FERROMAGNETIC JOSEPHSON JUNCTIONS CARRYING
SPIN-TRIPLET SUPERCURRENT FOR CRYOGENIC MEMORY
By
Joseph Allen Glick III

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics—Doctor of Philosophy
2017

ABSTRACT
FERROMAGNETIC JOSEPHSON JUNCTIONS CARRYING SPIN-TRIPLET
SUPERCURRENT FOR CRYOGENIC MEMORY
By
Joseph Allen Glick III
In this thesis we present the first experimental demonstration of phase-controllable Josephson junctions that carry long range spin-triplet supercurrent. These junctions exhibit groundstate phase shifts of either 0 or π and are of considerable interest for the development of
random access memory for energy efficient superconducting computers.
We demonstrate a scheme by which spin-triplet supercurrent in the junctions is generated through the ferromagnetic proximity effect using three magnetic layers with noncolinear
magnetizations. The central layer is a synthetic antiferromagnet with magnetization perpendicular to the plane, while the other two ferromagnetic layers have in-plane magnetization.
First, we establish that the supercurrent in these junctions is spin-triplet in nature by observing the characteristic slow decay of the critical current versus the central layer thickness when
compared to other junctions that do not have the in-plane layers and carry only spin-singlet
supercurrent. The phase state of the junctions is revealed by measuring the interference
between two such Josephson junctions in a Superconducting QUantum Interference Device
(SQUID) loop. By switching the magnetization of one of the layers by 180◦ without disturbing the other two layers, we show that the phase state of the Josephson junctions can
be controllably switched between 0 and π over a thousand times without error, opening
possibilities for their use in superconducting memory.
We also show that there are easier ways to make a phase-controllable cryogenic memory
device using spin-singlet supercurrent. We discuss how Josephson junctions containing only

two magnetic layers of appropriate thickness arranged into a spin-valve configuration exhibit
controllable 0-π switching, first demonstrated by the Birge group at Michigan State University in 2016 using a similar SQUID measurement scheme. I describe the main contributions I
made as a part of that effort, in particular the development of a general asymmetric SQUID
fitting program that provided the unambiguous proof that the devices switched between the
0 and π phase states.
We also discuss a number of material studies that served as stepping stones toward
the development and improvement of both of the previously mentioned phase-controllable
memory demonstrations. We use primarily Fraunhofer physics and SQUID magnetometry to
characterize the magnetic and superconducting properties of Josephson junctions containing
the ferromagnets: Ni, Ni81 Fe19 , Ni65 Fe15 Co20 , Pd97 Fe3 , and multilayers of Pd/Co. We
examine the relative advantages and disadvantages that each of these materials offer to
the development of future superconducting memory devices and compare the strengths and
weaknesses of the two phase-control memory schemes.

Copyright by
JOSEPH ALLEN GLICK III
2017

To my mother and father.

v

ACKNOWLEDGMENTS

I have been fortunate to work alongside many excellent people while in the Birge group at
Michigan State University (MSU). I thank Norman Birge who is truly the best advisor I
could have asked for, and likewise a fantastic person and friend. You are an exceptional
teacher, mentor, and a great wellspring of ideas from whom I learned new things everyday.
I appreciated your patience in allowing me to grow and develop as an experimental physicist, the great level of autonomy you gave me during our research pursuits, the incredible
opportunities you opened up to me to collaborate with industry and academic researchers,
and encouraging me to attend many academic conferences and workshops.
I thank Reza Loloee, for his inspiring dedication to students such as myself. Needless
to say, you saved a great number of my sputtering runs from certain disaster. I thoroughly
enjoyed working with you to fix vacuum systems and low-temperature probes, times when I
learned so much from your knack for creative troubleshooting. Your insightful suggestions
helped lead to the selection of some of the materials used in this work and the improvement
of many of our sample fabrication processes. You were also one of the first people that helped
me feel welcome at MSU, always filling the lab with laughter.
I thank William Pratt for teaching me so much about low-temperature physics and magnetic materials, and I appreciated his clear and systematic approach to physics. None of
the transport measurements in this work could have been done without your masterfully
designed probes and cryostats, in addition to several of your vacuum chambers for thin-film
deposition. I already miss your sagely wisdom, witty stories, and impeccable craftsmanship.
I appreciate the helpful advice of Baokang Bi regarding sample fabrication, especially in
electron-beam lithography, and for keeping the Keck microfabrication clean room facility at
vi

MSU running smoothly and efficiently, despite its many users.
To all of the former graduate students of the Birge group who I worked with: Yixing
Wang, Eric Gingrich, William Martinez, Mazin Khasawneh, and Bethany Niedzielski. Each
of you taught me parts of the fabrication and measurement process that made my own
research possible. I also benefited immensely from your own research efforts and progressed
only by building off of the foundation that you had laid. It was truly rewarding to see some
of our ideas become experimental realities.
I am also grateful to the current graduate students Victor Aguilar and Alex Madden. I
am glad to know that you both will be instrumental in carrying our group’s research forward.
Victor, I want to especially thank you for your help with LabVIEW programming, which
saved me untold days of measurement time in addition to working alongside me to develop
our technique to pattern magnetic bit arrays.
I enjoyed working with our postdocs and visitors to our group including Nathan Satchell,
Adel Gougam, and Demet Korucu as well as the many undergraduates, including Josh
Willard, Sam Edwards, Anna Osella, Thomas Bertus, and Kevin Werner who helped me
perform a number of experimental studies. Special thanks to Alex Cramer for his assistance
in writing code to fit I-V curves that account for thermal rounding.
Together all of the Birge group members, past and present, made for a collaborative team
that was both fun and intellectually stimulating.
I thank our collaborators at Northrop Grumman Corporation, especially Anna Herr,
Don Miller, Ofer Naaman, Tom Ambrose, Jackson Chang and Steve Van Campen. Working
with you has been an invaluable experience and your many insightful suggestions have aided
in my thinking and greatly benefited this work. I’ve learned much from our collaborators
at Arizona State University, Nate Newman and Nick Rizzo. Nick’s expertise in magnetic
vii

materials directly aided in the development of many of my samples. To our collaborators at
Sandia National Laboratory, namely Nancy Missert and Paul Kotula, your amazing Scanning
Transmission Electron Microscopy images of my samples were key in understanding some of
the issues regarding our sample fabrication process.
I am grateful for the funding from the Intelligence Advanced Research Projects Activity
(IARPA) in addition to the encouragement from Scott Holmes and Marc Manheimer. Further
financial support from the MSU physics department and Northrop Grumman Corporation
during my time at MSU were essential to the completion of this work.
I thank Megan Donahue, Johannes Pollanen, David Tomanek, and Scott Pratt for serving
on my thesis committee and for their many helpful comments. Scott, I want to especially
thank you as some of the biggest decisions I made during my time at MSU were strongly
influenced by the guidance you gave me.
I thank my parents Steve and Linda, my brother Aaron, and my sisters Hannah and
Emily, as well as my wife’s family for the incredible love and encouragement they have
always given me. You all inspired me to challenge and continually improve myself both as a
person and as a scientist, all the while helping me to counterbalance the stresses of graduate
school. F .c. Fermi and Nambu have also shown me how to take a more relaxed perspective
during vexing circumstances.
Finally, I thank my wife Jen for her steadfast love, perseverance, and fortitude that have
aided and replenished me everyday on this journey we took together. Besides this, she has
made countless indelible scientific contributions to this work.

viii

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Energy Efficient Superconducting Computing . . . . . . . . . . . . . . . . .
1.2 Chapters Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 2 Superconductivity . . . . .
2.1 The Meissner-Ochsenfeld Effect and
2.2 Ginzburg–Landau Theory . . . . .
2.3 BCS Theory . . . . . . . . . . . . .
2.4 The Josephson Effect . . . . . . . .
2.5 The RCSJ Model . . . . . . . . . .
2.5.1 Overdamped Junctions . . .
2.6 Flux Quantization and SQUIDs . .
2.6.1 Magnetic Flux Quantization
2.6.2 SQUIDs . . . . . . . . . . .

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Chapter 3 Ferromagnetism . . . . . . . . . . . . . . . . . . .
3.1 Historical Overview . . . . . . . . . . . . . . . . . . . . . .
3.2 Magnetic Domains . . . . . . . . . . . . . . . . . . . . . .
3.3 Magnetic Anisotropy . . . . . . . . . . . . . . . . . . . . .
3.3.1 The Stoner-Wohlfarth Model: Uniaxial Anisotropy
3.3.2 Types of Magnetic Anisotropy . . . . . . . . . . . .

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Chapter 4 Interplay Between Superconductivity and Ferromagnetism . .
4.1 The Proximity Effect at S/N Interfaces . . . . . . . . . . . . . . . . . . . . .
4.2 The Proximity Effect at S/F Interfaces . . . . . . . . . . . . . . . . . . . . .
4.3 π-Phase Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Fraunhofer Patterns in Ferromagnetic Josephson Junctions . . . . . . . . . .
4.4.1 The Josephson Effect in the Presence of Magnetic Flux . . . . . . . .
4.4.2 Adding a Ferromagnetic Layer . . . . . . . . . . . . . . . . . . . . . .
4.5 Generating Long-Range Spin-Triplet Correlations with Ferromagnetic Barriers

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Chapter 5 Experimental Techniques . . . . . . . . .
5.1 Sputtering . . . . . . . . . . . . . . . . . . . . . . .
5.2 Photolithography . . . . . . . . . . . . . . . . . . .
5.3 Electron-Beam Lithography . . . . . . . . . . . . .
5.4 Ion Milling . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Ion Mill and Thermal Evaporation Chamber

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5.5
5.6
5.7

Josephson Junction Sample Fabrication . . . . . . . . . . . . . . . . . . . . .
Measurement Probes and Low Noise rf SQUID Electronics . . . . . . . . . .
SQUID Magnetometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84
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Chapter 6 S/F/S Josephson Junctions: Materials Characterization .
6.1 Role of π-Junctions in Cryogenic Memory . . . . . . . . . . . . . . . .
6.2 Sample Fabrication and Characterization . . . . . . . . . . . . . . . . .
6.3 Critical Current Oscillations of Josephson Junctions . . . . . . . . . . .
6.3.1 Junctions Containing Ni65 Fe15 Co20 . . . . . . . . . . . . . . . .
6.3.2 Junctions Containing Permalloy . . . . . . . . . . . . . . . . . .
6.3.3 Junctions Containing Pd97 Fe3 . . . . . . . . . . . . . . . . . . .
6.3.4 Junctions Containing Ni . . . . . . . . . . . . . . . . . . . . . .
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 7 Controllable Phase SQUIDs: Spin-Singlet
7.1 Spin-Valve Josephson Junctions and SQUIDs . . . . .
7.2 Theory of the Asymmetric DC SQUID . . . . . . . .
7.3 SQUID Oscillation Data: Comparison to Theory . . .

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Chapter 8 Spin-Triplet Josephson Junctions Containing a PMA SAF . .
8.1 Magnetic Properties of Pd/Co Multilayers and Synthetic Anti-Ferromagnets
With Perpendicular Magneto-Anisotropy . . . . . . . . . . . . . . . . . . . .
8.1.1 Synthetic Anti-Ferromagnets . . . . . . . . . . . . . . . . . . . . . . .
8.2 Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.1 Sample Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Transport Measurements and Analysis . . . . . . . . . . . . . . . . . . . . .
8.3.1 Fraunhofer Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.2 Critical Current vs Thickness . . . . . . . . . . . . . . . . . . . . . .
8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153

Chapter 9 Controllable Phase SQUIDs: Spin-Triplet . . . . .
9.1 SQUID Sample Fabrication . . . . . . . . . . . . . . . . . . .
9.2 Measurement of SQUID Oscillations . . . . . . . . . . . . . .
9.3 SQUID Oscillation Data: Comparison to Theory . . . . . . . .
9.4 Repeated Switching Experiment . . . . . . . . . . . . . . . . .
9.5 SQUID Measurements Below 4.2 K in a Dilution Refrigerator
9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

x

LIST OF TABLES

Table 5.1: Sputtering parameters used for the various metallic targets . . . . . . . .

73

Table 5.2: Ion mill settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

Table 5.3: Ion mill conversion factors for various materials . . . . . . . . . . . . . . .

82

Table 6.1: Parameters obtained from fits of the Ivanchenko-Zil’berman function to the
data shown in Fig. 6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Table 6.2: Best-fit parameters from Eq. (6.5) corresponding to the data in Figs. 6.7(a)
and 6.11(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Table 7.1: Best-fit parameters for the four magnetic states of the SQUID using the
asymmetric SQUID theory . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Table 7.2: Inductances and critical currents extracted from the least-squares fits . . . 151
Table 8.1: Fitting the data in Fig. 8.8 to Eq. (8.2) yields the best-fit parameters . . . 173
Table 9.1: Best-fit parameters for the P and AP states for SQUID 2A-4 . . . . . . . 192
Table 9.2: Summary of the the seven spin-triplet SQUID samples measured . . . . . 194
Table 1:

Best-fit parameters for the P and AP states for SQUID 2A-1 . . . . . . . 208

Table 2:

Best-fit parameters for the P and AP states for SQUID 2A-2 . . . . . . . 209

Table 3:

Best-fit parameters for the P and AP states for SQUID 2A-3 . . . . . . . 211

Table 4:

Best-fit parameters for the P and AP states for SQUID 3A-3 . . . . . . . 214

Table 5:

Best-fit parameters for the P and AP states for SQUID 4A-1 . . . . . . . 217

Table 6:

Best-fit parameters for the P and AP states for SQUID 4A-2 . . . . . . . 219

xi

LIST OF FIGURES

Figure 2.1: The equivalent circuit for the RCSJ model of a physical Josephson junction 20
Figure 2.2: The tilted washboard potential of the RCSJ model. . . . . . . . . . . . .

22

Figure 2.3: The time-average voltage difference, Eq. (2.32), across the overdamped
junction in the RCSJ model (solid line). For |I| ≤ Ic0 , the voltage remains
zero, but approaches Ohm’s law (dashed line) when |I|  Ic0 . . . . . . .

23

Figure 2.4: The symmetric dc SQUID with total supercurrent I . . . . . . . . . . . .

28

Figure 2.5: Symmetric SQUID oscillations . . . . . . . . . . . . . . . . . . . . . . . .

32

Figure 3.1: Ferromagnetic domain structures . . . . . . . . . . . . . . . . . . . . . .

37

Figure 3.2: Schematic setup for the Stoner-Wohlfarth model . . . . . . . . . . . . . .

38

Figure 3.3: Magnetization reversal in the Stoner-Wohlfarth model . . . . . . . . . .

40

Figure 3.4: Hysteresis loops showing the magnetization M vs. the magnetic field H
applied along the easy axis of the ferromagnet . . . . . . . . . . . . . . .

42

Figure 3.5: Astroid diagram from the Stoner-Wolfarth model . . . . . . . . . . . . .

43

Figure 4.1: Proximity effect near the interface of a superconductor and a normal metal 48
Figure 4.2: Density of states at a superconductor/normal metal interface and Andreev
reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Figure 4.3: A simplified toy model of a normal metal and a ferromagnet . . . . . . .

52

Figure 4.4: Proximity effect near the interface of a superconductor and a ferromagnet

55

Figure 4.5: 0-π phase oscillations in an S/F/S Josephson junction . . . . . . . . . .

57

Figure 4.6: Geometry for calculating the Fraunhofer pattern in a Josephson junction
with rectangular cross-section of area LW . . . . . . . . . . . . . . . . .

59

Figure 4.7: Fraunhofer patterns for rectangular and circular Josephson junctions . .

62

Figure 4.8: Fraunhofer patterns for a ferromagnetic Josephson junction

63

xii

. . . . . . .

Figure 4.9: Schematic for the generation of long-range spin-triplet correlations using
noncolinear magnetizations . . . . . . . . . . . . . . . . . . . . . . . . .

65

Figure 4.10: One method for the generation of 0 and π phase junctions using long-range
spin-triplet correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

Figure 4.11: Schematic for the generation of long-range spin-triplet correlations using
noncolinear magnetizations . . . . . . . . . . . . . . . . . . . . . . . . .

69

Figure 5.1: 7-target sputtering chamber with active plasma . . . . . . . . . . . . . .

71

Figure 5.2: Bit arrays written with electron-beam lithography . . . . . . . . . . . . .

79

Figure 5.3: The ion milling and thermal evaporation vacuum chamber . . . . . . . .

84

Figure 5.4: Ferromagnetic Josephson junction fabrication steps, part 1 . . . . . . . .

86

Figure 5.5: Ferromagnetic Josephson junction fabrication steps, part 2 . . . . . . . .

87

Figure 5.6: Josephson junction sample design and images . . . . . . . . . . . . . . .

88

Figure 5.7: rf-SQUID comparator measurement circuit . . . . . . . . . . . . . . . . .

92

Figure 6.1: Cryogentic phase-sensitive memory cell proposed by Northrop Grumman
Corporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

Figure 6.2: A schematic representation of the cross-sectional structure of our S/F/S
Josephson junctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

Figure 6.3: Cross-sections through the center of our S/F/S Josephson junctions prepared with focused ion beam (FIB) milling and imaged with Scanning
Transmission Electron Microscopy (STEM) to validate the fabrication process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Figure 6.4: The voltage across an overdamped S/F/S Josephson junction containing
a 2 nm layer of NiFeCo, versus the applied current . . . . . . . . . . . . 106
Figure 6.5: The voltage across an S/F/S Josephson junction versus the applied current at applied magnetic fields of 18 mT (circles) and 36 mT (diamonds),
using two different measurement schemes . . . . . . . . . . . . . . . . . . 108
Figure 6.6: Fraunhofer patterns of Josephson junctions containing NiFeCo . . . . . . 110
Figure 6.7: Critical current oscillations and Fraunhofer pattern shifts in Josephson
junctions containing NiFeCo . . . . . . . . . . . . . . . . . . . . . . . . . 113
xiii

Figure 6.8: Switching field distribution for both the NiFeCo and NiFe based S/F/S
junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Figure 6.9: NiFeCo hysteresis loops of unpatterned films with dNiFeCo ranging from
1-5 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Figure 6.10: Fraunhofer patterns of Josephson junctions containing NiFe . . . . . . . 120
Figure 6.11: Critical current oscillations and Fraunhofer pattern shifts in Josephson
junctions containing NiFe . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Figure 6.12: Hysteresis loops of unpatterned films containing PdFe with thickness dPdFe
spanning 8-16 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Figure 6.13: Fraunhofer patterns of Josephson junctions containing PdFe . . . . . . . 125
Figure 6.14: Magnetic switching of PdFe Josephson junctions at small applied fields. . 127
Figure 6.15: Critical current oscillations and Fraunhofer pattern shifts in Josephson
junctions containing PdFe . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Figure 6.16: Critical current oscillations in Josephson junctions containing Ni . . . . . 131
Figure 7.1: Scheme to control the phase-state of a spin-valve Josephson junction . . 135
Figure 7.2: Spin-valve SQUID schematic and the configuration of the four magnetic
states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Figure 7.3: Oscillations in the critical current versus the current applied to the fluxline and the in-plane set field used to control the magnetic state of the
SQUID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Figure 7.4: Asymmetric SQUID theory, closed solutions . . . . . . . . . . . . . . . . 145
Figure 7.5: Asymmetric SQUID theory, open solutions . . . . . . . . . . . . . . . . . 147
Figure 7.6: Using asymmetric SQUID theory to fit the SQUID oscillation data from
the four magnetic states . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Figure 8.1: Hysteresis loops of a Pd/Co multilayer film measured at 5K using a SQUID
magnetometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Figure 8.2: Hysteresis loop measurements of synthetic antiferromagnet (SAFs) films
measured using a dc-SQUID magnetometer . . . . . . . . . . . . . . . . 158

xiv

Figure 8.3: A schematic representation of the vertical cross sectional structure of our
Josephson junctions (not to scale) . . . . . . . . . . . . . . . . . . . . . . 161
Figure 8.4: Vertical cross sections of the junctions described in Fig. 6.2 (b) prepared
using a focused ion beam (FIB) were investigated by high-resolution scanning transmission electron microscopy (STEM) and energy dispersive xray spectroscopy (EDX) . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Figure 8.5: The critical current, Ic , is plotted versus the applied in-plane field H for
junctions with the structure shown in Fig. 8.3 (a) (without F 0 and F 00 ) . 167
Figure 8.6: Critical current, Ic , vs. the applied in-plane field H, for a sample with F 0
= F 00 = Ni (1.6 nm), and n=1, similar to Fig. 8.3 (b) . . . . . . . . . . . 169
Figure 8.7: Ic is plotted vs. the applied in-plane field H, for junctions with the structure shown in Fig. 8.3 (b) with F 0 = Py, F 00 = Ni . . . . . . . . . . . . . 172
Figure 8.8: Comparison of the critical current decay in spin-triplet and spin-singlet
samples and their area-resistance products . . . . . . . . . . . . . . . . . 174
Figure 9.1: Spin-triplet Josephson junction structure and SQUID loop design . . . . 180
Figure 9.2: Flowchart describing the measurement process for probing the two main
aspects of the SQUID samples . . . . . . . . . . . . . . . . . . . . . . . . 183
Figure 9.3: A 3D plot of a complete minor magnetization loop, for SQUID 2A-4 . . . 185
Figure 9.4: The average critical current Ic,Avg oscillates with the current applied to
the flux line Iflux for the P and AP states of SQUID 2A-4 . . . . . . . . 187
Figure 9.5: [A 3D plot of a complete minor magnetization loop, for SQUID 2A-1 . . 188
Figure 9.6: Fitting the P and AP states from SQUID 2A-4 . . . . . . . . . . . . . . 192
Figure 9.7: Repeated switching between the P and the AP-states.

. . . . . . . . . . 195

Figure 9.8: Critical current versus temperature for SQUID 2A-2 in a dilution refrigerator197
Figure 9.9: Critical current versus flux-line current and temperature for SQUID 2A-1
in a dilution refrigerator . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Figure 1:

Repeated switching between the P and the AP-states from SQUID 2A-1.

Figure 2:

Fitting the P and AP states from SQUID 2A-1 . . . . . . . . . . . . . . 208

xv

207

Figure 3:

A 3D plot of a complete minor magnetization loop, for SQUID 2A-2 . . . 209

Figure 4:

Fitting the P and AP states from SQUID 2A-2 . . . . . . . . . . . . . . 210

Figure 5:

A 3D plot of a complete minor magnetization loop, for SQUID 2A-3 . . . 211

Figure 6:

Repeated switching between the P and the AP-states from SQUID 2A-3.

Figure 7:

Fitting the P and AP states from SQUID 2A-3 . . . . . . . . . . . . . . 213

Figure 8:

A 3D plot of a complete minor magnetization loop, for SQUID 3A-3 . . . 214

Figure 9:

Repeated switching between the P and the AP-states from SQUID 3A-3.

Figure 10:

Fitting the P and AP states from SQUID 3A-3 . . . . . . . . . . . . . . 216

Figure 11:

A 3D plot of a complete minor magnetization loop, for SQUID 4A-1 . . . 217

Figure 12:

Fitting the P and AP states from SQUID 4A-1 . . . . . . . . . . . . . . 218

Figure 13:

A 3D plot of a complete minor magnetization loop, for SQUID 4A-2 . . . 219

Figure 14:

Repeated switching between the P and the AP-states from SQUID 4A-2.

Figure 15:

Fitting the P and AP states from SQUID 4A-2 . . . . . . . . . . . . . . 221

xvi

212

215

220

Chapter 1

Introduction

1.1

Energy Efficient Superconducting Computing

Energy use by U.S. data and computing centers is projected to increase from 72 to
176 TWh from 2009 to 2020 [1]. Reducing data center energy consumption by a factor of
10 would save the United States approximately $15 billion per year in 2020, assuming an
energy cost of 0.1 $/kWh. The current push towards exascale computing and beyond with
conventional devices will soon require untenable power consumption, increasing the appeal
for energy-efficient solutions to these growing problems.
Superconducting computing has been recently targeted by the U.S. Department of Energy as a possible way to meet their energy dissipation goal for exascale high-performance
computing [2].† While it is difficult to project, at large scales superconducting computers
are thought to be orders of magnitude more energy efficient than conventional systems based
on Complementary Metal Oxide Semiconductor (CMOS) technologies [1, 3].
The gains in energy efficiency derive from a unique property of superconducting materials, namely that electrical current can be driven through them with zero energy loss from
Joule heating. In the superconducting state, the resistance of such a material is exactly zero,
provided it is below a certain critical temperature. Since the critical superconducting transition temperature Tc of most elemental superconductors (such as Niobium) is below 10 K,
† The Japanese government is also currently involved in an effort to develop large-scale superconducting
computers.

1

cryogenic cooling is required, typically by immersion in liquid Helium (≈ 4.2 K). While most
data centers today utilize some form of refrigeration for cooling, the energy gains of a superconducting computer would have to offset the additional costs of liquid Helium in order to
be economical assuming all other factors, such as performance, were equal.†
Some projections have shown that if the circuits for logic and memory operations as well as
the interconnects between them were made of superconductors, the gains in energy efficiency
would be 10-100 times that of current CMOS technologies, after accounting for the cost of
cryogenic cooling [1]. Also, researchers and industry partners alike are actively investigating
digital superconducting computers and circuits (near 4.2 K) as possible intermediate systems
that would interface above the cold space with quantum computers that operate in the mK
temperature regime.‡
Efforts to harness the advantages offered by superconductors for computing have been
waged since the 1950s with varying levels of success. An early example was the “cryotron”
switch invented by Dudley A. Buck at the Massachusetts Institute of Technology Lincoln
Laboratory, who wound a straight superconducting rod of Ta with an insulated superconducting Nb wire [4]. The current through the Ta rod could be switched on or off by driving
a current through the Nb wire to generate a magnetic field that would suppress the superconducting properties of the Ta. The device was later used to develop several different types
of logical circuits.
When the Josephson effect was discovered in 1962, it paved the way for superconducting
devices based on Josephson junctions, i.e. two superconductors separated by a thin barrier
† High-temperature superconductors (for example with T higher than 77 K, the temperature of liquid
c
nitrogen) could significantly reduce the cost of cooling, but the incompatibility of those materials with most
microfabrication techniques remains a barrier to their expanded use.
‡ Although some quantum computing architectures utilize superconducting elements, their operations are
performed on quantum bits (qubits), unlike the superconducting computers we discuss here, which operate
on classical bits.

2

material such as an insulator, normal metal, or ferromagnet. The decades that followed produced a number of technological advances towards the fabrication of more reliable Josephson
junctions. In 1985, a new logic family based on Josephson junctions called Resistive SingleFlux Quantum (RSFQ) logic was proposed by Konstantin Likharev, Oleg Mukhanov, and
Vasili Semenov at Moscow State University, who suggested a way to achieve fast switching
on picosecond time scales [5]. The general idea is to use brief picosecond Single Flux Quantum (SFQ) voltage pulses to transmit binary information, rather than dc voltages (as in
most semiconductor-based logic schemes). The SFQ pulses can be transmitted at nearly a
third of the speed of light along superconducting lines [6]. Circuits derived from more recent
classes of SFQ logic have been successfully demonstrated to operate at frequencies up to
770 GHz [7]. Major advances via the development of new superconducting logic families in
2011, for example Reciprocal Quantum Logic (RQL) invented at Northrop Grumman Corporation [8] and energy efficient RSFQ (eeRSFQ) by researchers at HYPRES, Inc. [9, 10],
have greatly improved the energy efficiency of logical superconducting circuits.
Several groups are now actively pursuing the development of scalable computers that
utilize superconducting elements in both the logic and memory portions of their circuits.
In 2005 the National Security Agency’s Superconducting Technology Assessment forecasted
that the limiting factor in the eventual development of a successful, fully superconducting
computer would be the memory and storage, and not the processor performance [11]. This
prediction rings true today, as memory development is still catching up with the impressive
advances in superconducting logic.
While there are many possible approaches to building a memory device, a viable system
would ideally need to be energy efficient, fast, non-volatile (retaining the memory information even while the device is turned “off”), and random-access (where the access time is not
3

impacted by the physical location of the requested bits within the memory bank). For example, Magneto-resistive Random-Access Memory (MRAM) has been utilized successfully
since 2006 in conventional computing. In the pursuit of a memory with these characteristics for superconducting computing many researchers today, including the Birge group at
Michigan State University, are studying Josephson junctions containing ferromagnetic layers [12–21]. Because superconductivity and ferromagnetism are inherently antagonistic forms
of order† combining and simultaneously controlling those effects in a ferromagnetic Josephson junction is not a faint-hearted endeavor. Nevertheless, it is precisely the juxtaposition
of these competing forms of order that opens a rich area of physics with new possibilities for
superconducting memory.
For example, if we can modify the amplitude, phase, or type of supercurrent inside a
ferromagnetic Josephson junction in a controllable manner, those degrees of freedom can be
used to represent the logical states of a superconducting memory device. The work presented
here falls under the general scheme proposed by researchers at Northrop Grumman Corporation called Josephson Magnetic Random-Access memory (JMRAM), a superconducting
energy-efficient variant of MRAM that operates by controlling the phase of the supercurrent
across a ferromagnetic Josephson junction [22].
In this thesis, we discuss the materials characterization, optimization, and testing that
culminated in the successful experimental implementation of two types of phase-controllable
superconducting memory devices using ferromagnetic Josephson junctions, both compatible
with JMRAM. In the first type of device, the phase state of the junction is controlled by
tuning the relative orientations of two different ferromagnetic layers in the center of the
junction [18]. In the second type, a series of three ferromagnetic layers with noncolinear
† As we will see in Ch. 4, ferromagnetic materials destroy or strongly suppress superconductivity.

4

magnetizations is utilized to generate a supercurrent carried by long-range, spin-triplet electron pairs. While the physics is much different from the first type of device, the phase state
of this junction is also controlled by modifying the relative magnetization direction of one of
the layers [21]. An overview of the remaining chapters in this thesis is provided below.

1.2

Chapters Overview

In Ch. 2, we give a brief review of superconductivity theory, explaining some relevant aspects of Josephson junctions and Superconducting Quantum Interference Devices (SQUIDs),
the latter of which is used to make low-noise voltage measurements and for phase-sensitive
detection of the memory devices.
In Ch. 3, we discuss the physics of ferromagnetism and some of the necessary considerations that go into the design of controllable magnetic memory bits, including magnetic
domains and different types of magnetic anisotropy.
In Ch. 4, we discuss how the interplay of superconductivity and ferromagnetism (via the
proximity effect) can be advantageously used in various types of ferromagnetic Josephson
junctions. We describe the schemes by which the supercurrent in junctions can be manipulated to produce spin-singlet or spin-triplet supercurrent and the methods for controlling
either the amplitude of the current or the phase across the junctions.
In Ch. 5, we describe the experimental methods, procedures, and equipment used to
fabricate and measure our Josephson junctions and SQUIDs. The basic principle of the
various techniques used such as sputtering, photolithography, electron-beam lithography, ion
milling, and SQUID magnetometry are also discussed.
In Ch. 6, we outline a scheme for using ferromagnetic junctions as a phase-sensitive

5

superconducting memory. As a precursor to that effort we present four separate experimental materials characterization studies of Nb-based micron-scale elliptically-shaped Josephson
junctions containing the central ferromagnetic layer of either Ni65 Fe15 Co20 , Permalloy (abbreviated as Py, the alloy Ni81 Fe19 ), Pd97 Fe3 , and Ni. We compare the results to theory
and discuss the comparative advantages and disadvantages that each of these materials offer
as potential components in cryogenic memory devices.
In Ch. 7, we discuss the first experimental demonstration that the phase state of a Josephson junction can be controllably switched between 0 and π, i.e. a phase-sensitive cryogenic
one-bit memory device using spin-singlet supercurrent. The phase-sensitive detection is
achieved by adding two ferromagnetic Josephson junctions with a “spin-valve” configuration
into a SQUID loop. Since that was the main work of Bethany Niedzielski and Eric Gingrich,
we will focus on my main contribution to that effort: the development of a general asymmetric SQUID fitting program that provided the unambiguous proof that their devices switched
between the phase values of 0 and π. That program is also used to fit the data in Ch. 9.
This type of spin-valve junction is currently being used as part of superconducting memory
cell by our collaborators at Northrop Grumman Corporation [23].
In Ch. 8, we present measurements of a different type of Josephson junction that generates spin-triplet supercurrent through the use of three ferromagnetic layers with noncolinear
magnetizations. The central layer is a synthetic antiferromagnet composed of Pd/Co multilayers on either side of a Ru spacer with magnetization perpendicular to the plane. The other
two ferromagnetic layers, either Py or Ni, have in-plane magnetization. We demonstrate that
the junctions carry spin-triplet supercurrent by observing the relative enhancement of the
critical current in those junctions compared to other junctions without the in-plane layers
that carry only spin-singlet supercurrent.
6

In Ch. 9, we verify experimentally one of the most salient predictions of the spin-triplet
theory, namely that a Josephson junction containing three magnetic layers with noncolinear
magnetizations exhibits a ground-state phase shift of either zero or π depending on the relative orientations of those magnetizations. We implement two of the same types of junctions
presented in Ch. 8 into a SQUID loop to achieve the phase-sensitive detection, similar to
the technique used in Ch. 7. We show that the phase can be reliably switched between
the zero or π states for seven different spin-triplet SQUID devices measured, a number of
which we were able to switch between the two states over a thousand times without error.
This type of phase-controllable junction could also be used as a novel memory element in a
superconducting computer.
In Ch. 10, we conclude by summarizing the results and discuss some remaining open
questions. In light of the results obtained we discuss the future outlook for energy efficient
superconducting memory.

7

Chapter 2

Superconductivity

It has been well over a hundred years since the discovery of superconductivity in 1911,
when the Dutch physicist Heike Kamerling Onnes cooled a sample of mercury below 4.2
K and watched its electrical resistance precipitously drop from about 0.1 Ω to an unmeasureably small value (or < 10−5 Ω, as precise as they could claim at the time) [24]. The
superconducting state, onset by a second-order phase-transition, was later realized to be
a manifestation of quantum mechanical phenomena expressed over macroscopic scales. It
has striking and magnificent physical consequences: electrical currents which can flow in a
material for years in the absence of a driving field, without transferring heat or dissipating
energy, the complete expulsion of magnetic flux within their interior, and an energy gap, just
to name a few. And though superconductivity first appeared to be a curious anomaly, it has
been found to be a rather ubiquitous phenomenon under the right conditions. A majority of
the metals in the periodic table, and many of their alloys, have been discovered to exhibit
superconductivity under the right conditions, for example, at a low enough temperature,
high enough pressures, assuming quality growth conditions, purity, etc. All told, superconductivity is an extremely diverse field of research, with many unsolved puzzles. Even today
its mechanism and behavior in many materials is almost entirely unknown, including many
with unusually high superconducting transition temperatures (Tc ).
In the experiments presented in this thesis, however, we only use one of the better understood superconductors, Niobium, which is heralded for its use in Josephson junctions,

8

possessing the highest Tc of any elemental superconductor (at atmospheric pressure), about
9.2 K. Even if we restrict our focus to the most commonly used and well understood superconductors, that subfield of superconductivity is still quite extensive. Therefore, in this chapter,
we only discuss some basic theoretical aspects of elemental superconductors, such as Nb,
and their applications in Josephson junctions and Superconducting QUantum Interference
Devices (SQUIDs).

2.1

The Meissner-Ochsenfeld Effect and London Theory

If a metal is cooled below its superconducting transition temperature, Tc , the magnetic
flux in its interior is expelled, an effect first discovered by Walther Meissner and Robert
Ochsenfeld in 1932 [25]. Moreover, they discovered that a superconductor is a perfect diamagnet, if it is cooled in the presence of a magnetic field (but not in an excessively strong
one) the field inside the superconductor will be spontaneously expelled.† At the point of the
phase transition, a superconductor will acquire surface currents (which are thermodynamically stable) that act to perfectly cancel the magnetic field inside of it.
The brothers Fritz and Heinz London proposed a model of a superconductor in which
Ohms’ Law (J = σE) was replaced with a relation that would hold for a perfect diamagnet [26, 27]. Since the superconducting charges were required to be dissipationless (as found
by Kamerling Onnes) they argued that one should instead use Newton’s law of motion,

∂J/∂t = E/Λ,

(2.1)

† That behavior is quite different from a perfect conductor: if cooled to T in zero field, a perfect conductor
c
will continue to have B = 0 in it’s interior, even after the application of a magnetic field. However, if a
perfect conductor is in the presence of a field while in the normal state and then cooled to Tc , it will not
expel the magnetic field.

9

where the time derivative of the current density, J, is related to the electric field, E, and
the factor, Λ ≡ m/(e2 ns ), with e the magnitude of the electron charge and and m it’s mass,
and ns the density of superconducting electrons.† This together with the Maxwell equation
∇ × E = −∂B/∂t, where B is the magnetic induction field, leads to

∂h
Bi
∇×J+
= 0.
∂t
Λ

(2.2)

The results of Meissner and Oschenfeld suggested that inside the superconductor, J = 0 and
B = 0, so in Eq. (2.2) the term in brackets must be conserved. Thus we arrive at what is
known as the London equation,
∇×J+

B
= 0.
Λ

(2.3)

This, together with the Maxwell relation, ∇ × B = µ0 J + µ0 0 (∂E/∂t), assuming that the
electric field E is approximately constant in time, implies (taking the curl of both sides and
using ∇ · B = 0),

Âľ
∇2 B = 0 B
Λ
Âľ
∇2 J = 0 J.
Λ

(2.4)
(2.5)

The prefactor determines the length scale over which the fields and currents can typically
† In Sec. 2.3, we’ll see that supercurrent is carried by pairs of electrons, but replacing the variables with
∗
m = 2m, e∗ = 2e, and n∗
s = ns /2, doesn’t change the London theory. The London theory assumed a socalled two fluid model, first described by Gorter and Casimir [28], where only a portion of the total number
of electrons in a superconductor contribute to the supercurrent, but it was later described by Lev Landau
that there aren’t two types of fluids or particles, but rather more like two kinds of motions, superfluid flow
versus normal flow.

10

exist on the surface of the superconductor, called the London penetration depth,

λL ≡

p

Λ/µ0 =

q

m/Âľ0 e2 ns .

(2.6)

Eq. (2.4) implies that the field will penetrate into a superconductor, decaying from it’s value
at the surface in an exponential manner to zero over a short distance ÎťL , which in most
superconductors is about 0.1 µm at zero temperature.† This decay is a signature of the
Meissner-Ochsenfeld effect.

2.2

Ginzburg–Landau Theory

A powerful yet simple way to explain many features of superconductivity from a phenomenological perspective, was developed by Vitaly Ginzburg and Lev Landau in 1950 [29],
now called Ginzburg-Landau theory. In Ginzburg-Landau theory, we are not overly concerned with the microscopic details yet can obtain predictions of physical properties with
high accuracy in many situations. Indeed, Ginzburg–Landau theory is in some sense a bridge
linking the more empirical ideas of the previous section (Meissner-Ochsenfeld effect) with
the full microscopic theory of the next section (BCS theory).
Ginzburg and Landau noted that while London’s theory explained some aspects of superconductivity, it couldn’t be used to understand why the superconducting state could be
destroyed by large fields or currents (as experiments had shown), nor could it predict the surface energy of a superconductor-normal phase boundary. Instead, they started from the key
insight that the transition to the superconducting state, at Tc , could be viewed analogously
to Landau’s theory of second-order phase transitions.
† The penetration depth changes as a function of temperature, being largest at the critical point, T .
c

11

In the theory of second-order phase transitions, the equilibrium state of a system on either
side of a phase-transition can be determined by minimizing its free energy, expressed in terms
of an order parameter, whose value changes from zero above the critical point to a nonzero
value below it.† In Ginzburg–Landau theory, the order parameter for superconductors is a
complex quantity, Ψ(r), related to the density of superconducting electrons (|Ψ|2 ∝ ns ), and
plays the role of an “effective,” or averaged, wavefunction.
In the absence of a magnetic field, we can expand the free energy density of the superconducting electrons, Fs , in even powers of the order parameter,

1
Fs = Fn + ι|Ψ|2 + β|Ψ|4
2

(2.7)

where Fn is the free energy density of the normal state, and the variables α = α(T − Tc ) and
β > 0 are phenomenological parameters, discussed in more detail later. The order parameter,
Ψ, is assumed to be small and not varying too rapidly in space.
In the presence of a magnetic field, B, since the order parameter is coordinate dependent,
we’ll need to also consider terms corresponding to it’s gradient, (∇Ψ)2 , and properly maintain
invariance under a change of gauge through the magnetic vector potential, A, where B =
∇ × A. Then, the free energy density takes the form,

1
1
1
Fs = Fn + ι|Ψ|2 + β|Ψ|4 +
|(−i~∇ + e∗ A)Ψ|2 +
|B|2 .
2
2m
8π
The procedure is then to minimize the total free energy, F =

R

(2.8)

drFs , first with respect

to the order parameter and then A. In particular, minimizing with respect to Ψ∗ yields a
† Landau used this same methodology to explain ferromagnetism, the subject of the next chapter.

12

nonlinear equation for Ψ, the first Ginzburg–Landau equation,

1
(−i~∇ + e∗ A)2 Ψ + αΨ + βΨ|Ψ|2 = 0,
2m∗

(2.9)

while minimizing with respect to the vector potential A yields an expression for the current
density, the second Ginzburg–Landau equation,

Js = −

∗2
i~
∗ ∇Ψ − Ψ∇Ψ∗ 
 − e |Ψ|2 A.
Ψ
2m∗
m∗

(2.10)

The two equations (2.9) and (2.10) are constrained by a boundary condition at the interface
between a superconductor and vacuum,

n · (−i~∇ + e∗ A) Ψ = 0,

(2.11)

where n is the unit vector normal to the boundary.
This set of equations, and their more generalized forms, can be used to understand a
great number of phenomenon, for instance, the first term in Eq.(2.10) leads to the Josephson
effect, while the second term leads to the Meissner effect. Beyond this they can be used to
derive flux quantization, the behavior of type I and II superconductors, the proximity effect,
and the surface energy of a superconductor/normal metal (S/N) interface, which determines
whether it’s favorable to form such a boundary. In type–II superconductors this is the
determining factor for the formation of vortices. For now, we will only use Ginzburg-Landau
theory to set the stage for the microscopic theory in the next section and use it to discuss
some important length scales and the key differences between type–I and II superconductors;
however we will return to it later.
13

As a simple case, consider a homogeneous superconductor without supercurrent, Eq. (2.9)
reduces to,
ιΨ + β|Ψ|2 Ψ = 0,

(2.12)

and has only two solutions: the trivial solution Ψ = 0, corresponding to the normal state,
or the solution where the order parameter is nonzero and constant in space, |Ψ|2 = −α/β.
The phenomenological parameters are given the restrictions, α = α(T − Tc ) and β > 0,
to ensure that this second solution is only valid when T < Tc . Note that if T > Tc , then
α(T − Tc )/β > 0 makes the entire right-hand side of Eq. (2.12) positive, so in that case the
only solution can be Ψ = 0. On the other hand, if T < Tc , the solution |Ψ|2 = −α/β is
valid.
There are two important two length scales that emerge from the Ginzburg–Landau equations: the London penetration depth λL , which we have already seen, and the coherence
length Ξ. The coherence length
s
Ξ=

~2
2m∗ |α|

(2.13)

characterizes the decay of the order parameter near a S/N boundary. The ratio of these
two length scales yields the dimensionless Ginzburg–Landau parameter, κ, the fundamental
characteristic of a superconducting material. Its value determines whether a superconductor
is type I or II. It turns out that Îş is related to the critical magnetic field Hc , above which
superconductivity vanishes,
Îť
κ = L ∝ Hc λ2L .
Ξ

(2.14)

In addition, κ is related to the surface tension at the boundary between a normal and super√
conducting state. For superconductors with Îş < 1/ 2, the surface tension is positive so that

14

boundaries are energetically unfavorable. Such materials are called type I superconductors.
√
On the other hand, the surface tension is negative when Îş > 1/ 2 allowing vortices to form
between a certain range of fields. This is the case for type II superconductors including Nb.
While the Ginzburg–Landau theory works well within its limitations, it is not able to
account for the microscopic origin of superconductivity, which is achieved using BCS theory.
However, Lev Gor’kov established in 1959 that Ginzburg–Landau theory can be derived
from BCS in certain regimes [30, 31]. The physical understanding extracted from Ginzburg–
Landau theory was instrumental in guiding the development of the full microscopic theory.

2.3

BCS Theory

Ginzburg-Landau theory is excellent at capturing many properties of superconductors,
but it is unable to answer some questions that only a microscopic theory can address. For
example, why can we describe some materials with a superconducting order parameter, Ψ,
while others are not superconducting at all? We can distinguish between type I and II
superconductors with the Ginzberg–Landau parameter, κ = λL /ξ, but what determines
ΝL and Ξ for one material versus another? What sets the precise values of the critical
temperature, Tc , and the critical field, Hc ? How do the individual electrons behave in a
superconductor? We won’t discuss all of these questions, but ask them merely to motivate
the need for a microscopic theory.
The first complete microscopic theory for one class of superconductors, which included
most elemental superconductors, including Nb and its diluted alloys, was put forth by John
Bardeen, Leon Cooper, and Bob Schrieffer in 1957, and is now known as BCS theory [32,33].
BCS were motivated, in part, by several experimental results that came before, which showed

15

that below Tc there existed an energy gap in the excitation spectrum near the Fermi level.
In 1950, C. A. Renolds et al. [34] and E. Maxwell [35] showed that the Tc in these materials
√
varied with the mass of its isotopes (Tc ∝ 1/ Miso ), which suggested superconductivity may
be related to the interactions with the lattice of a material.
In 1956, Cooper showed that two electrons with oppositely aligned spins near the Fermi
level will form a bound state if there is a weak attractive interaction between them [36].
Cooper assumed that the attractive potential felt by the electrons is only nonzero within
a narrow band in k-space. Under these conditions, two electrons will form a bound state,
called a Cooper pair, for any attractive interaction, regardless of how small.
An explanation for the physical origin of the attractive interaction was provided earlier
by Frölich in 1952 [37] and by Bardeen and Pines in 1955 [38]. In most metals, the electrons
interact with each other via the Coulomb force and with phonons, i.e. quantized lattice
vibrations in the crystal. As they showed, when two electrons exchange virtual phonons they
will generate a weak attractive interaction, usually of the order 10−3 eV. That attractive
electron-phonon interaction together with Cooper’s theory implied that Cooper pairs would
form in a superconductor, provided that it was at low enough temperature, so that the
weakly bound pairs could not be broken by thermal excitations.
The complete explanation for superconductivity came shortly thereafter. In 1957, Schrieffer, working with Bardeen and Cooper, proposed a macroscopic quantum state composed of
electron pairs with opposite spins. Extending Cooper’s two-particle results to an N -particle
system, Schrieffer was able to obtain the energy gap for excitations from the ground state.
In their model, the electron pairs begin to condense below the critical temperature Tc until
the binding energy for an additional pair becomes zero.
One of the major triumphs of BCS theory was the determination of the temperature
16

dependence of the energy gap ∆(T ). It can be obtained in the limit of weak coupling,
~ωc /kB Tc  1, where ~ωc is a cutoff energy: between the range −~ωc to ~ωc , around the
Fermi energy the interactions between the electrons is attractive, −V , while outside it is
zero. Under those conditions BCS theory yields the equation,

1=

V X tanh(Ek /(2kB T ))
,
2
Ek

(2.15)

k

where
q
Ek = ξk2 + ∆2

(2.16)

is the excitation energy of a quasiparticle of momentum ~k and ξk = k − µ is the single
particle energy measured relative to the Fermi energy Âľ (also called the chemical potential).
From the expression (2.15), we can determine the critical temperature Tc when ∆(T ) → 0.
Noting that in this limit Eq. (2.16) reduces to |Ξk | and replacing the sum in (2.15) with an
integral (solved numerically) yields,

kB Tc = 1.13~ωc e−1/N (0)V .

