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Hands-on guide for scientists and engineers on how to use SQUID technology
This practical book covers SQUID (superconducting quantum interference device) readout electronics and magnetometric systems. It illustrates their many practical applications in measuring extremely subtle magnetic fields and shows how the technique is developing into an enabling technology for many applications, such as biomagnetic imaging and geophysical prospecting. Clear and comprehensive, the book builds a bridge for scientists and engineers to fill in potential know-how gaps for all who work on SQUID systems and their practical applications. It helps make key words like readout electronics, flux quantization, Josephson effects, and noise contributions completely understandable to all who design and use simple and robust SQUID systems.
Beginning with an introduction to the subject, SQUID Readout Electronics and Magnetometric Systems for Practical Applications offers in-depth chapter coverage of: Josephson junctions; dc SQUID’s I-V characteristics and its bias modes; functions of the SQUID's readout electronics; direct readout scheme (DRS); SQUID magnetometry system and SQUID parameters; flux modulation scheme (FMS); and flux feedback concepts and parallel feedback circuit. Other sections examine: analyses of the "series feedback coil (circuit)" (SFC); weakly damped SQUID; two-stage and double relaxation oscillation readout schemes; and radio-frequency (rf) SQUID.
· Provides a unique view of how simplicity and robustness are crucial for practical SQUID systems in applications
· Focuses on the readout electronics of SQUID systems, particularly the advantages and disadvantages of the various systems
· Helps materials scientists, physicists, and engineers overcome various major know-how barriers in order to understand the important challenges and to design practical SQUID systems
· Largely documents the joint achievements accomplished in the cooperation between SIMIT and FZJ in the field of superconducting electronics
SQUID Readout Electronics and Magnetometric Systems for Practical Applications is an excellent book for all materials scientists, electrical engineers, and physicists who can benefit from SQUID systems and their applications. It will also be of great benefit to analytical laboratories in industry, manufacturers of laboratory equipment, and systems engineers.
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Veröffentlichungsjahr: 2020
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Table of Contents
Cover
Preface
Acknowledgments
1 Introduction
1.1 Motivation
1.2 Contents of the Chapters
References
2 Josephson Junctions
2.1 Josephson Equations
2.2 RCSJ Model
References
3 dc SQUID's
I–V
Characteristics and Its Bias Modes
3.1 SQUID's I–V Characteristics
3.2 An Ideal Current Source
3.3 A Practical Voltage Source
References
4 Functions of the SQUID's Readout Electronics
4.1 Selection of the SQUID's Bias Mode
4.2 Flux Locked Loop (FLL)
4.3 Suppressing the Noise Contribution from the Preamplifier
4.4 Two Models of a dc SQUID
References
5 Direct Readout Scheme (DRS)
5.1 Introduction
5.2 Readout Electronics Noise in DRS
5.3 Chain Rule and Flux Noise Contribution of a Preamplifier
5.4 Summary of the DRS
References
6 SQUID Magnetometric System and SQUID Parameters
6.1 Field‐to‐Flux Transformer Circuit (Converter)
6.2 Three Dimensionless Characteristic Parameters, βc, Γ, and βL, in SQUID Operation
References
7 Flux Modulation Scheme (FMS)
7.1 Mixed Bias Modes
7.2 Conventional Explanation for the FMS
7.3 FMS Revisited
7.4 Conclusion
References
8 Flux Feedback Concepts and Parallel Feedback Circuit
8.1 Flux Feedback Concepts and History
8.2 SQUID's Apparent Parameters
8.3 Parallel Feedback Circuit (PFC)
8.4 Quantitative Analyses and Experimental Verification of the PFC in Voltage Bias Mode
8.5 Main Achievements of PFC Quantitative Analysis
8.6 Comparison with the Noise Behaviors of Two Preamplifiers
References
9 Analyses of the “Series Feedback Coil (Circuit)” (SFC)
9.1 SFC in Current Bias Mode
9.2 The SFC in Voltage Bias Mode
9.3 Summary of the PFC and SFC
9.4 Combination of the PFC and SFC (PSFC)
References
10 Weakly Damped SQUID
10.1 Basic Consideration of Weakly Damped SQUID
10.2 SQUID System Noise Measurements with Different βc Values
10.3 Statistics of SQUID Properties
10.4 Single Chip Readout Electronics (SCRE)
10.5 Suggestions for the DRS
References
11 Two‐Stage and Double Relaxation Oscillation Readout Schemes
11.1 Two‐Stage Scheme
11.2 ROS and D‐ROS
11.3 Some Comments on D‐ROS and Two‐Stage Scheme
References
12 Radio‐Frequency (rf) SQUID
12.1 Fundamentals of an rf SQUID
12.2 Conventional rf SQUID System
12.3 Introduction to Modern rf SQUID Systems
12.4 Further Developments of the rf SQUID Magnetometer System
12.5 Multichannel High‐Tc rf SQUID Gradiometer
12.6 Comparison of rf SQUID Readout with dc SQUID Readout
12.7 Summary and Outlook
References
Index
End User License Agreement
List of Tables
Chapter 6
Table 6.1 Parameters of LTS SQUID magnetometers with
A
p
of 5 × 5 mm
2
.
