )V n , i.e. I V n = (2V n /Rd ) + (V n /Rp ). In this case, the I V n increases due to the increase in (I s )V n . δΦe consists of two independent components, δΦV n and δΦIn , i.e. δΦ2e = δΦ2V n + δΦ2In , where δΦV n = [V n /(Rd )PFC ]/(𝜕I/𝜕Φ) and δΦIn = I n /(𝜕I/𝜕Φ) in voltage bias mode. In fact, δΦe cannot be completely cancelled in the NC scheme: (i) The current noise I n of the preamplifier always exists and the PFC cannot change the SQUID’s transfer coefficient (𝜕I/𝜕Φ); thus δΦIn does not change with the PFC. (ii) The noise current I V n caused by V n is just reduced with the PFC. Under the NC critical condition, the total noise current (I n )total of the preamplifier flowing out from the current-to-voltage converter is denoted by (In )total = [(Vn ∕Rp )2 + In2 ]1∕2 , thus resulting in δΦe = (I n )total /(𝜕I/𝜕Φ) at the working point. Actually, the mechanism of the PFC in voltage bias mode (NC scheme) is replacing the original SQUID’s Rd with an apparent (Rd )PFC , thus reducing the noise contribution δΦV n in the case of (Rd )PFC > Rd : i.e. δΦV n = [Vn ∕(Rd )PFC ]∕(𝜕I∕𝜕Φ) Figure 8.11 schematically shows the I(Φ) characteristics with PFC, where the sign of V s leads to different noise contributions of the preamplifier’s voltage noise V n at both slopes of I(Φ), despite its symmetrical shape. 8.3.3

Brief Summary of Qualitative Analyses of PFC

A SQUID’s I–V characteristics periodically change with changing Φ. At any stage of the I–V characteristics in flux modulation region (see Figure 3.3), all three

97

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8 Flux Feedback Concepts and Parallel Feedback Circuit

IVb (Vm)

Figure 8.11 Schematic of the I(Φ) characteristics in the NC scheme, where the behaviors of δΦe at the two slopes are different because IVn is suppressed at the working point W due to a large (Rd )PFC .

W Φ/Φ0 0

1

2

parameters, I, V , and Rd , are functions of Φ. The V (Φ) characteristics are exhibited in current bias mode and the I(Φ) characteristics in voltage bias mode, as shown in Figure 4.1. Both characteristics reflect the behaviors of Rd (Φ). As a highlight in Section 8.3, we have experimentally proved the differential chain rule (equation) in SQUID operation, where the definition Rd = 𝜕V /𝜕I (e.g. at the working point) can be modified to Rd = (𝜕V /𝜕Φ)/(𝜕I/𝜕Φ), i.e. 𝜕V /𝜕Φ = Rd × (𝜕I/𝜕Φ) or 𝜕I/𝜕Φ = (𝜕V /𝜕Φ)/Rd , as mentioned in Chapter 5. It is very important that the PFC does not change 𝜕I/𝜕Φ in either bias mode. Therefore, 𝜕I/𝜕Φ can be regarded as a constraint condition for PFC operation, namely, 𝜕I/𝜕Φ = (𝜕V /𝜕Φ)/Rd = (𝜕V /𝜕Φ)PFC /(Rd )PFC . In essence, the PFC increases the apparent (Rd )PFC , where (Rd )PFC > Rd . So, the large (𝜕V /𝜕Φ)PFC is just exhibited in the V (Φ) characteristics. In fact, a SQUID shunted by PFC can be regarded as a new device with asymmetric apparent (I–V )PFC characteristics at the working point, e.g. Φ = (2n + 1)Φ0 /4, where (Rd )PFC depends on whether n is an odd number or an even number (see Figure 8.7). This new device can be operated in either (nominal) current bias mode (APF) or voltage bias mode (NC). Because (𝜕I/𝜕Φ) = (𝜕I/𝜕Φ)PFC , δΦIn remains. The noise contribution of δΦV n is suppressed in two ways: either by increasing (𝜕V /𝜕Φ)PFC in APF or by employing a large (Rd )PFC in NC. Actually, one obtains the following unified expression of δΦV n with PFC: δΦV n = Vn ∕(𝜕V ∕𝜕Φ)PFC = [Vn ∕(Rd )PFC ]∕(𝜕I∕𝜕Φ)

(in current bias mode) (in voltage bias mode)

where (Rd )PFC × (𝜕I/𝜕Φ) = (𝜕V /𝜕Φ)PFC . Generally, there is only one readout quantity, either a current or a voltage. In (nominal) current bias mode with PFC (APF), the device exhibits asymmetrical V (Φ) characteristics and an increased (𝜕V /𝜕Φ)PFC at the working point on the steeper slope. However, all current performances are hidden; in voltage bias mode with the PFC, namely, in the NC scheme, (𝜕V /𝜕Φ)PFC is latent, but the I(Φ) characteristics become readable. Here, the symmetry and the swing of the I(Φ) characteristics remain, or say, (𝜕I/𝜕Φ)PFC is unchanged. In fact, a suppression of (I n )V n is realized by the large apparent (Rd )PFC , i.e. δΦV n = [V n /(Rd )PFC ]/(𝜕I/𝜕Φ) at W. Here, the different noise contributions of the preamplifier’s V n at the two slopes of the I(Φ) characteristics are demonstrated in Figure 8.11. In fact, APF is not operated in ideal current bias mode, because the swing of V (Φ) is already decreased due to the shunt resistor Rp of the PFC. In contrast, in

8.4 Quantitative Analyses and Experimental Verification of the PFC in Voltage Bias Mode

NC scheme, the swing of I(Φ) remains because the bias voltage V b is connected to the parallel circuit of SQUID and PFC. In theory, PFC should perform better in NC rather than in APF mode. However, in practice, both schemes effectively suppress δΦV n . Note that the PFC can also be called a “voltage feedback circuit” [8, 9]. Although it was emphasized that the PFC has the same essence in both bias modes, we obtained the nonuniform critical conditions from the references: (1) Mp × (𝜕V /𝜕Φ) = (Rd + Rp ) for APF in current bias mode [3] and (2) Mp × (𝜕V /𝜕Φ) = Rp or Mp × (𝜕I/𝜕Φ) = (Rp /Rd ) for NC in voltage bias mode [4]. In the following quantitative analysis, we will make the critical condition uniform: One will see that (Rd )PFC in voltage bias mode may approach infinity when Mp × (𝜕V /𝜕Φ) = (Rd + Rp ) is fulfilled. Actually, the expression of (2) for a voltage-biased SQUID is not the real critical condition; instead, it only describes the intermediate state, where the noise current caused by V n disappears only in the SQUID branch, i.e. (I s )V n = 0 in Figure 8.10. Therefore, the puzzling problem described above is eventually solved. Furthermore, we should pay attention to the flux noise contribution from Rp .

8.4 Quantitative Analyses and Experimental Verification of the PFC in Voltage Bias Mode Zeng et al. quantitatively analyzed the PFC in sufficient detail and suggested some possible designs with practical recommendation ranges for suitable PFC parameter choices [10]. This analysis of PFC was performed in voltage bias mode, where two currents flowing through the SQUID branch and the PFC branch can be separated due to the bias voltage source V b with Rinter ≈ 0 (Rinter is the internal resistance), as shown in Figure 8.9. Furthermore, the flux noise δΦR introduced by Rp of the PFC branch may not be negligible when Rp is larger than a certain value. Generally, the total noise of the readout electronics with the PFC, ΣδΦ, may thus include both δΦe and δΦR , i.e. ∑ δΦ2 = δΦ2e + δΦ2R = δΦ2V n + δΦ2In + δΦ2R Because the PFC is known not to affect δΦIn , only the preamplifier’s voltage noise δΦV n is taken into account. The objective in the following quantitative PFC analyses is to achieve minimal δΦ2PFC = δΦ2V n + δΦ2R . Note again that δΦ2V n cannot represent δΦ2e at all, because in most cases, δΦ2In from the preamplifier cannot be neglected, as shown in Chapter 5. 8.4.1

The Equivalent Circuit with the PFC in Voltage Bias Mode

The readout configuration of a voltage-biased SQUID with a PFC has already been shown in Figure 8.9, where the preamplifier (op-amp) acts as a currentto-voltage converter. For noise analysis of δΦPFC , two equivalent circuits for the noise current i′n caused by V Rp (the Johnson noise of Rp ) and the noise current i′′n caused by V n of the preamplifier are shown in Figure 8.12a,b, respectively. The total noise current in is the sum of i′n in (a) and i′′n in (b), i.e. i2n = (i′n )2 + (i′′n )2 ,

99

100

8 Flux Feedback Concepts and Parallel Feedback Circuit

′ in1

″ in1

in′

Rd

Rp

V–

in2 ′

V+

Rg – +

VRp

Vs

Rd

Vm

(a)

Vs

in″ Rg

″ in2

V–

–

Rp

V+

+

Vm

Vn

(b)

Figure 8.12 There are two circuits for the noise analyses: (a) is equivalent to V Rp originating from the PFC resistor Rp (a), and (b) is equivalent to the noise contribution V n of the ′ ′′ preamplifier. Here, V s across the SQUID is induced by either in2 (V Rp ) or in2 (V n ) via Mp .

thus resulting in a noise voltage (V m )n = in × Rg at V m . Here, the noise current is divided into two paths, e.g. i′n splits into i′n1 and i′n2 , which denote the currents flowing in the SQUID branch and in the PFC branch, respectively. We first discuss the flux noise contribution δΦR from the Johnson voltage noise V Rp = (4k B TRp )1/2 , where k B denotes the Boltzmann constant and T denotes the temperature. In the equivalent circuit in Figure 8.12a, the noninverting input of the operational amplifier is directly grounded. For an ideal op-amp, the voltages at the inverting and noninverting inputs are the same, i.e. V + = V − = 0. In fact, the circuit of Figure 8.12a is a typical adder with i′n = i′n1 + i′n2

(8.1)

i′n1

(8.2)

= −Vs ∕Rd

i′n2 = −VRp ∕Rp = −(4kB T∕Rp )1∕2

(8.3)

In Eq. (8.2), the induced noise voltage V s equals (𝜕V ∕𝜕Φ) × Mp × i′n2 , so i′n1 is proportional to i′n2 (8.3), where Rp is regarded as Johnson noise current source. By substituting Eq. (8.3) into Eq. (8.2), we obtain the following expression for i′n1 : i′n1 = [(𝜕V ∕𝜕Φ) × Mp × (4kB T∕Rp )1∕2 ]∕Rd Note that the sign of i′n1 is always positive; i.e. its direction is maintained, as shown in Figure 8.12a. Here, we define the parameter Mdyn = 1/(𝜕I/𝜕Φ) as the SQUID’s dynamic inductance due to its inductance dimension. Using the two relations Mdyn = 1/(𝜕I/𝜕Φ) and (𝜕V /𝜕Φ)/Rd = (𝜕I/𝜕Φ), we obtain i′n = [(𝜕V ∕𝜕Φ) × Mp × (4kB T∕Rp )1∕2 ]∕Rd − (4kB T∕Rp )1∕2 = (4kB T∕Rp )1∕2 × (𝜕I∕𝜕Φ) × (Mp − Mdyn )

(8.4)

by 𝜕I/𝜕Φ, we obtain the following expression for the Johnson noise By contribution δΦR : dividing i′n

δΦR = (4kB T∕Rp )1∕2 × MPFC

(8.5)

whereby MPFC = (Mp − Mdyn ). The parameter MPFC is the difference between a static mutual inductance, Mp , and a dynamic inductance, Mdyn . In the physical

8.4 Quantitative Analyses and Experimental Verification of the PFC in Voltage Bias Mode

meaning of Eq. (8.5), the term (4k B T/Rp )1/2 represents the thermal noise current of Rp , and MPFC is considered the apparent mutual inductance of the parallel circuit. In fact, δΦR is the product of these two items. Interestingly, δΦR disappears at MPFC = 0, i.e. Mp = Mdyn . The equivalent circuit used to analyze δΦV n from the preamplifier’s V n is given in Figure 8.12b. In principle, the behavior of δΦV n and the “NC” critical condition have already been discussed in Section 8.3.2, so we summarize only the analysis of δΦV n with the newly defined parameters, e.g. Mdyn and MPFC . The equivalent circuit in Figure 8.12b includes two resistors, Rp and Rd , and two correlated voltage sources V n and V s . However, V n is applied across the parallel input circuit, where Rp is no longer regarded as a noise source. In this analysis of i′′n , the key aspect is that i′′n2 ≡ Vn ∕Rp does not change and is independent of the SQUID branch. Thus, i′′n2 generates a voltage V s across the SQUID via Mp , namely, Vs = i′′n2 × Mp × (𝜕V ∕𝜕Φ). Then, the total noise currents, i′′n , caused by V n flowing out from the preamplifier’s output V m are denoted by i′′n = i′′n1 + i′′n2 = Vn ∕{(Rd × Rp )∕[Rd + Rp − (𝜕V ∕𝜕Φ) × Mp ]} = Vn ∕{(Rd × Rp )∕[Rp − (𝜕V ∕𝜕Φ) × MPFC ]} = Vn ∕(Rd )PFC

(8.6)

where the chain rule (equation) of Rd = (𝜕V /𝜕Φ)/(𝜕I/𝜕Φ) in the square brackets is used. In Eq. (8.6), the expression (Rd )PFC = {(Rd × Rp )/[Rp − (𝜕V /𝜕Φ) × MPFC ]} represents the apparent resistance of this parallel circuit, as depicted in Figure 8.10. Therefore, the flux noise contribution of the preamplifier’s V n , δΦV n , is expressed as δΦV n = i′′n ∕(𝜕I∕𝜕Φ) = [Vn ∕(Rd )PFC ]∕(𝜕I∕𝜕Φ) = [Vn ∕(Rd )PFC ] × Mdyn

(8.7)

From Eq. (8.6), when the term of [Rp − (𝜕V /𝜕Φ) × MPFC ] is close to zero, (Rd )PFC can approach infinity, thus leading to δΦV n → 0 (see Eq. (8.7)). Therefore, we define the case of [Rp = (𝜕V /𝜕Φ) × MPFC ] as the critical condition of (Rd )PFC , at which δΦV n disappears. Note that this critical condition differs from the NC critical condition of Rp = (𝜕V /𝜕Φ) × Mp , as mentioned in Section 8.3. Although the conditions for disappearance of δΦR and δΦV n are different, i.e. MPFC = 0 for δΦR and MPFC = Rp /(𝜕V /𝜕Φ) for δΦV n , both of them involve the parameter MPFC . By comparing δΦV n (Eq. (8.7)) with δΦR (Eq. (8.5)), we obtain the following relations: For a given Rp , a small value of MPFC leads to a smaller δΦR but to a larger δΦV n (due to a small (Rd )PFC ). For a given MPFC , a small value of Rp leads to a larger δΦR but a smaller δΦV n . Obviously, MPFC > 0 (i.e. Mp > Mdyn ) must be fulfilled to suppress δΦV n . Note that 1/(𝜕I/𝜕Φ) is also called the “SQUID current sensitivity” in APF [3]. 8.4.2

Introduction of Two Dimensionless Parameters r and 𝚫

To characterize SQUID operation with PFC, we introduce two dimensionless parameters: r = Rp ∕Rd

101

102

8 Flux Feedback Concepts and Parallel Feedback Circuit

and Δ = (Mp ∕Mdyn ) − (Rd ∕Rp ) where Rd and Mdyn = 1/(𝜕I/𝜕Φ) are the original dynamic properties of the SQUID itself. Therefore, the relationship 𝜕V /𝜕Φ = Rd × (𝜕I/𝜕Φ) can be expressed as 𝜕V /𝜕Φ = Rd /Mdyn . For some assumed intrinsic SQUID parameters, we numerically analyze the dependence of δΦPFC noise components on r and Δ to determine some suitable ranges of r and Δ, thus reaching the feasible low δΦPFC . We use the two dimensionless parameters r and Δ to rewrite (8.5) and (8.7), thereby obtaining δΦR = [Mdyn × (4kB T∕Rd )1∕2 ] × [(Δ + r − 1)∕r1∕2 ]

(8.8)

δΦV n = [Vn ∕(𝜕V ∕𝜕Φ)] × [(1 − Δ)∕r] with Δ < 1

(8.9)

Additionally, from Eq. (8.5), (Rd )PFC can be expressed as (Rd )PFC = (Rd × r)∕(1 − Δ)

(8.10)

One immediately perceives that the introduction of the parameters r and Δ makes Eqs. (8.8–8.10) more complicated. In fact, both introduced parameters (Δ and r) have definite physical significance: (i) r = Rp /Rd is the noise suppression ratio at the so-called NC critical condition, as described in Section 8.3.2; (ii) for δΦV n , the sign of Δ marks the direction of i′′n1 in Figure 8.12b. We can divide all parameters in the equations into two groups: while Rd , Mdyn , and 𝜕V /𝜕Φ are inherent parameters of the SQUID itself, Δ and r depend on PFC. Before we numerically calculate the combination of δΦR and δΦV n , we should first return to the discussion about the critical condition of PFC in voltage bias mode. Now, we specifically discuss four cases of δΦV n with the equivalent circuit in Figure 8.12b: (1) Δ < 0 corresponds to i′′n1 > 0, i.e. V n > V s , while (Rd )PFC = Rp //(Rd )SB < Rp . In this regime, once (Rd )PFC becomes larger than Rd , δΦV n starts to decrease. (2) Δ = 0, i.e. V n = V s or (Rd )SB → ∞, leads to i′′n1 = 0 (denoted by (I s )V n in Figure 8.10) and (Rd )PFC = (Rd )SB //Rp = Rp . This case is the known NC critical condition [4], i.e. Mp × (𝜕V /𝜕Φ) = Rp , where δΦV n is suppressed by a factor of r = Rp /Rd , as mentioned above. (3) 1 > Δ > 0, i.e. V n < V s . Our main interest in this quantitative analysis focuses on this Δ range. Here, the relationship V n < V s results in i′′n1 < 0, which means that i′′n1 flows out from the SQUID branch into the PFC branch. In this case, i′′n1 is similar to the ring current in Figure 8.4 and Δ indicates the degree of reversal of i′′n1 . Actually, i′′n1 , instead of being a part of i′′n2 , flows into the PFC branch to establish V n across Rp , where Vn = i′′n2 × Rp must be maintained. In this case, i′′n2 contains two components, i′′n1 and (i′′n2 )conv , i.e. i′′n2 = i′′n1 + (i′′n2 )conv , because both are coherent. Here, only (i′′n2 )conv conventionally flows out from the current-to-voltage converter, for which i′′n1 is unseen. With the concept of apparent parameters, one obtains Vn ∕(i′′n2 )conv = (Rd )PFC > Rp . Thus, δΦV n can be suppressed more effectively in this case than at Δ = 0. As i′′n1 completely replaces i′′n2 , no current flowing out

8.4 Quantitative Analyses and Experimental Verification of the PFC in Voltage Bias Mode

from the readout electronics (the current-to-voltage converter) is needed, i.e. (i′′n2 )conv → 0 or (Rd )PFC → ∞. At this point, the voltage across the two branches generated by V s is just equal to V n , i.e. [V s /(Rp + Rd )] × Rp = V n . Using the relationship V s = (V n /Rp ) × Mp × (𝜕V /𝜕Φ), one obtains the critical condition: Mp × (𝜕V ∕𝜕Φ) = Rd + Rp This condition matches the critical condition of the PFC in current bias mode (APF), as described in Section 8.3.1. The unified critical condition further confirms the same nature of the PFC despite the different readouts, an infinite (𝜕V /𝜕Φ)PFC in current bias mode or an infinite (Rd )PFC in voltage bias mode. Furthermore, from the definition of Δ = (Mp /Mdyn ) − (Rp /Rd ) = 1, the above critical condition of Mp × (𝜕V /𝜕Φ) = Rp + Rd can be directly obtained. In brief, in the regime 1 > Δ > 0, δΦV n is effectively suppressed. (4) Δ ≥ 1, i.e. Mp × (𝜕V /𝜕Φ) > Rp + Rd , so (Rd )PFC becomes negative, thus resulting in an oscillation of the current-to-voltage converter. The dashed line in Figure 8.13 clearly demonstrates the negative (Rd )PFC in the I–V characteristics at Φ = (2n + 1)Φ0 /4. In fact, due to the influence of noise from the SQUID and the preamplifier, oscillation already occurs when Δ is close to unity [11]. In brief, we have accounted for the physical significance of the two dimensionless PFC parameters, r = Rp /Rd and Δ = (Mp /Mdyn ) − (Rp /Rd ). Both of them act as basic PFC parameters in the following numerical calculations for the readout electronics noise, where our primary focus is on how to obtain the minimum δΦ2PFC = δΦ2V n + δΦ2R . 8.4.3

Numerical Calculations

In fact, δΦ2R and δΦ2V n must be considered simultaneously to minimize δΦ2PFC . Here, we perform numerical calculations of δΦPFC by varying the PFC parameters Δ and r. For the SQUID characteristics, we take two sets of parameters typical for niobium SQUIDs: (1) Rd = 10 Ω, 𝜕V /𝜕Φ = 200 μV/Φ0 and Mdyn = 0.1 nH and (2) Rd = 10 Ω, 𝜕V /𝜕Φ = 100 μV/Φ0 and Mdyn = 0.2 nH, Figure 8.13 The I–V characteristics (photo) measured near Φ = (2n + 1)Φ0 /4 when Δ ≥ 1. An oscillation appears when (Rd )PFC < 0. The dashed line is recorded by using a low-pass filter with a corner frequency of f c ≈ 1 kHz.

I

Vb

V

103

8 Flux Feedback Concepts and Parallel Feedback Circuit

∂V/∂Φ = 100 μV/Φ0 Mdyn = 0.2 nH

∂V/∂Φ = 200 μV/Φ0 Mdyn = 0.1 nH

5

10 r = Rp/Rd

2.0

Δ = 0.5

III

1.5

0.5

I

(ii)

0.6 0.4

60 (iii)

(iii) (i)

0.0 0.6

1

30

II

10

r = Rp/Rd

(b) 1.0

120

0.6

90 III (ii) 60

I

0.4 0.2

II 30 (iii)

0.0 0.2

0.2

r = Rp/Rd

2

1.5 1.0 (iii)

0.5

10

(Rd)PFC 120

III

I

0.4 60

1 (iii) 1

(iii)

30

II (i)

10

r = Rp/Rd

120

1.5 1.0

I

III

0.0 0.2

(f)

90

(i) (ii) 1

Curvature

2.5

1.0 0.5 (iii)

r = Rp/Rd

0.0

1.5

II 30 (iii)

0.2

2.0

60

0.5

0.0

0.6

90

Δ = 0.9

2.0

0.1

0.8

Δ = 0.5

2.0

2.5

0.2

0.0 20

15

10 r = Rp/Rd

(ii)

0 0.6

0.0

(i) 1

5

(e)

Δ = 0.9

0.8

1

3

(Rd)PFC 0.8 120 90

1.0

Δ=0

Curvature

(d) 1.0

0.3

A

1

0

0.0 20

15

0.4

Curvature

1

(a)

δΦPFC (μΦ0/√Hz)

0.2

Curvature

0.0

(c)

Δ=0

Curvature

2

Curvature

1

0.5 A

0.5

δΦVn δΦR δΦPFC

(Rd)PFC

10

Curvature

Δ = 0.5

0.0

Flux noise (μΦ0/√Hz)

0.5

1.0

0.4

δΦPFC (μΦ0/√Hz)

1.0

Curvature

1.5

3

0.6

δΦVn δΦR δΦPFC

1.5

δΦPFC (μΦ0/√Hz)

2.0

(Rd)PFC

Flux noise (μΦ0/√Hz)

2.0

δΦPFC (μΦ0/√Hz)

104

0.0 10

Figure 8.14 Dependence of numerically calculated flux noise contributions, δΦPFC , δΦR , and δΦVn , on the resistance ratio r at Δ = 0 (first row, graphs (a) and (d)), 0.5 (second row, (b) and (e)), and 0.9 (third row, (c) and (f )).

where Mdyn = 𝜕Φ/𝜕I = Rd × (𝜕Φ/𝜕V ). In the following numerical calculations, √ we employ a preamplifier with V n = 0.35 nV/ Hz (f > 2 Hz), which was experimentally realized with parallel-connected bipolar transistors (PCBTs) in Chapter 5. In Figure 8.14, the three flux noises δΦPFC , δΦR (Eq. (8.8)), and δΦV n (Eq. (8.9)) are plotted as functions of r at three specific values of Δ = 0, 0.5, and 0.9, where the assumptive SQUID parameters mentioned above are used in our numerical calculations. The plots in the left column (graphs (a), (b), and (c)) are calculated based on a SQUID with parameters (1) and the ones in the right column (graphs (d), (e), and (f )) with parameters (2). In the upper row (graphs (a) and (d)), for Δ = 0 (NC critical condition, i.e. r = Rp /Rd = Mp /Mdyn ), there are four curves in one graph: δΦR (r), δΦV n (r),

8.4 Quantitative Analyses and Experimental Verification of the PFC in Voltage Bias Mode

δΦPFC (r), and its curvature. With increasing r, δΦR (r) (black line) increases monotonically, but δΦV n (r) (gray) decreases. Thus, δΦPFC (r) (light gray) should be similar to a parabola because δΦ2PFC = δΦ2R + δΦ2V n . In addition, we plot the curvature of δΦPFC (r) (light black), where the curvature peak “A” apparently differentiates the changes in the slope of δΦPFC . Left of A, δΦPFC changes very fast and δΦR is negligible, while right of A, δΦR begins to contribute to δΦPFC , which shows a gradual slope. Both asymptotes of δΦR (r) and δΦV n (r) to δΦPFC intersect at point A. Namely, when r ≪ rA , δΦPFC ≈ δΦV n , but when r ≫ rA , δΦPFC ≈ δΦR . From graphs (a) and (d), at r < 5, δΦV n dominates δΦPFC , while at r > 5, δΦR is the dominant component. Thus, the minimum δΦPFC value appears at r ≈ 5, which is independent of the √ SQUID parameters. However, the minimum √ δΦPFC value is found to be 0.6 μΦ0 / Hz for 𝜕V /𝜕Φ = 200 μV/Φ0 and 1.2 μΦ0 / Hz for 𝜕V /𝜕Φ = 100 μV/Φ0 . To suppress δΦPFC at Δ = 0, the condition Rp > Rd (r > 1) must be fulfilled. In fact, the minimum δΦPFC value also depends on Δ. The inset in (a) shows how these noise contributions, δΦPFC, δΦR , and δΦV n , behave at √ Δ = 0.5, where the minimum δΦPFC < 0.5 μΦ0 / Hz appears at r ≈ 3. Compared to those at Δ = 0 and r ≈ 5, δΦPFC is reduced. According to the values of (Rd )PFC , we define the three operating ranges marked as (i), (ii), and (iii) with different gray backgrounds in the following graphs: Range (i) denotes the recommended range for SQUID operation; range (ii) is the transition range; and (iii) indicates the ranges that are not recommended. The four graphs in (b), (e) (the second row), (c), and (f ) (the third row) have three ordinates: (Rd )PFC (r), δΦPFC (r), and its curvature. The graphs illustrate how δΦPFC (curve I) can be reduced and how (Rd )PFC (curve III) increases further with r. The peak of the curvature of δΦPFC (r) (curve II) is used to aim at the minimal recommended r, i.e. the left boundary of range (i). It is already known that in the regime 0 < Δ < 1, i′′n1 flows out of the SQUID branch into the PFC branch, as analyzed in Section 8.4.2. Here, we deliberately chose two typical values: Δ = 0.5 (second row), which is in the middle of the possible range, and Δ = 0.9 (third row), which is closer to the unsafe limit Δ → 1. In both cases, (Rd )PFC > Rp occurs. Therefore, we plot here the effective dynamic resistance (Rd )PFC (r) defined in Eq. (8.10), which is inversely proportional to δΦV n . All four δΦPFC (r) appear as parabolic functions, and the minima of δΦPFC (r) are located at r ≈ 3 for Δ = 0.5 (b, e) and at r ≈ 1 for Δ = 0.9 (c, f ). Although there are no clear physical boundaries for defining these three ranges in Figure 8.14, we utilize our experience based on our measurements of approximately 400 SQUIDs with various parameters, to suggest the following range division: (1) The recommended range (range (i)) is located between the r value at the peak of the curvature (called the minimum safe r below) and the r value of (Rd )PFC (r) ≈ 80 Ω. In our experiments, when (Rd )PFC ≤ 80 Ω, the SQUID system with PFC has always been operating stably. For Δ = 0.5, the minima of δΦPFC (r) appear at the bottom of the parabola, i.e. at r ≈ 3, which is independent of the SQUID’s parameters, as shown in the second row (graphs (b) and (e)). However, a large 𝜕V /𝜕Φ exhibits a broad range of (i) and reaches a small minimum of δΦPFC . The range width of r in

105

106

8 Flux Feedback Concepts and Parallel Feedback Circuit

(i) represents a tolerance of selectable parameters, which becomes increasingly narrow with increasing Δ. At Δ = 0.9 in the third row (graphs (c) and (f )), range (i) located at the left of the minima δΦPFC (r) (the bottom of the parabola), i.e. at r < 1, becomes very narrow. For suppressing δΦV n , Rp > Rd is required at Δ = 0, i.e. r > 1 is a prerequisite at NC condition. It is very interesting that at Δ = 0.9, δΦV n is suppressed more efficiently, but r returns to r ≤ 1. (2) In the transition range (range (ii)) of 80 Ω < (Rd )PFC < 150 Ω, the SQUID system operates in a semi-stable state. If a SQUID’s parameters are slightly changed by, e.g. reheating, the value of Δ may become close to unity, thus resulting in the oscillation of the readout electronics mentioned above. The minimal δΦPFC is located in range (ii), but δΦPFC increases again√with r due to the contribution from δΦR . The analyzed δΦPFC < 0.7 μΦ0 / Hz is still obtained at Δ = 0.5 (graphs√(b) and (e)). In this case, the SQUID intrinsic noise δΦs , assuming 1 μΦ0 / Hz, already dominates the system noise δΦsys . Here, one should not devote oneself to reaching the minimum δΦPFC value. It was experimentally confirmed that no significant further decrease in δΦPFC is observed, as (Rd )PFC reaches a certain value. It is necessary to operate a SQUID in range (ii) only for SQUIDs with a small 𝜕V /𝜕Φ and a very low δΦs (e.g. strongly damped SQUID). In fact, it is very difficult to operate a SQUID in range (ii) because of its narrow width, especially at Δ = 0.9 in (c) and (f ). (3) The range (iii) that is not recommended consists of two areas: the first area is located on the left side of (i), where δΦPFC (r) is not only relatively large but also very sensitive to r variations. The second area of the range (iii) is located on the right side of (ii). When (Rd )PFC > 150 Ω, not only δΦPFC increases, but an oscillation may also occur. Indeed, for a stable low-noise SQUID readout technique, we recommend operating a SQUID with a PFC in range (i), or possibly in the transition range (range (ii)). In fact, range (iii) is not suitable for operation of a SQUID with a PFC. From the graphs in Figure 8.14, we learned that the minimum δΦPFC value appears at r ≈ 5 for Δ = 0, r ≈ 3 for Δ = 0.5, and r ≈ 1 for Δ = 0.9. Moreover, the attainable value of δΦPFC decreases with increasing Δ; e.g. in the√left column of 𝜕V /𝜕Φ = 200 μV/Φ0 , δΦPFC √ may reach approximately 0.5 μΦ0 / Hz at Δ√= 0 (see Figure 8.14a), 0.9, δΦR should account for δΦPFC , but Δ > 0.93 is beyond the scope of our interest. (b) At r = 3 (graph (b)), the decreasing slope of δΦPFC (Δ) becomes increasingly gradual because the contribution of δΦR is clearly present when Δ > 0.6. Operating ranges (i) and (ii) located at 0.13 < Δ < 0.8 are widened √ significantly, while the minimum δΦPFC value of approximately 0.4 μΦ0 / Hz is located at Δ ≈ 0.8 (the right boundary of range (ii)). (c) At r = 5 (graph (c)), the bottom of the parabolic shape of δΦPFC (Δ) is located in range (ii) because δΦR contributes more strongly to δΦPFC as Δ increases. Operating ranges (i) and (ii) are √shifted left to 0 < Δ < 0.68, while the minimum δΦPFC increases to >0.5 μΦ0 / Hz at Δ ≈ 0.6. Indeed, the operating range width and the attainable minimum value of δΦPFC cannot be optimized simultaneously by varying Δ and r. In practice, one does not need to aim exactly for the lowest δΦPFC because the system √ noise δΦsys is already dominated by the SQUID intrinsic noise δΦs ≈ 1 μΦ0 / Hz, as √ δΦPFC < 0.7 μΦ0 / Hz. Note that a large Mp may introduce an adverse effect; e.g. the integrated coil Lp on the SQUID washer can act as a half-wavelength resonator. Here, the resonance and its harmonics may include the Josephson frequency of the Josephson junction (JJ), thus leading to additional noise due to the deteriorated SQUID performance [12–14]. According to the synopsis of the two analytical Figures 8.14 and 8.15 as well as our experience, the ranges of 1.5 < r < 3 and 0.3 < Δ < 0.6 may provide a useful design guidance for a class of practical magnetometric SQUIDs with PFC. Indeed, a numerical analysis approach could be employed to optimize PFC parameters. To verify our analysis, we experimentally characterized three niobium SQUIDs with PFC having suitably chosen r and Δ values. As shown in Section 8.4.4, all measured SQUID system flux noise Φsys values are actually comparable to the √ intrinsic noise of the SQUID, i.e. Φsys ≈ Φs ≈ 1 μΦ0 / Hz. 8.4.4

Experimental Results

In our experiments, we employed three niobium washer SQUIDs with Stewart-McCumber parameters 𝛽 c < 0.5 (strongly damped) and measured all parameters with and without PFC. The coil Lp was integrated within the pickup loop on the SQUID chip (but not on the SQUID washer), while some lumped resistors Rp were employed to construct the PFC with Lp . The values of Mp were determined experimentally. These strongly damped SQUIDs should exhibit a low intrinsic noise δΦs , which is much lower than δΦe (readout electronics noise) in a DRS without a PFC. Table 8.2 lists all important parameters of the SQUIDs: In the second column, the design values of the SQUID geometrical inductance Ls calculated according

8.4 Quantitative Analyses and Experimental Verification of the PFC in Voltage Bias Mode

Table 8.2 Measured parameters of SQUIDs with and without PFC.

SQUID no.

Ls (pH)

RJ a)(𝛀)

(Rd )PFC (𝛀)

Mdyn b) (pH)

(𝝏V/𝝏𝚽)PFC (mV/𝚽0 )

r

𝚫

#1

180

5

125 (7.3)c)

100 (97)

2.5 (0.15)

2.74

0.85

#2

125

7.5

40 (8.8)

67 (66)

1.2 (0.27)

1.39

0.70

#3

110

8.3

38.3 (10)

70 (70)

1.1 (0.28)

2.7

0.30

a) The junction shunt resistance of a SQUID. Note that the measured Rd of SQUIDs without the PFC is close to the value of RJ . b) Mdyn does not change with the PFC (constraint condition), and (Rd )PFC = Mdyn × (𝜕V /𝜕Φ)PFC . c) All data in brackets are the originals, i.e. without the PFC.

to Ref. [10] are listed in descending order. From the third column to the sixth column, the measured device and circuit parameters are enumerated. In the seventh and eighth columns, the values of r and Δ chosen for these experiments are given. All original values of the three SQUIDs without PFC are written in brackets. The dynamic mutual inductance Mdyn and the dynamic resistance (Rd )PFC were determined from the recorded I(Φ) and I–V characteristics at the working point with the PFC, respectively. It is known that Mdyn = (Mdyn )PFC remains unchanged and acts as a constraint condition for the PFC (see the fourth column). In fact, (𝜕V /𝜕Φ)PFC is still relevant to δΦV n in voltage bias mode because the relation δΦV n = [V n /(Rd )PFC ] × Mdyn = V n /(𝜕V /𝜕Φ)PFC is valid. Therefore, we listed the parameter (𝜕V /𝜕Φ)PFC in Table 8.2. For the PFC parameters, r values 2 Hz). Furthermore, in the case of SQUID #2 with PFC, the dynamic resistance increased from Rd = 9 Ω to (Rd )PFC ≈ 40 Ω, while Mdyn ≈ 66 pH remained almost constant, thus leading to a large (𝜕V /𝜕Φ)PFC of 1.2 mV/Φ0 . Analogous to Figure 8.15, the data in Table 8.2 have been employed to numerically calculate the two curves of δΦPFC (Δ) and (Rd )PFC (Δ) at r ≈ 1.4 for SQUID #2 (graph (b)) and r ≈ 2.7 for both SQUID #1 (a) and SQUID #3 (c) in Figure 8.16. Both r values are suggested by our above analyses. Once Rd , Mdyn , and r are determined, the values of Δ = (Mp /Mdyn ) − (Rp /Rd ) depend on only Mp . The values of both δΦPFC and (Rd )PFC calculated for the three SQUIDs are marked by solid (black) triangles in the graphs of Figure 8.16. For each of the three SQUIDs with PFC, the experimentally measured (Rd )PFC value agrees well with the value calculated from (Rd )PFC = (Rd × r)/(1 − Δ) (Eq. (8.10)) and plotted in the graphs, so we can evaluate the corresponding

109

8 Flux Feedback Concepts and Parallel Feedback Circuit

1.0

200 r = 2.74

0.8

150

Δ = 0.85 100

0.6 0.4

(iii)

50 (i)

0.2 0.00 (a)

(Rd)PFC (Ω)

δΦPFC (μΦ0/√Hz)

SQUID #1

(ii)

(iii)

0 1.00

0.75 0.25 0.50 Δ = Mp/Mdyn – Rp/Rd

1.2

Figure 8.16 Numerically calculated δΦPFC (Δ) and (Rd )PFC (Δ). The solid (black) triangles mark the values of both δΦPFC and (Rd )PFC under certain Δ and r values determined based on the experimental values of Rd , Rp , Mdyn , and Mp (see Table 8.2). Generally, the contribution of δΦR can be neglected in (b) and (c), where the marks are set on the decreasing slope of the parabola.

200 r = 1.4

0.8

150

Δ = 0.7 100

0.4

0.0 0.00 (b)

(iii)

(ii) (iii) (i)

(Rd)PFC (Ω)

δΦPFC (μΦ0/√Hz)

SQUID #2

50

0 1.00

0.25 0.50 0.75 Δ = Mp/Mdyn – Rp/Rd

200

0.6 SQUID #3 0.5

Δ = 0.3 0.4

100

0.3

50 (i)

0.2 0.00 (c)

150

r = 2.7

0.25

(ii)

0.75 0.50 Δ = Mp/Mdyn – Rp/Rd

(Rd)PFC (Ω)

δΦPFC (μΦ0/√Hz)

110

(iii)

0 1.00

δΦPFC with confidence. Taking the example of SQUID #2, the measured (Rd )PFC ≈ 40 Ω fits very well to the graphical value on the√curve of (Rd )PFC at Δ = 0.7. The corresponding calculated δΦPFC is 0.35 μΦ0 / Hz (see graph (b)). Here, one can see that SQUIDs #2 (graph (b)) and #3 (graph (c)) with PFC operate almost in the middle of the safe range (i), while SQUID #1(graph (a)) with the highest (Rd )PFC operates close to the right limit of the transition range (ii). However, SQUID #1 could still be operated stably in our experiments. Now, we compare the calculated δΦPFC (Δ) values marked by the solid (black) triangles in Figure 8.16 with the measured system noise δΦsys , where

8.4 Quantitative Analyses and Experimental Verification of the PFC in Voltage Bias Mode

Table 8.3 Noise data and analysis for SQUIDs with and without the PFC. SQUID no.

√ 𝛅𝚽sys (𝛍𝚽0 / Hz)

√ 𝛅𝚽PFC (𝛍𝚽0 / Hz)

√ 𝛅𝚽s (𝛍𝚽0 / Hz)

√ Vn∗ (Vn ) (pV/ Hz)

#1

1.1 (2.7)a)

0.45 (2.47)

1 (1.1)

68 (350)

#2

1.1 (1.78)

0.35 (1.4)

1.04 (1.1)

93 (350)

#3

0.85 (1.5)

0.4 (1.3)

0.75 (0.75)

114 (350)

a) All data in brackets are without the PFC.