(2.17)

The T = 0 density of states for electrons at the Fermi level, that are either spin-up or
spin-down, is N (0). Meanwhile, the energy gap turns out to be given by,

∆=

~ωc
≈ 2~ωc e−1/N (0)V
sinh[1/N (0)V ]

17

(2.18)

As a consequence of Eqs. (2.17)–(2.18), the energy gap at zero temperature is

∆(0) = 1.764kB Tc .

(2.19)

This result is consistent with many experimental tests with different materials, which have
been shown to have a ∆ between 1.5 kB Tc and 2.25 kB Tc [39].
The first direct observation of the energy gap predicted by BCS theory came with with
the work of Ivar Giaever in 1960. While at General Electric, he demonstrated that electrons
would quantum mechanically tunnel from a superconductor to a normal metal through an
insulating barrier [40]. His experiment, which used an Al/AlOx /Pb tunnel junction, showed
that when the temperature was lowered below the Tc of Pb (7.2 K), at low bias, an energy gap
appeared. He further realized that by measuring the differential conductance dI/dV versus
voltage V , where I is the current, one would find a curve that would mimic the density of
states predicted from BCS theory.

2.4

The Josephson Effect

In 1962 Brian Josephson predicted that a supercurrent Is could pass between two superconductors that were separated by a thin barrier (sometimes also called a “weak link”),
in what is now called a Josephson junction. The supercurrent can flow even without an
external bias, known as the dc Josephson effect,

Is = Ic sin φ,

18

(2.20)

provided that there is a non-zero phase difference φ across the junction and that the current
is less than the critical current, Ic .
Eq. (2.20) can be derived from BCS theory or Ginzburg–Landau theory assuming that the
coherence length is much larger than the thickness of the barrier, Ξ  l. The gauge-invariant
phase difference across the junction,

2π
φ ≡ ∆θ −
ÎŚ0

Z
A ¡ ds,

(2.21)

is computed by integrating the vector potential A along a path between the two superconducting electrodes. In the absence of magnetic fields, φ reduces to the phase difference ∆θ ≡
θ2 − θ1 of the complex Ginzburg-Landau order parameter, with form Ψ1,2 = |Ψ0 | exp(iθ1,2 ),
in the two electrodes.
If a finite voltage difference V exists across the junction, then the phase difference acquires
a time dependence known as the ac Josephson effect,

2eV
2π
dφ
=
=
V,
dt
~
ÎŚ0

(2.22)

where Φ0 = 2π~/(2e) is the magnetic flux quantum. As a result, the supercurrent Eq. (2.20)
oscillates at a frequency ν = 2eV /h and the energy of a Cooper pair traversing the junction
changes by an amount hν.
The energy stored in the junction, the Josephson energy, is found by integrating the
electrical work done by a current to change the phase,

Z
E=

~
Is V dt =
2e

Z



Ic sin φ dφ = EJ 1 − cos φ ,

19

(2.23)

where we used Eq. (2.22) to change the integration variable from time t to the phase φ, and
the Josephson coupling energy is
EJ =

2.5

~Ic
.
2e

(2.24)

The RCSJ Model

To describe a physical Josephson junction, we consider an ideal junction described by
Eqs. (2.20) and (2.22) that has been shunted by a resistance R, which accounts for dissipation
at finite voltages, and a geometric capacitance C between the two electrodes.

R

C
Figure 2.1: The equivalent circuit for the RCSJ model of a physical Josephson junction. An
ideal junction, at the center, is in parallel with a resistor R and capacitor C.

A bias current I applied to the junction is split into three branches, as in Fig. 2.1, yielding
an expression we can solve for the time dependence of the phase difference φ,

C

dV
V
+ + Ic0 sin φ = I.
dt
R

(2.25)

In practice, the critical current of the physical junction, Ic , can be different from that of
the ideal junction, Ic0 , in Eq. (2.20). Indeed, thermal fluctuations are known to surpress
the observed critical current, Ic < Ic0 [41]. Using Eq. (2.22) to eliminate V , we can recast
Eq. (2.25) in terms of φ to arrive at the second-order differential equation describing the

20

time dependence of the phase,






 2
~ 2 d2 φ
1 dφ
I
~
= EJ
− sin φ .
C 2 +
2e
2e R dt
Ic0
dt

(2.26)

Remarkably, this expression is analogous to the classical equation of motion for a particle of
 2
~
mass 2e
C moving in a potential,

U = −EJ

under a linear drag force



~
2e

2

1 dφ
R dt .


I
φ − cos φ ,
Ic0

(2.27)

The dimensionless Stewart and McCumber damping

parameter [42, 43], which is the square of the junction’s quality factor Q, is given by

βc = Q2 =

(Ic0 R)2 C
= (ωp RC)2 ,
EJ

(2.28)

where ωp is the plasma frequency.
When the bias current vanishes, the potential in Eq. (2.27) is a series of identical wells of
depth 2EJ as a function of the phase difference φ (see Fig. 2.2). As the current is increased,
the cosine begins to tilt and the barrier height shrinks, making it possible for the particle
to escape the well via thermal fluctuations. When I = Ic0 , the increasingly shallow minima
become inflection points—unstable equillibria that, if subjected to perturbations, guarantee
the particle will tumble down the washboard. In the absence of fluctuations, the solution
to Eq. (2.27) is static, as the particle is confined to a particular well. In contrast, no stable
equilibrium exists for a bias current higher than Ic0 , and the phase difference φ necessarily
becomes time dependent. According to Eq. (2.22), in such a case, the voltage V is no longer
zero.
21

0

-10
U
EJ

-20

I /Ic0
0
1/2
1
3/2

-30

-40
0

2π

4π

6π

8π

ϕ
Figure 2.2: The tilted washboard potential of the RCSJ model. Static solutions to Eq. (2.27)
exist for a bias current I less than or equal to the critical current Ic0 (solid, dashed, and
dot-dashed lines). As soon as I > Ic0 , only solutions time-dependent in the phase φ exist
(dotted line). This causes the voltage V to become nonzero, according to Eq. (2.22), and
the junction to go normal.
2.5.1

Overdamped Junctions

When the capacitance C is small (the quality factor Q  1), Eq. (2.26) for the phase
reduces to a first-order differential equation
I2 R
dφ
= c0
dt
EJ




I
− sin φ .
Ic0

(2.29)

Evidently, the time-dependent solutions for φ, which we saw previously occur when I > Ic0 ,
correspond to a positive dφ/dt.
We can compute the voltage averaged over one period T of the phase by integrating

22

4
2

V
Ic0 R

0
-2
-4
-4

-3

-2

-1

0

1

2

3

4

I /Ic0
Figure 2.3: The time-average voltage difference, Eq. (2.32), across the overdamped junction
in the RCSJ model (solid line). For |I| ≤ Ic0 , the voltage remains zero, but approaches
Ohm’s law (dashed line) when |I|  Ic0 .
Eq. (2.29),
Z 2π
0

Z T
dφ
=
I/Ic0 − sin φ
0

2R
Ic0
dt.
EJ

(2.30)

Solving this expression for the period T and inserting into Eq. (2.22),

2π
I V
= c0 ,
T
EJ

(2.31)

yields the time-average voltage whose magnitude scales with the bias current as:

|V | = R

q
2 .
I 2 − Ic0

(2.32)

The voltage, plotted in Fig. 2.3, remains zero for |I| ≤ Ic0 (as only static solutions exist for
the phase) and asymptotically approaches Ohm’s law, V = IR, for |I|  Ic0 .

23

2.6

Flux Quantization and SQUIDs

In this section we will discuss flux quantization and combine it together with the Josephson effect to describe Superconducting QUantum Interference Devices (SQUIDs). The experimental realization of flux quantization was observed independently in 1961 by the groups
of Deaver and Fairbank at Stanford [44] and Doll and Naubauer at the Walther-Meibner
Institute (WMI) [45]. They showed that since the macroscopic wavefunction must be singlevalued while going around a closed superconducting loop, the flux is quantized in multiples
of the flux quantum Φ0 ≡ h/2e ≈ 2.07 × 10−15 Wb. Normally, without any fields present
the phase difference φ is constant in the superconductor, but if the loop is threaded by a
magnetic field, φ will change in discrete steps of 2πn, with n being the number of flux quanta
in the loop.

2.6.1

Magnetic Flux Quantization

The macroscopic quantum model of superconductivity postulates the existence of a
macroscopic wavefunction describing an ensemble of Cooper pairs in a superconductor,

Ψ(r, t) =

p
n∗s (r, t) eiθ(r,t) .

The local density n∗s (r, t) = |Ψ(r, t)|2 yields the total number

(2.33)

R

Ψ∗ (r, t)Ψ(r, t)dV = Ns∗ of

Cooper pairs with mass m∗ and charge q ∗ = −2e. Subject to an electromagnetic field, the
wavefunction Eq. (2.33) obeys the Schrödinger equation,

1
∂Ψ(r, t)
=
i~
∂t
2m∗



2
~
∗
∇ − q A(r, t) Ψ(r, t) + q ∗ φ(r, t)Ψ(r, t),
i

24

(2.34)

for the vector potential A and the scalar potential φ(r, t) (not to be confused with the phase
difference φ). It follows from Eq. (2.33) and Eq. (2.34) that the corresponding macroscopic
quantum current density Js of the Cooper pairs is
 
 
−i~∇ − q ∗ A
∗
Ψ
Js =
Ψ
m∗


~
q∗
∗
∗
= q ns (r, t)
∇θ(r, t) − ∗ A(r, t) .
m∗
m
q ∗ Re

(2.35)

Evidently, the velocity of the Cooper pairs is vs = ~/m∗ ∇θ(r, t) − q ∗/m∗ A(r, t).
Expressing the current density Eq. (2.35) in terms of the London coefficient from Sec. 2.1,
m∗
Λ ≡ ∗ ∗2 ,
ns q

(2.36)

which assumes a constant Cooper pair density n∗s (r, t) = const, we can derive the quantization of magnetic flux enclosed by a superconducting ring. Indeed, we know from Stoke’s
theorem that an integral of the vector potential around a closed contour C can be converted
to an integral over a surface S to yield the magnetic flux ÎŚ,

I

Z

Z

A ¡ dl =
C

(∇ × A) · ds =

B ¡ ds = Ό,

S

(2.37)

S

for a magnetic field B piercing the surface. Replacing the vector potential in this expression
with the current density (2.35) leads to a quantity ÎŚ0 ,

ÎŚ0

I

~
= Φ + ΛJs · dl = ∗
q
C

termed the fluxoid.

25

I
∇θ · dl,
C

(2.38)

To show that Ό0 in Eq. (2.38) is quantized, we will study the integral of the phase θ. For
an arbitrary path from r1 to r2 , the integral of the gradient of a scalar function is simply
the difference of the function at the two locations
Z r
2
r1

∇θ · dl = θ(r2 , t) − θ(r1 , t),

(2.39)

which clearly vanishes for a closed loop, r1 = r2 . However, the phase of the wavefunction
Eq. (2.33) is only defined within modulo 2π,

θ(r, t) = θ0 (r, t) + 2πn,

(2.40)

for a θ0 (r, t) specified on [−π, π], and an integer n to ensure that the wavefunction remains
single valued. Therefore, an integral over a closed loop evaluates to 2πn. This finding gives
rise to the flux quantum,

ÎŚ0 =

h
h
=
≈ 2.067 × 10−15 Wb,
∗
|q |
2e

(2.41)

as the fluxoid, Eq. (2.38),
ÎŚ0 = n ÎŚ0 ,

(2.42)

becomes quantized.
When the superconducting ring thickness is much larger than the London penetration
depth ÎťL , our integration in Eq. (2.38) along a contour C well within the superconducting
ring has a negligibly small contribution from the current density Js . In this case, we find

26

that Eq. (2.38) reduces to
Z
ÎŚ=
S

B ¡ ds = n Ό0 ,

(2.43)

so that the flux Ό through the ring is an integer multiple of the flux quantum. The measurement of this effect, mentioned earlier (Deaver and Fairbank [44], Doll and Näbauer [45]),
served as further confirmation that superconducting electrons are paired according to BCS
theory. Shortly after these findings, William Little and Roland Parks clearly showed in their
experiment [46] that it is the fluxoid ÎŚ0 that is quantized in general, and that the flux ÎŚ is
only quantized when the contribution in Eq. (2.38) from the current density can safely be
ignored.

2.6.2

SQUIDs

At this point, a natural question we might ask is whether we can combine the quantization of flux in a superconducting loop with the Josephson effect to elicit novel phenomena.
The answer, of course, is that when a superconducting ring is interrupted by one or more
Josephson junctions macroscopic quantum interference is observed. This now famous discovery was the pioneering work of Robert Jaklevic, John Lambe, James Mercereau, James
Zimmerman, and Arnold Silver at the Ford Research Labs in the mid 1960s. The resulting
Superconducting QUantum Interference Device, or SQUID, has a wide range of applications
including superconducting electronics and quantum computing, and is so sensitive that it
can be used as a magnetometer to measure extremely small magnetic fields in biological
and medical research. In this thesis, we utilize SQUIDs in three different contexts to enable
the development of near-term cryogenic memory devices: as magnetometers to characterize
magnetic materials, for low-noise voltage measurements, and as phase-sensitive detectors.

27

I
C
Ic1 sinϕ 1

a
b

Bext
Is

d
c

Ic2 sinϕ 2

I
Figure 2.4: The symmetric dc SQUID with total supercurrent I. A superconducting loop
(light gray) is interrupted by two weak links (dark gray) to form two Josephson junctions connected in parallel. Integration around the closed contour C, shown in blue, yields Eq. (2.51)
for the relation between the gauge invariant phase differences φ1 and φ2 . The circulating supercurrent Is produces a screening flux that contributes, along with the flux from an external
magnetic field Bext , to the total flux in the loop, Eq. (2.52).
Although SQUIDs can be formed from as little as one Josephson junction (the rf SQUID),
a particularly useful variant is the dc SQUID invented in 1964. Here, two Josephson junctions
are connected in parallel via a superconducting loop as shown in Fig. 2.4. We will see that the
amplitude of the supercurrents through the two branches interfere, producing a modulation
in the maximum total supercurrent of the SQUID.
As we saw previously in Sec. 2.6.1, an integral of the gradient of the phase over a closed
contour is quantized. For the integration path shown in Fig. (2.4), we find
I
∇θ · dl = 2πn
C

(2.44)
= (θb − θa ) + (θc − θb ) + (θd − θc ) + (θa − θd ) + 2πn.

28

where we’ve simply added 0 mod 2π. Let’s consider each term in parentheses in Eq. (2.44).
The phase difference across the junctions can be written in terms of the gauge-invariant
phase difference as
Z
2π b
A ¡ dl,
θb − θa = φ1 +
ÎŚ0 a
Z
2π d
θd − θc = −φ2 +
A ¡ dl.
ÎŚ0 c

(2.45)
(2.46)

The negative sign appearing in front of φ2 is because we chose it to be defined in the opposite
direction of the integration path, unlike φ1 , which is in the same direction. For the remaining
two terms in Eq. (2.44), recall from Sec. 2.6.1 that the gradient of the phase is related to the
quantum current density Js through the expression

∇θ =

2π
(ΛJs + A) .
ÎŚ0

(2.47)

This means that we can write the last two expressions in Eq. (2.44) as
Z c

Z
2π c
θc − θb =
∇θ · dl =
(ΛJs · dl + A · dl) ,
ÎŚ0 b
b
Z d
Z
2π a
(ΛJs · dl + A · dl) .
θa − θd =
∇θ · dl =
ÎŚ0 d
a

(2.48)
(2.49)

Combining all four terms and replacing the integration of the vector potential around the
closed loop by the magnetic flux, we find
Z
Z
2π
2π c
2π d
φ2 − φ1 =
ÎŚ+
ΛJs · dl +
ΛJs · dl,
ÎŚ0
ÎŚ0 b
ÎŚ0 a

(2.50)

mod 2π. Furthermore, as we saw in Sec. 2.6.1 regarding the flux quantization in a su29

perconducting cylinder, if the thickness of the junction is much greater than the London
penetration depth ÎťL , then the current density is approximately zero. In such a case, the
final integrals can be neglected and we arrive at the simple relation for the difference in the
two gauge-invariant phase differences,

φ2 − φ1 =

2π
Φ (mod 2π),
ÎŚ0

(2.51)

which show that the phases are related via the total flux through the loop.
From the result of Eq. (2.51), we can compute how the current in the SQUID varies with
the total flux. In general, the flux ÎŚ appearing in Eq. (2.51) has a contribution from an
externally applied magnetic field Bext and from the inductance L of the loop,

ÎŚ = ÎŚext + ÎŚs = ÎŚext + LIs .

(2.52)

The circulating supercurrent Is that produces a screening flux ÎŚs in the loop is
I sin φ1 − Ic2 sin φ2
,
Is = c1
2

(2.53)

while the total current I in the SQUID is

I = Ic1 sin φ1 + Ic2 sin φ2 .

(2.54)

With Eq. (2.51) and assuming both junctions have identical critical currents, Ic1 = Ic2 = Ic ,

30

Eqs. (2.52) and (2.54) become


πΦ
sin φ1 +
I = 2Ic cos
ÎŚ0
 


πΦ
πΦ
Φ = Φext − LIc sin
cos φ1 +
.
ÎŚ0
ÎŚ0


πΦ
ÎŚ0





(2.55)
(2.56)

To find the maximum current Imax through the SQUID for a given ÎŚext , we maximize
Eq. (2.55) with respect to φ1 while accounting for the φ1 dependence of Eq. (2.56).
We can consider a simple limiting case of this approach. To start, let’s define a screening
parameter βL = 2LIc /Ό0 to be the ratio of the maximum screening flux, corresponding to
a maximum circulating current of Ic according to Eq. (2.53), to half a flux quantum ÎŚ0 /2.
The limit we will study is βL  1 so that the screening flux is negligibly small and Όext
dominates the total flux. After a straightforward calculation, we find that the maximum
current is
 



πΦ
ext
,
Imax = 2Ic cos

ÎŚ

(2.57)

0

for a given ÎŚext . The variation of Imax with ÎŚext is shown in Fig. 2.5.
Later, in Ch. 7, we will consider a more general version of the SQUID theory discussed
here. In particular, we will allow for the two Josephson junctions to have different critical
currents and accounting for possible geometric asymmetries in the SQUID whereby two
inductances are needed to specify the loop. We will also treat both large and small screening
parameters.

31

2

Imax
1
Ic

0
0

1/2

1

3/2

2

ÎŚext /ÎŚ0
Figure 2.5: Symmetric SQUID oscillations. The maximum current Imax in the SQUID
oscillates as a function of the externally applied flux Όext according to (2.57). Here, βL  1
so that Φ ≈ Φext .

32

Chapter 3

Ferromagnetism

Ferromagnetism is one of the most spectacular and readily observable physical phenomena. In this chapter we give a rapid-fire overview of ferromagnetic materials, insofar as to
understand some aspects of their behavior relevant to ferromagnetic Josephson junctions,
discussed later in Ch. 4.

3.1

Historical Overview

Magnetic materials are generally classified according to their magnetic susceptibility,

χm = M/H,

(3.1)

the dimensionless ratio of the magnetization, M, induced in a material and the magnetic
field, H, applied to it, as demonstrated by Pierre Curie in 1895. The sign of χm varies for
paramagnetic (χm > 0) and diamagnetic materials (χm < 0). Curie observed that excessive
thermal fluctuations will destroy magnetic ordering, reasoning that a paramagnetic or diamagnetic materials’ magnetic susceptibility was inversely proportional to its temperature,
now known as Curie’s law. Curie was well aware of ferromagnetic materials, classifying them
as a subset of paramagnetic materials with extremely large χ, though he could find no reason
why their magnetic susceptibility was measured to be thousands of times larger than other
typical paramagnets.

33

Then in 1906 Pierre Weiss, astonishingly, even without direct knowledge of the quantum
mechanical origins of ferromagnetism, put forth two foundational postulates that have since
dominated the theory of ferromagnetism for over a century: spontaneous magnetization and
division into magnetic domains [47]. He found that if a ferromagnet, such as magnetite
(Fe3 O4 ) was cooled below a certain temperature, TCurie , called the Curie temperature, it
would spontaneously develop an ordered magnetic state with a net magnetization, even in
the absence of an applied field. Wiess discovered that in an applied field (that is sufficiently
weak), ferromagnetic materials deviated from Curie’s law, following instead what is now
known as the Curie-Wiess law,
χm =

C
,
T − TCurie

(3.2)

where C is a constant, T is the temperature.† Above Tcurie the material will become paramagnetic. Wiess hypothesized that spontaneous magnetization in ferromagnetic materials was
due to an additional strong molecular field generated within the magnetic materials, but its
origin was somewhat of a mystery.
He tried to reconcile this theory with the well known fact that some ferromagnetic materials, such as single crystals of Fe, can often be found in a demagnetized state (clearly
not “self-saturating”), but when placed in a small magnetic field will subsequently produce a
extraordinarily large magnetic moment of (about 1.7 MJ/T per cubic meter), persisting long
after the field is removed. Wiess posited that these unmagnetized ferromagnetic materials
consisted of a scrambled network of many magnetic domain regions whose net magnetization
was very nearly zero. He argued that the act of magnetizing the material, by placing it in
a strong enough magnetic field, (which he discovered had to be applied in certain magneti† The Curie-Wiess law fails to describe the magnetic susceptibility of some materials very close to T =
TCurie . Rather, it follows χm ∝ (T − TCurie )−γ , where γ is a critical exponent.

34

zation directions, but not others), would then cause the domains whose magnetization’s are
aligned with the applied field to grow at the expense of the domains who are anti-aligned to
the field. The resulting contribution would then sum up to the very large values that had
been observed experimentally.
The origin of Weiss’s molecular field remained a mystery until the advent of quantum
mechanics, and was explained independently by Werner Heisenberg [48] and Paul Dirac in
1926 [49], who showed that ferromagnets (as well as other materials) are strongly influenced
by the exchange interaction. The exchange interaction is a consequence of Coulomb forces
combined with the spin-statistics theorem applied to a joint system of Fermions (electrons).
It results in an “effective” force (the exchange force) that causes the spins in a material to
align themselves along some preferred axis.†

3.2

Magnetic Domains

While the dipolar interactions between the magnetic moments in a ferromagnet result
in forces that are typically several orders of magnitude smaller than the strong exchange
forces, they often play a crucial role in the materials’ collective magnetic state. That is
because the exchange interaction, while powerful over a few atomic spacings, is fairly shortranged, decreasing exponentially with the distance between spins. In contrast, at large
distances, the magnet moments will behave more like dipoles, and the resulting energy of
those dipole-dipole interactions decreases as the inverse square of the distance between spins,
a quantity that can become quite large after summing over the contributions from a large
† In the parlance of Hartree-Fock theory for many-electron systems, to satisfy the anti-symmetry property
of the wavefunction of a Fermionic system, one must replace the trial wavefunction, originally a simple
product of single electron states, by a Slater determinant (which is anti-symmetric), resulting in an extra
term (the exchange term).

35

macroscopic collection. A ferromagnet can thus seek to reduce it’s dipolar energy by breaking
up into uniformly magnetized domains, each of whose magnetizations are oriented in different
directions. While conceptually proposed by Wiess, the first concrete theory to predict the
shape and size of magnetic domains is attributed to the work of Lev Landau and Evgeni
Lifshitz in 1935 [50].
A somewhat contrived model, but one that captures the essence of magnetic domains
in real materials, is to imagine a number of individual spins arranged in a regular threedimensional lattice with the restriction that the spins can only point along a few directions
(e.g. ±x̂ and ±ŷ). Then, one can write down the energy contributions associated with the
short range exchange forces (like in the Heisenberg model), together with both the longrange dipolar forces and the magnetic induction field B that is created by all the spins. If
the strength of the exchange coupling is large, and the dimensions (i.e. number of lattice
sites) of the system are small, the most energetically favorable configuration is to align all
of the spins into one large magnetic domain. In this limit the exchange field more than
R
offsets the large magnetostatic energy, which goes as out B 2 dV with integration spanning
the region outside the volume V of the solid. If, on the other hand, the strength of the
exhange coupling is small and the size of the system is large, it becomes more energetically
favorable for the system to separate into many domains so as to lower the magnetostatic
energy, while at the expense of creating many magnetic interfaces.
Landau and Lifshitz showed that the resulting individual domains can be macroscopic
in size and the collection of them are often arranged into rather complex structures. Some
examples of possible magnetic domain configurations are shown in Fig. 3.1, but these depend
strongly on the shape, size, and composition of the ferromagnet of interest. The boundary
between domains, refered to as domain walls, can occur rapidly over only a few lattice sites,
36

(a)

(b)

(c)

Figure 3.1: Ferromagnetic domain structures: A ferromagnetic material, seeking primarily
to balance the competition between exchange forces and electrostatic dipolar forces between
it’s electrons, will break up into regions of uniformly magnetized regions called domains.
Fig. (a) shows a schematic of a ferromagnet with two domains separated by a single domain
wall. Arrows indicate the direction of the magnetization in their respective regions. For
larger size samples, or those with weaker exchange forces the system will typically break up
into many domains as shown in (b). Many ferromagnets have more complicated magnetic
structures, for instance, where the change in the direction of the spins occurs gradually over
many lattice sites, as shown in (c), for a “C”-shaped configuration. Fig.(c) could also be a
zoom in of the domain wall in (a).
as in Fig. 3.1, or the spin reversal can occur gradually over a large number of adjacent spins.
In two-dimensions, for example, one will often find large-scale “C”-shaped domains, similar
to those shown in Fig. 3.1(c), or “S”-shaped domains. While those types of structures appear
complex at first glance, they turn out to be more energetically favorable than an abrupt
domain wall. Magnetizing a ferromagnet in a strong field is the process of “sweeping out”
domain walls to convert a multi-domain state into one with a single magnetic domain. As one
can infer from the M vs. H loop of Fig. 3.4(b), starting at the origin and tracing the curve
to the point Msat , this initial magnetization process is irreversible, in that if the applied field
is removed, the ferromagnet does not revert back to it’s original domain structure. Other
factors such as defects in the ferromagnet or on it’s boundaries could cause some magnetic
domains to become pinned regardless of the field applied, or revert back to their former

37

Easy
Axis

z
Îł

M

θ

H

Hard
Axis

x

Figure 3.2: Schematic setup for the Stoner-Wohlfarth model. We consider a single magnetic
domain with uniaxial anisotropy, shown in gray. The magnetization, M, makes an angle θ
with the easy axis and the applied magnetic field, H, makes an angle Îł with the easy axis.
orientations when the applied field is removed.

3.3

Magnetic Anisotropy

The magnetic properties of a ferromagnet are often directionally dependent, requiring the
least amount of energy to magnetize it in one particular spatial direction, concurrent with the
direction of spontaneous magnetization, commonly called the “easy axis” (as opposed to the
least energetically favorable direction, called the “hard axis”). Such a ferromagnet is said to
have magnetic anisotropy. The subject is of great practical interest, being exploited in many
magnetic technologies, including the experiments presented in Chs. 6-9. There are many
different types of magnetic anisotropy, but we’ll focus on only three: magneto-crystalline,
shape, and exchange anisotropies, which will be discussed in more detail later.

38

3.3.1

The Stoner-Wohlfarth Model: Uniaxial Anisotropy

To model the magnetization of a single ferromagnetic domain with magnetic anisotropy,
we’ll consider the case of a general uniaxial anisotropy, which for example, could be due to
magnetocrystalline or shape anisotropy. It is an example of the Stoner-Wohlfarth model,
developed by Edmund Clifton Stoner and Erich Peter Wohlfarth in 1948 [51]. In this model,
the ferromagnet of interest has one axis of high symmetry (the easy axis), which we will
arbitrarily designate to point along ẑ. The ferromagnet is placed in an external magnetic
field, H, that is applied in the xz-plane, as shown in Fig. 3.2.
A first order approximation is to write the energy per unit volume due to the anisotropy
as
uaniso = K(m2x + m2y ) = K sin2 θ,

(3.3)

expressed in spherical polar coordinates, where θ is the angle that the magnetization M makes
with the z-axis, and m = M/|M|, whose cartesian components obey m2x + m2y + m2z = 1, and
K is a constant called the anisotropy parameter. For convenience we’ll define K = βM 2 /2
where β is a dimensionless constant. The total energy density u is calculated by adding
Eq. (3.3) to the energy density due to the applied field, uext , or

u = uaniso + uext
β 2 2
M sin θ − µ0 M · H
2
β
= M 2 sin2 θ − µ0 M (Hx sin θ + Hz cos θ).
2

=

(3.4)

How does this uniaxial anisotropy influence the magnetic state of the ferromagnet? Consider the simple case where H k ẑ, so that u = (βM 2 /2) sin2 θ − µ0 M Hz cos θ. Initially, if
µ0 Hz = 0, the energy density is minimal at θ = 0 (and also degenerate with θ = π, but let
39

3
2
1
u
K

0

Îź0H
-1

0
-1
-2
-3

-2
-3
0

π/4

π/2

3π/4

π

θ
Figure 3.3: Magnetization reversal in the Stoner-Wohlfarth model. Energy density u, normalized to the anisotropy parameter, K, versus the angle of the magnetization to the easy axis
θ, and the applied field µ0 H. We plot Eq. (3.4), for the case of H k ẑ, for K/M = βM/2 = 1.
With the magnetization initially aligned with the easy axis (θ = 0), there is an energy minimum at small applied fields (solid line). As the magnitude of the field is increased in the
−ẑ direction, the energy minima at θ = 0 becomes metastable (when µ0 H = −βMẑ), and
eventually is unstable, with a new global energy minima at θ = π. The magnetization has
reversed direction.
us consider the system with the θ = 0 solution). Next, let’s say we wish to apply the field in
the opposite direction, −ẑ, to try and flip M from θ = 0 to θ = π. In Fig. 3.3 we plot how
the energy density changes with respect to θ for different values of the applied field.
It’s easy to see that at small values of |Hz | the solution at θ = 0 is a stable local minimum

2 
> 0). Further increasing |Hz | until it reaches µ0 Hz = −βM , now one will find that
( d 2u 
dθ θ=0

2 
the solution at θ = 0 becomes metastable ( d 2u 
= 0) and further increasing |Hz | will
dθ

θ=0

force the magnetization to the new global energy minimum at θ = π, as shown in Fig. 3.3.
Therefore, by this field the magnetization suddenly reverses its direction.
The response of the magnetization M versus the applied field H is known as a magne40

tization curve. In ferromagnets, the magnetization curve is nonlinear with H, rising until a
sufficiently large field where the curve saturates to a constant value Ms . Furthermore, reducing the applied field, the magnetization will not retrace over the same curve but remain
nonzero even when H is reduced to zero. In the last example (with H k ẑ) the field had to
be reduced to µ0 Hz = −βM to get the magnetization to reverse. A similar behavior is also
true sweeping the applied field back in the other direction.
The ferromagnet thus displays a hysteretic behavior, and in effect, retains some “memory”
of its prior configuration history. We will often refer to magnetization curves as “hysteresis
loops.” Examples of such curves are shown in Fig. 3.4 for the ideal ferromagnet (the example
of the previous section using uniaxial anisotropy follows the curve in Fig. 3.4(a)) and a
realistic ferromagnet, shown in Fig. 3.4(b). In realistic ferromagnets, the magnetization
curve averages over potentially many magnetic domains, each reversing their orientations
at slightly different fields, broadening and rounding the switching curve. In Fig. 3.4(b)
the quantity Mr denotes the non-vanishing remanent magnetization that remains after the
applied field is returned to zero. A coercive field field, Hc , must be applied in the opposite
direction to reverse the magnetization.
The magnetization curve can vary widely depending on the material, shape, and size of
the ferromagnet. A material is said to be “magnetically soft” if it can be magnetized to
saturation with a small applied field, typically a few mT or less. Some materials that fall
into this category, which will be studied at length in later Ch. 6, are Ni81 Fe19 (also known
as Permalloy, which we will abbreviate as Py), or Pd97 Fe3 . Other materials, also discussed
Ch. 6, such as Co, Ni, or Ni65 Fe15 Co20 , require much larger fields, typically tens or hundreds
of mT, and are referred to as “magnetically hard”.
Let us return to Eq. (3.4) and our discussion regarding the stability of the solutions for
41

a)

M

M

b)

Mr

Hc

H

Ms

H

Figure 3.4: Hysteresis loops showing the magnetization M vs. the magnetic field H applied
along the easy axis of the ferromagnet. a) The expected hysteresis curve of an ideal ferromagnet. b) The typical hysteresis curve of a real ferromagnetic material. When the applied
field is increased the magnetization increases until it saturates to a maximum value Ms . The
quantity Mr denotes the non-vanishing “remanent” magnetization that remains after the applied field is returned to zero. A “coercive field” field Hc must be applied in the opposite
direction to reverse the magnetization.
the magnetization. By extremizing the total energy density we can determine the conditions
for marginal stability, where ∂ 2 u/∂θ2 = 0. After a first derivative of Eq. (3.4) with respect
to the angle θ, we find
Âľ0 Hx Âľ0 Hz
−
= βM.
sin θ
cos θ

(3.5)

Taking another derivative of this expression yields

Âľ0 Hx Âľ0 Hz
+
= 0.
sin3 θ cos3 θ

(3.6)

Solving Eqs. (3.5) and (3.6) for the components of the external field, we find

¾0 Hx = βM sin3 θ,

(3.7)

µ0 Hz = −βM cos3 θ.

(3.8)

42

Hz
βM

- βM

βM

Hx

- βM

Figure 3.5: Astroid diagram from the Stoner-Wolfarth model. Extremizing the solutions to
the total energy leads to Eq. (3.9), whose geometric representation is an astroid. Hysteretic
solutions, of the type shown in Fig. 3.4, occur inside the asteroid, while nonhysteretic unique
solutions occur outside it.
These expressions correspond to the metastable solutions for the magnetization, where
∂ 2 u/∂θ2 = 0, and can be combined into a single equation,

(¾0 Hx )2/3 + (¾0 Hz )2/3 = (βM )2/3 .

(3.9)

Eq. (3.9) traces out an astroid, shown in shown in Fig. 3.5, and is a geometric representation
of the Stoner-Wohlfarth model, first proposed by John Slonczewski in 1956 [52]. The stable
hysteretic behavior of the magnetization, shown in Fig. 3.4, is found within the astroid,
while metastable solutions for M exist on the line, and non-hysteretic unique solutions occur
outside. At a given value of the field (Hx , Hz ), the magnetization will follow the vector lines
that connect that point to lines that lie tangent to the astroid curve. The stable solutions
which have tangent lines nearest to the easy axis cause M to align with H [52]. Traversing
across the astroid line by modifying the field direction or strength, results in discontinuous

43

changes to the magnetization.

3.3.2

Types of Magnetic Anisotropy

The underlying crystalline structure of a ferromagnet is one of the ways that can preferentially set the direction of its magnetization. Such a ferromagnet is then said to have
magnetocrystalline anisotropy, an effect which originates microscopically from spin-orbit
coupling in the following manner. In a crystal of closely spaced atoms, whose electronic
orbitals are anisotropically shaped, there will be more orbital overlap in certain directions
than in others. Then, if the direction of the spin of an electron is strongly coupled to the
orientation of those electronic orbitals (i.e. the crystal electric field), then strength of the
exchange field will then too become anisotropic in space. Considering the number of diverse
crystallographic structures found in the periodic table, it is evident there are many different
types of magneto-crystalline anisotropy.
Even if a ferromagnetic material has no magneto-crystalline anisotropy, its shape can still
influence its magnetization direction. For instance, an external field will equally magnetize
a sphere in any direction, whereas a prolate ellipsoid will be easier to magnetize along its
long axis than along its short axis. This shape anisotropy is due to the magnetostatic energy
arising from so-called demagnetizing fields, generated by the object’s own magnetization. The
demagnetizing field for an arbitrary shaped ferromagnet varies spatially, in a manner that
can be difficult to predict, but reduces to more simple forms for shapes such as ellipsoids,
which we will mainly study. For example, a uniformly magnetized sphere has a uniform
internal magnetic field,
H = Hext − nd M,

44

(3.10)

if placed in a uniform external magnetic field Hext . The second term in Eq. (3.10) corresponds
to the demagnetizing field, which points opposite to the direction of M, where nd is called
the “demagnetizing factor,” which for a sphere equals 1/3. In general, nd is a tensor. For
example, an extremely oblate ellipsoid, the components of the tensor along the principal axes
are nd = 1/2 â + 1/2 b̂ + 0 ĉ. In the notation of Sec. 3.3.1, this case is the same as taking
β/2 → (na − nc ) = 1/2.
Another type of magnetic anisotropy is exchange anisotropy, attributed to the exchange
interaction near the interfaces between ferromagnetic and antiferromagnetic layers, and will
be discussed briefly in Ch. 8. Among other features it can result in shifted magnetization
curves, to the left or to the right along the field axis.
Finally, we mention only in passing some other considerations that may need to be taken
into account when dealing with ferromagnetic samples. Stress applied to a ferromagnet can
change it’s magnetic properties. Stress can cause the shape of the sample to change either
mechanically, by thermal expansion or contraction, or by exposing the ferromagnet to a magnetic field (called magnetostriction), though the effects typically add small corrections to the
magnetization curves. However, even if a ferromagnet has a Curie temperature above room
temperature, it may have noticeably different magnetization curves at room temperature
compared to at cryogenic temperatures. Stress can also change the shape and behavior of
the magnetic domains in the sample.

45

Chapter 4

Interplay Between Superconductivity and
Ferromagnetism

At first glance, one may balk at the prospect of combining ferromagnets and superconductors, given that ferromagnetism and superconductivity are inherently antagonistic forms
of order. In a typical BCS-like superconductor the electrons bind into Cooper pairs with
anti-aligned spins, while in a ferromagnet the exchange field acts to align the electron spins
in one direction. Yet, combining and controlling the interplay between these two forms of
competing order has proven to be a fertile arena for rich interesting physics, particularly
near the interfaces of superconductors and ferromagnets. This situation stems from the fact
that superconducting correlations can extend beyond the boundary of a superconductor into
a non-superconducting material, called the superconducting proximity effect, which will be
discussed in the next section. Many intriguing phenomena then emerge from how those
superconducting correlations are modified in the presence of ferromagnetism.
In this chapter, after discussing the proximity effect and the physics of superconductor/ferromagnet (S/F ) interfaces, we will explain how those effects modify the behavior of
ferromagnetic Josephson junctions, for example S/F/S junctions. Next, we discuss how different shaped junctions of that type behave in the presence of an externally applied magnetic
field. Finally, we consider how using three ferromagnets (F 0 , F , and F 00 ) with noncolinear
magnetizations in Josephson junctions of the form S/F 0 /N/F/N/F 00 /S, where N is a normal
metal, can be used to convert between spin-singlet Cooper pairs and an altogether differ46

ent form of spin-triplet pairing. This chapter will establish the theoretical background for
the experiments that follow in Chs. 6–9, which involve each of the aforementioned types of
Josephson junctions.

4.1

The Proximity Effect at S/N Interfaces

It has been known for a long time that superconductivity can extend beyond the boundary
of a superconductor. For example, in 1932 R. Holm and W. Meissner found it was possible
for two superconductors separated by a thin film of normal metal (an S/N/S junction)
to have zero resistance, the first observation of the superconducting proximity effect [53].†
Further experiments in the 1960s on superconductor/normal metal (S/N ) bilayers confirmed
that superconducting pair correlations can extend into an adjacent non-superconducting
material [54].
The proximity effect is simplest to understand (and observe) at S/N interfaces, shown
schematically in Fig. 4.1, where it is possible for superconducting correlations to persist
deep into the normal metal. The superconducting pair-correlation function Ψ decays in
the normal metal, until coherence is lost via scattering and thermal fluctuations. For clean
metals like Cu, Ψ typically decays on the order of 100s of nanometers. Correspondingly,
on the superconducting side of the interface, the superconductivity is weakened due to the
presence of the normal metal.
This behavior may be perplexing considering the electronic density of states on the two
sides away from the interface. As we saw in Ch. 2, in a superconductor there is a finite
energy gap ∆ in the energy density of states near the Fermi energy quite unlike a normal
metal as shown in Fig. 4.2(a). For single electrons in the normal metal, whose energy is
† Intriguingly, 30 years prior to the prediction of the Josephson effect.

47

Ψ
S

N
x

Figure 4.1: Proximity effect near the interface of a superconductor and a normal metal. Pair
correlations from the superconductor (S) will penetrate into a normal metal (N ) over a
distance ΞN , typically 100s of nm for clean metals like Cu at cryogenic temperatures. In N
the superconducting pair-correlation function Ψ decays exponentially as thermal fluctuations
and scattering events destroy the pairing. Likewise, on the S side of the interface, the leakage
of Cooper pairs into the normal metal weakens the superconducting order parameter.
less than EF + ∆, transmission across the interface is forbidden due to the superconducting
energy gap and the filled levels below. How then is current transferred across the interface
(in either direction)?
The microscopic mechanism for the charge transfer process was explained by Alexander
F. Andreev in 1964 [55]. Andreev showed that a double-charge transfer process called Andreev reflection is permissible. In this process, a single electron in the normal metal, close
to the Fermi energy, can only be transferred into the superconductor when accompanied by
a second electron (of opposite spin) and together form a Cooper pair in the superconductor.
In the normal metal, the absence of this second electron is equivalent to a hole retro-reflected
at the S/N interface, as shown in Fig. 4.2(b). For incident electrons moving in the other
direction, due to time-reversal invariance the opposite process occurs, as shown in Fig. 4.2(c).
Andreev reflection is the primary mechanism for Cooper pairs to be transfered into or out
of the superconductor [56]. Perhaps the most important aspect of the Andreev reflection
(with regards to this work) is that it preserves the phase coherence of the electrons. Like48

(a)

(b)

E E

N

S

N

S

empty states

EF

θ

e
h

Δ
-Δ

(c)
lled states

N

S
θ

e

nN

nS

Cooper pair

h

Cooper pair

Figure 4.2: Density of states at a superconductor/normal metal interface and Andreev reflection. (a) The energy density of states on either side of a superconductor/ normal metal
interface (nS and nN , respectively), prohibits single electron transfer at energies E < EF +∆,
where ∆ is the superconducting gap and EF is the Fermi energy, with filled states at T = 0
shaded. Instead, charge transport happens though Andreev reflection (b), where an electron
near the Fermi level is converted into a retro-reflected hole in N and a Cooper pair in S.
The inverse Andreev reflection process is shown in (c) for transferring Cooper pairs out of
the superconductor. Both the Andreev processes conserve the total spin and the momentum
in the direction parallel to the interface, all the while maintaining phase coherence.
wise, the reflected hole will retain phase information about the macroscopic phase of the
superconductor.
In the Andreev reflection process, momentum is conserved in the directions parallel to the
interface. At low energies, when the S/N interfaces are very clean and there is no potential
energy barrier between them, practically all of the electrons near the interface are converted
via Andreev reflection and electrons at the Fermi energy have perfect retro-reflection, which
has been observed in experiments [56].
Finally, as seen in Fig. 4.2(b) Andreev reflection is dependent on the spin degree of
freedom. If a spin-up electron in the normal metal enters the superconductor it needs to
form a Cooper pair with a spin-down electron. Thus, to conserve the total spin in the
process the reflected hole is in the spin-down band. Inside a normal metal, where the bands

49

associated with spin-up and spin-down electrons are the same, the spin degree of freedom for
Andreev reflection is not usually important. However, the situation is drastically different
in ferromagnetic materials, as we will see in the next section, where the two spin bands are
mismatched.
These basic features of the proximity effect were described phenomenologically in terms
of Ginzberg-Landau theory by Pierre-Gilles de Gennes in 1969, which we only briefly mention
here, but is described in detail in several good references [54, 57]. Essentially, it amounts
to solving Eqs. (2.9)–(2.11), where in the case of an S/N interface, the right-hand side of
the boundary condition, Eq. (2.11), is replaced with (1/t)Ψ, where t is a material dependent
property that is proportional to the superconducting coherence length. †
In actuality, the proximity effect results from the combination of the Andreev reflection
process at the interface and how the superconducting correlations retain phase coherence
once inside the non-superconducting material. The length scale over which the electrons
keep their phase coherence varies with temperature and from material to material, and
depends strongly on the presence of impurities. Nonetheless, that distance can be calculated
in two important cases: the diffusive limit (sometimes called the “dirty limit”), where the
electrons propagate via disorder scattering, i.e. the mean-free path is smaller than the sample
dimensions, and the ballistic limit (sometimes called the “clean limit”), where the electrons
propagate in relatively straight lines and their mean-free path is comparable to the size of
the device.
When the electron transport is diffusive the propagation takes place over the distance
† We also note that there exist several more accurate and sophisticated theoretical quasiclassical methods
for gaining practical solutions to the physics near S/N or S/F interfaces, or in simplistic situations involving
diffusive motion or disorder, namely the Usadel and Eillenberger equations, but we will not dwell on them
here.

50

that the pairs can remain correlated, called the normal metal coherence length †
s
ΞN, diffusive =

~D
,
2πkB T

(4.1)

where D is the electron diffusion constant that depends on the material, kB is Boltzmann’s
constant, and T is the temperature.
Pure normal metals without disorder typically follow the ballistic limit, where the primary
cause of scattering is at the boundaries of interfaces. In this case the normal metal coherence
length takes the form
ΞN, ballistic =

~vF
,
2πkB T

(4.2)

where vF is the Fermi velocity. The value of ΞN is on the order of 100s of nanometers, typical
for diffusive Cu samples below 10 K.