Table 6.2 Parameters of LTS SQUID magnetometers with
A
p
of 10 × 10 mm
2
.
Table 6.3 SQUID parameters measured by the DRS.
Chapter 7
Table 7.1 Comparison of schematic diagrams of a DRS and the FMS.
a)
Table 7.2 Measured Δ
i
and
V
R
s
.
Chapter 8
Table 8.1 Feedback circuits in the two bias modes.
Table 8.2 Measured parameters of SQUIDs with and without PFC.
Table 8.3 Noise data and analysis for SQUIDs with and without the PFC.
Chapter 9
Table 9.1 Parameters of the SQUID magnetometer.
Table 9.2 Summary of PFC and SFC at working point W.
Table 9.3 Parameters of PSBC in experiment II.
Chapter 10
Table 10.1 Slew rates at different frequencies.
Chapter 12
Table 12.1 SQUID parameters and the measured system noise with
f
L
≈ 150 MHz.
Table 12.2 Characteristics of the tank circuit with an inductive coupling and...
Table 12.3 Parameters of different labyrinth resonators.
Table 12.4 SQUID loop size and corresponding ∂
B
/∂Φ.
Table 12.5 Voltage‐to‐flux coefficient (
V
/Φ
0
) of FLL measured with different ...
Table 12.6 Bandwidth in kHz, measured with a SQUID in FLL with the integrator...
List of Illustrations
Chapter 2
Figure 2.1 A Josephson element
J
, a capacitance
C,
and a resistance
R
J
are c...
Figure 2.2
I–V
characteristics for different
β
c
values: (a)
β
...
Figure 2.3 Schematic illustrations of the
I–V
characteristics with dif...
Figure 2.4
I
–
V
characteristics of two SIS junctions in the RCSJ model with d...
Chapter 3
Figure 3.1 Schematic diagram of a four‐pole (terminal) measuring method. Her...
Figure 3.2
I–V
characteristics of a resistor (a) and a diode (b), wher...
Figure 3.3 The SQUID's
I–V
characteristics are divided into three regi...
Figure 3.4 Principle circuits of current bias mode (a) and voltage bias mode...
Figure 3.5 Practical circuit of the current bias mode for measuring SQUID dy...
Figure 3.6 Two possible concepts for realizing voltage bias mode. (a) A curr...
Chapter 4
Figure 4.1 The
I–V
characteristics are the SQUID's essence, while
V
(Φ)...
Figure 4.2 (a) A linearly changing flux Φ(
t
) with time and the expected outp...
Figure 4.3 In dc SQUID magnetometry, the “head stage” consists of at least t...
Figure 4.4 A fundamental circuit of a SQUID's readout electronics with an FL...
Figure 4.5 Schematic diagram of working point selection from
V
(Φ) in a perio...
Figure 4.6 (a) The dynamic balancing process of the working point when a tra...
Figure 4.7 For readout electronics, a dc SQUID can be described with either ...