δΦ2sys ≈ δΦ2s + δΦ2PFC . Note that the noise contribution of the preamplifier δΦ2In is temporarily absent. If δΦsys ≈ δΦs (white noise), δΦPFC is effectively suppressed below δΦs . In our experiments, all noise measurements were performed in FLL operation, while the three SQUIDs were placed in a niobium shielding tube at liquid helium temperature. Table 8.3 summarizes the noise data (white noise) with and without PFC. The second column provides the experimentally determined δΦsys values, while the third column shows the numerically calculated δΦPFC values obtained from the marks in Figure 8.16. The fourth column lists the δΦs values derived from the relationship δΦ2s = δΦ2sys –δΦ2PFC . Note that the preamplifier’s current noise contribution δΦ2In in the white noise range is temporally neglected. Naturally, as δΦe < δΦs , the measured δΦsys with the PFC is close to the SQUID intrinsic √ noise δΦs : δΦsys ≈ δΦs ≈ 1 μΦ0 / Hz for all three SQUIDs listed in Table 8.3. In brief, Table 8.3 and Figure 8.16 confirm that the numerically calculated δΦPFC (Δ) with different r values can help us to design the PFC. Before we explain the last data column of Table 8.3, let us change perspective to discuss δΦPFC . As δΦsys = δΦs , one cannot further evaluate the real level of δΦPFC . Now, we introduce an equivalent voltage noise Vn∗ = δΦPFC × (𝜕V ∕𝜕Φ) for a “virtual preamplifier,” which is connected √ to the SQUID directly without PFC. Here, we should know that a Vn∗ of 100 pV/ Hz is quite a low value, which is equal to the thermal noise of a 43 Ω resistor at 4.2 K or of a 0.6 Ω resistor at 300 √ K. In the case of SQUID #1, the minimum Vn∗ = δΦPFC × (𝜕V ∕𝜕Φ) ≈ 68 pV∕ Hz, which may be compatible with the thermal noise of the connection wires between the SQUID at 4.2 K and the readout√ electronics at 300 K. Therefore, in the low-noise readout technique Vn∗ ≈ 70 pV∕ Hz may be close to the practical limit when the readout electronics is located at RT. In the last column of √ Table 8.3, the actual preamplifier voltage noise of the PCBT is V n = 350 √ pV/ Hz (in brackets). However, the equivalent voltage noise Vn∗ is ≤100 pV/ Hz in all cases of three SQUIDs with PFC. In brief, the intrinsic noise δΦs of strongly damped SQUIDs can be observed with the help of PFC in a DRS. 8.4.5

Noise Comparison and Interpretation

For SQUID #2 in Table 8.3, we performed a preliminary experimental comparison of the noise spectrum with PFC and in the two-stage scheme, which

111

8 Flux Feedback Concepts and Parallel Feedback Circuit

Figure 8.17 Flux noise spectra of SQUID #2 measured (I) with the PFC in the Research Centre Juelich and (II) with a two-stage scheme at the IPHT Jena.

SQUID #2 Flux noise (μΦ0/√Hz)

112

10 I

1 II 0.3 1

10

100 Frequency (Hz)

1k

10k

is widely considered the readout scheme with the lowest noise [15–17]. The principle of the two-stage scheme will be explained in Chapter 11. Figure 8.17 shows the two measured noise spectra of SQUID #2. The data of the plot labeled “I” was performed with PFC at the Research Centre Juelich, Germany, while plot labeled “II” was subsequently recorded in the two-stage scheme at the Leibniz-Institut fuer Photonische Technologien (IPHT) Jena, Germany. The white noise levels recorded with both readout schemes were practically the same, but the low-frequency noise below approximately 400 Hz was remarkably different and was higher with PFC than in the two-stage scheme [10]. To account for the difference between the two measurements in the lowfrequency range in Figure 8.17, we should take into account the total noise contribution from the current-to-voltage converter (preamplifier) in voltage bias mode, where the total noise from the readout scheme is expressed by ΣδΦ2 = δΦ2In + δΦ2PFC = δΦ2V n + δΦ2In + δΦ2R . From Figure 8.16b, we observed that the numerically calculated δΦPFC is dominated by δΦV n , thus leading to a total noise ΣδΦ2 ≈ δΦ2V n + δΦ2In in DRS with PFC, The large noise of δΦsys with PFC (curve I) in the frequency range of 10 Hz. Curve IV (gray) is the sum of curves II and III. Actually, the synthetic system noise δΦ′ 2sys = δΦ2s + δΦ2e (curve IV) is lower than curve I of the measured δΦsys with PFC in the frequency range of 10 Hz. 8.4.6

Two Practical Designs for PFC

As indicated in Figures 8.14–8.16, the two requirements of a large operating range, i.e. the width of ranges (i) and (ii), and low δΦPFC cannot be fulfilled simultaneously. However, for the strongly damped SQUIDs with small 𝜕V /𝜕Φ values, a very low δΦPFC is needed to suppress δΦV n . In practice, it is difficult for planar niobium thin film SQUIDs with an integrated PFC to achieve the designed δΦPFC because deviations of the parameters, e.g. Rd , Rp , and 𝜕V /𝜕Φ, cannot be avoided not only during the fabrication process but also upon cooling and reheating or after long-term storage. Occasionally, the parameter deviations make it very complicated to use PFC, thus seriously affecting the wide range of applications of the integrated chips. To solve this problem, there are two designs: (1) using a field-effect transistor (FET) acting as an adjustable Rp operated at 4.2 K, and (2) selectable Lp and Rp integrated into the layout on the chip. Design (1) is already employed in commercial multichannel SQUID systems for magnetoencephalography (MEG) measurements [18]. Design (2) was reported in Refs. [19, 20]. Design (1) begins at the original work on the “NC” scheme, where the authors first provided the concept and the critical condition in voltage bias mode. Here, the key technique is the use of the forward on-resistance between the drain and source of the FET, Rds , as an adjustable Rp at 4.2 K (see Figure 8.20), where the gate voltage V gs controls Rds and V gs can be adjusted at RT. By selecting V gs , one can always find a proper Rp (Rds of FET) to reach a very low δΦPFC while maintaining stability of operation. In brief, a simple readout electronics (DRS) in which V gs is easily adjusted suits the NC scheme to construct a multichannel SQUID system. However, for each channel, one more wire for setting V gs is needed to connect the readout electronics at RT. In the suggested design (2), both Mp and Rp of the PFC are selectable, as shown by the equivalent circuit in Figure 8.21. In practice, two selectable taps, K 1 and K 2 , are realized by interchangeable bond pads on the layout. By selecting Mp and Rp , one can obtain suitable PFC parameters to fit the original SQUID’s parameters of Rd and 𝜕V /𝜕Φ at the working point, thus realizing δΦsys ≈ δΦs . In other words, with resistance and inductance taps, one can compensate for the deviations in the SQUID parameters, thus meeting practical demands.

Rd

D S

G

Lp Mp

To readout electronics at RT

114

Figure 8.20 The Rp of the PFC is replaced by an FET, where V gs controls the forward on-resistance of the FET, Rds . The FET located at 4.2 K connects to the readout electronics at RT. Here, the feedback inductance Lf for FLL and its two connecting wires are omitted.

8.4 Quantitative Analyses and Experimental Verification of the PFC in Voltage Bias Mode

Figure 8.21 The equivalent circuit of design (2) with selectable Mp and Rp , where two taps, K 1 and K 2 , are employed. Two wires with arrows (Sa and Sb ) connect to the readout electronics at RT.

Sa K1 Rp

Rd Mp K2

Lp Sb

Rheat

FZJ-SIMIT

Rp

JJ

JJ

Lp 350 12III 22

Lin

Lf

Sb Sa″ (a)

(b)

Fa Lf Fb Lpick

JJ

Lse

Sb′

Sa

Lse

Sa′

Figure 8.22 An example of design (2): a niobium SQUID magnetometer layout with an area of 5 × 5 mm2 (a) and its corresponding equivalent circuit (b), where the dotted lines represent the bonding wires. The four-terminal element consists of a SQUID shunted by a PFC with terminals of Sa and Sb (S′b ) and flux feedback coil Lf with terminals of Fa and Fb . Here, the Sb and S′b of the SQUID connections are equivalent. Note that the two JJ symbols located at two sides of the upper portion are four test junctions for calibration purposes. The strip located at the bottom in (a) is sketched as two series feedback coils Lse (SFC) in (b). To insert Lse between the input terminal of the preamplifier (op-amp) and the SQUID shunted by the PFC, for example, one can select the bonding point S′a (or S′′a ) instead of Sa . This effect of the SFC will be discussed in Chapter 9.

For design (2), we take a planar SQUID magnetometer layout as an example to understand the selected PFC parameters [19, 20]. Figure 8.22 shows this practical layout with selectable Lp and Rp (a) and its equivalent arrangement (b). Here, Figure 8.22b is much easier to understand than (a); therefore, one can look at (a) against (b). A double-loop SQUID gradiometer is employed, where two input coils Lin with opposite winding directions are integrated on the gradiometer washer and coupled to the SQUID. Both Lin coils connect to a square pick-up loop with an area of approximately 5 × 5 mm2 , thus forming a sensitive magnetometer. In this layout, many coils and resistors are designed and

115

116

8 Flux Feedback Concepts and Parallel Feedback Circuit

integrated inside the pick-up loop. Therefore, all these coils indirectly couple to the SQUID loop via the square pick-up loop, Lpick , of the magnetometer. The planar resistors are designed with 10 Ω per square. Two flux feedback coils Lf with a spiral form (one of which may serve as a reserve) for FLL operation are located above the SQUID’s zone, while a heating resistor Rheat is situated on the top and outside Lpick . Two key elements of the PFC, Lp and Rp , are located between Lf and Rheat within Lpick . The large spiral-form coil Lp with five turns surrounds the planar Rp , which are arranged in an “Ω” form, i.e. 18 square resistors connected in series similar to a chain. One terminal (upper) of Lp connects to, e.g. the first resistor chain at the left, while two PFC terminals with the selected Rp and Lp , both of which are already arranged in series, shunt to the SQUID using bonding wires. In this layout in Figure 8.22, there are two elements that are not relevant to the operation of the SQUID with PFC: (i) two vertically arranged lines with two JJs spaced by the “Ω” form resistor chain act as references for characterizing the I–V curves of junctions and (ii) a strip located at the bottom acts as two SFCs Lse (left and right), the function of which will be explained in Chapter 9. In practice, the two original SQUID parameters Rd and 𝜕V /𝜕Φ should be first determined, as shown in Table 8.2. According to the numerical calculation described above, one can estimate the value of Rp to fit 1.5 < r < 3. To obtain a suitable (Rd )PFC , one can change the bonding positions along the broad lines of Lp (e.g. start from the third turn in Figure 8.22a) to match the optimal Mp . Generally, it is not difficult to find effective PFC parameters to reduce δΦPFC . One should not pursue the lowest δΦPFC in order to avoid system instability.

8.5 Main Achievements of PFC Quantitative Analysis The PFC is an important breakthrough in SQUID readout technique. In fact, PFC can suppress the preamplifier’s ΦV n contribution with two performances: (1) In current bias mode, a SQUID with PFC is connected to a normal forward amplifier. Here, asymmetric V (Φ) characteristics appear when a large (𝜕V /𝜕Φ)PFC occurs on the steep slope of V (Φ), thus reducing δΦV n = V n /(𝜕I/𝜕Φ)PFC . However, the original (𝜕V /𝜕Φ) across the parallel circuit is already reduced due to the Rp shunting effect in the APF scheme, as mentioned in Section 8.3. (2) In voltage bias mode, the suppression of δΦV n is directly realized by increasing (Rd )PFC due to the unchanged (𝜕I/𝜕Φ), where δΦV n = [V n /(Rd )PFC ]/ (𝜕I/𝜕Φ) = V n /(𝜕V /𝜕Φ)PFC (see Section 8.4). In the readout electronics, the preamplifier acts as a current-to-voltage converter. To simplify the quantitative analysis, we focused only on the performance of the voltage-biased parallel circuit, i.e. a SQUID shunted by PFC, where the two currents I s and I PFC are independent. Furthermore, we introduced two dimensionless parameters, r and Δ, which are useful in determining safe ranges for practical PFC operation. In cases with large values of r or Δ, the Johnson noise contribution δΦR from Rp cannot be neglected.

8.6 Comparison with the Noise Behaviors of Two Preamplifiers

The most important contribution of this quantitative analysis is finding the regime of 1 > Δ > 0, i.e. the regime between the critical condition of Mp × (𝜕V /𝜕Φ) = (Rd + Rp ) (Δ = 1) and the so-called “NC” condition of Mp × (𝜕V /𝜕Φ) = Rp (Δ = 0). In this Δ regime, the SQUID can operate stably, while the suppression of δΦV n becomes very effective because (Rd )PFC > Rp > Rd . By separately varying r and Δ, we numerically calculated the PFC scheme with the assumed and measured original SQUID parameters, Rd and 𝜕V /𝜕Φ. According to the value of (Rd )PFC , we distinguished three operating ranges: (i) For the safe operating range (i), parameter margins (tolerances) exist, while the preamplifier noise V n is largely suppressed and δΦR may be neglected. The measured δΦsys of the SQUID #2 with PFC reached comparable values to that in the two-stage scheme operated at low temperature, e.g. 4.2 K. (ii) In the transition range (ii), the suppression of δΦV n is most effective, so that the wish δΦV n < δΦsys is almost fulfilled. In this range, δΦR starts to influence δΦPFC and the parameter tolerances become narrower. (iii) In the unsafe operation range (iii), δΦR plays a role. If slight parameter changes occur, the SQUID operation may become unstable. The other achievement of our quantitative analyses is the finding that the two requirements for the PFC, a large operating range and a very low δΦPFC , cannot be fulfilled simultaneously. Finally, we verified the numerical calculations by experimentally characterizing the three SQUIDs listed in Tables 8.2 and 8.3. The measured and calculated dynamic resistance (Rd )PFC of SQUIDs with PFC were in good agreement. Experimentally, the intrinsic noise δΦs of the strongly damped SQUIDs can be observed; i.e. PFC provides the possibility that the measured δΦsys can be close to δΦs . For a SQUID magnetometric system, PFC can be used not only in a planar structure such as in Figure 8.22 but also in antenna configurations similar to that in Figure 6.1 [9]. Furthermore, based on the concept of the PFC, the chain rule of 𝜕V /𝜕I = (𝜕V /𝜕Φ)/(𝜕I/𝜕Φ) = Rd is proved in SQUID operation with two different bias modes. Essentially, the PFC increases (Rd )PFC at the working point, while (𝜕I/𝜕Φ) is maintained and acts as a constraint condition, i.e. (𝜕I/𝜕Φ) = (𝜕V /𝜕Φ)/(Rd ) = (𝜕V /𝜕Φ)PFC /(Rd )PFC . Increasing (Rd )PFC leads to an increase in (𝜕V /𝜕Φ)PFC , thus suppressing δΦV n = V n /(𝜕V /𝜕Φ)PFC , where V n is the voltage noise of the preamplifier. Note that V n and (𝜕V /𝜕Φ)PFC have the same weight of contribution to δΦV n . Thus, once δΦV n has been suppressed by the PFC, δΦIn is highlighted in δΦe , as demonstrated in Figure 8.18.

8.6 Comparison with the Noise Behaviors of Two Preamplifiers In all analyses and experiments mentioned above, we took only a PCBT as the preamplifier with the measured noise figures, V n and I n , shown in Figure 5.4. Now, let us observe the relation between δΦIn and δΦV n of an op-amp (preamplifier) in the case of the PFC, according to the measured noise data of AD797

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8 Flux Feedback Concepts and Parallel Feedback Circuit

30 Ω 100 Ω

0

102

Vn /(Rd)PFC 10

1

δΦ (μΦ0 /√Hz)

10

Vn /Rd

10 Ω

101

10

103

2

In

101

10 Ω

10

30 Ω

0

100 –1

(a)

10

0

10

1

2

10 10 Frequency (Hz)

3

10

–1

(b)

1

100

10–1

4

10

10 Vn /(Rd)PFC

In 10–1

102

Vn /Rd

δI (pA/√Hz)

103

2

δI (pA/√Hz)

10 δΦ (μΦ0 /√Hz)

118

10

0

10

1

2

10 10 Frequency (Hz)

3

10

104

Figure 8.23 Illustrations plotting the current noise δI spectra (right ordinate) and their corresponding flux noise δΦ (left ordinate) of AD797 (a) and PCBT (b) with and without the PFC in voltage bias mode. Here, a SQUID parameter of (𝜕I/𝜕Φ = 10 μA/Φ0 ) is assumed. The curves of In (black) and (V n /Rd ) (dark gray) represent the original noise spectra, δΦIn and δΦVn , without PFC, respectively, while the gray curves represent the noise spectra with PFC.

shown in Figure 5.2. To give the readers a specific impression, we illustrate the weights δΦIn and δΦV n of two commonly used preamplifiers with and without PFC, as shown in Figure 8.23, where the op-amp (AD797) data are plotted in (a) and PCBT (e.g. 3 × SSM2221) data in (b). Here, the current noise δI originates from the preamplifier’s I n and V n in voltage bias mode, both of which can be translated into flux noise with the assumed SQUID’s 𝜕I/𝜕Φ = 10 μA/Φ0 (or say, Rd = 10 Ω and 𝜕V /𝜕Φ = 100 μV/Φ0 ). In Figure 8.23a, (Rd )PFC increases with PFC, e.g. from an initial value of Rd = 10 Ω to (Rd )PFC = 30 or 100 Ω; thus δΦV n = [V n /(Rd )PFC ]/(𝜕I/𝜕Φ) = V n /(𝜕V /𝜕Φ)PFC proportionally decreases by a factor of 3 or 10. The three curves sketched in different gray tones marked with 10, 30, and 100 Ω indicate their corresponding (I n )V n and δΦV n caused by V n of AD797. However, the flux noise δΦIn (the lowest black curve) caused by the preamplifier’s I n remains without or with PFC. It means that for AD797 in panel (a), δΦV n always plays the dominant role in δΦe . Only when (Rd )PFC = 100 Ω, δΦIn surpasses δΦV n below f inter ≈ 1 Hz, where the curves of both I n and V n /(Rd )PFC are just intersecting. Therefore, in most applications of SQUIDs with PFC, the δΦIn of AD797 can still be neglected. Furthermore, when the readout electronics employs an LT1028 op-amp as preamplifier, it has the same V n level as OP797, but its I n is just halved. In this case, δΦIn plays no longer a role in δΦe . In principle, PCBT noise behavior has already been discussed in Figure 8.18. Here, we take three curves of the PCBT noise contributions shown in Figure 8.23b just for a better comparison with AD797. They are the curve of I n (black), and the curves of (I n )V n = V n /Rd (10 Ω) (dark gray) and (I n )V n = V n /(Rd )PFC (30 Ω) (gray). Comparing Figure 8.23a with (b), for this assumed SQUID, δΦV n at (Rd )PFC = 100 Ω for AD797 and δΦV n at (R √d )PFC = 30 Ω for PCBT are almost equal, but the current noise δΦIn ≈ 3.5 μΦ√0 / Hz at 1 Hz for AD797 (see panel a) is obviously lower than δΦIn ≈ 17 μΦ0 / Hz for PCBT (panel b). This result suggests that an op-amp (AD797) is better than a PCBT in PFC operation. However, in practice, the suppression factor of [(Rd )PFC /Rd ] = 10 with PFC is very difficult to realize because the width of the operating range is narrow, as

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formance of the dc SQUID bootstrap circuit with that of the standard flux modulation dc SQUID readout scheme. IEEE Transactions on Applied Superconductivity 21 (3): 501–504. Zappe, H.H. and Landman, B.S. (1978). Analysis of resonance phenomena in Josephson interferometer devices. Journal of Applied Physics 49 (1): 344–350. Tuckerman, D.B. and Magerlein, J.H. (1980). Resonances in symmetric Josephson interferometers. Applied Physics Letters 37 (2): 241–243. Enpuku, K., Sueoka, K., Yoshida, K., and Irie, F. (1985). Effect of damping resistance on voltage versus flux relation of a dc SQUID with large inductance and critical current. Journal of Applied Physics 57 (5): 1691–1697. Wellstood, F.C., Urbina, C., and Clarke, J. (1987). Low-frequency noise in dc superconducting quantum interference devices below 1 K. Applied Physics Letters 50 (12): 772–774. Foglietti, V., Giannini, M.E., and Petrocco, G. (1991). A double DC-SQUID device for flux locked loop operation. IEEE Transactions on Magnetics 27 (2): 2989–2992. Foglietti, V. (1991). Double dc SQUID for flux-locked-loop operation. Applied Physics Letters 59 (4): 476–478. Elekta Neuromag Oy. PO Box 68, FIN-00511 Helsinki, Finland. https://www.elekta.com/ (cited material from August 2005). Zhang, Y., Liu, C., Schmelz, M. et al. (2012). Planar SQUID magnetometer integrated with bootstrap circuitry under different bias modes. Superconductor Science and Technology 25 (12): 125007. Liu, C., Zhang, Y., Mück, M. et al. (2013). Statistical characterization of voltage-biased SQUIDs with weakly damped junctions. Superconductor Science and Technology 26 (6): 065002.

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9 Analyses of the “Series Feedback Coil (Circuit)” (SFC) To suppress the contribution of δΦIn from the preamplifier current noise in a SQUID system, one may introduce a series feedback coil (circuit) (SFC). The SFC consists of only one inductor Lse inserted, e.g. between the SQUID and ground, where the coil Lse couples to the SQUID with a mutual inductance Mse , as shown in Figure 8.1b. Note that the positions of the SQUID and Lse can be exchanged due to the series circuit. Usually, the Lse is a superconducting coil integrated with the SQUID chip in thin film technology. At the low-frequency limit (i.e. 𝜔Lse → 0), the introduction of Lse does not influence the bias mode. In this chapter, we will thoroughly analyze the effects of the SFC. In current bias mode, we begin with a model of “two voltages in series” to understand the principle of the SFC. In voltage bias mode, the SFC increases the apparent flux-to-current transfer coefficient (𝜕I/𝜕Φ)SFC of the SQUID, thus reducing the noise contribution of the preamplifier I n , i.e. δΦIn = I n /(𝜕I/𝜕Φ)SFC , directly. For both bias modes, the SFC essentially reduces the apparent dynamic resistance (Rd )SFC .

9.1 SFC in Current Bias Mode 9.1.1

Working Principle of the SFC in Current Bias Mode

In current bias mode, the readout quantity is the voltage signal of the SQUID. Here, our analysis of the SFC is based on Ref. [1]. In DRS, a current noise of the preamplifier, I n , inevitably flows out and passes through the SQUID, thus translating a noise voltage across the SQUID’s dynamic resistance Rd , i.e. V In = I n × Rd , where Rd is an original parameter of SQUID. This noise voltage V In is applied to the input terminal of the preamplifier, which is a voltage amplifier. With the SFC, an additional flux caused by I n × Mse is fed back into the SQUID loop and converted into a voltage V s = (I n × Mse ) × (𝜕V /𝜕Φ) across the SQUID so that these two voltages, V In and V s , are connected in series (see Figure 9.1). The input terminal of the preamplifier has a very high impedance, so the serially connected voltages, (V in )In , at the input terminal do not generate any current. In other words, both voltages V In and V s can also be regarded as two potentials. When V In and V s are opposite, (V in )In is reduced, i.e. (V in )In = (V In − V s ) < V In , as depicted in Figure 9.1. In the SFC, the apparent dynamic resistance (Rd )SFC is defined as (V in )In /I n . SQUID Readout Electronics and Magnetometric Systems for Practical Applications, First Edition. Yi Zhang, Hui Dong, Hans-Joachim Krause, Guofeng Zhang, and Xiaoming Xie. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

122

9 Analyses of the “Series Feedback Coil (Circuit)” (SFC)

In Rd

VIn = In × Rd

(Vin)In (Rd)SFC Vs = In × Mse × (∂V/∂Φ) Mse

Lse

Figure 9.1 The arrangement of the SFC, where the coil Lse and the SQUID are connected in series and Lse couples to the SQUID via a mutual inductance Mse . Here, the preamplifier’s current noise In flows through the SQUID and Lse , thus generating two voltages, V In and V s , connected in series, which are applied to the preamplifier input terminal.

In fact, the shape of the V (Φ) characteristics of the SQUID stays the same with or without Lse due to the constant bias current I b , as shown in Figure 9.2a. In contrast to the PFC, Lse (Mse ) does not affect 𝜕V /𝜕Φ of the SQUID in current bias mode, i.e. 𝜕V /𝜕Φ = (𝜕V /𝜕Φ)SFC = (Rd )SFC × (𝜕I/𝜕Φ)SFC . Here, (𝜕I/𝜕Φ)SFC is hidden in current bias mode, but (Rd )SFC must be changed. If (V in )In = V In − V s = 0 (the critical condition), the apparent (Rd )SFC at the working point (e.g. W2 ) on V (Φ) characteristics approaches zero, whereas (Rd )SFC ≈ 2Rd at the working point W1 . Obviously, the critical condition (V in )In ≈ 0, or (Rd )SFC ≈ 0 at W2 , is expressed as Mse × (𝜕V ∕𝜕Φ) = Rd or, Mse × (𝜕I∕𝜕Φ) = 1 Hereby, δΦIn disappears, although I n inevitably flows out from the preamplifier. Clearly, the mutual inductance Mse is the one and only adjustable parameter for a certain SQUID during SFC operation. The flux generated by the constant I b (without any noise) can be regarded as a dc flux off-set ΔΦoff in the V (Φ) characteristics via Mse only. Generally, the SQUID dynamic resistance Rd can be obtained from the I–V characteristics. Now, let us observe the evolution of Rd without and with SFC. There are three I–V characteristics in Figure 9.2b without SFC: both gray curves describe the I–V characteristics at the two flux limit states, Φ = nΦ0 and Φ = (n + 1/2)Φ0 , while the third (black curve) is located at Φ = (2n + 1)Φ0 /4, on which two working points denoted by W1 and W2 in Figure 9.2a are set on the same position W, where Rd ≈ 45 Ω is obtained. In Figure 9.2c (without the I–V characteristics at two flux limit states), W1 and W2 are set on two I–V characteristics at Φ = (n + 1/4)Φ0 and Φ = (n + 3/4)Φ0 , respectively, where the two I–V characteristics are separated due to the SFC. Thus, the two working points present different (Rd )SFC values, i.e. 80 Ω (W1 ) and 8 Ω (W2 ). The ratio 𝛾 = Rd /(Rd )SFC (at W2 ) ≈ 5.6 can be defined as the noise suppression factor 𝛾 of δΦIn with the SFC.

9.1 SFC in Current Bias Mode

30 20 V (μV)

10 W1

0

W2

–10 –20 –30

(a)

0

1 Φ/Φ0

8

2

8 Φ = nΦ0

6

Φ = (n + 3/4)Φ0

6

W1 Φ = (n + 1/2)Φ0 Rd = 45 Ω

2 0 (b)

I (μA)

I (μA)

W 4

10 20 30 40 50 60 70 80 V (μV)

0 (c)

Φ = (n + 1/4)Φ0

(Rd)SFC = 80 Ω or (Rd)SFC = 8 Ω

2

Φ = (2n + 1)Φ0 /4 0

W2

4

0

10 20 30 40 50 60 70 80 V (μV)

Figure 9.2 (a) The measured V(Φ) characteristics are unchanged with and without the SFC; (b) without SFC, two working points, W1 and W2 , are overlapping at W and present the same resistance Rd ≈ 45 Ω (original) obtained from the I–V characteristics at Φ = (2n + 1)Φ0 /4 (black line); (c) with the SFC, W1 and W2 present different (Rd )SFC , 80 Ω (W1 ) at Φ = (n + 1/4)Φ0 and 8 Ω (W2 ) at (n + 3/4)Φ0 , respectively.

9.1.2 Noise Measurements of a Weakly Damped SQUID (Magnetometer) System with the SFC To highlight the effect of the SFC on suppressing the noise contribution of δΦIn , we employed a weakly damped SQUID (magnetometer) with a large original Rd ≈ 45 Ω and PCBTs (e.g. 3 × SSM2220) with a large I n (shown in Figure 5.4) as the preamplifier, thus generating a large voltage noise, V In = I n × Rd . In the general case of Mse × (𝜕V /𝜕Φ) < Rd , the voltage contribution (V in )In caused by I n at the input terminal of the preamplifier is reduced with the SFC; in other words, (Rd )SFC is decreased, i.e. (Rd )SFC < Rd . For the following system noise δΦsys measurements, the parameters of the employed SQUID magnetometer are listed in Table 9.1. Figure 9.3 shows different frequency spectra of the SQUID system flux noise δΦsys : In curve I without the SFC, the noise contributed by I n of the PCBTs is clearly observed in the frequency regime of (𝜕I/𝜕Φ). The ratio of (𝜕I/𝜕Φ)SFC /(𝜕I/𝜕Φ) ≈ 5.7 yields the same noise suppression factor of 𝛾 = Rd /(Rd )SFC measured with the I–V characteristics of the SQUID described Figure 9.4 The equivalent circuit of a voltage-biased SQUID with SFC.

A I

Rd

Vb Lse

Mse

125

9 Analyses of the “Series Feedback Coil (Circuit)” (SFC)

1.5 1.0 W

0.5 I (μA)

126

0.0 W1

W2

–0.5 –1.0 –1.5 0

1 Φ/Φ0

2

Figure 9.5 In voltage bias mode, the I(Φ) characteristics without (dotted curve) or with the SFC (solid curve), where its period is given by a flux quantum Φ0 and the signal swing records the current change with a flux change of Φ0 /2. The I(Φ) characteristics with the SFC are asymmetrical, where two working points, W1 and W2 , are marked at the gradual and the steep slope, respectively. The experimentally measured values, Mse ≈ 0.23 nH, 𝜕I/𝜕Φ ≈ 7 μA/Φ0 , and Mse × (𝜕I/𝜕Φ) ≈ 0.8, determine the noise suppression factor of (𝜕I/𝜕Φ)SFC /(𝜕I/𝜕Φ) to be approximately 5.7 at W2 .

in Figure 9.2b,c. In fact, the term [1 ± Mse × (𝜕I/𝜕Φ)] can be determined experimentally. Note that the current amplitude I swing of I(Φ) does not change with or without the SFC in voltage bias mode (see Figure 9.5). With the SFC, the SQUID’s flux-to-current transfer coefficients (𝜕I/𝜕Φ)SFC at the working points are expressed as (𝜕I∕𝜕Φ)SFC = (𝜕I∕𝜕Φ)∕[1 ± Mse × (𝜕I∕𝜕Φ)]

(9.1)

At the critical condition of Mse × (𝜕I/𝜕Φ) → 1 (or Φse = I swing × Mse ≈ Φ0 /2), (𝜕I/𝜕Φ)SFC at W2 reaches infinity; thus, δΦIn at W2 disappears. Otherwise, (𝜕I/𝜕Φ)SFC at W1 reduces to (𝜕I/𝜕Φ)/2. In general cases of Mse × (𝜕I/𝜕Φ) < 1, an increased (𝜕I/𝜕Φ)SFC at W2 decreases the noise contribution of the preamplifier δΦIn = I n /(𝜕I/𝜕Φ)SFC , where I n does not depend on the existence of SFC. For the configuration of a SQUID with SFC, we recommend a range of 0.65 < Mse × (𝜕I/𝜕Φ) < 0.9; within this range, the corresponding δΦIn suppression factors of 𝛾 = Rd /(Rd )SFC are 3–10. As Mse × (𝜕I/𝜕Φ) > 1 (Mse ≈ 0.17 nH), hysteresis appears at the steep slope in the I(Φ) characteristics [2], while (𝜕I/𝜕Φ)SFC at W1 reduces to approximately 4 μA/Φ0 (see Figure 9.6). However, the linearity of the gradual slope was considerably improved, so the working point W1 can be employed for a small signal amplifier, as discussed below. For SFC operation in voltage bias mode, the constraint condition is that the V (Φ) of the SQUID and its (𝜕V /𝜕Φ) at working points remains unchanged, i.e. (𝜕V ∕𝜕Φ) = Rd × (𝜕I∕𝜕Φ) = (𝜕I∕𝜕Φ)SFC = (Rd )SFC × (𝜕I∕𝜕Φ)SFC

(9.2)

In voltage bias mode, V (Φ) or 𝜕V /𝜕Φ is not the readout quantity; i.e. it remains hidden, but their behaviors were already observed and analyzed in current bias

9.3 Summary of the PFC and SFC

2

I (μA)

1 W1

0 –1 –2 0

1

Φ/Φ0

2

3

Figure 9.6 Hysteresis appears in the measured I(Φ) characteristics, as Mse × (𝜕I/𝜕Φ) > 1, where the width of the gradual slope is large.

mode in Section 9.1. To keep the product of 𝜕V /𝜕Φ constant during SFC operation in voltage bias mode, a decrease in (Rd )SFC leads to an increase in (𝜕I/𝜕Φ)SFC at the working point on the steep slope. Integrating Eqs. (9.1) and (9.2), the (Rd )SFC with the SFC is expressed as (Rd )SFC = Rd × [1 − Mse × (𝜕I∕𝜕Φ)] Indeed, the preamplifier’s V n does not benefit from using the SFC in voltage bias mode because δΦV n = V n /(𝜕V /𝜕Φ)SFC = V n /(𝜕V /𝜕Φ). It is in accordance with Eq. (9.2). Experimental noise measurements with the SFC in voltage bias mode are omitted here because the results are the same as those demonstrated in current bias mode in Figure 9.3. For our design of planar SQUID magnetometers in Figure 8.22, we can select the bonding positions Sa′ (or, Sa′′ ) on the bottom strap to vary the Mse in the SFC. Generally, a suitable Mse can easily be found, so the spread (variation) of SQUID parameters typically occurring in production could be covered for suppressing δΦIn [1].

9.3 Summary of the PFC and SFC We attach Table 9.2 to summarize the behavior of the PFC and SFC in both bias modes. From Table 9.2, we learn that the functions of the PFC and SFC are different. Namely, the former suppresses δΦV n , while the latter reduces δΦIn . Furthermore, both noise suppressions are independent of the bias modes. In practice, the choice between the PFC and SFC depends not only on the preamplifier but also on the SQUID parameters. In case of a strongly damped SQUID with a small value of 𝜕V /𝜕Φ and intrinsic noise √ δΦs , in order to observe δΦs in a DRS, the flux noise δΦV n caused by V n ≈ 1 nV/ Hz of the preamplifier (op-amp) must at first be suppressed with PFC. Namely, increasing the apparent dynamic resistance (Rd )PFC and the transfer coefficient (𝜕V /𝜕Φ)PFC at the working point reduces δΦV n = V n /(𝜕V /𝜕Φ)PFC (see Chapter 8). However, for an intermediately damped SQUID, the current noise contribution I n of PCBTs must be suppressed by the SFC, especially in the low-frequency regime, as shown in Figure 9.3.

127

Table 9.2 Summary of PFC and SFC at working point W. PFC

SFC

Consequence

(Rd )PFC ↑, (𝜕V /𝜕Φ)PFC ↑, (𝜕I/𝜕Φ)PFC = 𝜕I/𝜕Φ

(Rd )SFC ↓, (𝜕I/𝜕Φ)SFC ↑, (𝜕V /𝜕Φ)SFC = 𝜕V /𝜕Φ

Critical condition

(Rd )PFC → ∞ @ Mp × (𝜕V /𝜕Φ) = Rd + Rp

(Rd )SFC → 0 @ Mse × (𝜕I/𝜕Φ) = 1

Bias mode

Current bias

Voltage bias

Current bias

Voltage bias

Characteristics

Asymmetric V (Φ)

Symmetric I(Φ)

Symmetric V (Φ)

Asymmetric I(Φ)

W

V swing ↓ (𝜕V /𝜕Φ)PFC ↑ (δΦV n )PFC Noise contribution from V n

Noise contribution from I n

= Vn ∕(𝜕V ∕𝜕Φ)PFC < δΦV n = Vn ∕(𝜕V ∕𝜕Φ)

W

W

W

I swing is unchanged

I swing is unchanged

V swing is unchanged

(𝜕I/𝜕Φ)PFC is unchanged

(𝜕V /𝜕Φ)SFC is unchanged

(𝜕I/𝜕Φ)SFC ↑

(δΦV n )PFC

(δΦV n )SFC

(δΦV n )SFC

= [Vn ∕(Rd )PFC ]∕(𝜕I∕𝜕Φ)PFC

= Vn ∕(𝜕V ∕𝜕Φ)SFC

= [Vn ∕(Rd )SFC ]∕(𝜕I∕𝜕Φ)SFC

= Vn ∕(𝜕V ∕𝜕Φ)PFC

= Vn ∕(𝜕V ∕𝜕Φ)

= Vn ∕(𝜕V ∕𝜕Φ)

< δΦV n = Vn ∕(𝜕V ∕𝜕Φ)

Unchanged

Unchanged

(δΦIn )PFC

(δΦIn )PFC

= [In × (Rd )PFC ]∕(𝜕V ∕𝜕Φ)PFC

= In ∕(𝜕I∕𝜕Φ)PFC

= In ∕(𝜕I∕𝜕Φ)

= In ∕(𝜕I∕𝜕Φ)

= δΦIn

= δΦIn

Unchanged

Unchanged

(δΦIn )SFC = [In × (Rd )SFC ]∕(𝜕V ∕𝜕Φ)SFC = In ∕(𝜕I∕𝜕Φ)SFC < δΦIn = In ∕(𝜕I∕𝜕Φ)

(δΦIn )SFC = In ∕(𝜕I∕𝜕Φ)SFC < δΦIn = In ∕(𝜕I∕𝜕Φ)

9.4 Combination of the PFC and SFC (PSFC)

To make a long story short, for PFC, (𝜕V /𝜕Φ)PFC and (Rd )PFC increase; for SFC, (𝜕I/𝜕Φ)PFC increases while (Rd )SFC decreases. For both, the unchanged transfer coefficients of (𝜕I/𝜕Φ) in current bias mode and (𝜕V /𝜕Φ) in voltage bias mode act as constraint conditions.

9.4 Combination of the PFC and SFC (PSFC) In fact, the total noise contribution of the preamplifier, δΦ2e , is the sum of its two noise sources, i.e. δΦ2e = δΦ2In + δΦ2V n , when δΦRp is negligible. Using a combination of the PFC and SFC to minimize the noise from the preamplifier, δΦe , should be a good idea. D. Drung and H. Koch first suggested and realized this combination, i.e. additional positive feedback (APF) and bias current feedback (BCF) [3]. It is known that APF changes the bias mode from a pure current bias mode to a mixed bias mode as discussed in Section 8.3, thus complicating the analysis of the combination. Therefore, our discussion about this combination will also be limited to voltage bias mode. In this book, the combination of the PFC and SFC in voltage bias mode can be abbreviated as PSFC, which was also called the SQUID bootstrap circuit (SBC) in Ref. [4]. The equivalent circuit of the PSFC is shown in Figure 9.7, where the preamplifier acts as a current-to-voltage convertor and V b is applied at its noninverting terminal to bias the SQUID. The PSFC has two branches, the PFC branch featuring a coil Lp and a resistor Rp and the SQUID branch consisting of a SQUID and a coil Lse (SFC) connected in series. We expect that the PSFC yields not only a large flux-to-current coefficient (𝜕I/𝜕Φ)SFC for suppressing the contribution from the preamplifier’s I n , i.e. δΦIn = I n /(𝜕I/𝜕Φ)SFC , but also a large (𝜕V /𝜕Φ)PFC for reducing the contribution from the V n of the preamplifier, where δΦV n = V n /(𝜕V /𝜕Φ)PFC . Thus, δΦe may be effectively suppressed. In the following, we analyze the practical possibility of the feedback combinations. 9.4.1

PSFC Analysis Under Independence Conditions

According to Kirchhoff ’s law, in principle, we can analyze the PSFC circuit to obtain its two transfer coefficients, (𝜕I/𝜕Φ)PSFC and (𝜕V /𝜕Φ)PSFC . In fact, a mutual Figure 9.7 The SQUID with the PSFC in voltage bias mode, where both the Lse and Lp coils are superconducting. A mutual inductance Mps between Lp and Lse is indicated by a gray color, while Lf and its mutual inductances with both coils, Lp and Lse , are omitted.