4.2

The Proximity Effect at S/F Interfaces

When the normal metal is replaced with a ferromagnet the Andreev reflection process
is strongly altered on account of the incoming electron pairs (or reflected holes) occupying
different spin bands. For instance, in the extreme example of a ferromagnet with perfect
spin polarization, the Andreev reflection process will be completely supressed [56, 57]. If
there is a large mismatch between the spin-up and spin-down bands along the relevant Fermi
wavevector in k-space it will result in a very short correlation length in the ferromagnet, as
we will show below.
Besides the rapid decay with thickness, the pair correlation function in a ferromagnet
† There are many interesting questions that can be explored here, regarding how the superconducting
correlations can endure the averaging over many disordered scattering events. So-called weak localization
effects, or other signatures such as coherent backscattering, can be investigated in S/N interfaces.

51

a)

b)

ÎľF

ÎľF
2Eex

Figure 4.3: A simplified toy model of a normal metal and a ferromagnet. Consider a model
where the Bloch bands are assumed to have a parabolic dispersion relation (Energy E vs.
wavevector k). In a normal metal (a) the spin-up and spin-down bands are degenerate, while
in a ferromagnet (b) the spin bands (blue and yellow lines) are shifted with respect to one
another by twice the exchange energy. In the ferromagnet, oppositely aligned spin pairs that
enter those respective bands near the Fermi energy EF will have a nonzero difference between
their respective Fermi wavevectors, so the pair acquires a net center-of-mass momentum.
That leads to a spatial oscillation of the pair correlation function, as described in Eq. (4.6).
A similar situation occurs in real ferromagnets, although the band structure is typically more
complex.
is strongly modified in another manner that is dependent on the particular structure of
the spin-bands. Accurately modeling the band structure in realistic ferromagnets can be
quite complex. However, we can discuss some of the more salient features by considering
a “toy model” of a ferromagnet, similar to that described by E. Demler, G. Arnold, and
M. Beasley [58]. Consider a situation in which the spin bands are parabolic in k-space,
separated by twice the exchange energy Eex , shown in Fig. 4.3. Via Andreev reflection,
oppositely aligned spin pairs enter the ferromagnet near the Fermi level, but occupy different
↑

↓

spin bands. The correlated pairs thus have different Fermi wavevectors, kF and −kF , which
we will initially consider to be perpendicular to the S/F interface. Their difference is,

↑

↓

(kF − kF ) ≡ Q ≈ 2Eex /~vF .

(4.3)

Thus the pair acquires a net center-of-mass momentum, ~Q. Quantum mechanically, accord52

ing to the anti-symmetry property of fermions, in the wavefunction |Ψi we are also required
to consider the case where the particles are interchanged. That is a spin pair with the state
|↑↓i will acquire a center-of-mass momentum ~Q, while a pair with the state |↓↑i will acquire
a center-of-mass momentum −~Q. Arranging these two pairs in a spin-singlet configuration
(as imposed by the memory of their pairing from the BCS-type superconductor) we write
the spin and center-of-mass contributions to the wavefunction as,


1 
|Ψi = √ |↑↓i eiQX − |↓↑i e−iQX ,
2

(4.4)

where X is the coordinate describing the center-of-mass of the electron pair. Rearranging
Eq. (4.4) with Euler’s formula yields




i
1 h
|↑↓i − |↓↑i cos(QX) + i |↑↓i + |↓↑i sin(QX) ,
|Ψi = √
2
= |0, 0i cos(QX) + i |1, 0i sin(QX),

(4.5)
(4.6)

where the states |s, mi are labeled according to their total spin quantum number s and
projection m. From Eq. (4.6) we see that the pair-correlation function in a ferromagnet
contains two terms: the first corresponding to the spin-singlet correlations, and the second
corresponding to the spin-triplet correlations with m = 0, which each oscillate in the direction
perpendicular to the interface. The oscillation of the pair-correlation function was first
predicted, not in S/F systems, but for bulk superconductors in the presence of an exchange
field by Fulde and Ferrell in 1964 and independently by Larkin and Ovchinnikov a year later,
but nonetheless is often referred to as the FFLO state in their namesake [58–61].
Note that Eq. (4.4) did not account for the full angular dependence of the pair-correlation

53

function. In general, we ought to consider the case where the incident electrons make an angle
θ with the interface normal, so in the ballistic limit the difference in the Fermi wavevectors
is (k ↑ − k ↓ ) cos(θ) = Q cos(θ) = 2Eex /(~vF ). For example, if we consider only the amplitude
of the first term in Eq. 4.5 and integrate over all angles we find that the spin-singlet paircorrelation function decays algebraically in X,
Z 1
d(cos θ) cos
0

 2E X  sin(X/Ξ )
ex
F
≈
~vF cos θ
X/ΞF

(4.7)

where we have defined,
ΞF, ballistic =

~vF
.
2Eex

(4.8)

which corresponds to the coherence length in the ferromagnet [58–61] and the integration in
Eq. (4.7) is only valid for X  ΞF . A similar calculation for the second term in Eq. (4.6),
shows that the spin-triplet correlations (with m = 0), being composed of oppositely aligned
spin pairs, also oscillate and decay rapidly with thickness.
In the diffusive limit, the spin-singlet pair correlation function oscillates and decays exponentially with distance [57]

|Ψi ∝ exp (−x/ξF ) sin (X/ξF ),

(4.9)

where the coherence length is
r
ΞF 1, diffusive =

~DF
,
Eex

(4.10)

and DF is the diffusion constant in the ferromagnet. In Fig. 4.4 we plot Eq. (4.9) vs. X. Due
to the large exchange energy in typical ferromagnets, the coherence length is several orders

54

Ψ
S

F
x

Figure 4.4: Proximity effect near the interface of a superconductor and a ferromagnet. Pair
correlations from the superconductor (S) penetrate into a ferromagnet (F ), quickly decaying
over a distance ΞF , which in weak ferromagnets like Cu48 Ni52 is on the order of 10 nanometers
at cryogenic temperatures. Since ΞF depends on the strength of the exchange energy, in
strong ferromagnets, like Fe or Co, the decay is on the order of a nanometer. In addition to
the decay, the pair-correlation function oscillates spatially in the direction perpendicular to
the interface.
of magnitude smaller in S/F systems than in S/N systems. Taking the expressions for the
ballistic limit, Eq. 4.2 and Eq. 4.8, and assuming the Fermi velocities are similar in the two
materials, and Eex ≈ kB TCurie to be proportional to the ferromagnet’s Curie temperature,
we find the ratio of the two coherence lengths to be

ΞN /ΞF =

T
Eex
≈ Curie .
πkB T
πT

(4.11)

For example, consider Co, which has TCurie = 1388 K [62]; if the sample temperature is T = 4
K, then ξN /ξF ≈ 110. In typical ferromagnets ξF is usually on the order of a nanometer.
The spatial oscillation of the pair-correlation function can be detected experimentally
through several different means. For instance, it can cause measurable oscillations in the
critical temperature of S/F bilayers [63,64]. It can also be measured by probing the tunneling
density of states, via differential conductance measurements, in a S/F/I/N structure, where
I is an insulator [65, 66]. In those experiments the thickness of the F-layer was varied until
55

the pair correlation function changed sign, resulting in an inversion of the measured density
of states. Another method, which we will discuss in the next section, as well as in Ch. 6, is
to measure the oscillation of the critical current in S/F/S Josephson junctions versus the
ferromagnetic layer thickness.

4.3

π-Phase Josephson Junctions

The oscillation of the pair-correlation function can be combined with the Josephson effect
to make a so-called π − junction, i.e. an S/F/S junction in which the equilibrium phase
difference between the two superconducting electrodes is π. This effect was predicted in
1977 by Bulaevskii et al. who considered what would happen if magnetic impurities were
inserted into the central layer of an S/I/S Josephson junction [67]. The first experimental
verification of π-junction physics was not achieved until the early 2000’s with the works of
V. Ryazanov et al. and T. Kontos et al. [65, 68].
The absolute value of the pair correlation function is related to the critical current in a
Josephson junction. As we found in Eq. (2.23), the energy stored in a conventional Josephson
junction is
E=

~Ic
(1 − cos φ),
2e

(4.12)

and is normally minimized when the phase difference across the junction is φ = 0. In an
S/F/S junction, if one can set the thickness of the F -layer such that the oscillating pair
correlation function is negative, then

E=

~Ic
(1 − cos(φ − π)),
2e

56

(4.13)

Ic
0

π

0

dF
Figure 4.5: 0-π phase oscillations in an S/F/S Josephson junction. The critical current
Ic oscillates as a function of the ferromagnetic layer thickness dF . The minima signal the
transition between the 0 and the π-phase states, as the labels indicate.
which forces the minimal energy state to occur at φ = π. Emphasis should be placed on the
important feature that in these types of junctions only two distinct values of φ are allowed,
0 or π, as showed by Buzdin et al. in 1982 [61].
Experimentally, by measuring Ic for many S/F/S junctions, each with a different F -layer
thickness, dF , we can directly trace out the oscillations in the pair-correlation function as
shown in Fig. 4.5, and from those measurements extract the coherence length, the period
of the oscillations, and the critical current magnitude for an F-layer of interest. We will
discuss experiments of this type in Ch. 6 for a number of different ferromagnets with both
long and short coherence lengths. Experiments demonstrating the 0 − π oscillations have
been performed on a wide expanse of materials by many groups, including ours, in the last
two decades [19, 69–77].

57

4.4

Fraunhofer Patterns in Ferromagnetic Josephson Junctions

A key method for probing both the ferromagnetic and superconducting properties of
Josephson junctions is to measure them in the presence of an external magnetic field applied perpendicular to the direction of the supercurrent. In small ferromagnetic Josephson
junctions, one can simultaneously measure both the Josephson physics of Ch. 2 and the
Stoner-Wohlfarth physics of Ch. 3. In this section, first we return to the Josephson effect and examine how standard Josephson junctions of finite area behave in the presence
of magnetic flux. Then, we discuss how the situation changes for ferromagnetic Josephson
junctions.

4.4.1

The Josephson Effect in the Presence of Magnetic Flux

We consider the case of “short” Josephson junctions, the relevant limit for all the experiments that follow, where the distance between the superconducting electrodes d, is small
compared to the lateral dimension of the junction. We assume that the thickness t of each
superconducting electrode is larger than the London penetration depth ÎťL , and that the
critical current density Jc through the junction is homogeneous. First, we will assume that
there is no contribution from the magnetization M of the material between the electrodes (if
any), so that an externally applied field Âľ0 Hext equals the field induced inside the junction
B.
Recall from Eq. (2.51) that the difference in the gauge invariant phase differences φ between two points along a closed path is proportional to the flux threading the loop. Consider
two sets of such points P and Q shown in Fig. 4.6, which are separated by an infinitesimal
distance dz on the z-axis and lie along a closed path (Q1 , Q2 , P3 , P4 ) that extends deep

58

I

S1

x

ÎťL
d
ÎťL

z

y
Q1

P4

Q2

P3

B

L

S2
W

I

Figure 4.6: Geometry for calculating the Fraunhofer pattern in a Josephson junction with
rectangular cross-section of area LW . The superconducting electrodes are separated by a
distance d, which for example could be the thickness of a normal metal spacer. The integration path is shown with a yellow dotted line and extends deep into each superconductor,
taking into account the London penetration depth ÎťL , near both interfaces. The field B in
the junction is assumed to have only a component along ŷ.
( ÎťL ) into the superconducting electrodes. Following a calculation analogous to what we
did for Eq. (2.51), we arrive at
ÎŚ
,
φP − φQ = 2π
ÎŚ0

(4.14)

where φQ is the gauge invariant phase difference from point Q1 to Q2 and φP is from from P3
to P4 . There are no contributions to this expression from the current density Js since both
integrations along the z-direction vanish deep in the superconductor and the integrations
along the x-direction cancel the contributions from the adjacent paths dz away.

59

The flux penetrating the closed yellow path in Fig. 4.6

dÎŚ = B t dz,

(4.15)

where the magnetic field inside the junction B = B ŷ is only along the y-direction, the
distance t = d + 2ÎťL is the sum of the separation between the electrodes d and the London
penetration depth of each electrode, and dz is the width of the integration path along x̂.
Combining Eqs. (4.14) and (4.15), we can write φP − φQ = ∆φ as

∆φ =

Bt
∂φ
dz = 2π
dz,
∂z
ÎŚ0

(4.16)

so that ∂φ/∂z = 2πBt/Φ0 .† Integrating this expression over z we see that the gauge invariant
phase difference varies linearly along z as

φ(z) =

2πtB
z + φ(0).
ÎŚ0

(4.17)

As a result of Eq. (4.17), the supercurrent density is

Js (z) = Jc sin φ(z)
(4.18)
= Jc sin(kz + φ(0)),
where k = 2πtB/Φ0 describes the period 2π/k of the supercurrent density oscillations in the
z-direction.‡ Integrating the supercurrent density over the junction area A yields the total
† In general, we can write the gradient of φ as ∇φ = 2πt/Φ B× x̂, where x̂ is the unit vector perpendicular
0
to the junction area and, in this case, opposing the direction of the current flow.
‡ For one complete oscillation of the supercurrent density along the z-axis, the corresponding flux enclosed
is precisely equal to a flux quantum: Φ = (2π/k)tB = Φ0 .

60

supercurrent,
Z
Is =

Js (z)dA.

(4.19)

We now consider two specific junction shapes—rectangular and circular—and determine how
the maximum supercurrent (at a given flux) Icmax varies with the total flux ÎŚ through the
junction.
The schematic of a Josephson junction with a rectangular cross section of length L and
width W is shown in Fig. 4.6. Integrating the supercurrent density in Eq. (4.18) along y and
x yields the total supercurrent,
Z L/2

Z W/2



Jc sin kz + φ(0) dydz

Is =
z=−L/2 y=−W/2



sin kL/2
= Jc W L
sin φ(0),
kL/2

(4.20)

where k was defined previously in Eq. (4.18). The total flux penetrating the junction is
Φ = B(d + 2λL ). With the replacements kW/2 = πΦ/Φ0 and Jc W L = Ic , it follows that
the maximum supercurrent at a given value of the flux is


 sin πΦ/Φ 
 

0 
Ic (ÎŚ) = Ic0 
,
 πΦ/Φ0 

(4.21)

which traces out a Fraunhofer pattern † with respect to the total magnetic fux. This function
is shown in Fig. 4.7, normalized to the maximum critical current Ic0 .
For a junction with a circular cross section of radius R, the integration of the supercurrent
density is performed in polar coordinates z = r cos Îą and y = r sin Îą. Namely, Eq. (4.19) is
† This is directly analogous to the Fraunhofer diffraction pattern in optics for light incident on a slit of
finite width.

61

Ic
Ic0

Figure 4.7: Fraunhofer patterns for rectangular and circular Josephson junctions. The critical
current Ic (ÎŚ) oscillates as a function of the applied flux ÎŚ. Rectangular shaped junctions
(solid) follow a Sinc function (Eq. (4.21)), while circular junctions (dashed) follow and Airy
function (Eq. (4.23)) The current axis is normalized to the maximum critical current Ic0 and
the flux axis to the flux quantum ÎŚ0 .
written as
Z R Z 2π
Is =



Jc sin kr cos α + φ(0) rdrdα

r=0 Îą=0

(4.22)

J (kR)
= 2Jc πR2 1
sin φ(0).
kR
Comparing to Eq. (4.20), we see that instead of a sine function, which appeared for the
rectangular junction, we have a first order Bessel function of the first kind, J1 (kR). With
kR = πΦ/Φ0 and Jc πR2 = Ic , the maximum supercurrent for a circular junction is


 J (πΦ/Φ ) 

0 
Ic = 2Ic0  1
.
 πΦ/Φ0 

(4.23)

This function, which traces out an Airy pattern, is also shown in Fig. 4.7 for comparison
to the result obtained for a rectangular junction. In the Chs. 6-9, we fabricate Josephson
62

Ic
Ic0

Figure 4.8: Fraunhofer patterns for a ferromagnetic Josephson junction. The critical current
Ic follows Fraunhofer patters that are shifted with respect to the applied flux ÎŚ, by an amount
corresponding to the flux added (or subtracted) to the junction due to the magnetization of
the ferromagnetic layer. The behavior of the ferromagnet is hysteretic with the applied flux,
so the solution jumps between the two Fraunhofer patterns when the applied flux corresponds
to the ferromagnet’s coercive field.
junctions with elliptical cross-sections which also follow Eq. (4.23). We will refer to all curves
of this type in general, as “Fraunhofer patterns”, even when they correspond to Airy functions
or some other similar looking function.

4.4.2

Adding a Ferromagnetic Layer

We will now examine how the Fraunhofer physics is modified by the presence of a ferromagnetic layer in an S/F/S Josephson junction. In addition to the assumptions discussed
in the previous section, we consider the ideal case of an elliptical junction in which the ferromagnetic layer consists of a single magnetic domain, and follows the Stoner-Wohlfarth model
discussed in Ch. 3.

63

The total magnetic flux in the junction is

ÎŚtot = ÎŚ + Âľ0 M wdF ,

(4.24)

where w is the width of the junction (minor axis) and dF is the thickness of the ferromagnetic
layer. The magnetization of the ferromagnet, M , which is assumed to be uniform and initially
point along the same direction as ÎŚ, produces a flux (the second term in Eq. 4.24) that adds
to the external flux in the junction. In Eq. 4.24 we neglect the small contributions to ÎŚtot
from the uniform demagnetizing field and any magnetic field from the ferromagnetic layer
that returns inside the junction. From Eq. 4.24 it is clear that the Fraunhofer pattern will
be shifted along the field axis by an amount,

Hshift = −

M dF
.
2ÎťL + d

(4.25)

If the external flux starts at a large positive value and is slowly reduced through zero, the
Fraunhofer pattern traces out the blue curve shown in Fig. 4.8. Due to the hysteresis in the
magnetization curve, shown previously in Fig. 3.4, once the applied flux becomes sufficiently
negative to exceed the coercive field of the ferromagnetic layer the magnetization will change
sign. Thus the solution abruptly switches to another Fraunhofer curve that is shifted in
the opposite direction (the red curve in Fig. 4.8). Increasing the applied flux from a large
negative value towards positive flux produces the opposite effect, as in the Stoner-Wohlfarth
model.

64

SL N F' N

M'

F

N F'' N SR

θ

θ''

M

M''

Figure 4.9: Schematic for the generation of long-range spin-triplet correlations using noncolinear magnetizations. The angles θ and θ00 lie in the same plane parallel to the interfaces
and describe the orientation of the magnetizations of the second and third ferromagnets.

4.5

Generating Long-Range Spin-Triplet Correlations with Ferromagnetic Barriers

What would happen if the electrons inside a Josephson junction could somehow be paired
with the same spin-orientation, that is |↑↑i or |↓↓i? In that case, both electrons enter the
same spin band and can remain correlated in the ferromagnet over much longer distances.
In a Josephson junction there is thus an overall slower decay in the critical current and no
oscillation of the supercurrent with the thickness of the ferromagnet [78]. While spin-triplet
superconductivity is not commonly found in nature, it was predicted in the early 2000s by S.
Bergeret, M. Eschrig and others that it could be engineered in multi-layered ferromagnetic
systems having noncolinear magnetizations or magnetic inhomogeneity [78–82]. Since then,
demonstrations of spin-triplet proximity effects have been reported by many groups using a
variety of experimental techniques [83–104].
Our group has focussed on spin-triplet Josephson junctions (JJs) of the form suggested
by Houzet & Buzdin [105], containing three magnetic layers S/F 0 /N/F/N/F 00 /S [85,90,106,
107]. A schematic of a junction of this type is shown in Fig. 4.9, in which all the ferromagnets,

65

F 0 , F , and F 00 have in-plane magnetization (parallel to the layer interfaces). Normal metal
layers are inserted between the various layers to prevent them from magnetically coupling
to one another in an undesireable manner. Such a Josephson junction converts between
spin-singlet and long-range spin-triplet supercurrent in the following manner [108]:
1. Spin-singlet pairs from the superconductor on the left (SL ) enter the first ferromagnet
(F 0 ) and acquire a net center-of-mass momentum, generating a short-range (m = 0)
spin-triplet component, as previously described in Eq. (4.6), which we repeat here for
convenience
|Ψi = |0, 0i cos(QX) + i |1, 0i sin(QX).

(4.26)

Both the spin-singlet and the m = 0 spin-triplet terms decay rapidly and oscillate with
the thickness of the F 0 layer.
2. The electron pairs enter a second ferromagnet (F ) with magnetization noncolinear to
the first and the m = 0 spin-triplet states undergo a basis rotation as the quantization
direction (set by M) changes. The spin-singlet state |0, 0i is rotationally invariant,
however the |1, 0i state is not. Expression of the |1, 0i state in the rotated basis includes
the long-range (m = Âą1) spin-triplet states. For example, if the angle between F 0 and
F is θ then in the new basis,

sin(θ)
sin(θ)
|1, 0i = cos(θ) |1, 0iθ + √ |1, 1iθ − √ |1, −1iθ
2
2

(4.27)

where |s, miθ denotes the basis rotated by θ. The conversion to the s = 1 spintriplet states, |1, 1i = |↑↑i and |1, 1i = |↓↓i, is maximized when the magnetizations
are perpendicular (e.g. θ = π/2 or 3π/2). The spin-singlet states and m = 0 spin66

triplet states decay quickly in F , as described previously, but the (m = Âą1) states
can propagate much further, and thus are refered to as long-range triplet correlations
(LRTC). Since the spin pairs are aligned in these states both electrons enter the same
spin band and have no center-of-mass momentum, implying that the pair correlation
function will not oscillate in space. Analogous to that of a normal metal, the correlation
length for the LRTC is much larger than it would be inside a single ferromagnet with
constant magnetization. In the dirty limit it has the same form as Eq. (4.1),
s
ΞF, LRTC =

~DF
2πkB T

(4.28)

where DF is the diffusion constant in the central ferromagnet.
3. Since the final superconducting electrode (SR ) can only accept spin-singlet states,
a third ferromagnetic layer (F 00 ) is needed to convert the long-range triplet states
back into spin-singlet. That occurs similarly through another basis rotation and that
conversion process is optimal when the magnetization of F 00 is perpendicular to that
of F .
The phase across such a spin-triplet junction will take on different values depending
on the relative orientation of the three magnetizations. The simplest case is when each of
the adjacent ferromagnetic layers have their magnetization directions at 90 degrees to one
another, for example in Fig. 4.10, where θ = π/2 and θ00 is either 0 or π. When the direction
of rotation between the F 0 and the F layers is the same as that between the F and the F 00
layers (i.e. θ00 = π, Fig. 4.10(a)), then the phase across the junction will be 0. On the other
hand, if the relative direction of rotation reverses from the F 0 and the F layers versus the F
and the F 00 layers (i.e. θ00 = 0, Fig. 4.10(b)) then the phase across the junction will be π.
67

SL N F' N

F

N F'' N SR

(a)

M'
SL N F' N

M
F

M''
N F'' N SR

(b)

M'

M

M''

Figure 4.10: Schematic for the generation of 0 and π phase junctions using long-range spintriplet correlations. When the chirality of the rotation angle between the F 0 and F layers is
the same as between the F and F 00 layers, shown in (a), the phase across the junction is 0.
When the relative rotation angle is reversed, as in (b), the phase across the junction is π.
In the past, our group has made spin-triplet Josephson junctions in which all the magnetic
layers have in-plane magnetization [85], as in Fig. 4.10, and when some of the layers have
out-of-plane magnetization, such as the geometry shown in Fig.4.11 [21, 106].
By fixing the magnetization direction of two of the ferromagnetic layers and controllably
toggling the direction of the the third, one can conceivably switch between the 0 and the
π phase states, a proposition that was partially realized by our group in 2014, spearheaded
by the work of Eric Gingrich [109]. In Ch. 9 we will discuss a variant of this controllable
switching experiment.
Recently, another Birge group member William Martinez successfully implemented another scheme to manipulate the LRTC states, whereby rotating the magnetization of the F 00
layer so that it was either || or ⊥ to that of the F layer, the long-range triplet supercurrent
could be controllably toggled “on” or “off” as evidenced by large amplitude changes in the

68

SL N F' N

F

N F'' N SR
θ

M'

M

M''

z

y
x

Figure 4.11: Schematic for the generation of long-range spin-triplet correlations using noncolinear magnetizations The angle θ with respect to the z-axis lies in the y − z plane and
describes the orientation of the magnetization of the third ferromagnet.
critical current [107]. This experiment was an excellent display of spin-triplet physics, and
the “on” and “off” states could serve as the basis for a superconducting memory. However,
when the current control scheme is in the “off” state, the device is in the resistive state. If
one’s aim is to build devices with high energy efficiency the more advantageous method is
to control the phase of the junction, all the while keeping it in the superconducting state so
as to prevent resistive energy losses.
Finally, for spin-triplet Josephson junctions of the type discussed in this section to work
as advertised, one must contend with a whole host of challenges beyond just controlling the
orientation of the magnetic layers. Some of those issues, many of which can be resolved
with the proper selection of materials and fabrication techniques, will be discussed in the
remaining chapters. One notable consideration, for instance, is the proper matching of the
band structure between the adjacent ferromagnetic materials in a junction. This will be
discussed further in Ch. 8.

69

Chapter 5

Experimental Techniques

In this chapter we discuss the experimental methods used in the fabrication and measurement of Josephson junctions and SQUIDs for the experiments presented in the following
chapters.

5.1

Sputtering

We use sputtering almost exclusively for the deposition of uniform metallic thin-films
during sample fabrication. Sputtering is a kinetic process where energetic particles, in our
case ionized Ar atoms, are bombarded into a metallic target material of interest, ejecting
a vapor of metallic atoms onto a nearby sample. The deposition rate can be made appreciable by confining the Ar ions using magnetrons that produce strong magnetic and electric
fields. The ions undergo helical cyclotron motion along the magnetic field lines, leading to
a cascading chain of ionization events with the surrounding gas, which at moderately low
pressures (< 10−2 Torr) can form a sustained plasma.
Our main sputtering chamber, shown in Fig. 5.1 contains up to four large DC triode
magnetron guns, each of which can be equiped with 2.25 inch diameter sputtering targets,
in addition to three small DC magnetron guns that carry 1 inch diameter targets. In total,
one can sputter any combination of layers composed of up to seven different materials in a
single run. The chamber also contains a small 1 cm ion mill, which will be discussed later in
the next section.

70

Figure 5.1: 7-target sputtering chamber with active plasma. (top) The interior of the sputtering chamber showing: the four large DC magnetron guns loaded with the targets (starting at
2 o’clock, moving clockwise) Nb, Al, Co, Pd; the three small DC magnetrons loaded with the
targets (starting at 3 o’clock, moving clockwise) Au, Ru and Cu. There is also a small ion
mill located at the 12 o’clock position. Examples of the foil chimney collimators can be seen
on top of the Nb and Al targets. (bottom) The sputtering process in action, showing the
ignited plasma (purple and blue in color) over the Al (front) and Nb (far back) targets. One
can see the sample holder plate and film thickness monitor on top, the shutter plate wrapped
in Al foil in the center, and on the bottom the chimney around the Al target as observable
from the front viewing window. The images were taken during the run to fabricate one of
the sets of samples in Ch. 8
71

A photograph showing the sputtering process in action can be found in Fig. 5.1, showing
the glowing plasmas above Al and Nb targets. The system was originally designed by William
Pratt and has been extensively modified and maintained by Reza Loloee.
The sputtering deposition rate is controlled by changing the strength of the negative
voltage bias of the target, which attracts the positively charged Ar ions. The density of
the plasma near the triode magnetrons, which is also related to the sputtering rate, can be
controlled by modifying the current driven through a Tungsten filament used to ionize the Ar
atoms. The sputtering rate is measured using a crystal film thickness monitor attached to the
sample plate accurate to ±0.1 Å/s, and calibrated using a tooling factor dependent on the
density of target material of interest. The target voltage, filament current, and corresponding
sputtering rates for all the materials used in this thesis are listed in Table 5.1.

To produce high-quality films the sputtering process needs to take place in an extremely
clean environment, thus the sputtering chamber is equipped with a cryopump (assisted by
a mechanical roughing pump) used to reduce the pressure in the chamber before the run to
≈ 2×10−8 Torr. Those low pressures are achieved by baking the entire chamber to ≈ 100 ◦ C
(using thermal tapes attached to the outside of the chamber) for eight hours, releasing water
molecules trapped on the inner walls to be captured by the cryopump.† Afterward, the
cryopump continuously pumps on the chamber, in total for around 48 hours. Furthermore,
to reduce the partial pressure of contaminant water molecules, just prior to the run the
vacuum chamber is cooled by circulating liquid nitrogen through a Meissner trap housed in
the lid. The sputtering run is typically performed with the Argon pressure near 1.3 × 10−3
Torr and the sample temperature held between −30 ◦ C and −20 ◦ C. The partial pressure of
† Attached to the cold head of the cryopump is a sorption pump of activated charcoal pellets to assist in
the adsorbtion of gases.

72

Table 5.1: Sputtering parameters used for the various metallic targets. The target voltage,
filament current and the approximate sputtering rate used for all the materials used in this
thesis. The 2.25 in targets are used with the DC triode magnetrons guns, and the 1 in targets
in the small DC magnetron guns. The sputtering is performed in the presence of Argon gas
at 1.3 × 10−3 Torr.
Sputtering Target (2.25 in)

Target Voltage (V )

Filament Current (A)

Sputtering Rate (Å/s)

Nb

600

0.60

4.1

Al

400

0.50

2.1

Pd

130

0.44

1.9

Co

250

0.35

1.1

Ni

350

0.30

1.8

Ni81 Fe19 (Py)

500

0.50

4.5

Ni65 Fe15 Co20

500

0.50

3.0

Pd97 Fe3

245

0.45

4.9

Sputtering Target (1 in)

Target Power (W )

Sputtering Rate (Å/s)

Cu

18

3.4

Ru

18

1.2

Au

19

5.0

73

various contaminant materials can be monitored using a residual gas analyzer attached to
the chamber.
To limit cross contamination, the chamber contains an aluminum foil wrapped shutter
plate attached to a stepper motor, controlled to open or close off access between the samples
above and the sputtering targets below as shown in Fig. 5.1. Also, around each gun a
chimney wrapped with aluminum foil helps collimate the sputtered material toward the
opening in the properly aligned shutter plate. The sputtered material is deposited on 0.5×0.5
in2 Si substrates that are firmly isolated in stainless steel sample holders each with their
own individual shutters. The sample shutters can be manually manipulated from outside
the vacuum chamber wall via a wobble stick with flexible bellows that allows for linear
and angular motions along a cone of movement. After manually opening the individual
sample shutters, the timing and movement sample plate, the shutter plate, and the film
thickness monitors with respect to each of sputtering targets is computer controlled using
the LabVIEW software package. In total, the system can be programed to produce a different
layer sequence for each of the possible 16 samples that can be supported during a single run.
In between sputtering runs, the sample holders and sample shutter parts are thoroughly
cleaned in a 1:3 solution of deionized (DI) water and nitric acid for approximately 10 minutes,
followed by a manual scrubbing with steel brushes to remove all remaining residual metal
films. Next, the parts are separately agitated in an ultrasonic bath while immersed in acetone,
isopropanol, and then DI water, each for 10 minutes, and blown dry with an industrial heat
dryer.

74

5.2

Photolithography

Photolithography is one of the techniques we use to define the circuit layout by shining
ultraviolet light through a geometric mask onto a photosensitive material coating the sample in order to reveal a pattern. In this process, we use a viscous light-sensitive chemical
substance called photoresist that can be spin-coated onto the sample substrates. It is subsequently baked, driving off the solvent in the photoresist to form a firm uniform coating.
For “positive” photoresists, which we use most frequently, when this hardened material is
exposed to an ultraviolet light source it becomes soluble in a chemical “developer.” On the
other hand, unexposed regions are left untouched (“negative” photoresists work in the opposite manner). A pattern with fine features down to several microns can be achieved when
the sample is pressed close against a glass plate with a chrome-plated pattern of the desired
circuit geometry and is exposed to the ultraviolet light on the other side.
For photolithography, we use an ABM mask aligner that exposes the sample to light of
wavelengths peaked near 405 and 365 nm. A weak vacuum holds down the sample to a
stage that can be adjusted with a set of micrometers to achieve the necessary three cardinalaxis and rotational alignment with the glass photomask. In the corners and center of the
photomask we incorporate Vernier marking features in order to align the top and bottom
wiring layers with one another. Alignment markings are also included on the bottom wiring
layer photomask for electron-beam lithography, described in the next section.
For high-quality lithography, it is essential to use very clean substrates during the photolithography process. In addition to performing all of the lithography steps in a class-100
cleanroom, we typically clean the bare Si substrates in an ultrasonic bath of acetone, isopropyl alcohol, and DI water for ten minutes each. The samples are subsequently blown

75

dry with a nitrogen gun. The positive photoresist that we typically use is called Microposit
S1813 (hereafter refered to as S1813) and is spun onto the substrates at 5000 rpm for 50
seconds. It is then immediately baked at 110 ◦ C for two minutes. The photoresist is exposed
to an ultraviolet lamp on the mask aligner for 11 seconds. We configure the resist to have
a slightly undercut edge profile using the following process: we immerse the exposed sample
into chlorobenzene for five minutes, which acts to harden the top surface of the previously
unexposed regions of photoresist, and blow dry with a nitrogen gun. Then, we immediately
immerse the sample in a developer (called 352 solution) for 45 seconds, rinse in DI water for
at least 30 seconds, and finally blow dry with a nitrogen gun. The regions that had already
been exposed to the ultraviolet light are dissolved away in the developer.
The resulting photolithographically-defined pattern can act as a mask during sputtering.
Metals deposited onto the surface of the sample adhere to the bare regions of the substrate,†
while the material deposited on top of the photoresist is removed later using a lift-off process.
During lift-off the sample is immersed in acetone and the hardened photoresist dissolves away
along with the metal on top of it. The undercut profile of the resist previously mentioned is
useful in obtaining a clean lift-off of the resist.
For photomasks with very small features of only a few microns, or with long thin wire
lines, such as those used in the SQUID samples that will be discussed in Ch. 9, a few steps are
modified in the photolithography process. After the same cleaning procedure the sample is
baked in an oven at 110 ◦ C for 15 minutes to remove any water on the surface of the substrate.
Next, the sample is coated in an HMDS adhesion primer, which forms a hydrophobic surface
that improves the wetting of the resist to the substrate surface. It is important to cover the
† Some materials such as Cu adhere poorly to Si, however we typically use Nb as the first sputtered layer,
which does not suffer from that issue.

76

entire substrate and beyond, breaking the surface tension with the edge of the chip before
spinning the sample at 5000 rpm for 50 seconds. This is immediately followed by the S1813
spin-coating process, again completely coating the entire substrate followed by the baking
procedure.
Finally, we note that for the so-called “Anacostia” SQUID mask used for the samples
in Ch. 9, which was designed by Eric Gingrich at Northrop Grumman Corporation, after
applying the HMDS procedure the S1813 photoresist is only exposed for 6 seconds, followed
by the treatment in the chlorobenzene for 5 minutes and developed in the 352 solution for
only 35 seconds.

5.3

Electron-Beam Lithography

To define even smaller features with dimensions less than a few microns, a much more
accurate form of lithography using electron beams is utilized. Analogous to photolithography,
an electron-sensitive resist coated on a sample can be exposed to a fine electron beam which
the user can trace or raster over an area to make a pattern of interest. We typically use
the “negative” electron beam resist ma-N 2401, which when exposed to an electron beam is
hardened, while the unexposed resist can be dissolved away in a chemical developer, revealing
the pattern.
Several electron-beam lithography systems have been assembled and maintained in the
Keck Microfabrication Facility at Michigan State University by Baokang Bi, who outfitted a
commercial scanning electron microscope (SEM) with a beam blanker and computer control
using the Nanometer Pattern Generation System (NPGS) software developed by Joe C.
Nabity.

77

During the course of the work presented in this thesis two different SEM systems were used
for lithography: a legacy JEOL 840A (no longer in operation at Michigan State University)
and a new Hitachi SU5000 Schottky Field Emission SEM. The latter SEM, which we also use
for fine resolution imaging as shown in Fig. 5.2, can be programmed to pattern individual
junctions of any desired shape using the following process.
The ma-N 2401 is spin-coated onto the substrates at 3000 rpm for 40 seconds and baked
for two minutes at 90◦ C. After transferring the samples into the SEM, the beam current
is set to 68 pA (using aperture 4, spot size 1, and intensity 30) and the z-height of the
stage to a working distance of 15 mm. Next, the focus and stigmation needs to be properly
adjusted, which is typically done in a non-critical region of the sample at high magnification
e.g. 300,000X. Returning to a low magnification setting (200X), we locate the desired writing
area on the sample by finding the electron-beam alignment markings present on the bottom
wiring layer. So as to not expose the resist in the area of interest the electron beam is
blocked by manipulating the beam blanker with the NPGS software, except for a small
window of interest used to align the beam while at higher magnification (1000X). Next, at
this magnification elliptical Josephson junction patterns with ≈ 0.5 µm2 area are written
onto the resist on the bottom wiring layer via an area dose of 550 ÂľC/cm2 . In addition,
a pinwheel or wagon wheel shaped feature is written off to the side using a line dose of
2.5 nC/cm, which is used to monitor the quality of the stigmation and focus adjustments
previously mentioned. Finally, the samples are removed from the SEM and developed in AZ
MIF solution for 30 seconds and washed with DI water for 30 seconds before being blown
dry with a nitrogen gun.
The electron beam resist can be used as an ion mill mask, discussed in the next section.
Afterward, a lift-off process is performed by immersing the samples in PG remover for at
78

least 15 minutes. Depending on the type of process, especially with Josephson junctions, the
lift-off at this stage is the most unreliable part of the fabrication process. These issues will
be discussed in more detail later.

(a)

(b)

(c)

(d)

Figure 5.2: Bit arrays written with electron-beam lithography. We pattern sub-micron scale
features for SQUID magnetometry or Josephson junction experiments using electron-beam
lithography and the resist ma-N 2401. In the bottom two images we show images of ≈
0.5 Âľm2 area elliptical bits taken with a scanning electron microscope. Images (c) and
(d) are taken at 90 degrees and 60 degrees to the sample plane, respectively. For SQUID
magnetometry experiments, we have developed a method to pattern 100 µm × 100 µm sized
arrays containing 99 × 39 bits, shown in (b), which are then repeatedly stamped over a 0.45
cm × 0.55 cm area, shown in (a). The total number of bits is ≈ 9.5 million, necessary for
an appreciable signal in the SQUID magnetometer.

Additionally, as shown in Fig. 5.2, Victor Aguilar and I developed a method for the
patterning of large arrays of magnetic bits onto a single substrate. By programming the

79

stage of the Hitachi SU5000, one can repeatedly stamp a large pattern (e.g 99 × 39) of 0.5
µm2 area elliptical bits densely over the surface of a substrate. We showed that a 0.45 cm ×
0.55 cm area can be automated and written overnight in the SEM over the course of about
9 hours. The total number of bits is ≈ 9.5 million to achieve the necessary signal to noise
for measurement and characterization in our SQUID magnetometer system, discussed later.
The spacing of the adjacent stamped patterns needs to account for the drift and backaction of the stage. The most severe of those effects can be prevented by ensuring that the
motion of the stage does not reverse its direction by 180◦ at any point during the writing
process. Additional developments are currently underway by Josh Willard to lessen the
discrete edge effects caused by the rapid rastering of the electron beam, possibly by using a
finer line spacing during the raster.

5.4

Ion Milling

Ion milling is a process used both for the etching of metal layers to define our Josephson
junctions, or for cleaning the surface of our films prior to sputtering. Ion milling is essentially
the opposite of sputtering, and consists of bombarding Argon atoms directly into the sample
to remove atoms on the top surface. In the ion milling process, ionized Argon atoms are
produced by passing a current through a Tungsten cathode filament and are then transported
across the inner part of the ion mill (by the “beam voltage”) and then linearly accelerated
out of the ion mill by a graphite grid (called the “accelerator”) that is biased with a negative
voltage. Another filament, called the “neutralizer,” is held about a centimeter above the
accelerator and injects electrons back into the outgoing ion beam to reduce spreading caused
by the otherwise mutual repulsion of the ions. The emerging Argon atoms collide with the

80

Table 5.2: Ion mill settings. Depending on the purpose listed, we use one of two ion mills in
the presence of Argon gas at a pressure of 1.9 × 10−4 Torr with the following settings. The
voltage of the ion beam, and that applied to the accelerator and the anode (discharge) are
Vbeam , Vacc , and Vdis , respectively. The currents applied to the cathode and the neutralizer
filaments are Ibeam and Ineut , respectively.
Ion Mill

Purpose

Vbeam (V )

Vacc (V )

Vdis (V )

Ibeam (mA)

Ineut (mA)

3 in

Deep etching

300

-30

39

9.0

9.5

1 in

Pre-clean

175

-50

40

1.3

1.8

sample positioned directly above the ion mill and slowly remove the atoms on the top surface.
Similar to sputtering, the etching or “milling” rate depends on the chamber pressure,
the cathode current, and the accelerator voltage, each of which we keep fixed run to run
for our two ion mills. In the etching process, used for Josephson junction design, we use
a large 3 inch ion mill from Commonwelth Scientific in a separate vacuum chamber, while
for pre-cleaning prior to sputtering we use a smaller 1 inch ion mill, shown in Fig. 5.1. The
various voltage and current settings for typical use are listed in Table 5.2.
The ion mill etching rate is highly material dependent. Beforehand, extensive testing
was performed by myself and other group members over the years to accurately calibrate
the etching rates of the various materials shown in Table 5.3, which are listed relative to the
standard etching rate of Au. Those tests comprise of the following process:
1. Sputter many samples of a given material or a repeated multilayer of materials to a
specific thickness.† Also, fabricate several independent control Au film samples in the
same run.
2. For each of the samples, ion mill for a set time interval using the standard condi† We’ll often cap the test samples with a thin layer of Au to avoid the formation of an oxide after exposure
to air. Oxides will typically mill much slower than the metals themselves, modifying the extracted etching
rate.

81

Table 5.3: Ion mill conversion factors for various materials. The unitless inverse ratio of
the etching rates for various materials to the standard rate of Au (k-factors). One can see,
for example, that Nb etches ≈ 6.71 times slower in the ion mill than Au, whereas a material
such as Cu etches only about twice as slowly as Au. The listed numbers are obtained using
the calibration procedure described in the text. A repeated [Pd/Co] multilayer, used in
Chs. 8–9, was also calibrated, with the individual layer thicknesses (in parentheses) listed in
nanometers.

k-factor

Nb

Co

Ni

Ni81 Fe19

Ni65 Fe15 Co20

Cu

Au

Ru

Pd97 Fe3

[Pd(0.9)/Co(0.3)]n

6.71

3.5

3.905

2.695

3.64

1.9

1.0

3.5

1.89

1.63

tions listed in Table 5.2 to form a narrow series of channels, typically defined by a
photolithographic ion mill mask.
3. Accurately measure the depth of the etched channel(s) using, for example, atomic force
microscopy. Repeat for a number of samples, preferably some of which have been ion
milled for different amounts of time (e.g. 1, 2, and 3 minutes).
4. From the measured depth and the known mill time, calculate the effective mill rate.
Determine the conversion factor or “k-factor,” defined as the inverse ratio of the etching
rate for a given material relative to that for Au.
Once all the “k-factors,” shown in Table 5.3, are known we can accurately ion mill down
to a desired depth even for very complicated samples, such as Josephson junctions with a
large number of layered materials. This is achieved by depositing Au onto a crystal film
thickness monitor, placing it directly over the ion mill, and measuring the etching rate of
Au. Then, we simply multiply the thickness of each layer in the sample of interest by the
corresponding “k-factors,” and divide by the measured Au milling rate to determine the
milling time. We typically measure the etching rate of Au with the film thickness monitor
before and after the sample is ion milled and average those rates to improve the accuracy.
Evidence of the precision of our milling process, even in complex samples, using this “on82

the-fly” measurement scheme can be obtained afterward by more accurate techniques such
as Transmission Electron Microscopy (TEM), as shown in Chs. 6 and 8.

5.4.1

Ion Mill and Thermal Evaporation Chamber

The deposition chamber we use primarily for ion mill etching is equipped with a continuously operating turbomolecular pump (and a supporting mechanical pump) that allows
us to reduce the chamber pressure to ≈ 2 × 10−8 Torr. The primary volume is kept at
high vacuum, while samples can be easily added or removed from the chamber via a load
lock, a small volume that can be independently sealed, vented to atmosphere, loaded with a
sample, and evacuated via a second mechanical pump. The sample can then be transfered
into the primary chamber volume via a transfer arm, without adversely effecting the high
vacuum of the system. In this manner, up to five samples can be transferred into the main
chamber during a single run. Usually within an hour or two the turbomolcular pump will
have the system back down at base pressure (2 × 10−8 Torr), so that the ion milling process
previously described can begin. The vacuum chamber is equipped with the 3 inch ion mill
and one Au sputtering gun, enabling us to perform the ion mill calibration scheme. The
chamber was designed by William Pratt and has been modified and maintained over the
years by Reza Loloee.
In addition, for the fabrication of Josephson junctions discussed further in the next section, the chamber also allows us to thermally evaporate SiOx onto the samples, which we can
immediately deposit after ion milling to electrically isolate the circuit of interest and protect
the sidewalls of the remaining material. The thermal evaporation process simply amounts to
passing a current through a small baffle boat filled with SiOx pellets. When the SiOx pellets
get hot enough to sublimate a vapor of the oxide is deposited on to the samples positioned
83

Figure 5.3: The ion milling and thermal evaporation vacuum chamber. The vertical transfer
arm (top) is attached to a small load lock used for transferring samples in and out of the
system. The main volume of the chamber (bottom) contains a 3 in ion mill, a Au sputtering
gun, and a baffle boat for SiOx thermal evaporation (not shown). The turbomolecular pump
and the main gate valve is also visible (left).
directly above the boat. The deposition rate can be actively monitored with a film thickness
monitor, whose tooling factor adjusts for the throw distance of SiOx particles from the boat,
to reflect the true deposition rate. The sample is rotated back and forth during the SiOx
deposition to help with film uniformity.

5.5

Josephson Junction Sample Fabrication

In this section, we give a brief outline of our scheme for fabricating Josephson junctions.
The fabrication details for each specific experiment are provided in Chs. 6–9.