Chapter 5
Figure 5.1 The test circuit of an op‐amp functioning as a voltage amplifier ...
Figure 5.2 The measured noise spectra of
I
n
(left vertical axis) and
V
n
(rig...
Figure 5.3 Test circuit of a PCBT as a voltage amplifier with
G
= 1000 for c...
Figure 5.4 The measured noise spectra of
I
n
(left vertical axis) and
V
n
(rig...
Figure 5.5 The voltage noise
δ
V
e
of an AD797 acting as a voltage preamp...
Figure 5.6 The total voltage noise
δ
V
e
of a PCBT acting as a voltage pr...
Figure 5.7 (a) Schematic diagram showing the test circuit, where the same pr...
Figure 5.8 Noise measurements of a strongly damped SQUID with
β
c
≈ 0.3 ...
Chapter 6
Figure 6.1 (a) Schematic “head stage” of a SQUID magnetometric system, consi...
Figure 6.2 Measured
V
(Φ) curves of current‐biased SQUIDs with
L
s
≈ 350 pH: (...
Figure 6.3 Three simulated
I–V
characteristics of a symmetric SQUID wi...
Figure 6.4 Geometrical images of three typical
β
L
values,
β
L
≈ 0 (...
Figure 6.5 The tendencies of the three electrically readable SQUID values wi...
Chapter 7
Figure 7.1 The input circuit (head stage) is a SQUID shunted by an element
Z
Figure 7.2 A schematic diagram of the FMS. The SQUID is biased by a constant...
Figure 7.3 (a) A linearly changing flux Φ(
t
) results in a periodic SQUID's
V
Figure 7.4 Φ(
t
),
V
s
(Φ), and
V
M1
(
t
) in a modulation period for the case of a ...
Figure 7.5 The left column shows the positions of the modulation points, W
M
...
Figure 7.6 Working principle diagram of FMS demonstrated with Φ(
t
),
V
s
(Φ), a...
Figure 7.7 If ΔΦ < Φ
0
/4 (see point 1) or ΔΦ > Φ
0
/4 (see point 2) in
T
M
/2, th...
Figure 7.8 The frequency spectrum of
V
M1
. The peaks present the Fourier comp...
Figure 7.9 SQUID input circuit of FMS with a step‐up transformer and preampl...
Figure 7.10 The simplified input circuit by replacing
Z
s
with a variable res...
Figure 7.11 Test circuit for determining
V
out
and Δ
i
of the SQUID (SQ1) with...
Figure 7.12 Measured SQUID's
I–V
characteristics at Φ =
n
Φ
0
and Φ = (
n
Figure 7.13 The test circuit simulates the SQUID's input circuit for measuri...
Figure 7.14 Complex transfer characteristics measured by a frequency respons...
Figure 7.15 A square wave signal with dashed line (right axis) passing throu...
Figure 7.16 The
V
(Φ) characteristics of a SQUID with a loop inductance
L
s
= ...
Chapter 8
Figure 8.1 The SQUID's “head stage” with the PFC and SFC. Here, the PFC is a...
Figure 8.2 Four schematic diagrams of a modified DRS at the “head stage” wit...
Figure 8.3 Schematic diagram of a circuit containing a resistor
R
and two ba...
Figure 8.4 In current bias mode, the equivalent circuit of a SQUID shunted b...
Figure 8.5 Two measured SQUID's
I
–
V
characteristics: the original one with t...
Figure 8.6 The SQUID's
V
(Φ) characteristics without (step (i)) and with the ...
Figure 8.7 The experimentally recorded data: the dashed lines represent the ...
Figure 8.8 The
V
(Φ) characteristics with the PFC (APF) are schematically sho...
Figure 8.9 (a) A SQUID shunted by a PFC (NC scheme) connects to the invertin...
Figure 8.10 The equivalent circuit of
V
n
of an op‐amp (preamplifier) in volt...
Figure 8.11 Schematic of the
I
(Φ) characteristics in the NC scheme, where th...
Figure 8.12 There are two circuits for the noise analyses: (a) is equivalent...