I Mp

Rg

Rp – +

Lp Mse Lse

Mps

Vb

Vm

129

130

9 Analyses of the “Series Feedback Coil (Circuit)” (SFC)

inductance Mps between Lp and Lse cannot be avoided, so the two branches shown in Figure 9.7 are no longer independent. In fact, the value of Mps depends on the arrangement of two coils, i.e. on the layout of the planar structures. In addition, the two coils also have different mutual inductances coupling to the Lf coil during FLL operation (Lf is omitted in Figure 9.7). More than two unwanted mutual inductances make the analysis very complicated. We believe that most readers have no interest in the lengthy and complex circuit analyses because such analysis is not clear at a glance in terms of results. For simple PSFC analysis, the two branches should be independent. Therefore, our analyses are limited under the following independence conditions: (i) Mps = 0, and Lf is absent. There are two references about numerical simulations, which describe the behavior of the PSFC under this condition [5, 6]; (ii) the NC condition of Mp × (𝜕V /𝜕Φ) = Rp is just fulfilled, where the noise current (I n )V n caused by the V n of the preamplifier flows through only the PFC branch, i.e. (I s )V n = 0. Thus, two branches of the PFC and SQUID with the SFC are separated for δΦV n . So, all above analyses in Chapter 8 and Section 9.2 are still valid without any amendment. For the PSFC in voltage bias mode, we focus on two parameters: (i) the transfer coefficient (𝜕I/𝜕Φ)PSFC (signal) and (ii) the preamplifier noise currents ΣI n consisting of two noise components from the preamplifier, V n /(Rd )PSFC and I n (noise). Namely, we aim to determine the signal-to-noise ratio of the system with the PSFC. The four graphs in Figure 9.8 show the analytical equivalent circuits of the PSFC, where some graphs consist of two parts: the gray part is deactivated due to the independence conditions, whereas the black part is active. In

Rd

Mp

Rd

Rp Vn

Mse

Lp

Inp

Mse

Lse

Lp Lse

(b)

Rd

Mp

(Rd)PSFC

Rp

Mp

Vb Mse (c)

Rp

Mp

Vs

(In)Vn

(a)

Ins

Lp Lse

Rp Lp

Ioff = Vb/Rp

Mse

Vb

Lse

(d)

Figure 9.8 Four equivalent circuits of the PSFC under the independence conditions in voltage bias mode.

9.4 Combination of the PFC and SFC (PSFC)

Graphs (a) and (b) describe the behaviors of V n and I n from the preamplifier, respectively. Graph (a) presents the current noise caused by V n , (I n )V n . For the noise voltage V n of the preamplifier, the SQUID branch is opened due to (Rd )SB → ∞ at Mp × (𝜕V /𝜕Φ) = Rp , so no current generated by V n flows through the SQUID branch, i.e. (I s )V n = 0. Therefore, the δΦV n = (V n /Rp )/ (𝜕I/𝜕Φ) = V n /(𝜕V /𝜕Φ)PFC is suppressed due to Rp > Rd (see Section 8.3.2). Graph (b) describes the behavior of I n . The inevitable noise current I n flowing out from the preamplifier divides into I ns and I np . Thus, two additional fluxes of I ns × Mse and I np × Mp are delivered to the SQUID loop, thus generating a voltage (V s )In across the SQUID. The (V s )In generates a ring current, which can change the distribution of I ns and I np ; nevertheless, the ring current cannot influence I n flowing out from the preamplifier. In brief, I n is not changed with the PFC and SFC. Consequently, δΦIn only depends on (𝜕I/𝜕Φ)PSFC , i.e. δΦIn = I n /(𝜕I/𝜕Φ)PSFC . Graphs (c) and (d) present the different functions of the two branches for the SQUID signal: (c) The biased voltage V b generates a constant current I off flowing through the PFC branch, thus generating a dc offset flux, Φoff = (V b /Rp ) × Mp , in the SQUID’s I(Φ). (d) A SQUID with SFC is operated under V b , thus resulting in the asymmetric I(Φ) described above in Section 9.2. At the working point W2 set on the steep slope of I(Φ), a large (𝜕I/𝜕Φ)PSFC is obtained, so δΦIn = I n /(𝜕I/𝜕Φ)PSFC is suppressed because (𝜕I/𝜕Φ)SFC /(𝜕I/𝜕Φ) > 1. Note that in graph (d), the SQUID with the PFC can be regarded as a modified (new) sensor, which presents a large (𝜕V /𝜕Φ)PFC and (Rd )PFC , whereas the original (𝜕I/𝜕Φ) remains constant. In voltage bias mode, (𝜕V /𝜕Φ)PFC is hidden, and only (Rd )PFC appears. Then, the SFC modifies the new sensor further, i.e. (𝜕I/𝜕Φ) → (𝜕I/𝜕Φ)SFC and (Rd )PFC → (Rd )PSFC , but (𝜕V /𝜕Φ)PFC remains. Under the independence conditions, the SQUID with the PSFC is characterized by the following parameters: (𝜕I/𝜕Φ)SFC and (Rd )PSFC , where (𝜕V /𝜕Φ)PFC = (𝜕I/𝜕Φ)SFC × (Rd )PSFC . Generally, once the PSFC circuit is used, as shown in Figure 9.7, all parameters should use the subscript PSFC, e.g. (𝜕I/𝜕Φ)SFC is denoted by (𝜕I/𝜕Φ)PSFC , and (𝜕V /𝜕Φ)PFC is denoted by (𝜕V /𝜕Φ)PSFC . Actually, the original SQUID parameters, (𝜕V /𝜕Φ) and (𝜕I/𝜕Φ), are changed only once with the PSFC, whereas Rd is changed twice. Here, (Rd )PSFC is neither (Rd )PFC = (𝜕V /𝜕Φ)PFC /(𝜕I/𝜕Φ) nor (Rd )SFC = (𝜕V /𝜕Φ)/(𝜕I/𝜕Φ)SFC ; therefore, we should define the apparent (Rd )PSFC as (Rd )PSFC = (𝜕I∕𝜕Φ)PSFC ∕(𝜕I∕𝜕Φ)PSFC = [(𝜕V ∕𝜕Φ)PSFC ∕(𝜕I∕𝜕Φ)]∕[(𝜕I∕𝜕Φ)PSFC ∕(𝜕I∕𝜕Φ)] = (Rd )PFC ∕[Rd ∕(Rd )SFC ] i.e. (Rd )PSFC = [Rp × (Rd )SFC ]∕Rd or, (Rd )PSFC = Rp × [1 − Mse × (𝜕I∕𝜕Φ)]

131

132

9 Analyses of the “Series Feedback Coil (Circuit)” (SFC)

In brief, the sensor in graph (d) is characterized by (Rd )PSFC and (𝜕I/𝜕Φ)PSFC under two independent conditions. For PSFC operation, the polarities of Lse and Lp should be taken into account; i.e. a large (𝜕V /𝜕Φ)PSFC and a large (𝜕I/𝜕Φ)PSFC should appear at the working point W2 on the same slope of I(Φ). Then, the preamplifier noise of V n and I n at W2 can be suppressed simultaneously, as desired: δΦV n = Vn ∕(𝜕V ∕𝜕Φ)PSFC = Vn ∕(𝜕V ∕𝜕Φ)PFC = (Vn ∕Rp )∕(𝜕I∕𝜕Φ) δΦIn = In ∕(𝜕I∕𝜕Φ)PSFC = In ∕(𝜕V ∕𝜕Φ)SFC In brief, PSFC operation can be regarded as SFC affecting a new SQUID with the parameters of (Rd )PFC and (𝜕V /𝜕Φ)PFC . We hope that the qualitative analyses in (d) can also outline the basic trend of PSFC operation when the independence conditions are not fulfilled. 9.4.2

PSFC Experiments and Results

Now, we introduce two experiments of the PSFC from our early works: experiment I is the first demonstration of the PSFC (SBC) in voltage bias mode, which proves the above analysis under two independent conditions. Experiment II is employed to observe the different behavior of PSFC at working points W1 and W2 in general cases where the independent conditions are not fulfilled. In both experiments with the PSFC, a helium-cooled planar SQUID magnetometer with an inductance Ls ≈ 350 pH was used. Here, a pickup loop of 6 × 6 mm2 was connected to a five-turn input coil Lin , which was integrated with the SQUID washer, thus constructing a flux transfer system. The field-to-flux transfer coefficient 𝜕B/𝜕Φ of the magnetometer was 1.25 nT/Φ0 [7]. Here, three coils, Lse , Lp , and Lf , made from niobium wire were individually coupled to the SQUID via the pickup loop. In experiment I, we assumed that there are no mutual inductances between them. In practice, we employed an op-amp (AD797) as the preamplifier in a DRS. Furthermore, some data and statements will be pointed out when they are different from those in the original publications. In the first report of the PSFC in voltage bias mode [4], i.e. experiment I, the original parameters of the strongly damped SQUID were determined as Rd ≈ 10 Ω and 𝜕I/𝜕Φ ≈ 9 μA/Φ0 (i.e. 𝜕V /𝜕Φ ≈ 90 μV/Φ0 ). The measured asymmetric I(Φ) characteristic of the PSFC in voltage bias mode is shown in Figure 9.9. According to the asymmetry of I(Φ) with a (𝜕I/𝜕Φ)PSFC of 35 μA/Φ0 at W2 , we obtained the value of Mse ≈ 0.15 nH and an SFC factor of [1 − Mse (𝜕I/𝜕Φ)] ≈ 0.22. If the NC condition is just matched, Mp ≈ 0.6 nH1 (without Mse ) is thus obtained for the Rp = 27 Ω used in this experiment. Here, Mp is much larger than Mse . The measured I–V curves of the SQUID with the PSFC near the working points are shown in the inset of Figure 9.10, where we were able to extract (Rd )PSFC of approximately 6 Ω at the working point W2 .2 The (Rd )PSFC measured at W2 fits our analytical value (Rd )PSFC = Rp × [1 − Mse (𝜕I/𝜕Φ)] = 27 × 0.22 Ω = 5.9 Ω well. 1 The estimated Mp was approximately 0.77 nH in the original. 2 W1 and W2 were reversed in the original.

9.4 Combination of the PFC and SFC (PSFC)

Figure 9.9 Working point W2 is set on the steep slope of the asymmetric I(Φ) function with the PSFC (photo).

I 4.2 μA W2

W1

Φ/Φ0

100

100

1 RW d =6Ω

10

Rd 2 = 30 Ω

10 μV

Rd = 10 Ω V

10

Field noise (fT/√Hz)

W

3 μA

Flux noise (μΦ0/√Hz)

I

1 1

10

100

1000

Frequency (Hz)

Figure 9.10 The flux noise spectrum and the corresponding field sensitivity of a helium-cooled SQUID magnetometer with Ls ≈ 350 pH were measured at W2 in the PSFC configuration. The inset shows the corresponding I–V characteristics near both working points W2 at Φ ≈ (2n + 1)Φ0 /4, thus resulting in the different (Rd )PSFC values. The original Rd = 10 Ω of the SQUID is illustrated by the dashed line.

In a system noise δΦsys measurement, two flux noise contributions from the preamplifier, δΦV n = (V n /Rp )/(𝜕I/𝜕Φ) and δΦIn = I n /(𝜕I/𝜕Φ)SFC , were simultaneously suppressed. The white δΦsys (>20 Hz) of the SQUID magnetometer at √ working point W2 reached approximately √ 3 μΦ0 / Hz in the FLL, corresponding to a field sensitivity of δB ≈ 4 fT/ Hz, as shown in Figure 9.10. This noise measurement was performed in a magnetically shielded room and inside a Nb shielding tube inside a helium can. Indeed, experiment I of the PSFC should be considered a proof of our PSFC analyses under two independence conditions (see Section 9.4.1). In experiment II, the PSFC is operated in the general case where the independent conditions are not fulfilled [2]. We used the same configuration as in experiment I; i.e. the PSFC was constructed of some discrete components.

133

9 Analyses of the “Series Feedback Coil (Circuit)” (SFC)

2

Mse = 0

1 I (μA)

134

W

0

W2

W1

–1 Mse = –0.09 nH

–2 0

Mse = 0.09 nH 1 Φ/Φ0

2

Figure 9.11 The measured I(Φ) characteristics for three different values of Mse . The three working points W2 , W, and W1 are set in the middle of the positive slopes.

In the general case, the noise suppression factors for δΦV n and δΦIn can be separately determined. It is known that the PFC does not influence the shape of I(Φ), and vice versa. To demonstrate the function of SFC in PSFC, we chose Mse = 0 and two moderate values Mse = ±0.09 nH and plotted their corresponding I(Φ) in Figure 9.11. The different flux-to-current transfer coefficients (𝜕I/𝜕Φ)PSFC obtained at working points W, W1 , and W2 were 11.8 μA/Φ0 (W), 7.2 μA/Φ0 (W1 ), and 28 μA/Φ0 (W2 ). It means that δΦIn will be suppressed by a factor of 2.4 = (28/11.8) at W2 . In order to clearly observe the PFC effect in the PSFC operation, we chose a rather high Mp of 3.2 nH and an appropriately high Rp of 250 Ω, thus resulting in a high value of (𝜕V /𝜕Φ)PSFC . However, the dynamic resistances (Rd )PSFC in the PSFC operation were determined from the slope of the I–V curve (not shown here), which increased from the original Rd of 12 Ω to (Rd )PSFC values of 230 Ω at W1 , 120 Ω at W, and 40 Ω at W2 . Here, the (Rd )PSFC of 120 Ω at W is approximately half of Rp = 250 Ω. It means that the above second independence condition is not fulfilled; in other words, the two branches become dependent. Thus, the analysis in Section 9.4.1 is no longer valid. However, in the three cases of Mse in Figure 9.11, all measured (𝜕V /𝜕Φ)PSFC reached >1 mV/Φ0 , when the SQUID with the PSFC was temporally operated in current bias mode. For example, at working point W (without Mse ), the operation of the PSFC is simplified as that of the PFC; thus, (𝜕V /𝜕Φ)PSFC → (𝜕V /𝜕Φ)PFC = (𝜕I/𝜕Φ) × (Rd )PFC = (11.8 μA/Φ0 ) × 120 Ω = 1.4 mV/Φ0 . Thus, in the three cases of Mse , the δΦV n of the preamplifier must be suppressed effectively, although the NC condition is not fulfilled. To summarize experiment II, we list the values of our experimentally determined PSFC parameters at working points W, W1 , and W2 in Table 9.3. The parameters of the original SQUID (without the PSFC) are also given under the bottom line of Table 9.3. Because the independence conditions are not fulfilled, (𝜕V /𝜕Φ)PSFC cannot remain constant as the working point changes from W to W1 and W2 , as analyzed above. According to the three cases in Table 9.3, the SQUID system noise δΦsys with the PSFC was measured in the FLL, as plotted in Figure 9.12. The system noise √ δΦsys of 2.5 μΦ0 / Hz (white) was achieved in all three cases, corresponding to a

9.4 Combination of the PFC and SFC (PSFC)

Table 9.3 Parameters of PSBC in experiment II. Parameters

W

W1

Mse

0

Mp

3.2

(Rd )PSFC

a)

W2

Unit

0.09

−0.09

nH

3.2

3.2

nH

120

230

40

Ω

(𝜕I/𝜕Φ)PSFC b)

11.8

7.2

28

μA/Φ0

(𝜕V /𝜕Φ)PSFC c)

1.4

1.7

1.1

mV/Φ0

a) Rd(orig.) = 12 Ω. b) 𝜕I/𝜕Φ(orig.) = 11.8 μA/Φ0 . c) 𝜕V /𝜕Φ(orig.) = 0.14 mV/Φ0 .

10 10 Mse = 0.09 nH

Mse = 0 Mse = –0.09 nH 1 100

101

Field noise (fT/√Hz)

Without PSFC Flux noise (μΦ0 /√Hz)

Figure 9.12 Comparison of the system flux and field noise spectra measured with the PSFC at working points W (Mse = 0), W1 , and W2 . Compared to that without the PSFC, the system noise with the PSFC is suppressed by a factor of 3. Here, a small (𝜕I/𝜕Φ)PSFC at W1 (Mse = 0.09 nH) is not sufficient to suppress δΦIn in the low-frequency regime.

With PSFC

102 103 Frequency (Hz)

104

√ field sensitivity of δB ≈ 3 fT/ Hz for the SQUID magnetometer. We believe that the δΦV n of the preamplifier (op-amp, e.g. AD797) was completely suppressed below δΦs because of the large (𝜕V /𝜕Φ)PSFC . However, we observed slightly higher low-frequency noise at W1 . It may be contributed by δΦIn from I n of the preamplifier due to the smaller (𝜕I/𝜕Φ)PSFC at W1 than at W2 . It is expected that the slightly higher low-frequency noise of δΦIn measured at W1 may be improved when using the LT1028 op-amp instead of the AD797 op-amp. Interestingly, an operating advantage at W1 appears; i.e. the wide linearity regime of the I(Φ) characteristic exceeds 0.5 Φ0 , as illustrated in Figure 9.13. Here, the upper plot is one of the curves (Mse = 0.09 nH) from Figure 9.11, while the lower curve is the derivative of the upper plot; i.e. (𝜕I/𝜕Φ)PSFC vs. Φ/Φ0 . It is clear that the linear regime of Φlin is larger than Φ0 /2. As we further increased Mse from 0.09 to 0.17 nH, a hysteresis at W2 appeared on the steep slope of the I(Φ) curve, and the (𝜕I/𝜕Φ)PSFC at W1 was reduced to well below 5 μA/Φ0 (see Figure 9.6). In the case of Mse = 0.17 nH, the measured system noise δΦsys remained as shown in Figure 9.6 (Mse = 0.09 nH) because the (𝜕V /𝜕Φ)PSFC ≈ 1 mV/Φ0 at W1 was large enough to suppress δΦV n . However, the

135

9 Analyses of the “Series Feedback Coil (Circuit)” (SFC)

Figure 9.13 Plots of I(Φ)PSFC and (𝜕I/𝜕Φ)PSFC vs. Φ/Φ0 at Mse = 0.09 nH. The width of the linear flux regime Φlin at W1 (between the two dashed lines) is 0.54 Φ0 , where (𝜕I/𝜕Φ)PSFC is practically constant.

I (μA)

2 W1

W2

0

–2 30 (∂I/∂Φ)PSFC (μA/Φ0)

136

20 10 0

Φlin

–10

W1 0

1

Φ/Φ0

2

3

linearity regime Φlin of the I(Φ) characteristics exceeds >0.8 Φ0 , so the working point W1 can be arbitrarily set on the gradual slope of I(Φ). A wide linear flux regime Φlin improves not only the system linearity but also the system slew rate Φ̇ f because Φ̇ f ∝ Φlin [8]. Indeed, the system noise measurements shown in Figure 9.12 expand our idea; i.e. the working point can be set on a gradual slope of I(Φ) in the PSFC operation for some special applications. In PSFC experiment II in the general case, the functions of the PFC and SFC are separately demonstrated in PSFC operation. Here, the PFC always increases (𝜕V /𝜕Φ)PSFC to suppress δΦV n . Furthermore, the behavior of Mse is discussed in three cases; the negative Mse increases the (𝜕I/𝜕Φ)PSFC at W2 to suppress δΦIn and the positive Mse increases the linear flux regime Φlin at W1 in I(Φ)PSFC . 9.4.3

Conclusion of the PSFC

In this study, we first qualitatively analyzed PSFC operation under two independence conditions: (i) Mps = 0, and Lf is absent; (ii) the NC condition of Mp × (𝜕V /𝜕Φ) = Rp is just fulfilled, where two branches of the PSFC are separated in voltage bias mode. An important consequence of the analysis is that a SQUID with the PSFC can be regarded as a new sensor with the parameters of (Rd )PSFC and (𝜕I/𝜕Φ)PSFC = (𝜕I/𝜕Φ)SFC , as shown in Figure 9.8d. In this case, instead of the original resistance Rd , the (Rd )PSFC is modified twice and plays the actual role during PSFC operation. Experimentally, we examined the principle of the PSFC under independence conditions in voltage bias mode (experiment I) and expanded our idea to set the working point W1 at a gradual slope of I(Φ) during PSFC operation to enhance the system linearity and slew rate in the general case where the two independence conditions are not fulfilled (experiment II). The application of PSFC in biomagnetism was reported in [9]. In practice, the two flux feedbacks via Mse and Mp are not independent, so

References

adjusting Mse can also change Mp of the PFC, and vice versa [10, 11]. Therefore, reaching the designed mutual inductances of Mse and Mp is difficult. According to our experience, we do not recommend employing the PSFC in the readout scheme for simultaneously suppressing δΦV n and δΦIn from the preamplifier. Using an op-amp as a preamplifier, the PFC always plays the main role in suppressing δΦV n , while compared to δΦV n , δΦIn is not critical. In contrast, the SFC plays the main role in suppressing δΦIn in the low-frequency range when using PCBTs as the preamplifier, as discussed in Chapter 5. In brief, for general SQUID applications of SQUID magnetometry, we suggest two practical concepts with flux feedback in a DRS: (1) op-amp (preamplifier) with PFC only and (2) PCBTs with SFC only. Actually, both concepts are equivalent when noise suppression factors of 𝛾 = 3 are obtained, where 𝛾 = (𝜕V /𝜕Φ)PFC /(𝜕V /𝜕Φ) (in PFC) or 𝛾 = (𝜕I/𝜕Φ)SFC /(𝜕I/𝜕Φ) (in SFC). Thus, both preamplifiers with flux feedbacks yield √ √ the same equivalent noise of Vn∗ ≈ 0.33 nV∕ Hz(> 2 Hz) and In∗ ≈ 1 pA∕ Hz (white noise). In practice, the first choice should be concept (2), which needs only one adjustment of Mse in the SFC. Furthermore, a small Mse (≪Mp ) is easily integrated with a planar layout (see Figure 8.22).

References 1 Zhang, G.F., Zhang, Y., Hong, T. et al. (2015). Practical dc SQUID system:

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devices and electronics. Physica C: Superconductivity and Its Applications 518: 73–76. Zhang, G.F., Zhang, Y., Dong, H. et al. (2011). An approach to optimization of the superconducting quantum interference device bootstrap circuit. Superconductor Science and Technology 24 (6): 065023. Drung, D. and Koch, H. (1993). An electronic second-order gradiometer for biomagnetic applications in clinical shielded rooms. IEEE Transactions on Applied Superconductivity 3 (1): 2594–2597. Xie, X.M., Zhang, Y., Wang, H.W. et al. (2010). A voltage biased superconducting quantum interference device bootstrap circuit. Superconductor Science and Technology 23 (6): 065016. Wang, Y.L., Xie, X.M., Dong, H. et al. (2011). Voltage biased SQUID bootstrap circuit: circuit model and numerical simulation. IEEE Transactions on Applied Superconductivity 21 (3): 354–357. Wang, H.W., Wang, Y.L., Dong, H. et al. (2012). Noise behavior of SQUID bootstrap circuit studied by numerical simulation. Physics Procedia 36: 127–132. Zhang, Y., Liu, C., Schmelz, M. et al. (2012). Planar SQUID magnetometer integrated with bootstrap circuitry under different bias modes. Superconductor Science and Technology 25 (12): 125007. Drung, D. (1996). Advanced SQUID read-out electronics. In: SQUID Sensors: Fundamentals, Fabrication and Applications, vol. 329 (ed. H. Weinstock), 63–116. Dordrecht, The Netherlands: Kluwer Academic Publishers.

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9 Kong, X.Y., Zhang, Y., Xie, X.M., and Jiang, M.H. (2013). Novel supercon-

ducting quantum interference device bootstrap circuit and its application in biomagnetism. IEICE Transactions on Electronics E96c (3): 320–325. 10 Zhang, G.F., Zhang, Y., Dong, H. et al. (2012). Parameter tolerance of the SQUID bootstrap circuit. Superconductor Science and Technology 25 (1): 015006. 11 Zhang, G.F., Zhang, Y., Krause, H.J. et al. (2013). A SQUID bootstrap circuit with a large parameter tolerance. Chinese Physics Letters 30 (1): 018501.

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10 Weakly Damped SQUID 10.1 Basic Consideration of Weakly Damped SQUID Generally, there are two noise sources in a SQUID system with a direct readout scheme (DRS), the intrinsic SQUID noise δΦs and preamplifier noise δΦe , both yielding the system noise, δΦ2sys = δΦ2s + δΦ2e . In a DRS, the preamplifier’s noise contribution to V n and I n can be suppressed by the parallel feedback circuit (PFC) (Chapter 8) and the SFC (Chapter 9), respectively. Actually, the SQUID fabrication process often fails to yield the identical properties we need, e.g. I c , RJ , and C of the SQUID, and Rp and Mp of the PFC, so that feedback amounts of both the PFC and SFC are often higher or lower than the predetermined design values. The former case (higher than the design value) might lead to system instability (oscillation), while in the latter, the preamplifier’s noise contribution cannot be suppressed effectively. Therefore, an adjustable feedback is introduced to compensate for the deviations of SQUID parameters after fabrication. For example, the commercial multi-channel magnetoencephalography (MEG) system (e.g. the early version of Elekta Neuromag Oy) employs field-effect transistors (FETs) acting as the Rp of the PFC in the noise cancellation (NC) readout scheme [1, 2], as discussed in Chapter 8. During the time of system use, the SQUID’s I c could be changed due to the flux trapping in the junctions, and occasionally, it cannot return to its original value after reheating. In principle, FETs in the NC readout scheme provide the possibility of re-optimizing the system noise without major problems. Nevertheless, note that noise evaluation of each channel and readjusting the gate voltage of an FET must still be completed by professionals, not a staff nurse. To overcome the complexity of fitting the feedback amount to SQUID parameters, one might look for another way to tolerate large deviations of SQUID parameters while ensuring suitable system noise δΦsys without any feedbacks. In addition, simple readout electronics are always required, especially for multichannel systems. In 2012, Liu et al. first reported the concept of a weakly damped SQUID system [3]. In many applications, the main ambition for a SQUID system is not achieving ultimate sensitivity but rather obtaining a SQUID system with simplicity, user-friendliness, robustness, large capacity of resisting disturbance, good stability, and a suitable system noise δΦsys . In this way, we should abandon the traditional ideas, i.e. to pursue a lower δΦe than δΦs , which is also very low, SQUID Readout Electronics and Magnetometric Systems for Practical Applications, First Edition. Yi Zhang, Hui Dong, Hans-Joachim Krause, Guofeng Zhang, and Xiaoming Xie. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

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10 Weakly Damped SQUID

e.g. employing a strongly damped SQUID. In contrast, tolerating a relatively large SQUID intrinsic noise δΦs to achieve a suitable δΦsys is our suggested practical approach. To build a weakly damped SQUID system with a DRS, one should first choose a low-noise op-amp employed as a preamplifier, e.g. selecting an op-amp such as LT1028 or AD797, the noise contribution δΦe of which has been analyzed in Chapters 5 and 6. It has been discussed that the main noise contribution from the above op-amps is the voltage noise source, V n , while the contribution of I n can be neglected in most cases. According to the SQUID’s differential model in current bias mode shown in Figure 4.7b, it is also known that the three SQUID readable quantities at working point W, the SQUID intrinsic noise δΦs , the transfer coefficient of 𝜕V /𝜕Φ, and the SQUID dynamic resistance Rd , increase with increasing nominal 𝛽 c , as demonstrated in Figure 6.5. Generally, a small SQUID nominal 𝛽 c leads not only to small δΦs but also to small 𝜕V /𝜕Φ (i.e. to large δΦe ), and vice versa. Thus, for an op-amp, δΦe = δΦV n = V n /(𝜕V /𝜕Φ) reduces√with increasing 𝛽 c . From Table 6.3, the minimum system noise δΦsys ≈ 4 μΦ0 / Hz occurs at √ 𝛽 c ≈ 3, while δΦs ≈ δΦe ≈ 3 μΦ0 / Hz for a SQUID inductance Ls ≈ 350 pH [3, 4]. Indeed, our new paradigm for setting up a simple SQUID system is to strive for equal noise contributions from SQUID and electronics, i.e. a relationship of δΦs ≈ δΦe (e.g. at 𝛽 c ≈ 3 for weakly damped SQUIDs). Such a SQUID system exhibits not the very best δΦsys but rather a δΦsys suitable for applications. We believe that a weakly damped SQUID system with large tolerable SQUID parameters, simple readout electronics, and suitable δΦsys has a high potential for applications, especially for constructing a practical multichannel SQUID system.

10.2 SQUID System Noise Measurements with Different 𝛃c Values Table 6.3 exhibits two important consequences: (1) the SQUID intrinsic noise δΦs increases with increasing 𝛽 c . This has been experimentally proved by Wang et al. who measured SQUID’s δΦs with different 𝛽 c using a two stage readout scheme [5]; and (2) the minimum δΦsys appeared at 𝛽 c ≈ 3 with op-amp in DRS. However, when the new paradigm of weakly damped SQUIDs is considered as a preferable scheme for SQUID systems, consequence (2) needs more experimental support. In this subsection, we systematically perform SQUID system noise measurements in which the employed SQUIDs with different 𝛽 c values were obtained by using different junction shunt resistors RJ in the fabrication [3, 5, 6]. Here, we always try to keep a critical current I c ≈ Φ0 /2Ls to obtain the optimal SQUID screening parameter of 𝛽 L ≈ 1. For Ls ≈ 350 pH of our standard low-T c SQUID, the SQUID’s optimal critical current 2I c should be approximately 6 μA. Assuming that the junction capacitance C and its I c remain constant, the nominal 𝛽 c is then proportional to the R2J given by the layout, based on which the value of RJ is determined by the resistance per square in design. Although we want to ensure process repeatability during fabrication, SQUID parameter deviations cannot be avoided, thus influencing the SQUID’s nominal 𝛽 c .

10.2 SQUID System Noise Measurements with Different βc Values

100

Flux noise (μΦ0 /√Hz)

I

Lin Lpick

II 10

100

Lp Lse

III 10

Field noise (fT/√Hz)

Lf

Ls =

Gradiometer 350 pH SQUID

IV 2

100

101 102 Frequency (Hz)

103

3 104

Figure 10.1 System noise δΦsys measurements (right ordinate gives the field noise Bsys of SQUID magnetometers) with different RJ values were performed in FLL using an op-amp with √ V n ≈ 1 nV/ Hz acting as a current-to-voltage converter (preamplifier), i.e. in voltage bias mode. The inset shows the equivalent circuit of the measured SQUID magnetometers described in Chapter 8, where Lpick is the inductance of the pickup loop (5 × 5 mm2 ), Lf is the flux feedback coil for FLL, and LP and Lse are employed for the PFC and SFC. In the operation of weakly damped SQUID, LP and Lse are deactivated.

In subsequent noise experiments, planar SQUID magnetometers were employed and operated in voltage bias mode. In fact, the layout of the magnetometer has been shown in Figure 8.22. Here, the feedback coil Lf was utilized for flux-locked loop (FLL), while we consider the case that neither PFC nor SFC is activated for the operation of weakly damped SQUIDs (see inset of Figure 10.1). Simple SQUID magnetometers are integrated on a 5 × 5 mm2 chip with a field-to-flux transfer coefficient of 1.5 nT/Φ0 . We investigated the dependence of SQUID’s Rd and 𝜕I/𝜕Φ on the junction shunt resistor RJ with different design values of 5, 10, 30, and 40 Ω. Here, the SQUID’sI(Φ) characteristics as well as 𝜕I/𝜕Φ at the working point were observed at the output of the current-to-voltage converter (preamplifier) shown in Figure 3.6b, while the dynamic resistance Rd at the working point was obtained from I–V characteristics, as described in Figure 3.1. During noise measurements, the SQUID magnetometers were placed in a niobium tube at 4.2 K. Figure 10.1 adapted from Ref. [3] shows system noise δΦsys of three typical SQUIDs: curve I for RJ = 5 Ω (SQUID #1), (II) for 10 Ω (#2), and (III) for 30 Ω (#3). There, the system noise δΦsys in the white noise region decreased from √ √ √ 17 μΦ0 / Hz (I) to 13 μΦ0 / Hz (II) √ and 4 μΦ0 / Hz (III), corresponding to a field resolution (noise) δBsys ≈ 6 fT/ Hz. In case (III), a nominal 𝛽 c ≈ 3 of SQUID #3 was obtained with an estimated junction capacitance C and the measured values for I c as well as for RJ . Here, the product 𝜕V /𝜕Φ = (𝜕I/𝜕Φ) × Rd ≈ 350 μV/Φ0 should √ be achieved, so that δΦ √e (i.e. δΦV n here) is suppressed to be δΦV n ≈ 3 μΦ0 / Hz. Then, δΦs ≈ 3 μΦ0 / Hz can be derived from δΦsys . All in all, it is proved that the new paradigm of weakly damped SQUID with the nominal 𝛽 c ≈ 3 can be considered as a preferable readout scheme of SQUID

141

10 Weakly Damped SQUID

system when using an op-amp as a preamplifier. Furthermore, the system noise δΦsys remains almost the same as the values of 𝛽 c vary between 2 and 4, which will be described in detail in Section 10.3. When 𝛽 c > 5 (e.g. RJ ≈ 40 Ω), the measured δΦsys (not shown here) increased again in spite of the increased SQUID’s √ 𝜕V /𝜕Φ > 700 μV/Φ0 . In this case, the measured noise level of δΦsys ≈ 8 μΦ0 / Hz approaches curve II. It means that the large SQUID intrinsic noise δΦs indeed dominates δΦsys . In contrast, for SQUID #2, δΦe caused by the preamplifier’s V n dominates. We then used the PFC, which is already present in the SQUID layout (see the inset of Figure 10.1), to further reduce δΦsys of strongly damped SQUID #2 √ √ to ≤3 μΦ0 / Hz (i.e. Bsys < 4.5 fT/ Hz), as shown by curve IV in Figure 10.1. This result demonstrates that a strongly damped SQUID with PFC can achieve a lower δΦsys than the weakly damped one, if the same readout electronics is used. Indeed, the four measured noise curves in Figure 10.1 greatly benefit the understanding of the concept of a weakly damped SQUID system; i.e. its simplicity originates from the 25% sacrifice in δΦsys . However, in most applications, this noise increase by 25% is not a relevant factor. Furthermore, massive confusions caused by the parameter inconsistency of SQUID and PFC, as discussed in Chapter 8, do not appear in the operation of the weakly damped SQUID. Another experiment on the weakly damped SQUID was to compare the system noise δΦsys measured under two typical bias modes shown in Figure 10.2, where a SQUID with 𝛽 c ≈ 3.5 (2I c ≈ 9.5 μA, RJ ≈ 23 Ω) was employed [6]. The SQUID system noise δΦsys measured in current bias mode (curve I) and in √ voltage bias mode (curve II) exhibits a good consistency with δΦsys ≈ 4 μΦ0 / Hz √ and δBsys ≈ 6 fT/ Hz. This result once again indicates that there is no difference between the two bias modes regarding the system noise, as analyzed in Chapters 5 and 8. The inset shows the SQUID’s V (Φ) and I(Φ) characteristics and their working points, W. The value of (𝜕V /𝜕Φ) at the working point W reached up 102

Current bias mode

V

W

Φ

II

Voltage bias mode

I I

101

W

Φ

III

101

2

Field noise (fT/√Hz)

102 Flux noise (μΦ0 /√Hz)

142

3 100

101 102 Frequency (Hz)

103

Figure 10.2 Noise characteristics of the SQUID with 𝛽 c ≈ 3.5. The inset shows the photos of V(Φ) and I(Φ) curves. W denotes the working point.

10.3 Statistics of SQUID Properties

to 440 μV/Φ0 , while a dynamic resistance Rd ≈ 50 Ω was obtained from the I–V characteristics (not shown here). In this case, we can derive the preamplifier noise √ contribution as√δΦe ≈ δΦV n = V n /(𝜕V /𝜕Φ) ≈ 2 μΦ0 / Hz at W, thus estimating δΦs ≈ 3.5 μΦ0 / Hz derived from the measured δΦsys . In fact, the ultralow noise preamplifier (parallel-connected bipolar transistor, PCBT) with a voltage noise √ V n ≈ 0.35 nV/ Hz shown in Figure 5.3√can be used to directly observe the intrinsic SQUID noise δΦs of 3.5 μΦ0 / Hz with a negligible δΦe , shown as curve III.√In fact, the difference between the two measured noise values of 4 and 3.5 μΦ0 / Hz is very small, as displayed in Figure 10.2. √ To highlight the low V n of the PCBT preamplifier, the minimum δΦsys ≤ 3 μΦ0 / Hz may be reached at 𝛽 c ≈ 1–2, where the optimal criterion of δΦs ≈ δΦe is still valid. Generally, two preamplifiers, op-amp and PCBT, are suitable for different SQUID’s 𝛽 c [7]. In brief, the above works conformably lead us to aim at a nominal 𝛽 c ≈ √ 3 for weakly damped SQUIDs. In our experiments, a minimum δΦsys ≈ 4 μΦ0 / Hz for Ls = 350 √ pH at 4.2 K was reached when a preamplifier (op-amp) with V n ≈ 1 nV/ Hz was utilized. Furthermore, the system noise δΦsys is independent of the SQUID bias mode. In fact, the advantages of the weakly damped SQUID system concept are not only its “ease of making” but also, more importantly, its parameter robustness; i.e. values of 𝛽 c within certain deviations may be tolerated for still reaching the minimum δΦsys in a DRS.

10.3 Statistics of SQUID Properties People often publish their best results of SQUID noise measurements to perform a scientific analysis. In practice, the averaged system noise δΦsys is more relevant, e.g. for setting up a multichannel SQUID system. In that case, a high yield of SQUID fabrication is required. However, it is difficult to find publications about the statistical yield of SQUID-chip production. In this subsection, we show that the weakly damped SQUID concept tolerates relatively large parameter deviations in statistics, thus enhancing the yield of SQUID fabrication. Furthermore, the feasibility of employing weakly damped SQUIDs in a multichannel SQUID system is demonstrated by measuring 101 magnetometer samples [3]. The measured average data of the 101 SQUID magnetometers and their deviations are illustrated in Figure 10.3 for the case of four different design values of RJ . Here, all SQUIDs were measured only in voltage bias mode. As RJ increases from 5 to 40 Ω in four steps, the mean value of the SQUID current swing I swing (i.e. I c,max –I c,min ) modulated by Φ0 /2 decreases from 4.6 to 2.9 μA (Figure 10.3a), which corresponds to the 𝜕I/𝜕Φ at the working point decreasing from 14 to 9 μA/Φ0 (the I(Φ) characteristics and the working point are not shown here), while the dynamic resistance Rd (mean value) increases from 6.5 to 44 Ω (Figure 10.3b). In these cases, the statistical maximum-to-minimum ratios of the two physical quantities I swing and Rd were rather large. For example, when RJ = 30 Ω, the ratios were approximately 2 for I swing and 3 for Rd . These ratios can be marked as the

143

10 Weakly Damped SQUID

7 50 Rd (Ω)

(22)*

4

(53)* (19)*

5

30 20

(7)*

2 0

Mean Minimum Maximum

40

5

3

(a)

60

Mean Minimum Maximum

6 Iswing (μA)

144

10

20 30 RJ (Ω)

10 0

40 (b)

0

5

10

20 30 RJ (Ω)

40

Figure 10.3 Statistical data of Iswing (a) and Rd (b) of 101 SQUIDs as a function of the designed RJ [3]. Here, the SQUID’s Iswing is measured in voltage bias mode, while Rd is yielded from I–V characteristics at Φ ≈ (2n + 1)Φ0 /4. Note that (..)* in (a) denotes the sample number. Source: Liu et al. 2012 [3]. Reproduced with permission of American Institute of Physics.

quality of fabrication processes, where the uncertainties depend on the fabrication equipment, the processes, the clean-room class, etc. Note that the niobium SQUID magnetometers employed in these measurements were produced under very simple conditions. Nevertheless, the large deviations of the SQUID parameters are beneficial for understanding the parameter tolerance range. In the following, we focus on the behaviors of 53 weakly damped SQUID magnetometers having a design value RJ = 30 Ω, where SQUIDs with the nominal 𝛽 c ≈ 3 may occupy a large proportion [8]. In fact, all 53 SQUIDs could be operated stably in both bias modes without oscillation (hysteresis). Figure 10.4a illustrates the distribution of current swing I swing modulated by Φ0 /2. Here, I swing varies seriously from 2.5 to 4.8 μA, and the mean value of I swing is 3.7 μA because most SQUIDs have an I swing value located between 3.5 and 4 μA. Here, an I swing of 3.7 μA corresponds to a 𝜕I/𝜕Φ of 11 μA/Φ0 at the working point (not shown); Figure 10.4b shows two distributions: the values of the actually fabricated RJ (with the filled circles) were obtained by measuring RJ /2 in region III of Figure 3.3, where the SQUID’s I–V characteristics were no longer modulated by Φ. In other words, the RJ values shown in Figure 10.4b were yielded at a large I b , which is much larger than the SQUID’s I c at Φ = nΦ0 . However, the dynamic resistances Rd were obtained from the I–V characteristics at the working point W (e.g. Φ = Φ0 /4). Here, the two mean values are denoted by 23 Ω (RJ ) and 35 Ω (Rd ). The deviation range of RJ , which varies from 20 to 60 Ω, is obviously smaller than that of Rd because Rd depends on both RJ and I c , as demonstrated in Figure 2.4. Most measured RJ values are located between 20 and 23 Ω, but most of the Rd values are located between 28 and 38 Ω. In brief, Rd is always larger than RJ , as discussed in Chapter 2. In the interval between sample #25 and #40, the RJ spread is very narrow, but the spread in Rd is wide. Therefore, in principle, no correlation exists between these two resistance distributions. Finally, the distribution in the product (𝜕I/𝜕Φ) × Rd = 𝜕V /𝜕Φ, the flux-to-voltage transfer coefficient, is shown in Figure 10.4c, where 𝜕V /𝜕Φ spreads from 150 to 860 μV/Φ0 . In the

10.3 Statistics of SQUID Properties

Iswing (μA)

5 4

(a)

2

3

60 Rd and RJ (Ω)

Figure 10.4 Statistical characterization of 53 SQUIDs with nominal RJ = 30 Ω: the distribution of Iswing (a), RJ (marked by filled circles), Rd at the working point (b), and 𝜕V/𝜕Φ (c) [8]. The dashed lines indicate the mean values, and the lower dashed line in (b) indicates the mean value of RJ . Source: Liu et al. 2013 [8]. Reproduced with permission of IOP Publishing.