84

The Josephson junction fabrication proceeds as follows. Photolithography is first used
to define the bottom wiring layer of a circuit. Then, as shown in Fig. 5.4, sputtering is
used to deposit the base superconducting electrode followed by a series of normal metal and
or ferromagnetic layers. The last layer is typically a thin film of Au to prevent oxidation.
After removing the photoresist with a liftoff process, a second series of electron-sensitive
negative resist is spun and baked onto the sample. After being exposed to an electron
beam in the pattern of the desired Josephson junction (e.g. a sub-micron ellipse with a
given aspect ratio) the sample is developed, revealing a lithographic mask that protects the
material below. Next, ion milling is used to etch down through the ferromagnetic materials
of interest, followed by an in situ deposition of SiOx .
As previously mentioned, the lift-off process following the SiOx deposition is a critical
step. We have found that excessive exposure to the ion mill or the thermal evaporation
processes can cause heat and damage to photoresist or electron beam resist. The edges of
the resist can begin to droop, melt, or disturb the desired undercut profile, and sometimes
prevent the resist from lifting off at all. As a general rule of thumb we never ion mill or
deposit SiOx onto a sample for more than 120 seconds at a time, instead we break up the
process into several steps with cooldown stages if necessary.

85

ma-N 2401

F
Si

Top Down
View:

2.5 aspect ratio
area ~0.5 Îź m2

Figure 5.4: Josephson junction fabrication steps, part 1. (a) Sputtering is used to deposit the
superconducting base layer (violet), the ferromagnetic layers (blue), and the normal metal
spacers (orange and gold) onto a Si substrate (dark blue). (b) A negative tone resist ma-N
2401 (green) is spun onto the sample and baked. (c) The resist is exposed to an electron
beam and after a subsequent development process forms a resistant cap in the pattern of the
desired Josephson junction. (d) Ion milling etches back the metal layers that are not covered
by the resist. We ion mill completely though the ferromagnetic layers to form elliptical
nanomagnets, utilizing shape anistropy to orient the magnetization along a preferred axis.
Prior to the first metalization step (not shown) the bottom lead circuit geometry is patterned
with photolithography (similar to the process shown in Fig. 5.5 (h)-(i)).

86

Figure 5.5: Josephson junction fabrication steps, part 2. Following the processes shown in
Fig. 5.4, the nanopillar is electrically isolated (e) with a thermally evaporated layer of SiOx .
(f) To break through the sidewall of the nanopillar we rotate the sample and ion mill at
grazing angle (3◦ ) of incidence. (g) That allows the chemical liftoff-process to access and
dissolve the resist. (h) The top lead circuit geometry is patterned with photolithography
using a positive tone resist (S1813), that is hardened in chlorobenzene and developed to
create an undercut in the resist profile. (i) Sputtering deposition is used to deposit the top
superconducting electrode and Au protective cap. (j) The resist is lifted-off chemically and
the completed ferromagnetic Josephson junction is ready for measurement.

87

JJ 1,2

JJ 3,4

½ in

JJ 5,6

JJ 1

½ in

JJ 2

Figure 5.6: Josephson junction sample design and images. The left sketch shows an overlay
of the two photolithography masks used to define the bottom and top leads of the Josephson
junction circuit. The bottom lead consists only of the vertical line down the center of the
image, the two rectangular contact pads attached to it, and several alignment marks that are
not electrically connected. The top leads consist of all the lines with a horizontal component
and the associated contact pads. The mask can accommodate up to six independent Josephson junctions, which reside at the intersection of the two lead patterns (JJ 1-6). The optical
images on the right show magnified views (red and green boxes) of a finished sample made
with the mask. In those images, the Au of the top leads stands out as a bright yellow, the
surrounding SiOx layer is reddish brown in color (at this thickness of 50 nm), and the bottom
leads underneath it appear light brown. In the highest magnification image in the lower right
(green box) one can see the L-shaped alignment marks used for electron-beam lithography.
In that image the large ellipse is written above JJ-1 to help tell if the electron-beam resist
lifted off properly. The total chip size is 0.5 × 0.5 in2 .

To improve the possibility of a successful lift-off, next we perform two additional ion
milling steps (2 minutes each) at a grazing angle of incidence of 3◦ to the sample plane,
88

as shown in Fig. 5.5. The purpose of this “side-mill” step is to use the ion mill to break
through any SiOx that may have been deposited onto the sidewall of the electron-beam
resist (or metal redeposited during the ion milling process). Breaking through the side wall
allows the lift-off solution to dissolve the resist. The circuit geometry of the underlying
bottom wiring layer must be considered when deciding the direction to side-mill the samples
from. Generally, one should avoid side-milling at 90◦ to edges in the bottom wiring layer
because it can cause those layers to become electrically shorted to overlapping regions of the
top electrode (sputtered later). If the lift-off process fails, one can very carefully swipe the
junction area a few times with a cotton Q-tip while immersed in the lift-off solution. While
this procedure is not “ideal,” the mechanical force from the Q-tip is usually enough to break
through the side-wall, allowing for a successful lift-off and saving an otherwise ruined sample.
Finally, as shown in Fig. 5.5, a second photolithography process is used as a pattern
for the top wiring layer. Prior to depositing the top electrode, any residual hydrocarbons
are removed from the surface of the junctions using a commercial March3 oxygen plasma
etcher. After inserting the samples into the sputtering chamber, we use the small 1 cm ion
mill to clean the top surface of the samples again immediately before sputtering the top
superconducting electrode. A final lift-off process removes the photomask, leaving a finished
sample. To fabricate one full set of samples (16 substrates) the entire process usually takes
between 2-3 weeks.
In Fig. 5.6 we show images from an optical microscope of a completed sample containing
six Josephson junctions, alongside the superimposed outline of the mask layout for both the
top and the bottom electrodes. In Fig. 5.6, some of the Vernier and electron-beam alignment
markings used in both photo and electron-beam lithography are visible.

89

5.6

Measurement Probes and Low Noise rf SQUID Electronics

Our samples are typically measured at 4.2 K by immersing the sample into a Dewar
of liquid helium using low-temperature measurement probes built by William Pratt. The
simplest of these probes can be used to make four-terminal measurements of the I-V characteristic curves of our Josephson junction samples. The workhorse probe used for most
of the measurements in this thesis is called Quick Dipper I (QD-I) and is equipped with
several additional improvements. To measure extremely small voltages down to 10’s of pV,
the probe contains a rf SQUID comparitor circuit, shown in Fig. 5.7.
The scheme works as follows: the sample represented by the resistor Rs is connected in
a loop with an inductor and a reference resistor, Rref = 126 µΩ, shown at the bottom of
Fig. 5.7. That loop is inductively coupled through a transformer to an rf SQUID, which
consists of a single Josephson junction in a superconducting loop. The goal is to try and
sense a small change in the voltage of the sample. For example, if the sample is a Josephson
junction (or a DC SQUID) it will begin to develop a voltage when a current comparable to
the sample’s critical current is driven through it. A dynamic change in the voltage across
the sample will induce a current in the rf SQUID loop. The resulting flux change in the rf
SQUID loop is detected by the SQUID electronics attached to the top of the probe outside
the Dewar. The SQUID electronics system is a commercial Quantum Design 2010 SQUID
control circuit and amplifier. After sensing a change in the rf SQUID, the SQUID electronics
system outputs a feedback voltage Vout which is reduced by a voltage divider composed of
Rref and a feedback resistor RFB = 2 kΩ. The feedback in the SQUID control circuit acts to
maintain a voltage across Rref that matches the voltage across Rs . Since the voltage across
the sample and reference resistor is balanced, we can infer the voltage across the sample

90

using the simple formula for a voltage divider with the feedback and reference resistors:†

Vs = Vout




Rref
.
Rref + RFB

(5.1)

Plugging in the values for Rref and RFB , we see that Vs can be very small in principle even
for appreciable Vout values (the gain is on the order of 107 ), while the rf SQUID is sensitive
to changes in flux on the order of Φ0 = 2.067 × 10−15 Wb. This causes the system to have
extremely good rms voltage noise, which we have measured to be approximately 6 pV per
10 power line cycles.
QD-I has several other features. In addition to the current and voltage wiring leads, the
probe contains several extra sets of wire leads. Those leads can be used, for example, to
control an on-chip flux line for SQUID experiments as done in Chs. 7 and 9.
Furthermore, around the sample, at the end of the probe, there is a solenoid coil that can
be used to apply a uniform magnetic field parallel to the plane of the sample. To achieve a
very stable magnetic field value, the solenoid is made of superconducting wire and is shorted
by a controllable superconducting persistent switch. If a current is present in the solenoid
and the persistent switch is closed, the supercurrent will “persist” in the loop made by the
short across the solenoid, producing a very stable magnetic field. To change the value of
the magnetic field, a resistive heater nearby the persistent switch can warm the persistent
switch and drive it into the normal state, breaking the short. The circuit to control the
current supplied to the solenoid coil, the persistent switch heater, the current supplied to
the sample, and the voltage detected from the SQUID electronics is all automated using
the LabVIEW package. Significant improvements to the computer control software were
† In Eq. (5.1) we have neglected the ≈ 0.06 % correction to V due to the finite open-loop gain of the
s
feedback circuit.

91

SQUID
Electronics

SQUID
Vout

R FB

Current
Source

Rs

R ref

Inside Dewar
Figure 5.7: rf-SQUID comparator measurement circuit. An rf SQUID, composed of a single Josephson junction is inductively coupled through a transformer to an LC circuit that
contains the sample (Rs ) and a reference resistor Rref . Commercial SQUID electronics,
described in the text sense the state of the rf SQUID and provide a voltage Vout passed
through a feedback resistor RFB . The system allows for large amplification of very small
voltage changes in the sample, with extremely low rms voltage noise.
developed by Victor Aguilar, which enabled us to measure I-V curves at a much faster rate.
Victor also improved the code to allowed us to iterate various magnetic field sweep sequences.

5.7

SQUID Magnetometry

We used two different SQUID magnetometers for our measurements. Both are Magnetic
Property Measurement Systems (MPMS) built by Quantum design, one an older model built
in the 1980s and the other a more state of the art system, the MPMS3. The MPMS3 can
measure over temperature ranges of 1.8 - 400 K and is equipped with a magnet capable of
92

generating fields up to 7 T, with accuracy of Âą0.005 mT.
In addition to acting as a DC SQUID magnetometer, the MPMS3 can also act as a
Vibrating Sample Magnetometer (VSM), in which the sample is mechanically vibrated in
a field while measuring the induced voltage in a pickup coil that is proportional to (magnetic moment)×(vibration frequency)×(vibration amplitude). This method is combined with
SQUID magnetometry to achieve more sensitive measurements.
As previously shown in Fig. 5.2, we have developed a process for patterning arrays of
nanomagnets using electron-beam lithography for magnetic characterization in the SQUID
magnetometer. As opposed to measuring large continuous films of various magnetic materials
whose magnetic behavior is dominated by domain wall motion, the magnetic bit arrays will
allow us to better characterize the behavior of magnetic bits of the actual dimensions that
are used in our Josephson junction and SQUID experiments. This allows us to observe the
spread in the magnetic switching and quantify the effects of shape anisotropy.

93

Chapter 6

S/F/S Josephson Junctions: Materials
Characterization

Josephson junctions containing ferromagnetic materials (S/F/S junctions) have been under intense study for the past 15 years. As discussed in Ch. 4, the experimental breakthrough
that triggered such intense interest was the demonstration that the ground-state phase difference across the junction can be either 0 or π, depending on the thickness of the ferromagnetic
layer(s) in the junction [65, 68]. That result has been confirmed by numerous groups since
the initial discovery, using a wide variety of weak and strong ferromagnetic materials [69–75].
In this spirit, with this chapter we present four separate experimental studies of Nb-based
micron-scale elliptically-shaped Josephson junctions containing central ferromagnetic layer
of either Ni65 Fe15 Co20 , Py (the alloy Ni81 Fe19 ), Pd97 Fe3 , and Ni. Throughout this chapter,
we’ll examine the comparative advantages and disadvantages that these materials offer as
potential components in cryogenic memory devices. This chapter encompasses Refs. [19, 20]
where these results were first published, including some additional findings.
Before using a new magnetic material in a memory device, one would like to know how it
behaves by itself inside a Josephson junction. Most importantly, at what F -layer thickness,
dF , does the junction transition from the 0 state to the π state? Secondly, how does Ic decay
as dF increases? Thirdly, how much does the field generated by the magnetization affect the
junction properties? And lastly, does the material exhibit single-domain switching behavior
when it is patterned into a micron-scale elliptical nanomagnet? All of those questions can
94

be answered by fabricating and measuring micron-scale S/F/S junctions containing only a
single magnetic layer [77].
By applying an external magnetic field, the critical current of the junctions are found
to follow characteristic Fraunhofer patterns. The high quality of the Fraunhofer patterns
typically enables us to extract the maximum value of the critical current even when the
peak is shifted significantly outside the range of the data due to the magnetic moment of
the ferromagnetic layer. Indeed, we observe that the maximum value of the critical current
oscillates as a function of the ferromagnetic barrier thickness, indicating transitions in the
phase difference across the junction between values of zero and π. We compare the data to
previous work and to models of the 0-π transitions based on existing theories.

6.1

Role of π-Junctions in Cryogenic Memory

Josephson junctions that exhibit a ground-state phase shift of π for certain ranges of
ferromagnetic layer thickness are of considerable interest for the development of practical
cryogenic memory and superconducting qubits. There have been several proposals to use
π-junctions as new components in either classical or quantum information processing circuits
[110–116]. Our current work in this area focuses on the development of cryogenic random
access memory [18, 77].
Numerous ideas have been presented in the literature regarding how S/F/S junctions
might be used as practical memory devices [12, 14, 16, 17, 22, 117, 118]. The ferromagnetic
layer influences the properties of the junction both through the magnetic field and the exchange field it generates, and ideas have been presented using either of those mechanisms.
In addition, either the critical current magnitude, Ic , or the ground-state phase difference

95

across the junction, φ, can be used as the physical quantity associated with information
storage. Without trying to summarize the whole field, we can nonetheless make two general
observations:
1. Proposals that rely on the internal magnetic field of the junction tend to become less
viable as junction size decreases, since the relevant physical parameter defining the
effect of the field is the magnetic flux, ÎŚ, threading the junction area. If the junction
area is reduced to the point where Φ << Φ0 = h/2e = 2.07×10−15 Tm2 , then the
magnetic field has negligible effect on the junction properties. For that reason, we
have chosen to emphasize the effect of the exchange field in our work.
2. Proposals that rely on the magnitude of Ic invariably require that the S/F/S junction
switch from the zero-voltage state into the voltage state when the memory is read.
That switching process takes a time of order τswitch ≈ ~/(eIc RN ), where RN is the
junction resistance in the voltage state. For standard S/F/S junctions, τswitch is much
too long to be useful for memory applications. One can shorten τswitch somewhat
by increasing RN via the introduction of an insulating barrier to make an S/I/F/S
junction [14, 72]. Suppression of Ic by the insulating layer can be mitigated by adding
a thin auxiliary nearly-superconducting (s) layer, to form an S/I/s/F/S junction with
very large values of Ic RN [15, 119–121].
An alternative scheme, proposed by workers at Northrop Grumman Corporation, shown
in Fig. 6.1, envisions a memory cell consisting of a SQUID loop that contains a phasecontrollable Josephson junction and two conventional S/I/S junctions with much smaller
Ic ’s [22, 23]. The critical current of the SQUID loop is determined by the phase state of the
controllable junction, either 0 or π, which correspond to the logical 0 or 1 of the memory.
96

Figure 6.1: Cryogentic phase-sensitive memory cell proposed by Northrop Grumman Corporation. The memory cell is composed of a phase-controllable ferromagnetic Josephson
junction together with two S/I/S junctions in a SQUID loop. The critical current of the
SQUID will oscillate as a function of the flux threading the loop. Depending on the state
of the phase-controllable junction, it will contribute either an additional 0 or π-phase into
the loop, resulting in a larger or smaller total critical current, respectively, in the SQUID.
The S/I/S junctions are added to facilitate a fast switching time (set by their large Ic RN
product) while reading out the state of the memory. Figure taken from [18].
During the read operation, only the S/I/S junctions may switch into the voltage state,
providing a fast τswitch , while the controllable junction remains in the superconducting state.
As a result, high Ic RN is not a critical requirement for the controllable junction in the
Northrop Grumman design. This is the memory design we are currently pursuing.
What we want, then, is a Josephson junction whose ground-state phase difference can be
controllably switched between the 0 and π states. It is advantageous that one or both of the
nanomagnets in such a junction be single-domain, so that the magnetizations are uniform
and magnetic switching is clean and reproducible. One method of accomplishing these goals
is to make a junction containing a “spin valve”, i.e. two F layers whose relative magnetization
directions can be switched between parallel and antiparallel [12,16,17]. These type of memory
devices contain a spin valve with one “hard” magnetic layer and one controllable “soft” layer.
We demonstrated such a controllable 0-π junction, which is the subject of Ch. 7, using
Ni and Ni81 Fe19 (refered to as NiFe or Py) as the two ferromagnetic materials [18]. Thin

97

Ni films are magnetically “hard”, whereas NiFe is “soft”, hence it was possible to reverse the
magnetization direction of the NiFe without changing that of the Ni. However, due to the
poor magnetic properties of Ni, discussed later in this chapter, we have been searching for a
material to replace it as the hard magnetic layer. In addition, we have been exploring other
possible alternatives for the soft layer as well.

6.2

Sample Fabrication and Characterization

Our S/F/S Josephson junctions are fabricated using high-vacuum sputtering deposition
on 0.5×0.5 in2 silicon chips. The geometry of the bottom leads was defined via optical
photolithography and the positive photoresist S1813, using the liftoff process.
Before the sputtering deposition, the chamber was baked for eight hours and reduced
to a base pressure of 2×10−8 Torr with a cryopump. The chamber was then cooled by
circulating liquid nitrogen though a Meissner trap to reduce the partial pressure of water in
the chamber to < 3 × 10−9 Torr as confirmed by an in-situ residual gas analyzer. The films
were deposited via dc sputtering in an Argon plasma with pressure 1.3 × 10−3 Torr. During
deposition the sample temperature was held between −30 ◦ C and −20 ◦ C measured with a
thermocouple affixed to the back of one of the substrates.
In a single run, a rotating sample plate and shutter system passes up to 16 chips over
a sequence of triode sputtering guns containing 2.25-inch diameter targets of Nb, Al, NiFe,
NiFeCo, PdFe or Ni and magnetron guns containing 1-inch diameter targets of Au and
Cu. The thicknesses of the various deposited materials were controlled by measuring the
deposition rates (accurate to Âą0.1/s) using a crystal film thickness monitor and a computer
controlled stepper motor that operates the position of the shutter and sample plate.

98

Figure 6.2: A schematic representation of the cross-sectional structure of our S/F/S Josephson junctions. The F layer is either Ni65 Fe15 Co20 , Ni81 Fe19 , Pd97 Fe3 , or Ni. The thickness
dF ranges from 0.8 to 3.8 nm for the Ni65 Fe15 Co20 , Ni81 Fe19 , and Ni samples. The much
weaker ferromagnet, Pd97 Fe3 , is thicker, ranging from 9 to 36 nm. The ferromagnets are
grown on a smooth superconducting Nb/Al base electrode, set between Cu spacer layers.
The lateral shape of the junctions are elliptical with aspect ratio of 2.5 and area of either
0.1, 0.25 or 0.5 Âľm2 . The patterned region is isolated by SiO. All thicknesses listed in nm.
Figure taken from [19].

99

Figure 6.3: Cross-sections through the center of our S/F/S Josephson junctions prepared with
focused ion beam (FIB) milling and imaged with Scanning Transmission Electron Microscopy
(STEM) to validate the fabrication process. In (a) the STEM image shows where our [Nb/Al]
bottom electrode meets the left edge of the patterned junction area and SiO barrier. Energy
dispersive x-ray spectroscopy (EDX) was used to map out the elemental composition of the
individual layers within. In (b) we show the EDX analysis for the region outlined by the
green square, which clearly shows a continuous NiFeCo layer (dF =1.6 nm). Figure taken
from [19], courtesy of P. Kotula and N. Missert, Sandia National Laboratory.

100

The bottom wiring layer, which was deposited without breaking vacuum, consists of the
sequence [Nb(25)/Al(2.4)]3 /Nb(20)/Cu(5)/F (dF )/Cu(5)/Nb(5 or 20)/Au(10), with thicknesses given in nanometers. A schematic of the sample structure is shown in Fig. 6.2.
In order to verify the fabrication process, for the junctions containing NiFeCo, crosssectional areas of the junctions were investigated by high-resolution scanning transmission
electron microscopy (STEM) and energy dispersive x-ray spectroscopy (EDX). Since we do
not have these capabilities at Michigan State University, we sent samples to our collaborators
Paul Kotula and Nancy Missert at Sandia National Laboratories to help us with the STEM
preparation and imaging.
The cross-sections were prepared using a FEI Helios focused ion beam (FIB) with a Ga
ion source, and transferred to a Ti grid for imaging in a FEI Titan G2 80-200 aberrationcorrected STEM operated at 200kV and equipped with four silicon drift X-ray detectors.
The high-angle annular dark field STEM image in Fig. 6.3(a) shows the left hand side of one
of the NiFeCo junction stacks.
To achieve quality magnetic switching it is crucial to grow the ferromagnets on a smooth
underlayer. The low surface roughness of the [Nb/Al] multilayer [122–124] provides a smooth
template for subsequent growth of the Cu spacer and ferromagnetic layer, where the Al is
thin enough to be superconducting through the proximity effect with Nb. We independently
measured the roughness of the [Nb/Al] multilayer to be ≈ 2.3 Å using atomic force microscopy (AFM), smoother than a single Nb(100) film (> 5 Å). Significant diffusion of Al
along Nb grain boundaries was not observed. The superconducting transition temperature of
[Nb(25)/Al(2.4)]3 /Nb(20) films, was measured independently with a SQUID magnetometer
to be 8.0 K, well above the temperature at which we measure our S/F/S junctions (4.2 K).
Due to the crystal lattice mismatch between the fcc ferromagnetic materials and the
101

bcc Nb we add a 5-nm Cu spacer on either side of the F layer. STEM diffraction patterns
show that the Cu layer grows with a [111] orientation on Nb [011]. Grains with favorable
orientation relative to the beam direction show lattice fringes extending through the entire
Cu/ferromagnetic layer/Cu thickness. In comparison to films grown on only Nb the spacer
improves the magnetic properties of the F layers: decreasing the coercive field, and increasing
the sharpness of hysteresis loops. Also, smooth normal metal spacer layers will be used in
cryogenic memory to magnetically decouple the multiple F layers.
EDX phase maps were created by performing a multivariate statistical analysis of the
spectra from each individual pixel, and color-coding pixels containing the same spectrum [125].
The phase map shown in Fig. 2b corresponds to the area within the green square in Fig. 2a.
The 1.6 nm NiFeCo layer is clearly uniform and continuous, consistent with the magnetic
behavior discussed below.
To set the direction of the magnetocrystalline anisotropy of the ferromagnetic alloys, the
NiFeCo and PdFe samples were sputtered in a magnetic-field of ≈ 80 mT (whose direction
points along what will become the major axis of our elliptical junctions) generated by small
NdFeB magnets affixed directly on the back of the substrates. The NiFe and Ni samples,
made in separate sputtering runs, were sputtered in a magnetic-field of ≈ 50 mT. Finally
the samples were capped with a thin layer of Nb and Au to prevent oxidation.
The junctions were patterned by electron-beam lithography followed by ion milling in
Argon. We use the negative e-beam resist ma-N2401 as the ion mill mask. The junctions
are elliptical in shape with an aspect ratio of 2.5 and area of either 0.1, 0.25 or 0.5 Âľm2 ,
all sufficiently small to ensure that the magnetic layer is single domain (not including the
junctions containing Ni). Elliptically-shaped junctions have the advantages that the modulation of the critical current through the junction versus the applied magnetic field, known
102

as a Fraunhofer pattern, follows an analytical formula while the (small) demagnetizing field
is nearly uniform when the magnetization is uniform.†
Outside the mask region, we ion milled though the capping layer, the F layer, and nominally half-way into the underlying Cu spacer layer. Figure 6.3 confirms our patterning of
the F layer, though it is clear this sample was slightly over-milled; the step edge extends
through the second Cu layer and just into the underlying Nb. After ion milling, a 50 nm
thick SiO layer was deposited by thermal evaporation to electrically isolate the junction and
the bottom wiring layer from the top wiring layer. During thermal evaporation the sample
was rotated to improve the uniformity of the oxide and reduce pinhole formation.
To prevent the e-beam resist from over-heating during the ion milling and SiO deposition
the back of the substrate is placed against a Cu heatsink with a small drop of vacuum grease
or silver paste to improve thermal contact. A capping layer containing 20 nm Nb was used
in some of the NiFe-based samples, but was later reduced to 5 nm in the remaining NiFe
and NiFeCo-based samples for two main reasons: i) During ion milling a veil of Nb can be
backsputtered onto the edge of the e-beam resist, preventing the e-beam resist from lifting-off
properly, or at all. Reducing the Nb thickness reduces the extent of the veil. ii) Since Nb has
the slowest etching rate of all our materials, reducing the Nb thickness drastically reduces
the total time required to ion mill. Reducing the Nb thickness in the capping later improved
our lift-off success rate, likely due to less damage and distortion of the resist under the heat
of the ion mill and during SiO deposition.
Finally, the top Nb wiring layer was patterned using the same photolithography and liftoff process as the bottom leads. Residual photoresist on the surface is cleaned with oxygen
† Strictly speaking, the field is uniform only inside a uniformly magnetized ellipsoid. Because the elliptical
nanomagnets in our junctions are very thin, there is very little difference between an ellipse and an ellipsoid.

103

plasma etching followed by in-situ ion milling in which 2 nm of the top Au surface is removed
prior to sputtering. We deposited top leads of Nb(150 nm)/Au(10 nm), ending with the Au
to prevent oxidation.

6.3
6.3.1

Critical Current Oscillations of Josephson Junctions
Junctions Containing Ni65 Fe15 Co20

The alloy Ni65 Fe15 Co20 is a promising candidate for the “hard layer”: like NiFe, it has a
small magneto-crystalline anisotropy whose direction can be set by depositing the material
in the presence of a magnetic field, but its anisotropy is somewhat larger than that of NiFe.
Using NiFeCo as the hard layer should allow us to use initialization fields of only a few tens
of mT.
First we describe the measurement scheme in detail for the junctions containing NiFeCo,
but the process is similar for all of the S/F/S samples that follow in this chapter. Each device
was connected to the wire leads of a dip-stick probe with pressed indium. The samples were
immersed at 4.2 K in a liquid-He dewar outfitted with a Cryoperm magnetic shield and placed
inside a shielded room to reduce noise from external sources of electromagnetic radiation. The
dipping probe is equipped with a superconducting solenoid used to apply uniform magnetic
fields over a range of -60 to 60 mT along the long-axis of the elliptical junctions in the plane
of the sample. The current-voltage characteristics of the junctions were measured at 4.2 K
in a four-terminal configuration using one or both of the following methods: 1) a Yokogawa
current source provides a bias current to the Josephson junction while voltage measurements
were made with a Keysight nanovoltmeter or 2) a commercial Quantum Design rf SQUID in
a self-balancing potentiometer comparator circuit (described in Ch. 5) measures the voltage

104

across the junction, while the measurement current is provided by a battery-powered ultralow noise programmable current source [126]. Data taken independently from the two setups
agree with one another, however the rf SQUID comparator scheme has lower RMS voltage
noise of only 6 pV compared to 11 nV for the commercial nanovoltmeter, with both systems
measuring over 10 power line cycles. Typical I-V curves, shown in Figs. 6.4 and 6.5, have
the expected behavior of overdamped Josephson junctions [127]. Figure 6.4 shows how the
I-V curve shape changes while being measured in applied magnetic fields ranging from 0-50
mT. The entire data collection process is automated using the LabVIEW software package.
The sample resistance in the normal state RN was determined by the slope of the linear
region of the I-V curve when |I|  Ic . While the sensitivity of the rf SQUID measurement
system allows us to measure junctions with small Ic , it operates only over a restricted voltage
range. Thus, depending on the resistance of the sample, one may be limited in the extent
to which the linear tail of the I-V curve can be measured. In these cases, independent
measurements using both measurement schemes are necessary to accurately determine both
Ic and RN . Measurements of the area-resistance product in the normal state yield consistent
values of ARN = 5-10 fΩ-m2 for junctions of different areas – an indicator of the reproducible
high quality interfaces. This total specific resistance is close to twice the Nb/F interface
resistance, determined in separate current-perpendicular-to-plane giant-magnetoresistance
studies [128].
The critical current Ic was extracted by fitting the I-V curves to a square root function
of the form,
q
V = RN I 2 − Ic2 , I ≥ Ic .

(6.1)

We occasionally observe that Ic is slightly different in the positive and negative current

105

Figure 6.4: The voltage across an overdamped S/F/S Josephson junction containing 2 nm
layer of NiFeCo, versus the applied current. The data are measured via standard fourterminal measurement in an external magnetic field of 0 - 50 mT as indicated. The critical
current Ic , extracted from I-V curves above, is used to produce the Fraunhofer pattern
shown in Fig. 6.6 (b). For clarity, each successive curve is shifted along the voltage axis in
steps of 5 ÂľV. Figure taken from [19].

106

directions. That does not violate any symmetry given the presence of the ferromagnetic
material in the junctions, but we find it puzzling given the rather small value of mutual
inductance between the electrical leads and the junction proper. In those cases we define Ic
to be the average value of the critical currents in the two directions.
When the critical current of the junctions is small, there is a noticeable amount of rounding of the I-V curves as I approaches Ic . Ivanchecko and Zil’berman (IZ) and Ambegaokar
and Halperin developed a theory to fit such data when the rounding is due to thermal fluctuations [129,130]. (In the “tilted-washboard” potential of the RCSJ model from Ch. 2, thermal
fluctuations cause the particle to escape out of the potential wells when I < Ic .) When the
rounding is caused by fluctuations in the electromagnetic environment coupled to the junction (usually from the measurement apparatus), the temperature in the IZ model becomes
an effective temperature, which can be much larger than the actual sample temperature. In
the IZ model the I-V curve has the analytical solution [129]


V (Ic , RN , Teff ) = Ic RN

where IÂą =

I

(1Âąi)Îł (Îłc )
2iIÂąiÎł (Îłc )




I
− I− + I+ , I ≥ 0,
Ic

(6.2)

, γ = I~/(2ekB Teff ) and γc = Ic ~/(2ekB Teff ). Iν (z) are modified

Bessel functions of the first kind with ν a non-integer complex number, where e is the electron
charge, kB is the Boltzmann constant, and Teff is the effective noise temperature.
Figure 6.5 shows fits of the IZ function to data from a sample at magnetic fields where Ic is
rather small, hence the rounding is apparent. Data are presented both for the nanovoltmeterbased measurement system and for the rf-SQUID-based system. The noise temperature, Teff ,
is ≈ 95 K for the former, versus ≈ 37 K for the latter, as shown in Table 6.1. The table
shows that the values of Ic extracted from the fits are comparable despite the difference in
107

Figure 6.5: The voltage across an S/F/S Josephson junction versus the applied current
at applied magnetic fields of 18 mT (circles) and 36 mT (diamonds), using two different
measurement schemes. The orange and purple colored data points are measured with a
commercial nanovoltmeter while the blue and red colored data points are measured with
an rf SQUID in a self-balancing potentiometer comparator circuit. The data are fit to the
Ivanchenko-Zil’berman function, Eq. (6.2), which accounts for noise-induced rounding and
allows us to extract the noise temperature of our measurement systems. The upper set
of curves (shifted along the voltage axis by 1 ÂľV for clarity) compares the two different
fitting methods: to the square root function (Eq. (6.1), dashed lines) and to the IvanchenkoZil’berman function (Eq. (6.2), solid lines). The dotted black line represents Ohms’ law for
the measured normal state resistance. This junction contains a 1 nm layer of NiFeCo. Figure
taken from [19].

108

Table 6.1: Parameters obtained from fits of the Ivanchenko-Zil’berman function to the data
shown in Fig. 6.5. The normal-state resistance RN was measured to be 20.5 mΩ, and was
not used as a free fitting parameter.

Method

Field (mT)

Ic (ÂľA)

Teff (K)

nanovoltmeter

18

31.0 Âą 0.1

98.6 Âą 4.8

36

12.6 Âą 0.3

85.3 Âą 12.9

18

32.6 Âą 0.05

37.4 Âą 0.7

36

15.9 Âą 0.1

36.5 Âą 2.8

rf SQUID

Teff . However, due to it’s much lower RMS voltage noise, the SQUID-based measurement
system was used for samples whose maximum value of Ic is less than about 10 ÂľA.
Measuring Ic as a function of the applied magnetic field, we map out so-called “Fraunhofer” diffraction patterns, shown in Fig. 6.6. To compare junctions with different crosssectional areas we normalized our data by multiplying Ic by RN . For elliptical junctions the
functional form is an Airy function [127], as described in Eq. (4.23) but is repeated here for
convenience,
Ic = Ic0 |2J1 (πΦ/Φ0 ) / (πΦ/Φ0 )| ,

(6.3)

where J1 is an unmodified Bessel function of the first kind (whose order is a real integer,
unlike the modifed Bessel functions of Eq. (6.2)) and Ic0 is the maximum critical current
and ÎŚ0 = h/2e is the flux quantum. The magnetic flux through the junction is [76]

ÎŚ = Âľ0 Hw(2ÎťL + 2dN + dF ) + Âľ0 M wdF ,

(6.4)

where H, w, ÎťL , dN and dF are the applied field, the width of the junction (minor axis), the
London penetration depth of the Nb electrodes, and the thicknesses of the normal metal and

109

3

a)

2

d

NiFeCo

= 1 nm

1

4

b)
d

NiFeCo

= 2 nm

2

c

I R

N

(

V)

0

0
0.2

c)

d

NiFeCo

= 3 nm

0.1

0.0
-60

-30

0
0

30

60

H (mT)

Figure 6.6: Fraunhofer patterns of Josephson junctions containing NiFeCo. Critical current
times the normal state resistance, Ic RN , is plotted versus the applied field H, for three
samples with dNiFeCo = (a) 1 nm, (b) 2 nm, (c) 3 nm. The data before Hswitch , the field at
which the NiFeCo magnetization reverses direction (solid markers), and the corresponding
fits (lines) to Eq. (6.3) show excellent agreement for both the positive (red, dashed) and
negative (blue) field sweep directions. The hollow circles are the corresponding data points
after Hswitch . The Fraunhofer patterns display magnetic hysteresis and are increasingly
shifted with larger dNiFeCo . Figure taken from [19].

110

F layers. The last term in Eq. (6.4) arises from the magnetization M of the single-domain
ferromagnet, and is valid only if M is uniform and points along the same direction as the
applied field H. Equation (6.4) neglects the small contributions to ÎŚ from the uniform
demagnetizing field and any magnetic field from the nanomagnet that returns inside the
junction. From Eq. (6.4) it is clear that the Fraunhofer pattern will be shifted along the field
axis by an amount Hshift = −M dF /(2λL + dF + 2dCu ).
The data shown in Fig. 6.6 were acquired as follows: we first applied a field of -60 mT
to fully magnetize the nanomagnet, then the field was slowly ramped to +60 mT in steps
of typically 1.5 mT. At a critical value Hswitch > 0, the ferromagnet undergoes a rapid
reversal of it’s magnetization direction. The data then transition to another Fraunhofer
pattern shifted in the opposite direction. The applied magnetic field was then swept in the
reverse orientation to observe the magnetic hysteresis. Similar magnetic hysteresis loops in
Josephson junctions have been previously studied [17, 77], and are well understood.
We fit the data starting from the initialization field to Hswitch with Eq. (6.3), where
Ic0 and Hshift are fitting parameters. Allowing the sample width, w, to be a free fitting
parameter gives rise to large uncertainty in Hshift for data sets with large values of Hshift .
Hence we fixed w to its nominal value in all the fits presented here. We keep ÎťL fixed at
85 nm, as determined by data obtained in our group over many years [76]. In Fig. 6.6 the
corresponding fits (lines) show excellent agreement with the fitted data (solid markers), for
the positive (red) and negative (blue) sweep directions. The hollow data points after Hswitch ,
whose fits are not plotted for clarity, match well with the Fraunhofer pattern from where
the field is swept in the opposite direction. The nodes in the Fraunhofer patterns closely
approach Ic = 0, which illustrates that there are no shorts in the SiO. This and the lack of
distortion in the pattern at large field values indicates that there is little if any trapped flux
111

in the junction electrodes.
The excellent quality of the Fraunhofer patterns starting at high field and extending
past zero field to Hswitch indicates that the remanent magnetization in the junctions is
uniform, suggesting that the ferromagnetic layers are probably single-domain. In many cases,
the magnetic switching is abrupt, which also supports that interpretation. The excellent
quality of the fits allows us to extract the peak value of Ic even in cases where the peak is
inaccessible in the data because it lies beyond the field where the magnetic layer switches.
The uncertainties in peak height and position in such cases, are, of course, larger than when
the peak is directly visible in the data.
Some Fraunhofer patterns, such as those shown in Figs. 6.6 (b)-(c), show that the reversal
of the magnetization occurs over a range of field values, implying that our junctions do not
strictly follow the abrupt switching behavior predicted by the Stoner-Wolfarth model. For
those samples, magnetization reversal takes place through a non-uniform intermediate state,
for example an “S-shaped” state or a multi-domain state. This could be exacerbated by
a number of factors including nonuniform magnetocrystalline anisotropy, surface or edge
roughness, or magnetostriction.
The thickness at which the junctions transition from a 0 to π-phase state is determined
by plotting Ic RN for many samples spanning a range of ferromagnet thicknesses dF , and
looking for deep local minima where Ic theoretically passes through zero. This is shown in
Fig. 6.7(a) where Ic denotes the maximum critical current obtained from the Fraunhofer
pattern fits. The transition from a 0 to π-phase state occurs at a thickness of dNiFeCo =1.2
nm.
Theoretical predictions for the behavior of Ic RN versus thickness of the ferromagnetic
layer describe an oscillating function with either an algebraic decay for ballistic transport [61]
112

a)

1

c

I R

N

(

V)

10

0.1

10

0

H

shift

(mT)

b)
20

0

1

2
d

NiFeCo

3

(nm)

Figure 6.7: Critical current oscillations and Fraunhofer pattern shifts in Josephson junctions
containing NiFeCo. a) The maximal Ic times RN is plotted versus dNiFeCo for many samples,
with the error bars determined by the goodness of fit parameters of the individual Fraunhofer
patterns. The data are fit to Eq. (6.5), and the best fit parameters are shown in Table 6.2.
The first minima indicates that at a critical NiFeCo thicknesses of 1.15 Âą 0.02 nm there is a
transition between the 0 and π-phase states. b) The Fraunhofer pattern field shift Hshift is
shown to increase with dNiFeCo . The fit to Eq. (6.6) yields a magnetization of 855 Âą 81 kA/m
and an x-intercept which corresponds to no discernible magnetic dead layer (ddead =0.06 Âą
0.17 nm). Figure taken from [19].

113

or an exponential decay for diffusive transport [131]. The crossover from the ballistic to the
diffusive limit has also been addressed in several theoretical works [78, 132]. In the diffusive
limit the behavior is governed by the Usadel equations [57] in cases where the majority and
minority spin bands have nearly identical properties. For strong ferromagnetic materials,
the Usadel equations are not adequate [78]. In principle, one should take into account the
Fermi-surface mismatch at each interface, as well as the different densities of states, mean
free paths, and diffusion constants for the majority and minority spin bands. Microscopic
calculations based on the Bogoliubov-deGennes equations and taking into account the finite
interface transparency, have been performed for ballistic systems [133, 134], and could, in
principle, be extended to systems with disorder. But there have been no theoretical calculations of the supercurrent that take into account the complex band structure of transitionmetal ferromagnetic materials such as those discussed in this chapter. Nevertheless, several
previous experimental works have used existing theoretical formulas to fit data from strong
ferromagnetic materials. For instance, the ballistic form was used by Robinson et al. to
fit data from junctions containing elemental ferromagnets, Ni, Co, and Fe, but data from
junctions containing NiFe appeared to show a crossover from ballistic to diffusive behavior
at a thickness of about 2 nm [73, 74]. We also attempted to fit our data to the ballistic limit
used in Ref. [74], but did not find good agreement.
The data shown in Figs. 6.7(a) roughly follow an exponential decay, which is not surprising
given the short mean free paths of minority carriers in NiFeCo [135]. It should be emphasized
however that with our thin F layers, the concept of a mean free path may not be a valid
notion when considering that the dominant scattering occurs at the F-layer/Cu interfaces.
We estimate that ratio of interfacial minority-majority scattering for our F-layer materials

114

Table 6.2: Best-fit parameters from Eq. (6.5) corresponding to the data in Figs. 6.7(a) and
6.11(a).

F-layer

d0−π (nm)

V0 (ÂľV)

ΞF 1 (nm)

ΞF 2 (nm)

NiFeCo

1.15 Âą 0.02

30 Âą 6

1.11 Âą 0.16

0.48 Âą 0.03

NiFe

1.76 Âą 0.05

69 Âą 19

1.50 Âą 0.38

0.58 Âą 0.10

to be ≈ 6 [135]. Regardless, we fit the data to the diffusive limit case with the function,

Ic RN = V0





e−dF /ξF 1 sin



dF − d0−π
ΞF 2



,

(6.5)

where ΞF 1 and ΞF 2 are length scales that control the decay and oscillation period of Ic
with dF , and d0−π is the 0-π transition thickness. The simplest model of a diffusive S/F/S
p
Josephson junction based on the Usadel equation [57] predicts that ΞF 1 = ΞF 2 = ~DF /Eex
where DF and Eex are the diffusion constant and exchange energy of F, respectively, and
has an overall phase offset φ ≡ (d0−π /ξF 2 ) − π/2 = π/4. In the presence of spin-orbit or
spin-flip scattering [136], or when the F-layer contains domain walls [137], one expects to
find ΞF 1 < ΞF 2 . For strong ferromagnetic materials with large exchange energy, however,
one often finds that ΞF 1 > ΞF 2 [74], a result that can be explained from a semi-ballistic
calculation starting from the Eilenberger equation [78, 132]. In addition, the phase offset,
φ, has been shown by Heim et al. [138] to vary sensitively with the type and thickness of
normal-metal spacer layers or insulating barriers included in the junction.
Equation (6.5) fits the data reasonably well our NiFeCo based junctions. Table 6.2 lists
the best-fit parameters for the data in Fig. 6.7(a), and shows that, in spite of the significant
scatter in the data, the thickness corresponding to the first 0-π transition can be extracted
with reasonable precision. Despite this, Eq. (6.5) does not fit so well for the the thickest

115

subset of the NiFeCo samples, where one would have expected a better fit in this morediffusive regime. The junctions appear to have ΞF 1 > ΞF 2 .
In Fig. 6.7(b), for each junction we plot the average of the Hshift values obtained from
Fraunhofer pattern fits in each sweep direction versus the F layer thickness. Indeed, Hshift
vs. dF increases proportional to the magnetic flux in the junction contributed by the uniform
magnetization of the ferromagnet. The trend is approximately linear due to the fact that
our ÎťL  dF . We fit these data to:

Hshift = M (dF − ddead )/(2λL + dF + 2dCu ),

(6.6)

with M and ddead as free parameters. The resulting fits for NiFeCo give M = 855 Âą 81 kA/m
with no discernable dead layer, ddead = 0.06 Âą 0.17 nm.
In Fig. 6.8 we plot the average of the switching field Hswitch for the two sweep directions
from each Fraunhofer pattern versus dF . In general NiFeCo has a larger Hswitch than NiFebased junctions (to be discussed in the next section) and their difference increases as dF
approaches 1 nm. Stoner-Wohlfarth theory predicts that Hswitch should grow linearly with
dF , though that trend is clearly violated by the NiFeCo data at small dF . The large scatter
in the data as well as that violation, are probably the result of extrinsic factors such as
surface roughness, magnetostriction, or defects in the film. The large switching field could
be advantageous if NiFeCo is used as a fixed layer, since a sufficient difference between the
switching fields of the free and the fixed F layers is important for controlling a cryogenic
memory device.
To further characterize the magnetic properties of our NiFeCo we fabricated samples for
SQUID magnetometry measurements. Unpatterned thin films of Nb(25)/Cu(5)/NiFeCo(dF )

116

12
NiFeCo
NiFe

0

H

switch

(mT)

8

4

0

1

2
d

F

3
(nm)

Figure 6.8: Switching field distribution for both the NiFeCo and NiFe based S/F/S junctions.
The average of the switching field Hswitch for the two sweep directions from each Fraunhofer
pattern versus ferromagnet thickness dF . The large increase in Hswitch for NiFeCo at small
values of dF is due to extrinsic factors such as strain, surface roughness, or defects. Figure
taken from [19].

117

/Cu(5)/Au(5), with thicknesses in nanometers, were sputtered under the same conditions
as for the S/F/S junctions. The samples were measured at 10 K in a Quantum Design
SQUID magnetometer, with the applied magnetic field parallel to the plane the films. The
hysteresis loops of films with dNiFeCo = 1-5 nm are shown in Fig. 6.9. When accounting for
the F layer thickness the saturization magnetization per unit volume is nearly constant for
all samples 934 Âą 8 kA/m and is similar to the results in Figs. 6.7(b), while the x-intercept
shows ddead = -0.06 Âą 0.03 nm. Note that these unpatterned films contain many magnetic
domains that switch predominantly by domain-wall motion, and should not be viewed as a
direct comparison to the single-domain nanomagnets in our S/F/S junctions. Nonetheless,
as dNiFeCo is reduced from 5 nm to 1 nm, the coercive field increases from ≈ 2 mT to ≈ 6
mT. The Curie temperature of NiFeCo was measured to be >400 K, so our NiFeCo samples
should not require cooling in a field to set their magnetization direction.