Figure 8.13 The
I
–
V
characteristics (photo) measured near Φ = (2
n
+ 1)Φ
0
/4 w...
Figure 8.14 Dependence of numerically calculated flux noise contributions, δ...
Figure 8.15 Dependence of δΦ
PFC
on Δ at
r
= 1 (a),
r
= 3 (b), and
r
= 5 (c) ...
Figure 8.16 Numerically calculated δΦ
PFC
(Δ) and (
R
d
)
PFC
(Δ). The solid (black...
Figure 8.17 Flux noise spectra of SQUID #2 measured (I) with the PFC in the ...
Figure 8.18 Current noise spectra of the employed PCBT preamplifier: curves ...
Figure 8.19 The measured spectra δΦ
sys
: curves I (PFC) and II (two‐stage sch...
Figure 8.20 The
R
p
of the PFC is replaced by an FET, where
V
gs
controls the ...
Figure 8.21 The equivalent circuit of design (2) with selectable
M
p
and
R
p
, ...
Figure 8.22 An example of design (2): a niobium SQUID magnetometer layout wi...
Figure 8.23 Illustrations plotting the current noise δ
I
spectra (right ordin...
Chapter 9
Figure 9.1 The arrangement of the SFC, where the coil
L
se
and the SQUID are ...
Figure 9.2 (a) The measured
V
(Φ) characteristics are unchanged with and with...
Figure 9.3 Measured system noise δΦ
sys
without (I) and with the SFC (II) and...
Figure 9.4 The equivalent circuit of a voltage‐biased SQUID with SFC.
Figure 9.5 In voltage bias mode, the
I
(Φ) characteristics without (dotted cu...
Figure 9.6 Hysteresis appears in the measured
I
(Φ) characteristics, as
M
se
×...
Figure 9.7 The SQUID with the PSFC in voltage bias mode, where both the
L
se
...
Figure 9.8 Four equivalent circuits of the PSFC under the independence condi...
Figure 9.9 Working point W
2
is set on the steep slope of the asymmetric
I
(Φ)...
Figure 9.10 The flux noise spectrum and the corresponding field sensitivity ...
Figure 9.11 The measured
I
(Φ) characteristics for three different values of
Figure 9.12 Comparison of the system flux and field noise spectra measured w...
Figure 9.13 Plots of
I
(Φ)
PSFC
and (∂
I
/∂Φ)
PSFC
vs. Φ/Φ
0
at
M
se
= 0.09 nH. The...
Chapter 10
Figure 10.1 System noise δΦ
sys
measurements (right ordinate gives the field ...
Figure 10.2 Noise characteristics of the SQUID with
β
c
≈ 3.5. The inset...
Figure 10.3 Statistical data of
I
swing
(a) and
R
d
(b) of 101 SQUIDs as a fun...
Figure 10.4 Statistical characterization of 53 SQUIDs with nominal
R
J
= 30 Ω...
Figure 10.5 (a) δ
B
sys
classification of the 53 voltage‐biased SQUID magnetom...
Figure 10.6 Schematic diagram of SCRE with a current‐biased SQUID. The AD797...
Figure 10.7 Flux noise (left scale) and field noise (right scale) measured i...
Figure 10.8 The equivalent electronic circuits of SCRE in FLL (shown in Figu...
Figure 10.9 MCG signal of an adult male subject recorded in an MSR.
Figure 10.10 Schematic real‐time TEM signal (a) and its enlargement (b) reco...
Chapter 11
Figure 11.1 In principle, the voltage‐biased circuit needs an ammeter “A” to...
Figure 11.2 Different concepts of FLL operation in the two‐stage scheme: two...
Figure 11.3 Suggestion of a practical SQUID magnetometric system with a two‐...
Figure 11.4
V
(Φ) characteristics at
V
out
, where the real characteristics of ...
Figure 11.5 Hysteretic
I–V
characteristics of an un‐shunted SQUID. The...
Figure 11.6 The equivalent circuit of ROS. The virtual voltmeter and ammeter...
Figure 11.7 Simulated
I
1
(upper) and
V
s
(lower) of an ROS in the time domain...