45 30

(b) 15

∂V/∂Φ (mV/Φ0)

1.0

0.5

0.0 0 (c)

10

20

30

40

50

SQUID No. (1–53)

statistics, approximately 30% of the SQUIDs with 𝜕V /𝜕Φ < 300 μV/Φ0 do not belong in the category of weakly damped SQUIDs, while 15% of the SQUIDs with 𝜕V /𝜕Φ > 500 μV/Φ0 may be called ultraweakly damped SQUIDs, where the intrinsic noise δΦs is higher than δΦe . Fortunately, over 50% of SQUIDs with 300 μV/Φ0 < 𝜕V /𝜕Φ < 500 μV/Φ0 (i.e. near the mean values of 380 μV/Φ0 marked with a dashed line in Figure 10.4c) indeed meet the requirements of weakly damped SQUIDs, e.g. 𝛽 c ≈ 3. Therefore, from the √ perspective of statistical meaning, a readout electronics noise δΦe of 2.5 Φ0 / Hz could be achieved at 𝜕V /𝜕Φ = 380 μV/Φ0 when using an op-amp as a preamplifier. A SQUID magnetometric system is characterized by its field sensitivity δBsys , which is a product of δΦsys × (𝜕B/𝜕Φ). Both factors can be separately determined. Once the value of (𝜕B/𝜕Φ) = 1.5 nT/Φ0 is determined by the layout on the 5 × 5 mm2 chip, the SQUID system noise δΦsys becomes our ultimate focus. Figure 10.5 shows the statistical value of the square of δBsys using the 53 SQUID magnetometers at a nominal RJ = 30 Ω, where a DRS with a commercial op-amp as a preamplifier is utilized in the noise measurements. Depending on the field sensitivity δBsys in the white noise region, the mag√ netometers were divided into five classes with a step of 1 fT/ Hz: (I) δBsys ≤ √ √ √ √ √ 6 fT/ Hz; (II) 6 fT/ Hz < δBsys ≤ 7 fT/ Hz; (III) 7 fT/ Hz < δBsys ≤ 8 fT/ Hz;

145

10 Weakly Damped SQUID

25

Number of SQUIDs

20 15 10 5 0 ≤6

6–7 7–8 8–9 Field sensitivity (fT/√Hz)

>9

≤6

6–7 7–8 8–9 Field sensitivity (fT/√Hz)

>9

(a) 14

Figure 10.5 (a) δBsys classification of the 53 voltage-biased SQUID magnetometers. (b) δBsys improvement of 23 SQUID magnetometers utilizing a PFC [8]. Note that the planar SQUID magnetometers with a field-to-flux transfer coefficient of 1.5 nT/Φ0 were integrated on 5 × 5 mm2 chips; i.e. the √ measured δΦsys of 4 μΦ0 / Hz corresponds to a field √ sensitivity δBsys ≈ 6 fT/ Hz. Source: Liu et al. 2013 [8]. Reproduced with permission of IOP Publishing.

12 Number of SQUIDs

146

10 8 6 4 2 0

(b)

√ √ √ (IV) 8 fT/ Hz < δBsys ≤ 9 fT/ Hz, and (V) δBsys > 9 fT/ Hz. Statistically, the √ δBsys values of approximately 85% of the SQUIDs were below 8 fT/ Hz, √ √ 51% were below 7 fT/ Hz (or δΦsys < 4.5 μΦ0 / Hz), and 8% were below √ √ √ √ 6 fT/ Hz (δΦsys ≤ 4 μΦ0 / Hz). If δBsys < 7 fT/ Hz (i.e. δΦsys < 4.5 μΦ0 / Hz) is taken as the selection criterion, more than 50% of the 53 measured SQUID magnetometers were directly qualified. We randomly took 23 SQUID magnetometers from the above five classes and connected them with the PFC that was already integrated on the magnetometer chip (its equivalent circuit is shown in the inset of Figure 10.1). In Chapter 8, it has been shown in detail that PFC can enhance (𝜕V /𝜕Φ)PFC in both bias modes. Figure 10.5b illustrates the δBsys improvement using the PFC; e.g. two of the seven SQUIDs that originally belonged to class III in (a) were now grouped into class I, and two others were grouped into class II, while the class of the remaining three did not change. Indeed, in approximately 65% of all cases investigated, the SQUID noise was improved: the δBsys of all SQUIDs showed a noise lower than

10.4 Single Chip Readout Electronics (SCRE)

√ √ 8 fT/ Hz; when using the PFC, 87% of these were below 7 fT/ Hz, and 61% were √ below 6 fT/ Hz. However, this improvement arising from using the PFC is not our original intention of weakly damped SQUIDs, because the weakly damped SQUID system does not aim for extreme system noise. We want to reiterate only that there is a possibility of improving δBsys , as further shown in Figure 10.5b. Here, the correct method is to improve the SQUID’s fabrication process, thereby increasing the yield of highly sensitive magnetometers with the concept √ of weakly damped SQUIDs. Occasionally, a difference of approximately 2 fT/ Hz between a weakly damped SQUID magnetometer and a strongly damped one with a PFC does not play a big role. In this subsection, we describe the behaviors of a weakly damped SQUID, namely, high Rd and a large 𝜕V /𝜕Φ, referring to the original δBsys (δΦsys ) classification of the 53 SQUID magnetometers shown schematically in Figure 10.5a. With the statistics, we further proved that the nominal 𝛽 c plays a role in regulating both δΦs and δΦe . For example, 𝜕Φ2sys = 𝜕Φ2e + 𝜕Φ2s = √ 16 pΦ20 ∕Hz (i.e. δΦsys = 4 μΦ0 / Hz) at a SQUID inductance of Ls = 350 pH can contain different combinations of both 𝜕Φ2e and 𝜕Φ2s , e.g. 42 ≈ 2.82 + 2.82 ≈ 3.52 + 22 ≈ 22 + 3.52 ≈ 42 + 12 ≈ 12 + 42 , in the range of 2 < 𝛽 c < 4. Namely, to find a proper 𝛽 c , say 𝛽 c ≈ 3, is the key point of the weakly damped SQUID system. In brief, the statistical results experimentally prove that the weakly damped SQUIDs have the wide parameter tolerance that we need to establish a multichannel system. Here, the statistical product 𝜕V /𝜕Φ = (𝜕I/𝜕Φ) × Rd of a SQUID with Ls ≈ 350 pH is calculated to be approximately 380 μV/Φ0 at RJ ≈ 30 Ω, √ thus leading to δΦsys ≈ 4 μΦ0 / Hz, corresponding to a field sensitivity of √ δBsys ≈ 6 fT/ Hz. Indeed, such noise levels are acceptable for a simple SQUID system in most applications.

10.4 Single Chip Readout Electronics (SCRE) We have discussed in detail the behavior of weakly damped SQUIDs from a statistical perspective and noted that this concept is well suited for setting up a multichannel SQUID magnetometric system or a simple system in many applications. For this purpose, simple readout electronics is also required. In 2014, Chang et al. developed dc SQUID readout electronics in FLL mode without a conventional integrator and with only one op-amp, which was thus called “single chip readout electronics” (SCRE) [9]. A weakly damped niobium-SQUID magnetometer and SCRE resulted in the simplest SQUID system. We characterized such a system and demonstrated its applicability to magnetocardiography (MCG) and the transient electromagnetic (TEM) method in geophysical measurements. SCRE not only simplifies the readout scheme but also demonstrates the system stability. With its equivalent circuit, the gain factor of the readout electronics in the FLL operation mode of the SQUID system could be intuitively achieved, so that one can easily analyze the features of the electronics, i.e. the dynamic range,

147

148

10 Weakly Damped SQUID

the bandwidth, and the slew rate. The difference between SCRE and a conventional DRS (preamplifier + amplifier + integrator) is also discussed. 10.4.1

Principle of SCRE and Its Performance

SCRE is probably the simplest example of a DRS to date. The schematic diagram in Figure 10.6 depicts its principle: a current-biased SQUID operated in FLL mode. The head stage (a weakly damped SQUID) is cooled to, e.g. liquid helium temperature (4.2 K), and the op-amp (e.g. AD797) is located at room temperature (RT). The SQUID voltage signal, V s , at the noninverting terminal of the op-amp is generated by the bias current I b flowing though the SQUID dynamic resistance; i.e. V s equals the product of I b and Rd at the working point. To maintain stable operation of SCRE, the voltage V w at the op-amp’s inverting terminal must be adjusted to equal V s . In other words, V w determines the SQUID working point when I b is already selected, as discussed in Chapter 4. Any voltage change, ΔV s , across the SQUID caused by external flux will be compensated by an opposite feedback flux, which is the product of V out /Rf (namely, the feedback current I f ) and Mf . Clearly, the SCRE scheme with an open-loop amplifier, which replaces the amplifier and the integrator of conventional readout electronics, has a sufficiently high gain to fulfill the requirements of FLL. To test the performance of the SCRE scheme, we employed a weakly damped planar niobium-SQUID magnetometer. The SQUID exhibits a large 𝜕V/𝜕Φ ≈ 380 μV/Φ0 and a large Rd ≈ 40 Ω obtained from the I–V characteristics at the working point, i.e. Φ = (2n √ + 1)Φ0 /4, thus reducing the preamplifier noise contribution δΦe to 500 kHz. To demonstrate the large slew rate of the SCRE, a real-time voltage response to a square wave flux with the amplitude of 0.3 Φ0 was recorded in the inset of Figure 10.7. In the conventional readout scheme with an integrator as the output stage, instead of the amplifier’s intrinsic gain–bandwidth product, the system bandwidth is generally limited by the RC time constant of the integrator. One could also introduce a two-pole integrator to obtain both a high slew rate of up to 30 Φ0 /μs at low frequency and a large bandwidth of 5 MHz [10]. Generally, the SQUID system bandwidth and its slew rate are influenced by the nonlinear I f -to-Φ transfer function via Mf . In principle, SQUID systems, e.g. with different sample-holder lengths and different kinds of feedback-coil Lf , have their own transmission characteristics, so the discussion of their effects is omitted because of their non-universality. Table 10.1 Slew rates at different frequencies. Test frequency (kHz)

Slew rate (𝚽0 /𝛍s)

1

0.8

10

0.9

100

1

500

2

1000

2

151

152

10 Weakly Damped SQUID

In brief, the system bandwidth and system slew rate are decided by both the flux feedback character and the electronic features of SCRE. Nevertheless, the latter can be easily analyzed with the equivalent circuit, as shown in Figure 10.8b. 10.4.3 Differences Between the Conventional Version of Readout Electronics with an Integrator and SCRE SCRE is a simplified version of conventional SQUID readout electronics. We now discuss two differences between SCRE and conventional readout electronics consisting of amplifiers (preamplifier + main amplifier) and an integrator. At first, we take the integrator in FLL mode as an example. If the op-amp output voltage V out ≠ 0, a steady-state error V error = V out /Gopen always exists at its input. Here, V error describes the influence of the output voltage of the integrator at its input; it can be avoided when the open-loop gain of the integrator approaches infinity. Furthermore, V error plays no role, as the input signal of the integrator is large enough. In fact, the amplifiers in conventional readout electronics have two functions: to separate the SQUID from the integrator (i) and to provide a large signal for the integrator (ii). In short, V out does not influence the SQUID working point at the input terminal of the preamplifier. This is the reason why people never take V error into account in conventional readout electronics. In SCRE, the op-amp acts as both an amplifier and an integrator. The open-loop gain is approximately 2 × 107 (see datasheet of AD797), and the steady state error V error at the input terminal of SCRE may reach into the microvolts range when V out = 10 V, thus influencing the SQUID working point. However, in the case of small signals, we did not observe any noise change when varying Rf from 1.44 kΩ to 1 MΩ (see Figure 10.7). We also did not observe any negative effects on the slew rate or on the system distortion, which is characterized by a measured total harmonic distortion (THD) of 5 × 10−5 at the output signal for an applied sinusoidal signal flux of 30 Hz. Generally, a THD of 1 × 10−6 can be reached with conventional readout electronics. The difference of a factor of two in THD may be caused by V error . Secondly, the dynamic range of the SQUID readout electronics depends on the feedback resistance Rf , while the required gain G in SCRE is automatically adjusted. In fact, the equivalent circuit in Figure 10.8 also suits conventional readout electronics. Here, one should understand that the required total gain G is a product of G1 , G2 , and G3 , which represent the gains of the preamplifier, amplifier, and integrator, respectively. However, the gain G1 × G2 of both amplifiers (preamplifier + amplifier) located before the integrator is usually set to a fixed gain value in the range of 103 –104 , which is occasionally much higher than the required total gain G of, e.g. approximately 40 at Rf = 1.44 kΩ in the above analyses. In this case, the integrator must be operated with a gain G3 of ≪1 to fulfill the required total gain G ≈ 40. This counterproductive action, first setting a high G1,2 and then reducing G3 , probably leads to instability at the integrator. Usually, one increases the RC integration time constant to improve the stability. Therefore, some commercial readout electronics are designed with an adjustable gain (G1 × G2 ) to solve this problem (http://ez-squid.de/ Home.htm). In our SCRE, this problem does not exist, so the stability is well maintained. Note that a large dynamic range, bandwidth, and slew rate are

10.4 Single Chip Readout Electronics (SCRE)

not always required for practical applications, whereas the system stability and the system noise are always important with respect to the performance of the SQUID system. 10.4.4

Two Applications of SCRE

We performed MCG and the TEM measurements to demonstrate the applicability of SCRE to the fields of biomagnetism and geomagnetic exploration. (1) Biomagnetism measurement: To suppress external disturbances, an MCG measurement was performed in a magnetically shielded room (MSR) by using an electronic gradiometer consisting of two SCRE systems with a baseline length of 7 cm. At Rf = 1 MΩ, the cardiogenic QRS wave of an adult human subject reached approximately 0.5 Vpp at the output of the SCRE, corresponding to a field change at the sensing SQUID of approximately 70 pTPP (see Figure 10.9). Because of the large voltage output of the SCRE signals, an A/D converter can be directly connected without any additional amplifier. Here, a digital low-pass filter with a cutoff frequency of 100 Hz and a 50 Hz notch filter were used for subsequent data processing. (2) Geomagnetic exploration: At Hengsha island (Shanghai), we conducted TEM explorations using SCRE with Rf = 1.44 kΩ to reach a large dynamic range, a wide bandwidth, and a high slew rate, as discussed in Section 10.4.2. A 100 × 100 m2 wire loop with a bipolar pulse current of approximately 14 A was applied as the transmitter on the ground to produce TEM signals, which were detected by the SQUID magnetometer (SCRE) located at the center of the loop. Figure 10.10a shows the real-time TEM signals containing primary and secondary fields measured by SCRE. Here, a transient signal with amplitudes of approximately ±110 nT (primary field) was recorded without problem, although this experiment requires a large dynamic range and high slew rate, as mentioned above. Figure 10.10b shows the enlarged transient signal in the time interval A∼B of (Figure 10.10a). Its response signal (secondary) could still be observed when it decayed by over 3.5 orders of magnitude at 50 ms after the primary field was switched off. In brief, SCRE is particularly well suited to read out weakly damped SQUIDs. Generally, SCRE benefits the simplicity, stability, and user-friendliness of the SQUID system. It also intrinsically yields a large bandwidth and slew rate. 40 20 0 –20

0.2 0.0 –0.2 –0.4 0.0

–40 0.5

1.0

1.5

2.0 2.5 Time (s)

3.0

3.5

Mag. field (pT)

Out voltage (V)

0.4

4.0

Figure 10.9 MCG signal of an adult male subject recorded in an MSR.

153

10 Weakly Damped SQUID

150

100 Amplitude (nT)

A

100 Amplitude (nT)

154

50 B

0 –50

(a)

1 0.1

0.01

–100 –150

10

0

2

4 Time (s)

6

1E-3 0.1 (b)

1

10 Time (ms)

100

Figure 10.10 Schematic real-time TEM signal (a) and its enlargement (b) recorded for the interval A ∼ B. The black curve in (b) is the measured real-time decay curve, and the other curve (gray) is the smoothed curve.

10.5 Suggestions for the DRS From the discussion of DRS in Chapters 8–10, we learn that there are two kinds of preamplifiers (op-amp and PCBT), three kinds of SQUIDs with different damping degrees, and two feedback circuitries (PFC and SFC). According to different needs in applications, people can build different systems with three typical combinations: Combination #I to build a very sensitive system: For strongly damped SQUIDs with low δΦs and small 𝜕V /𝜕Φ (or (𝜕I/𝜕Φ) × Rd ), the suppression of δΦe below δΦs (i.e. δΦe < δΦs ) may become possible only by means of the PFC and SFC. Using PCBT preamplifier, we successfully observed the intrinsic SQUID noise √ δΦs , i.e. δΦsys ≈ δΦs = 1.2 μΦ0 / Hz at Ls = 200 pH, as demonstrated in Figure 8.17. For our standard SQUID with Ls = 350 pH, we used√a commercial op-amp (AD797) and PFC to yield the measured δΦsys ≤ 3 μΦ0 / Hz in the white noise region, as shown by curve IV in Figure 10.1. However, understanding, designing, and evaluating the feedback circuitry of PFC requires professional experience with the SQUID system technique. The merits and drawbacks of PFC with adjusting elements have also been discussed at the beginning of this chapter. In fact, the required noise suppression may not be reached with PFC. Combination #II for a system with medium sensitivity: There are two possibilities: (1) the combination of a PCBT and an intermediately damped SQUID at √ 𝛽 c ≈ 1 may reach a system noise δΦsys ≤ 3 μΦ0 / Hz for Ls = 350 pH. Therefore, this combination is a good compromise between strongly and weakly damped SQUIDs. Here, the SFC is only an option to improve system noise performance at low frequencies. However, a PCBT amplifier is not commercially available yet; therefore, people still have to make their own; (2) using the same SQUID with PFC and an op-amp (preamplifier), the preamplifiers present √ √ the same equivalent noise of Vn∗ ≈ 0.33 nV∕ Hz (>2 Hz) and In∗ < 1 pA∕ Hz (white noise), when PFC suppresses δΦV n with a factor of 3. We believe that both setups will exhibit the same system noise δΦsys .

References

Combination #III for a simple system: The measured system noise δΦsys ≈ 4 μΦ0 / √ Hz of the weakly damped SQUID with Ls = 350 pH and 𝛽 c ≈ 3 exhibits a δΦsys increase of approximately 25–30%, as compared to Combination #I. This result was applicable to our statistics in Section 10.3. However, this noise level is sufficient to meet the requirements of many practical applications. Owing to the convenience of a simple readout (e.g. SCRE [9]) and a large SQUID parameter tolerance, the weakly damped SQUID scheme is a very promising choice to construct a multichannel SQUID system. Here, no professionals are required for the operation and routine services of the SQUID system. In fact, the principles of combination #II (1) and combination #III are the same, i.e. to reach the minimum δΦsys at δΦe ≈ δΦs . Nevertheless, the two noise components, δΦe and δΦs , of #II (1) are smaller than those of #III. For combination #II, we suggest that the designed noise suppression factors of δΦV n (for op-amp with PFC) or δΦIn (for PCBT with SFC) do not exceed 4. Our basic consideration for DRS is that the three different damped SQUIDs should be suited for their own preamplifier and feedback circuits. Here, we note again that not all systems pursue extremely low noise. Overall, the paradigm that the requirement of δΦe < δΦs must be fulfilled for a readout scheme has been rebutted. Indeed, the strongly damped SQUID system (combination #I) is recommended only for special applications, e.g. ultralow field nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) [11]. Recently, using sub-micrometer sized cross-type Josephson tunnel junctions in niobium technology [12, 13] and DRS, it was shown that SQUID magnetometers can yield a small parameter 𝛽 c and large √ 𝜕V /𝜕Φ at W, thus obtaining low magnetic field noise, as low as 0.3 fT/ Hz [14]. As a practical approach, the intermediately damped (combination #II) and weakly damped SQUIDs (combination #III) use the nominal 𝛽 c deploying the distribution of δΦe and δΦs to reach an acceptable level of system noise δΦsys . Thus, users can easily realize such systems by themselves in applications. Furthermore, simple SQUID systems with a large parameter tolerance are always popular. In fact, in biomagnetism, e.g. MCG and MEG, Qiu et al. demonstrated that weakly damped SQUIDs are beneficial to multichannel systems [15, 16].

References 1 Elekta Neuromag Oy., PO Box 68, FIN-00511 Helsinki, Finland,

https://www.elekta.com/ (cited material from August 2005). 2 Kiviranta, M. and Seppä, H. (1995). DC-SQUID electronics based on the

noise cancellation scheme. IEEE Transactions on Applied Superconductivity 5 (2): 2146–2148. 3 Liu, C., Zhang, Y., Mück, M. et al. (2012). An insight into voltage-biased superconducting quantum interference devices. Applied Physics Letters 101 (22): 222602.

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4 Zeng, J., Zhang, Y., Mück, M. et al. (2013). High intrinsic noise and absence

5

6

7

8

9

10

11

12

13

14

15 16

of hysteresis in superconducting quantum interference devices with large Steward-McCumber parameter. Applied Physics Letters 103 (4): 042601. Wang, H., Wang, Y.L., Kong, X.Y. et al. (2016). Dependence of SQUID intrinsic flux noise on Stewart-McCumber parameter beta(c) of Josephson Junction. IEEE Transactions on Applied Superconductivity 26 (5): 1601705. Zeng, J., Zhang, Y., Qiu, Y. et al. (2014). Superconducting quantum interference devices with different damped junctions operated in directly coupled current- and voltage-bias modes. Chinese Physics B 23 (11): 118501. Wang, H., Chen, H., Kong, X.Y. et al. (2017). Study on noise matching between SQUID sensor and its readout electronics. IEEE Transactions on Applied Superconductivity 27 (4): 1601104. Liu, C., Zhang, Y., Mück, M. et al. (2013). Statistical characterization of voltage-biased SQUIDs with weakly damped junctions. Superconductor Science and Technology 26 (6): 065002. Chang, K., Zhang, Y., Wang, Y.L. et al. (2014). A simple SQUID system with one operational amplifier as readout electronics. Superconductor Science and Technology 27 (11): 115004. Drung, D., Matz, H., and Koch, H. (1995). A 5-MHz bandwidth SQUID magnetometer with additional positive feedback. Review of Scientific Instruments 66 (4): 3008–3015. Storm, J.H., Hommen, P., Hofner, N., and Korber, R. (2019). Detection of body noise with an ultra-sensitive SQUID system. Measurement Science and Technology 30 (12): 125103. Anders, S., Schmelz, M., Fritzsch, L. et al. (2009). Sub-micrometer-sized, cross-type Nb–AlOx –Nb tunnel junctions with low parasitic capacitance. Superconductor Science and Technology 22 (6): 064012. Schmelz, M., Stolz, R., Zakosarenko, V. et al. (2012). Sub-fT/Hz(1/2) resolution and field-stable SQUID magnetometer based on low parasitic capacitance sub-micrometer cross-type Josephson tunnel junctions. Physica C: Superconductivity and Its Applications 482: 27–32. Schmelz, M., Stolz, R., Zakosarenko, V. et al. (2011). Field-stable SQUID magnetometer with sub-fT Hz(−1/2) resolution based on sub-micrometer cross-type Josephson tunnel junctions. Superconductor Science and Technology 24 (6): 065009. Qiu, Y., Liu, C., Zhang, S.L. et al. (2014). A SQUID gradiometer module with large junction shunt resistors. Chinese Physics B 23 (8): 088503. Qiu, Y., Li, H., Zhang, S.L. et al. (2015). Low-T-c direct current superconducting quantum interference device magnetometer-based 36-channel magnetocardiography system in a magnetically shielded room. Chinese Physics B 24 (7): 078501.

157

11 Two-Stage and Double Relaxation Oscillation Readout Schemes In the previous chapters, we discussed the most often used dc SQUID readout schemes, e.g. flux modulation scheme (FMS), flux feedback circuitries (parallel feedback circuit [PFC] and SFC), and direct readout scheme (DRS) with weakly damped SQUID, and analyzed their merits and shortcomings. In this chapter, we will introduce two special dc SQUID readout schemes, i.e. a two-stage scheme and a double relaxation oscillation (D-ROS), suitable for observation of the SQUID intrinsic noise δΦs . Here, the two-stage scheme is employed for reading the small δΦs of a strongly damped SQUID (𝛽 c ≪ 1), while D-ROS is for the un-shunted SQUIDs (𝛽 c → ∞) with large δΦs . In these readout schemes, the conventional requirement for the readout electronics is fulfilled; i.e. the δΦe is suppressed below δΦs . In 1984, the basic idea of a two-stage scheme, two SQUIDs in one system, was presented by Wellstood et al. [1]. The first (sensing) SQUID operating in voltage bias mode is an ordinary flux sensor required by all SQUID systems. A superconductive coil is inserted into the voltage-biased (sensing) SQUID circuit and is inductively coupled to the second (reading) SQUID, where the coil and the (sensing) SQUID act as an ammeter A, as shown in Figure 3.6a. Because the SQUID ammeter A is very sensitive, the measured system noise δΦsys is usually dominated by the intrinsic δΦs of the sensing SQUID, i.e. δΦsys ≈ δΦs > δΦe . In 1968, the first observation of a relaxation oscillation (ROS) in a Josephson junction was reported [2]. Approximately 20 years later, Gudoshnikov et al. first presented ROS and D-ROS readout schemes with un-shunted SQUIDs (RJ → ∞) in current bias mode, by which the flux is translated to a dc voltage [3]. Here, the initial motivation was to achieve a high flux-to-voltage transfer coefficient 𝜕V /𝜕Φ, e.g. in the 10 mV/Φ0 region, thus simplifying the readout electronics and improving the slew rate. As a consequence of the D-ROS scheme, the measured system noise δΦsys is dominated by the intrinsic SQUID noise δΦs . However, δΦs is very high in the D-ROS scheme due to the parameter 𝛽 c → ∞ of an un-shunted SQUID. At first glance, this consequence looks strange because nobody wants to employ a SQUID with a large δΦs . In fact, the system noise δΦsys of D-ROS is still acceptable for recording signals of, e.g. human biomagnetism. Furthermore, the large δΦs (≈δΦsys ) improves the system robustness. For example, high-resistance wires can be taken to connect the SQUIDs to room temperature (RT) electronics, thus reducing the evaporation rate of liquid helium in low-T c systems. There, SQUID Readout Electronics and Magnetometric Systems for Practical Applications, First Edition. Yi Zhang, Hui Dong, Hans-Joachim Krause, Guofeng Zhang, and Xiaoming Xie. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

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11 Two-Stage and Double Relaxation Oscillation Readout Schemes

the additional thermal noise of the wires equivalent to a small part of δΦsys plays a small role. In brief, the readout electronics noise δΦe can be neglected, and the contributions from other noise sources are relatively small due to the large 𝜕V /𝜕Φ ≥ 1 mV/Φ0 in the D-ROS scheme. Therefore, the major requirements of a multichannel system may be fulfilled. For magnetocardiography (MCG) and magnetoencephalography (MEG), commercial multichannel SQUID systems are made using the D-ROS scheme, e.g. the systems manufactured by KRISS (www.kriss.re.kr.) in R.O. Korea [4]. In brief, the two-stage scheme is employed for research on the SQUID intrinsic noise, while D-ROS is well suited for multichannel SQUID systems for human biomagnetism measurements.

11.1 Two-Stage Scheme Actually, the two-stage readout scheme consists of a voltage-biased sensing SQUID and an ammeter with the reading SQUID. Here, the latter is the so-called SQUID current sensor (ammeter), which has already been studied for multiple purposes for many√ years [5–9]. Such SQUID ammeters can reach a sensitivity in the region of 3 fA/ Hz [10]. In the two-stage scheme, the ammeter in Figure 3.6a is realized with the help of a superconducting input coil La coupled to a reading SQUID, where the coil La is inserted into the voltage-biased (sensing) SQUID circuit. Figure 11.1 shows the evolution from the principle (left) to a real circuit (right). Unlike a conventional ammeter with two terminals, the SQUID ammeter “A” (inside of the dashed square) has four terminals. Here, two terminals are for La, and the other two, A1 and A2 , are the connecting electrodes of the reading SQUID. In fact, the SQUID ammeter “A” is an ideal ammeter because its internal resistance approaches zero. We assume that the voltage-biased sensing SQUID has an intrinsic noise (δΦs )Sensing and a transfer coefficient (𝜕I/𝜕Φ)Sensing . The approach of the two-stage scheme is to read out (δΦs )Sensing with the SQUID ammeter. In practice, the SQUID ammeter with its readout electronics at RT is an ordinary SQUID system, and the system noise of the reading SQUID can be regarded as the readout noise δΦe for the sensing SQUID. Here, we try to achieve δΦe < (δΦs )Sensing . In fact, the real trick of the two-stage scheme is the “flux amplifier,” where the reading SQUID measures only the “amplified flux.” In Figure 11.1(left), the loop current ΔI = ΔΦ × (𝜕I/𝜕Φ)Sensing is changed by the measured flux ΔΦ. There, ΔI flows through La , thus generating a reading flux, ΔI × Ma , for the reading SQUID, where Ma is the mutual inductance between La and the reading SQUID, as shown in Figure 11.1(right). So, the flux of ΔI × Ma at the reading SQUID is amplified by a factor of [(𝜕I/𝜕Φ)Sensing × Ma ], denoted by GF . In fact, ΔI × Ma includes the information about not only the measured flux ΔΦ but also the intrinsic noise (δΦs )Sensing from the sensing SQUID. When GF > 1, the two-stage scheme realizes readout electronics noise δΦe (i.e. the reading SQUID system noise) below the product of GF × (δΦs )Sensing . If ΔΦ = Φ0 is applied at the sensing SQUID, we assume that the voltage-biased sensing SQUID with its (𝜕I/𝜕Φ)Sensing generates a reading flux of ΔI × Ma = 10 Φ0

11.1 Two-Stage Scheme

ΔΦ

ΔΦ

A

Vb ΔI

Sensing Ma

A1

La

Vb

Reading

ΔI

A2 ΔI × Ma

A

Figure 11.1 In principle, the voltage-biased circuit needs an ammeter “A” to read out the voltage-biased sensing SQUID signal, ΔI (left part); the basic idea of the cascaded “two-stage” concept is sketched (right part). Here, the current change ΔI is read out by a (reading) SQUID ammeter, consisting of a superconducting input coil La and a reading SQUID, where both are coupled with a mutual inductance Ma . However, no mutual inductance between the sensing SQUID and La exists.

at the reading SQUID (i.e. GF = 10) so that δΦe (the reading SQUID system noise with semiconducting electronics at RT) is equivalently reduced by a factor of 10 for measuring (δΦs )Sensing . Consequently, the total system noise δΦsys of the two-stage scheme is close to (δΦs )Sensing , even though the sensing SQUID is a strongly damped SQUID with very small (δΦs )Sensing . An effective way of obtaining a large GF of the “flux amplifier” is to increase the mutual inductance Ma , e.g. by integrating the coil La on the washer of the reading SQUID. Note that a large Ma leads to additional problems; e.g. rf resonances appear, thus influencing dc SQUID characteristics and then limiting the sensitivity of the two-stage scheme, as discussed in Refs. [11, 12]. Therefore, the GF of the “flux amplifier” is mostly designed to be approximately 5–10. In a SQUID magnetometric system, the periodically flux-changed SQUID signal should be linearized with the flux-locked loop (FLL) circuit, as discussed in Chapter 4. In the two-stage scheme, there are two SQUIDs; i.e. the reading SQUID ammeter reads out the I(Φ) characteristics of voltage-biased sensing SQUID. Nevertheless, the reading SQUID has two choices to select its bias modes. Therefore, the cascaded system leads to an interesting question: how does the FLL work in the two-SQUID system? Of course, the selection of the working (i.e. locking) points for the two SQUIDs are also involved in this question. In fact, three typical concepts to solve the problem have been proposed, as shown in Figure 11.2. In 1991, V. Foglietti et al. reported “a double dc-SQUID device for flux locked loop operation” with two FLL operations (see Figure 11.2a) [13]. Here, the current-biased reading SQUID has its own independent system and is an ordinary SQUID system using a DRS, where a preamplifier “P1” and an integrator “I1” are employed and the flux feedback loop is closed with the help of the feedback circuit of Rf,r and Lf,r (Mf,r ). For a DRS, the adjustments of I b and V w are described in detail in Chapter 5. Here, the four terminal SQUID ammeter in Figure 11.1 is independent and complete; i.e. the reading SQUID system acts as a current(ΔI)-to-voltage(ΔV out ) converter. Consequently, the output voltage of the SQUID ammeter, V out , shows a linear relationship with the loop current ΔI generated by the voltage-biased sensing SQUID. Note that the system noise Φe of the reading SQUID does not differ from that of a common SQUID system

159

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11 Two-Stage and Double Relaxation Oscillation Readout Schemes

Lf,s –

Vw

Ma

+

I1

P1

Ib

La

Vb

Vc

Rf,s

Vm

– +

If

Vout

I2

P2

Rf,r

ΔI Lf,r

(a)

Lf,s

Ma

Ib

C Vout

P

La

Vb

If

Rf,s

M

T

ΔI

G

Lf,r

(b)

I

RM Rf,s If

Lf,s

Ma

Ib

RM

C

P

La

Vb ΔI

T

M

I

Vout

G

(c)

Figure 11.2 Different concepts of FLL operation in the two-stage scheme: two FLLs (a) and one FLL only for the sensing SQUID (b) and (c). Here, (a) is operated in DRS; (b) and (c) use the FMS, which consists of a step-up transformer T, a preamplifier P, a multiplier M, an audio-frequency oscillator G, an integrator I, etc. The modulation flux is applied either at the reading SQUID (b) or at the sensing SQUID (c). However, all input circuits, that is, the voltage-biased sensing SQUID coupled to the reading SQUID via Ma , remain unchanged in all configurations. The circuits for adjusting the original dc flux states of both SQUIDs are omitted here.

using a DRS, as discussed in Chapter 5 for strongly damped SQUIDs and in Chapter 10 for weakly damped SQUIDs. Also, the original sensing SQUID’s characteristics, i.e. the periodical I(Φ), can be observed at V m (i.e. the integrator I1 output) while selecting a V b . To obtain a linear relationship between the measured flux ΔΦ at the sensing SQUID and the output V out of the integrator “I2,” the second flux feedback loop needs to be closed with the help of amplifier “P2,” integrator “I2,” Rf,s , and Lf,s (Mf,s ). Here, the compensation voltage V c applied at the inverse input of “I2” is used to select the working point W of the voltage-biased sensing SQUID, i.e. setting W at a quarter of Φ0 of the SQUID’s I(Φ) characteristics displayed at V m . Now, ΔV out represents the measured flux ΔΦ at the sensing SQUID in FLL. Here, the calibration of ΔΦ/ΔV out is omitted. The measured total system noise Φsys of the sensing SQUID at V out should be close to the sensing SQUID’s intrinsic noise δΦs with a small

11.1 Two-Stage Scheme

parameter 𝛽 c . In brief, each of the two SQUIDs used its own FLL in (Figure 11.2a), where the reading SQUID system is independent. However, the two FLLs in the concept of Figure 11.2a may not be necessary. In 1999, I. Jin et al. used only one FLL to complete the linear transformation from ΔΦ applied at the sensing SQUID to ΔV out of the reading SQUID system [14]. In the concept of Figure 11.2b and c, instead of the DRS in the concept of Figure 11.2a, the standard FMS was employed, where the modulation flux can be applied either at the reading SQUID (Figure 11.2b) or at the sensing SQUID (Figure 11.2c). The principle of the FMS has been exhaustively discussed in Chapter 7. We believe that there is no obvious difference between DRS and FMS readout in the two-stage scheme because the total system noise suppression is realized by GF of the “flux amplifier,” not by the readout scheme of the reading SQUID. The important aspect of the concept of Figure 11.2b and c with only one FLL is that the two SQUIDs and the readout electronics can be regarded as one complete system. Taking the concept of Figure 11.2b as an example, the readout electronics seem to be a normal FMS; nevertheless, the current I f generated by the output voltage V out of the integrator “I” does not feed back to the reading SQUID in the concept of "I1" (see Figure 11.2a), but it passes the coil Lf,s coupled to the sensing SQUID via the resistor Rf,s , thus directly closing the flux feedback loop of the system. In fact, the reading SQUID in (c) is operated in an open loop, where the coil La , the reading SQUID, and the RT electronics can be regarded as a current (ΔI)-to-voltage (ΔV out ) converter. Actually, the two-stage scheme is often employed to measure the intrinsic noise δΦs in magnetically well shielded environments, especially for strongly damped SQUIDs with 𝛽 c < 1. Some measurements yielded a record low system noise δΦsys , where both a low δΦs of the sensing SQUID and a low δΦe of the reading SQUID system noise are required. However, the use of the two-stage scheme has rarely been reported in practical SQUID applications. According to our analysis, there are three main reasons that discourage experimentalists from employing the two-stage scheme in applications: (1) it is a challenge to implement in practice that the reading SQUID is blind to the measured flux, i.e. the reading SQUID reads out only the flux of ΔI × Ma via input coil La , that is, the current ΔI generated by the voltage-biased sensing SQUID; (2) flux cross-talk between the two SQUIDs must be completely avoided; and (3) system operation requires four adjustments, e.g. in the concept of Figure 11.2a, the bias voltage V b for the sensing SQUID and I b for the reading SQUID as well as two SQUID working points (V w and V c ) for the respective FLLs need to be set. Occasionally, performing these adjustments is too cumbersome for users, especially when operating the system in an unshielded environment. Most likely, condition (3) can be resolved by training, but conditions (1) and (2) must be perfected by design. For the two-stage scheme, we suggest a practical SQUID magnetometric system, e.g. with a superconducting wire-wound pickup antenna. Overall, one FLL is operated at the sensing SQUID, as realized in concepts (b) and (c), while a very simple SQUID system including a weakly damped SQUID with single chip readout electronics (SCRE), as described in Chapter 10, is used as the reading SQUID system. Here, we recommend a layout for the two-stage scheme to

161

162

11 Two-Stage and Double Relaxation Oscillation Readout Schemes

Ma

Reading SQUID (Second-order gradiometer)

Ro

I

K1 II

+ Ib

Vb′ G1

Vout

–

Rb

Vb IM

SCRE

Vw

La

If

Lf

II I

Lin Sensing SQUID

K2

Rf

4.2 K

Ro

G2

Ro′ Vcom to pickup antenna

Figure 11.3 Suggestion of a practical SQUID magnetometric system with a two-stage scheme.

improve conditions (1) and (2) mentioned above. However, the operator of the system is required to visually set the working point one by one. Figure 11.3 shows the arrangement of a practical SQUID system with a two-stage readout scheme, where only one magnetic sensing element should be utilized for measuring the ambient field. The core idea of our suggestion is that the reading SQUID is constructed as a second-order gradiometer, which is insensitive to environmental flux changes. Actually, the second-order gradiometer is composed of two symmetrical first-order gradiometers, where the second-order gradient coil La with four loops is integrated on the washer of the reading SQUID (second-order gradiometer). Because the polarities are complementary for La and the second-order gradient SQUID, both form a sensitive SQUID ammeter. Two SQUIDs, a resistor Rb , and three coils (Lin , La , and Lf ) are integrated on the same chip (inside the dashed square). The two SQUIDs should be spaced with the largest possible distance to minimize the flux cross-talk. Furthermore, the chip of the two-stage structure is located in a shielding (niobium) tube at 4.2 K to further reduce the influence of environmental flux changes, so that the SQUID system with two-stage readout shown in Figure 11.3 may become practical. For the suggested SQUID magnetometric system with a two-stage scheme, the sensing SQUID is operated in standard voltage bias mode, while a flux feedback coil Lf and an input coil Lin are integrated on the washer of the sensing SQUID, where Lin connects the superconducting pickup antenna (not shown here). The voltage-biased circuit is realized as a parallel circuit, where one branch is a small resistor Rb ≤ 1 Ω regarded as the inner resistance of the bias voltage source, and the other branch is the sensing SQUID connecting the ammeter’s input coil La in series, inductively coupled to the reading SQUID. A current generated by Vb′ (or G1) is flowing through the parallel circuit via a large resistance Ro , thus leading to two consequences: V b ≈ 0 and V b ≠ 0, where V b is the voltage across this parallel circuit. At V b ≈ 0, the current of Vb′ ∕Ro is smaller than the SQUID’s critical current of 2I c ; i.e. the SQUID keeps its superconducting state, where most of

11.1 Two-Stage Scheme

the current Vb′ ∕Ro flows through the SQUID branch. In the other case, the bias voltage V b ≠ 0 is quasi-constant because Rb ≪ Rd of the sensing SQUID. Another important factor for practical applications is user-friendliness; thus, the adjustment (condition (3)) of our suggested two-stage scheme should be intuitive. The “adjustment sequence” is recommended as follows: (1) First, one sets the working point of the reading SQUID. The SCRE should be temporally switched to a linear amplifier with a gain of, e.g. 1000 (not shown here; see Ref. [15]), while K1 and K2 are switched to position “I.” Meanwhile, the circuit of Lf should be deactivated by setting V com and V G2 to zero. The oscillator G1 with a low frequency (e.g. 30 Hz) generates a modulation current I M via Ro (current-limiting resistor), which flows through the parallel circuit of Rb ≤ 1 Ω and La connecting the sensing SQUID in series (La + the sensing SQUID). When I M < 2I c (the critical current of the sensing SQUID), i.e. V b = 0, I M contributes to the modulation flux for the reading SQUID via Ma . Then, the adjustment of the reading SQUID can follow the usual procedure, i.e. to select the suitable bias current I b and the working point voltage V w in the SCRE scheme. At this point, the optimally (positively–negatively) symmetric V (Φ) curve of the reading SQUID appears at the output of the amplifier V out . Now, the adjustment of the reading SQUID is ready for measuring the sensing SQUID signal, but the FLL is not yet closed. (2) Then, one sets the working point of the voltage-biased sensing SQUID. The voltage of the oscillator (G2) is increased to apply a modulation flux for the sensing SQUID. Switch K1 is set to position “II,” where the modulation flux for the reading SQUID is switched off. Now, the reading SQUID can see only the flux of ΔI × Ma , as discussed above. When adjusting the voltage Vb′ to an optimal value, one should find the irregularly periodic V (Φ) characteristics at V out , such as the example shown in Figure 11.4, which can be displayed on an oscilloscope operated in X–Y mode. From Figure 11.4, it can be seen that when one flux quantum Φ0 is applied at the sensing SQUID, the real characteristics of the sensing SQUID, I(Φ), are concealed (dotted line). However, a multi-periodical V (Φ) appears at V out of

1

Vout

Figure 11.4 V(Φ) characteristics at V out , where the real characteristics of the sensing SQUID, I(Φ), are concealed (dotted line), but irregularly periodic V out reflects the 𝜕I/𝜕Φ of the sensing SQUID. Here, the small period of Φ/Φ0 represents a large 𝜕I/𝜕Φ of the sensing SQUID.