6.3.2

Junctions Containing Permalloy

To maximize energy-efficiency in a memory application, it is desirable to use a free layer
whose magnetization direction can be controllably switched by a very low applied field.
Permalloy (NiFe) has very sharp magnetic switching at low magnetic fields (< 2 mT), high
Curie temperature, and other advantageous properties, thus it is standardly used in magnetic
memory technologies such as magnetoresistive random access memory (MRAM). Similarly,
we envision using it as a switchable layer in future controllable JJ-based memory devices.
While NiFe has already been studied by other groups [16, 74], there is no guarantee that we
can safely rely on those previous results. The 0-π transition thickness of NiFe was reported
to be 1.2 nm by Robinson et al. [74] and 2.3 nm by Qader et al [16]. Since we continue to
use NiFe in our spin-valve junctions, it is important for us to characterize the NiFe grown in
118

2500

m/A (

2

J/Tm )

5000

0

1 nm

-2500

1.4 nm
2.5 nm
5 nm

-5000
-10

-5

0

5

10

H (mT)

Figure 6.9: NiFeCo hysteresis loops of unpatterned films with dNiFeCo ranging from 1-5 nm.
Plotted is the sample magnetic moment divided by the area versus the applied field, measured
using SQUID magnetometry. Note that dividing the ordinate by the thickness in nm will
give magnetization in kA/m or emu/cm3 . As dNiFeCo is reduced, the switching field increases
and the squareness decreases. Figure taken from [19].
our lab.
A set of samples with varying NiFe thickeness was fabricated identically as described
in Section 6.2 over the course of several three separte runs by myself, Mazin Khasawneh,
Bethany Niedzielski and Eric Gingrich. The bottom wiring layer sequence consisted of
[Nb(25)/Al(2.4)]3 /Nb(20)/Cu(5)/ NiFe(dF )/Cu(5) /Nb(5)/Au(15), with top-leads of Nb(150)/
Au(10), as described in Fig. 6.2.
Measurements of the resulting area-resistance product in the normal state yielded consistent values of ARN = 5-10 fΩ-m2 . Following the same procedure as before, we then
measured Fraunhofer patterns, shown in Fig. 6.10, by first applying a field of -60 mT to fully
magnetize the nanomagnet, then slowly ramped the field to +60 mT in steps of typically 1.5

119

15

a)

d

NiFe

= 1.2 nm

10
5

20

b)
d

NiFe

= 2.4 nm

10

c

I R

N

(

V)

0

0
2

c)
d

= 3.4 nm

Area = 0.25

1
0
-60

NiFe

-30

0
0

30

m

2

60

H (mT)

Figure 6.10: Fraunhofer patterns of Josephson junctions containing NiFe. Ic RN vs applied
field H, varying dNiFe from (a) 1.2 nm, (b) 2.4 nm, and (c) 3.4 nm. The nominal junction
area in (a) and (b) is 0.5 µm2 , and in (c) is 0.25 µm2 , as evidenced by (c)’s comparatively
wider Fraunhofer pattern. Figure taken from [19].
mT.
In Fig. 6.10 the corresponding fits (lines) show excellent agreement with the fitted data
(solid markers), for the positive (red) and negative (blue) sweep directions. The hollow data
points after Hswitch , whose fits are not plotted for clarity, match well with the Fraunhofer
pattern from where the field is swept in the opposite direction.
The 0-π transition plot for NiFe, extracted from the Fraunhofer patterns, is shown in
Fig. 6.11(a) where Ic denotes the maximum critical current obtained from the Fraunhofer
pattern fits. The transition from a 0 to π-phase state occurs at thickness of dNiFe =1.76 ±
0.05 nm. That thickness is between the values found by Robinson et al. [73,74] and by Qadar
et al. [16]. As in the NiFeCo junctions, the data shown in Fig. 6.11(a) roughly follow an

120

100

10

c

I R

N

(

V)

a)

1
20
15
10

0

H

shift

(mT)

b)

5
0

1

2
d

NiFe

3

(nm)

Figure 6.11: Critical current oscillations and Fraunhofer pattern shifts in Josephson junctions containing NiFe. a) The maximal Ic times RN is plotted versus dNiFe for many samples,
with the error bars determined by the goodness of fit parameters of the individual Fraunhofer
patterns. The data are fit to Eq. (6.5), and the best fit parameters are shown in Table 6.2.
The first minima indicates that at a critical NiFe thicknesses of 1.76 Âą 0.05 nm the junctions
transition between the 0 and π-phase states. b) Despite some scatter the Fraunhofer pattern
field shift Hshift increases with dNiFe . The fit to Eq. (6.6) yields a magnetization of 711 Âą
144 kA/m and no discernible magnetic dead layer (ddead = −0.05 ± 0.55 nm). Figure taken
from [19].

121

exponential decay. We fit the data to the diffusive limit case with Eq. (6.5).
Equation (6.5) fits the data reasonably well for both our NiFe based junctions. Table
6.2 lists the best-fit parameters for the data in Fig. 6.11(a), and shows that, in spite of
the significant scatter in the data, the thickness corresponding to the first 0-π transition
can be extracted with reasonable precision. Similar to the NiFeCo samples, the NiFe-based
junctions appear to have ΞF 1 > ΞF 2 . In the case of NiFe, the values are similar to those found
by Robinson et al. (ΞF 1 = 1.4 nm, ΞF 2 = 0.46). Combining our results on NiFe and NiFeCo
with those of Robinson et al., one might conclude that ΞF 1 > ΞF 2 is a generic condition for
Josephson junctions containing strong transition-metal ferromagnetic materials. That is not
true, however, as the thickness dependence of Ic RN in junctions containing NiFeMo was fit
very well by Eq. (6.5) but with ΞF 1 < ΞF 2 , presumably due either to the very short mean
free path or very short spin diffusion length in that material [77].
In Fig. 6.11(b), for each junction we plot the average of the Hshift values obtained from
Fraunhofer pattern fits in each sweep direction versus the F layer thickness. We fit these
data to Eq. (6.6), with M and ddead as free parameters. The resulting fits for NiFe give M
= 711 ± 143 kA/m and ddead = −0.05 ± 0.55 nm. As previously mentioned, in Fig. 6.8
we plot the average of the switching field Hswitch for the two sweep directions from each
Fraunhofer pattern versus dF .

6.3.3

Junctions Containing Pd97 Fe3

To achieve low switching fields, it is advantageous for the free layer to have low magnetization and low magnetocrystalline anisotropy. Another candidate class of materials that
fulfills these criteria is the Pd1−x Fex alloy system with low Fe concentrations. We present
studies of micron-scale elliptically-shaped Josephson junctions containing Pd97 Fe3 layers of
122

varying thickness.
Dilute PdFe alloys have been known for several decades to have very low magnetocrystalline anisotropy [139], and previous work from our group on the alloy with 1.3% Fe concentration found it to have a spin diffusion length of 9.6 Âą 2 nm [140]. Josephson junctions
containing PdFe with a lower Fe concentration of ≈1% have already been studied by other
groups [13–15] with an eye toward applications in cryogenic memory. In the past our group
has tried using Pd98.7 Fe1.3 as the free layer in controllable spin-triplet Josephson junctions [109], but the poor magnetic switching characteristics of the PdFe led to unreliable
control over the phase state of the junctions. That work provided the main motivation for
studying PdFe alloys with somewhat higher Fe concentrations.
We first characterized the magnetic properties of unpatterned continuous Pd97 Fe3 films
via SQUID magnetometry. Thin films of Nb(5)/Cu(5)/PdFe(dF )/Cu(5)/Nb(5), with thicknesses in nanometers, were deposited via dc sputtering under the same conditions as described
in Section 6.2.
The samples were measured using a Quantum Design SQUID magnetometer at 5 K,
with the applied magnetic field parallel to the film plane. The hysteresis loops of films with
dPdFe = 8-16 nm are shown in Fig. 6.12. The saturation magnetization per unit volume is
nearly constant for the three samples. Plotting the saturation magnetic moment divided by
the sample area versus dPdFe and fitting to a straight line gives a slope which corresponds
to a magnetization of M = 90 Âą 9 kA/m. Meanwhile, the x-intercept ddead =2.8 Âą0.9 nm,
indicates the thickness over which the PdFe has no net magnetization, known as the magnetic
dead layer. Note that these unpatterned films contain many magnetic domains so that the
switching mechanism is governed by domain-wall motion; hence the film results should not
be directly compared to the switching behavior of the nanomagnets in our S/F/S junctions,
123

m/A (

2

J/Tm )

1000

500

0

-500

8 nm
12 nm
16 nm

-1000

-8

-4

0

4

8

H (mT)

Figure 6.12: Hysteresis loops of unpatterned films containing PdFe with thickness spanning
8-16 nm. Plotted is the magnetic moment divided by the sample area versus the applied
field, measured using SQUID magnetometry. For the three samples the magnetization is
approximately constant, M = 90 Âą 9 kA/m. The data are slightly shifted along the field
axis due to a small amount of trapped flux in the solenoid of the SQUID magnetometer.
From the data we extract a magnetic dead layer thickness of ddead =2.8 Âą0.9 nm, discussed
in the text. Figure taken from [20].
discussed later.
In a separate sputtering run we fabricated S/F/S Josephson junctions containing PdFe
using the same techniques described in Section 6.2. The bottom wiring layer consists of the
sequence [Nb(25)/Al(2.4)]3 /Nb(20)/ Cu(5)/PdFe(dF )/Cu(5) /Nb(5)/Au(15), and the top
wiring layer is Nb(150)/Au(10) as shown in Fig. 6.2. The junctions have an aspect ratio of
2.5 and area of 0.5 Âľm2 .
The samples I-V curves were measured at 4.2 K using the same methods as in Sections
6.3.1 and 6.3.1. The sample resistance in the normal state, RN , measured from the I-V
curves and was independently confirmed using a lock-in amplifier. Measurements of the
area-resistance product in the normal state yielded consistent values of ARN = 11 Âą 1
124

50
d

a)

PdFe

= 9 nm

25

0
10
d

PdFe

= 15 nm

5

c

I R

N

(

V)

b)

0
10

c)
d

PdFe

= 24 nm

5
0
-60

-30

0
0

30

60

H (mT)

Figure 6.13: Fraunhofer patterns of Josephson junctions containing PdFe. Critical current
times the normal state resistance, Ic RN , is plotted versus the applied field H, for three
samples with dPdFe equal to (a) 9 nm, (b) 15 nm, and (c) 24 nm. The data before Hswitch ,
the field at which the PdFe magnetization vector reverses direction (solid markers), and
the corresponding fits (lines) to Eq. (6.3) show good agreement for both the positive (red,
dashed) and negative (blue) field sweep directions. The hollow circles are the corresponding
data points after Hswitch . The Fraunhofer patterns display magnetic hysteresis and are
increasingly shifted with larger dF . Figure taken from [20].
fΩ-m2 , an indicator of reproducible high quality interfaces.
To acquire the Fraunhofer patterns in Fig. 6.13, first we fully magnetized the nanomagnet
with an applied field of -60 mT along the long-axis of the elliptical junctions, then ramped
the field to +60 mT in steps of 2.5 mT, measuring Ic at each step.
The data follow the expected Airy function from Eq. (6.3) from the initialization field
up to the beginning of a small field range, Hswitch > 0, during which the ferromagnet
switches the direction of its magnetization vector. Beyond Hswitch the data jump to another
Fraunhofer pattern that is shifted in the opposite direction. The magnetic flux through
125

the junction is given by Eq. (6.4). The Fraunhofer pattern will be shifted by an amount
Hshift = −M dF /(2λL + dF + 2dCu ) along the field axis due to Eq. (6.4). To measure the
magnetic hysteresis, we then swept the applied field in the opposite orientation.
The data prior to the magnetic switching event were fit to Eq. (6.3) with Ic0 and Hshift as
fitting parameters, while w is fixed at the nominal sample width of 0.5 Âľm. In Fig. 6.13, for
both the positive (red) and negative (blue) sweep directions, the corresponding fits (lines)
show excellent agreement with the data (solid markers). The hollow markers denote the
data after Hswitch , and closely correspond to the Fraunhofer pattern in which the field is
swept in the opposite orientation. The excellent nature of the Fraunhofer patterns allow us
to obtain a value for Ic0 , albeit with a larger uncertainty, even when the central peak in the
Fraunhofer pattern is shifted far outside the range of the data as shown in Figs. 6.13(b)-(c).
The nodes in the Fraunhofer pattern nearly approach Ic = 0, indicating a robust SiO barrier
around the junction. The data typically follow the Airy function through zero field before
the relatively sharp magnetic switching event, but for a few samples did not. Therefore, it
appears that not all of the samples are single-domain near zero field.
The switching characteristics of the PdFe layer were maintained even when smaller initialization fields were used. After returning the field to zero, we measured the Fraunhofer
pattern again, sweeping the field from only Âą 5 mT in both directions at finer field steps
of 0.5 mT, as shown in Fig. 6.14 (green and orange points), where we have zoomed-in on
the central peak. It is clear that the junctions switch the direction of their magnetization
over a range of field values. To characterize the magnetic switching we use two parameters:
Hswitch,1 , denoting the beginning of the switching event, is the field at which Ic begins to
deviate from the initial Airy function, and Hswitch,2 , denoting the end of the switching event,
is the field at which Ic joins the corresponding shifted Airy function. Across the range of
126

8
60 to -60 mT
5 to -5 mT
-60 to 60 mT

c

I R

N

(

V)

-5 to 5 mT

6

-6

-3

0
0

3

6

H (mT)

Figure 6.14: Magnetic switching of PdFe Josephson junctions at small applied fields. Ic RN
is plotted versus the applied field H for the same Josephson junction shown in Fig. 6.13(b),
zoomed-in on the central peak. Separate measurements using small initialization fields of
Âą5 mT and finer step size (green and orange data points) show the behavior of the magnetic
switching. The reversal of the PdFe magnetization direction for the two sweep directions
begins at Hswitch,1 = 1.0 mT (orange) and -0.5 mT (green) and ends at Hswitch,2 = 2.5
mT (orange) and -2.0 mT (green). During the switching event the data deviate from the
expected Fraunhofer pattern fit. As the field approaches Hswitch,2 the data converge with
the corresponding measurements from Fig. 6.13(b) where much larger Âą60 mT initialization
fields were used (blue and red points). Lines connect the adjacent finer spaced data for
clarity. Figure taken from [20].
thicknesses studied, on average the junctions began to switch at a very low field |Hswitch,1 | =
0.4 mT with standard deviation 0.6 mT, and completed the switching process at |Hswitch,2 |
= 2.4 mT with standard deviation 0.9 mT. The value of |Hswitch,1 | for PdFe is smaller
than found in Ni81 Fe19 -based junctions of similar construction measured by our group [19],
however |Hswitch,2 | is comparable. The low Fe concentration in the Pd97 Fe3 alloy may give
rise to this gradual switching behavior. Prior work on an alloy with lower Fe concentration,
Pd99 Fe1 , showed that the ferromagnetic behavior of thin films is controlled by the presence
of weakly coupled ferromagnetic clusters [141].
Repeating the measurement at even lower initialization fields (3 mT) sometimes caused
127

irregular and irreproducible changes to Ic and Hshift . We surmise that too low of an initialization field allows domain walls to form within the junction, which disturb the magnetic
switching. Hence, if Pd97 Fe3 layers are used in cryogenic memory, an initialization field of
at least 5 mT would be necessary to reproducibly magnetize the nanomagnet.
In Fig. 6.15(a) we plot Ic RN for many samples of varying ferromagnet thicknesses dF .
The junctions transition from a 0 to π-phase state at the value of dF where the first deep
local minimum occurs. In Fig. 6.15(a) Ic denotes the maximum critical current obtained
from the Fraunhofer pattern fits. The data are of sufficient quality to determine the location
of the the 0-π transition with high precision, as we will show below.
As discussed earlier, theoretical predictions describe the behavior of Ic RN versus dF
as an oscillating function with either an exponential decay for diffusive transport or an
algebraic decay for ballistic transport [57]. Bergeret et al. have shown that ΞF 1 > ΞF 2 is
a persistent feature in the semi-clean limit where ΞF 1 = le , the mean free path [78]. We
find that the diffusive limit agrees best with our Pd97 Fe3 data in Fig. 6.15, after fitting the
points to Eq. (6.5). The best-fit parameters are: V0 = 102 ¹ 12 ¾V, ΞF 1 = 16.2 ¹ 1.4 nm,
ξF 2 = 7.2 ± 0.6 nm, and d0−π = 16.3 ± 0.2 nm, which corresponds to φ = 0.7 ± 0.1. The fits
show that the junctions have ΞF 1 > ΞF 2 , which was also the case for a PdNi alloy studied
previously [76].
In Fig. 6.15(b), we plot the average of Hshift from the Fraunhofer pattern fits for each
sweep direction versus dF . We fit these data to Eq. (6.6), with M and ddead used as fitting
parameters. The fit yields M = 85 Âą 9 kA/m and ddead = -0.6 Âą 2.2 nm. The values of both
the magnetization and the dead layer thickness, obtained from Fig. 6.15(b) are consistent
with those from the SQUID magnetometry data shown in Fig. 6.12.

128

100

10

0

c

I R

N

(

V)

a)

1
20
15
10

0

H

shift

(mT)

b)

5
0

0

5

10

15
d

PdFe

20

25

30

35

(nm)

Figure 6.15: Critical current oscillations and Fraunhofer pattern shifts in Josephson junctions containing PdFe. a) The maximal Ic times RN is plotted versus dPdFe for many
samples, with the error bars determined by the goodness of fit parameters of the individual Fraunhofer patterns. For many data points the uncertainty is smaller than the symbol
size. The minimum indicates the critical PdFe thickness at which the junctions transition
between the 0 and π-phase states. The solid red line is a fit to the data using Eq. (6.5).
b) The Fraunhofer pattern field shift Hshift increases with dPdFe . The blue line is the fit
to Eq. (6.6), which yields M = 85 Âą 9 kA/m and ddead = -0.6 Âą 2.2 nm. Figure taken
from [20].

129

6.3.4

Junctions Containing Ni

We also attempted to carry out a study of S/F/S junctions containing Ni, using the same
fabrication process as described previously. The layer configuration was sputtered without
breaking vacuum as follows, [Nb(25)/Al(2.4)]3 /Nb(20)/ Cu(5)/Ni(dF )/Cu(5)/ Nb(20)/Au(15),
with dF ranging from 1.2 to 3.8 nm in samples from three different sputtering runs. The
samples have top electrode of Nb(150)/Au(10), as shown in Fig. 6.2.
Measuring the samples I-V curves at 4 K, we found the area resistance product of the
samples to be slightly larger than usual with a moderate amount of scatter, A×RN = 15 ± 14
fΩm2 . Measuring the samples in the presence of an applied external field oriented along the
long-axis of the junctions ranging from Âą 60 mT, we found the resulting Fraunhofer patterns
to be of generally poor quality, thought to be attributed to the Ni layer containing multiple
magnetic domains for each of the three junction sizes measured. We subsequently attempted
to magnetize the junctions in a large field of 260 mT along the long axis of the junctions. To
remove any possible flux trapped in the Nb electrodes, we subsequently raised the sample
probe in the dewar until it was just above the liquid helium level, above the Tc of Nb, as
monitored via changes in the sample resistance using a multimeter. After the magnetization
procedure we again measured the Fraunhofer patterns, which improved somewhat, but were
still of less than ideal quality. Over a year later, we found that other groups use even larger
fields (> 400 mT) to magnetize their junctions containing Ni [142].
Despite some scatter in the measured values of Ic RN and a relatively small number of
data points (due in part to low yield from this sample set), we nontheless can map out a
rough picture of how Ic RN decays with thickness, shown in Fig. 6.16. Perhaps the most
important observation is that the Ic RN values are quite large in comparison to most of

130

Figure 6.16: Critical current oscillations in Josephson junctions containing Ni. The maximal Ic times RN is plotted versus dNi for many samples, with the error bars determined by
the goodness of fit parameters of the individual Fraunhofer patterns. The data are compatible with a minima between a Ni thickness of 3.0-3.6 nm, as measured later by groups at
NIST [143] and Northrop Grumman Corporation [23]. Overall, Ic RN does not appear decay
substantially with increasing Ni thickness.
the other materials presented in this chapter. Also, the critical current in the Ni junctions
appears to decay quite slowly with the Ni thickness. Although the magnetic properties
evidenced in the Fraunhofer patterns are of less than desired quality, Ni appears to be one of
the rare ferromagnetic materials that both switches at a large magnetic field and transmits
large critical currents in Josephson junctions. While the data is by no means definitive it is
compatible with a minima between a Ni thickness of 3.0-3.6 nm, which was later measured
by groups at the National Institute of Standards and Technology (NIST), Boulder [143] and
Northrop Grumman Corporation [23]. Given the large scatter we did not fit the data in
Fig. 6.16 to the previously discussed theoretical predictions.

131

6.4

Conclusion

This section provides a study of the magnetic and transport behaviors of S/F/S Josephson junctions containing the ferromagnets NiFeCo, NiFe, PdFe, and Ni. Systematic studies
of the material properties of such junctions, including the 0-π transition thickness, switching fields, and shifts in the Fraunhofer patterns versus F -layer thickness are crucial for the
advancement of cryogenic memory technologies. To develop phase-controlled S/F/N/F 0 /S
memory devices one would need to fix either the F or F 0 layer to a thickness near it’s respective 0-π transition, while the other layer could be kept much thinner to avoid substantial
decay in Ic . Then, by tuning the relative orientation of the F and F 0 magnetization vectors
between parallel and anti-parallel configurations, the junction can be controllably toggled
between the π and 0 phase states [18].
We have shown that NiFeCo is a potential candidate for such a fixed layer. If positioned
near the first 0-π transition at 1.2 nm, it has a reasonable switching field, Hswitch ≈ 7 mT,
which is somewhat larger than NiFe, but not as unwieldy as Ni (260 mT). However, from
Fig. 6.8, it is not clear if the relatively small difference in the switching fields of NiFeCo
and NiFe would allow them to be successfully used together as the fixed and free layers in a
single device. One could attempt to use another material with a lower switching field than
NiFe, or possibly tune the relative concentrations of Ni, Fe, and Co to optimize the switching
characteristics for the situation desired.
In addition we have studied the magnetic and transport behavior of Josephson junctions
containing Pd97 Fe3 . If used as a “free” magnetic layer in cryogenic memory, Pd97 Fe3 is
advantageous in that its 0-π transition occurs at a thickness of ≈16.5 nm, much greater
than for NiFe, making Pd97 Fe3 much less sensitive to small thickness variations. Meanwhile,

132

junctions with Pd97 Fe3 maintain a relatively low switching field |Hswitch,2 | = 2.4 mT (with
standard deviation 0.9 mT). As a “free” layer Pd97 Fe3 has some disadvantages– the magnetic
switching can occur over a range of fields, possibly due to the existence of weakly coupled
ferromagnetic clusters. For reproducible magnetic switching, the junctions had to be magnetized at an initialization field of 5 mT or greater. Preliminary magnetometry measurements
on Pd1−x Fex films with x = 7-13 % suggest that those alloys with higher Fe concentration
may have better magnetic properties. In the future, we plan to test the behavior of those
alloys in Josephson junctions.
Looking forward, the addition of extra magnetic layers could pose a number of new
complications: i) The surface roughness may grow with the number of layers; one would
need to ensure that both the F and F 0 layers remain single domain. ii) An additional spacer
layer would be required between the F and F 0 layers to keep them magnetically decoupled.
As shown by Heim et al. [138], spacer layers could cause modifications to the precise value of
dF at which the 0-π transition occurs. iii) Magnetostriction, edge roughness and other effects
should be minimized so that the magnetic switching behavior of the junction is dominated
primarily by shape anisotropy.

133

Chapter 7

Controllable Phase SQUIDs: Spin-Singlet

Here we discuss the first experimental demonstration that the phase state of a Josephson
junction can be controllably switched between 0 and π, i.e. a phase-sensitive cryogenic
one-bit memory device [18]. That feat was achieved by adding two ferromagnetic Josephson
junctions to a superconducting loop to form a Superconducting QUantum Interference Device
(SQUID), which is sensitive to the relative phase between the two junctions. While the
sample fabrication and measurement of these samples was undertaken primarily by Bethany
Niedzielski and Eric Gingrich, the former whose dissertation [144] explains the experiment
and the development thereof in greater detail, my main contribution to that effort was to
extract from the data unambiguous proof that the junction(s) had indeed switched between
the 0 and π phase states. As described below, this confirmation required that the data
from the various states of the junctions be self-consistently fit to a general SQUID theory
accounting for asymmetries in both the critical current and the geometric inductance on
either side of the loop.
To avoid treading over the same ground, after outlining the experimental approach and
relevant SQUID physics, in this chapter we focus mainly on the methods of the data analysis. The techniques described here will lay the groundwork for Chapter 9, where they
will be utilized again to analyze controllable phase-sensitive SQUIDs that carry spin-triplet
supercurrent.

134

Figure 7.1: Scheme to control the phase-state of a spin-valve Josephson junction. The critical
current will oscillate as a function of the phase accumulated across the junction φ; the
ground-state phase of the junction can be either 0 or π. If the layer thicknesses of the two
ferromagnets are properly adjusted, the ground state phase of the junction will be π if the
two layers are parallel (φP = φ1 + φ2 ), where φ1 and φ2 are the phases accumulated in
each of the two ferromagnetic layers respectively, and 0 if the two layers are antiparallel
(φAP = φ1 − φ2 ). In this simplified picture, the decay of the supercurrent versus the
ferromagnetic layer thickness is ignored. Figure taken from [18].

7.1

Spin-Valve Josephson Junctions and SQUIDs

Consider a Josephson junction, similar to those discussed in Chapter 6, but with two different ferromagnetic layers, F1 and F2 whose magnetizations are in-plane and either parallel
or antiparallel with respect to one another, arranged in the configuration S/F1 /N/F2 /S [12,
16, 17, 145], called a spin-valve Josephson junction. As discussed in Ch. 4.3, in a ferromagnetic Josephson junction the pair correlation function will oscillate and decay as a function
of the thickness of a ferromagnet. The pair correlation function will accumulate a total phase

135

shift of,

φ = φ1 ± φ2 ,
(7.1)
= (Q1 × dF1 ) ± (Q2 × dF1 ),
where φ1 = Q1 × dF1 and φ2 = Q2 × dF1 are the individual phase shifts acquired from
traversing the F1 and F2 layers. In Eq. (7.1), the thicknesses of F1 and F2 are dF1 and dF2 ,
respectively, and Q1 and Q2 are related to the center-of-mass momentum of the spin pairs,
given by,

↑

↓

↑

↓

Q1 = (kF − kF )
1
1
Q2 = (kF − kF ),
2
2
↓

↑

where kF

1,2

and kF

1,2

(7.2)
(7.3)

are the Fermi wavevectors in the majority and minority bands, re-

spectively. The presence of the minus sign in Eq. (7.1) originates from the fact that if the
magnetizations of F2 and F1 are antiparallel, then from one ferromagnet to the next the
majority and minority bands will reverse their roles, leading the pair correlation function to
acquire the opposite phase, −φ2 .
Controlling the phase state of these types of junctions depends crucially on the precise
layer thicknesses of F1 and F2 . As shown schematically in Fig. 7.1, if we choose the thickness
of F1 such that φ1 is close to π/2 and the thickness of F2 such that φ2 ≤ π/2, then when
the layers are parallel, φ = φP = φ1 + φ2 , the junction is in the π-state, whereas when
the layers are antiparallel, φ = φAP = φ1 − φ2 , the junction is in the 0-state. Or, in the
language of Ch. 6, we need to fabricate the two ferromagnetic layers such that one is near
its 0-π transition thickness while the other is far away from its 0-π transition thickness. It
136

is because of this restriction that we went to such lengths to characterize the 0-π transition
thicknesses of the various ferromagnetic materials in Ch. 6.
The free layer in both junctions was chosen to be Ni81 Fe19 (Py) of thickness 1.5 nm,
which switches at a low field and puts the junction close to the 0-π transition, as indicated
by Fig. 6.11 [19]. The fixed layer in both junctions was chosen to be Ni of thickness 1.2 nm,
which according to Fig. 6.16, is away from its 0-π transition thickness. Moreover the Ni should
not significantly decay the critical current and should add or subtract a small component to
the phase [17, 69, 73]. The ferromagnetic layers are prevented from coupling magnetically by
a Cu spacing layer. The full structure of the junctions is: [Nb(40)/Al(2.4)]3 /Nb(20)/Cu(5)/
NiFe(1.5)/Cu(10)/ Ni(1.2)/Cu(5)/Nb(20)/Au(15)/ Nb(150)/Au(10). For all other details
about the sample fabrication, including work on the development and study of smooth normal metal spacers, studies regarding the smoothness of the bottom Nb/Al electrodes, and
how those layers effect the magnetic properties of the ferromagnetic layers, see Bethany’s
dissertation [144].
While only a single junction is required for the phase change to occur, to observe such
a change requires a phase-sensitive measurement. One way to accomplish that is to set up
an interference experiment between two spin-valve junctions in a SQUID loop, as shown in
Fig. 7.2. The SQUID loop was fabricated to have inner dimensions of 10 µm × 10 µm, and
the wire leads composing the loop were 5 Âľm wide. The junctions was fabricated to have
elliptical cross-sections, each with different aspect ratios (2.2 and 2.8), to help control the
coercive fields of the magnetic layers via shape anisotropy. In this way the more eccentric
junction has a larger coercive field than the less eccentric junction.
The idea is to set up the following scenario: The sample is initialized by applying an
in-plane magnetic field using a solenoid coil around the sample to align the magnetizations
137

flux
flux

set
set

set

set

set

Figure 7.2: Spin-valve SQUID schematic and the configuration of the four magnetic states.
The circuit layout for the asymmetric SQUID (top) is composed of a superconducting loop
intersected by two spin-valve Josephson junctions (small yellow ellipses). The critical current
in the SQUID (probed by the measurement current Is ) oscillates as a function of the flux
inside the loop, provided in part by a current Iflux through a nearby superconducting fluxline to add a field Bflux to the loop. The magnetic state of the junctions is controlled via
an external in-plane magnetic field Hset . The (bottom) diagrams show the four possible
orientations of the F1 and F2 magnetizations, whose configuration sets the phase across the
junctions to be either 0 or π. The junctions are elliptical in shape, one with a smaller aspect
ratio (JJ-1) and the other, with a larger aspect ratio (JJ-2), each with area 0.5 Âľm2 . The
length of the purple arrows indicate the magnitude of Hset for each state. Figure modified
from [18].
138

of all the magnetic layers in both junctions. Next, a small set field Hset is applied in the
reverse direction, which will cause the “free”-layer in the less eccentric junction to reverse
its magnetization direction relative to that of the “fixed”-layer. If Hset is not too large, then
both magnetic layers in the more eccentric junction will remain parallel. The net result is a
relative total phase shift of π between the two junctions in the SQUID.
The phase shift between the two magnetic states was detected by adding flux to the
SQUID loop, accomplished by driving a small current Iflux through a nearby superconducting
line, which we refer to as the “flux-line.” The total critical current of the SQUID oscillates as
a function of the applied flux, as shown in Fig. 7.3, allowing for the detection of phase shifts
by comparing the curves before and after the magnetic switching. The sudden phase shifts
in Fig. 7.3 are also generally accompanied by sudden shifts in the critical current (mainly
because the P and AP states do not have perfect alignment about φ = π/2, as in Fig. 7.1).
After applying Hset the external field is always returned to zero before measuring the
SQUID oscillations versus Iflux . Further increasing Hset will eventually cause the Py layer
in the more eccentric junction to also reverse its magnetization, returning the total phase
shift in the SQUID to 0. Reversing the direction of the set field, one can then trace out a
full magnetic major-loop, as shown in Fig. 7.3.
From a careful inspection of the current paths through the SQUID in Fig. 7.2, one will
notice that the path length traversed by the current in the two arms the SQUID is not equal.
Current exiting one of the junctions into the straight bottom electrode must then travel
(in the bottom electrode) between the “fork” opening in the top electrode, whereas current
coming from the other junction does not. This leads to an asymmetry in the geometrical
inductance of the two arms of the SQUID that distorts the resulting oscillations of the critical
current versus the applied flux.
139

Moreover, the magnitude of the critical current may divide in an asymmetric manner
through the two junctions. These two features were not fully appreciated by our group
until high-resolution SQUID data was obtained and the shape of the resulting oscillations
was investigated. As it turns out, the two effects mentioned above complicate the analysis
by contributing new “effective” phase shifts to the resulting SQUID oscillations, but can be
properly accounted for using the theory that follows.

140

mT

.

se

t

.
.
.

Îź

0

flux

T

se
t

.

m

.

Îź

0

flux

Figure 7.3: Oscillations in the critical current versus the current applied to the flux-line and
the in-plane set field used to control the magnetic state of the SQUID. After initializing all of
the magnetic layers in the SQUID to have parallel alignment using a large negative in-plane
field, next a small positive set field Hset is momentarily applied to the sample. Current is
then applied to the flux line, causing the positive critical current Ic+ to oscillate versus the
applied flux, corresponding to the lower purple colored state in (a). The SQUID oscillations
are always measured in zero external field. Further increase of Hset will cause the free
magnetic layer in one of the junctions to change its magnetization, resulting in a π-phase
shift relative to the other junction, the green colored state in (a). At even larger values
of Hset , the free layer in the second junction will flip the direction of its magnetization
and a second phase shift occurs, the red colored state in (a). Reversing the direction of
Hset , one can switch the two junctions back to their original orientations, as shown in (b),
completing a magnetic major-loop. The sudden jumps in the relative phase correspond to
the four magnetic states, whose layer configurations are shown in Figure 7.2. The sudden
phase changes are also accompanied by changes in the magnitude of Ic+ . Figure modified
from [18].
141

7.2

Theory of the Asymmetric DC SQUID

We now return to the theory of the DC SQUID, but consider it in a more general sense,
one that accounts for asymmetries in both the critical currents in the two junctions and in
the geometric inductance of the two arms of the SQUID loop. We follow and extend upon a
clever method first described by W.T. Tsang and T. Van Duzer in 1975 [146]. Consider the
SQUID shown in Fig. 7.2, containing two junctions, which we assume are small enough to
contain only a small fraction of a flux quantum. As in Eq. (2.54), the total current in the
SQUID is the sum of the currents in the two arms according to the Josephson effect,

I(φ1 , φ2 ) = I1c sin(φ1 ) + I2c sin(φ2 ),

(7.4)

where φ1 and φ2 are the gauge-invariant phase differences across Josephson junctions 1 and
2, respectively, with I1c and I2c their critical currents. The phase differences are subject to
the constraint of Eq. (2.51),
φ2 − φ1 =

2π
Φ (mod 2π).
ÎŚ0

(7.5)

The goal is to determine the dependence of the total current in the SQUID on an externally
applied flux.
As we saw previously in Eq. (2.52), the total magnetic flux ÎŚ in the SQUID is the sum of
the contributions from an externally applied magnetic field, ÎŚa , and from the self-inductance
L of the loop,
ÎŚ = ÎŚ a + ÎŚs .

(7.6)

Generalizing to an asymmetric inductance, we define L1 and L2 to be the inductances of
the two arms of the SQUID, such that L = L1 + L2 . Thus, we rewrite the second term in
142

Eq. (7.6) to be
Φs (φ1 , φ2 ) = L1 Ic1 sin(φ1 ) − L2 Ic2 sin(φ2 ).

(7.7)

The phase difference between the two junctions, Eq. (2.51), becomes

φ1 − φ2 = θa + α1 sin(φ1 ) − α2 sin(φ2 ),

(7.8)

where for convenience we have defined three parameters corresponding to the applied flux,
θa , and the flux generated by the flow of the two currents, ι1 and ι2 , defined to be:
2π
ÎŚa ,
ÎŚ0
2π
α1 ≡ L1 Ic1 ,
ÎŚ0
2π
α2 ≡ L2 Ic2 .
ÎŚ0
θa ≡

(7.9)
(7.10)
(7.11)

The problem of finding how the current I(φ1 , φ2 ) relates to the external flux, θa , amounts to
finding the maxima and minima of Eq. (7.4) with respect to the constraint imposed on φ1
and φ2 by Eq. (7.8). The solution proposed by Tsang and Van Duzer is to apply the method
of Lagrange multipliers by introducing a new parameter Îť to Eq. (7.4),

I(φ1 , φ2 , λ) =I1c sin(φ1 ) + I2c sin(φ2 )
(7.12)
+ λ[φ2 − φ1 + θa + α1 sin φ1 − α2 sin φ2 ],
which is permissible as the term in square brackets, from Eq (7.8), is equal to zero. We
then solve for the critical points of Eq. (7.12) with respect to φ1 , φ2 , and λ. By finding
the partial derivatives ∂I/∂φ1 , ∂I/∂φ2 , ∂I/∂λ, and setting them each to zero, the resulting

143

three equations can be used to eliminate Îť, leaving us with two equations that relate the
critical values of φ1 and φ2 (denoted with subscript c’s):

φ2c = cos−1




−1
,
Ic2 /(Ic1 cos(φ1c )) + α

θa = φ2c − φ1c + α2 sin φ2c − α1 sin φ1c ,

(7.13)

(7.14)

where Îą = Îą2 + Îą1 (Ic2 /Ic1 ). To obtain the critical current at a fixed value of the applied
flux, θa , one needs to find the set of values (φ1c , φ2c ), that satisfy Eqs. (7.13) and (7.14). In
order to quickly find the critical current over the entire θa axis, first we solve for all of the
(φ1c , φ2c ) pairs that satisfy Eq. (7.13): a locus of solutions in the φ1 –φ2 plane. Examples
of these solutions are shown in the left panel of Fig. 7.4. The shape of the locus of solutions
varies depending on the input physical parameters Ic1 , Ic2 , Îą1 , Îą2 , forming either discrete
loops or continuous oscillations in the φ1 –φ2 plane due in part to the argument of the cos−1
in Eq. (7.13) being bounded by ±1, discussed later. Next, we insert the (φ1c , φ2c ) pairs into
Eq. (7.14) to find θa , while the same pairs are correspondingly inserted into Eq. (7.4) to find
Ic (θa ). This results in the curves shown in the middle panel of Fig. 7.4. The perodic SQUID
oscillations shown in the right pannel of Fig. 7.4 are obtained by shifting the Ic (θa ) solutions
by integer multiples of π and finding the intersections of the resulting curves.
Closer inspection of Eq. (7.13) reveals that for certain values of the parameters Îą and
the ratio Ic2 /Ic1 , the locus of solutions will become an open continuous curve along the φ1
or φ2 axis, such as in Fig. 7.5, when either one of the following two conditions is satisfied:

(Ic2 /Ic1 + α ≤ 1)
144

(7.15)

Figure 7.4: Asymmetric SQUID theory, closed solutions. The left panel shows the locus of
solutions to Eq. (7.13), where the independent variable is taken to be either φ1 (red) or φ2
(blue). For the set of the input parameters, listed below, the locus of solutions are discrete
closed rings in the φ1 –φ2 plane. Inserting the (φ1 , φ2 ) pairs into Eq. (7.14) and Eq. (7.6),
yields solutions for the critical current Ic versus the flux applied to the SQUID, θa , shown
in the middle panel. Shifting the I(θa ) solution by integer multiples of π, and finding the
intersections of the resulting curves, and restoring the physical units, determines the SQUID
oscillation curves versus the applied flux current, Iflux , shown in the right panel. Units of the
flux current are in mA for a SQUID where Iflux = 1.7 mA puts a flux of ÎŚ0 in the SQUID
loop. Input parameters (top): Ic1 = 1.1, Ic2 = 0.9, Îą1 = 1.209, Îą2 = 0.989, (bottom):
Ic1 = 0.748, Ic2 = 1.252, Îą1 = 0.205, Îą2 = 0.344.
(Ic2 /Ic1 − α ≥ 1).

(7.16)

The nature of these “open” solutions presents a minor additional complication. For instance, as φc1 → 0, the cos−1 will become imaginary when the condition (Ic2 /Ic1 ) + α < 1
is satisfied. In this case, we rearrange Eq. (7.13), solving for φ1c ,

φ1c = cos−1




−(Ic2 /Ic1 )
,
(1/ cos(φ2c )) + α)

145

(7.17)

and take φ2c to be the independent variable. Likewise, as φc2 → 0, Eq. (7.17) will become
imaginary when (Ic2 /Ic1 ) − α > 1, so we use Eq. (7.13), where φ1c is the independent
variable. What’s more is that very close to the boundary of either of the two conditions
(Ic2 /Ic1 ) ± α ≈ 1 the solution for Ic (θa ) does not increase monotonically. Figure 7.5 shows
the shape of the locus of solutions and the resulting solutions for the SQUID oscillations for
all the cases of the open solutions and near the edge of the discontinuity.
From this general construction, we can analyze several important features present in the
SQUID ocillation curves. As previously discussed in Ch. 2, the depth of the modulation
decreases as the total inductance or critical current in the SQUID increases, defined by the
screening parameter, which now becomes,

βL =

(L + L2 )(Ic1 + Ic2 )
LIc
= 1
.
ÎŚ0
ÎŚ0

(7.18)

In addition, the asymmetries due to the inductance and critical current cause the curves to
have several new features. The inductance asymmetry distorts the oscillations into a tilted
“ratchet”-like shape. The extent of the skewness increases according to the parameter

ÎąL =

L2 − L1
.
L2 + L1

A ratchet leaning to the right indicates that the SQUID has ÎąL > 0, or L2 > L1 .

146

(7.19)

Figure 7.5: Asymmetric SQUID theory, open solutions. Similar to Fig. 7.4, but using input
parameters, listed below, which result in the locus of solutions in the φ1 –φ2 plane that are
continuous (“open”) curves of either φ1 or φ2 . Close to the cross over point between open
and closed solutions, as shown in the 2nd and 4th rows, the solutions for Ic (θa ) do not
always increase monotonically with θa , as mentioned in the text. Nonetheless the SQUID
oscillations can still be computed. Input parameters: (1st row): Ic1 = 0.682, Ic2 = 1.318,
Îą1 = 0.157, Îą2 = 0.317; (2nd row): Ic1 = 0.748, Ic2 = 1.252, Îą1 = 0.200, Îą2 = 0.336; (3rd
row): Ic1 = 0.682, Ic2 = 1.318, Îą1 = 0.157, Îą2 = 0.317; (4th row): Ic1 = 1.48, Ic2 = 0.52,
Îą1 = 0.046, Îą2 = 0.016. Units of the flux current are in mA, for a SQUID where Iflux = 1.7
mA puts a flux of ÎŚ0 in the SQUID loop.

147

Moreover, the critical current asymmetry causes the oscillations for the positive (Ic+ )
and negative (Ic− ) critical current curves to become shifted with respect to one another in
opposite directions along the flux axis, according to the parameter†

I − Ic1
.
ÎąI = c2
Ic2 + Ic1

(7.20)

The origin of those shifts can be understood from the following description, which was first
pointed out to us by our collaborator, Don Miller from Northrop Grumman Corporation.
Consider the peak value of the critical current, from Eq. (7.4), I = Ic1 + Ic2 , which occurs
when the phase across each junction is φ1 = φ2 = π/2. The maximum self-induced flux
peak,+

through the SQUID loop, from Eq. (7.7), is ÎŚs

= L2 Ic2 − L1 Ic1 , a quantity that must

be balanced by the externally applied flux. Thus, the peak of the maximum positive critical
peak,+

current Ic+ occurs at a flux Φpeak,+ = −Φs

. On the other hand, the maximum negative

critical current occurs at a flux Φpeak,− = −Φpeak,+ , and hence the positive and negative
critical current curves are shifted with respect to each other by an amount

∆Φpeak ≡ Φpeak,+ − Φpeak,− = 2(L2 Ic2 − L1 Ic1 ).

(7.21)

Since the critical current is periodic with respect to flux the relative phase shift, ∆Φpeak ,
can only be determined modulo ÎŚ0 . However, it is still possible to determine which value
of ∆Φpeak corresponds to the physically relevant value by self-consistently fitting the data
from the four magnetic states. From the best-fit parameters one can then observe that
after a magnetic switching event, one of the junction’s critical currents will change, while
the other’s remains approximately constant. Finally, between the four magnetic states one
† The value of α can also cause changes in the depth of the modulation.
I

148

should extract a self-consistent set of parameters for the inductances L1 and L2 , which are
set by the sample geometry. In general, if one finds the wrong value of ∆Φpeak , it typically
results in an unphysical value of L1 and L2 .
These considerations allowed us to write a generalized nonlinear least-squares SQUID
fitting program using the Mathematica software package that that can quickly compute the
solutions for any combination of input parameters. For the shift, we define a parameter φshift
that describes how far the nearest peak of the Ic+ curve is shifted away from zero. In the
fitting program we work in terms of the dimensionless set of parameters just described (βL ,
αL , αI , φshift ), but one can equivalently use the set (Ic1 , Ic2 , α1 , α2 ). For reference, the
conversion between the two sets of parameters is given by:

peak,+

with peak shift, ÎŚs

α = πβL (1 + αI ),

(7.22)

α1 = πβL (1 − αL )(1 − αI )/2,

(7.23)

α2 = πβL (1 + αL )(1 + αI )/2,

(7.24)

= L2 Ic2 − L1 Ic1 = βL (αL + αI )Φ0 /2.

The code can perform a simultaneous global fit of both the positive (Ic+ ) and negative
(Ic− ) critical current data from the experiment. The user can choose, depending on the
nature of the data, to use any number of the parameters listed above either as fixed inputs
or as free parameters to be optimized in the fit, and find the standard errors on each of the
free parameters. Using the code, the fitting of typical SQUID oscillation data takes between
2-10 minutes depending on the step size and accuracy desired (and the specifications of the
machine used).

149

Table 7.1: Best-fit parameters for the four magnetic states using the asymmetric SQUID
theory. The uncertainties are extracted via the “NonlinearModelFit” routine in Mathematica,
and do not account for possible correlations between the fitting parameters. The difference
between the individual phase shifts φshift for adjacent rows in the table is ≈ ±π (the values
in the last two columns are listed in units of 2π). For convenience we define a global phase
shift, φglobal = -0.1271, which is the average of the values (φshift − 0.5) in the (0-π) and (π-0)
states.
State

βL

ÎąL

ÎąI

Ic+ (ÂľA)

|Ic− | (µA)

φshift

φshift −
φglobal

(π-π)

4.207 Âą 0.023

0.330 Âą 0.006

-0.147 Âą 0.006

506.0 Âą 0.2

510.7 Âą 0.3

-0.1195 Âą 0.0007

0.0076

(0-π)

6.296 Âą 0.030

0.335 Âą 0.005

-0.472 Âą 0.005

771.2 Âą 0.2

762.8 Âą 0.2

+0.3517 Âą 0.0006

0.4788

(0-0)

8.198 Âą 0.033

0.345 Âą 0.004

-0.150 Âą 0.004

985.5 Âą 0.2

987.8 Âą 0.2

-0.1360 Âą 0.0005

-0.0089

(π-0)

5.957 Âą 0.020

0.339 Âą 0.004

+0.176 Âą 0.004

707.7 Âą 0.2

719.6 Âą 0.2

+0.3955 Âą 0.0004

0.5226

7.3

SQUID Oscillation Data: Comparison to Theory

For the same SQUID shown in Fig. 7.3, higher resolution data in each of the four magnetic
states was measured and fit using the general SQUID theory described above. As shown in
Fig. 7.6(a), the theoretical fits for the positive (Ic+ ) and negative (Ic− ) critical current
oscillations versus the applied flux show excellent agreement with the measured data.
Those oscillations clearly show the tilted ratchet shape of the curves due to the inductance
asymmetry of the SQUID.† The effective phase shifts between the curves make it hard to
discern by eye whether or not a relative 0 or π-phase shift has occurred between two of the
magnetic states, but can be unequivocally extracted from the best-fit parameters.
As shown in Table 7.1, the analysis produces only one consistent set of best-fit parameters, with rather small error bounds. When these variables are converted into the physical
parameters for the inductances and the critical currents, shown in Table 7.2, the values are
in agreement with those expected from the design geometry based on simulations using the
† We verified that rewiring the sample, so that the flux current flows in the opposite direction, reverses
the direction that the ratchet leans, as expected by the theory.