Figure 11.8 Schematic arrangement of D‐ROS where the output voltage drops ac...
Chapter 12
Figure 12.1 The magnetic flux relationship inside and outside the SQUID loop...
Figure 12.2 (a) Readout principle of an rf SQUID: the rf SQUID is coupled to...
Figure 12.3 Illustration of the rf SQUID behavior in dissipative mode (
β
...
Figure 12.4 rf SQUID behavior in dispersive mode (
β
e
≪ 1): (a) the rf S...
Figure 12.5 At
β
e
≈ 1, both effects on the tank circuit caused by the r...
Figure 12.6 Block diagram of the 30 MHz version of rf SQUID readout electron...
Figure 12.7 Schematic illustration of a mixture of technologies, i.e. a mode...
Figure 12.8 The improved parts of rf SQUID readout electronics that are diff...
Figure 12.9 Modified conventional LC tank circuit with inductive coupling pl...
Figure 12.10 Schematics of the first planar tank circuit for rf SQUID operat...
Figure 12.11 (a) Schematic diagram of the
λ
resonator integrated with a...
Figure 12.12 (A) Schematic of three flux concentrator/coplanar resonator lay...
Figure 12.13 (a) In the rf SQUID tank circuit, the schematic rf EM field dis...
Figure 12.14 (a) Two arrangements of a practical encapsulated SQUID magnetom...
Figure 12.15 Noise spectrum of a YBCO rf SQUID magnetometer with a SrTiO
3
di...
Figure 12.16 (a) Measured
V
rf
(Φ) characteristics of an rf SQUID coupled with...
Figure 12.17 Typical
V
rf
(Φ) characteristics of rf SQUIDs in the planar labyr...
Figure 12.18 Schematic arrangements of two cylindrical coils,
L
in
and
L
T
, in...
Figure 12.19 Layout of the rf SQUID magnetometer with an rf labyrinth resona...
Figure 12.20 System flux noise δΦ
sys
of a high‐
T
c
rf SQUID magnetometer with...
Figure 12.21 Schematic of the modern rf SQUID readout electronics.
Figure 12.22 (a) Top: amplification as a function of frequency; bottom: rf p...
Figure 12.23 (a) SQUID signal as a function of attenuator voltage (which tra...
Figure 12.24 (a) Schematic sketch of a four‐channel electronic gradiometer, ...
Figure 12.25 Four‐channel gradient fMCG signals from a fetus at a gestationa...
Figure 12.26 Averaged fMCG data with a 75 seconds duration.
Figure 12.27 The transfer characteristics of the tank circuit in SQUID opera...
Figure 12.28 Reflected power and phase as a function of pumping frequency fo...
Guide
Cover
Table of Contents
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SQUID Readout Electronics and Magnetometric Systems for Practical Applications
Yi Zhang
Hui Dong
Hans‐Joachim Krause
Guofeng Zhang
Xiaoming Xie
Authors
Prof. Dr. Yi Zhang
Forschungszentrum Jüelich (Retired)
Institute of Biological Information Processing
Wilhelm‐Johnen‐Straße
52428 Jüelich
Germany
Prof. Dr. Hui Dong
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
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.
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Print ISBN: 978‐3‐527‐34488‐8
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Weinheim, Germany
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 Tcradio‐frequencysuperconducting quantum interference devices (high temperature superconducting [HTS] rfSQUIDs), 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 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.
Xiaoming XieShanghai, China
November 2019.
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.
1Introduction
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 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 Vn and the current noise In. 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‐Tc SQUID, usually operated at 4.2 K (liquid helium temperature); and the high‐temperature superconducting (HTS) SQUID, also called high‐Tc SQUID, usually operated at 77 K (the liquid nitrogen temperature). The LTS material is typically niobium and HTS material is yttrium barium copper oxide (YB2Cu3O7−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‐Tc) dc SQUID and HTS (high‐Tc) 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
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 βc 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 βc 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 Ic. Generally, JJs without hysteresis are suitable for SQUID operation. In fact, one habitually transforms the parameters βc 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 Vout(Φ) of the readout electronics; i.e. Vout 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 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, Vn and In, 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, βc, Γ, and βL. Note that the definitions of βc 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 βc values were characterized. Here, a reasonable interpretation of the observed absence of hysteresis in the SQUID's I–V characteristics at high βc is given. For SQUID operation, the dimensionless parameter βL particularly describes the modulation depth of the SQUID. Importantly, βL ≈ 1 imposes a design condition on the product LsIc – namely, all electrically readable values of SQUID parameters increase with increase in the SQUID's nominal βc.