0

–1 –0.5

0 Φ/Φ0

0.5

163

164

11 Two-Stage and Double Relaxation Oscillation Readout Schemes

the current-biased reading SQUID, where the periods of unequal density at V out reflect the different 𝜕I/𝜕Φ of the sensing SQUID’s I(Φ). From the number of periods, we can estimate that the gain GF of the “flux amplifier” is approximately 7; i.e. V (Φ) at V out presents 7 periods. The period-dense region should be selected as the working point for the sensing SQUID. With a dc compensation current generated by the adjustable voltage V com via R′o , the V (Φ) characteristics can be shifted to the left or to the right, as indicated by the arrows. This shift should be done until the region of dense periods of the V (Φ) in Figure 11.4 is located at the center. Thereby, the working point W of the sensing SQUID, where 𝜕I/𝜕Φ of the voltage-biased sensing SQUID reaches its maximum, is selected. In brief, once I b and V b are selected, the curve at V out shown in Figure 11.4 can be moved up and down by adjusting V w , or right and left by adjusting V com . Thus, all adjustments are visualized in our suggested two-stage scheme. (3) Finally, one closes the FLL by switching K2 to position “II,” where G2 is deactivated. Now, the sensing SQUID measures the applied flux picked up by the antenna because the FLL is just working via La , the second-order gradiometer SQUID, SCRE, and Rf and Lf of the sensing SQUID. Here, the output voltage V out and the measured magnetic field ΔB exhibit linear relationships. At this point, our practical SQUID magnetometric system with the two-stage readout is complete. We hope that a practical measurement using such a system with the two-stage readout in an unshielded environment will be reported in the near future.

11.2 ROS and D-ROS It is well known that an un-shunted Josephson junction exhibits hysteretic I–V characteristics (see Figure 2.2b). Intuitively, SQUIDs with two such junctions maintain hysteresis, and their I–V characteristics should contain information on the applied flux. Similarly to the normal SQUID shunted by RJ , the un-shunted SQUID’s critical current can be modulated by the flux between I c at integer Φ0 and Ic′ at half-integer Φ0 , as shown in Figure 11.5. The readout schemes of ROS and D-ROS are based on such hysteretic I–V characteristics [2, 3, 16, 17]. The so-called “hysteresis” means that the paths of current rise and fall are different in the I–V characteristics, thus forming a hysteresis loop with two vertical lines (at V s = 0 and V s = V gap ) and two horizontal dotted lines. In Figure 11.5, the three typical horizontal dotted lines indicated at I r , Ic′ , and I c form two loops, consisting of an upper line at I c and a lower line at I r , or a middle line at Ic′ and a lower line at I r . For these dotted lines, the arrows represent one-way voltage transition processes within picoseconds, so that these transitions cannot be observed by recording the I–V characteristics, which are displayed on an oscilloscope operated in X–Y mode, as shown in Figure 3.1. In fact, an arbitrary Ic′′ between I c and Ic′ and the lower line at I r can also form a hysteresis loop. The value of Ic′′ reflects the flux state threading the SQUID loop. Here, the parameter 𝛽 c of the “hysteretic” SQUID approaches infinity.

11.2 ROS and D-ROS

Figure 11.5 Hysteretic I–V characteristics of an un-shunted SQUID. The top and the middle dotted line denote the SQUID’s critical currents at integer and half-integer Φ0 , respectively. The diagonal dashed line, which starts at Ib and ends at Ib Rro , is the operating load line. Here, the SQUID’s shielding parameter 𝛽 L is approximately unity.

I

Ib Integer Φ0

Ic

Ic″

Half-integer Φ0

Ic′

Load line Ir 0

IbRro Vr

V

Vgap

Let us observe the current paths when the bias current varies from 0 up to I b (I b > I c ) and then returns, in the case of a SQUID with un-shunted junctions. Here, the current rises along the left vertical solid line at zero voltage up to I c marked in Figure 11.5, where the first voltage transition from left to right (upper dotted line) occurs; i.e. the voltage across the SQUID, V s , jumps from zero to V gap (V gap ≈ 2.8 mV for Nb–AlOx –Nb junctions at 4.2 K). The current further increases up to its maximum at I b along the right vertical solid line at V gap and then returns. As the current decreases along this right line down to I r , the second voltage transition from V r (≈V gap ) to zero happens. Ultimately, the remaining current falls again to the origin at the left vertical line (V s = 0). During SQUID operation, the first (forward) voltage transitions can occur at different current values of Ic′′ when I c (Φ) is modulated by the applied flux. Nevertheless, there is only one pathway for the second (backward) transition (lower dotted line). Thus, the current paths always run along a loop, whereas the forward voltage transitions can be different. Generally, the return current I r and the return voltage V r depend on the junction’s features and quality. If one requires more knowledge about tunnel junctions, the book edited by A. Barone and G. Paterno is recommended [18]. A ROS SQUID is based on a hysteretic SQUID shunted with a circuit consisting of Lro –Rro in series (see Figure 11.6). This shunted circuit is no longer a stranger Figure 11.6 The equivalent circuit of ROS. The virtual voltmeter and ammeter represent our observables.

Ib I1

I2

V

Lro M=0 A

Rro

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11 Two-Stage and Double Relaxation Oscillation Readout Schemes

to us. Unlike the PFC in Chapter 8, the Lro is not coupled to the SQUID; i.e. the mutual inductance between them is zero (M = 0). When a constant current I b above I c biases the parallel circuit, the ROS becomes active to oscillate. In the parallel circuit in Figure 11.6, during ROS operation, I b remains constant and the equation I b = I 1 + I 2 is always valid. Now, we discuss the oscillation condition of the parallel circuit, first without considering the relaxation effect of Lro ; i.e. Lro is shorted. After the forward voltage transition from zero to V gap occurs at I c , due to I b > I c (the upper dotted line in Figure 11.6), the voltage of the parallel circuit should instantaneously appear at V gap . However, this voltage transition cannot be smoothly realized because the condition I b × Rro < V r < V gap is designed, as marked in Figure 11.6. At this moment of the voltage transition, I 1 rapidly decreases to I 1 = I r , and then the second voltage transition (lower dotted line) occurs. Namely, the SQUID is again in the superconducting state, and I 1 starts to rise up to I c , so that a ring current I between two branches permanently runs along the hysteretic loop. In other words, the voltage of the parallel circuit appears intermittently. In brief, the ROS oscillates under the necessary condition of (I b × Rro ) < V r ; i.e. its load line (dashed line) is inclined, as in Figure 11.5. For the completed ROS with Lro , the relaxation effect (charging and discharging) will be considered, so that a stable voltage across the parallel circuit may intermittently occur in the time domain. In Figure 11.6, the voltmeter and ammeter represent our main interest in observing the behaviors of I 1 and of the voltage V s across the SQUID. We take the simulation of Ref. [17] and show the results in Figure 11.7. The current I 1 rises and falls during the t 0 and t v intervals, while the relaxation effect of Lro in I 1 and V s can be clearly observed. Here, V s = V gap appears during the whole interval of t v , thus leading to a time-averaged voltage across the SQUID, V avg ≠ 0, during the periods t 0 and t v . The intervals t 0 (at the zero voltage) and t v (at the V gap voltage) are expressed as t0 = (Lro ∕Rro ) ln[Ib ∕(Ib − Ic (Φ))]

(11.1) Figure 11.7 Simulated I1 (upper) and V s (lower) of an ROS in the time domain, where both should be monitored with ammeter A and voltmeter V, as indicated in Figure 11.6.

t0 + tv

I1

Ic

Vgap Vs

166

t0 0

20

tv 40 t (ns)

60

80

11.2 ROS and D-ROS

and tv = (Lro ∕Rro ) ln[Rro Ic (Φ)∕(Vgap − Ib Rro ) + 1]

(11.2)

respectively. In fact, t 0 and t v depend on I c (Φ); i.e. the time-averaged dc voltages V avg reflect I c (Φ). In the case of using ROS, one can take V avg as the readout signals, which corresponds to the SQUID’s flux state. From Eqs. (11.1, 11.2), the dc V avg and the relaxation frequency f ro can be easily derived: Vavg = [tv ∕(t0 + tv )] × Vgap and fro = 1∕(t0 + tv ) Owing to the function I c (Φ), the intervals are changed with the SQUID’s flux state, thus changing V avg and f ro . According to the above simulation, two important consequences are obtained: (1) for an ROS with 𝛽 L = 1 (its I–V characteristics shown in Figure 11.5), at I b = 1.1I c , the V avg modulation for ΔΦ = Φ0 /2, i.e. ΔV swing ≈ 0.42I c Rro ≈ 1 mV, can be achieved; (2) at the working point of ±Φ0 /4, the transfer coefficient (𝜕V /𝜕Φ)ro can be estimated to be 0.9(Rro /Ls ), where Ls is the SQUID inductance. To obtain a high (𝜕V /𝜕Φ)ro of ROS, the value of Rro should be large, but it is limited by the condition I b × Rro < V r . D-ROS, originally called balanced ROS in Ref. [3], consisting of two hysteretic dc SQUIDs in series, was developed to further increase the transfer coefficient (𝜕V /𝜕Φ). In the original work, the signal flux was applied to both SQUIDs; i.e. the roles of the two SQUIDs, where one is reading and the other is sensing, can be exchanged. In contrast to balanced ROS, the functions of the two SQUIDs are well defined in D-ROS; i.e. one is the sensing SQUID, and the other is the reading SQUID (see Figure 11.8a), similar to the two-stage scheme. Here, we assume that the sensing SQUID has a critical current I c1 and the reading SQUID has a critical current I c2 . In fact, the reading SQUID is blind to the measured flux and works only at a selected constant flux (e.g. Φ0 /4) for adjusting I c2 , so that the reading SQUID actually acts as an adjustable current comparator. However, I c1 is modulated by the measured flux. If I c1 < I c2 , the voltage across the reading SQUID is zero. If I c1 > I c2 , the time-averaged V read appears. In the latter case, V read can be employed to measure the flux change of the sensing SQUID and to achieve a large transfer coefficient of (𝜕V /𝜕Φ) > 50 mV/Φ0 . In this subsection, we want to omit more descriptions because two references from Adelerhof elaborate on these principles and on the noise mechanisms and experimental data [17, 19]. Actually, an important improvement brings the D-ROS scheme into the practical stage. Lee et al. utilized a reference junction instead of the reading SQUID to act as a current comparator (see Figure 11.8b) [20, 21]. Here, the critical current I c,J of the junction J must be somewhat smaller than the critical current I c1 of the sensing SQUID at Φ = Φ0 . In that case, the readout voltage, V J , drops across J. The key factor is that during the voltage transitions from zero to V J , a very large dynamic resistance Rd appears at J, thus enlarging the transfer coefficient 𝜕V /𝜕Φ of the sensing SQUID in a certain flux region selected by I c,J . At I c.J ≈ 0.8 I c1 at

167

168

11 Two-Stage and Double Relaxation Oscillation Readout Schemes Ib Rro Φ Lro

Ib Sensing SQUID

Rro

To electronics at RT Reading SQUID

Φ Lro

Sensing SQUID

VJ Vgap (avg.)

VJ J0.8Ic1

W

To electronics at RT

0

Φ0/4

(a)

(b)

Φ0

Φ

(c)

Figure 11.8 Schematic arrangement of D-ROS where the output voltage drops across the reference (reading) SQUID (a), which can be replaced by a reference junction J (b), and its V(Φ) characteristics (c).

Φ = Φ0 , the V (Φ) characteristics of the sensing SQUID look like a square wave, as schematically shown in (c). Here, the large 𝜕V /𝜕Φ at working point W can sufficiently suppress δΦe below δΦs . Previously, we asserted that the flux-to-voltage transfer coefficient 𝜕V /𝜕Φ at W is proportional to the value of Rd in current bias mode (see Chapters 8 and 10). For 𝜕V /𝜕Φ = 10 mV/Φ0 , the value of Rd at W can possibly reach up to 1–2 kΩ. Note that the value of Rd is also time-averaged in D-ROS. Actually, the current noise contribution to (Rd × I n ) from the preamplifier may be reached in the range √ of nV/ Hz. However, a 𝜕V /𝜕Φ of several mV/Φ0 is suitable for suppressing the δΦe in the D-ROS scheme below the δΦs of a SQUID with 𝛽 c → ∞. Note that setting the working point W on a very steep slope (see Figure 11.8c) may lead to instability of FLL operation. Overall, a unilateral pursuit of a high 𝜕V /𝜕Φ is not the goal for D-ROS.

11.3 Some Comments on D-ROS and Two-Stage Scheme In this chapter, we have discussed two different readout schemes, the two-stage scheme and D-ROS, and have demonstrated their respective advantages. There is a common feature, i.e. δΦsys is close to δΦs . According to our experience in setting up a practical SQUID magnetometric system, only one FLL should be employed for the sensing element (SQUID). In this sense, the improved D-ROS with a junction J as a current comparator (see Figure 11.8b) fulfills the above requirement. For the two-stage scheme, our suggestion is the use of a second-order gradiometer acting as the reading SQUID (see Figure 11.3); i.e. the reading SQUID is blind to the measured field. However, we want to indicate the following facts: It is impossible for the reading SQUID to read out only the desired signal from the sensing SQUID without any environmental influence or without cross-talk between both SQUIDs. In practice, one can only reduce such influences but not avoid them completely. Here, D-ROS operates in the extreme case where the parameter 𝛽 c of the dc SQUID approaches infinity, thus leading to a large intrinsic SQUID noise δΦs . On other hand, an unusually large 𝜕V /𝜕Φ appears at the working point W, as shown

References

in Figure 11.8c. In principle, the readout electronics for D-ROS and for weakly damped SQUID are the same; i.e. both use a DRS. Therefore, we speculate that the system noise δΦsys of D-ROS is probably compatible with that of a weakly damped SQUID. However, the two philosophies of system noise distribution are different: in a D-ROS system, δΦ2sys ≈ Φ2s , whereas in a weakly damped SQUID, δΦ2sys ≈ Φ2s + Φ2e , where Φs ≈ Φe [22]. In fact, the concept of a weakly damped SQUID is simpler and more direct than the concept of a D-ROS system. In a certain sense, the SQUID in D-ROS operation can be regarded as the limit of a weakly damped SQUID, while its Φs should be the estimated asymptotic upper limit of Φs shown in Figure 6.5. For a multichannel setup, one is free to choose between the two schemes. For a strongly damped SQUID, the two-stage scheme makes it possible to read out the intrinsic SQUID noise Φs , where Φsys ≈ Φs . However, this feature becomes difficult in the case of a very low Φs of the SQUID and small 𝜕V /𝜕Φ values at, e.g. 𝛽 c < 0.3. Nevertheless, the two-stage scheme is not user-friendly due to the need for adjusting two SQUIDs, thus hindering its application.

References 1 Wellstood, F.C., Urbina, C., and Clarke, J. (1987). Low-frequency noise in DC

2 3

4

5

6

7

8

9

superconducting quantum interference devices below 1-K. Applied Physics Letters 50 (12): 772–774. Vernon, F.L. and Pedersen, R.J. (1968). Relaxation oscillations in Josephson junctions. Journal of Applied Physics 39 (6): 2661–2664. Gudoshnikov, S.A., Maslennikov, Y.V., Semenov, V.K. et al. (1989). Relaxation-oscillation-driven DC-SQUIDS. IEEE Transactions on Magnetics 25 (2): 1178–1181. Lee, Y.H., Yu, K.K., Kwon, H. et al. (2009). A whole-head magnetoencephalography system with compact axial gradiometer structure. Superconductor Science & Technology 22 (4): 045023. Muhlfelder, B., Johnson, W., and Cromar, M.W. (1983). Double transformer coupling to a very low-noise SQUID. IEEE Transactions on Magnetics 19 (3): 303–307. Gay, F., Piquemal, F., and Geneves, G. (2000). Ultralow noise current amplifier based on a cryogenic current comparator. Review of Scientific Instruments 71 (12): 4592–4595. Granata, C., Vettoliere, A., and Russo, M. (2011). An ultralow noise current amplifier based on superconducting quantum interference device for high sensitivity applications. Review of Scientific Instruments 82 (1): 013901. Luomahaara, J., Kiviranta, M., and Hassel, J. (2012). A large winding-ratio planar transformer with an optimized geometry for SQUID ammeter. Superconductor Science & Technology 25 (3): 035006. Schmelz, M., Zakosarenko, V., Schonau, T. et al. (2017). A new family of field-stable and highly sensitive SQUID current sensors based on sub-micrometer cross-type Josephson junctions. Superconductor Science & Technology 30 (7): 074010.

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10 Zakosarenko, V., Schmelz, M., Stolz, R. et al. (2012). Femtoammeter on the

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18 19

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22

base of SQUID with thin-film flux transformer. Superconductor Science & Technology 25 (9): 095014. Koch, R.H., Foglietti, V., Rozen, J.R. et al. (1994). Effects of radio-frequency radiation on the DC SQUID. Applied Physics Letters 65 (1): 100–102. Enpuku, K., Sueoka, K., Yoshida, K., and Irie, F. (1985). Effect of damping resistance on voltage versus flux relation of a DC SQUID with large inductance and critical current. Journal of Applied Physics 57 (5): 1691–1697. Foglietti, V., Giannini, M.E., and Petrocco, G. (1991). A double DC-SQUID device for flux locked loop operation. IEEE Transactions on Magnetics 27 (2): 2989–2992. Jin, I. and Wellstood, F.C. (1999). Continuous feedback operation of a two-stage DC SQUID system. IEEE Transactions on Applied Superconductivity 9 (2): 2931–2934. Chang, K., Zhang, Y., Wang, Y.L. et al. (2014). A simple SQUID system with one operational amplifier as readout electronics. Superconductor Science & Technology 27 (11): 115004. Gutmann, P. (1979). DC SQUID with high-energy resolution. Electronics Letters 15 (13): 372–373. Adelerhof, D.J., Nijstad, H., Flokstra, J., and Rogalla, H. (1994). (Double) relaxation oscillation SQUIDS with high flux-to-voltage transfer – simulations and experiments. Journal of Applied Physics 76 (6): 3875–3886. Barone, A. and Paterno, G. (1982). Physics and Applications of the Josephson Effect. New York: Wiley. Adelerhof, D.J., Nijstad, H., Flokstra, J., and Rogalla, H. (1993). Relaxation oscillation SQUIDS with high delta V/delta Phi. IEEE Transactions on Applied Superconductivity 3 (1): 1862–1865. Lee, Y.H., Kim, J.M., Kwon, H.C. et al. (1995). 3-Channel double relaxation oscillation SQUID magnetometer system with simple readout electronics. IEEE Transactions on Applied Superconductivity 5 (2): 2156–2159. Lee, Y.H., Kwon, H., Kim, J.M. et al. (1999). Noise characteristics of double relaxation oscillation superconducting quantum interference devices with reference junction. Superconductor Science & Technology 12 (11): 943–945. Liu, C., Zhang, Y., Mück, M. et al. (2012). An insight into voltage-biased superconducting quantum interference devices. Applied Physics Letters 101 (22): 222602.

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12 Radio-Frequency (rf) SQUID 12.1 Fundamentals of an rf SQUID Structurally speaking, a radio-frequency (rf) superconducting quantum interference device (SQUID) is a superconducting ring interrupted by a Josephson junction [1, 2]. To read out rf SQUID signals, an rf resonance tank circuit is usually needed. A conventional rf tank circuit consists of two lumped elements, a coil LT (inductive element) and a capacitor √ C T (capacitive element). The tank circuit resonates at a frequency f0 = 1∕(2𝜋 LT CT ) or an angular frequency 𝜔0 = √ 1∕( LT CT ) with a quality factor Q0 . When an rf SQUID is inductively coupled to the LT with a mutual inductance Mrf , the rf SQUID’s behavior influences the characteristics of the tank circuit, i.e. the resonance frequency f L and the quality factor Q of the tank circuit [3]. In other words, the rf SQUID changes the resonance frequency from the unloaded f 0 to the loaded f L and the quality factor from the unloaded Q0 to the loaded Q. Here, these changes depend on the flux threading into the SQUID loop. Usually, f L > f 0 is valid because of the superconductive shielding effect of the rf SQUID, and Q0 > Q because of the rf SQUID’s damping effects. In the rf SQUID readout technique, Δf L and ΔQ of the tank circuit are always taken as the readout criteria. Indeed, a conventional rf SQUID system is operated in an rf range of approximately 30 MHz. In fact, the biased rf component is only a carrier (in other words, a supporter), but the envelope of the rf component resembles the SQUID’s magnetometric signal. Similar to the screening parameter 𝛽 L of the dc SQUID, the dimensionless parameter 𝛽 e = 2𝜋Ls I c /Φ0 of the rf SQUID normalizes not only the product of Ls I c but also the rf SQUID operation modes, which are divided, in the theoretical description of rf SQUID operation, into (i) hysteretic (dissipative) mode for 𝛽 e > 1 [4–6] and (ii) nonhysteretic (dispersive) mode for 𝛽 e < 1 [7–10]. In the physical picture, the parameter 𝛽 e describes the magnetic flux relationship inside and outside the SQUID loop. Because of the shielding effect and the flux quantization effect of the rf SQUID, the flux inside the SQUID loop, Φloop , serpentines up when the outside (applied) flux Φ increases linearly (see Figure 12.1). Depending on the steepness and curvature, the serpentine curve Φloop can lead to two different results: a single-valued function of Φloop (Φ) when 𝛽 e < 1 [11] or a multivalued function when 𝛽 e > 1 [12]. SQUID Readout Electronics and Magnetometric Systems for Practical Applications, First Edition. Yi Zhang, Hui Dong, Hans-Joachim Krause, Guofeng Zhang, and Xiaoming Xie. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

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12 Radio-Frequency (rf) SQUID

2

Φloop /Φ0

2

1

Φloop /Φ0

1

1

Φ/Φ0 0 (a)

1

2

Φloop /Φ0

2

Φ/Φ0 0 (b)

1

Φ/Φ0

2

0

1

2

(c)

Figure 12.1 The magnetic flux relationship inside and outside the SQUID loop is illustrated for different 𝛽 e , (a) 𝛽 e < 1, (b) 𝛽 e > 1, and (c) 𝛽 e ≫ 1, where the dashed straight line represents Φloop = Φ at 𝛽 e = 0, while flux quantum transitions appear at the inflection points of the function Φloop (Φ) in the case where 𝛽 e > 1 (b) and 𝛽 e ≫ 1 (c).

In fact, at all inflection points of the function Φloop (Φ) in the case where 𝛽 e > 1 shown in Figure 12.1b,c, flux quantum transitions occur, so the parts with negative slopes in these curves are not implemented. Such quantum transitions dissipate the energy in the tank circuit, i.e. dampen the value of Q. Here, at 𝛽 e > 1 (Figure 12.1b), one value of Φ corresponds to two values of Φloop , whereas at 𝛽 e ≫ 1 (Figure 12.1c), one value of Φ corresponds to more than two values of Φloop . To date, all rf SQUID readout electronics have operated in current bias mode. In contrast to the dc SQUID, the rf bias current I rf with a certain frequency cannot be directly injected into the rf SQUID but rather into only the LT C T tank circuit, across which the rf voltage V rf is read out by an rf voltmeter (see Figure 12.2a). The V rf contains the rf SQUID signals via a mutual inductance Mrf . In most practical rf SQUIDs, the coil LT simultaneously acts as a flux feedback coil Lf for flux-locked loop (FLL) operation. The function of Lf in the FLL has been described in Chapter 4. In a magnetometric system, an rf SQUID with tank circuit can be regarded as a two-terminal element, while a dc SQUID with Lf is considered a four-terminal element. Indeed, the rf SQUID signals are also based on I rf –V rf characteristics, although the characteristics are quite different from those of a dc SQUID. Here, I rf is proportional to Φrf because of the constant Mrf , while V rf across the tank circuit increases monotonically with increasing I rf . To reflect the nonlinear function of Φloop (Φrf ) in Figure 12.1, the I rf –V rf curves across the tank circuit develop “steps and risers” at 𝛽 e > 1. At the two-flux limit states (i.e. Φ = Φ0 and Φ = (n + 1/2)Φ0 ), two I rf –V rf curves mutually intersect to form a series of parallelograms (see Figure 12.2b). Usually, an appropriate bias I rf,bias is set on the middle of the first parallelogram to obtain the optimal V rf (Φ) characteristics at which the V rf swing achieves its maximum. A detailed interpretation of the I rf –V rf curves and the parameter 𝛼 can be found in [4, 13], where the dimensionless parameter 𝛼 = ΔV s /ΔV o reflects the rf SQUID’s intrinsic noise at 𝛽 e > 1. Indeed, the value of 𝛼 plays a minor role in V rf swing (V rf,swing ) in the case where 𝛼 ≪ 1. To increase V rf,swing , one should substantially enhance the pumping frequency 𝜔rf , which will be discussed later. For rf SQUID operation, a certain I rf,bias , marked in Figure 12.2b, is usually chosen at the middle of the first plateau in the I rf –V rf characteristics. Furthermore, the coupling condition of k 2 Q > 1 is

12.1 Fundamentals of an rf SQUID

Figure 12.2 (a) Readout principle of an rf SQUID: the rf SQUID is coupled to a tank circuit consisting of two lumped elements, a capacitor C T (capacitive element) and a coil LT (inductive element), via a mutual inductance Mrf . A bias current Irf is injected through the tank circuit, thus generating a voltage V rf , which is our readout quantity; (b) Irf –V rf curves of an rf SQUID with 𝛽 e ≈ 1, where the bias Irf,bias for rf SQUID operation and two inflection points A and B are marked. The dimensionless parameter 𝛼 = ΔV s /ΔV o is denoted with its geometric definition.

Mrf

Irf LT

CT

Vrf

(a) Vrf

Φ = nΦ0

Φ = (n + 1/2)Φ0 ΔVo B Vrf,swing A 0

ΔVs

Irf

Irf,bias

(b)

very important [3, 14–18], where the coupling factor k is given by k 2 = Mrf2 ∕Ls LT . Here, we try to explain this condition geometrically with the help of Figure 12.2b. To obtain a complete parallelogram, the inflection point A on the I rf –V rf curves at Φ = (2n + 1)Φ0 /2 must be located further to the right (i.e. at higher I rf values) than the inflection point B at Φ = nΦ0 (see Figure 12.2b). Here, the position of A depends on the product k 2 Q. Namely, point A moves to the right with increasing k 2 Q. When the condition k 2 Q > 1 is fulfilled, a parallelogram is formed and point A lies on the right side of B [15]. Because of the thermal noise, all inflection points of “steps and risers” in the rf SQUID’s I rf –V rf curves are rounded, which is similar to the I–V characteristics of a JJ and dc SQUID discussed in Chapters 2 and 6. Especially for high-T c SQUIDs, the inflection points, e.g. points A and B, become strongly blurred. In fact, Chesca [19] discussed in detail the relation between the junction thermal parameter Γ and this rounding effect on I rf –V rf characteristics. However, to ensure that the parallelogram depicted in Figure 12.2b appears, the requirement k 2 Q > 1 needs to be tightened to a condition k 2 Q ≫ 1 for high-T c SQUIDs [19]. Now, we give two equivalent circuits to explain two typical SQUID operation modes, i.e. 𝛽 e ≫ 1 (dissipative mode) and 𝛽 e ≪ 1 (dispersive mode): (1) In the dissipative mode (𝛽 e ≫ 1), plateaus appear in the I rf –V rf curves because the energy in the tank circuit is dissipated by the transitions between quantum states because of the multivalued function of Φloop (Φrf ) [3]. Therefore, the rf SQUID can be regarded as a damping element for the tank circuit; i.e. a simple resistance R(Φ) is inserted into the resonance tank circuit (see Figure 12.3a). In other words, the dissipated energy in the tank circuit can be equivalent to resistive damping. An increase in the tank circuit resistance R(Φ) leads to a reduction in its quality factor, Q ≈ 𝜔0 LT /R(Φ), thus yielding a reduced voltage V rf across the tank circuit, as illustrated in Figure 12.3b.

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12 Radio-Frequency (rf) SQUID

LT

Vrf

Vrf

R(Φ)

Φ = nΦ0

CT

Vswing

Φ = nΦ0 Φ = (n + 1/2)Φ0

Φ = (n + 1/2)Φ0

0 (a)

(b)

f0,bias

f

D

βe ≫ 1

Irf

0 (c)

Figure 12.3 Illustration of the rf SQUID behavior in dissipative mode (𝛽 e ≫ 1): (a) the rf SQUID coupled to the tank circuit can be considered equivalent to a resistance R(Φ) inserted into the LT C T resonance circuit; (b) two schematic resonance curves with different quality factors Q at applied fluxes of nΦ0 and (n + 1/2)Φ0 ; (c) Irf –V rf curves of an rf SQUID at 𝛽 e ≫ 1, where the separation point D is located much higher than the point at the origin for 𝛽 e ≈ 1 shown in Figure 12.2b.

The function of the resistance R(Φ) in rf SQUID dissipative mode (𝛽 e ≫ 1) can be analogous to that of the dynamic resistance, Rd (Φ), in dc SQUID operation. Here, R(Φ) and Rd (Φ) determine the SQUID signal swings in rf and dc SQUID operation, respectively. In practice, the position of the separation point D of the two limiting I rf –V rf curves (i.e. Φ = Φ0 and Φ = (n + 1/2)Φ0 ) represents the value of 𝛽 e . For 𝛽 e > 1, point D is not located at the origin of the I rf –V rf curves but shifted toward higher bias current, where the segment of the I rf –V rf curves from the origin to point D is a straight line (see Figure 12.3c). The longer the straight line from the origin to D is (in other words, the higher the location of point D is), the greater the value of 𝛽 e [20]. When the value of 𝛽 e > 1 decreases to 𝛽 e ≈ 1, point D falls along the line segment and is sitting just at the origin (i.e. D → 0), as shown in Figure 12.2b. Then, the separation point D remains at the origin when 𝛽 e < 1. Interestingly, the discussion about the position of the separation point D in the I rf –V rf curves of an rf SQUID can be correlated with the I–V curve behavior of a dc SQUID for different values of the parameter 𝛽 L , as shown in Figure 6.4. (2) In dispersive mode (𝛽 e ≪ 1), the rf SQUID can be regarded as a flux-sensitive inductor L(Φ) participating in the L′T CT tank circuit via a mutual inductance Mrf [8]. The equivalent circuit is shown in Figure 12.4a. Thus, the resonance frequency f L of the equivalent circuit is modulated by the applied flux, while its quality factor Q remains almost constant (see Figure 12.4b). Here, two resonance curves of a tank circuit, marked with the resonance frequencies f L and fL′ , represent the two flux limits, nΦ0 and (n + 1/2)Φ0 . If the operating (pumping) frequency is aimed at f L , which is the resonance frequency of the tank circuit at Φ = nΦ0 , then the readout voltage swing V rf,swing records the voltage difference between the two resonance curves at f L , as shown in Figure 12.4b. If one selects fL′ as the operating (pumping) frequency, which aims at the resonance frequency of the tank circuit at Φ = (2n + 1)Φ0 /2, a reversed SQUID signal is obtained. Therefore, at 𝛽 e ≪ 1, one has two choices

12.1 Fundamentals of an rf SQUID

Vrf

Vrf L(Φ)

Φ = (n + 1/2)Φ0

Φ= (n + 1/2)Φ0

Φ = nΦ0

Vrf,swing

CT LT

D

Φ = nΦ0 (a)

0 (b)

fL′

fL

f

Irf

0 (c)

Figure 12.4 rf SQUID behavior in dispersive mode (𝛽 e ≪ 1): (a) the rf SQUID coupled to the tank circuit can be considered equivalent to a flux-sensitive inductor L(Φ) inserted into the L′T CT resonance circuit; (b) two schematic resonance curves with different f L values at applied fluxes of nΦ0 and (n + 1/2)Φ0 ; and (c) the Irf –V rf curves of an rf SQUID directly reflect the relationship between Φ and Φloop , as can be seen in Figure 12.1a, whereas the separation point D is always located at the origin.

for the operating frequency. At both frequencies, f L and fL′ , V rf,swing reaches a maximum. The polarity of these two maxima, however, is inverted, as can be seen in Figure 12.4b. At the lower frequency maximum fL′ , the voltage V rf at half integer flux is higher than at integer flux, whereas at the higher frequency maximum f L , the voltage at integer flux is higher. In dispersive mode, the I rf –V rf curves look roughly like the curve in dissipative mode, but point D is always sitting at the origin. Here, the plateaus in the I rf –V rf curves disappear because no transitions between quantum states occur because of the single-valued function of Φloop (Φrf ). The two limiting I rf –V rf curves are symmetrically twisted at 𝛽 e ≪ 1, as schematically shown in Figure 12.4c. Indeed, the V rf,swing of the rf SQUID decreases with decreasing 𝛽 e because the Δf L between the two flux limits decreases. As 𝛽 e → 0, the SQUID does not work and its signal disappears, as illustrated by the I rf –V rf curve denoted by a dashed line. According to the operation modes, the rf SQUID intrinsic noise 𝛿Φs also has different mechanisms: (i) for 𝛽 e > 1, 𝛿Φs is caused by the distribution function of quantum transitions, 𝜎, thus leading to a corresponding voltage noise across the tank circuit. Usually, δΦ2s is proportional to L2s ∕𝜔rf [13]; (ii) for 𝛽 e < 1, 𝛿Φs no longer depends on the pumping frequency 𝜔rf , while 𝛿Φs may be limited by the thermal noise of the junction shunt resistor RJ , i.e. δΦ2s ∝ L2s ∕RJ [8]. In dissipative mode (𝛽 e > 1), the SQUID intrinsic noise 𝛿Φs leads to two abnormalities in the I rf –V rf curve: (i) except the thermal noise, the I rf –V rf curves near all inflection points (steps-to-risers or risers-to-steps) are further rounded because of the distribution function of 𝜎 and (ii) the plateaus (steps) in the I rf –V rf curves become tilted. This tilt is described with a dimensionless (step tilting) parameter 𝛼 = ΔV s /ΔV o , where ΔV s is the voltage rise along a plateau and ΔV o is the voltage difference between the ends of a plateau (see Figure 12.2b). Indeed, one can estimate the rf SQUID intrinsic noise 𝛿Φs from 𝛼 [3]. However, in dispersive mode (𝛽 e < 1), the slope of the I rf –V rf curves is not related to 𝜔rf and 𝛿Φs . Here, the SQUID acts as a flux-sensitive inductor L(Φ), resulting in symmetrically twisted I rf –V rf curves at 𝛽 e ≪ 1.

175

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Figure 12.5 At 𝛽 e ≈ 1, both effects on the tank circuit caused by the rf SQUID, ΔQ and Δf , are combined, thus resulting in a maximum swing V rf,swing .

Vrf Φ = nΦ0

Vswing

Φ = (n + 1/2)Φ0

0

f0,bias

f

In practice, most rf SQUIDs are operated neither completely in dissipative mode nor completely in dispersive mode. Indeed, the two operating modes have no obvious dividing line because of the thermal noise. Actually, rf SQUIDs are operated in a mixed mode, where the effects of both ΔQ and Δf L on the tank circuit coexist [21]. Generally, at 𝛽 e ≈ 1 of a low-T c (e.g. niobium) rf SQUID (or at 𝛽 e ≈ 3 of a high-T c [YBCO] rf SQUID [19]), the rf SQUID signal swing can yield a maximum because the effects of both ΔQ and Δf L together determine the signal swing, as outlined in Figure 12.5. If 𝛽 e ≥ 1, the rf SQUID operates more in dissipative mode than in dispersive mode. Conversely, as 𝛽 e ≤ 1, it operates more in dispersive mode. In 2007, R. Kleiner et al. investigated the characteristics and noise performance of rf SQUIDs by solving the corresponding Langevin equations numerically and optimizing the model parameters with respect to noise energy. After introducing the basic concepts of the numerical simulations, they gave a detailed discussion of the performance of the SQUID as a function of all relevant parameters. The best performance was obtained in the crossover region between the dispersive and dissipative regimes, characterized by an inductance parameter 𝛽 e ≈ 1 [22]. The tank circuit parameters, ΔQ and Δf , were experimentally measured for the SQUIDs with three typical 𝛽 e values (𝛽 e ≫ 1, 𝛽 e ≪ 1, and 𝛽 e ≈ 1) [20]. Furthermore, the optimal value of 𝛽 e ≈ 1 determines the product Ls I c for designing an rf SQUID, where Ls and I c are the two basic design parameters of an rf SQUID. In this sense, the rf SQUID’s parameter 𝛽 e plays the same role as the dc SQUID’s 𝛽 L . However, one should be careful not to confuse 𝛽 e and 𝛽 L because they differ by a factor of 2𝜋.