150

Table 7.2: Inductances and critical currents extracted from the least-squares fits. After fitting
the four magnetic states with the asymmetric SQUID theory, the best fit parameters from
Table 7.1 were converted into the physical parameters below. Note that the two inductances
in the two arms of the SQUID L1 and L2 are very consistent between the four states. Also,
between adjacent states in the table the critical current noticeably changes in the junction
whose magnetic state was modified, whereas in the other junction the critical current is
approximately constant.
State

L1 (pH)

L2 (pH)

Ic1 (ÂľA)

Ic2 (ÂľA)

(π-π)

5.73 Âą 0.05

11.38 Âą 0.08

291.6 Âą 1.5

216.8 Âą 1.5

(0-π)

5.64 Âą 0.04

11.33 Âą 0.07

564.5 Âą 1.8

202.5 Âą 1.8

(0-0)

5.63 Âą 0.03

11.56 Âą 0.06

567.3 Âą 2.0

419.3 Âą 2.0

(π-0)

5.71 Âą 0.03

11.56 Âą 0.05

294.0 Âą 1.3

419.7 Âą 1.3

FastHenry software package. The most important finding of the fitting analysis is that the
phase shifts between the adjacent magnetic states in the Table 7.1 all differ by ≈ ±π.
Another way to better visualize the relative phase shifts between the magnetic states
avg

is to compare the plots of their average critical current Ic

= (Ic+ − Ic− )/2 versus the
avg

applied flux, as shown in Fig. 7.6(b). The particular shape of the Ic

curves is determined
avg

by the misalignment between the Ic+ and Ic− curves, but clearly shows that the Ic
transition between maxima and minima for each of the four magnetic states.

151

curves

mT

mT

mT

.

flux

mT

flux

Figure 7.6: Using asymmetric SQUID theory to fit the SQUID oscillation data from the four
magnetic states. In (a) the measured positive and negative critical current data (Ic+ and
Ic− ) are plotted versus the flux-line current, Iflux , for the four magnetic states. The states
are labeled π − π, 0 - π, etc., according to the phase states of JJ-1 and JJ-2, as in Fig. 7.2.
The lines are the theoretical least-squares fits to the data. In b) the average critical current,
avg
Ic = (Ic+ − Ic− )/2 is plotted versus the applied flux to more clearly see the relative π
phase-shifts between the four magnetic states. The solid lines in (b) are determined by the
best-fit curves from (a). At a fixed value of the flux current (dashed line) one can see that
the states transition from maxima to minima. The labels denote the switching field for each
magnetic state. Figure modified from [18].

152

Chapter 8

Spin-Triplet Josephson Junctions Containing a PMA
SAF

In this chapter we present measurements of Josephson junctions containing three magnetic layers with noncolinear magnetizations carrying spin-triplet supercurrent [21]. The
junctions are of the form S/F 0 /N/F/N/F 00 /S, where S is superconducting Nb, F 0 is either
a thin Ni or Permalloy layer with in-plane magnetization, N is the normal metal Cu, F is a
synthetic antiferromagnet (SAF) with magnetization perpendicular to the plane, composed
of Pd/Co multilayers on either side of a thin Ru spacer, and F 00 is a thin Ni layer with inplane magnetization. The supercurrent in these junctions decays more slowly as a function
of the F -layer thickness than for similar spin-singlet junctions not containing the F 0 and F 00
layers. The slower decay is the prime signature that the supercurrent in the central part
of these junctions is carried by spin-triplet pairs. The junctions containing F 0 = Permalloy
are suitable for experiments, such as those discussed in Ch. 9, where either the amplitude of
the critical current or the ground-state phase difference across the junction is controlled by
changing the relative orientations of the magnetizations of the F 0 and F 00 layers.
As we discussed in Ch. 4, the theory of spin-triplet Josephson junctions predicts that the
ground-state phase difference across a junction of the form described above depends on the
relative orientations of the three magnetizations [80–82, 105, 147, 148]. Spin-triplet junctions
where the magnetization of all three ferromagnetic layers are coplanar exhibit complementary
0 and π-phase states dependent only on whether the outer two magnetizations are parallel
153

or antiparallel.
In principle, there are many ways to design a Josephson junction to test that prediction.
We have focused on designs where the magnetizations of the outer F 0 and F 00 layers both
lie in-plane; the junctions are given an elliptical shape to set the directions of those in-plane
magnetizations by shape anisotropy. But the elliptical junction shape makes it difficult to
achieve non-colinear magnetization in the central F layer, if it is also in-plane. A solution is to
use out-of-plane magnetization for F , which is easily accomplished using a magnetic material
with strong perpendicular magnetic anisotropy (PMA) [106]. Then, one can utilize shape
anisotropy to preferentially orient the magnetization direction of the F 0 and F 00 layers, all the
while preserving the optimal 90 degree relative magnetization angle between each successive
ferromagnetic layer. Previous efforts by our group, primarily undertaken by Eric Gingrich, to
detect the Josephson junction phase change using such a design were only partially successful,
however [109]; while a π phase change appeared in some experiments, the magnetic behavior
of the junctions was poor and irreproducible. We suspected at the time that stray fields
from the domain walls in the PMA F layer penetrated the F 0 and F 00 layers and ruined their
magnetic properties. A possible solution to that problem is to replace the central PMA layer
with a PMA synthetic antiferromagnet (SAF), in which each magnetic domain in the lower
half of the SAF is coupled to a domain with opposite-pointing magnetization in the upper
half of the SAF. Such a system should produce minimal stray fields in the F 0 and F 00 layers
that are located above and below the SAF [149].
The main result of this chapter is that the critical current in these junctions decays more
slowly with increasing thickness of the PMA SAF than it does in junctions that do not
contain the F 0 and F 00 layers. That result represents strong evidence that the supercurrent
in the central part of these junctions is carried by spin-triplet pairs, whereas it is carried
154

only by the short-range components in the control samples. The results of this chapter lay
the groundwork for Ch. 9, where the spin-triplet Josephson junctions presented here will be
implemented into phase-controllable SQUID devices.

8.1

Magnetic Properties of Pd/Co Multilayers and Synthetic AntiFerromagnets With Perpendicular Magneto-Anisotropy

Thin multilayers of Pd and Co can be grown to have perpendicular magnetic anisotropy,
i.e. with magnetization perpendicular to the sample plane, as reported by Chang et al. [150].
Moreover, when two such Pd/Co multilayers are separated by a thin normal metal spacer
(such as Ru, Rh, Ir, or Cu), they may couple via the exchange interaction to form a SAF,
in which the magnetizations on either side of the spacer align in an antiparallel fashion.
The outstanding PMA and SAF properties of such layers have attracted interest for their
use in spin-transfer-torque magnetic random access memories [151] and other applications.
Spurred by the initial suggestion of Mike Schneider at the National Institute of Standards
and Technology, Boulder, we decided to investigate if PMA SAFs could be advantageous for
spin-triplet Josephson junctions with phase control. For the PMA SAF to serve as the central
F layer and optimize the generation of long-range spin-triplet supercurrent its magnetization
needs to remain pinned perpendicular to the sample plane over the range of measurement
fields used in the experiments.
To characterize the Pd/Co multilayers and verify that they have PMA, we sputtered films
of: Nb(5)/Cu(5)/[Pd(dPd )/Co(0.3)]n /Pd(dPd )/Cu(5)/Nb(5), where the layer thicknesses in
nanometers are shown in parentheses and the sequences in brackets are repeated n = 10
or 20 times. Similar to Chang et al. [150] we fix the Co thickness to 0.3 nm while the Pd

155

m area

m area

Figure 8.1: Hysteresis loops of a Pd/Co multilayer film measured at 5K using a SQUID
magnetometer. The data are expressed as total magnetic moment m per unit area. With
the applied magnetic field perpendicular to the sample plane (black, left axis), the loop is
square-like indicating the Pd/Co multilayer has perpendicular magento-anisotropy (PMA),
with a large coercive field of over 160 mT. With the applied field parallel to the sample
plane (red, right axis) the hysteresis is only slightly discernable, thus the magnetization has
a very small in-plane component. Also, note the comparatively smaller scale on the right
axis. The sample is composed of Nb(5)/Cu(5)/[Pd(0.9)/Co(0.3)]10 /Pd(0.9)/Cu(5)/Nb(5)
with thicknesses in nanometers. Lines are to guide the eye.
thickness, dPd , was varied from 0.8 - 1.0 nm. All the samples throughout this chapter were
fabricated using high-vacuum sputtering deposition, using the same procedure and growth
conditions as used in Ch. 6.2.
We measured the films’ magnetic moment m per unit area vs. the applied magnetic
field (M -H loop), using a Quantum Design dc-SQUID magnetometer at 5 K. Sweeping an
applied magnetic field that is perpendicular to the sample plane results in square-like M -H
loops, as shown in Fig. 8.1 (black data points), confirming that the Pd/Co multilayer has
PMA. The sample with dPd = 0.9 nm had the best magnetic properties: a coercive field of
over 160 mT and the largest squareness. With H applied parallel to the sample plane, the
M -H loops show only a slight hysteresis with small remanent magnetization, indicating a
very small in-plane moment (red data points). Note that dividing the m/area values by the

156

total Co thickness of 3.0 nm gives a saturation magnetization of about 2.7×106 A/m (=2700
emu/cm3 ) which is about twice the saturation magnetization of bulk Co. This is because
the Co partially polarizes the surrounding Pd layers [152].

8.1.1

Synthetic Anti-Ferromagnets

Next, we characterized the magnetic behavior of two such Pd/Co multilayers arranged
on either side of a thin Ru spacer to form a SAF with PMA. The coupling of the Pd/Co
multilayer into a SAF structure arises due to interlayer exchange coupling (IEC) between the
two multilayers. The energy density of the IEC can be modulated by tuning the thickness
of the spacer layer and depends strongly upon which material(s) it forms interfaces with.
We arranged the Ru spacer to have adjoining Co layers on either side, similar to Chang et
al. [150].
To optimize the antiferromagnetic coupling in the SAF, we sputtered a set of samples with
an“unbalanced”SAF configuration of the form: Nb(5)/Cu(5)/[Pd(0.9)/Co(0.3)]12 /Ru(dRu )/
[Pd(0.9)/Co(0.3)]10 /Cu(5)/Nb(5), varying the Ru thickness between dRu =0.7, 0.8, ..., 1.1
nm. We measured the samples’ magnetic response in a dc-SQUID magnetometer with H
perpendicular to the sample plane. As shown in Fig. 8.2, at a Ru thickness of 0.7 nm the
Pd/Co multilayers are ferromagnetically coupled, since only a single (slightly distorted) loop
is observable. As the Ru thickness increases, the M -H loops have an intermediate step with
a flat plateau in the magnetization, indicating stable antiferromagnetic exchange coupling
at applied fields less than Âą 250 mT. The maximum width of the intermediate plateau, and
hence the maximum antiferromagnetic coupling measured, was obtained for the samples with
dRu = 0.9 and 1.0 nm (not shown). Therefore, in the Josephson junction samples presented
in the next section we choose to fix dRu = 0.95 nm.
157

m area

Figure 8.2: Hysteresis loop measurements of synthetic antiferromagnet (SAF) films measured
using a dc-SQUID magnetometer. The samples are measured at 5K with the applied field
perpendicular to the sample plane, with the data expressed as the total magnetic moment
m per unit area. The SAF is composed of Pd(0.9 nm)/Co(0.3 nm) multilayers separated by
a thin Ru spacer, whose thickness, dRu , was varied from 0.7 to 1.1 nm (0.7-0.9 nm shown).
At a Ru thickness of 0.7 nm (blue) the Pd/Co multilayers are ferromagnetically coupled outof-plane, since only a single loop is observable. However, as the Ru thickness increases, the
out-of-plane M -H loops have an intermediate step with a flat plateau in the magnetization
(yellow, red). The width of the intermediate plateau is maximal near dRu = 0.9 (red),
indicating stable antiferromagnetic exchange coupling at applied fields less than Âą 250 mT.
Lines are to guide the eye.

158

Note that the extra two repeats of the Pd/Co multilayer in the data presented in Fig. 8.2
are added merely to accentuate the separation between the two corresponding hysteresis
loops, allowing us to more easily determine the optimal Ru thickness. In spin-triplet Josephson junctions, it is desirable to maximize the flux cancelation within the PMA SAF. Thus,
in the experiments that follow, we used a balanced SAF structure which has an equal total
thickness of Pd and Co on either side of the Ru spacer.
We briefly mention that we tested another, similar type of PMA SAF which was composed
of Ni/Co multilayers of the form: [Co(0.3)/Ni(0.6)]n /Co(0.3)/ Ru(dRu )/[Co(0.3)/Ni(0.6)]m
/Co(0.3), where n = 4 and m = 3 or 4. The magnetic behavoir of these PMA SAFs was
quite similar to the data in Fig. 8.2, with strong antiferromagnetic pinning, PMA, and
square-shaped M -H loops. However, after a visitor to our group Demet Korucu fabricated
them into Josephson junctions, we found the Ni/Co SAFs to be rather unsatisfactory from
a device perspective in that they suffered from extremely small critical current. The same
was not true for the Pd/Co based PMA SAFs which will be discussed in the next section.

8.2

Josephson Junctions

We next seek to address the following questions: 1) how does the Pd/Co PMA SAF
structure behave as a barrier to current transport in Josephson junctions (JJs)? 2) can a
PMA SAF of this nature be utilized in a JJ device with ferromagnetic layers to generate spintriplet supercurrent? Both questions can be answered by measuring how the critial current
in these type of junctions varies with the number of Pd/Co layers on either side of the Ru
spacer. To this end we fabricated three sets of JJs. The first is a series of control samples
that contain only the Pd/Co PMA SAF, shown in Fig. 8.3 (a), and are meant to measure

159

the decay of the short-range spin-singlet supercurrent. The second and third series of JJs
are designed to carry long-range spin-triplet supercurrent. They contain a Pd/Co PMA SAF
centered between two additional ferromagnetic layers with in-plane magnetization, shown in
Fig. 8.3 (b). In the second set both the bottom ferromagnet, F 0 , and the top ferromagent,
F 00 , are Ni with thickness 1.6 nm. The third set of JJ’s is similar, but the F 0 layer is
Permalloy (Py = Ni81 Fe19 ) with thickness 1.25 nm and the F 00 layer is Ni(1.6 nm), as shown
Fig. 8.3 (b).
Due to Permalloy’s sharp magnetic switching at low magnetic field, we envision using it in
controllable JJs [17,107,153]. However, from previous experience we know that Py’s magnetic
properties degrade if grown on a rough surface. Thus, in the third series of JJ’s, the Py layer
is intentionally placed near the bottom of the stack where it will be the least effected by
upward-propagating surface roughness, as discussed in more detail later in Fig. 8.4. Nickel,
while harder to control magnetically, is the ferromagnetic material we and others have found
to be the least detrimental to the propagation of supercurrent [143]. Thus, it was used as
both F 0 and F 00 in the second set of samples in case the critical currents in the JJs with Py
were too small to measure. The role of the other layers in Fig. 8.3 will be discussed later.

8.2.1

Sample Fabrication

The sample fabrication proceeds similarly to other nanopillar junctions made by our
group [19, 77], such as in Ch.6, with a few noted differences. The geometry of the bottom
leads was defined via optical photolithography and a lift-off process. The bottom electrode
is a sputtered Nb/Al multilayer of form [Nb(25)/Al(2.4)]3 /Nb(20), which is much smoother
than a continuous Nb layer [122–124], and is capped with a thin 2 nm layer of Au to prevent
oxidation. We then had to break vacuum and exchange sputtering targets. Ideally one would
160

b)

a)

Nb
Nb

}

Nb

F''
F

}

F

Au
Cu
[Co/Pd] n
Ru
[Pd/Co] n
Cu

F'

Au
Cu
Ni
Cu
[Co/Pd] n
Ru
[Pd/Co] n
Cu
Py or Ni
Cu

Nb

Figure 8.3: A schematic representation of the vertical cross sectional structure of our Josephson junctions (not to scale). The central F layer is composed of two sets of n [Pd(0.9
mm)/Co(0.3) nm] bilayers with perpendicular magnetic anisotropy (PMA), on either side of
a Ru(0.95 nm) spacer to form a synthetic antiferromagnet (SAF). (a) With only the PMA
SAF in the center, the supercurrent is carried by short-range spin-singlet pairs. (b) When
combined with the two other ferromagnets, F0 and F00 , with in-plane magnetization, the
supercurrent in F is carried by long-range spin-triplet pairs. In this study, the F 0 layer is
either Ni(1.6 nm), which maximizes the spin-triplet supercurrent, or Py (1.25 nm), which
can act as a “free” layer, switching it’s magnetization at a low field.

161

sputter the entire stack in situ, however, we are limited to seven sputtering targets in our
chamber. During the target exchange, which takes less than 10 minutes, the samples are
contained in a bag filled with continuously flowing N2 gas to limit contamination. After
another bakeout, pump down, and liquid nitrogen cooling, we ion mill away the protective
Au layer before continuing the sputtering process. All the ferromagnetic layers are then
deposited in situ in the following sequence: for the “spin-singlet” samples of Fig. 8.3(a),
Cu(4)/PMA-SAF/Cu(4)/Au(2); and for the “spin-triplet” samples of Fig. 8.3 (b) we sputter,
Cu(2)/[Ni(1.6) or Py(1.25)]/Cu(4)/ PMA-SAF /Cu(4)/Ni(1.6)/Cu(7)/Au(2), where PMASAF = [Pd(0.9)/Co(0.3)]n / Ru(0.95)/ [Co(0.3)/Pd(0.9)]n . Due to the crystal lattice mismatch between the fcc ferromagnetic materials and the bcc Nb we add a Cu(2) spacer before
the (Py or Ni) F 0 layer. Meanwhile, between the F 0 , F , and F 00 layers, Cu(4) buffers are
inserted to prevent them from coupling magnetically. Finally, the stack is capped with a
thin layer of Cu and Au to prevent oxidation.
The junctions were patterned by electron-beam lithography followed by ion milling in
Argon. We use the negative e-beam resist ma-N2401 as the ion mill mask. The junctions are
elliptical in shape with an aspect ratio of 2.5 and area of 0.5 Âľm2 , sufficiently small for the
Py layers to be mostly single domain [19]. Elliptically-shaped junctions have the advantage
that the Fraunhofer patterns follow an analytical formula while the (small) demagnetizing
field is nearly uniform when the magnetization is uniform.
Outside the mask region, we ion milled from the capping layer through the F 0 layer, and
nominally half-way into the underlying Cu spacer layer. After ion milling, a 50-nm-thick
SiO layer was deposited by thermal evaporation to electrically isolate the junction and the
bottom wiring layer from the top wiring layer.
Finally, the top Nb wiring layer was patterned using the same photolithography and
162

(a)

~ •

SiO
(black)

(d)
1
2
3
54
ò

(c)

7
8

Figure 8.4: Vertical cross sections of the junctions described in Fig. 6.2 (b) prepared using a
focused ion beam (FIB) were investigated by high-resolution scanning transmission electron
microscopy (STEM) and energy dispersive x-ray spectroscopy (EDX). Panel (a) shows an
STEM image of the full extent of the junction, including the smooth Nb/Al bottom electrode.
Expanded views of the individual ferromagnetic layers are shown in panels (b) and (c) near
the center and side of the junction, respectively. The EDX phase map shown in panel (d)
corresponds to the area within the orange square in panel (c). The multivariate statistical
analysis of the spectra from each individual pixel are color coded and numbered in the figure
as follows: Au (yellow, 1), Cu (blue, 2), Ni + Fe (cyan, 3 and 6), Pd + Co (magenta, 4), Co
+ Ru (white, 5), Nb (red, 7), Al (green, 8), SiO (black). Figure courtesy of P. Kotula and
N. Missert, Sandia National Laboratory.
lift-off process as the bottom leads. The surface is cleaned with oxygen plasma etching
followed by in-situ ion milling in which 1 nm of the top Au surface is removed immediately
before sputtering. We deposited top leads of Nb(150 nm)/Au(10 nm), ending with the Au
to prevent oxidation.
In order to verify the fabrication process, vertical cross sections of the junctions were investigated by high-resolution scanning transmission electron microscopy (STEM) and energy
dispersive x-ray spectroscopy (EDX) by our collaborators Paul Kotula and Nancy Missert at
Sandia National Laboratories. The cross sections were prepared using a FEI Helios focused

163

ion beam (FIB) with a Ga ion source, and transferred to a Ti grid for imaging in a FEI
Titan G2 80-200 aberration-corrected STEM operated at 200kV and equipped with four
silicon drift X-ray detectors.
The high-angle annular dark field STEM image in Figs. 8.4(a)-(c) show cross sections of
a junction with the same structure as Fig. 8.3 (b) with n = 3. The Nb/Al bottom electrode,
seen at the bottom of Fig. 8.4(a), is a smooth and continuous surface that provides a good
growth template for the layers grown on top. Fourier transforms of regions of the highresolution STEM image show that the Cu layer directly above grows with a [111] orientation
on Nb [011]. Grains with favorable orientation relative to the beam direction show lattice
fringes extending through the entire Cu/ferromagnetic layer/Cu thicknesses. In the top Cu(7
nm) layer there appear to be three to four isolated regions (width ≈ 5 nm) with lower Cu
density than the rest of the layer (one such dark patch can be clearly seen in Fig. 8.4(c)). The
origin of those low-density Cu regions is unknown. The individual Pd and Co layers inside
the SAF, shown in Fig. 8.4(b), near the center of the junction, appear relatively smooth and
continuous. Furthermore, the STEM image shows that the ion milling procedure used in the
sample fabrication to define the junction area is accurately calibrated to mill down to the
desired depth.
Identifying the elemental composition of the layers is achieved through EDX phase maps,
created by performing a multivariate statistical analysis of the spectra from each individual
pixel, and color-coding pixels containing the same spectral shape [154]. The phase map
shown in Fig. 8.4(d) corresponds to the area within the orange square in Fig. 8.4(c). The
Py layer (cyan, labeled as layer 6) is clearly uniform and continuous. The individual layers
inside the SAF are not distinguishable due to their sub-nanometer thickness and the lower
spatial resolution of EDX compared to STEM. However, we clearly show a difference between
164

the component Pd/Co X-ray peaks from the outer regions of the SAF (magenta, labeled as
layer 4) compared to it’s center, where component Ru/Co peaks are more prominent (white,
labeled as layer 5). The Ni layer (cyan, labeled as layer 3), while continuous, has some
observable roughness, consistent with the magnetic behavior discussed in the next section.

8.3

Transport Measurements and Analysis

Each device was connected to the wire leads of a dip-stick probe and measured at 4.2 K
using the rf SQUID comparator circuit, setup in the same manner as in Ch. 6. Typical I-V
curves have the expected behavior of overdamped Josephson junctions [127]. The critical
current Ic was extracted by fitting the I-V curves to a square root function, Eq. (6.1), where
the sample resistance in the normal state RN was determined by the slope of the linear
region of the I-V curve when |I|  Ic .
Measurements of the area-resistance product in the normal state typically yield consistent
values of ARN , with a median value of 22.5 fΩ-m2 , an indicator of the reproducible high
quality interfaces. The ARN values for the full set of samples are shown in Fig. 8.8(b).
The junction area typically varies by less than 10% from the nominal value of 0.5 Âľm2 , and
can be accurately extracted from the Fraunhofer pattern measurements discussed later. It
is thought that RN is dominated by the interfacial resistance between the various layers.
It is therefore noteworthy that, although these junctions contain many interfaces, the ARN
products are (on average) only about twice those of similarly-sized junctions containing only
a single ferromagnetic layer [19, 77], which had 5-10 fΩ-m2 . The junctions with F 0 = F 00 =
Ni with n = 3 or 4 Pd/Co repeats have slightly larger ARN than those with n = 1 or 2.
Otherwise, the average value of ARN does not appear to be correlated with n.

165

When Ic is less than a few ÂľA, the I-V curves exhibit noticeable rounding due to thermal
effects and instrumental noise. Such rounding is accommodated by the theories of Ivanchenko
and Zil’berman (IZ) and of Ambegaokar and Halperin [129,130]. Fitting the I-V curves with
the IZ function, Eq. (6.2) instead of the square-root function of Eq. (6.1) results in values of
Ic that are somewhat larger – typically 30% for Ic ≈ 1µA [19]. However, fitting every I-V
curve with the IZ function is computation-intensive and not practical, so we used the simpler
square-root fits for the Fraunhofer data shown in Figs. 8.5-8.7. For the summary shown in
Fig. 8.8, we used the values obtained from fitting the IZ function to the data near the peaks
of the individual Fraunhofer patterns.

8.3.1

Fraunhofer Patterns

Measuring Ic as a function of the applied magnetic field, we map out “Fraunhofer” diffraction patterns, shown in Figs. 8.5-8.7 for the three JJ types described in Fig. 8.3.
Fig. 8.5 shows data from three spin-singlet samples with n = 1, 2, and 3. Those data
were acquired by applying a field of 60 mT then slowly ramping the field to -60 mT in steps
of typically 2 mT (blue data points). We then repeated this procedure in the other field
direction (red data points) and observed very little magnetic hysteresis since the F layer’s
magnetizations are aligned perpendicular to the applied field.
For the spin-triplet samples with the additional F 0 and F 00 layers, we first measured
the critical current near zero field to monitor the “virgin” or as-grown state of the various
nanomagnets. Initially sweeping the field from (Âą 60 mT) the critical current was typically
small and any semblance of “Fraunhofer patterns” were rather irregular, as shown in Fig. 8.6
(a). Successively expanding the applied field sweep range (Âą60, Âą90, Âą120, Âą150 mT) to
help align the F 0 and F 00 layers resulted in significant improvements in the Fraunhofer quality
166

a)

b)

c)

Figure 8.5: The critical current, Ic , is plotted versus the applied in-plane field H for junctions
with the structure shown in Fig. 8.3 (a) (without F 0 and F 00 ). The supercurrent is carried
primarily by spin-singlet pairs. Increasing the number of Pd/Co layer repeats: (a) n = 1,
(b) n = 2, (c) n = 3, causes Ic to decay rapidly. Since the magnetization of the SAF is
perpendicular to the plane there are only slight horizontal shifts in the Fraunhofer patterns
and very little magnetic hysteresis. The corresponding fits to Eq. (6.3) (lines) show excellent
agreement for both the positive (red, dashed) and negative (blue) field sweep directions.

167

and enhanced the peak value of Ic (near zero field), shown in Fig. 8.6 (b)-(c). After initializing
the samples at Âą150 mT the peak value of Ic appears to saturate, and the Fraunhofer pattern
closely follows the theoretical curve described later in Eq. (6.3). This detailed “initialization”
behavior was reproducible on five separate junctions from various chips. Therefore, for the
data presented in Figs. 8.6-8.7, we determined that a large initialization field of 150 mT
was required to help set the initial orientation of the Ni layer(s). Ideally, an even larger
initialization field would be beneficial to fully magnetize the Ni [142], however, too large
a field might disturb the magnetic properties of the PMA SAF. Based on the results of
Figs. 8.1-8.2, as a precautionary measure any external magnetic fields were kept at 150 mT
or below.
After the initialization procedure, we removed any flux trapped in the junction or in the
Nb leads by lifting the dip-stick probe slightly in the Dewar until the sample lay just above
the liquid Helium bath, such that the Nb ceased to be superconducting as monitored by
an Ohmmeter connected to the sample leads. After reinserting the sample into the liquid
Helium we next applied a field of 90 mT and slowly ramped the field to -90 mT, in steps
of 2 mT. Finally, the field was slowly swept in the opposite direction, after reinitializing the
sample at -150 mT, measuring I-V curves with the field from -90 mT to 90 mT, observing
any hysteretic effects from the in-plane ferromagnets.
Fraunhofer pattern measurements such as these contain information about the magnetic
state of the in-plane ferromagnetic layers, the behavior of the critical current, and the dimensions of the junction. For elliptical junctions the functional form of the Fraunhofer patterns
are the Airy functions from Eq. (6.3). Since the magnetization of the F -layer is parallel to the
current flow, it does not contribute to the flux ÎŚ in Eq. (6.3). The in-plane magnetizations
of F 0 and F 00 do contribute, however, and cause shifts in the Fraunhofer pattern along the
168

a)

b)

c)
up

down

Hswitch

Hswitch

Figure 8.6: Critical current, Ic , vs. the applied in-plane field H, for a sample with F 0 =
F 00 = Ni (1.6 nm), and n=1, similar to Fig. 8.3 (b). If the initialization field is too small,
e.g. (a) Hinit = 60 mT or (b) Hinit = 90 mT, and the magnetizations of the two Ni layers are
not aligned, the “Fraunhofer” patterns are of poor quality. With an initialization field of 150
mT (c) the Ni layers are fully magnetized, and the data before Hswitch (solid markers) show
good agreement with the expected form for both the positive (red) and negative (blue) field
sweep directions. Solid lines are fits to Eq. (6.3). Hence we initialized all our spin-triplet
samples at 150 mT. In (c) the hollow markers represent the data points after Hswitch and the
dashed lines are only to guide the eye. The Ni layers, while amenable to large supercurrents,
contain multiple magnetic domains and switch magnetization over a broad field range.

169

field axis [17, 19, 76, 77, 153, 155]. Hence the horizontal Fraunhofer pattern shifts will differ
for the three types of samples outlined in Fig. 8.3.
If the in-plane F 0 and F 00 layers are uniform and colinear with H, and the F layer is
assumed to be perpendicular to H, then similar to the discussion in Ch. 6 the magnetic flux
through the junction is [76, 153],

ÎŚ = Âľ0 Hw(2ÎťL + dtot ) + Âľ0 w(M 0 dF 0 + M 00 dF 00 ),

(8.1)

where H, w, and ÎťL are the applied field, the width of the junction (minor axis), and
the London penetration depth of the Nb electrodes. The total thickness between the Nb
electrodes is dtot = dN + dF 0 + dF + dF 00 , where dN is the total thickness of all the normal
metal layers (including the Ru in the SAF) and dF 0 , dF , and dF 00 are the thicknesses of the
three ferromagnetic layers F0 , F, and F00 , respectively (dF being the total thickness of all
the Pd/Co multilayers). The last term in Eq. (8.1) arises from the in-plane magnetization’s
M 0 and M 00 corresponding to the ferromagnets F 0 and F 00 , respectively. Eq. (8.1) neglects
the small contributions to Φ from the uniform demagnetizing field’s and any magnetic field
from the nanomagnets that returns inside the junction. From Eq. (8.1) it is clear that the
Fraunhofer pattern will be shifted along the field axis by an amount Hshift = −(M 0 dF 0 +
M 00 dF 00 )/(2λL + dtot ) in the direction opposite of the junctions’ net in-plane magnetization.
For samples without the F 0 and F 00 layers there should be very little shift, resulting
only from any canting of the [Pd/Co] perpendicular magnetization into the plane. That
expectation is born out by the data shown in Fig. 8.5. For samples with the F 0 and F 00
layers the shifts will be more pronounced. The shifts should be largest when M 0 and M 00 have
the same sign (parallel alignment) and smaller when they have opposite signs (antiparallel

170

alignment) [153].
The complex nature of these Fraunhofer pattern shifts, combined with the possibility
that the Ni layers do not switch abruptly or behave as a single magnetic domain, make
a comprehensive analysis difficult. For the samples with F 0 = F 00 , the two Ni layers may
not switch at the same field due to the upward propagating surface roughness, making it
impractical to fit all the data points in the Fraunhofer patterns shown in Figs. 8.6 and 8.7.
Therefore we only attempt to fit the clean sections of the Fraunhofer patterns before the first
magnetic switching event occurs. We fit Eq. (6.3) to the data starting from the initialization
field to Hswitch . The free parameters in the fit are Ic0 , the junction width transverse to the
field direction, and the field shift of the central peak. In Figs. 8.5 - 8.7 the corresponding
fits show excellent agreement with the data, for the positive (red) and negative (blue) sweep
directions. In Figs. 8.6(c) and 8.7 the hollow data points denote the data after Hswitch .
Those data show that the reversal of the Ni magnetization occurs over a range of fields,
consistent with the behavior of Ni seen in previous work by us and others [17, 153]. Most
of the junctions display full magnetic remanence, continuing to follow Eq. (6.3) though zero
applied field before Hswitch .

8.3.2

Critical Current vs Thickness

In Fig. 8.8(a) we plot the maximum measured Ic times RN on a log scale versus the
number of Pd/Co repeats n on either side of the Ru spacer. In Fig. 8.8 (b) we plot the arearesistance product for the entire data set. The Ic RN products of the spin-singlet samples
(blue circles) decay much more rapidly with increasing n than do the spin-triplet samples
with F 0 = Ni (orange triangles) and those with F 0 = Py (red diamonds). The three data

171

a)

b)

c)

Figure 8.7: Ic is plotted vs. the applied in-plane field H, for junctions with the structure
shown in Fig. 8.3 (b) with F 0 = Py, F 00 = Ni. Increasing the number of repeats of Pd/Co:
(a) n=1, (b) n=2, (c) n=3, causes Ic to decay, but more slowly than without the F 0 and
F 00 layers (Fig. 8.5). The horizontal shifts in the Fraunhofer patterns are indicative of the
magnetic state of the in-plane ferromagnets. The data before Hswitch , the field at which the
Py magnetization reverses direction (solid markers), and the corresponding fits to Eq. (6.3)
(lines) show excellent agreement for both the positive (red, dashed) and negative (blue) field
sweep directions. The hollow circles are the corresponding data points after Hswitch . The
Py typically switches abruptly at low fields (< 2 mT), whereas the Ni is thought to contain
multiple magnetic domains and gradually switches over a broad range of fields.

172

Table 8.1: Fitting the data in Fig. 8.8 to Eq. (8.2) yields the best-fit parameters:

Sample Set

A0 (ÂľV)

n̄

Spin-singlet, only F

89 Âą 28

0.70 Âą 0.04

14 Âą 3

1.38 Âą 0.07

4.1 Âą 0.6

1.53 Âą 0.07

Spin-triplet, F 0 =F 00 = Ni
Spin-triplet, F 0 =Py, F 00 =Ni

sets are fit to the total number of bilayers (2n) with a simple exponential decay,

Ic RN (n) = A0 ∗ e−(2n/n̄) .

(8.2)

The best-fit parameters are listed in Table 8.1. The decay length expressed as a number of
[Co/Pd] bilayers is 1.38Âą0.07 and 1.53Âą0.07 for the spin-triplet samples, whereas it is only
0.70Âą0.04 for the spin-singlet samples. That observation is the main result of this work.
It is instructive to compare the data in Fig. 8.8(a) with data from the only other study of
spin-triplet Josephson junctions containing PMA layers, namely ref. [106]. In that work the
central F layer consisted of a [Ni/Co] multilayer with strong PMA, but not a SAF. In those
junctions the decay of the spin-triplet samples was much slower than the decay of the control
samples that did not contain the F 0 and F 00 layers. (The ratio of the spin-triplet to spinsingle decay lengths in that work was about 4.5, whereas it is only about 2.1 in the present
work.) Later, our group measured a series of junctions containing [Ni/Co] PMA SAFs, and
found not only that they carried much smaller critical currents, but that the decay with the
number of repeats was also much steeper than in the non-SAF [Ni/Co] junctions [156]. To
explain the sharp decay of the spin-triplet supercurrent in the [Ni/Co] SAFs, we propose the
following explanation. It is known from giant magnetoresistance (GMR) studies [157] that
the [Ni/Co] interface has strong spin-scattering asymmetry – i.e. minority-band electrons are
173

a)

b)

Figure 8.8: Comparison of the critical current decay in spin-triplet and spin-singlet samples
and their area-resistance products. (a) The maximum measured Ic times RN is plotted vs.
the number of Pd/Co repeats n in the PMA SAF. The Ic RN of the spin-singlet samples
(blue circles) decays more rapidly with increasing Pd/Co thickness than do the spin-triplet
samples with F 0 = Ni (orange triangles) and those with F 0 = Py (red diamonds). The
data are fit to the total number of Pd/Co bilayers according to Eq. (8.2), with the best-fit
parameters shown in Table 8.1. (b) The area-resistance product of the junctions vs. n does
not appear to be correlated to the number of Pd/Co interfaces, and is on average 11.4 fΩm2
across all the datasets, with the exception of the F 0 =Ni chips with n = 3 and 4, which have
slightly larger resistances.

174

scattered more strongly than majority-band electrons. Hence in a [Ni/Co]n multilayer, the
current will become more strongly spin-polarized as the number of repeats n increases. In a
[Ni/Co] SAF, the majority electrons in one half of the SAF become minority electrons in the
other half and vice versa. Hence all electrons passing through the SAF are strongly scattered
at some point. This effect increases with n, hence causing a steep decay of critical current
as a function of n. We note that this mechanism affects both spin-singlet and spin-triplet
supercurrents; but the former already face a steep decay due to the standard S/F physics
discussed in the introduction, whereas for the spin-triplet supercurrent the presence of the
SAF becomes the dominant decay mechanism.
Our decision to use [Pd/Co] multilayers was a direct result of the discussion above.
Unfortunately, the degree of spin-scattering asymmetry at the [Pd/Co] interface has not
been measured, as far as we know. (Our own attempts to do so using GMR techniques were
thwarted by our inability to achieve reproducible in-plane magnetic states for any thickness
combination in the [Pd/Co] system.) From comparing the results shown in Fig. 8.8(a) with
our unpublished data on junctions containing [Ni/Co] PMA SAFs, we infer that the spinscattering asymmetry in [Pd/Co] is weaker than in [Ni/Co]. Nonetheless, we believe that the
spin-scattering asymmetry in [Co/Pd] underlies the fact that the decay of the spin-triplet
supercurrent is only a factor of two less steep than the decay of the spin-singlet supercurrent.
An alternative explanation for the unusually steep decay of the spin-triplet supercurrent
in our samples is the strong spin-orbit interaction in the Co/Pd system [158]. It should
be possible to distinguish between these two explanations by comparing the decay of the
supercurrent in spin-triplet JJs containing [Pd/Co] plain multilayers in the center, with the
decay we observe in our JJs containing [Pd/Co] multilayer SAFs. Our group intends to carry
out such a study in the near future.
175

The next question to address is, at what value of n can we be certain that the spin-triplet
component of the supercurrent in the spin-triplet samples is the dominant contribution? A
straightforward interpretation of the data shown in Fig. 8.8(a) might lead one to conclude
that the point where the spin-triplet curves cross the spin-singlet curve is the point where the
spin-singlet and spin-triplet contributions to the supercurrent are equal in the spin-triplet
samples. Those crossings occur approximately at n = 1 for the Ni-Ni samples and at n = 2
for the Ni-Py samples. But that interpretation is wrong. The magnitude of the spin-singlet
supercurrent in the spin-triplet samples is surely far less than the supercurrent we measure
in the spin-singlet samples, because the spin-singlet supercurrent will be further suppressed
when it has to pass through the additional F 0 and F 00 layers. That suppression would
effectively shift the entire spin-singlet curve down vertically, thus the blue data set can only
be considered as a generous upper bound on the magnitude of the spin-singlet supercurrent
that can pass through the spin-triplet samples. We do not know how large that suppression
is, but we can guess that the suppression is roughly three times greater in the Py-Ni samples
than in the Ni-Ni samples, from the vertical offset between the two spin-triplet curves.

8.4

Conclusion

In conclusion, we have measured the critical current in Josephson junctions of the form
S/F 0 /N/F/N/F 00 /S, where F is a synthetic antiferromagnets consisting of [Pd/Co] multilayers with perpendicular anisotropy. The critical currents in those junctions decay less steeply
with the number of [Pd/Co] bilayers than in junctions without the F 0 and F 00 layers. That
result represents strong evidence that the 3-layer junctions carry spin-triplet supercurrent.
Furthermore, by choosing F 0 to be a soft magnetic material such as Permalloy, while F 00 is a

176

hard magnetic material such as Ni, the relative magnetization directions of those two layers
can be controlled. Such junctions are utilized in Ch. 9 to make cryogenic memory devices
in which the ground-state phase difference across the junctions is controlled on-demand by
changing the magnetic configuration of the ferromagnetic layers.

177

Chapter 9

Controllable Phase SQUIDs: Spin-Triplet

It is now well established that spin-triplet supercurrent can be generated from conventional spin-singlet superconductors using ferromagnetic layers with an inhomogeneous magnetization profile [85–87, 90–98, 100–104], resulting in several important consequences, such
as the extension of superconducting properties deep into strongly ferromagnetic materials,
as discussed in Ch. 4. Surprisingly, one of the most salient predictions of the spin-triplet
theory has yet to be verified experimentally, namely that a Josephson junction containing
three magnetic layers with noncolinear magnetizations should exhibit a ground-state phase
shift of either 0 or π depending on the relative orientations of those magnetizations.
In this chapter, we verify that prediction experimentally, using the same type of junctions
presented in Ch. 8. The junctions are of the form S/F 0 /N/F/N/F 00 /S, which contain three
ferromagnetic layers, F 0 and F 00 both with in-plane magnetization, and the central layer F
with magnetization perpendicular to the plane. Detection of the ground-state phase across
these spin-triplet Josephson junctions is achieved using similar techniques to those employed
in Ch. 7, in which we measure the relative phase-shift between two such junctions in a SQUID
loop.
We demonstrate a scheme by which the magnetization direction of the F 0 layer can be
controllably switched by 180◦ without disturbing the F and F 00 layers, thereby allowing
the phase state of the junction to be set to 0 or π on demand. We show that the phase
can be reliably switched between the zero or π states for seven different spin-triplet SQUID

178

devices measured, several of which we were able to switch between the two states over a
thousand times without error. Finally, we study the temperature dependence of the critical
current in these types of junctions below 4.2 K using a dilution refrigerator. This type of
phase-controllable junction could be used as a memory element in a fully-superconducting
computer.

9.1

SQUID Sample Fabrication

The samples are of similar construction to the individual spin-triplet junctions from
Ch. 8 where the in-plane ferromagnets are F 0 = Py and F 00 = Ni and the central F layer
is a Pd/Co-based synthetic antiferromagnet with perpendicular magnetic anisotropy. Thus,
the spin-triplet SQUID sample fabrication proceeds similarly to that detailed in Ch. 8, aside
from a few notable changes discussed below.
We designed a new SQUID layout, shown schematically in Fig. 9.1, which has nominally
equal inductances on either side of the loop, unlike the asymmetric SQUIDs discussed in
Ch. 7. This modification was aimed at simplifying the data analysis, in the sense that the
output SQUID oscillations will be symmetric (not “ratchet” shaped), though the critical
currents in the two junctions may still be asymmetric. The design uses a pitchfork shaped
bottom wiring layer, which contains the bottom electrode and all of the magnetic layers,
intersected by a “T-shaped” superconducting top wiring layer, making a SQUID loop with
an inner area of 6 × 6 µm2 . There is an on-chip superconducting flux line, similar to that
used in the SQUIDS from Ch. 7, through which we pass current to add flux to the SQUID
loop. The design layout accommodates up to four independent and identical SQUIDs on a
single 0.5 × 0.5 in2 chip.

179

Au
Cu
Ni
Cu

F''

}

F

[Co/Pd]n
Ru
[Pd/Co] n
Cu
Py
Cu

F'

Iflux
M Py,1

H set

Is ,Vs

Is ,Vs

H flux
M Py,2

Figure 9.1: Spin-triplet Josephson junction structure and SQUID loop design. (Top) A
schematic cross-section of the central layers in our Josephson junctions (not to scale). The
central F layer is composed of two sets of n alternating layers of Pd (0.9 nm) and Co (0.3 nm),
with perpendicular magnetic anisotropy (PMA), on either side of a Ru (0.95 nm) spacer to
form a synthetic anti-ferromagnet (SAF). When combined with another ferromagnetic layer,
F 0 (Py (1.25 nm)), with in-plane magnetization, the PMA SAF provides the 90 degree basis
rotation to generate long-range spin-triplet paring. The third ferromagnet, F 00 (Ni (1.6 nm)),
converts the spin-triplet pairs back to spin-singlet supercurrent to be accepted by the top
Nb electrode. One junction has an elliptical cross-section (aspect ratio 2.0) to make its
F 0 layer switch at a low field, while the other is an elongated hexagon (aspect ratio 3.0),
both have area 0.5 Âľm2 . (Bottom) The superconducting electrodes and the junctions are
arranged into a SQUID loop, used to measure the relative phase between the two junctions.
An external field Hset is used to set and control the magnetization directions of the F 0 layers
inside the junctions, while the current Is through the SQUID and voltage Vs across it are
typically measured after returning Hset = 0. The flux current Iflux is driven through a
nearby superconducting line to inject flux into the SQUID loop, causing the total critical
current in the SQUID to modulate with the flux current.

180

The geometry of the bottom leads was defined via optical photolithography using the
HMDS and S1813 procedure described at the end of Section 5.2. The bottom electrode is
a sputtered Nb/Al multilayer of form [Nb(25)/Al(2.4)]3 /Nb(20), where all thicknesses are
listed in nanometers, and is capped with a thin 2 nm layer of Au to prevent oxidation. As in
Section 8.2.1, we then had to break vacuum and exchange sputtering targets, due to the limited number of sputtering guns in our chamber. During the target exchange, which typically
takes about 10 minutes, the samples are contained in a bag filled with continuously flowing
N2 gas to limit contamination. After another bakeout, pump down, and liquid nitrogen
cooling, we ion mill away the protective Au layer before continuing the sputtering process.
All the ferromagnetic layers are then deposited in situ in the following sequence, shown at
the top of Fig. 9.1: Cu(2)/Py(1.25)/Cu(4)/ PMA-SAF/ Cu(4)/Ni(1.6)/Cu(7)/Au(2), where
PMA-SAF =[Pd(0.9)/Co(0.3)]n /Ru(0.95)/[Co(0.3)/Pd(0.9)]n , and n = 2 or 3, corresponding to a total number of [Pd/Co] bilayers of 4 or 6. The stack is capped with a thin layer of
Cu and Au to prevent oxidation.
Next, using electron-beam lithography and ion milling in Argon, we patterned 0.5 Âľm2
area junctions onto each of the arms of the forked bottom lead. The goal is to be able
to control the magnetic state of the “soft” layer (F 0 =Py) in the first junction with a small
external field, while the magnetic state of the other layers in the first junction, and all
the layers in the second junction, are left unperturbed. To accomplish that we patterned
the two junctions to have different shapes and aspect ratios, to take advantage of shape
anisotropy. The “controllable” first junction, is elliptical in shape with an aspect ratio of
2.0. Our colleague, Nick Rizzo at Arizona State University, suggested the stability of the
“fixed” second junction could be improved by making its shape an elongated “hexagon” (i.e.
a diamond with the edges chopped off, hereafter referred to as a “hex” bit) rather than an
181

ellipse. His idea was backed-up with micromagnetics simulations using the Landau-LifshitzGilbert equations, which showed that a hex bit of a 0.5 Âľm2 would, in theory, have a
somewhat larger switching field than an ellipse of the same size and aspect ratio. Following
his suggestion, we patterned the second junction into a hex shape and with a larger aspect
ratio of 3.0. The junctions are placed on either side of the SQUID loop, as shown in Fig. 9.1,
with their long-axes aligned parallel to one another.
We used the negative e-beam resist ma-N 2401 as the ion mill mask and outside the
mask region, we then ion milled from the capping layer through the F 0 layer, and nominally
half-way into the underlying Cu spacer layer. After ion milling, a 50 nm thick SiO layer
was deposited by thermal evaporation, followed by a liftoff process, to electrically isolate the
junction area. Finally, the top Nb wiring layer was patterned using the same photolithography process as the bottom leads. The surface was cleaned with oxygen plasma etching
followed by in-situ ion milling, in which 1 nm of the top Au surface was removed immediately before sputtering. We deposited top leads of Nb(150 nm)/Au(10 nm), ending with the
Au to prevent oxidation.