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 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 Vn. Separately, the large (∂I/∂Φ)SFC reduces the noise contribution from the preamplifier's In. 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.
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 βc 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 βc < 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 Ib above the SQUID's Ic flows through the parallel circuit, the D‐ROS becomes active 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 βc → ∞. 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 βe 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 . Conventionally, the pumping (resonance) frequency f0 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 f0 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 = 2πf0LTQ appears across the LTCT tank circuit.
Two main achievements have been attained in HTS rf SQUID research: (i) Instead of the conventional LTCT tank circuit, superconducting planar resonators or substrate resonators were developed for rf SQUID operation. This new kind of resonator possesses a high resonance frequency f0 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 Vrf(Φ) characteristics can be observed, thus changing the shapes of Vrf(Φ), where its slopes become steeper. Consequently, a large transfer coefficient (∂Vrf/∂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 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 Applications
. 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 magnetometers for low‐frequency applications.
Journal of Low Temperature Physics
76 (5–6): 287–386.
6
Simmonds, M.B. and Giffard, R.P. (1983). Apparatus for reducing low frequency noise in dc biased SQUIDs. US Patent 4, 389, 612.
7
Dutta, P. and Horn, P.M. (1981). Low‐frequency fluctuations in solids – 1‐F noise.
Reviews of Modern Physics
53 (3): 497–516.
8
Weissman, M.B. (1988). 1/F noise and other slow, nonexponential kinetics in condensed matter.
Reviews of Modern Physics
60 (2): 537–571.
9
Lam, S.K.H. (2006). Noise properties of SQUIDs made from nanobridges.
Superconductor Science & Technology
19 (9): 963–967.
10
Cleuziou, J.P., Wernsdorfer, W., Bouchiat, V. et al. (2006). Carbon nanotube superconducting quantum interference device.
Nature Nanotechnology
1 (1): 53–59.
2Josephson 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
where δ is the phase difference in the macroscopic wave functions of the two superconducting electrodes. For each JJ, a critical current Ic exists. When I < Ic, 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 > Ic), 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:
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 Ic, is connected in parallel with the junction capacitance C and a shunt resistance RJ as shown inFigure 2.1. When Idc > Ic, 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) Ic 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 CV2/2, 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 Idc 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.
Figure 2.1 A Josephson element J, a capacitance C, and a resistance RJ are connected in parallel to form the RCSJ model.
Usually, two parameters, βc and Γ, are introduced to characterize the features of a JJ. In the RCSJ model, the Stewart‐McCumber parameter βc is denoted as
which describes the junction's hysteresis behavior, or, the damping classes. Figure 2.2 sketches the I–V characteristics of junctions for different βc values. Generally, there are two extreme cases: (i) for βc ≪ 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. βc → ∞ (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, βc = 1 is the boundary between the nonhysteretic and hysteretic regimes in the junction's I–V characteristics. For βc = 2, for example, hysteresis including a local loop clearly appears in the I–V characteristics (see Figure 2.2c).
Figure 2.2I–V characteristics for different βc values: (a) βc ≪ 1 (RSJ model); (b) βc ≫ 1, (CSJ model); (c) βc = 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.
where kB 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 2.3
a). 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 to a smaller dynamic resistance
R
d
. 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.
Figure 2.3 Schematic illustrations of the I–V characteristics with different Γ values at βc ≪ 1 (RSJ model) (a) and at βc = 2 with Γ = 0 and 0.1 (b).
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 βc and Γ values. For example, a conventional dc SQUID is operated at βc < 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).