12.2 Conventional rf SQUID System 12.2.1 Block Diagram of rf SQUID Readout Electronics (the 30 MHz Version) Similar to dc SQUID systems, an rf SQUID system also detects a flux signal Φ(t) that can change slowly over time. Before we discuss the modern rf SQUID systems, let us roughly understand the conventional rf SQUID readout electronics with a pumping frequency f L of approximately 30 MHz [23]. The conventional standard block diagram of this rf SQUID system with readout electronics

12.2 Conventional rf SQUID System

V~ A

Mrf

C′ CT

LT

P

D C′

M

I

R

CT

Vout

C

Rf

Figure 12.6 Block diagram of the 30 MHz version of rf SQUID readout electronics. At the cryogenic temperature (CT) surrounded by the dashed line, a coil LT coupled to the rf SQUID via a mutual inductance Mrf can be regarded as a two-terminal magnetometric element. Here, LT acts as the flux feedback coil Lf in FLL, but the circuitry of the flux modulation scheme (FMS) is omitted. Usually, the CT is 4.2 K for low-T c SQUIDs or 77 K for high-T c SQUIDs.

is shown in Figure 12.6. Here, we call this system “the 30 MHz version,” even though it can be operated with f L values between 10 and 40 MHz. In the conventional 30 MHz version, the rf SQUID system consists of five blocks: (i) the SQUID and a coil LT of a tank circuit as the two fundamental elements make up the “head stage” with two terminals located at cryogenic temperature (CT), e.g. 4.2 K (or 77 K), where the LT is coupled to the rf SQUID with a mutual inductance Mrf . This LT and an adjustable capacitor C T at room temperature (RT) comprise a tank circuit, thus providing an adjustable resonance frequency f L . As the rf bias current I rf,bias oscillating with frequency f L flows through the tank circuit, it generates an rf voltage, V rf , across the tank circuit, where V rf is amplitude modulated by the SQUID signal. In fact, the rf component is only a carrier, and just the envelope is the SQUID magnetometric signals; (ii) an (rf ) amplifier consisting of a preamplifier (P) and a main amplifier (M), a diode detector (D), as well as a low-pass filter consisting of a resistor R in parallel to a capacitor C are employed to strip the envelope of the rf signals for reading the SQUID signal. Here, the input impedance of the preamplifier (P) should be much higher than that of the tank circuit so that the quality factor Q of the tank circuit is damped by the SQUID only. Usually, a low-noise field-effect transistor (FET) is employed as (P). Furthermore, the amplified V rf should be larger than the forward voltage of the diode (D). Therefore, the total gain G of the amplifiers, consisting of preamplifier (P) and main amplifier (M), should be designed to fulfill this requirement. With the detection diode (D) and the subsequent RC low-pass filter for rf components, the rf SQUID signal is demodulated, so the SQUID magnetometric signal becomes visible; (iii) the appropriate I rf,bias is provided by an rf voltage generator (V ∼ ) via an adjustable rf attenuator (A). In early readout electronics, the attenuator (A) was realized by a mechanically adjustable capacitor. Here, the generator (V ∼ ) oscillated with a fixed frequency, e.g. 30 MHz, and the resonance frequency f L of the tank circuit should be aimed at 30 MHz using the adjustable C T ; (iv) the FLL operation is the same as that for a dc SQUID: it consists of an integrator (I), a feedback resistor Rf , and a mutual inductance Mrf (Mf ). Here, LT plays a dual role; i.e. it serves both as the inductive element LT of the resonant tank circuit and as a current-to-field converter for the flux feedback coil Lf for FLL operation. By means of the FLL technique, the integrator (I) restores the Φ(t) signal at its output, V out ; (v) as an

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12 Radio-Frequency (rf) SQUID

option, flux modulation scheme (FMS), where a flux modulator with a frequency of, e.g. f M = 100 kHz and a demodulator (multiplier) can be employed, can also be used in an rf SQUID system. For clarity, the FMS is omitted in Figure 12.6. In this rf SQUID section, we will not discuss FMS again because it has been covered in depth in Chapter 7 for the dc SQUID. In principle, the difference between rf- and dc-SQUID readout electronics is not big because the rf bias circuitry ((V∼ ) + (A)) and the rf amplifiers ((P) + (M)) have counterparts employed in dc SQUID readout electronics. Only the diode detector (D) and the subsequent RC low-pass filter are unique to rf SQUID readout electronics. In fact, rf SQUID readout electronics exhibit two potential advantages over dc SQUID electronics in the direct readout scheme (DRS). The rf signals at the tank circuit are transferred to the main amplifier (M) with capacitive coupling C ′ (see the block diagram of Figure 12.6) so that the drift of the dc voltage component at the input of the integrator (I) is strongly eliminated. Such drift caused by temperature changes may mistakenly be treated as a detected SQUID signal. Simultaneously, the 1/f noise of the preamplifier plays no role because the capacitive coupling C ′ provides a path only for signals with f > MHz. Therefore, the introduction of the FMS in rf SQUID readout electronics may be superfluous [24]. Actually, our experiments did not show an obvious difference in system noise 𝛿Φsys in the low-frequency range when using our “homemade” readout electronics with and without FMS. 12.2.2

rf SQUID System Noise in the 30 MHz Version

In the 30 MHz version, one should pay attention to the tank circuit consisting of a coil LT at 4.2 K (CT) and a capacitor C T at 300 K (RT), as well as including two transmission lines between CT and RT, where the distributed capacitance and the distributed inductance of a sample holder (two transmission lines) inevitably contribute to the resonance circuit. This system leads to three consequences: (i) f L is generally limited to approximately 30 MHz, corresponding to the length of the sample holder and (ii) the quality factor of the tank circuit is small. Usually, the Q value is only approximately 30 and (iii) the effective operating temperature T e,T of the tank circuit is not 4.2 K, but it is speculated to be approximately 200 K [3]. Generally, an rf SQUID system has three noise sources: the SQUID intrinsic noise 𝛿Φs , the thermal noise 𝛿ΦT of the tank circuit, and the preamplifier’s noise 𝛿Φe . Indeed, the rf SQUID system noise, 𝛿Φsys , is the sum of three independent noise sources, i.e. δΦ2sys = δΦ2s + δΦ2T + δΦ2e

(12.1)

In the standard analysis method [3, 25], the three noise components in Eq. (12.1) are first unified with the voltage noise across the tank circuit. Here, 𝛿Φe is mainly caused by the voltage noise V n of the preamplifier. It seems that for rf SQUID readout electronics, the preamplifier’s current noise I n has not been considered separately yet. An equivalent noise temperature, T N , is often introduced in order to calibrate the preamplifier noise contribution [26]. The thermal voltage

12.2 Conventional rf SQUID System

noise, 𝛿V T , appears at the tank circuit and depends on the effective operating temperature T e,T . For 𝛿Φs , according to the theoretical explanation, there are two different mechanisms, which were introduced in Section 12.1. Actually, 𝛿Φs presents a voltage noise 𝛿V rf,s across the tank circuit. Then, the three flux noises in Eq. (12.1) are separately converted from the respective voltage noises with the flux-to-voltage transfer coefficient 𝜕V rf /𝜕Φ, which is expressed as 𝜕Vrf ∕𝜕Φ = 𝜔rf LT ∕Mrf or Vrf,swing = (𝜔rf LT ∕Mrf ) × (Φ0 ∕2)

(12.2)

where V rf,swing is the rf SQUID’s voltage signal swing across the tank circuit. Equation (12.2) is derived by rf circuit analysis [27]. In contrast to the corresponding coefficient of the dc SQUID, the coefficient 𝜕V rf /𝜕Φ of the rf SQUID is mainly determined by the external circuitry. Once the ratio of LT /Mrf is fixed, all three flux noise parts in Eq. (12.1) are inversely proportional to the pumping frequency 𝜔rf . A detailed introduction to and explanation of the three noise expressions can be found in Refs. [3, 27]. To reduce the rf SQUID system noise 𝛿Φsys and even to observe the rf SQUID intrinsic noise 𝛿Φs , increasing 𝜔rf is always preferred. We believe that similar to the dc SQUID, the rf SQUID’s intrinsic noise 𝛿Φs is also innate and independent of the readout electronics. In fact, the alleged 𝛿Φs in relation to 𝜔rf can be regarded as the “readable” 𝛿Φs only. Generally, the system noise 𝛿Φsys in the 30 MHz version is dominated by 𝛿ΦT and 𝛿Φe . The “readable” 𝛿Φs was first highlighted by using a preamplifier operated at 4.2 K and with a high pumping frequency, e.g. f L ≈ 430 MHz [28]. Now, we compare two rf SQUID experiments with different 𝜔rf values in dissipative mode: (1) For the 30 MHz version of a low-T c rf SQUID with √ Ls ≈ 1 nH and 𝛼 ≈ 0.2, −5 the measured system noise 𝛿Φsys was 8 × 10 Φ0 / Hz. Here, the tank circuit noise 𝛿ΦT was approximately three times larger than the SQUID’s intrinsic noise 𝛿Φs because T e,T ≈ 200 K, while 𝛿Φe was twice as large as 𝛿Φs , where an RT preamplifier with a noise temperature, T N , of 50 K was employed [3]. In brief, the ratio of the three noise parts was approximately 3(𝛿ΦT ) : 2(𝛿Φe ) : 1(Φs ). Therefore, the tank circuit noise 𝛿ΦT dominates the system noise 𝛿Φsys because of its high T e,T (≈200 K). Upon increasing the pumping frequency 𝜔rf (in other words, the resonance frequency f L of the tank circuit), 𝛿ΦT and 𝛿Φe decrease because δΦ2T ∝ 𝛼 2 ∕𝜔rf and δΦ2e ∝ 𝛼∕𝜔rf . (2) Long et al. increased the pumping frequency up to f L ≈ 430 MHz. In that work, a preamplifier with T N ≈ 5.5 K was realized by a GaAs FET operated at 4.2 K, with its gate directly connected to the tank circuit without long transmission lines between 4.2 and 300 K. Thus, the real operating temperature T op,T of the tank circuit was 4.2 K and no longer T e,T ≈ 200 K as in the 30 MHz √ version. With this setup, an rf SQUID system noise of 𝛿Φsys ≈ 3 𝜇Φ0 / Hz was measured, where 𝛼 ≈ 0.088, Q ≈ 50, and Ls ≈ 0.5 nH [28]. Here, the ratio of the three noise parts is approximately 1(𝛿ΦT ) : 1.3(𝛿Φe ) : 1.4(Φs ). This work

179

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12 Radio-Frequency (rf) SQUID

clearly demonstrated that increasing 𝜔rf leads to a larger 𝜕V rf /𝜕Φ, thus further reducing 𝛿ΦT and 𝛿Φe . Moreover, the effective temperature of the tank circuit T e,T = T op,T = 4.2 K, so the tank circuit noise 𝛿ΦT becomes the smallest component among the three noise parts in the system noise 𝛿Φsys . 𝛿Φe and Φs approximately contributed equally to the total noise in 𝛿Φsys . In brief, Long’s experiment was a major achievement in increasing 𝜔rf in rf SQUID operation and partly substantiated Kurkijärvi’s noise theory [28]. Now, we discuss the noise behavior in rf SQUID dispersive mode (𝛽 e < 1). It also contains three noise parts, similar to the dissipative mode behavior (𝛽 e > 1), but the value of 𝜕V rf /𝜕Φ should be corrected. Erné and Luther gave the following expression for the rf SQUID voltage swing at 𝛽 e < 1 [29]: Vrf,swing = 𝛽e (𝜔rf LT ∕Mrf ) × (Φ0 ∕2)

(12.3)

Comparing Eq. (12.3) with Eq. (12.2) reveals that 𝛽 e acts as a correction factor. Because no quantum transitions appear in this mode, the SQUID intrinsic noise 𝛿Φs,limit can be close to the Johnson noise limit [30], i.e. (𝛿Φs,limit )2 = (1.22∕𝛽e )2 (4kB TL2s ∕RJ ) ≈ (4kB TL2s ∕RJ ),

at k 2 Q > 1

(12.4)

By taking typical SQUID parameters, e.g. Ls = 100 pH and RJ = 10 Ω, an intrinsic √ noise limit 𝛿Φs,limit of the low-T c rf SQUID of (1.24/𝛽 e ) × 10−7 Φ0 / Hz could be reached. However, because of the small value of 𝜕V rf /𝜕Φ at the input terminal of the preamplifier, 𝛿Φsys is normally dominated by either 𝛿Φe or 𝛿ΦT . In brief, a helium-cooled GaAs FET used as a preamplifier at f L = 430 MHz in Long’s experiment is only suited for low-T c rf SQUID noise studies because the preamplifier (GaAs FET) located at CT brings about many additional problems, e.g. boiling of liquid helium and the vibrations caused by it. Furthermore, more transmission lines between the RT part of the electronics and the “head stage” at CT are needed. Actually, Long’s experiment tells us how to reduce 𝛿ΦT in rf SQUID operation.

12.3 Introduction to Modern rf SQUID Systems For half a century, SQUID advances have mainly been divided into two categories according to their manufacturing methods: conventional bulk SQUIDs and modern thin-film SQUIDs. With the revolutionary improvement achieved by Jaycox and Ketchen in 1981 [31–33], it became possible (i) to produce two identical junctions in dc SQUIDs and (ii) to produce identical input coils coupled to a SQUID with a large mutual inductance Mf for setting up SQUID magnetometers. In other words, compared to bulk SQUIDs, thin-film dc SQUIDs (magnetometers) have greatly improved performance and practicality. Immediately afterward, the first commercial 31-channel device KRENIKON (Siemens AG, Erlangen, Germany) was first run for magnetocardiography (MCG) in 1988 [34]. However, compared to thin-film dc SQUIDs, the modern (thin-film) rf SQUIDs fell behind following this revolutionary transition because the rf SQUIDs necessitate consideration of more issues, for example, obtaining a planar tank circuit, designing a planar rf

12.3 Introduction to Modern rf SQUID Systems

SQUID magnetometer with an integrated input coil, increasing the pumping frequency f L , transmitting the rf SQUID signal between CT and RT, matching the impedances between the tank circuit and the rf preamplifier, stabilizing the rf working point for a planar thin-film rf SQUID, and many more. Shortly after the discovery of high-T c materials, e.g. YBCO, naturally occurring grain boundary weak links in machined bulk constrictions and patterned thin-film microbridges came up [35, 36]. In the early 1990s, we demonstrated bulk BiPbSrCaCuO rf SQUID operating up to 101 K [37] and YBCO thin-film rf SQUID [38]. Actually, the developments of high-T c thin-film dc SQUIDs went along that of low-T c ones. However, the progress of high-T c thin-film rf SQUID magnetometric systems was not as fortunate because no mature technology could directly be copied from their low-T c counterparts. For a modern rf SQUID system, we can imagine the following: (i) the rf SQUID and its tank circuit should be made as planar structures; (ii) a high pumping frequency 𝜔rf ≫ 30 MHz should be realized; (iii) only one coaxial cable is utilized to connect the two parts at RT and CT, where the cable is no longer a part of the tank circuit to reduce the tank circuit operating temperature; (iv) the parameter 𝛽 e ≈ 1 for a low-T c rf SQUID or 𝛽 e ≈ 3 for a high-T c rf SQUID [39] should be maintained, thus yielding a large 𝜕V rf /𝜕Φ to reduce 𝛿Φe and 𝛿ΦT , as discussed above. In brief, the noise 𝛿Φsys of modern rf SQUID systems should be dominated by the SQUID intrinsic noise 𝛿Φs , even though it is close to the thermal noise limit 𝛿Φs,limit of high-T c SQUIDs. 12.3.1 Magnetometric Thin-Film rf SQUID and a Conventional Tank Circuit with a Capacitor Tap In 1992, the first low-noise high-T c (YBCO) thin-film rf SQUID magnetometer with step-edge junction was demonstrated at 77 K (the liquid nitrogen temperature) [40–42]. Two serious challenges were met: (i) How to make a sensitive thin-film SQUID magnetometer? (ii) How to increase the resonance frequency f L of the tank circuit while keeping an RT preamplifier? The first version of the thin-film rf SQUID was a mixture consisting of a modern (thin-film) rf SQUID and conventional tank circuit with lumped elements, as shown in Figure 12.7. According to these two parts, we separately explain their highlights: (1) It is known that in a low-T c planar dc SQUID magnetometer system, the multiturn input coil of the flux transfer system is usually integrated on the SQUID washer where a multilayer structure is required, as shown in Figure 4.3b. However, in the beginning of the 1990s, it was impossible to fabricate such a three-layer input coil of YBCO for a high-T c SQUID magnetometer system. Therefore, a large washer SQUID was the only remaining alternative to construct a sensitive high-T c SQUID magnetometric system [43]. Here, the effective pickup area, Aeff , of the thin-film washer SQUID magnetometer depends on the product of the loop and washer size, i.e. Aeff = lloop × lwasher

(12.5)

181

182

12 Radio-Frequency (rf) SQUID

To readout electronics 50 Ω cable C2 LT

C1

Figure 12.7 Schematic illustration of a mixture of technologies, i.e. a modern (thin-film) rf SQUID with a conventional tank circuit consisting of lumped elements, which connects to an RT preamplifier (bipolar transistor) via a 50 Ω coaxial cable. The coil LT simultaneously acts as a part of the LC tank circuit and as the flux feedback coil Lf for FLL operation. In use, a choke (inductor) shunted to C 1 (not shown here) provides a path for the feedback current If (quasi-dc current).

where lloop is the side length of the SQUID loop and lwasher is length of the washer, assuming that both are square [44]. In fact, the measured A′eff in [43] is smaller than Aeff in Eq. (12.5), namely, A′eff ≈ 0.8 × Aeff . (2) In the high-T c rf SQUID version, the conventional tank circuit consisting of two lumped elements, LT and C T , remained, whereas the wire-wound coil LT was placed on top of a modern thin-film SQUID (see Figure 12.7) instead of being located in the cylindrical hole of a bulk rf SQUID [27, 35]. Compared to that of the bulk version, the mutual inductance Mrf should be at least halved because only one end of the spiral coil LT couples to the thin-film rf SQUID. The capacitance C T was provided by two capacitors, C 1 and C 2 , connected in series for impedance matching. In our early works, a resonance frequency f L ≈ 150 MHz of the tank circuit was selected. To transmit the rf signal with high 𝜔rf , for impedance matching, both ends of a 50 Ω coaxial cable should connect to elements with 50 Ω impedance to avoid rf energy reflection. Here, the cable links the input of a preamplifier in the RT readout electronics and the two-terminal element (the tank circuit coupled to the SQUID) at CT. Here, a bipolar transistor with low-input impedance is employed as the preamplifier instead of a FET as in the 30 MHz version. Note that the SQUID signal transmission attenuation in the cable is neglected in the following discussion. Now we discuss the 50 Ω impedance matching of the tank circuit in rf SQUID operation. Because the tank circuit of f L ≈ 150 MHz usually possesses an impedance of >50 Ω, the use of an inductor or capacitor tap can complete the impedance matching between the tank circuit and the coaxial cable. In our first works, we initially selected a capacitor tap because of the ease of changing capacitors (see Figure 12.7). In fact, the capacitor tap indeed acts as an rf voltage divider as the impedance changes. Namely, the voltage swing of V rf,swing = (𝜔rf LT /Mrf ) × (Φ0 /2) across the tank circuit reduces to V rf,input at the tap position connected to the input terminal of the preamplifier at RT via a cable.

12.3 Introduction to Modern rf SQUID Systems

Here, the tap voltage ratio 𝜅 is defined as 𝜅 = Vrf,input ∕Vrf,swing

(12.6)

where 𝜅 is 30 MHz, some of the conventional analyses deal with this problem because of rf SQUID signal transmission from CT to RT [3]. Unfortunately, the matching among the 50 Ω coaxial cable, the tank circuit impedance, and the input impedance of the preamplifier are not contained in these analyses. With a planar thin-film rf SQUID, we first demonstrated the capacitor tap of the tank circuit to match a 50 Ω coaxial cable instead of two transmission lines (wires) in the 30 MHz version, thus increasing f L up to 150 MHz in our early works. In addition, the coaxial cable has good rf shielding protection, which is also very important for applications using SQUID magnetometric systems. Above all, we finally found a method to reduce the effective operating temperature, T e,T , of the tank circuit. The value of T e,T decreases with increasing 𝜔rf (in other words, with decreasing 𝜅) in this impedance-matching system. However, the technique did not improve the RT electronic noise 𝛿Φe , so the electronics noise 𝛿Φe should play a major role in 𝛿Φsys when 𝜔rf is increased sufficiently high.

183

184

12 Radio-Frequency (rf) SQUID

For high-T c rf SQUID studies in the early 1990s, the configuration shown in Figure 12.7 (the 150 MHz version) made large contributions to rf SQUID development. Two important achievements, i.e. adult magnetocardiograms (MCG) acquired in an unshielded environment and human brain activity (auditory evoked field) (magnetoencephalography [MEG]) recorded in a magnetically shielded room (MSR), were realized using these improved readout electronics and an additional high-T c bulk flux concentrator [45, 46]. 12.3.2

Improved rf SQUID Readout Electronics

To adapt the rf SQUID for high-𝜔rf operation, i.e. f 0 ≫ 30 MHz, the readout scheme should be changed accordingly. Compared to the 30 MHz version shown in Figure 12.6, the improved readout scheme should be suitable for low-impedance matching and for a large adjustable frequency range to allow for f L values ranging from dozens of MHz up to several GHz. A block diagram of the improved readout electronics is shown in Figure 12.8, where we take the capacitor tap at the “head stage” (the part at CT) as an example. Actually, some modern rf components are introduced. It is difficult to adjust the resonance frequency f L of the LT C T tank circuit located at CT to obtain a given oscillation frequency 𝜔rf of the rf generator (oscillator) (V ∼ ), as described in the 30 MHz version. Therefore, in the improved readout electronics, a voltage-controlled oscillator (VCO), i.e. an rf oscillator (V ∼ ) with adjustable 𝜔rf via a control voltage, is used to match in reverse the given resonance frequency f L of the tank circuit. Usually, a VCO changes 𝜔rf in a certain range, e.g. 150 ± 50 MHz, thus finding an oscillation frequency to aim for the f L of the tank circuit. To set up rf SQUID readout electronics, the output power of a VCO, which is typically +7 dBm, should be attenuated to less than −100 dBm to optimally bias an rf SQUID. Instead of mechanical attenuators in the 30 MHz version, this large attenuation range is accomplished by two commercial components: (i) fixed attenuators first reduce the VCO output signal to −80 dBm; (ii) a voltage-controlled attenuator (VCA) with an adjustment range, e.g. 0 to

Mrf

rf SQUID

LT

VCA

CT

DC

C1 C2

c b

(VCO)

V+

a 50 Ω cable

V~

A

C′

D

M C′ Zco

V+

“I” to input of integrator

Figure 12.8 The improved parts of rf SQUID readout electronics that are different from those in the 30 MHz version are sketched, while the identical parts are omitted. Here, six major changes should be mentioned: a 50 Ω coaxial cable as a transfer line, an additional directional coupler (DC), a bipolar transistor as a preamplifier, a mixer (multiplier) as an rf detector instead of a diode, a voltage-controlled oscillator (VCO) delivering f 0 ≥ 150 MHz, and a voltagecontrolled attenuator (VCA). Here, C ′ denotes the capacitive coupling.

12.3 Introduction to Modern rf SQUID Systems

−30 dBm, further reduces the rf power and is adjusted to attain the optimal rf bias current, I rf,bias , for the selected rf SQUID. Here, one should pay considerable attention to shielding the rf radiation of the VCO because its radiation energy can pervade a large space, preventing the actual rf bias power from being effectively attenuated. This effect will be discussed in Section 12.3.4.3. Here, a directional coupler (DC) is introduced at the input part of the readout electronics, as shown in Figure 12.8, to preserve the particular rf SQUID advantage that one cable transfers both the rf signal and feedback current I f for the FLL. This directional coupler was not present in the 30 MHz version. The attenuated I rf,bias first passes through the directional coupler and then flows through the tank circuit via the cable. Usually, the directional coupler is a three-terminal component. The attenuated forward rf current I rf,bias flows from terminal “c” to “a” to bias the rf SQUID (the tank circuit) via a 50 Ω coaxial cable. Generally, terminals “c” and “b” are isolated from each other. The backward SQUID voltage signal from the tap point of the tank circuit along the same cable passes through the directional coupler from “a” to “b” to connect to the preamplifier. In contrast to the 30 MHz version shown in Figure 12.6, this improved version of the rf SQUID readout electronics has a bipolar transistor with low input impedance acting as a preamplifier instead of a FET. Thus, the rf SQUID signals are transported from the tap point of the tank circuit to the input of the preamplifier via the 50 Ω cable. Namely, the SQUID’s signal at C 2 (tap-point), V rf,C2 = V rf,input = 𝜅V rf,swing , will be preamplified by the bipolar transistor, where the transistor base connects to the end of the 50 Ω cable at RT. The preamplified voltage across the load Zco at its collector is denoted by V rf,co . Generally, the gain of the preamplifier, V rf,co /V rf,input , should be at least 3.3 (i.e. the gain of the preamplifier is 10 dB). Subsequently, this V rf,co will be further enlarged by the main amplifier (M). Usually, the gain of M is designed to be 40 dB. With the help of modern rf components, we took two commercial four-terminal amplifiers with a fixed gain of 20 dB each. To obtain the SQUID magnetometric signal, we employed a mixer as the detector (D). Mixer D is a three-terminal component that operates as a multiplier of the two inputs. Here, the two inputs of mixer D separately connect to the VCO and to the output of the main amplifier M. The output voltage of mixer D depends not only on the rf SQUID amplitude but also on the phase difference 𝜑rf between the rf SQUID signal and VCO, where both are modulated by the flux Φ at SQUID’s 𝛽 e ≈ 1 [21]. Namely, the output voltage of mixer D can indicate not only the rf SQUID V (Φ) characteristics at 𝜑rf ≈ 0 (in-phase) but also the phase change, Δ𝜑rf (Φ), at 𝜑rf ≈ 𝜋/2. The latter is very useful when studying the rf SQUID’s behavior in dispersive mode. The output of mixer D connects to the integrator (I), which acts as the last stage in FLL operation. In the 30 MHz version, one uses a diode as the detector to obtain the envelope of the amplified V rf signals without any information on 𝜑rf . Instead of a diode, the mixer is used as a demodulator, and the tilt factor 𝛼, which couples with the SQUID intrinsic noise in dissipative mode, is generally invalid because 𝜑rf ≠ 0. In fact, the I rf –V rf curves of the rf SQUID, e.g. those in Figure 12.3c, can be rotated clockwise at the origin with Δ𝜑rf . Thus, the tilt factor 𝛼 already loses its geometrical meaning, as shown in Figure 12.2.

185

186

12 Radio-Frequency (rf) SQUID

In the one-cable version, one measures only the reflected rf energy (rf SQUID signal) from the tap point of the tank circuit, which differs from the tank circuit transfer function (resonance curve), e.g. V rf (f L ) in the 30 MHz version, as discussed above. Actually, the reflected wave is caused by impedance changes of the tank circuit because there is no reflected rf energy in the 50 Ω impedance-matching technique. Here, the bias rf current I rf,bias can be regarded as an incident wave, and the rf SQUID’s signal can be regarded as the reflected wave. An rf SQUID operated at f L varies the impedance at C 2 of the tank circuit, ZC2 (Φ), with the applied flux. If ZC2 = 50 Ω exactly at f L , and if it does not change with the flux, then no energy will be reflected; i.e. the input V rf,input of the preamplifier is zero. In the one-cable version, two antisymmetric SQUID signals (e.g. I rf –V rf curves) usually appear at f L ± Δf , where ZC2 ≠ 50 Ω causes the reflected energy (in other words, V rf,input ) to change with the rf SQUID signal. Experimentally, with the I rf –V rf and V rf (Φ) curves as well as the measured system noise, we did not find an obvious difference between using the transfer function (not shown here) and using the reflected signal in Figure 12.8 as the readout criterion. Note that (i) two cables are needed when measuring the transfer function, which can directly record the quality factor Q(Φ) and f L (Φ) changes of the tank circuit during rf SQUID operation, and (ii) the reflected function of V rf,input (𝜔rf ) indicates only the impedance changes and does not contain any information about Q. In fact, the configuration in Figure 12.8 is not the so-called 50 Ω impedance-matching system in the traditional sense because the two impedances at the two ends of the 50 Ω cable are not exactly 50 Ω. Here, one impedance of the bipolar transistor depends on its working point, while the other impedance at the tap point of the tank circuit is modulated by the flux changes. In addition, the noise temperature T N of a commercial low-noise preamplifier is calibrated with strict 50 Ω impedance matching at the input and output. In this sense, the rf SQUID in Figure 12.8 is not operated in a 50 Ω impedance-matching system. According to our experience, the actual noise contribution of such commercial preamplifiers with very low T N values is not significantly lower than that of a √ common low-noise bipolar transistor at RT with voltage noise of V n ≈ 0.5 nV/ Hz in the improved readout electronics because of the uncertain impedances. Therefore, the rf SQUID in Figure 12.8 is rather called a low-impedance system, where only the cable impedance is 50 Ω. Using this mixture version (shown in Figure 12.7) and the improved readout electronics (the 150 MHz version shown in Figure 12.8), we systematically studied the high-T c thin-film SQUID system noise with different SQUID loop inductances Ls , ranging from 30 up to 600 pH [43]. In this work, it was pointed out for the first time that a large washer SQUID with a single-layer structure could be used for a sensitive high-T c thin-film SQUID magnetometer. As a historical record, the SQUID parameters and the measured system noise are listed in Table 12.1. The experimental results listed in Table 12.1 lead us to a discussion of why the swing reduction of rf SQUID signals (in other words, 𝜕V rf /𝜕Φ) with increasing Ls is not as fast as that in the case of dc SQUIDs. As far as we know, almost no complete data of high-T c dc SQUIDs with Ls > 250 pH at 77 K have been reported to

12.3 Introduction to Modern rf SQUID Systems

Table 12.1 SQUID parameters and the measured system noise with f L ≈ 150 MHz. Inductancea)

(Ls )

(pH)

75

2

2

230 2

300 2

450

600

2

4002

(d × d)

(𝜇m )

50

150

200

300

Washer area

(L × L)

2

(mm )

2

6

2

6

2

6

2

6

62

Transfer function

(𝜕V /𝜕Φ)

(𝜇V/Φ0 )

100

80

60

30

18

Flux noise

(𝛿Φsys )

√ (𝜇Φ0 / Hz)

45

70

100

180

380

(measured/ calculated)

(%)

60

70

79

64

71

Field/flux transfer coefficient

(𝜕B/𝜕Φ)

(nT/Φ0 )

10.5

3

2

1.7

1.1

Field resolution

(𝛿Bn )

√ (fT/ Hz)

470

210

200

310

420

Hole area

A′eff ∕Aeff

b)

a) In the original reference, the SQUID inductance Ls was wrongly reduced by a factor of 1.2. b) The ratio of the measured area A′eff and its calculated area Aeff .

date [47]. However, an rf SQUID signal can be clearly observed with a SQUID inductance Ls ≈ 600 pH at 77 K. Here, we compare the expressions of SQUID signal swing to find the differences between an rf SQUID and a dc one: (1) For an rf SQUID, the expression of V rf, swing = (𝜔rf LT /Mrf ) × (Φ0 /2) in Eq. (12.2) can be rewritten as √ √ Vrf,swing = [𝜔rf LT ∕(k Ls )] × (Φ0 ∕2) or

√ √ QLT ∕ Ls ) × (Φ0 ∕2) at k 2 Q = 1 (12.7) √ Namely, V rf,swing is inversely proportional to Ls at k 2 Q = 1, but it increases √ with the product (𝜔rf QLT ), which is not directly dependent on operating temperature T and does not involve the SQUID parameters. Usually, V rf,swing of a high-T c rf SQUID should be smaller than that of a low-T c rf SQUID. Conceivably, the large intrinsic noise of a high-T c rf SQUID strongly rounds its I rf –V rf curves, thus leading to an additional reduction in V rf,swing . (2) For a current-biased dc SQUID [48], Vrf,swing = (𝜔rf

Vdc,swing = (RN ∕Ls ) × (Φ0 ∕2)

at 𝛽L ≈ 𝛽c ≈ 1

(12.8)

Therefore, the value of V dc,swing is inversely proportional to Ls . Furthermore, the RN of high-T c dc SQUIDs at 77 K is much smaller than that of low-T c SQUIDs at 4.2 K [49], thus leading to a drastic reduction in V dc,swing . How to increase RN at 77 K is a major challenge for high-T c dc SQUIDs. Additionally, the strong rounding effect on I–V curves caused by thermal noise at 77 K further reduces V dc,swing . Comparing Eq. (12.8) with Eq. (12.7) qualitatively reveals why the reduction in V dc,swing with increasing Ls is much faster than that in V rf,swing . In brief, V dc,swing

187

188

12 Radio-Frequency (rf) SQUID

is completely determined by the SQUID’s parameters, RN and Ls , while V rf,swing mostly depends on the external circuitry. In other words, to obtain the maximum Ls,max with a sufficiently readable SQUID signal at 77 K, the Ls,max of an rf SQUID is much larger than that of a dc SQUID. In this sense, a sensitive single-layer magnetometer is easier to realize as an rf SQUID with a large Ls , and a large washer can be employed in order to enhance the effective area of the rf SQUID magnetometer. 12.3.3

Tank Circuit Operating Up to 1 GHz with Inductive Coupling

A merit of rf SQUID systems is the need for just one coaxial cable to connect the RT readout electronics to the two-terminal element (SQUID + tank circuit) at CT. In the capacitor tap shown in Figure 12.7, the capacitor C 1 is practically shunted by a choke (inductor) LD , which provides a feedback current, I f , for the FLL, so one transmission cable suffices. Actually, LD mixing into the LC resonant circuit leads to three negative consequences: (i) the reduction of the Q of the tank circuit, (ii) the limit of f L because of the large distributed capacitance of LD , and (iii) the introduction of rf interference from the environment because of the antenna effect. To replace the capacitor tap (with LD ) of the conventional tank circuit shown in Figure 12.7, He et al. realized an inductive coupling with two separate coils, i.e. LT and Lcou , sketched in Figure 12.9 [50]. Here, the coil Lcou has two functions: (i) Lcou provides the bias current I rf for the LT C T tank circuit via the mutual inductance Mrf′ between LT and Lcou , where LT C T further couples the SQUID via Mrf and (ii) Lcou simultaneously couples the SQUID with Mf to act as the flux feedback coil Lf for FLL operation. In this inductive coupling configuration, the distance d between Lcou and LT can be adjusted to match the cable impedance. In practice, the dimensions of both wire-wound coils are small, where Lcou is only a few turns in a diameter of 1–2.5 mm and LT with the same diameter is less than 10 turns. Furthermore, the two coils are coaxially spaced with a distance d of approximately a few millimeters, as sketched in Figure 12.9. Using this inductive coupling, the rf SQUID could be operated with a pumping frequency f L of up to 1 GHz, which Figure 12.9 Modified conventional LC tank circuit with inductive coupling placed on a thin-film rf SQUID, where the coil Lcou couples to the LT C T tank circuit with M′rf and separately to the SQUID with Mf . Note that the mutual inductance Mrf between the rf SQUID and the tank circuit is not given.

To readout electronics 50 Ω cable

Lcou Mrf′

d Mf LT

CT

12.3 Introduction to Modern rf SQUID Systems

Table 12.2 Characteristics of the tank circuit with an inductive coupling and the SQUID system noise. C T (pF)

18

6.8

3.3

2.2

1.5

1

0.5

f 0 (MHz)

170

280

380

450

530

620

730

Q0

210

230

220

230

230

220

220

f L (MHz)

220

350

490

590

690

790

950

Q

50

50

50

60

60

65

60

23

20

19

17

15

15

15

√ 𝛿Φsys (𝜇Φ0 / Hz)

was first realized using a conventional lumped LC tank circuit connected with RT readout electronics via one 50 Ω transmission cable. Because of historical reasons, we have only obtained the measurement results for high-T c rf SQUIDs in the inductive coupling version. In the rf SQUID fabrication processes, four washer SQUIDs with a 3.5 mm diameter were patterned on a LaAlO3 substrate of 10 × 10 mm2 and subsequently cut to form four individual SQUIDs on 5 × 5 mm2 substrates. The SQUID loop was 100 × 100 mm2 or 10 × 500 mm2 , corresponding to SQUID inductances Ls of 150 or 260 pH and field-to-flux transfer coefficients 𝜕B/𝜕Φ of 10 or 6.5 nT/Φ0 , respectively. The preparation of the YBCO SQUIDs with step-edge junctions has been described before [43]. In the experimental results listed in Table 12.2, only one wire-wound tank circuit coil LT consisting of four turns with a diameter of 2.5 mm was utilized. The coupling (matching) coil Lcou had only two turns with the same diameter. Impedance matching can easily be realized by changing the distance, d, between them. Note that the so-called improved readout electronics in Figure 12.8 can adjust the pumping frequency from 150 MHz up to 1 GHz upon exchanging some rf components, e.g. the VCO. The transfer functions of the tank circuit were measured to determine the quality factors. In Table 12.2, the resonance frequency f 0 and the unloaded quality factor Q0 were measured without the SQUID at 77 K. The resonance frequency f 0 was tuned by changing the capacitor C T of the tank circuit. The quantities with the SQUID in place are labeled as f L and Q. As C T reduces from 18 to 0.5 pF, the resonance frequency f L was enhanced from 220 MHz up to approximately 1 GHz, while the value of the loaded quality factor Q undulated between 50 and 60. Note that only one SQUID with an inductance Ls ≈ 150 pH and one low-noise bipolar transistor at RT were utilized for acquiring the data in Table 12.2. The 𝛿Φsys (white flux noise) of the high-T c rf SQUID system was measured at 77 K. The best value of 𝛿Φsys for the high-T c SQUID was √ 15 𝜇Φ0 Hz down to 10 Hz in the spectrum at f L ≈ 950 MHz. In fact, 𝛿Φsys did not change much, as f L ≥ 600 MHz. Compared with 𝛿Φsys in the capacitor tap configuration (with a choke and f L ≈ 150 MHz) listed in Table 12.1, this noise level is clearly improved. Analyses about amplitude-frequency characteristics for rf SQUID can be found in [51, 52]. These 𝛿Φsys measurements were performed in a three-layer 𝜇-metal shield 100 times on average.

189

190

12 Radio-Frequency (rf) SQUID

In principle, there is no difference between the inductive coupling and the capacitor tap for SQUID operation. However, the choke (inductor) LD , which is present in the capacitor tap configuration in Figure 12.7, limits the operating frequency f L and the rf SQUID system noise 𝛿Φsys listed in Table 12.2. Actually, the inductive coupling configuration with two coils, LT and Lcou , turned out to be suitable for easy self-production and could very easily be adjusted to obtain an optimum V rf,swing . Conversely, the sizes of both coils are very small, thus leading to a small electromagnetic (EM) field distribution in space, which has the benefit of good stability of the SQUID’s rf bias, which will be discussed below. Furthermore, it was proved that the inductive coupling version exhibits good resistance against external high-frequency interference.

12.3.4 12.3.4.1

Modern rf SQUID System Microstrip Resonator

For rf SQUID operation, we have solved the problem of increasing the pumping frequency f 0 to values exceeding 30 MHz with RT readout electronics. By using the capacitor tap or the inductive coupling versions to realize a low-impedance system, the rf SQUID signals can be transported between the tank circuit and readout electronics. However, the above two coupling (tap) versions are mixtures. In the process of rf SQUID modernization, the planarization of the tank circuit should be on the agenda. In 1992, M. Mück and C. Heiden first utilized a half-wavelength (𝜆/2) microstrip resonator with f L ≈ 3 GHz as the tank circuit for an rf SQUID [53, 54]. There, a niobium microstrip was patterned on a dielectric substrate with a metal ground plane (sandwich structure), while the rf SQUID, one superconducting loop interrupted by a Josephson junction, was integrated at the microstrip center, as shown in Figure 12.10. The maximum rf voltages appear at the two ends of the resonator: at one end, the SQUID signal is coupled out with a small capacitor and further transmitted to the RT readout electronics via a 50 Ω coaxial cable. In fact, the SQUID signal at the input of the preamplifier is strongly reduced, i.e. V rf,input = 𝜅V rf,swing . In contrast to the rf voltage distribution in the microstrip, the maximum rf current appears at its center to couple with the SQUID. In fact, the rf coupling between the SQUID and microstrip (resonator) becomes very complicated because it can be considered as a “compound”; i.e. one part of the coupling is galvanic and the other is inductive. In a conventional rf SQUID system, the coupling between the SQUID and the tank circuit is purely inductive. In this microstrip version, both the length of the microstrip and the relative permittivity 𝜀 (formerly called dielectric constant) determined f L ≈ 3 GHz, which is 2 orders of magnitude higher than that of the conventional version (30 MHz) and almost 1 order of magnitude higher than the 430 MHz version used in Long’s experiment [28]. For a niobium rf SQUID with √Ls = 150 pH, (i.e. a loop size of 100 × 100 𝜇m2 ), a system noise 𝛿Φsys of 4 𝜇Φ0 / Hz was measured at 4.2 K [53]. A readout SQUID signal of 𝜕V rf /𝜕Φ ≈ 160 𝜇V/Φ0 at the input of the preamplifier was not large enough to suppress the preamplifier’s noise 𝛿Φe

12.3 Introduction to Modern rf SQUID Systems

Figure 12.10 Schematics of the first planar tank circuit for rf SQUID operation, i.e. a superconducting halfwavelength (𝜆/2) microstrip resonator patterned on a dielectric substrate and integrated with an rf SQUID structure at the center. Here, a metal ground plane is necessary.