9.2

Measurement of SQUID Oscillations

The sample was connected to the end of a dipping probe and immersed into a Dewar of
liquid helium that is outfitted with a Cryoperm magnetic shield and placed inside a shielded
room to reduce exposure to external sources of electromagnetic radiation. As in Ch. 8 the
junctions are initialized with a -150 mT field, to align the magnetizations of all the inplane ferromagnetic layers. Any flux trapped in the Nb leads is subsequently removed by
lifting the probe in the dewar until the sample temperature rises just above the Tc of Nb,

182

Measurement Flowchart
1

Magnetize F' and F'' layers
with large eld (~150 mT),
remove trapped flux

2

Return to zero eld

3

Measure I-V curves vs. I flux
to observe oscillations in the
SQUID I c vs. applied flux

4*

Apply a small set eld H set

*

Each time increment H set
and watch for phase shifts
(when M' in the elliptical JJ
reverses direction)

Figure 9.2: Flowchart describing the measurement process for probing the two main aspects
of the SQUID samples. (1) oscillations in the SQUID’s maximum Ic versus the flux in the
SQUID loop (proportional to the current applied to the flux line), and (2) the magnetic state
of the two junctions in the SQUID as modified by an external field. The relative phase shifts
in the SQUID oscillations associated with the states before and after a magnetic switching
event can be used to characterize the device as a magnetic memory.

183

monitored with an ohmmeter attached across the leads of the flux line. Re-immersing the
sample in the liquid He, we measure the voltage across the SQUID (Vs ) with the rf SQUID
comparator circuit discussed in Ch. 5.6, while varying the current supplied to the sample (Is )
using a battery-powered ultra-low noise programmable current source [126]. The resulting
I-V curves are representative of overdamped Josephson junctions and are fit to either the
square root function (Eq. (6.1)) or the Ivanchenko-Zil’berman function (Eq. (6.2)), the latter
of which accounts for thermal rounding. Either of those methods allows us to extract the
normal state resistance RN of the SQUID and the total critical current in the SQUID at
positive or negative bias, Ic+ and Ic− , respectively. Ic,Avg is defined to be the average of
Ic+ and Ic− .
The procedure to measure the relative phase of the two junctions in the SQUID is outlined in Fig. 9.2. A separate Yokogawa current source drives a current Iflux through the flux
line, which produces a very small out-of-plane field that induces a magnetic flux ÎŚ through
the SQUID loop. The Ic data exhibit oscillations as a function of Iflux with a period corresponding to one flux quantum ÎŚ0 = h/2e through the SQUID loop. A flux line current of
Âą3 mA is sufficient to observe just over three periods in Ic . We call the initial state of the
SQUID at zero field after being magnetized the “P-state”, since the magnetizations of the
Py layers in the two junctions have parallel alignment. The measured oscillations in Ic (Iflux )
for one representative sample are shown in Fig. 9.3 and 9.4 and many others are presented
in the Appendix.
A variable in-plane “set field” µ0 Hset is used to control the magnetization directions of
the F 0 layers in the junctions, however the SQUID oscillations, Ic (Iflux ), are measured with
the junctions in their remnant magnetization state to avoid convolving the SQUID physics
with the Fraunhofer effect. Therefore, after applying a field Âľ0 Hset = 0.5 mT in the positive
184

Figure 9.3: A 3D plot of a complete minor magnetization loop, for SQUID 2A-4. We show
the oscillations in Ic,Avg versus the flux line current, Iflux , measured after applying different
values of the set field, Âľ0 Hset . All the SQUID oscillations are measured at zero field. Starting
at zero field and increasing Âľ0 Hset in the positive direction, the SQUID switches from the
P to the AP state at +2.4 mT, and will remain in that state even if Âľ0 Hset is increased or
decreased slightly in either direction. Next, returning to zero field, Âľ0 Hset is applied in the
negative field direction until -2.4 mT, where the SQUID switches from the AP state back
into the P state. The phase of the SQUID can thus be controllably set to either 0 or π on
demand, and that behavior is reproducible over a large number of magnetic switches.

185

direction, the field is returned to zero before measuring Ic (Iflux ). The process is repeated
increasing Âľ0 Hset in steps of 0.5 mT, as described in Fig. 9.2.
Initially, if Âľ0 Hset is less than the coercive field of the Py layers in either junction,
the data shows that the state of the SQUID is unchanged by Âľ0 Hset . Then, when Âľ0 Hset
exceeds the coercive field of the Py layer in the elliptical junction, µ0 Hset ≈ 2.4 mT, that
layer suddenly reverses its magnetization direction and the resulting SQUID oscillation curve
shifts in phase (relative to the P-state) by almost exactly Φ0 /2. The half period (or “π”) phase
shift is immediately recognizable in the data for either either Ic,Avg , Ic+ , and Ic− . We refer
to the state after the phase-shift the “AP-state”, since the magnetizations of the Py layers
in the junction are now aligned anti-parallel to each other. Across the full set of samples,
shown in the Appendix, the SQUIDs typically switch into the AP-state for values of Hset
between 1.6–2.8 mT, which agrees well with separate SQUID magnetometry measurements
of the coercive field of Py, for junctions of this size, made in our lab [144].
As shown in Fig. 9.3, we observe that the Ic amplitude of the AP-state is typically
slightly larger than that in the P-state, and is sometimes accompanied by a vertical offset in
Ic . However, the difference between the amplitudes of the P and AP state critical currents is
considerably smaller in these spin-triplet SQUIDs than in the samples from Ch. 7 that carry
only spin-singlet supercurrent. According to the theory of spin-triplet Josephson junctions
discussed in Ch. 4, the pair correlation function describing the long range spin-triplet pairs in
the F layer, unlike the spin-singlet pair correlation function, does not oscillate in sign versus
the accumulated phase shift through the junction. Furthermore, if the F 0 layer magnetization
reverses direction, the theory predicts that the critical currents in the P and the AP states
will be nearly identical.
Indeed, we did observe nearly identical critical currents in the very first run of the first
186

■■■
■ ■

Ic,Avg (ÎźA)

10

6

●
■
●

0

■
■

●

●

●●
●

-2

■

■

■■

-1

■

■

■
●●
●
■● ●
■
●
●
■
●
■
●
●
●
■ ●
● ■
●
■
■
●
●
●
●
■
■
●●
●●
■
■
■
■

●●
● ●●
■●
■
●
●
●
■
■
●

●
●■

4
2

■

■

8●

■■
■ ■

■■■
■
■
■
■

●●● ■
■● ●
●
■
●
●
■
●

0

1

●
■

Îź0Hset
0 mT
2.8 mT

2

Iflux (mA)
Figure 9.4: The average critical current Ic,Avg oscillates with the current applied to the flux
line Iflux for the P and AP states of SQUID 2A-4. After initializing the sample with an
in-plane field of -150 mT, the SQUID oscillation corresponds to the P-state (blue). Applying
successively larger set fields Hset the SQUID will switch abruptly into the AP-state (yellow)
with a relative π-phase shift. The points are the values of Ic,Avg extracted from the I-V
curves using a simple square root fit, Eq. (6.1), while the dashed lines are only to guide the
eye.
SQUID we measured (2A-1), as shown in Fig. 9.5. But in all of the other samples, and
in subsequent runs of that sample, the critical currents were slightly different in the P and
AP states. We do not understand the source of that small difference. We initially surmised
that there might be a small amount of spin-singlet supercurrent traversing the samples, so
that the total observed supercurrent would be the superposition of the spin-singlet and spintriplet parts. If only the latter switches sign, then the total amplitudes would be different in
the two states.
To test that hypothesis, in our lab Victor Aguilar is currently studying a set of junctions,
similar to those described here, but with the order of the F and F 00 layers exchanged. This
“shuffled” configuration does not permit the conversion to long range spin-triplet supercurrent
and thus will be a direct measurement of the spin-singlet contribution to the supercurrent.
We observe that the AP-state is magnetically stable even if Âľ0 Hset is slightly increased

187

Figure 9.5: A 3D plot of a complete minor magnetization loop, for SQUID 2A-1. We show
the oscillations in Ic,Avg versus the flux line current, Iflux , measured after applying different
values of the set field, Âľ0 Hset . All the SQUID oscillations are measured at zero field. Starting
at zero field and increasing Âľ0 Hset in the positive direction, the SQUID switches from the
P to the AP state at +2.0 mT, and will remain in that state even if Âľ0 Hset is increased or
decreased slightly in either direction. Next, returning to zero field, Âľ0 Hset is applied in the
negative field direction until -1.6 mT, where the SQUID switches from the AP state back into
the P state. However, in this SQUID the amplitude of the critical current is approximately
the same in the two states. The phase of the SQUID can thus be controllably set to either 0
or π on demand, and that behavior is reproducible over a large number of magnetic switches.

188

or decreased in either direction. Assuming that the Py layers behave similarly to the StonerWohlfarth model discussed in Ch. 3, the magnetic state of the SQUID should remain stable
over a range of fields between Âľ0 HPy1 < Âľ0 Hset < Âľ0 Hc,Py2 , where Âľ0 Hc,Py1 , and Âľ0 Hc,Py2
are the coercive fields of the Py layer in the elliptical and hex junctions, respectively.† We
observe that after switching into the AP-state, the state is stable over a narrow range of
fields, typically around 1-2 mT, but if Âľ0 Hset is increased further (> Âľ0 Hc,Py2 ) then a
second switching event is observed, evidenced by another abrupt phase shift. We infer that
the range of fields over which the AP-state is stable is representative of the difference between
the coercive fields of the Py layers in the two junctions (µ0 Hc,Py2 − µ0 Hc,Py1 ). The value
of Âľ0 Hc,Py1 and Âľ0 Hc,Py2 may vary slightly from sample to sample, due to differences in
lithography, surface roughness, or defectivity during growth, so it is important to characterize
each SQUID individually and determine the mean value and variance of these two fields.
To further demonstrate that the state of the SQUID is magnetically controllable, after
setting the SQUID into the AP state we apply Âľ0 Hset in the negative direction to switch
the Py magnetization back to its original orientation, completing a minor magnetic loop.
The data shows that SQUID oscillations remain stable in the AP-state until µ0 Hset ≈ −2.4
mT, at which point the magnetization of the Py in the elliptical junction is reversed and the
original P-state SQUID oscillation curve is recovered. For comparison we plot the complete
set of SQUID oscillations, for both directions of the set field in the 3D plot of Fig. 9.3. One
can see that the SQUID oscillations in the P and AP-states for both positive and negative
values of Âľ0 Hset are functionally equivalent, but we often observe that the values of Âľ0 Hc,Py1
on either side of zero are not always symmetric. Typically, we observe that the coercive field
† The tacit assumption being that the coercive field of the Ni layers in either of the junctions exceeds that
of the Py layers in both of the junctions.

189

in the negative direction (or the initial direction of the magnetization) is slightly larger in
magnitude than the coercive field in the positive direction.
In total, nine SQUIDs were measured from four chips (two chips with n = 2 and two chips
with n = 3, corresponding to a total number of [Pd/Co] bilayers of 4 or 6, respectively).
The full set of those data can be found in the Appendix, where the sample numbering
convention (e.g. 2A-4) is listed by the chip name (2A, 3A, etc.) followed by the number
of the SQUID (each chip contains four SQUIDs). As we will discuss in the next section,
in seven of the SQUIDs we measured complete minor loops with reproducible 0-π state
switching between the P and AP states, confirmed by fitting the data to SQUID theory
from Ch. 7. The average normal state resistance of those samples was RN = 13.4 ± 0.9 mΩ,
about half the resistance of the individual spin-triplet junctions from Ch. 8, a reasonable
number given that the SQUID samples have twice the total area (two junctions per SQUID).
Minor loops were measured in most of those seven samples multiple times and displayed 0
π-phase changes even after different thermal cycles, i.e. bringing the sample up to room
temperature and subsequently reinserting it into the Dewar and repeating the measurement
process as previously described. Several of the SQUIDs, while having the same qualitative
behavior (complete minor loops, oscillations in Ic , etc.), were measured to have slightly
different magnetic switching fields Âľ0 Hc,Py1 or Ic amplitudes after different thermal cycles.
We surmise that these small changes over different thermal cycles are attributable to the
Ni layers not being completely magnetized by the -150 mT field in the same way for each
measurement.†
Overall the SQUID oscillation data from samples with n = 3 are very similar to those
† As discussed in Ch. 9, we kept the initialization field below ±150 mT, even though it is known that this
field may not be sufficient to completely magnetize all of the Ni domains at this thickness.

190

with n = 2, aside from a universal reduction in the critical current that is consistent with the
decay from the single junctions observed in Fig. 8.8. The n = 3 SQUIDs were also measured
to have reproducible 0-π switching, however the overall decrease in amplitude of the SQUID
oscillations typically makes the data noisier.
The last two of the nine samples measured did not behave as expected. One of those
samples exhibited SQUID oscillations, but had poor magnetic behavior with the phase appearing to move continuously rather than switching abruptly. The final sample had very large
resistance of > 1Ω and very small Ic , presumably due to a liftoff issue during fabrication.

9.3

SQUID Oscillation Data: Comparison to Theory

The critical current data analyzed using the square root function (e.g. Fig. 9.4) can be
easily fit to the SQUID theory from Ch. 7, to confirm the relative 0-π phase shifts between
the P and AP states. However, we find that due to the relatively small critical current
and noticeable thermal rounding in the I-V curves of these samples, more accurate critical
current values can be extracted using the Ivanchenko-Zil’berman function, Eq. (6.2).
The SQUID oscillations produced by fitting the fits to the Ivanchenko-Zil’berman function, shown in Fig. 9.6, generally have less scatter in Ic+ and Ic− and allow us to extract the
values of the thermal noise temperature Teff . Fitting all the data in a full three dimensional
plot such as Fig. 9.3 with Eq. (6.2) is rather time-intensive, so we fit the P and AP states
at only two values of Âľ0 Hset . As shown in Fig. 9.6, the resulting SQUID oscillations have
larger amplitude and maximum critical current compared with the data fit using the square
root function (Fig. 9.4). Typically Teff ≈ 40-80 K in the Ivanchenko-Zil’berman fits, which is
slightly larger than the values measured in individual Josephson junctions discussed in Ch. 6

191

15

✶

✶✶✶

✶

10

●

●
●

✶

Ic+ , Ic- (ÎźA)

✶

✶

✶

●

●
●

5

✶

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●

0✶

✶

●
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✶

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-2

✶

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✶

✶

●

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●
✶

●

●

✶

✶
✶
✶

✶

0
Iflux (mA)

✶

●

●

●

●

✶

✶

●

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✶

✶
✶

-1

✶

✶

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-15

✶

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-10

● ● ●
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✶✶✶

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-5

✶

✶

✶✶✶

1

●

● ●

●

●

✶
✶

✶

2

Figure 9.6: Fitting the P and AP states from SQUID 2A-4. At a given applied flux current
Iflux , each data point represents the positive or negative critical current values extracted
from a measured I-V curve that was fit to the Ivanchenko Zil’berman function Eq. (6.2)
for the P (circles) and AP (stars) states. The resulting oscillation data has good agreement
with the SQUID theory from Ch. 7, where the lines represent the nonlinear least-squares fits
for the P (blue) and AP (yellow) states. The associated best-fit parameters and errors are
listed in Table 9.1, which confirms the relative π-phase shift between the two states. The P
and AP states shown were measured after applying set fields, Âľ0 Hset , of 0 mT and 2.8 mT,
respectively, after the sample was initialized in a set field of -150 mT.
or Ch. 8, varying somewhat sample to sample.

Table 9.1: Best-fit parameters for the P and AP states for SQUID 2A-4. The value of ÎąL
in the fit is fixed at zero and the total inductance L is fixed at the nominal value of 9 pH,
thus determining βL . The phase difference between the two states is ≈ π (φshift is listed in
units of 2π).
βL

ÎąI

Ic+ (ÂľA)

|Ic− | (µA)

φshift

P

4.67 ×10−2

(1.50 ± 0.06)×10−1

5.20 Âą 0.02

5.52 Âą 0.02

0.329 Âą 0.001

AP

6.27×10−2

(-1.60 ± 1.75)×10−2

7.04 Âą 0.06

7.34 Âą 0.06

-0.180 Âą 0.002

State

AP
φP
shift − φshift

0.509 Âą 0.003

The SQUID oscillations are then fit to SQUID theory described in Ch. 7, to confirm the
0-π phase changes between the P and AP states and to extract quantitative information

192

about the critical currents in each of the junctions. Best-lines fits are shown in Fig. 9.6 with
the corresponding fit parameters and their uncertainties listed in Table 9.1. Similar plots for
each of the other six SQUIDs measured can be found in the Appendix.
As expected from the design geometry, the SQUID oscillations are symmetric due to
balanced inductances in the two arms of the SQUID. We find that better quality fits are
obtained if we fix the inductance asymmetry parameter (Eq. (7.19)) to ÎąL = 0, reducing the
number of free fitting parameters. Thus in all the fits that follow we set ÎąL = 0. In both the
P and AP states, the fits produce a very small screening parameter, for example βL ≈ 0.047
for the data shown in Fig. 9.6, allowing the depth of the modulation to be very deep. As
the data indicates, often the value of Ic at the oscillation minima are very close to zero.
Compared to the spin-singlet SQUID data from Ch. 7, the fits to the spin-triplet SQUIDs
also have a much smaller critical current asymmetry parameter ÎąI . Due to the small values
of βL and αI the Ic+ and Ic− data are hardly shifted with respect to each other.
Most importantly, the relative phase shift between the P and AP state is almost exactly π
within the margin of error, as shown in the last column of Table 9.1 where the phase difference
is listed in units of 2π. Table 9.2 contains a summary of all seven samples measured, each of
which shows a π phase difference between the P and the AP states. Table 9.2 also shows that
when most of the samples transition from the P to the AP state, one of the two junctions
critical current changes, while the critical current of the other junction stays essentially
constant. However, for SQUIDs 2A-2 and 4A-4 the current in both of the junctions appears
to change. We do not fully understand this issue, given the very good agreement between
the measured data and the fits.

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Table 9.2: Summary of the the seven spin-triplet SQUID samples measured. Fitting the data
to the SQUID theory from Ch. 7, and fixing the total inductance to the nominal value of
9 pH, we find that the phase difference between the P and the AP states in each of the
SQUID is ≈ π (φshift is listed in units of 2π). Moreover, from the fit parameters we can
extract the values of the critical current in each of the two junctions. Typically, we find that
between the two states only the critical current in the elliptical junction (Ic1 ) changes in
magnitude, while the critical current in the other junction (Ic2 ) is approximately constant.
The value 2n is the total number of [Pd/Co] bilayers in the F layer.
SQUID Name

2n

2A-1

4

2A-2

2A-3

2A-4

3A-3

4A-1

4A-2

9.4

4

4

4

6

6

6

State

Ic1 (ÂľA)

Ic2 (ÂľA)

P

6.65 Âą 0.08

4.20 Âą 0.08

AP

6.90 Âą 0.12

4.18 Âą 0.12

P

5.66 Âą 0.10

5.61 Âą 0.10

AP

4.09 Âą 0.14

4.02 Âą 0.14

P

4.53 Âą 0.12

4.56 Âą 0.12

AP

6.88 Âą 0.07

4.67 Âą 0.07

P

4.56 Âą 0.04

6.16 Âą 0.04

AP

7.30 Âą 0.15

7.08 Âą 0.15

P

0.80 Âą 0.30

0.79 Âą 0.30

AP

1.99 Âą 0.02

1.34 Âą 0.02

P

0.60 Âą 0.33

3.16 Âą 0.33

AP

1.45 Âą 0.03

3.87 Âą 0.03

P

1.32 Âą 0.04

2.41 Âą 0.04

AP

3.01 Âą 0.01

1.61 Âą 0.01

AP
φP
shift − φshift

0.491 Âą 0.005

-0.458 Âą 0.004

0.480 Âą 0.004

0.509 Âą 0.003

-0.481 Âą 0.010

0.618 Âą 0.005

-0.507 Âą 0.004

Repeated Switching Experiment

To gather statistics on the reproducibility of the magnetic switching of these devices, and
as a preliminary test of their robustness as a magnetic memory, we fixed the value of Iflux
near where the SQUID oscillation is at a minimum in the P-state (or a maximum in the AP194

Âľ0

:

-3.2 mT

2.8 mT

Figure 9.7: Repeated switching between the P and the AP-states. The SQUID was repeatedly
cycled between the P and the AP states a thousand times. To achieve a large Ic contrast
between the two states, the current in the flux line was fixed at -0.2 mA, where Ic is at a
maximum in the P-state (and a minimum in the AP-state), as shown in Fig. 9.4. We plot
Ic,Avg , obtained from fitting the measured I-V curves to the square root function (Eq. (6.1)),
versus the number of times the junction has been switched (left). No major switching errors
occur after a thousand cycles, with narrow spreads in Ic for each of the two states. We plot
a histogram where the Ic values have been binned into units of 0.2 ÂľA (right).
state), for example at -0.2 mA in Fig. 9.4. Then, we repeatedly alternated Âľ0 Hset between
two field values just beyond Âľ0 Hc,Py1 in the positive and negative directions, to switch the
SQUID back and forth between the P and AP states. In Fig. 9.7(a), we plot the value of
Ic,Avg (fit using the square root function) versus the number of switches. Over the course
of 1000 repeated switches the SQUID successfully alternated back-and-forth between the P
and the AP states without any major errors. We arbitrarily define the “error” threshold to
be when the value of a given Ic , in either the P or the AP states, at a given Iflux , deviates
by more than 20 % of its average value in that given state.
To get an idea for the spread in distribution of Ic ’s, in Fig. 9.7(b) we bin the Ic values
with even and odd switch numbers into two histograms, where the size of each bin represents
195

0.2 ÂľA. The distributions of Ic in each state are very tight, and the two do not overlap so
the P and AP states can, in principle, serve as the logical 0 and 1 states of the memory.

9.5

SQUID Measurements Below 4.2 K in a Dilution Refrigerator

We independently measured the spin-triplet SQUID samples at temperatures below 4.2
K in an Oxford dilution refrigerator, due to several motivating factors. First, the I-V curves
from the previous set of measurements all appeared to be thermally rounded near the critical
current, particularly those with very small Ic < 1 ÂľA, resulting in somewhat larger Teff . Second, the maximum Ic values in the spin-triplet SQUIDs were slightly smaller than expected,
based on the individual junction results from Ch. 8. We suspected that additional electromagnetic interference, either from the SQUID itself, the additional measurement equipment
(such as the flux line current source), or external sources, might be limiting the Ic of the
devices. The low noise rf-SQUID based electronics system previously described in Ch. 5, has
superb RMS voltage noise (≈ 6 pV measuring over 10 power line cycles), but the measurement probe used in conjunction with it has no rf-filtering.
The Oxford dilution refrigerator unit in our lab has rf-filtered lines, comparatively better
isolation, and can measure over a range of temperatures from 40 mK - 5 K. On the other
hand, the dilution refrigerator system is not equipped with the sensitive rf-SQUID electronics.
Instead, we used a lock-in amplifier to measure ∂V /∂I versus the measurement current, which
can be integrated to obtain an I-V curve.
We wished to address three main questions: What is the temperature dependence of
Ic ? Is Ic in the spin-triplet SQUIDs larger while measured in a system with better rffiltering? And finally, would the I-V curves have less thermal rounding? To address these

196

●

●
●

8

●

Ic (ÎźA)

●
●

6

●
●

4

●

●

2
0
2.0

2.5

3.0
3.5
4.0
Temperature (K)

4.5

5.0

Figure 9.8: Critical current versus temperature for SQUID 2A-2 in a dilution refrigerator.
The critical current decreases inversely with temperature, but is not substantially larger at
1-2 K compared to the previous measurements at 4.2 K. The lines connecting the data points
are only to guide the eye.
questions we measured the temperature dependence of the critical current, in addition to
several oscillations versus the applied flux for two different SQUID samples. Unfortunately,
due to a small helium leak in the dilution refrigerator we were only able to measure the
SQUIDs from about 5 K down to just above 1 K.
To answer the first question is that the value of the critical current only increases modestly
with decreasing temperature. Figure 9.8 shows the temperature dependence of Ic at a fixed
value of Iflux for a different SQUID sample than previously discussed (SQUID 2A-2, other
plots from this sample can be found in the Appendix). We measured that the critical current
increases with decreasing temperature, but is only about 30 % larger at 2 K than at 4.2 K.
The answer to the second question seems to be no: for two different samples the critical
current at 4.2 K measured while using the dilution refrigerator system is almost identical to
the value measured using the rf-SQUID comparator measurement system. To address the
third question, we fit some of the I-V curves for SQUID 2A-1, shown in Fig. 9.9, to the

197

Ivanchenko-Zil’berman function. Even at 2.0 K we find that the I-V curves display similar
thermal rounding in the dilution refrigerator system as in the rf SQUID comparator system.
The temperature dependence of Ic for that sample at zero flux current shown in Fig. 9.9
behaved similarly to that of SQUID 2A-2, shown in Fig. 9.8.

Figure 9.9: Critical current versus flux-line current and temperature for SQUID 2A-1 in a
dilution refrigerator. The measured ∂V /∂I curves are converted into I-V curves by numerical
integration. With the sample temperature near 2.0 K the I-V curves shown in (a) display
similar thermal rounding compared to junctions measured in the low noise rf-SQUID system.
In (a) each I-V curve corresponds to the different values of the applied flux current (ranging
from Âą1 mA) and are successively offset by 0.25 ÂľV for clarity. The lines represent fits to the
Ivanchenko-Zil’berman function, Eq. (6.2). In (b) we plot the critical current at 2.0 K (fit
using the square root function) versus the applied flux current, where the dashed line is only
to guide the eye. In (c) we fix the flux current at 0 mA and show that the critical current
increases with decreasing temperature, but is not substantially larger at 1-2 K compared to
the previous measurements at 4.2 K.

198

9.6

Conclusion

In this chapter we have shown the first demonstrations of phase-controllable Josephson
junctions that carry long range spin-triplet supercurrent. In seven different SQUID devices,
we measured that the ground-state phase of the junction can switched between the zero or
π-phase states. Many of the junctions can be reliably switched between the two states on
the order of 1000 times without error.
Such junctions could be useful for superconducting single-flux quantum circuits [116]; in
particular, we envision a superconducting memory based on such a device [18, 23]. There
are easier ways to make a phase-controllable memory device; as shown in Ch. 7, the simpler
spin-valve Josephson junction containing only two magnetic layers of appropriate thicknesses
can also exhibit controllable 0-π switching. In those devices, the physical mechanism of the
0-π phase shift is different; it relies on the accurate tuning of the thicknesses of the two
magnetic layers so that the total phase shift acquired by an electron pair traversing the
sample is closer to an even or odd multiple of π when the two magnetizations are parallel or
antiparallel.
In the spin-triplet devices presented in this chapter, the 0-π switching is caused by spin
rotations rather than phase accumulation, so the behavior is less sensitive to the exact
thicknesses of the F 0 and F 00 layers (as long as those layers are not close to the minima
in their own 0-π transition curves). We believe that is one explanation for the high degree
of consistency between the seven spin-triplet SQUID samples measured, compared to the
spin-valve SQUIDs discussed in Ch. 7. In addition, the central SAF geometry used in the
spin-triplet junctions may help improve the reliability of the device, canceling stray magnetic
fields that would otherwise be detrimental to the other in-plane ferromagnetic layers.

199

We also demonstrated that making the shape of the fixed junction in the SQUID a large
aspect ratio “hex” geometry instead of an ellipse could potentially be an important feature
of future test SQUIDs. Note that in cryogenic memory applications such as JMRAM, only
one controllable ferromagnetic junction is required, so the hex-shaped junction would not be
used. Also, the hex junctions still require further experimental study; we can only indirectly
guess that the hex shape played a role in the relative stability of these junctions. Detailed
magnetometry studies using magnetic bit arrays of the type discussed in Ch. 5 should be
performed on elliptical and hex junctions of different aspect ratios to substantiate the above
claim.
One potential downside to these spin-triplet junctions compared to those containing spinvalves is that the critical currents are considerably smaller. There are several different means
by which the spin-triplet supercurrent could possibly be enhanced using the scheme presented
in this chapter. For example, it is not clear that we have completely maximized the conversion
from spin-singlet to the long range spin-triplet correlations. If the Ni layers have multiple
domains and are not completely aligned by the 150 mT initialization field, it could lead to
a decreased critical current. Remeasuring these samples with larger initialization fields to
better align the Ni layers would be one simple way of testing this hypothesis (we would not
want to disturb the out-of-plane magnetization F layer, but the Âą 150 mT used here may
have been somewhat conservative). One could also try replacing the F 00 = Ni layer with
another ferromagnet that is more likely to be single domain. Also, in the F layer we used
Ru as the coupling layer in the center of the SAF, which we have found decreases the critical
current in S/N/F/N/S junctions by roughly a factor of two compared to Cu [144]. Using
a different coupling layer in the center of the SAF may also be worth exploring. Finally,
using a magnetically weaker ferromagnetic material for F 0 instead of Py could also help in
200

generating larger amplitude critical currents.

201

Chapter 10

Conclusion

In this final chapter we summarize the results presented in this thesis, revisit some open
questions to be explored in future work, and discuss the outlook for energy efficient superconducting memory.
After reviewing the basic theoretical concepts behind superconductivity and ferromagnetism in Chs. 2 and 3, in Ch. 4 we discussed how the the interplay of superconductivity
and ferromagnetism can be used in various types of ferromagnetic Josephson junctions. We
described various schemes by which ferromagnetic Josephson junctions can be manipulated
to carry spin-singlet or spin-triplet supercurrent, theoretical concepts that underpin all of
the experiments presented in the later chapters. After discussing the experimental methods,
procedures, and equipment in Ch. 5 that were used to fabricate and measure the Josephson
junction and SQUID samples in our lab, in Ch. 6 we briefly outlined a scheme for using
ferromagnetic Josephson junctions as a phase-sensitive superconducting memory.
In Ch. 6 we also presented four separate experimental materials characterization studies
of Josephson junctions containing a central ferromagnetic layer of either Ni65 Fe15 Co20 , Py,
Pd97 Fe3 , or Ni. Systematic studies of the material properties of such junctions, including
the 0-π transition thickness, switching fields, and shifts in the Fraunhofer patterns versus
F -layer thickness were examined.
Of the materials studied, we eventually settled upon Py and Ni as the respective free
and fixed layers for both our spin-valve and spin-triplet junctions. In general, we value Py

202

primarily for its excellent magnetic properties and controllable switching at a low magnetic
fields. Meanwhile, Ni, despite it’s unfavorable magnetic properties was chosen largely because
of it’s ability to permit large critical currents to pass through Josephson junctions.
In Ch. 6 we showed that NiFeCo is another potential candidate alloy for a fixed layer.
That work led to several follow-up studies by our group to examine its applicability in
cryogenic memory devices.† Also, we examined the enticing possibility of using much weaker
ferromagnetic layers for the free layer, such as dilute alloys of PdFe, which would allow us
to make the free layer much thicker yet permit large critical currents. While the Josephson
junctions containing Pd97 Fe3 showed those layers had comparable switching fields to Py, and
with relatively small magnetization, unfortunately the magnetic switching was not as sharp
as desired. This has since led to other studies in our group (still in progress) in which we
are increasing the Fe concentration of the alloy to improve its magnetic properties.
In Ch. 7, we discussed the first experimental demonstration that the phase state of a
Josephson junction can be controllably switched between 0 and π, i.e. a phase-sensitive
cryogenic one-bit memory device using spin-singlet supercurrent. The phase-sensitive detection was achieved by adding two ferromagnetic Josephson junctions with a “spin-valve”
configuration into a SQUID loop. My main contribution to that effort was the development
of a general asymmetric SQUID fitting program necessary for the unambiguous proof that
the devices switched between the phase values of 0 and π. The Ni/Py spin-valve junctions
similar to those discussed in Ch. 7 have since gone on to be a central part of an operational
superconducting memory cell by our collaborators at Northrop Grumman Corporation [23].
In Ch. 8, we presented magnetic and transport measurements on a new configuration
† NiFeCo was later implemented into spin-valve SQUID samples by Bethany Niedzielski, replacing Ni
as the hard layer, with mixed success. Those Py/NiFeCo-based spin-valve junctions suffered from some
variability in the magnetic switching and overall had smaller critical current compared to the Py/Ni based
spin-valves. Refer to Ref. [144] for more details.

203

of materials by which we are able to generate spin-triplet supercurrent through the fine
control of three ferromagnetic layers with noncolinear magnetizations. The central layer
is a synthetic antiferromagnet (SAF) composed of [Pd/Co] multilayers on either side of a
Ru spacer with magnetization perpendicular to the plane. The other two ferromagnetic
layers, either Py or Ni, have in-plane magnetization. Contrary to other approaches, we
demonstrated the viability of our scheme in which the SAF has perpendicular magnetic
anisotropy, which could be an important step for future devices. Our approach allows the
junctions to be patterned into non-circular geometries to take advantage of shape anisotropy,
meanwhile the SAF helps cancel the stray fields that were thought to cause problems with
other out-of-plane spin-triplet control experiments [109]. We demonstrated that the electron
pairs in the central layer of the junctions in Ch. 8 predominantly have long range spin-triplet
correlations. This was demonstrated by observing the characteristic slow decay of the critical
current in those junctions versus the layer thickness compared to other junctions without
the in-plane layers that carry only spin-singlet supercurrent. Work is underway by Victor
Aguilar to characterize the extent to which the spin-singlet supercurrent decays in these type
of junctions.
The junctions in Ch. 8 were then implemented in Ch. 9 in SQUIDs carrying spin-triplet supercurrent. We verified experimentally one of the most salient predictions of the spin-triplet
theory, namely that a Josephson junction containing three magnetic layers with noncolinear
magnetizations should exhibit a ground-state phase shift of either zero or π depending on
the relative orientations of those magnetizations. We showed in seven different spin-triplet
SQUID devices that the phase can be reliably switched between the zero or π states. A
number of the SQUIDs measured were able to switch between the two phase-states on the
order of a thousand times without error, demonstrating that this type of phase-controllable
204

junction could also be used as a novel memory element in a superconducting computer.
The future for superconducting computing is bright. As highlighted in Ch. 1, recent
advances in superconducting logic and the potential for energy-efficient solutions to growing energy problems, as well as the rise of quantum computing, have helped reinvigorate
the field. Superconducting memory using ferromagnetic Josephson junctions, such as those
presented in this thesis, still need major development before being implemented in largescale superconducting computers. However, in the last six years significant proof-of-concept
demonstrations by many groups including our own, such as those presented in Ch. 7 and
Ch. 9 have been realized experimentally.
Simultaneously controlling superconductivity and ferromagnetism in Josephson junction
based devices is difficult, given the inherent material science and fabrication challenges,
as well the relative difficulty in theoretically examining junctions with complex magnetic
multilayers. Hopefully the experiments presented in this thesis, beyond their interest to basic
science, have highlighted some of the features that ferromagnetic Josephson junctions bring
to the field of superconducting computing, in addition to exposing some of the challenges that
need to be solved for progress to continue. Novel approaches to superconducting memory
such as spin-triplet based devices are rich for exploration and it will be exciting to see the
role they may play in the future development of the field.

205

APPENDIX

206

Spin-triplet SQUID data

In this appendix we provide the complete data series of measurements and analysis of
SQUID samples carrying spin-triplet supercurrent from the experiments described in Ch. 9.
The sample numbering convention (e.g. 2A-4) is to first list the chip name (2A, 3A, etc.)
followed by the number of the SQUID (each chip contains four SQUIDs). The plots and
analysis for SQUID 2A-4 can be found in Ch. 9.

SQUID 2A-1
Âľ0

:

-10 mT

2.4 mT

Figure 1: Repeated switching between the P and the AP-states from SQUID 2A-1. We plot
Ic,Avg , obtained from fitting the measured I-V curves to the square root function (Eq. (6.1)),
versus the number of times the junction has been switched (left). No major switching errors
occur after a 500 switches, with narrow spreads in Ic for each of the two states. We plot a
histogram where the Ic values have been binned into units of 0.2 ÂľA (right).

207

Table 1: Best-fit parameters for the P and AP states for SQUID 2A-1. The value of ÎąL in
the fit is fixed at zero. The phase difference between the two states is ≈ π (φshift is listed in
units of 2π).
AP
φP
shift − φshift

βL

ÎąI

Ic+ (ÂľA)

|Ic− | (µA)

φshift

P

4.47×10−2

(2.27 ± 0.01)×10−1

5.25 Âą 0.04

5.60 Âą 0.04

0.156 Âą 0.002

AP

4.80×10−2

0.15)×10−1

5.34 Âą 0.06

5.74 Âą 0.06

-0.335 Âą 0.003

State

(2.45 Âą

10

✶✶✶

●

✶

●
●

✶

✶

●

●

Ic+ , Ic- (ÎźA)

5

●

●

✶

●

✶

✶

●

●

✶

✶
✶

✶

●

✶

●

●

●

✶

✶

✶

● ● ●

✶

●

✶

●

✶

●

●

●

✶

✶

✶

●

✶

●

●

● ●

●

●

✶

●

✶

● ●

●

✶

●

✶
●

✶

✶

●

✶

●

✶

●

●

✶✶

● ●

●

●

0

✶
●

✶

-5

✶

●

✶

●

●

●

✶

✶
✶✶✶

-2

✶

● ●

●

●

●
●

●
● ● ●

-1

✶
✶

✶

✶

●

✶

✶

●

✶

●
●

✶

●

✶

●
●
● ●

0
Iflux (mA)

●

✶

✶

●

●

●

●

●

✶

✶

✶

●

✶
●

●

●

✶

✶

✶
✶

●
●

✶

●
●

-10

✶

●

✶

●

✶✶✶

✶

●
●

✶

✶

✶✶✶

0.491 Âą 0.005

✶

✶

✶
✶✶✶

1

✶

●

✶

✶
●

●

✶
●

●
●

● ●

✶

●

✶

✶✶

✶

2

Figure 2: Fitting the P and AP states from SQUID 2A-1. At a given applied flux current
Iflux , each data point represents the positive or negative critical current values extracted
from a measured I-V curve that was fit to the Ivanchenko Zil’berman function Eq. (6.2)
for the P (circles) and AP (stars) states. For a few I-V curves the Ivanchenko Zil’berman
fits did not converge, thus there are some missing data points in the plot. The resulting
oscillation data has good agreement with the asymmetric SQUID theory from Ch. 7, where
the lines represent the nonlinear least-squares fits for the P (blue) and AP (yellow) states.
The associated best-fit parameters and uncertainties are listed in Table 1, which confirms the
relative π-phase shift between the two states. The P and AP states shown were measured
after applying set fields, Âľ0 Hset , of 0 mT and 2.3 mT, respectively, after the junctions were
initialized in a -150 mT set field.

208

SQUID 2A-2

Figure 3: A 3D plot of a complete minor magnetization loop, for SQUID 2A-2. We show
the oscillations in Ic,Avg versus the flux line current, Iflux , measured after applying different
values of the set field, Âľ0 Hset . All the SQUID oscillations are measured at zero field. Starting
at zero field and increasing Âľ0 Hset in the positive direction, the SQUID switches from the
P to the AP state and will remain in that state even if Âľ0 Hset is increased or decreased
slightly in either direction. Next, returning to zero field, Âľ0 Hset is applied in the negative
field direction until the SQUID switches from the AP state back into the P state. The phase
of the SQUID can thus be controllably set to either 0 or π on demand.

Table 2: Best-fit parameters for the P and AP states for SQUID 2A-2. The value of ÎąL in
the fit is fixed at zero. The phase difference between the two states is ≈ π (φshift is listed in
units of 2π).
βL

ÎąI

Ic+ (ÂľA)

|Ic− | (µA)

φshift

P

4.84×10−2

(-0.37 ± 1.51)×10−2

5.56 Âą 0.04

5.71 Âą 0.03

-0.194 Âą 0.001

AP

3.53×10−2

(-0.79 ± 3.08)×10−2

4.01 Âą 0.04

4.11 Âą 0.04

0.265 Âą 0.003

State

209

AP
φP
shift − φshift

-0.458 Âą 0.004

●

10
✶✶

Ic+ , Ic- (ÎźA)

● ●

●

✶

✶
●
●

✶

✶

●

●

✶

●

✶

✶

✶

-5

✶

✶

●

✶

●
✶

✶✶✶

✶

●

●

●

●

✶✶✶

●

-1

✶✶✶

●
●

● ●

✶

●
● ●

0
Iflux (mA)

✶

●

✶

✶
●

✶

✶

●

●

✶

●
●

✶✶
✶

✶

✶

●

✶

✶

●

●

✶

●

●

✶

●

●

●

✶

✶

●

✶

✶
✶

●

✶✶

●

✶
●

●

●

-2

✶

✶
✶

●

✶

✶
●

●
✶
●
✶

✶

●

✶

●

✶

✶

✶✶
✶

●

●

●

●

✶

●

✶

✶
✶✶ ✶

✶
✶

●

●
●

●

✶

●
●

-10

●

● ●

●

●

✶

✶

●

●

●

●
✶

●

✶

●

0

✶

✶✶✶

●

✶

✶

● ●

●

●

●

✶✶

5

●

●

●

●

✶

●

✶

✶

●

✶

✶
●

✶

●

✶

✶
●

●

✶

✶✶✶

✶

●
●

●
●
●

1

●

●

●

● ●

●

2

Figure 4: Fitting the P and AP states from SQUID 2A-2. At a given applied flux current
Iflux , each data point represents the positive or negative critical current values extracted
from a measured I-V curve that was fit to the Ivanchenko Zil’berman function Eq. (6.2)
for the P (circles) and AP (stars) states. The resulting oscillation data has good agreement
with the asymmetric SQUID theory from Ch. 7, where the lines represent the nonlinear
least-squares fits for the P (blue) and AP (yellow) states. The associated best-fit parameters
and uncertainties are listed in Table 2, which confirms the relative π-phase shift between the
two states. The P and AP states shown were measured after applying set fields, Âľ0 Hset , of
0 mT and -2.8 mT, respectively, after the junctions were initialized in a -150 mT set field.

210

SQUID 2A-3

Figure 5: A 3D plot of a complete minor magnetization loop, for SQUID 2A-3. We show
the oscillations in Ic,Avg versus the flux line current, Iflux , measured after applying different
values of the set field, Âľ0 Hset . All the SQUID oscillations are measured at zero field. Starting
at zero field and increasing Âľ0 Hset in the positive direction, the SQUID switches from the
P to the AP state and will remain in that state even if Âľ0 Hset is increased or decreased
slightly in either direction. Next, returning to zero field, Âľ0 Hset is applied in the negative
field direction until the SQUID switches from the AP state back into the P state. The phase
of the SQUID can thus be controllably set to either 0 or π on demand.

Table 3: Best-fit parameters for the P and AP states for SQUID 2A-3. The value of ÎąL in
the fit is fixed at zero. The phase difference between the two states is ≈ π (φshift is listed in
units of 2π).
βL

ÎąI

Ic+ (ÂľA)

|Ic− | (µA)

φshift

P

3.95×10−2

(-0.33 ± 2.20)×10−2

4.50 Âą 0.04

4.59 Âą 0.04

0.069 Âą 0.002

AP

5.02×10−2

(1.91 ± 0.09)×10−1

5.73 Âą 0.04

5.82 Âą 0.03

-0.411 Âą 0.002

State

211

AP
φP
shift − φshift

0.480 Âą 0.004

Âľ0

:

1.5 mT

-7 mT

Figure 6: Repeated switching between the P and the AP-states from SQUID 2A-3. We plot
Ic,Avg , obtained from fitting the measured I-V curves to the square root function (Eq. (6.1)),
versus the number of times the junction has been switched (left). When the window of fields
in which (Âľ0 Hc,Py1 < Âľ0 Hset < Âľ0 Hc,Py2 ) is small, the error rates increase. In switching
to the AP state the the error rate was ≈ 16 %. We presume that the variability in the set
field occasionally will cause the Py layer in both junctions to switch). However, the SQUID
always switched back into the P-state. We plot a histogram where the Ic values have been
binned into units of 0.2 ÂľA (right).

212

10✶ ✶

✶✶✶✶

✶
✶

Ic+ , Ic- (ÎźA)

●

5

●

✶

✶

●

✶

✶

✶

✶

●

✶

● ●

●

✶

●
●

✶
✶

●

✶✶

●

✶

●

●

✶

●

✶✶

●

●

●
●

● ●

●

✶

●

●

✶

✶

✶

●

●

✶

✶

●

●
●

●
● ● ●

●
✶

✶

✶

✶
✶

-1

●

✶✶✶

● ● ●

●

●

✶

●

●

✶

✶

0
Iflux (mA)

✶

✶
✶

✶
✶✶✶

1

●

✶

✶

●

●

✶

●

●
●

●

✶

✶

●

✶

●
●

●

✶

✶

●

✶✶✶✶

-2

✶✶

●

●

✶

✶

✶

✶

●

✶✶✶

✶

●

●

●
●

✶

●

●

✶

✶

●

●
●

●

-10

✶

✶

●

●

✶

✶

-5

●

●

●
●

●

●
✶

●
✶

✶

✶

●

●

✶✶✶

✶

● ● ●

✶

✶

●

✶

✶

●

●

0

●

✶

●

●

● ●

●

●

●

✶✶✶

✶

✶

●
●
✶

●
●

● ●

✶

●

✶

✶

✶

✶✶

2

Figure 7: Fitting the P and AP states from SQUID 2A-3. At a given applied flux current
Iflux , each data point represents the positive or negative critical current values extracted from
a measured I-V curve that was fit to the Ivanchenko Zil’berman function Eq. (6.2) for the P
(circles) and AP (stars) states. The resulting oscillation data has good agreement with the
asymmetric SQUID theory from Ch. 7, where the lines represent the nonlinear least-squares
fits for the P (blue) and AP (yellow) states. The associated best-fit parameters and errors
are listed in Table 3, which confirms the relative π-phase shift between the two states. The
P and AP states shown were measured after applying set fields, Âľ0 Hset , of 0 mT and 1.6
mT, respectively, after the junctions were initialized in a -150 mT set field.