Superconducting micro-strip

Dielectric substrate

SQUID λ/2

Metal groundplane

√ because of the small tap voltage ratio 𝜅. In other words, the 𝛿Φsys of 4 𝜇Φ0 / Hz was clearly dominated by the readout electronics noise 𝛿Φe . In this 3 GHz version, the calculated to √ tank circuit (microstrip) noise was already reduced √ −7 𝛿ΦT ≈ 6 × 10 Φ0 / Hz. An intrinsic SQUID noise 𝛿Φs of 1 𝜇Φ0 / Hz was estimated. Here, the ratio of the three noise parts in the microstrip SQUID system was changed to 0.6(𝛿ΦT ) : 1(𝛿Φs ) : 4(𝛿Φe ). Interestingly, 𝛿ΦT < 𝛿Φs appears in the microstrip version. According to Long’s experiment, we can assume that the effective temperature of the tank circuit, T e,T , may be close to 4.2 K because of a small tap ratio 𝜅, so the tank circuit noise 𝛿ΦT becomes the smallest component in the total system noise 𝛿Φsys . The readout SQUID signal of 𝜕V rf /𝜕Φ ≈ 160 𝜇V/Φ0 is still at least five times smaller than the value required to suppress the preamplifier’s noise 𝛿Φe below the SQUID noise 𝛿Φs . In brief, the measured noise level of 𝛿Φsys at 3 GHz was lower than that at 30 MHz but √ still higher than the 𝛿Φsys of 3 𝜇Φ0 / Hz for a bulk rf SQUID with Ls ≈ 500 pH obtained with a GaAs FET as a preamplifier located at 4.2 K in Long’s experiment. In fact, in the comparison of 𝛿Φsys with different SQUID inductances Ls , one indeed can directly compare their SQUID energy resolutions (in units of J/Hz). A similar microstrip of 𝜆/2 with f L ≈ 3 GHz was also integrated into a high-T c version [55], and a 𝜆-ring resonator integrated with an rf SQUID was simultaneously developed [56]. In the latter, the planar ring formed with a wavelength 𝜆 plays a dual role, i.e. as an rf resonator (tank circuit) and a dc flux pickup loop, in forming a SQUID magnetometer. The fundamental idea and the layout of the 𝜆-ring resonator are shown in Figure 12.11. In brief, the microstrip version constituted the first demonstration of a tank circuit in a planar structure without lumped elements. Here, the superconductive resonators yield a very high impedance because of the high Q and f L (e.g. 3 GHz); therefore, the value of the tap voltage ratio 𝜅 must be very small to match a low-impedance cable. In this case, an ultralow noise of the tank circuit, 𝛿ΦT , is obtained, thus supporting our above analysis in Section 12.3.1. However, the readout electronics (preamplifier) noise 𝛿Φe completely dominates the SQUID system noise 𝛿Φsys in this version. Furthermore, the microstrip version of a 𝜆/2 or 𝜆 resonator is unsuitable for the purpose of a magnetometric system because the metal ground plane in the microstrip version is

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12 Radio-Frequency (rf) SQUID

4.5 0.3

Ring resonator

5 1.12 SQUID

1 YBCO

(mm) (a)

(b)

(c)

Figure 12.11 (a) Schematic diagram of the 𝜆 resonator integrated with an rf SQUID; (b) layout of the ring resonator with an integrated rf SQUID operating at 3 GHz; and (c) magnified SQUID area, where the SQUID loop is a slit.

a non-negligible noise source because of eddy currents in the metal, which increase with decreasing operating temperature T [57]. Although one can reduce the thickness of the metal ground plane to reduce the eddy currents [58], the quality factor of the resonator, Q, also decreases. Nevertheless, the microstrip version serves as our inspiration for setting up a practical SQUID magnetometer using a planar tank circuit. The other shortcoming of the 3 GHz version is that one needs an additional wire to provide a dc current I f path for FLL operation because of the capacitive coupling between the 𝜆/2 (𝜆) microstrip and the cable for rf signal transmission with low-impedance matching. To recover the ability to use only one cable for transmissions of rf signals and feedback current (I f ), a readout loop (inductive coupling) was developed for a planar resonator, where a 𝜆/2 microstrip surrounds a large-area SQUID washer as the flux concentrator [59]. Here, we do not want to give more information about this development because a similar resonator in a coplanar structure was subsequently found. 12.3.4.2

Coplanar Resonator

To avoid eddy currents from the metal ground plane, taking two superconducting microstrips supplemented with each other to form a coplanar resonator can replace the metal ground plane [60, 61]. In the coplanar resonator, the resonant current in one microstrip and its mirror current in the other microstrip are supplementing each other. Only when the coplanar resonator (in the coplanar version) serving as the tank circuit was employed in the rf SQUID magnetometric system, did the advantage of a planar tank circuit became meaningful for applications. In the development of high-T c SQUID magnetometric systems, a single-layer structure has always been preferred. We first arranged a high-T c coplanar resonator surrounded by a large flux concentrator, where a small SQUID chip with a size of 5 × 5 mm2 was located at its center in the flip-chip configuration to construct a sensitive magnetometer [60]. Actually, the flux concentrator with the

12.3 Introduction to Modern rf SQUID Systems

(a)

ϕ1.5

(b)

θ

(d) 50 Ω cable

0.1

Readout loop

(c) 0.1 0.2

(A)

d

0.1 ϕ14.4 (mm)

s (B)

Figure 12.12 (A) Schematic of three flux concentrator/coplanar resonator layouts for resonance frequencies of (a) 850 MHz, (b) 650 MHz, and (c) 600 MHz. (B) An rf washer SQUID 3.5 mm in diameter is placed onto the flux concentrator to form a large washer rf SQUID with an integrated coplanar resonator. Here, a copper wire loop (readout loop) couples to the resonator with M′rf and simultaneously provides a feedback flux ΦF via Mf for FLL operation of the washer SQUID.

SQUID chip can be regarded as a large washer SQUID. Here, the coplanar resonator inductively couples to the washer SQUID via an rf mutual inductance, Mrf . Without doubt, the key aspect in this version is the coplanar resonator. The shapes of the coplanar resonators can be either square or circular. Figure 12.12A schematically shows some examples of the integration of a circular coplanar resonator with a flux concentrator. All superconducting structures were patterned from 200 nm epitaxial YBCO on 1 mm-thick LaAlO3 substrates. Furthermore, the relative opening positions of two microstrips can regulate the resonance frequency. The three layouts in (a), (b), and (c) shown in Figure 12.12A differ only by the mutual angular position 𝜃 of the resonator line (microstrip) slits. With increasing 𝜃, f 0 decreases from (a) 850 MHz and (b) 650 MHz to (c) 600 MHz. The unloaded quality factors, Q0 , measured without the standard rf electrical shielding (metal box), were in the range between 4000 and 5000 at 77 K for all three layouts. With a high Q0 and high f 0 , a high impedance at the resonator (the tank circuit) was obtained. One can find more information about the coplanar resonators in Ref. [61]. An important development was the use of a copper wire loop (readout loop) to inductively couple the resonator. This coupling method was directly adopted from the planar version [59] to the coplanar version, as shown in Figure 12.12B, to perform impedance matching from the tank circuit at 77 K to the preamplifier input (low impedance) at 300 K. With adjustments of the wire loop diameter S or the distance d between the coplanar resonator and the readout loop, one can easily achieve optimal low-impedance matching, where the maximum 𝜕V rf /𝜕Φ appears at the input of the preamplifier. In the coplanar resonator version, a loaded Q of several hundred was usually achieved at frequencies f L above several hundred MHz [60, 61]. Thus, a low effective operating temperature of the tank circuit, T e,T , can be achieved because of a small tap voltage ratio 𝜅.

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Meanwhile, a feedback current I f flowing through the readout loop generates a feedback flux ΦF via Mf between the washer SQUID (in flip-chip configuration) and the readout loop for FLL operation, so the superiority of just one cable is regained in the modern rf SQUID version with the coplanar resonator (tank circuit). Note that the two mutual inductances, Mrf′ and Mf , are quite different. Here, the condition k 2 Q > 1 can easily be fulfilled using a coplanar resonator. With this innovation, a truly modern rf SQUID magnetometric system has been developed. The system includes a thin-film rf SQUID with a large washer area, a coplanar resonator, improved readout electronics at RT, and just one 50 Ω cable with a readout loop as connection between CT and RT. Using the arrangement of the coplanar resonator version shown in Figure 12.12B and two rf SQUIDs with different Ls values, two important achievements are again exhibited here: (i) for the first time, a measured rf SQUID system noise 𝛿Φsys in √ √ the 𝜇Φ0 / Hz range with Ls ≈ 150 pH has been achieved, 6.5 𝜇Φ0 / Hz (white noise), which is by more than a factor of 2 lower than the best value in Table 12.2 obtained using the lumped element tank circuit with an inductive coupling, and (ii) using a circular coplanar resonator with a diameter of 12.4 mm and a a resonance frequency f 0 ≈ 650 MHz (see the layout in Figure 12.12A(b)), √ high-T c SQUID magnetometer achieved a field sensitivity of 16 fT/√ Hz in the white-noise regime, corresponding to a flux noise 𝛿Φsys of 8.5 𝜇Φ0 / Hz for the SQUID with Ls ≈ 260 pH. Here, both SQUIDs employed were operated at 77 K and 𝛽 e ≈ 1. In the 1990s, these results represented a remarkable achievement. Compared to the first microstrip version shown in Figure 12.10, the coplanar version makes three great improvements for a practical magnetometric system: (i) the metal ground plane is removed, so a major noise source is eliminated. (ii) The readout loop, as a kind of inductive coupling, provides not only the bias rf power for the tank circuit (resonator) but also the feedback flux ΦF for FLL operation. Thus, the system can operate with one coaxial cable. (iii) The coupling between the SQUID and the tank circuit is purely inductive, similar to the conventional configuration. 12.3.4.3

Instability of rf Bias Current

Using coplanar resonators as tank circuits, a new problem emerges, i.e. the stability of the rf bias flux, which is generated by the bias current, I rf,bias , via Mrf′ and Mrf . Generally, the rf radiation of the tank circuit cannot be avoided. In the coplanar version, the rf energy of I rf,bias does not focus onto the rf SQUID directly; instead, the readout loop first transmits the rf energy to the coplanar resonator with Mrf′ . Then, the coplanar resonator provides the rf energy for the rf SQUID via Mrf . Because of the large diameters of both the readout loop (transmitting antenna) and the resonator (receiving antenna) and the distance d between them (see Figure 12.12), the rf EM field distribution occupies a large space. Furthermore, the high-Q coplanar resonator strengthens the EM field. As schematically shown in Figure 12.13a, inside this large EM space, any movements of metals and dielectric materials can change the EM field distribution, thus changing the rf bias flux threading the SQUID’s loop. In practice, such disturbances cannot be controlled. This instability of rf SQUID bias flux will

12.3 Introduction to Modern rf SQUID Systems

Metal

irf

ib

Dieletric (a)

(b) Niob screw

(c)

Lin

LT

Figure 12.13 (a) In the rf SQUID tank circuit, the schematic rf EM field distribution is changed by any movements of metals and dielectric materials, thus causing an instability of the bias current Irf,bias . Especially, in the coplanar resonator version, the EM field occupies a large space. (b) In dc SQUID operation, the bias current Ib changes only when a resistor shunts to the SQUID. (c) The profile along the diameter of a toroidal bulk niobium rf SQUID (the simple vertical view is not shown). Two wire-wound coils, LT and Lin , are separately set inside the inner ring and outer ring. Note that Lin must be superconducting for the flux transfer coil system in a SQUID magnetometric system.

be translated into additional voltage noise across the coplanar resonator, that is, additional SQUID flux noise in the low-frequency regime at least. In some cases, e.g. if a large metal plate is moving near the SQUID outside the dewar, the SQUID’s V rf (Φ) characteristics are distorted because of the deviation from the optimal biasing condition of the rf SQUID. In contrast, a planar dc SQUID never suffers from the problem of bias current I b instability because I b is provided through a galvanic connection. Only when a small resistor shunts to SQUID, does I b change (see Figure 12.13b). For rf SQUID operation, this instability of rf bias condition did not occur for bulk rf SQUIDs in the 30 MHz version, where the EM field was relatively concentrated in the bulk SQUID hole, in which the wire-wound coil LT of the tank circuit was placed (see Figure 12.13a). In other words, the stability of rf bias condition in bulk rf SQUIDs was intrinsically given. In particular, the toroidal niobium-bulk rf SQUID was a very successful design [62, 63]. Here, its profile along the diameter is shown in Figure 12.13c. The rf tank coil LT is located inside the inner groove (ring), so the EM field distribution is effectively limited because of the superconducting toroidal shield.

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For thin-film rf SQUIDs, using tank circuits with lumped elements in the capacitor tap or the inductive coupling version, this instability of I rf,bias is still not serious because the EM fields are concentrated inside the small diameter of the spiral coil LT . The next step of rf SQUID development should solve the problem of how to stabilize rf bias condition for the planar resonator configurations. 12.3.5

Substrate Resonator

Generally, the requirement of k 2 Q > 1 for the coupled resonator version is not difficult to fulfill. For example, using a coplanar resonator with a high Q, as k 2 Q ≈ 1, the coupling coefficient k between its equivalent inductance LT and the SQUID’s Ls can be very small. Therefore, most EM energy from the resonator does not couple to the SQUID but instead diffuses into space. The new challenge is how to limit the EM spatial distribution into a small space for thin-film rf SQUID operation at high 𝜔rf using a modern planar tank circuit with a high Q. In 1995, we found that the dielectric substrate (SrTiO3 ) resonator concept may be suitable for high-T c thin-film rf SQUID operation and demonstrated the V rf (Φ) characteristics of a YBCO rf SQUID with this concept [64]. At that time, we did not find a good method for coupling the SQUID’s rf signals from the resonator. In 2002, the SrTiO3 substrate resonator concept was perfected for a practical SQUID magnetometric system [65] while maintaining the high Q of the resonator at f L > 500 MHz and 50 Ω transmission line technique. Now, we explain its highlights. The use of a substrate resonator as the tank circuit was an important milestone in rf SQUID development. Quasi-total reflection occurs when the EM wave travels from one medium with a high relative permittivity 𝜀 to another medium with a low 𝜀. Namely, one can make dielectric resonators (high 𝜀) in air medium (low 𝜀) by utilizing some canonical forms, e.g. a sphere, a cylinder, or a cuboid. Usually, dielectric resonators can achieve very high quality factors Q [66, 67]. In fact, epitaxial YBCO thin films are usually grown on SrTiO3 or LaAlO3 substrates [36, 49, 68, 69]. The relative permittivity 𝜀 of SrTiO3 is larger than 2000 at 77 K, thus providing the possibility of using a SrTiO3 substrate directly as a resonator (tank circuit). The resonance frequency f 0 of a bare SrTiO3 substrate with a size of 1 × 10 × 10 mm3 (standard) was measured to be approximately 2 GHz with an unloaded Q0 of a thousand in air medium at 77 K [64]. When one of the two main sides with an area of 10 × 10 mm2 in the cuboid-substrate resonator is covered with a superconducting YBCO thin film, f 0 reduces down to 1. The readout electronics with adjustable f L (≈ 650 MHz) are located at RT. In addition, a heater consisting of a thin-film carbon strip is thermally connected to the SQUID chip. There is an option to enlarge the area of the concentrator, thus enhancing the field sensitivity 𝛿B of the magnetometer (see Figure 12.14b). Here, the flux concentrator consists of two stages. The SrTiO3 substrate with the first flux concentrator acts as a resonator coupled to the rf SQUID via Mrf . The SQUID resonator package is further arranged on a second concentrator with a larger area, e.g. 18 mm in diameter, in flip-chip configuration. A hole 8 mm in diameter is located at the center of the second concentrator substrate, which is typically made of LaAlO3 . Note that the second concentrator does not play a role in the resonator

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12 Radio-Frequency (rf) SQUID

102

102

101

100

101

102

103

Magnetic field noise (fT/√Hz)

Flux noise (μΦ0 /√Hz)

198

101 104

Frequency (Hz)

Figure 12.15 Noise spectrum of a YBCO rf SQUID magnetometer with a SrTiO3 dielectric substrate resonator, where Ls ≈ 150 pH, i.e. a SQUID loop of 100 × 100 𝜇m2 . The measured √ system flux noise 𝛿Φsys is conceivably located in the range of 𝜇Φ0 / Hz.

because of its small 𝜀. The SQUID resonator package is placed in the hole of the second concentrator substrate. In practical use, the oxidation resistance of thin-film SQUIDs is also important, especially for high-T c YBCO. All concepts above, e.g. the conventional tank circuit with an inductive coupling, a coplanar resonator, and a substrate resonator, should be encapsulated with glass fiber, where only two bonding pads (two terminals, not including the heater) are exposed. Using the basic setup of the substrate resonator in Figure 12.14a, for a high-T c YBCO rf SQUID with Ls ≈√150 pH (i.e. with a loop size of 100 × 100 𝜇m2 ), a system noise 𝛿Φsys of 7 𝜇Φ0 / Hz was measured in the white-noise regime, corre√ sponding to a field sensitivity of 𝛿B ≈ 24 fT/ Hz, as shown in Figure 12.15. Here, a flux-to-field transfer coefficient of 𝜕B/𝜕Φ ≈ 3.2 nT/Φ0 was measured [65]. The noise measurement shown in Figure 12.15 is very striking. Now, we utilize this measured data to analyze the rf SQUID system noise 𝛿Φsys consisting of three parts,√i.e. 𝛿Φe , 𝛿Φs , and 𝛿ΦT : (i) Assuming a preamplifier voltage noise of V n ≤ 1 nV/ Hz and an rf SQUID transfer coefficient of (𝜕V rf /𝜕Φ) ≈ 320 𝜇V/Φ0 at the input of preamplifier at RT (this value will be given below in Section √ 12.4.1), the flux noise from the preamplifier, 𝛿Φe = V n /(𝜕V rf /𝜕Φ) ≤ 3.2 𝜇Φ0 / Hz, is estimated. (ii) The intrinsic noise 𝛿Φs of the YBCO rf SQUID at 77 K should be 4.5 times larger than that of the low-T c SQUID at 4.2 K because of the thermal noise. The classic noise limit 𝛿Φs,limit (see Eq. (12.4)) of the rf SQUID with Ls ≈ 150 pH at √ 77 K should be 5 𝜇Φ0 / Hz, which is generated by the junction thermal noise of, e.g. RN ≈ 1 Ω. In fact, there is no junction shunt resistance RJ , but the high-T c step-edge junction exhibits only RN , which is the junction’s normal resistance as I ≫ I c . However, it is impossible to determine the RN of an rf SQUID nondestructively because of the closed superconducting loop. Most high-T c junctions exhibited RN ≤ 1 Ω at 77 K in the early stage of YBCO research. Here, we assume

12.3 Introduction to Modern rf SQUID Systems

√ the real 𝛿Φs ≈ 𝛿Φs,limit ≈ 5 𝜇Φ0 / Hz at RN ≈ 1 Ω. (iii) The effective operating temperature T e,T of the tank circuit (resonator) could be close to 77 K because the lead to a small tap voltage ratio 𝜅. Indeed, high Q and high f L of the resonator √ the measured 𝛿Φsys of 7 𝜇Φ0 / Hz in Figure 12.15 is the sum of 𝛿Φe , 𝛿Φs , and √ 𝛿ΦT , thus obtaining 𝛿ΦT ≈ 3.2 𝜇Φ0 / Hz. Namely, the ratio of the three noise parts becomes (𝛿Φe ) : (𝛿ΦT ) : (𝛿Φs ) ≈ 1 : 1 : 1.6. This result is very exciting: 𝛿Φs clearly dominates the system noise 𝛿Φsys of an rf SQUID. The white noise values measured in the substrate resonator version and in the coplanar resonator version are almost the same. However, compared to the coplanar resonator version, the substrate resonator version improves the low-frequency noise because of its confined EM distribution. Because glass fiber encapsulation can well protect YBCO thin films, one encapsulated rf SQUID magnetometer with this basic setup in Figure 12.14a worked for more than 14 years at Research Center Juelich (FZJ) in Germany. During that time, the SQUID was often thermally cycled between 77 K and RT and was usually kept in ambient air. At the end of the section, we estimate the effective inductance LT,e of the coplanar resonator and of the substrate resonator because there is no lumped inductive element. In principle, the value of LT,e of the resonator may be calculated from its geometrical sizes and the substrate relative permittivity 𝜀, but the current distribution in the resonators for coupling to an √ rf SQUID is very complicated. Thus, the effective mutual inductance Mrf = k LT,e Ls , in other words, the coupling factor k, is difficult to be determined. In practice, we can estimate the product of k 2 Q according to the inflection positions A and B in the I rf –V rf curves of the rf SQUID, as discussed in Figure 12.2, where Q is measurable. The rf voltage swing across the resonator should still maintain the following expression (see Eq. (12.2)): Vrf,swing = (𝜔rf LT,e ∕Mrf ) × (Φ0 ∕2) √ where Mrf = k LT,e Ls and LT,e is the effective inductance of a resonator. √ √ At k 2 Q ≈ 1, Eq. (12.2) can be modified to read Vrf,swing = (𝜔rf QLT,e ∕ Ls ) × (Φ0 ∕2) (see Eq. (12.7)). Namely, the value of LT,e can be estimated from the retrievable parameters V rf,swing , 𝜔rf , Q, and Ls . In the different resonator versions, the product of 𝜔rf × Q can be very high, thus leading to a large V rf,swing across the resonator with a high impedance of Zreso ≈ 𝜔rf LT,e Q. However, the large V rf,swing cannot be transmitted directly, as discussed above. In fact, because of the low-impedance matching, a higher Zreso of the resonator becomes a lower impedance at the preamplifier because of the tap voltage ratio 𝜅 ≈ 50 Ω/Zreso , where 50 Ω is just a representation of the low impedance. However, the SQUID’s voltage swing at the input of the preamplifier, V rf,input = 𝜅V rf,swing , is measurable. Therefore, based on Eq. (12.7), we can estimate the effective inductance of the resonator LT,e at the condition of k 2 Q ≈ 1, according to the formula: LT,e = 4(Ls ∕Q) × [Vrf,swing ∕(𝜔rf Φ0 )]2

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12.3.6

Regarding the rf SQUID’s Thermal Noise Limit

In fact, there are three potential reasons that the intrinsic noise 𝛿Φs of an rf SQUID should be smaller than that of a dc SQUID: (i) instead of two noise sources (two junction shunt resistors) in the dc SQUID, only one noise source exits in the rf SQUID because it has just one junction, thus further avoiding the problem of “out-of-phase” noise caused by the two junctions in the dc SQUID [70, 71]. (ii) According to the operation principle of an rf SQUID, the shielding current irf surrounding the SQUID loop will never surpass the critical current I c of the junction, so no high-frequency radiation from the junction exists. This radiation leads to additional noise for dc SQUID operation, as mentioned in Section 6.1. (iii) In normal dispersive mode (𝛽 e < 1), no quantum transitions between the flux quantum states occur, so the rf SQUID’s intrinsic noise 𝛿Φs should be close to the thermal noise limit [30]. Unfortunately, the SQUID’s V rf,swing is too small at 𝛽 e ≪ 1, thus leading to difficulty in reaching a sufficiently low readout electronics noise 𝛿Φe to read small 𝛿Φs values. In fact, this problem was thoroughly discussed in the context of dc SQUID systems, e.g. Chapters 5 and 6. One always dreams that the measured system noise 𝛿Φsys approaches the SQUID’s thermal noise limit, 𝛿Φs,limit [30, 72, 73]. In the different resonator versions, suppression of the readout electronics noise 𝛿Φe becomes the top priority. In 1975, Danilov and Likharev predicted that very high 𝜕V rf /𝜕Φ at working point W will appear when (i) the rf SQUID is operated in dispersive mode; (ii) the operation frequency 𝜔rf is kept well below the frequency limit, i.e. 𝜔rf ≪ RJ /Ls ; (iii) a high Q of the tank circuit is realized for maintaining the high impedance of the resonator; and (iv) the condition k 2 Q𝛽 e > 1 is observed [8, 74, 75]. We call these four requirements the “D–L requirements.” In this case, it is possible that a high 𝜕V rf /𝜕Φ at W can be obtained in the range of mV/Φ0 . In 1982, Dmitrenko et al. carried out measurements of the current–voltage, current–phase, amplitude–frequency, phase–frequency, and signal characteristics of a low-T c rf SQUID operating at a pumping frequency of 30 MHz at 4.2 K, where the above D–L requirements were fulfilled [76]. In spite of the unchanged swing V rf,input , higher harmonic components in the V rf (Φ) transfer function appeared, thus dramatically enhancing the 𝜕V rf /𝜕Φ up to several mV/Φ0 at working point W. Such phenomena were observed during dc SQUID operation when its parameter 𝛽 c > 1 (see Figure 6.2). Generally, the high 𝜕V rf /𝜕Φ obviously reduces not only the readout electronics noise 𝛿Φe but also the tank circuit noise 𝛿ΦT . Unfortunately, no measured data of system noise 𝛿Φsys were reported in that work. Most likely, the tank circuit noise 𝛿ΦT still dominated 𝛿Φsys , as the readout scheme in the 30 MHz version (see Figure 12.6) was employed. The tank circuit at CT and preamplifier input at RT, both of which exhibit high impedance, were connected with transmission lines. It is expected that such high harmonic components of V (Φ) also appear at the low-impedance input of the preamplifier in the different resonator versions. Thus, an unusually large 𝜕V rf /𝜕Φ at working point W may suppress 𝛿Φe below 𝛿Φs . Eventually, the resonator concepts can achieve this dream that the measured system noise approaches the SQUID thermal noise limit, 𝛿Φsys ≈ 𝛿Φs,limit .

12.4 Further Developments of the rf SQUID Magnetometer System

12.4 Further Developments of the rf SQUID Magnetometer System For further developments of rf SQUID magnetometer systems, two works cannot be ignored: (i) increasing the transfer coefficient 𝜕V rf /𝜕Φ to reduce the readout electronics noise, 𝛿Φe , as just mentioned and (ii) integrating a multiturn input coil with a small field-to-flux transfer coefficient of 𝜕B/𝜕Φ, thus forming a magnetometer with high field sensitivity. 12.4.1 Achievement of a Very Large 𝝏V rf /𝝏𝚽 in a Low-Impedance System To reduce 𝛿Φe , it has been emphasized many times that the value of 𝜕V rf /𝜕Φ at the working point W indeed is of utmost importance and needs more attention than the voltage swing of the rf SQUID at the preamplifier input, V rf,input . Now, we supplement the above D–L experiments with the properties of our high-T c rf SQUIDs and our resonator concepts. In our experiments, the D–L requirements were fulfilled, and the high-T c rf SQUID’s V rf (Φ) characteristics at the input of the preamplifier did not exhibit a conventional quasi-sinusoidal form or a triangular form. Instead, they contain several higher harmonic components. Namely, the V rf (Φ) characteristics become a kind of quasi-square wave, thus greatly enhancing 𝜕V rf /𝜕Φ at working point W. With the improved readout electronics shown in Figure 12.8, the mixer used as a detector enriched the amplitude and phase of the demodulated SQUID’s signal. Figure 12.16a shows the completed V rf (Φ) characteristics of an rf SQUID coupled with a substrate resonator at 77 K using the basic setup shown in Figure 12.14a. Furthermore, the excellent system noise data displayed in Figure 12.15 were based on this V rf (Φ) characteristics. Figure 12.16b schematically demonstrates the influence of three main waveforms on 𝜕V rf /𝜕Φ at working point W. Under a certain voltage swing V rf,input , e.g. approximately 60 𝜇V, at the

(a)

Φ0

Φ

W

(b)

∂V/∂Φ = 120 μV/Φ0 188 μV/Φ0 314 μV/Φ0

60 μV

V

60 μV

V

Φ

Figure 12.16 (a) Measured V rf (Φ) characteristics of an rf SQUID coupled with a substrate resonator at 77 K; (b) the different 𝜕V rf /𝜕Φ values at working point W under a given swing of V rf,input ≈ 60 𝜇V. In the interval of Φ0 , the black line (solid) represents the experimentally recorded curve, while the gray line and the dotted line denote sinusoidal and triangularshaped signals, respectively.

201

202

12 Radio-Frequency (rf) SQUID

input of the preamplifier, different 𝜕V rf /𝜕Φ values appear. Here, 𝜕V rf /𝜕Φ values of 120, 190, and 320 𝜇V/Φ0 are obtained at working point W for a triangular signal, for a sinusoidal signal, and for the measured quasi-rectangular signal, respectively. Obviously, the higher harmonic components of the measured signal in Figure 12.16a enlarged the 𝜕V rf /𝜕Φ at working point W by about a factor of 2 compared to that of a purely sinusoidal signal. Let us take a look on the origin of the higher harmonic components in the V rf (Φ) characteristics of an rf SQUID. We start from a theoretical formula for the SQUID characteristics derived in the early days of SQUID theory. Hansma [7, 77] derived an analytical description of the rf SQUID voltage in the limit of a very small 𝛽 e at zero temperature. Based on this formulation, the dimensionless squared voltage across the tank circuit of an rf SQUID in dispersive mode can be described as [78]: ( ) I2 2𝜋 •ID −2I k 2 Q𝛽e •J sin[tan−1 (𝜉)] cos(2𝜋𝜙) UD2 = 2 D + 2 D • 1 𝜉 +1 𝜉 +1 𝜋 𝜉2 + 1 ⏟⏟⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ K

p •K

( ) k Q𝛽e 2𝜋 •ID 1 2 cos2 (2𝜋𝜙) + J 𝜋 𝜉2 + 1 1 𝜉2 + 1 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ (

2

)2

(12.9)

q •K

Here, the dimensionless voltage U D is a normalization of the SQUID characterM M Q istics given by UD = 𝜔L rfΦ Vrf (Φ), the dimensionless current I D is ID = Φrf Irf , the T 0 ( 0 ) Δf frequency variable 𝜉 = 2Q 𝜔𝜔 − 1 = 2Q f , the normalized flux 𝜙 = ΦΦ , and the 0 0 0 Bessel function of first kind, J 1 (x). ( ) ID2 2k 2 Q𝛽 2𝜋 •I Using the three coefficients K = 𝜉 2 +1 , p = − 𝜋 •I e •J1 √ 2 D sin[tan−1 (𝜉)], and 𝜉 +1 D ( )2 ( ) 2𝜋 •ID 1 k 2 Q𝛽e 2 √ marked by parentheses in Eq. (12.7), the dimensionless J1 q = I2 𝜋 𝜉 2 +1 D tank circuit voltage takes the simple form √ √ (12.10) UD = K 1 + p cos(2𝜋𝜑) + q cos2 (2𝜋𝜑) Recently, we performed an expansion of Eq. (12.10) to the first three harmonic components [79], yielding ⎡ ⎢ ) ( ) √ ⎢( p 3pq 3p3 q p2 + cos(2𝜋𝜑) − + UD = A ⎢ 1 + − 4 16 2 16 64 ⎢ ⎢⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⎣ A0 A1 ⎤ ⎥ ( ) ) ⎥ q p2 p3 pq + cos(4𝜋𝜑) + cos(6𝜋𝜑) + · · ·⎥ − − 4 16 64 16 ⎥ ⏟⏞⏞⏞⏟⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ ⎥ ⎦ A2 A3 (

(12.11)

12.4 Further Developments of the rf SQUID Magnetometer System

From this equation, the occurrence of the second and third harmonic terms can be seen directly. Harmonic terms of even higher order will emerge if the expansion is continued. The optimum working point, i.e. the global extreme of 𝜕U D /𝜕𝜙, 𝜕2 U is found as one of the zeros of the second derivative, 𝜕𝜙2D = 0. Transforming this equation to a polynomial in cos(2𝜋𝜙) using trigonometric identities and using the replacement variable u = cos(2𝜋𝜙) leads to the following third-order polynomial for the zeros u: 36A3 u3 + 8A2 u2 + (A1 − 27A3 )u − 4A2 = 0. The real zeros of the cubic polynomial, 𝜙i = (cos−1 ui )/2𝜋, denote the locations of the regionally steepest gradients. At least one real zero exists, but there may be three zeros. The 𝜙i that yields the maximum value of |𝜕U D /𝜕𝜙| is the optimum working point of the rf SQUID. Of course, a SQUID experimentalist never solves such complicated mathematical equations in order to find the optimum working point. The only aim of presenting this mathematical derivation is to explain the existence of higher harmonics in the rf SQUID characteristics. In practice, it is very simple to experimentally find the value of steepest ascend in the characteristic V rf (Φ) and set the working point there. In practice, it is usually sufficient to just carry the first three harmonic coefficients, as we did in Eq. (12.11). In this case, an rf SQUID signal can be characterized by just five numbers, namely the first three harmonic amplitudes, the offset voltage, and the flux-to-voltage coefficient. If the SQUID signal is very steep, as shown in Figures 12.16a and 12.17, further harmonic components up to fifth or seventh order should be included in order to achieve a better agreement between experiment and theory. Experimentally, the harmonic analysis proved favorable for approximation of the measured flux-to-voltage characteristics of the rf SQUIDs in dispersive mode, i.e. in a Levenberg–Marquardt fit of Eq. (12.11) to the measured characteristics V rf (Φ). The harmonic analysis is valid for well-defined flux-to-voltage characteristics obtained with a fixed choice of the adjustable parameters of the rf SQUID electronics, such as rf frequency, attenuator voltage, phase shift, and offset voltage. Of course, all these parameters have to be adjusted first before a harmonic analysis can be carried out. However, the optimization process for finding the optimum values for rf frequency, attenuator voltage, phase shift, and offset voltage can be V

∂V/∂Φ = 330 μV/Φ0@W

W

Φ0

Vrf,input

Figure 12.17 Typical V rf (Φ) characteristics of rf SQUIDs in the planar labyrinth resonator version, where Ls ≈ 210 pH and 𝛽 e ≤ 1 [76]. In this measurement with a pumping frequency of f L ≈ 460 MHz, a mixer is used as a demodulator that detects the rf SQUID’s amplitude and phase changes.

203

204

12 Radio-Frequency (rf) SQUID

facilitated if the harmonic analysis is automatically carried out in the background by software while performing changes in these four parameters. The relationship between the settings of the adjustable parameters of the electronics and the ensuing flux-to-voltage characteristics was experimentally examined in Ref. [80]. Actually, the V rf (Φ) characteristics of rf SQUIDs in superconducting planar resonator versions can exhibit various forms if the D–L requirements are fulfilled. Figure 12.17 shows the rf SQUID’s V rf (Φ) characteristics with V rf,input ≈ 40 𝜇V using the planar labyrinth resonator described in Section 12.4.2. Because of the abundant higher harmonic components, compared to that of a sinusoidal shape, the 𝜕V rf /𝜕Φ ≈ 330 𝜇V/Φ0 at the input of the preamplifier increases by approximately three times. Conceivably, a value of (𝜕V rf /𝜕Φ)′ at working point W across the resonators on the order of 10 mV/Φ0 may be achieved, where (𝜕V rf /𝜕Φ)′ ≈ (330 𝜇V/Φ0 )/𝜅 should occur across a high-impedance resonator if 𝜅 ≈ 0.03. In brief, two experimental results in Figures 12.16a and 12.17 confirmed that (i) the D–L requirements for unusually large 𝜕V rf /𝜕Φ at W are valid in the different resonator versions using an inductive coupling technique to match the low-impedance input of the preamplifier via a 50 Ω cable and (ii) based on the high-T c rf SQUID system noise 𝛿Φsys measurements in Figure 12.15, the occurrence of higher harmonic components in the V rf (Φ) characteristics greatly increases the value of 𝜕V rf /𝜕Φ at working point W up to the sub-mV/Φ0 range at the input of the preamplifier. In fact, we reported that a system flux noise 𝛿Φsys √ of 8.5 𝜇Φ0 / Hz was measured for Ls ≈ 260 pH at 77 K in the coplanar resonator version [61]. Consequently, we believe that the dream of achieving the SQUID thermal noise limit, i.e. 𝛿Φsys ≈ 𝛿Φs,limit , has been almost realized with the help of resonators and improved readout electronics, assuming the normal resistance of the step-edge junction RN ≤ 1 Ω. At the same time, realizing RN > 1 Ω is a challenge for high-T c junction preparation because of the large thermal parameter Γ. 12.4.2 Multiturn Input Coil for a Thin-Film rf SQUID Magnetometer with a Planar Labyrinth Resonator In a complete magnetometric SQUID system, an input coil Lin connects to a receiver loop (flux antenna) to pick up the measured signals and couples it to the SQUID via a mutual inductance Min . Note that Lin must be superconducting so that the flux transfer coil system can detect the quasi-dc flux signals and does not contribute additional thermal noise. Regarding this flux transformer, there is no difference between a dc and an rf SQUID system. In the age of bulk rf SQUIDs, three solutions were commonly utilized to place a niobium wire-wound multiturn input coil Lin : (i) in the toroidal rf SQUID configuration shown in Figure 12.13c, Lin is embedded inside the outer ring groove [62, 63]. (ii) In the double-hole configuration (see Figure 12.18a), the cylindrical input coil Lin is put into one hole of the rf SQUID, while the spiral tank coil LT is arranged in the other hole [27, 81]. (iii) In the single-hole configuration (see Figure 12.18b), LT is placed inside the hole, while Lin is wound around the SQUID body. This arrangement was usually suited for cylindrical low-T c

12.4 Further Developments of the rf SQUID Magnetometer System

Min Lin

(a)

Min

Mrf

LT Both inside loop

CT

Mrf

Lin

(b)

LT Outside loop

CT

inside loop

Figure 12.18 Schematic arrangements of two cylindrical coils, Lin and LT , in bulk rf SQUIDs in (a) double-hole and (b) single-hole configurations. Here, Lin must be superconducting.

thin-film rf SQUIDs, e.g. the commercial products: Sensors SQS5C, SQS6C of Oxford Instruments (1992) and Model 20 rf SQUID of Quantum Design (2002). For a modern thin-film rf SQUID system, the lack of a good solution for integrating a planar multiturn input coil Lin into a planar thin-film rf SQUID in the low-T c version is a long-standing problem. In typical modern dc SQUIDs, this type of Lin is mostly arranged on the washer with one SQUID hole (loop) to yield a large mutual inductance Min . The sandwich structure indeed leads to a large capacitance between the Lin coil strip and the SQUID washer. This capacitance and Lin do not directly influence the dc SQUID operation. However, they prevent the rf flux (energy) from the tank circuit from entering the planar thin-film SQUID hole (loop) so that the rf mutual inductance Mrf is dramatically reduced. In fact, all the tank circuit configurations mentioned above, e.g. the LC circuit with lumped elements, the microstrip resonator, the coplanar resonator, and the substrate resonator, cannot couple to a one-hole (loop) rf SQUIDs if an input coil Lin is integrated on its washer. We believe that the arrangement of Figure 12.18a can be realized in the thin-film rf SQUID configuration. To date, two concepts have been reported, and both of them have been realized with high-T c YBCO rf SQUIDs [82, 83]. Here, we focus only on one magnetometric concept, called a “labyrinth resonator,” √ which demonstrated an excellent field sensitivity of approximately 10 fT/ Hz with a pickup area of 1 cm2 . The bulk double-hole rf SQUID configuration shown in Figure 12.18a comes to mind immediately. In fact, we transformed this bulk rf double-hole SQUID magnetometer into the planar thin-film rf SQUID version. To implement this planar thin-film rf SQUID magnetometer, we specifically developed a novel coplanar labyrinth resonator. To reduce the EM radiation (in other words, the EM distribution space), as discussed above, the size of the labyrinth resonator should be as small as possible. Meanwhile, its resonance frequency f L should not be too high, e.g. f L ≤ 600 MHz. The layout of the magnetometer with a labyrinth resonator is shown in Figure 12.19, where a small thin-film rf SQUID gradiometer with two holes is schematically shown on the bottom right. The thin-film magnetometer consists of three parts: (i) The labyrinth resonator serves as a tank circuit. Here, we introduce a small rf single-loop input coil into part of the outer ring to couple one loop of the SQUID gradiometer via Mrf . (ii) The flux transfer is composed of a pickup loop of approximately 1 × 1 cm2 connected to a multiturn Lin (dc input

205

206

12 Radio-Frequency (rf) SQUID

Labyrinth resonator Shielding ring rf input coil

1 cm

SQUID

Pickup loop

dc input coil

Figure 12.19 Layout of the rf SQUID magnetometer with an rf labyrinth resonator and a multiturn dc input coil Lin . Here, a small SQUID gradiometer with two holes is positioned on two input coils in flip-chip configuration. In fact, the rf part is the labyrinth resonator acting as a tank circuit, while the dc part consists of the pickup loop and the dc input coil for collecting the flux signal into the SQUID. Here, the two parts couple to the respective loops of the SQUID gradiometer. Table 12.3 Parameters of different labyrinth resonators. Number of rings

3

4

5

3

4

5

Outer diameter of the resonator (mm)

5.5

5.5

5.5

5

5

5

Resonance frequency f L (MHz)

850

550

470

910

609

523

coil) to couple the other loop of the SQUID gradiometer via Min . (iii) The outer diameters of both the rf input coil and the multiturn coil Lin were 1.5 mm to fit the SQUID gradiometer’s washer size. Here, the thin-film SQUID gradiometer is aimed at both coils, face to face in flip-chip configuration at one corner of the whole layout. Thus, the rf and dc fluxes passing through the SQUID loops have their own independent paths, and there is a shielding ring between them to clearly separate the rf zone from the dc zone. All parts in the layout were structured in a 200 nm epitaxial YBCO film on a LaAlO3 substrate. The labyrinth resonator consists of several coaxial rings with one slit opening. Adjacent rings have their slits on opposite sides. In addition, shorts between adjacent rings are introduced at positions rotated 90∘ against the slits. Table 12.3 shows the dependence of the resonance frequency on the number of rings and on the outer diameter. Here, both the widths of the strip and the line-to-line distances were 100 𝜇m. The unloaded Q0 values of all bare resonators without the outer dc flux transfer circuit structure and shielding ring in Figure 12.19, as listed in Table 12.3, were measured to be larger than 10 000 at 77 K. In principle, the labyrinth resonator is a modification of a coplanar resonator. In the dc part of the flux transfer, the line width of the pickup loop is 1 mm. Here, the dc input coil Lin has 17 turns, with a line width of 20 𝜇m and a line-to-line distance of 15 𝜇m. Using pulsed laser-deposited films, the structure of Lin was fabricated in a three-layer structure, i.e. YBCO–LaAlO3 –YBCO, where the insulation layer was made of the substrate material LaAlO3 . For the magnetometer design, all measured values of 𝜕B/𝜕Φ ≤ 1 nT/Φ0 were obtained.

12.4 Further Developments of the rf SQUID Magnetometer System

Table 12.4 SQUID loop size and corresponding 𝜕B/𝜕Φ.

Sample no.

1 2 a)

SQUID loop size (𝜇m ) SQUID inductance Ls 𝜕B/𝜕Φ (nT/Φ0 )

(pH)b)

20 × 750

2

3

4 2

3 × 712 + 752

10 × 750

3 × 725 + 50

210

200

225

235

0.85c)

0.88

0.9

0.82

10–3

103

10–4

102

10–5 10

100 1000 Frequency (Hz)

Field noise (fT/√Hz)

Flux noise (Φ0 /√Hz)

a) For the two-loop gradiometer, the SQUID loop size is given for one of two identical loops, which comprises the size of one slit plus the size of one square loop (option) (see the lower right corner of Figure 12.19). b) The inductance Ls denotes the total inductance of the gradiometer SQUID. c) This value may be disturbed by the short circuit between turns of the dc input coil.

10 10 000

Figure 12.20 System flux noise 𝛿Φsys of a high-T c rf SQUID magnetometer with a multiturn flux transformer (dc part) and a labyrinth resonator (rf part) and its corresponding field sensitivity (noise) 𝛿B.

With different SQUID loop sizes, the field-to-flux coefficients, 𝜕B/𝜕Φ, of the magnetometer were experimentally determined, as shown in Table 12.4. In the noise measurements below, the D–L requirements are fulfilled. Here, a readout loop (inductive coupling) connected to the RT readout electronics via a 50 Ω cable, as used in the coplanar resonator and substrate resonator versions, is also employed in the labyrinth resonator version. Figure 12.20 shows the system flux noise 𝛿Φsys and its corresponding field sensitivity (noise) 𝛿Bsys of sample #1, measured in 𝜇-metal shielding. In the late 1990s, it was very exciting to obtain √ 𝛿Bsys ≈ 10 fT/ Hz (white noise) at 77 K with a high-T c thin-film rf SQUID magnetometer with a pick-up area of 1 cm2 . Notably, a system flux noise 𝛿Φsys of √ approximately 10 𝜇Φ0 / Hz is close to the noise limit 𝛿Φs,limit of an rf SQUID with Ls ≈ 210 pH at 77 K, assuming a normal resistance of RN ≤ 1 Ω of the YBCO step-edge junction. Overall, a very sensitive magnetometer with a planar resonator and an integrated multiturn input coil Lin was demonstrated with a high-T c rf SQUID for the first time; therefore, the concept of a labyrinth resonator and its measured

207

208

12 Radio-Frequency (rf) SQUID

k1

C Input

DC

Mixer

LNA b

M

a c VCA k2

Amp.

D

Vout

Voffset

A

ATT

Rtest

R

φ

Synth.

LP

VPhase Rf

VCO

k1′

Mod. in

Figure 12.21 Schematic of the modern rf SQUID readout electronics.

noise data should be recorded in the annals of rf SQUIDs. Because the 𝜇-metal shielding was not good enough to perform noise measurements with these very sensitive magnetometers, one should not pay too much attention to the noise performance in the low-frequency range, as shown in Figure 12.20. 12.4.3

Modern rf SQUID Electronics

Recently, a modern electronics for rf SQUID readout was developed [80], which is schematically depicted in Figure 12.21. It consists of a programmable rf synthesizer for generating the rf frequency. The synthesizer includes a fractional-n divider and a phase-locked loop (Texas Instruments LMX2572RHAT) for regulating a VCO (ROS-1300+ from Mini Circuits) to a selected frequency with a rational number quotient p/q to the frequency of a reference quartz with a stability of a few ppm. The rf output power is low pass (LP) filtered to suppress harmonics. Its power is adjusted by means of a VCA (attenuator [ATT]). The attenuated rf current is then sent to the tank circuit of the SQUID through directional coupler (DC). The reflected rf signal modulated by the SQUID response signal passes through the DC to the low noise preamplifier (LNA). For detection, the signal is preamplified and then demodulated from the rf carrier in a mixer (D) (TUF-5+ from Mini Circuits). In order to allow adjustment of the reference phase for demodulation and thus to compensate for the phase of the reflected signal because of variable length of the transmission line to the SQUID, a phase shifter (𝜑) adjusted by V Phase (JSPS-661+ from Mini Circuits) is being inserted between synthesizer and mixer. The electronics is operated and remote-controlled by means of a microcontroller that can be remotely operated from a computer. It allows adjustments of the parameters as frequency (VCO), rf power (ATT), demodulation phase (V Phase ), and dc-offset (V offset ). In addition, the values of the integrator capacitor C and of the feedback resistor Rf can be adjusted in seven or eight steps by electronic switches. The Rf determines the flux-to-voltage coefficient of FLL and thus the dynamic range, whereas the integrator capacitance C, together with the feedback resistance Rf , determines the frequency bandwidth of the feedback loop. The electronics is also equipped with a current source for SQUID heating (not shown here). In case of large changes of the external magnetic field, magnetic flux may be trapped inside the superconducting thin films of the SQUID or the

100 95 –60 –80 –100 –120

(a)

500

600 700 800 Frequency (MHz)

900

ATT (V) 5 4 3 2 1.5 1.25 1.0 0.9 0.8 0.7 0.6 0.5 0

–40 –50 –60 rf power (dBm)

rf power (dBm)

Amplitude (dB)

12.4 Further Developments of the rf SQUID Magnetometer System

–70 –80 Frequency (MHz) 450 600 750 900

–90 –100 –110 –120 0

(b)

1

2 3 Attenuator (V)

4

5

Figure 12.22 (a) Top: amplification as a function of frequency; bottom: rf pumping power (bottom) as a function of frequency for different values of the attenuator voltage and (b) rf pumping power as a function of the attenuator voltage for different frequencies.

substrate resonator. It results in spontaneous signal jumps because of flux vortex hopping within the film, the so-called “shot noise,” thus impairing SQUID operation. Warming up the SQUID and releasing the trapped flux solves this problem. A heating current of 100 mA can be applied for the selected heating time. At the beginning of the heating procedure, the SQUID is put into TEST mode because heating leads to unlocking of the SQUID. Usually, a heating time of three seconds suffices. The total amplification factor of the SQUID electronics can be easily measured by supplying an input signal, e.g. −100 dBm, with variable frequency from a HP 8657A signal generator to the input of the DC. By this means, the total amplification including preamplifier, demodulation, and output amplifier is measured. Then, the synthesizer of the electronics is set to the same nominal frequency as the external signal generator. The amplitude of the beat signal at the electronics output in TEST mode is measured with an oscilloscope. The beat frequency was typically in the range of a few hundreds of Hz, thus proving that the frequency precision of the electronics truly lies in the ppm range. Figure 12.22a (top) depicts the measured amplification as a function of the frequency. It varied from 102 dB at 450 MHz to 95 dB at 900 MHz. In addition, the range of rf power that the electronics can supply to a SQUID was measured at different frequencies by varying the ATT. After preamplification with a Trontech W162H amplifier, the power was measured by a HP 8596E spectrum analyzer. Figure 12.22a depicts the measured amplification factor as a function of the frequency for different attenuator voltage (ATT) supplied to the attenuator. Figure 12.22b shows the rf power as a function of ATT. An adjustment range of 60 dB allows to adapt the electronics to nearly all SQUIDs with their different critical currents. The demodulation phase of the electronics was also measured. It varies from 0∘ to approximately 210∘ when varying the control voltage from 0 to 12.5 V. This adjustment range is sufficient to compensate for phase changes introduced by the coaxial cable connecting to the SQUID’s coupling coil. The modern rf SQUID electronics was tested with an rf SQUID with a square 100 × 100 𝜇m2 loop and a circular 3.5 mm diameter washer, placed on a 10 × 10 mm2 substrate resonator in flip-chip configuration, as sketched in Figure 12.14a. Figure 12.23a shows the measured I–V characteristics of the rf

209

12 Radio-Frequency (rf) SQUID Signal (mV) 0

2.0 600

80 400

(n + 1/2).Φ0

1.5

200 0

(a)

240 320 400

1.0

480

n.Φ0

–200 –400

160

ATT (V)

SQUID signal (mV)

210

–110

–100 –90 –80 rf power (dBm)

560 0.5 565

–70

570 575 Frequency (MHz)

(b)

580

640

Figure 12.23 (a) SQUID signal as a function of attenuator voltage (which translates into rf current according to Figure 12.22) for integer and half-integer number of flux quanta for f = 573.75 MHz and (b) modulation depth, i.e. difference of the SQUID signal at integer and half-integer flux states as a function of frequency and attenuator voltage. Table 12.5 Voltage-to-flux coefficient (V/Φ0 ) of FLL measured with different values of the feedback resistor Rf and resultant dynamic range (Φ0 ). Feedback resistor, Rf (kΩ)a) Voltage-to-flux coefficient, V /Φ0 Dynamic range, Φ0

1

1.5

2

3

4

6

10

20

42

60

78

119

155

225

380

760

±238

±167

±128

±84

±64

±44

±26

±13

a) A resistor of 80 Ω is always connected with Rf in series.

SQUID for two different flux states of integer and half integer number of flux quanta, thus proving that the SQUID operates in dispersive mode. The symmetrical twist yielding the different steps that were introduced in Section 12.1 and illustrated in Figure 12.4c could be observed experimentally. Figure 12.23b shows the difference between the two curves, i.e. the modulation depth between the two flux states of the SQUID, as a function of both frequency and attenuator value. The dynamic range of FLL is determined by the value of the feedback resistor Rf , which converts the output voltage into the feedback current I f . The voltage-to-flux coefficients of the FLL were measured with a mutual inductance of 55 pH between SQUID and the readout loop in resonator versions. They are listed in Table 12.5, together with the dynamic ranges of the SQUIDs in FLL. With decreasing Rf , the dynamic range increases. The bandwidth in Table 12.6 was measured by applying a magnetic field of variable frequency and measuring the output voltage of the electronics in FLL. The excitation signal and the measurement of the response were carried out with a lock-in amplifier (SR 830 from Stanford Research). The amplitude and phase of the transfer characteristics as a function of frequency were measured for different settings of the feedback resistor Rf and the integrator capacitance C. The results were fitted with the transfer function of a first-order low-pass filter. Table 12.6 lists the low-pass corner frequencies of the low-pass characteristics of the transfer functions. The bandwidth reduces with increasing values of C and Rf .

12.5 Multichannel High-Tc rf SQUID Gradiometer

Table 12.6 Bandwidth in kHz, measured with a SQUID in FLL with the integrator capacitance C and the feedback resistor Rf . Rf (kΩ)

C (nF) 1

1.5

2

3

4

6

10

20

0.33

—

—

—

—

183

133

90

52

1

—

—

—

126

98

72

49

28

2.2

—

168

134

96

78

61

38

21

4.7

163

117

94

65

55

39

24

13

10

135

99

79

58

46

35

22

11

22

114

83

67

49

39

32

19

10

100

98

72

58

43

34

27

16

8

The missing values indicate that with the respective Rf –C combination, the FLL could not be locked.

Experimentally, it was found that the FLL is the more stable and the larger the integrator’s capacitance C is, the larger the feedback resistance Rf is. Therefore, to attain a compromise between performance and stability, C and Rf should be chosen only as low as the application requires. In summary, the novel electronics for high-T c rf SQUID features a frequency synthesizer with excellent stability, a large frequency range from 450 to 900 MHz, and a broad range of rf power from −47 to −119 dBm. Variable length of the transmission line between SQUID and electronics can be compensated for by an adjustable demodulation phase. It allows to operate rf SQUIDs with a large range of parameters under a multitude of application conditions.

12.5 Multichannel High-T c rf SQUID Gradiometer In 1974, fetal magnetocardiogram (fMCG) was first described by Kariniemi et al. [84]. For low-T c SQUID systems in a MSR, the train of peaks can reliably be derived from a gestational age (GA) of 20 weeks onward [85]. With a high-T c YBCO rf SQUID first-order gradiometer (single channel) in 2003, the real-time fMCG at GA of 37 weeks was recorded for the first time [86]. In this work, two rf SQUID magnetometers with the basic substrate-resonator setup with a pickup area of 1 cm2 shown in Figure 12.14a were arranged with a vertical baseline of 18 cm, and the measured QRS peak (main spike of heart activity due to ventricular contraction) was approximately 4 pT, corresponding to a signal-to-noise ratio of 2, where the video bandwidth was 90 Hz. To demonstrate the practicability and sensitivity of the version with the dielectric substrate resonator and an additional flux concentrator 18 mm in diameter shown in Figure 12.14b, we set up a multichannel high-T c rf SQUID electronical gradiometer system to measure the fMCG at a GA of approximately 30 weeks [87]. The multichannel first-order gradiometer system was constructed with four

211

212

12 Radio-Frequency (rf) SQUID

– +

Z

b = 20 cm c = 4 cm

R2

R b S2

c

S1

(a)

X

S3

R1

K3

– +

– +

K2

– +

S4 Y

0

K4

– +

R

R

K1

S1 S2 S3 S4 (b)

R

G1

G2

G3

G4

Figure 12.24 (a) Schematic sketch of a four-channel electronic gradiometer, consisting of four sensing SQUID magnetometers (S1 , S2 , S3 , and S4 ) and one reference SQUID magnetometer R with a baseline b of 20 cm and (b) electronic compensation circuit for delivering the gradient signals Gi by forming the operations Gi = (Si − K i R), where the subscript i takes values of 1, 2, 3, and 4 for dividable channels.

sensing SQUID magnetometers and only one common reference magnetometer, arranged as sketched in Figure 12.24a. The four sensing magnetometers (S1 , S2 , S3 , S4 ) are located at the bottom of the dewar on the four corners of a square with c = 4 cm. The z-axis, along which the reference magnetometer R is mounted, runs through the center of the square and is perpendicular to it. The reference R is located in the top plane. The spacing of the two planes (baseline b) is 20 cm [88, 89].√The field sensitivities of all five magnetometers were measured to be 100 K. The second significant difference between rf and dc SQUIDs is the occurrence of higher harmonic components in the flux-to-voltage characteristics, V (Φ). In dc SQUID operation, higher harmonic components of V (Φ) increasingly appear as the parameter 𝛽 c increases, while the SQUID intrinsic noise 𝛿Φs also increases with 𝛽 c (see Figure 6.5). In rf SQUID operation, when the D–L requirements are fulfilled, higher harmonic components of V rf (Φ) appear in dispersive mode (𝛽 e < 1). By this feature, the 𝜕V rf /𝜕Φ at the working point W can be dramatically increased. Thus, the rf SQUID’s intrinsic noise 𝛿Φs can be close to the SQUID thermal noise limit 𝛿Φs,limit , as demonstrated above. Furthermore, the Josephson radiation during dc SQUID operation leads to additional noise because of the resonance effect via the SQUID structure, especially when a multiturn input coil is integrated on the washer of planar SQUID magnetometer chips [91–93]. However, when an rf SQUID operates in dispersive mode, the SQUID should exhibit low intrinsic noise 𝛿Φs , and no radiation is

12.7 Summary and Outlook

emitted from the junction. Here, the noise measurement shown in Figure 12.20 can be regarded as a proof. Generally, dc SQUIDs are more widely adopted than rf SQUIDs in magnetometric systems, although the noise mechanism and structure of rf SQUIDs are clearer and simpler than those of dc SQUIDs. Furthermore, a dc SQUID having two junctions may lead to many problems, e.g. because of different properties of the junctions and because they form two independent thermal noise sources of RJ (RN ) with in-phase and out-phase behavior. An important reason is based on the readout electronics. The dc SQUID’s readout electronics, e.g. DRS, are intuitive and simple so that they can be easily understood and produced. However, building a readout electronics for an rf SQUID is complicated and difficult. The most challenging design requirement is to reduce the rf radiation from its generator (see Figure 12.8). Care should be taken so that no stray EM radiation is emitted into the environment, as it could adversely influence the rf SQUID bias current, I rf,bias . Furthermore, the EM distribution is not easy to control because of the environmental influence discussed in Figure 12.13.

12.7 Summary and Outlook Now, we summarize the rf SQUID readout criteria. Using a diode as the detector D in the 30 MHz version, the main readout quantity of an rf SQUID is the voltage V rf across the tank circuit, which is coupled to the rf SQUID via the mutual inductance Mrf . Thus, the voltage ΔV rf changes with flux change. Figure 12.27 shows the transfer characteristics of a tank circuit (resonator) coupled to a high-T c rf SQUID with Ls ≈ 100 pH and 𝛽 e ≈ 1 at 77 K, recorded by a network analyzer. Here, two cables were needed: one for the rf emitter and the other

910.6

Φ = (n + 1/2)Φ0 Φ = nΦ0 Φ = (n + 1/2)Φ0

911.6 Frequency (MHz)

Phase (30°/div)

Amplitude (3 dB/div)

Φ = nΦ0

912.6

Figure 12.27 The transfer characteristics of the tank circuit in SQUID operation at 𝛽 e ≈ 1, i.e. the curves of amplitude (V rf ) vs. frequency (f ) (upper two curves) and phase (Δ𝜑) vs. frequency (f ) (lower two curves) for the two limiting flux states. Here, the bias power (in other words, Irf,bias ) was optimally selected, and a dielectric SrTiO3 resonator and a SQUID with Ls ≈ 100 pH were employed. In this measurement, two rf cables were utilized, one for rf emission and one for reception. Source: Zhang et al. 1995 [64]. Reproduced with permission of American Institute of Physics.

215

ϕe = (n + 1/2)ϕ0 ϕe = (n + 1/4)ϕ0

457.5

460 Frequency (MHz)

ϕe = nϕ0

Phase (30°/div)

12 Radio-Frequency (rf) SQUID

Power (5 dB/div)

216

462.5

Figure 12.28 Reflected power and phase as a function of pumping frequency for three different flux states recorded at 77 K with a network analyzer (HP 8752A). Here, a high-T c rf SQUID of Ls ≈ 210 pH was employed. In the measurement, the bias power was approximately −90 dBm, which just reached the first optimum plateau for rf SQUID operation (see Figure 12.2). Source: Zhang et al. 1999 [83]. Reproduced with permission of IEEE.

for the receiver. The transfer characteristics consist of the amplitude (ΔV rf ) vs. frequency (f ) (upper) and the phase (Δ𝜑) vs. frequency (f ) (low), where Δ𝜑 can be considered as the phase change between the SQUID signal at the tank circuit and the rf bias signal, e.g. the output voltage of rf generator employed by the improved readout electronics shown in Figure 12.8. In brief, the voltage change at the output of the detector D results from a combination effect of changing quality factor ΔQ of the tank circuit, a resonance frequency shift Δf , and a phase change Δ𝜑. Generally, ΔQ typically determines ΔV rf in dissipative mode, whereas Δf L and Δ𝜑 typically play the main role in dispersive mode [10, 11, 94]. Therefore, instead of a diode, a mixer (multiplier) employed as the detector is necessary in the improved electronics. In fact, the transfer characteristics of the tank circuit determines the readout criterions for different rf SQUID operation modes. Here, the Figure 12.27 represents the SQUID’s behavior, e.g. at 𝛽 e ≈ 1, where ΔV rf and Δ𝜑 exhibit large changes. In order to preserve the particular advantage of an rf SQUID, i.e. the need for only one coaxial cable as transmission line, a directional coupler (DC) is introduced into the improved electronics (see Figure 12.8). Then, the behavior of reflected SQUID signals (power) can be observed at the input of the preamplifier. In the upper half of Figure 12.25, the maximum power can be regarded as the total reflection. However, the minimum of reflected rf power indicates the impedance change, ΔZtap , at the tap point of the resonator, which couples to the rf SQUID. Note that there is no reflected power if Ztap is exactly 50 Ω. The top three curves describe the reflected power caused by the impedance changes Ztap (Φ) across the resonator at three typical fluxes, where the minimum values shifting left from Φ = Φ0 to Φ = Φ0 /2 mirror the resonance frequency f L changes of the resonator. The bottom half of Figure 12.28 (also three curves) depicts the phase change Δ𝜑 between the incoming wave and the reflected wave with ΔΦ. At the minimum Ztap , the phase 𝜑 is drastically changed. In fact, the information of ΔQ, Δf L , and Δ𝜑 in the transfer characteristics of the tank circuit (in Figure 12.27) remains in the reflection system, but all are modified. We point out again that the reflection measurements do not contain any information about Q of the resonator directly.

12.7 Summary and Outlook

In fact, Figures 12.27 and 12.28 represent the rf SQUID’s readout criteria. According to the pumping frequency f L , the rf SQUID system including readout electronics can be divided into two different versions: (1) Conventional (bulk) rf SQUID systems from the 1970s are based on a high-impedance circuit, i.e. their tank circuit with high impedance is directly connected to the gate of a FET, serving as a preamplifier at RT. Here, the transmission lines connecting the tank circuit at CT and the readout electronics at RT are a part of the tank circuit, thus leading to three consequences: the limitation of the frequency to about f L ≈ 30 MHz, a low Q, and a high effective temperature T e,T of the tank circuit. For typical low-T c rf SQUID systems from the 1970s, the highest noise contribution came from the tank circuit and the readout electronics yielded the second largest noise, both larger than the intrinsic SQUID noise. Typically, the ratio of three noise contributions was 3(𝛿ΦT ) : 2(𝛿Φe ) : 1(𝛿Φs ). The highest contribution 𝛿ΦT from the tank circuit corresponds to a noise temperature of approximately T e,T ≈ 200 K. The block diagram of the readout electronics for 30 MHz version is shown in Figure 12.6, where the rf SQUID’s readout criteria are indicated in Figure 12.27. (2) In contrast to the 30 MHz rf SQUID systems from the 1970s, modern rf SQUID systems consist of a thin-film rf SQUID, a planar tank circuit (resonator) without lumped elements, and an improved (modern) readout electronics, as shown in Figure 12.8 (Figure 12.21), connected via a 50 Ω transmission cable and a readout loop. Here, improved readout electronics with adjustable frequency f L from 150 MHz up to 3 GHz have been developed, where the rf SQUID’s readout criteria are described in Figure 12.28. For instance, by using inductive coupling, the high impedance across the resonator can be transformed to a low impedance at the input of a bipolar transistor (as a preamplifier) at RT via a 50 Ω cable and the readout loop. Resonator versions with high Q and f L lead to a small tapping ratio 𝜅 to match the low impedance, thus significantly reducing T e,T of the tank circuit. Consequently, the tank circuit noise 𝛿ΦT can be suppressed below the SQUID’s intrinsic noise 𝛿Φs . However, the SQUID’s signal swing, V rf,input at the input terminal of the preamplifier is not benefitting from the increase of the resonance frequency f L and Q because of the voltage divider formed by the tapping ratio 𝜅. In modern rf SQUID systems, the readout electronics noise 𝛿Φe may become the main noise source in the total system noise 𝛿Φsys when the voltage swing V rf,input is not large enough. Fortunately, when the D–L requirements are fulfilled, the occurrence of higher harmonic components in V rf (Φ) function greatly increase the value of 𝜕V rf /𝜕Φ at the working point W, where the voltage swing V rf,input remains. It has been convincingly demonstrated that a high-T c rf SQUIDs with Ls ≈ 150 and 260 pH operated in dispersive mode (𝛽 e ≤ 1) with planar resonators yields > 300 𝜇V/Φ0 at W, thus leading to a very low system noise a large 𝜕V rf /𝜕Φ √ 𝛿Φsys in the 𝜇Φ0 Hz range, when the improved electronics at RT is utilized. Interestingly, such system noise 𝛿Φsys can come close to the thermal noise limit 𝛿Φe,limit determined by the junction’s normal resistance RN of the rf SQUID at 77 K [30].

217

218

12 Radio-Frequency (rf) SQUID

For historical reasons, most developments of the modern rf SQUID systems were performed with high-T c materials such as YBCO. It is a pity that the developments and the results obtained with high-T c rf SQUID have never been transferred to conventional low-T c (e.g. niobium) rf SQUIDs, e.g. with resonator versions operated at 4.2 K. We expect that low-T c rf SQUID systems operated in dispersive mode can reach the classical noise limit 𝛿Φs,limit , too. It should be fea√ sible that a system noise 𝛿Φsys in the 10−7 × Φ0 / Hz range can be reached for an rf SQUID with Ls ≈ 100 pH. All in all, we do not regard the rf SQUID as being a “living fossil.” In fact, it is a top sensor for magnetometric systems when the D–L requirements are fulfilled for rf SQUID operation, provided that planar resonators, improved readout electronics, and impedance matching are employed. Furthermore, our experimental data with high-T c rf SQUIDs at 77 K verified the occurrence of the maximum transfer function 𝜕V rf /𝜕Φ and the minimum system noise 𝛿Φsys at a value of 𝛽 e ≈ 1–3.

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46 47

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53 54 55

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Index a absolute temperature (T) 11, 34 ac 4, 5, 9, 10, 63–67, 71, 72, 74–76, 90 additional positive feedback (APF) 5, 61, 85, 129 air medium 196 amplitude and phase-to-frequency transfer characteristics 64, 210 angular velocity 10 applied flux 17, 18, 23, 51, 53, 55, 74, 92, 149, 150, 164, 165, 171, 174, 175, 186

b bandwidth 71–73, 79, 148–153, 208, 210, 211, 213 bias current feedback (BCF) 5, 87, 129 biomagnetism 7, 136, 153, 155, 157, 158 bipolar transistor 7, 143, 182, 184–186, 189, 217 Boltzmann’s constant (k B ) 11 bulk SQUID 180, 195

c capacitive coupling 178, 184, 192 capacitive displacement current 10 capacitively shunted junction (CSJ) 3, 9, 10, 12 capacitor tap 181–184, 188–190, 196 cardiogenic QRS wave 153 chain rule 5, 39–43, 98, 101, 117 charge of an electron, e 9 choke LD 188 clamp potential 94, 95

coaxial cable 181–185, 188, 190, 194, 197, 209, 216 communication signal 47 complex transfer characteristics 78, 79 coplanar resonator 192–196, 198, 199, 204–207 coupling factor k 173, 199 critical condition 5, 91–93, 97, 99, 101–104, 114, 117, 122, 125, 126 critical current I c 3, 9, 17, 49, 52, 54, 140, 200, 214 cryogenic temperature (CT) 5, 20, 25, 33, 61, 64, 85, 177 current bias mode 3, 18, 24, 26, 30, 33, 34, 40, 55, 62, 73, 74, 78, 85, 89–94, 98, 103, 125, 127, 129, 140, 142, 157, 172 current density 9 current noise source I n 38, 72 current-to-voltage (I-V) characteristics 1–3, 10–12, 15, 23, 24, 29, 49–51, 53, 74, 77, 90–92, 97, 103, 122, 125, 132, 141, 143, 144, 148, 173, 174, 187, 209

d demodulated (restored) SQUID signal 64 demodulator 64, 178, 185, 203 detector (D) 94, 177, 178, 185, 201, 215, 216 direct current (dc) 1, 2, 9, 12, 15, 23, 29, 45, 49, 92, 147, 172, 174, 176, 178, 180, 186, 200, 205, 214

SQUID Readout Electronics and Magnetometric Systems for Practical Applications, First Edition. Yi Zhang, Hui Dong, Hans-Joachim Krause, Guofeng Zhang, and Xiaoming Xie. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

226

Index

directional coupler (DC) 184, 185, 208, 216 distributed point D 174 distribution function of σ 175 D–L requirements 200, 201, 204, 207, 214, 217, 218 double-pole double-throw (DPDT) switch 40 double relaxation oscillation SQUID (D-ROS) 61 down mixing effect 53 dynamic inductance 100 dynamic range 147, 150–153, 208, 210 dynamic resistance Rd 1–3, 12, 15, 18, 19, 23, 29, 37, 78, 87, 88, 94, 105, 109, 117, 121, 122, 124, 125, 127, 134, 140, 141, 143, 144, 148, 174

e effective operating temperature T e,T 178, 179, 183, 193, 199 effective pickup area 181 electromagnetic (EM) field 190, 194, 195 electronic gradiometer 153, 212 equivalent circuit 6, 18, 41, 89, 94, 96, 99–102, 114, 115, 125, 129, 141, 146, 147, 149, 150, 152, 174

f fetal magnetocardiogram (fMCG) 211 field-effect transistor (FET) 114, 139, 177 field-to-flux transfer coefficient (𝜕B/𝜕Φ) 45, 48, 132, 141, 189, 201 field-to-flux transformer circuit (converter) 1, 4, 45–48 first-order gradiometer 46, 162, 211 flux amplifier 6, 158, 159, 161, 164 flux concentrator 184, 192, 193, 197, 211 flux noise of the readout electronics, δΦe 37 flux quantization 1, 15, 18, 49, 171 flux-sensitive inductor 174, 175 focused gas field ion source 13

Fourier-component 73, 79 frequency response analyzer

79

g GaAs FET 179, 180, 191 gain bandwidth 151 gain factor 35, 147, 150 geomagnetic exploration 153 grain-boundary 181

h half wavelength (λ/2) 48, 190, 191 half-wavelength resonator 108 head stage 5, 25, 26, 29, 46, 61, 62, 64, 66, 74, 85–88, 148, 150, 177, 180, 184 helium ion microscope 13 higher harmonic components 80, 81, 200–202, 204, 214, 217 high-frequency disturbance 47 high-frequency energy 56 high-frequency protection 48 high-impedance op-amp 62 hysteresis 3, 4, 10, 12, 15, 49–53, 55, 93, 126, 135, 144 hysteretic (dissipative) mode 171

i impedance matching 81, 182–184, 186, 189, 192, 193, 197, 199, 214, 218 inductance matching 46, 48 inductor-tap 7 insulating layer 9 intermediately damped 127, 154, 155

j Josephson coupling energy 11 Josephson effect ac Josephson equation 9 dc Josephson equation 9 Josephson element 9, 10 Josephson equations 9, 12 Josephson frequency 48, 108 Josephson junctions 1, 3, 9, 116, 157, 164, 171, 190

Index

Josephson radiation 214 Josephson tunneling 15, 18 junction capacitance 9, 49, 56, 140, 141 junction field effect transistor 37

k kinetic energy 10 Kirchhoff’s law 62, 129 Kurkijärvi’s noise theory 180

l LaAlO3 substrate 189, 193, 196, 206 labyrinth resonator 203–208 λ-ring resonator 191 LC resonant circuit 188 liquid helium-cooled SQUID 46 lumped elements 171, 173, 181, 182, 191, 194, 196, 205, 217

m magnetically shielded room (MSR) 17, 133, 153, 184 magnetic field noise δBsys 45, 148, 155 magnetic flux quantum Φ0 9 magnetocardiography (MCG) 6, 147, 158, 180 magnetoencephalography (MEG) 7, 114, 139, 158, 184 magnetometer 1, 7, 45–48, 56, 115, 123, 124, 132, 133, 135, 141, 143, 146–148, 153, 181, 186, 188, 191, 194, 197–199, 201, 206, 207, 212, 214 magnitude-frequency characteristic 72, 78, 79 MCG-mapping 212 30 MHz version 176, 177, 179, 182–185, 190, 195, 200, 215, 217 150 MHz version 184, 186 microbridge 12, 181 microstrip version 190–192, 194 minimal observable noise (δΦs )min 53 mixed bias mode 5, 61–63, 74, 75, 78, 81, 87, 129 mixture version 186 modern rf SQUID 176, 180–200, 208–211, 217, 218

modulation depth 4, 54, 55, 210 modulation flux ΦMO 64, 69, 70 modulation frequency f M 66, 71, 73 modulation point 67–69, 71 modulation readout scheme (FMS) 56 multichannel first-order gradiometer 211 multiplier 64, 68, 69, 71, 72, 178, 184, 185, 216 multiturn spiral input coil Lin 46 μ-metal shielding 207 mutual inductance Mrf 171, 174, 177, 182, 188, 193, 199, 205

n network analyzer 149, 215, 216 niobium (low Tc) SQUID 42 noise cancellation (NC) 5, 87, 94, 97, 139 noise spectrum 36, 37, 111, 133, 198 noise temperature T N 178, 179, 186 nominal 𝛽 c 49–51, 53, 55, 56, 143 nominal Γ 52 nominal Γ* 53 nonhysteretic (dispersive) mode 171

o Ohm’s law 39, 76, 90 output voltage of the readout electronics V out (Φ) 3

p parallel feedback circuit (PFC) 5, 40, 61, 85, 89–99, 139, 157 pendulum 10 phase difference δ 9 phase-frequency characteristics 71, 78, 79 phase shifter Δ𝜙 64, 71 pickup antennas Lpickup 45 pickup area 47, 48, 181, 197, 205, 211 planar resonators 7, 217, 218 planar thin-film techniques 48 Planck’s constant 9 primary field 153 pumping frequency 172, 175, 176, 179, 181, 188–190, 200, 203, 216, 217

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Index

q QRS peak 211, 213 quality factor Q 7, 171, 173, 174, 177, 178, 186, 189, 192, 216 quantum transition 172, 175, 180, 200 quasi-sinusoidal 42, 50, 80, 201

r radio-frequency (rf ) SQUID 171–218 random switching noise 52, 53 readout electronics 2, 6, 7, 13, 15, 18, 23, 33, 45, 49, 54, 63, 66, 68, 71, 72, 74, 75, 81, 85, 88, 92–94, 99, 103, 106, 108, 111, 114, 116, 118, 125, 139, 142, 145, 147, 152, 172, 176–179, 182, 184–186, 188–191, 194, 197, 200, 201, 207, 208, 215, 217, 218 receiving antenna 48, 194 reference magnetometer 212 reflected rf power 216 reflection system 216 relaxation oscillation SQUID (ROS) 12, 61 resistively and capacitively shunted junction (RCSJ) 3, 9, 10 resistively shunted junction (RSJ) 10–12 resistive microstrip 48 resonance frequency 7, 171, 174, 177, 179, 181, 182, 184, 189, 193, 194, 196, 205, 206, 216, 217 rf bias current I rf,bias 177 rf radiation 48, 185, 194, 215 rf resonance 48, 159, 171 room temperature (RT) 7, 20, 33, 64, 88, 94, 148, 157, 177 rounding effect 11, 12, 52, 53, 173, 187

shunt resistance (resistor) 9, 10, 49, 56, 76, 198 single chip readout electronics (SCRE) 6, 147–154, 161 square wave voltage generator 64, 65 SQUID bootstrap circuit (SBC) 87, 129 steady-state error 152 step-edge junction 181, 189, 198, 204, 207 step-up transformer 4, 5, 61, 63, 66, 69, 71, 74, 78–80, 85 primary winding (PW) 74, 76, 78, 80, 81 secondary winding (SW) 66 Stewart–McCumber parameter 𝛽 c 3, 10, 18, 49 strongly damped junction 43 substrate resonators 7, 196, 198, 201, 205, 207, 209, 211, 212 superconducting electrodes 9 superconducting quantum interference device (SQUID) 9 current sensitivity 101 impedance Zs 5, 61, 74 magnetometer system 1, 4, 13, 23, 25, 29, 33, 45, 47, 48, 67, 85, 117, 123, 145, 147, 181, 183, 192, 195, 196 signal swing 54, 174, 176, 187 system noise power δΦ2 sys 45 voltage swing V swing 50, 77 superconducting shield 47, 48 superconducting transmission line 48 superconductor-insulator-superconductor (SIS) tunnel contact 9 supercurrent 9, 10 symmetric SQUID 53, 186

s secondary field 153 second-order gradiometer 46, 162, 164, 168 sensing SQUID 6, 63, 153, 157–164, 167, 212 series feedback coil (circuit) (SFC) 85, 121–137

t tank circuit 2, 7, 171–186, 188–196, 198–200, 202, 205, 206, 208, 214–217 tank circuit noise δΦ T 179, 191, 200, 217 tap voltage ratio κ 183, 191, 193, 199

Index

thermal energy 11 thermal noise limit 7, 181, 200, 204, 217 thermal noise parameter Γ 11, 18, 52, 55 thin-film SQUID 180–182, 186, 198, 205, 206 three-layer construction 10 three-layer structure 7, 206 tilting parameter α 175 toroidal rf SQUID 204 torque 10 total harmonic distortion (THD) 152 total reflection 216 transfer characteristics 5, 61, 64, 78, 210, 215, 216 transient electromagnetic (TEM) 6, 147 two-stage scheme 6, 61, 63, 111–113, 117, 158–164, 168–169

v varying thickness bridge (V.T.B.) 13 virtual preamplifier 111 voltage bias mode 2, 18–21, 30, 33, 34, 40–42, 55, 61, 62, 76, 87, 94, 98, 99 voltage-controlled attenuator (VCA) 184 voltage-controlled oscillator (VCO) 184, 208 voltage noise source V n 85, 140 V rf,input at the tap position 182 V rf swing (V rf,swing ) 172

w weakly damped SQUID 6, 29, 56, 61, 63, 123–125, 139 white noise 35, 36, 39, 47, 52, 109, 111, 112, 137, 141, 145, 154, 194, 199, 207 wire-wound axial pickup antennas 45

u ultralow field magnetic resonance imaging 47, 155 unshielded/lightly shielded environment 47 un-shunted tunnel contact 10

y yttrium barium copper oxide (YBCO) 11, 13, 46, 176, 181, 189, 193, 196, 198, 199, 205–207, 211, 214, 218

229