213

SQUID 3A-3

Figure 8: A 3D plot of a complete minor magnetization loop, for SQUID 3A-3. We show
the oscillations in Ic,Avg versus the flux line current, Iflux , measured after applying different
values of the set field, Âľ0 Hset . All the SQUID oscillations are measured at zero field. Starting
at zero field and increasing Âľ0 Hset in the positive direction, the SQUID switches from the
P to the AP state and will remain in that state even if Âľ0 Hset is increased or decreased
slightly in either direction. Next, returning to zero field, Âľ0 Hset is applied in the negative
field direction until the SQUID switches from the AP state back into the P state. The phase
of the SQUID can thus be controllably set to either 0 or π on demand.

Table 4: Best-fit parameters for the P and AP states for SQUID 3A-3. The value of ÎąL in
the fit is fixed at zero. The phase difference between the two states is ≈ π (φshift is listed in
units of 2π).
βL

ÎąI

Ic+ (ÂľA)

|Ic− | (µA)

φshift

P

0.69×10−2

(0.16 ± 3.77)×10−1

0.84 Âą 0.03

0.75 Âą 0.03

-0.072 Âą 0.009

AP

1.45×10−2

(1.97 ± 0.09)×10−1

1.67 Âą 0.01

1.66 Âą 0.01

0.409 Âą 0.002

State

214

AP
φP
shift − φshift

0.481 Âą 0.010

Âľ0

:

-10 mT

2.4 mT

Figure 9: Repeated switching between the P and the AP-states from SQUID 3A-3. We plot
Ic,Avg , obtained from fitting the measured I-V curves to the square root function (Eq. (6.1)),
versus the number of times the junction has been switched (left). After 1000 cycles, in
switching to the P state the the error rate was ≈ 4 %. We plot a histogram where the Ic
values have been binned into units of 0.2 ÂľA (right).

215

✶

✶

3

✶✶

✶
✶

Ic+ , Ic- (ÎźA)

1

✶

●

●

✶

●
●

✶✶

●

●

●

●

✶✶
●

✶
●

●

✶
✶

✶

✶

-2

●

✶

✶✶
✶

✶
●

●

✶

✶✶

✶

●

●

● ● ●

✶

✶

●
●

✶✶

●

✶

✶

●

●
●

●

● ● ●
●

●

✶✶
●

●

●

✶

✶
✶

✶

✶✶

●

✶

✶

✶
●

● ● ● ● ●

●

-2

● ● ●

✶

● ●

●

-1

●

●
●

✶

✶
✶

●
✶

✶

✶✶

✶

✶

● ● ● ●

-3✶ ✶

✶

✶

●

0

✶

✶

✶

✶

2

✶✶✶

✶
●

●

✶
✶

✶
●

✶✶

-1

✶✶
●

✶
● ● ●

●

✶
✶
✶

✶✶

✶

✶

0
Iflux (mA)

✶

✶✶

●

●
●

● ● ●

✶
✶

✶
✶

✶

✶
✶

●

●

● ●

●

●
✶

✶

✶
✶

●

●

✶

✶

●
● ● ● ● ●

●
●

✶

✶

1

✶✶✶

✶

2

Figure 10: Fitting the P and AP states from SQUID 3A-3.At a given applied flux current
Iflux , each data point represents the positive or negative critical current values extracted
from a measured I-V curve that was fit to the Ivanchenko Zil’berman function Eq. (6.2) for
the P (circles) and AP (stars) states. The Ivanchenko Zil’berman fits occasionally do not
converge, typically at small values of the critical current, thus there are some missing data
points in the plot. The resulting oscillation data has good agreement with the asymmetric
SQUID theory from Ch. 7, where the lines represent the nonlinear least-squares fits for the
P (blue) and AP (yellow) states. The associated best-fit parameters and errors are listed
in Table 3, which confirms the relative π-phase shift between the two states. The P and
AP states shown were measured after applying set fields, Âľ0 Hset , of 0.0 mT and 2.4 mT,
respectively, after the junctions were initialized in a -150 mT set field.

216

SQUID 4A-1

Figure 11: A 3D plot of a complete minor magnetization loop, for SQUID 4A-1. We show
the oscillations in Ic,Avg versus the flux line current, Iflux , measured after applying different
values of the set field, Âľ0 Hset . All the SQUID oscillations are measured at zero field. Starting
at zero field and increasing Âľ0 Hset in the positive direction, the SQUID switches from the
P to the AP state and will remain in that state even if Âľ0 Hset is increased or decreased
slightly in either direction. Next, returning to zero field, Âľ0 Hset is applied in the negative
field direction until the SQUID switches from the AP state back into the P state. The phase
of the SQUID can thus be controllably set to either 0 or π on demand.

Table 5: Best-fit parameters for the P and AP states for SQUID 4A-1. The value of ÎąL in
the fit is fixed at zero. The phase difference between the two states is ≈ π (φshift is listed in
units of 2π).
βL

ÎąI

Ic+ (ÂľA)

|Ic− | (µA)

φshift

P

1.72×10−2

(6.81 ± 0.45)×10−1

1.82 Âą 0.02

1.95 Âą 0.02

0.220 Âą 0.025

AP

2.31×10−2

(4.55 ± 0.06)×10−1

2.57 Âą 0.02

2.74 Âą 0.01

0.455 Âą 0.005

State

217

AP
φP
shift − φshift

0.618 Âą 0.005

6

✶
✶

4✶
Ic+ , Ic- (ÎźA)

●

✶✶✶✶

✶
●

✶✶

✶
✶

●

●

●

2

●

●

●

●

✶✶

✶

✶

✶

✶
●

✶

●
✶

●

✶
✶

● ● ● ● ●

✶

✶✶

●

✶

●

● ●

●

●

✶

✶

✶
●

✶

✶✶

●

●

●

● ● ● ●
✶

●

✶

✶

●

●

●

✶✶

● ●

✶✶✶
✶

●

✶
●

● ●

✶
●

●

✶

●

✶

✶

✶
●
✶

●

✶

●

●

●

✶✶

0
-2
● ● ●
● ●

-4✶ ✶
●

●

●

✶

●

✶

-6

✶✶
●

✶
✶

✶✶✶

-2

✶

✶
●

●

✶
✶

● ● ●

● ●

●

●

● ●
● ●
●

✶

✶

✶

✶

✶

✶✶

✶

✶
●

✶

✶

-1

✶✶
●

●

● ●

✶
●

✶

0
Iflux (mA)

●

●

●

●

●

● ● ●

●

✶
✶

✶
✶

✶
✶

✶
●

✶✶✶

1

●

✶✶
✶
● ● ● ✶ ●
●

●

●

●

● ●

✶
✶

✶

✶

✶✶✶

2

Figure 12: Fitting the P and AP states from SQUID 4A-1.At a given applied flux current
Iflux , each data point represents the positive or negative critical current values extracted
from a measured I-V curve that was fit to the Ivanchenko Zil’berman function Eq. (6.2) for
the P (circles) and AP (stars) states. The Ivanchenko Zil’berman fits occasionally do not
converge, typically at small values of the critical current, thus there are some missing data
points in the plot. The resulting oscillation data has good agreement with the asymmetric
SQUID theory from Ch. 7, where the lines represent the nonlinear least-squares fits for the
P (blue) and AP (yellow) states. The associated best-fit parameters and errors are listed
in Table 3, which confirms the relative π-phase shift between the two states. The P and
AP states shown were measured after applying set fields, Âľ0 Hset , of 0.8 mT and 2.0 mT,
respectively, after the junctions were initialized in a -150 mT set field.

218

SQUID 4A-2

Figure 13: A 3D plot of a complete minor magnetization loop, for SQUID 4A-2. We show
the oscillations in Ic,Avg versus the flux line current, Iflux , measured after applying different
values of the set field, Âľ0 Hset . All the SQUID oscillations are measured at zero field. Starting
at zero field and increasing Âľ0 Hset in the positive direction, the SQUID switches from the
P to the AP state and will remain in that state even if Âľ0 Hset is increased or decreased
slightly in either direction. Next, returning to zero field, Âľ0 Hset is applied in the negative
field direction until the SQUID switches from the AP state back into the P state. The phase
of the SQUID can thus be controllably set to either 0 or π on demand.

Table 6: Best-fit parameters for the P and AP states for SQUID 4A-2. The value of ÎąL in
the fit is fixed at zero. The phase difference between the two states is ≈ π (φshift is listed in
units of 2π).
βL

ÎąI

Ic+ (ÂľA)

|Ic− | (µA)

φshift

P

0.16×10−1

(2.93 ± 0.13)×10−1

1.84 Âą 0.02

1.90 Âą 0.02

-0.116 Âą 0.003

AP

0.20×10−1

(-3.05 ± 0.04)×10−1

2.28 Âą 0.01

2.34 Âą 0.01

0.391 Âą 0.001

State

219

AP
φP
shift − φshift

-0.508 Âą 0.004

Âľ0

:

-4 mT

2.4 mT

Figure 14: Repeated switching between the P and the AP-states from SQUID 4A-2. We plot
Ic,Avg , obtained from fitting the measured I-V curves to the square root function (Eq. (6.1)),
versus the number of times the junction has been switched (left). After 1000 cycles, switching
between the P and AP states there was only one major error, though there is some spread
in the critical current value in the AP state. We plot a histogram where the Ic values have
been binned into units of 0.2 ÂľA (right).

220

✶✶

4

✶✶

✶

✶
● ●
● ●

✶
●

Ic+ , Ic- (ÎźA)

●

2

✶

✶

✶
✶✶

●
●

●

✶

●

✶
●

●

✶

✶

✶✶

✶
✶

✶

●

●

●

●

✶

●

✶
✶
✶

✶
●

✶

✶

●

●

✶✶

✶✶

✶

● ●

●

●

✶

✶

●

●
●

●

✶✶

●

✶

✶

●

●
●
●
●

●

●

0

●

-2

● ●
●

✶

✶
●

●

●

✶✶✶✶

✶

✶
●

✶

●

✶✶✶

✶

●

●

✶

●

✶

●

✶

●

✶

-4✶

✶
✶✶✶

●

● ●
● ●

✶

●

✶

✶

✶

✶

-1

✶

●

●

●
●

● ● ●

●

●

✶

✶

0
Iflux (mA)

✶✶✶

✶

●
●

✶
●
●

●

✶

✶
●

✶

●

●

●

✶

✶

✶

✶

●

✶

✶
●

✶

● ●
●

✶

●

✶✶✶

-2

✶✶
✶

●

●

●
✶

✶

●

●

●

●
● ●

●
●

✶

✶

●

✶

✶
✶✶✶

1

✶

✶

2

Figure 15: Fitting the P and AP states from SQUID 4A-2.At a given applied flux current
Iflux , each data point represents the positive or negative critical current values extracted
from a measured I-V curve that was fit to the Ivanchenko Zil’berman function Eq. (6.2) for
the P (circles) and AP (stars) states. The Ivanchenko Zil’berman fits occasionally do not
converge, typically at small values of the critical current, thus there are some missing data
points in the plot. The resulting oscillation data has good agreement with the asymmetric
SQUID theory from Ch. 7, where the lines represent the nonlinear least-squares fits for the
P (blue) and AP (yellow) states. The associated best-fit parameters and errors are listed
in Table 3, which confirms the relative π-phase shift between the two states. The P and
AP states shown were measured after applying set fields, Âľ0 Hset , of 0 mT and 2.0 mT,
respectively, after the junctions were initialized in a -150 mT set field.

221

REFERENCES

222

REFERENCES

[1] D. S. Holmes, A. L. Ripple, and M. A. Manheimer. Energy-efficient superconducting computing—power budgets and requirements. IEEE Trans. Appl. Supercond.,
23(3):1701610, 2013.
[2] M. A. Manheimer. Cryogenic computing complexity program: Phase 1 introduction.
IEEE Trans. Appl. Supercond., 25(3):1–4, 2015.
[3] D. S. Holmes, A. M. Kadin, and M. W. Johnson. Superconducting computing in
large-scale hybrid systems. Computer, 48(12):34–42, 2015.
[4] D. A. Buck. The cryotron-a superconductive computer component.
44(4):482–493, 1956.

Proc. IRE,

[5] K. K. Likharev, O. A. Mukhanov, and V. K. Semenov. Resistive single flux quantum
logic for the Josephson-junction digital technology. In SQUID’85, pages 1103–1108.
Walter de Gruyter, Berlin, 1985.
[6] K. K. Likharev and V. K. Semenov. RSFQ logic/memory family: A new Josephsonjunction technology for sub-terahertz-clock-frequency digital systems. IEEE Trans.
Appl. Supercond., 50(1), 1991.
[7] W. Chen, A. V. Rylyakov, V. Patel, J. E. Lukens, and K. K. Likharev. Rapid single
flux quantum T-flip flop operating up to 770 GHz. IEEE Trans. Appl. Supercond.,
9(2):3212–3215, 1999.
[8] Q. P. Herr, A. Y. Herr, O. T. Oberg, and A. G. Ioannidis. Ultra-low-power superconductor logic. J. App. Phys., 109(10):103903, 2011.
[9] O. A. Mukhanov. Energy-efficient single flux quantum technology. IEEE Trans. Appl.
Supercond., 21(3):760–769, 2011.
[10] D. E. Kirichenko, S. Sarwana, and A. F. Kirichenko. Zero static power dissipation
biasing of RSFQ circuits. IEEE Trans. Appl. Supercond., 21(3):776–779, 2011.
[11] F. Bedard, N. Welker, G. R. Cotter, M. A. Escavage, and J. T. Pinkston. Superconducting technology assessment. National Security Agency, Office of Corporate Assessments,
2005.

223

[12] C. Bell, G. Burnell, C. W. Leung, E. J. Tarte, D.-J. Kang, and M. G. Blamire. Controllable Josephson current through a pseudospin-valve structure. Appl. Phys. Lett.,
84(7):1153–1155, 2004.
[13] V. V. Ryazanov, V. V. Bol’ginov, D. S. Sobanin, I. V. Vernik, S. K. Tolpygo, A. M.
Kadin, and O. A. Mukhanov. Magnetic Josephson junction technology for digital and
memory applications. Physics Procedia, 36:35–41, 2012.
[14] T. I. Larkin, V. V. Bol’ginov, V. S. Stolyarov, V. V. Ryazanov, I. V. Vernik, S. K.
Tolpygo, and O. A. Mukhanov. Ferromagnetic Josephson switching device with high
characteristic voltage. Appl. Phys. Lett., 100(22):222601, 2012.
[15] I. V. Vernik, V. V. Bol’ginov, S. V. Bakurskiy, A. A. Golubov, M. Y. Kupriyanov, V. V.
Ryazanov, and O. A. Mukhanov. Magnetic Josephson Junctions With Superconducting
Interlayer for Cryogenic Memory. IEEE Trans. Appl. Supercond., 23(3):1701208, 2013.
[16] M. Abd El Qader, R. K. Singh, S. N. Galvin, L. Yu, J. M. Rowell, and N. Newman.
Switching at small magnetic fields in Josephson junctions fabricated with ferromagnetic
barrier layers. Appl. Phys. Lett., 104(2):022602, 2014.
[17] B. Baek, W. H. Rippard, S. P. Benz, S. E. Russek, and P. D. Dresselhaus. Hybrid
superconducting-magnetic memory device using competing order parameters. Nat.
Commun., 5:3888, 2014.
[18] E. C. Gingrich, B. M. Niedzielski, J. A. Glick, Y. Wang, D. L. Miller, R. Loloee,
W. P. Pratt Jr., and N. O. Birge. Controllable 0 − π Josephson junctions containing a
ferromagnetic spin valve. Nat. Phys., 12(6):564–567, 2016.
[19] J. A. Glick, M. A. Khasawneh, B. M. Niedzielski, R. Loloee, W. P. Pratt Jr., N. O.
Birge, E. C. Gingrich, P. G. Kotula, and N. Missert. Critical current oscillations of
elliptical Josephson junctions with single-domain ferromagnetic layers. J. App. Phys.,
122(13):133906, 2017.
[20] J. A. Glick, R. Loloee, W. P. Pratt, and N. O. Birge. Critical current oscillations
of Josephson junctions containing PdFe nanomagnets. IEEE Trans. App. Supercond.,
27(4):1–5, 2017.
[21] J. A. Glick, S. Edwards, D. Korucu, V. Aguilar, B. M. Niedzielski, R. Loloee, W. P.
Pratt Jr., N. O. Birge, P. G. Kotula, and N. Missert. Spin-triplet supercurrent in
Josephson junctions containing a synthetic antiferromagnet with perpendicular magnetic anisotropy. arXiv:1710.07247, 2017.
[22] A. Y. Herr, Q. P. Herr, and O. Naaman. Phase hysteretic magnetic Josephson junction
memory cell, Dec. 8, 2015. US Patent 9,208,861 B2.

224

[23] I. M. Dayton, T. Sage, E. C. Gingrich, M. G. Loving, T. F. Ambrose, N. P. Siwak,
S. Keebaugh, C. Kirby, D. L. Miller, A. Y. Herr, and Naaman O. Experimental
demonstration of a Josephson magnetic memory cell with a programmable π-junction.
arXiv:1711.01681, 2017.
[24] H. K. Onnes and K. Hof. Commun. Phys. Lab. Univ. Leiden., 120b, 122b, 124c, 1911.
[25] W. Meissner and R. Ochsenfeld. Ein neuer effekt bei eintritt der supraleitfähigkeit.
Naturwissenschaften, 21(44):787–788, 1933.
[26] F. London and H. London. The electromagnetic equations of the supraconductor. Proc.
Royal Soc. A, 149(866):71–88, 1935.
[27] F. London. Macroscopical interpretation of supraconductivity. Proc. Royal Soc. A,
152:24–34, 1935.
[28] C. J. Gorter and H. B. G. Casimir. The thermodynamics of the superconducting state.
Phys. Z., 35:963–966, 1934.
[29] V. L. Ginzburg and L. D. Landau. On the theory of superconductivity. Zh. Eksp. Teor.
Fiz., 20:1064–1082, 1950.
[30] L. P. Gor’kov. Microscopic derivation of the Ginzburg-Landau equations in the theory
of superconductivity. Sov. Phys. JETP, 9(6):1364–1367, 1959.
[31] L. P. Gor’kov. Theory of superconducting alloys in a strong magnetic field near the
critical temperature. Sov. Phys. JETP, 10:998, 1960.
[32] J. Bardeen, L. N. Cooper, and J. R. Schrieffer. Microscopic theory of superconductivity.
Phys. Rev., 106(1):162, 1957.
[33] J. Bardeen, L. N. Cooper, and J. R. Schrieffer. Theory of superconductivity. Phys.
Rev., 108(5):1175, 1957.
[34] C. A. Reynolds, B. Serin, W. H. Wright, and L. B. Nesbitt. Superconductivity of
isotopes of mercury. Phys. Rev., 78(4):487, 1950.
[35] E. Maxwell. Isotope effect in the superconductivity of mercury. Phys. Rev., 78(4):477,
1950.
[36] L. N. Cooper. Bound electron pairs in a degenerate Fermi gas. Phys. Rev., 104(4):1189,
1956.
[37] H. Fröhlich. Interaction of electrons with lattice vibrations.
215(1122):291–298, 1952.
225

Proc. Roy. Soc. A,

[38] J. Bardeen and D. Pines.
99(4):1140, 1955.

Electron-phonon interaction in metals.

Phys. Rev.,

[39] M. Tinkham. Introduction to Superconductivity. McGraw-Hill, 2nd edition, 1996.
[40] I. Giaever. Energy Gap in Superconductors Measured by Electron Tunneling. Phys.
Rev. Lett., 5:147–148, 1960.
[41] T. A. Fulton and L. N. Dunkleberger. Lifetime of the zero-voltage state in Josephson
tunnel junctions. Phys. Rev. B, 9:4760–4768, 1974.
[42] W. C. Stewart. Current-voltage characteristics of Josephson junctions. App. Phys.
Lett., 12(8):277–280, 1968.
[43] D. E. McCumber. Effect of ac impedance on dc voltage-current characteristics of
superconductor weak-link junctions. J. App. Phys., 39(7):3113–3118, 1968.
[44] B. S. Deaver and W. M. Fairbank. Experimental Evidence for Quantized Flux in
Superconducting Cylinders. Phys. Rev. Lett., 7:43–46, 1961.
[45] R. Doll and M. Näbauer. Experimental Proof of Magnetic Flux Quantization in a
Superconducting Ring. Phys. Rev. Lett., 7:51–52, 1961.
[46] W. A. Little and R. D. Parks. Observation of Quantum Periodicity in the Transition
Temperature of a Superconducting Cylinder. Phys. Rev. Lett., 9:9–12, 1962.
[47] P. Weiss. La variation du ferromagnétisme avec la température.
143:1136–1139, 1906.

Compt. Rend.,

[48] W. Heisenberg. Mehrkörperproblem und resonanz in der quantenmechanik. Zeitschrift
für Physik, 38(6):411–426, 1926.
[49] P. A. M. Dirac. On the theory of quantum mechanics. Proc. R. Soc. London A,
112(762):661–677, 1926.
[50] L. Landau and E. Lifshitz. On the theory of the dispersion of magnetic permeability
in ferromagnetic bodies. Phys. Zeitsch. der Sow, 8(153):153–169, 1935.
[51] E. C. Stoner and E. P. Wohlfarth. A mechanism of magnetic hysteresis in heterogeneous
alloys. Phil. Trans. R. Soc. Lond. A, 240(826):599–642, 1948.
[52] A. Thiaville. Extensions of the geometric solution of the two dimensional coherent
magnetization rotation model. J. Magn. Magn. Mater., 182(1):5–18, 1998.
[53] R. Holm and W. Meissner. Messungen mit hilfe von flüssigem helium. xiii. Zeitschrift
für Physik, 74(11):715–735, 1932.
226

[54] G. Deutscher and P. G. De Gennes. Superconductivity, volume 2. Marcel Dekker, New
York, 1969.
[55] A. F. Andreev. The thermal conductivity of the intermediate state in superconductors.
Sov. Phys. JETP., 46(5):1823–1828, 1964.
[56] B. Pannetier and H. Courtois. Andreev reflection and proximity effect. J. Low Temp.
Phys., 118(5-6):599–615, 2000.
[57] A. I. Buzdin. Proximity effects in superconductor-ferromagnet heterostructures. Rev.
Mod. Phys., 77:935–976, 2005.
[58] E. A. Demler, G. B. Arnold, and M. R. Beasley. Superconducting proximity effects in
magnetic metals. Phys. Rev. B, 55:15174–15182, 1997.
[59] P. Fulde and R. A. Ferrell. Superconductivity in a strong spin-exchange field. Phys.
Rev., 135:A550–A563, 1964.
[60] A. I. Larkin and Yu. N. Ovchinnikov. Inhomogeneous State of Superconductors. Zh.
Eksp. Teor. Fiz., 47:1136, 1964. [Soviet Physics J. Exp. Theor. Phys., 20, 762, (1965)].
[61] A. I. Buzdin, L. N. Bulaevskii, and S. V. Panyukov. Critical-current oscillations as a
function of the exchange field and thickness of the ferromagnetic metal (F) in an S-F-S
Josephson junction. Pis’ma Zh. Eksp. Teor. Fiz., 35(4):147–148, 1982. [J. Exp. Theor.
Phys. Lett., 35, 4, 20 (1982)].
[62] C. Kittel. Introduction to solid state physics. Wiley, 2005.
[63] T. Mühge, N. N. Garif’yanov, Yu. V. Goryunov, G. G. Khaliullin, L. R. Tagirov,
K. Westerholt, I. A. Garifullin, and H. Zabel. Possible origin for oscillatory superconducting transition temperature in superconductor/ferromagnet multilayers. Phys. Rev.
Lett., 77(9):1857, 1996.
[64] C. Strunk, C. Sürgers, U. Paschen, and H. v. Löhneysen. Superconductivity in layered
Nb/Gd films. Phys. Rev. B, 49(6):4053, 1994.
[65] T. Kontos, M. Aprili, J. Lesueur, F. Genêt, B. Stephanidis, and R. Boursier. Josephson
Junction through a Thin Ferromagnetic Layer: Negative Coupling. Phys. Rev. Lett.,
89:137007, 2002.
[66] K. M. Boden, W. P. Pratt Jr., and N. O. Birge. Proximity-induced density-of-states oscillations in a superconductor/strong-ferromagnet system. Phys. Rev. B, 84(2):020510,
2011.
[67] L. N. Bulaevskii, V. V. Kuzii, and A. A. Sobyanin. Superconducting system with weak
coupling to the current in the ground state. Sov. JETP Lett., 25(7):290–294, 1977.
227

[68] V. V. Ryazanov, V. A. Oboznov, A. Yu. Rusanov, A. V. Veretennikov, A. A. Golubov,
and J. Aarts. Coupling of Two Superconductors through a Ferromagnet: Evidence for
a π Junction. Phys. Rev. Lett., 86:2427–2430, 2001.
[69] Y. Blum, A. Tsukernik, M. Karpovski, and A. Palevski. Oscillations of the superconducting critical current in Nb-Cu-Ni-Cu-Nb junctions. Phys. Rev. Lett., 89:187004,
2002.
[70] H. Sellier, C. Baraduc, F. Lefloch, and R. Calemczuk. Temperature-induced crossover
between 0 and π states in S/F/S junctions. Phys. Rev. B, 68:054531, 2003.
[71] V. Shelukhin, A. Tsukernik, M. Karpovski, Y. Blum, K. B. Efetov, A. F. Volkov,
T. Champel, M. Eschrig, T. Löfwander, G. Schön, and A. Palevski. Observation
of periodic π-phase shifts in ferromagnet-superconductor multilayers. Phys. Rev. B,
73:174506, 2006.
[72] M. Weides, M. Kemmler, E. Goldobin, D. Koelle, R. Kleiner, H. Kohlstedt, and
A. Buzdin. High quality ferromagnetic 0 and π Josephson tunnel junctions. Appl.
Phys. Lett., 89:122511, 2006.
[73] J. W. A. Robinson, S. Piano, G. Burnell, C. Bell, and M. G. Blamire. Critical current
oscillations in strong ferromagnetic π junctions. Phys. Rev. Lett., 97:177003, 2006.
[74] J. W. A. Robinson, S. Piano, G. Burnell, C. Bell, and M. G. Blamire. Zero to π
transition in superconductor-ferromagnet-superconductor junctions. Phys. Rev. B,
76:094522, 2007.
[75] A. A. Bannykh, J. Pfeiffer, V. S. Stolyarov, I. E. Batov, V. V. Ryazanov, and M. Weides. Josephson tunnel junctions with a strong ferromagnetic interlayer. Phys. Rev. B,
79:054501, 2009.
[76] T. S. Khaire, W. P. Pratt Jr., and N. O. Birge. Critical current behavior in Josephson
junctions with the weak ferromagnet PdNi. Phys. Rev. B, 79:094523, 2009.
[77] B. M. Niedzielski, E. C. Gingrich, R. Loloee, W. P. Pratt Jr., and N. O. Birge. S/F/S
Josephson junctions with single-domain ferromagnets for memory applications. Supercond. Sci. Technol., 28(8):085012, 2015.
[78] F. S. Bergeret, A. F. Volkov, and K. B. Efetov. Josephson current in superconductorferromagnet structures with a nonhomogeneous magnetization. Phys. Rev. B,
64:134506, 2001.
[79] A. Kadigrobov, R. I. Shekhter, and M. Jonson. Quantum spin fluctuations as a source
of long-range proximity effects in diffusive ferromagnet-super conductor structures.
EPL (Europhysics Letters), 54(3):394, 2001.
228

[80] M. Eschrig, J. Kopu, J. C. Cuevas, and G. Schön. Theory of half-metal/superconductor
heterostructures. Phys. Rev. Lett., 90:137003, 2003.
[81] A. F. Volkov, F. S. Bergeret, and K. B. Efetov. Odd triplet superconductivity in
superconductor-ferromagnet multilayered structures. Phys. Rev. Lett., 90:117006, 2003.
[82] F. S. Bergeret, A. F. Volkov, and K. B. Efetov. Odd triplet superconductivity and related phenomena in superconductor-ferromagnet structures. Rev. Mod. Phys., 77:1321–
1373, 2005.
[83] R. S. Keizer, S. T. B. Goennenwein, T. M. Klapwijk, G. Miao, G. Xiao, and A. Gupta.
A spin triplet supercurrent through the half-metallic ferromagnet CrO2. Nature,
439(7078):825–827, 2006.
[84] I. Sosnin, H. Cho, V. T. Petrashov, and A. F. Volkov. Superconducting phase coherent
electron transport in proximity conical ferromagnets. Phys. Rev. Lett., 96:157002, 2006.
[85] T. S. Khaire, M. A. Khasawneh, W. P. Pratt Jr., and N. O. Birge. Observation of
Spin-Triplet Superconductivity in Co-Based Josephson Junctions. Phys. Rev. Lett.,
104:137002, 2010.
[86] J. W. A. Robinson, J. D. S. Witt, and M. G. Blamire. Controlled Injection of SpinTriplet Supercurrents into a Strong Ferromagnet. Science, 329(5987):59–61, 2010.
[87] D. Sprungmann, K. Westerholt, H. Zabel, M. Weides, and H. Kohlstedt. Evidence
for triplet superconductivity in Josephson junctions with barriers of the ferromagnetic
Heusler alloy Cu2 MnAl. Phys. Rev. B, 82:060505, 2010.
[88] M. S. Anwar, F. Czeschka, M. Hesselberth, M. Porcu, and J. Aarts. Long-range
supercurrents through half-metallic ferromagnetic CrO2 . Phys. Rev. B, 82:100501,
2010.
[89] J. Wang, M. Singh, M. Tian, N. Kumar, B. Liu, C. Shi, J. K. Jain, N. Samarth, T. E.
Mallouk, and M. H. W. Chan. Interplay between superconductivity and ferromagnetism in crystalline nanowires. Nat. Phys., 6(5):389–394, 2010.
[90] C. Klose, T. S. Khaire, Y. Wang, W. P. Pratt, N. O. Birge, B. J. McMorran, T. P.
Ginley, J. A. Borchers, B. J. Kirby, B. B. Maranville, and J. Unguris. Optimization
of spin-triplet supercurrent in ferromagnetic Josephson junctions. Phys. Rev. Lett.,
108:127002, 2012.
[91] P. V. Leksin, N. N. Garif’yanov, I. A. Garifullin, Ya. V. Fominov, J. Schumann,
Y. Krupskaya, V. Kataev, O. G. Schmidt, and B. Büchner. Evidence for triplet superconductivity in a superconductor-ferromagnet spin valve. Phys. Rev. Lett., 109:057005,
2012.
229

[92] V. I. Zdravkov, J. Kehrle, G. Obermeier, D. Lenk, H.-A. Krug von Nidda, C. Müller,
M. Yu. Kupriyanov, A. S. Sidorenko, S. Horn, R. Tidecks, and L. R. Tagirov. Experimental observation of the triplet spin-valve effect in a superconductor-ferromagnet
heterostructure. Phys. Rev. B, 87:144507, 2013.
[93] N. Banerjee, C. B. Smiet, R. G. J. Smits, A. Ozaeta, F. S. Bergeret, M. G. Blamire,
and J. W. A. Robinson. Evidence for spin selectivity of triplet pairs in superconducting
spin valves. Nat. Commun., 5:3048, 2014.
[94] N. Banerjee, J. W. A. Robinson, and M. G. Blamire. Reversible control of spinpolarized supercurrents in ferromagnetic Josephson junctions. Nat. Commun., 5:4771
EP, 2014.
[95] A. Iovan, T. Golod, and V. M. Krasnov. Controllable generation of a spin-triplet
supercurrent in a Josephson spin valve. Phys. Rev. B, 90:134514, 2014.
[96] X. L. Wang, A. Di Bernardo, N. Banerjee, A. Wells, F. S. Bergeret, M. G. Blamire,
and J. W. A. Robinson. Giant triplet proximity effect in superconducting pseudo spin
valves with engineered anisotropy. Phys. Rev. B, 89:140508, 2014.
[97] A. A. Jara, C. Safranski, I. N. Krivorotov, C. Wu, A. N. Malmi-Kakkada, O. T. Valls,
and K. Halterman. Angular dependence of superconductivity in superconductor/spinvalve heterostructures. Phys. Rev. B, 89:184502, 2014.
[98] M. G. Flokstra, T. C. Cunningham, J. Kim, N. Satchell, G. Burnell, P. J. Curran, S. J.
Bending, C. J. Kinane, J. F. K. Cooper, S. Langridge, A. Isidori, N. Pugach, M. Eschrig, and S. L. Lee. Controlled suppression of superconductivity by the generation of
polarized Cooper pairs in spin-valve structures. Phys. Rev. B, 91:060501, 2015.
[99] A. Singh, S. Voltan, K. Lahabi, and J. Aarts. Colossal Proximity Effect in a Superconducting Triplet Spin Valve Based on the Half-Metallic Ferromagnet CrO2 . Phys.
Rev. X, 5:021019, 2015.
[100] J. Linder and J. W. A. Robinson. Superconducting spintronics. Nat. Phys., 11(4):307–
315, 2015.
[101] M. Eschrig. Spin-polarized supercurrents for spintronics: a review of current progress.
Rep. Prog. Phys., 78(10):104501, 2015.
[102] A. Di Bernardo, S. Diesch, Y. Gu, J. Linder, G. Divitini, C. Ducati, E. Scheer, M. G.
Blamire, and J. W. A. Robinson. Signature of magnetic-dependent gapless odd frequency states at superconductor/ferromagnet interfaces. Nat. Commun., 6:8053, 2015.
[103] Z. Feng, J. W. A. Robinson, and M. G. Blamire. Out of plane superconducting
Nb/Cu/Ni/Cu/Co triplet spin-valves. Appl. Phys. Lett., 111(4):042602, 2017.
230

[104] C. Cirillo, S. Voltan, E. A. Ilyina, J. M. Hernandez, A. Garcia-Santiago, J. Aarts, and
C. Attanasio. Long-range proximity effect in Nb-based heterostructures induced by a
magnetically inhomogeneous permalloy layer. New J. Phys., 19(2):023037, 2017.
[105] M. Houzet and A. I. Buzdin. Long range triplet Josephson effect through a ferromagnetic trilayer. Phys. Rev. B, 76:060504, 2007.
[106] E. C. Gingrich, P. Quarterman, Y. Wang, R. Loloee, W. P. Pratt Jr., and N. O. Birge.
Spin-triplet supercurrent in Co/Ni multilayer Josephson junctions with perpendicular
anisotropy. Phys. Rev. B, 86:224506, 2012.
[107] W. M. Martinez, W. P. Pratt Jr., and N. O. Birge. Amplitude Control of the SpinTriplet Supercurrent in S/F/S Josephson Junctions. Phys. Rev. Lett., 116:077001,
2016.
[108] M. Eschrig. Spin-polarized supercurrents for spintronics. Phys. Today, 64:43, 2011.
[109] E. C. Gingrich. Phase control of the spin-triplet state in S/F/S Josephson junctions.
PhD thesis, Michigan State University, 2014.
[110] E. Terzioglu and M. R. Beasley. Complementary Josephson junction devices and circuits: a possible new approach to superconducting electronics. IEEE Trans. Appl.
Supercond., 8:48–53, 1998.
[111] L. B. Ioffe, V. B. Geshkenbein, M. V. Feigel’man, A. L. Fauchere, and G. Blatter. Environmentally decoupled sds-wave Josephson junctions for quantum computing. Nature,
398:679–681, 1999.
[112] G. Blatter, V. B. Geshkenbein, and L. B. Ioffe. Design aspects of superconductingphase quantum bits. Phys. Rev. B, 63:174511, 2001.
[113] A. V. Ustinov and V. K. Kaplunenko. Rapid single-flux quantum logic using π-shifters.
J. App. Phys., 94(8):5405–5407, 2003.
[114] T. Yamashita, K. Tanikawa, S. Takahashi, and S. Maekawa. Superconducting π qubit
with a ferromagnetic Josephson junction. Phys. Rev. Lett., 95:097001, 2005.
[115] M. I. Khabipov. A single flux quantum circuit with a ferromagnet-based Josephson
π-junction. Supercond. Sci. Technol., 23:045032, 2010.
[116] A. K. Feofanov, V. A. Oboznov, V. V. Bol’ginov, J. Lisenfeld, S. Poletto, V. V.
Ryazanov, A. N. Rossolenko, M. Khabipov, D. Balashov, A. B. Zorin, P. N.
Dmitriev, V. P. Koshelets, and A. V. Ustinov. Implementation of superconductor/ferromagnet/superconductor π-shifters in superconducting digital and quantum
circuits. Nat. Phys., 6(8):593–597, 2010.
231

[117] A. Y. Herr and Q. P. Herr. Josephson magnetic random access memory system and
method, Sept. 18, 2012. US Patent 8,270,209.
[118] E. Goldobin, H. Sickinger, M. Weides, N. Ruppelt, H. Kohlstedt, R. Kleiner, and
D. Koelle. Memory cell based on a φ Josephson junction. App. Phys. Lett., 102(24),
2013.
[119] S. V. Bakurskiy, N. V. Klenov, I. I. Soloviev, V. V. Bol’ginov, V. V. Ryazanov, I. V.
Vernik, O. A. Mukhanov, M. Yu. Kupriyanov, and A. A. Golubov. Theoretical model
of superconducting spintronic SIsFS devices. App. Phys. Lett., 102(19), 2013.
[120] N. Ruppelt, H. Sickinger, R. Menditto, E. Goldobin, D. Koelle, R. Kleiner, O. Vavra,
and H. Kohlstedt. Observation of 0 − π transition in SIsFS Josephson junctions. Appl.
Phys. Lett., 106(2):022602, 2015.
[121] S. V. Bakurskiy, V. I. Filippov, V. I. Ruzhickiy, N. V. Klenov, I. I. Soloviev, M. Yu.
Kupriyanov, and A. A. Golubov. Current-phase relations in SIsFS junctions in the
vicinity of 0 − π transition. Phys. Rev. B, 95:094522, 2017.
[122] Y. Wang, W. P. Pratt Jr., and N. O. Birge. Area-dependence of spin-triplet supercurrent in ferromagnetic Josephson junctions. Phys. Rev. B, 85:214522, 2012.
[123] C. D. Thomas, M. P. Ulmer, and J. B. Ketterson. Superconducting tunnel junction
base electrode planarization. J. App. Phys., 84(1):364–367, 1998.
[124] H. Kohlstedt, F. König, P. Henne, N. Thyssen, and P. Caputo. The role of surface
roughness in the fabrication of stacked Nb/Al–AlOx /Nb tunnel junctions. J. App.
Phys., 80(9):5512–5514, 1996.
[125] Paul G. Kotula, Michael R. Keenan, and Joseph R. Michael. Tomographic Spectral
Imaging with Multivariate Statistical Analysis: Comprehensive 3D Microanalysis. Micros. and Microanal., 12:36–48, 2 2006.
[126] D. L. Edmunds, W. P. Pratt Jr., and J. A. Rowlands. 0.1 ppm four-terminal resistance
bridge for use with a dilution refrigerator. Rev. Sci. Instrum., 51(11):1516–1522, 1980.
[127] A. Barone and G. Paternò. Physics and applications of the Josephson effect. Wiley,
1982.
[128] L. Vila, W. Park, J. A. Caballero, D. Bozec, R. Loloee, W. P. Pratt Jr., and J. Bass.
Current perpendicular magnetoresistances of NiFeCo and NiFe “Permalloys”. J. App.
Phys., 87(12):8610–8614, 2000.

232

[129] Y. M. Ivanchenko and L. A Zil’berman. The Josephson Effect in Small Tunnel Contacts.
Zh. Eksp. Teor. Fiz., 55:2395–2402, 1968. [Soviet Physics J. Exp. Theor. Phys., 28, 6,
(1969)].
[130] V. Ambegaokar and B. I. Halperin. Voltage Due to Thermal Noise in the dc Josephson
Effect. Phys. Rev. Lett., 22:1364–1366, 1969.
[131] A. I. Buzdin and M. Yu. Kupriyanov. Josephson junction with a ferromagnetic layer.
Pis’ma Zh. Eksp. Teor. Fiz., 53(6):308–312, 1991. [J. Exp. Theor. Phys. Lett., 53, 6,
321 (1991)].
[132] N. G. Pugach, M. Yu. Kupriyanov, E. Goldobin, R. Kleiner, and D. Koelle.
Superconductor-insulator-ferromagnet-superconductor Josephson junction: From the
dirty to the clean limit. Phys. Rev. B, 84:144513, 2011.
[133] Z. Radović, N. Lazarides, and N. Flytzanis. Josephson effect in double-barrier
superconductor-ferromagnet junctions. Phys. Rev. B, 68:014501, 2003.
[134] P. H. Barsic, O. T. Valls, and K. Halterman. Thermodynamics and phase diagrams of
layered superconductor/ferromagnet nanostructures. Phys. Rev. B, 75:104502, 2007.
[135] J. Bass. CPP magnetoresistance of magnetic multilayers: A critical review. J. Magn.
Magn. Mater., 408:244 – 320, 2016.
[136] M. Fauré, A. I. Buzdin, A. A. Golubov, and M. Yu. Kupriyanov. Properties of superconductor/ferromagnet structures with spin-dependent scattering. Phys. Rev. B,
73:064505, 2006.
[137] S. V. Bakurskiy, A. A. Golubov, N. V. Klenov, M. Yu. Kupriyanov, and I. I. Soloviev.
Josephson effect in SIFS tunnel junctions with domain walls in the weak link region.
Pis’ma Zh. Eksp. Teor. Fiz., 101(11):863–868, 2015. [J. Exp. Theor. Phys. Lett., 101,
11, 765–771 (2015)].
[138] D. M. Heim, N. G. Pugach, M. Yu. Kupriyanov, E. Goldobin, D. Koelle, R. Kleiner,
N. Ruppelt, M. Weides, and H. Kohlstedt. The effect of normal and insulating layers
on 0 − π transitions in Josephson junctions with a ferromagnetic barrier. New J. Phys.,
17(11):113022, 2015.
[139] S. Senoussi, I. A. Campbell, and A. Fert. Evidence for local orbital moments on Ni
and Co impurities in Pd. Solid State Commun., 21(3):269 – 271, 1977.
[140] H. Z. Arham, T. S. Khaire, R. Loloee, W. P. Pratt Jr., and N. O. Birge. Measurement of
spin memory lengths in PdNi and PdFe ferromagnetic alloys. Phys. Rev. B, 80:174515,
2009.

233

[141] L. S. Uspenskaya, A. L. Rakhmanov, L. A. Dorosinskii, A. A. Chugunov, V. S. Stolyarov, O. V. Skryabina, and S. V. Egorov. Magnetic patterns and flux pinning in
Pd0.99 Fe0.01 -Nb hybrid structures. Pis’ma Zh. Eksp. Teor. Fiz., 97(3):176–179, 2013.
[J. Exp. Theor. Phys. Lett., 97, 3, 155–158 (2013)].
[142] B. Baek. Private communication. One may need 400 mT to initialize the magnetization
direction of a thin Ni nanomagnet, whereas we used only 260 mT.
[143] B. Baek, M. L. Schneider, M. R. Pufall, and W. H. Rippard. Phase Offsets
in the Critical-Current Oscillations of Josephson Junctions Based on Ni and Ni(Ni81 Fe19 )x Nby Barriers. Phys. Rev. Appl., 7:064013, 2017.
[144] B. M. Niedzielski. Phase Sensitive Measurements of Ferromagnetic Josephson Junctions for Cryogenic Memory Applications. PhD thesis, Michigan State University,
2017.
[145] B. Baek. Spin-transfer torque switching in nanopillar superconducting-magnetic hybrid
Joseph junctions. Phys. Rev. Appl., 3:011001, 2015.
[146] W.-T. Tsang and T. Van Duzer. DC analysis of parallel arrays of two and three
Josephson junctions. J. App. Phys., 46(10):4573–4580, 1975.
[147] A. F. Volkov and K. B. Efetov. Odd spin-triplet superconductivity in a multilayered
superconductor-ferromagnet Josephson junction. Phys. Rev. B, 81:144522, 2010.
[148] L. Trifunovic and Z. Radović. Long-range spin-triplet proximity effect in Josephson
junctions with multilayered ferromagnets. Phys. Rev. B, 82:020505, 2010.
[149] M. Schneider, October 2015. Private communication.
[150] Y.-J. Chang, A. Canizo-Cabrera, V. Garcia-Vazquez, Y.-H. Chang, and T.-H. Wu. Perpendicular magnetic tunnel junctions with synthetic antiferromagnetic pinned layers
based on [Co/Pd] multilayers. J. App. Phys., 113(17):17B909, 2013.
[151] J.-B. Lee, G.-G. An, S.-M. Yang, H.-S. Park, W.-S. Chung, and J.-P. Hong. Thermally
robust perpendicular Co/Pd-based synthetic antiferromagnetic coupling enabled by a
W capping or buffer layer. Sci. Rep., 6:21324, 2016.
[152] D. Smith, V. Parekh, E. Chunsheng, S. Zhang, W. Donner, T. R. Lee, S. Khizroev,
and D. Litvinov. Magnetization reversal and magnetic anisotropy in patterned Co/Pd
multilayer thin films. J. App. Phys., 103(2):023920, 2008.
[153] B. M. Niedzielski, T. J. Bertus, J. A. Glick, R. Loloee, W. P. Pratt Jr., and N. O.
Birge. Spin-valve Josephson junctions for cryogenic memory. arXiv:1709.04815, 2017.

234

[154] P. G. Kotula and M. R. Keenan. Application of multivariate statistical analysis to
STEM X-ray spectral images: Interfacial analysis in microelectronics. Micros. and
Microanal., 12(6):538–544, 2006.
[155] V. V. Ryazanov. Josephson superconductor-ferromagnet-superconductor π-contact as
an element of a quantum bit (experiment). Phys. Usp., 42(8):825, 1999.
[156] D. Korucu. unpublished, 2016.
[157] H. Y. T. Nguyen, R. Acharyya, E. Huey, B. Richard, R. Loloee, W. P. Pratt Jr.,
J. Bass, S. Wang, and K. Xia. Conduction electron scattering and spin flipping at
sputtered Co/Ni interfaces. Phys. Rev. B, 82:220401, 2010.
[158] S. D. Pollard, J. A. Garlow, J. Yu, Z. Wang, Y. Zhu, and H. Yang. Observation
of stable Néel skyrmions in cobalt/palladium multilayers with Lorentz transmission
electron microscopy. Nat. Commun., 8:14761, 2017.

235