ABSTRACT
Title of dissertation: INVESTIGATION OF TUNNELING IN
SUPERCONDUCTORS USING A MILLIKELVIN
SCANNING TUNNELING MICROSCOPE
Wan-Ting Liao, Doctor of Philosophy, 2019
Dissertation directed by: Professor Christopher J. Lobb
Department of Physics
In this thesis, I discuss my use of a millikelvin scanning tunneling microscope
(STM) to investigate tunneling phenomena in superconductors. As part of an ef-
fort to construct an STM to measure the superconducting phase difference, I first
describe how I modified a dual-tip scanning tunneling microscope by electrically con-
necting the two tips together with a short (3 mm) strip of flexible 25 µm thick Nb
foil. I also discuss the technique I developed for keeping each tip in feedback when
only the total tunnel current through both tips can be measured. I then describe
simultaneous room-temperature imaging with both tips on samples of Au/mica and
highly oriented pyrolytic graphite (HOPG).
Next, I report single-tip results from scanning tunneling microscopy of 25 nm
and 50 nm thick films of superconducting TiN at 0.5 K. I found large variations
in the tip-sample conductance-voltage characteristics in these samples. At some
locations the characteristics showed a clear superconducting gap, as expected for
superconductor-normal (S-I-N) tunneling through a high barrier height. At other
locations there was a distinct zero-voltage conductance peak, as expected for S-N
Andreev tunneling through a low barrier height. I compare the data to the Blonder-
Tinkham-Klapwijk (BTK) theory and the Dynes model of tunneling into a super-
conductor with broadened density of states. I find that the BTK model provides
better fits and reveals a remarkable correlation between the superconducting gap
∆, the temperature T and the barrier height Z. Possible causes for this correlation,
including local heating and surface contamination, are discussed.
Finally, I describe measurements of I(V) characteristics of a Josephson junction
formed by a scanning tunneling microscope with a Nb sample and a Nb tip at 50 mK
and 1.5 K. To better understand the physics of this system, I generalized the multiple
Andreev reflection (MAR) theory of Averin and Bardas to describe junctions having
electrodes with different superconducting gaps. For tunneling resistance Rn between
10 MΩ and 100 kΩ, there was no observable supercurrent at 50 mK or 1.5 K. For Rn
between 100 kΩ and about 10 kΩ, the junctions showed hysteretic behavior, with the
forward-sweep switching current Is larger than the reverse-sweep retrapping current
Ir. In this regime, the critical current I0 was suppressed and the current-voltage
characteristics showed a relatively small non-zero resistance R0 at V = 0 that scaled
with R2n. For Rn less than the quantum resistance (∼ 12 kΩ), the I-V characteristics
deviate from single channel MAR theory. In this limit, the tip makes contact with
the sample, as revealed by the dependence of the junction conductance curves on
the tip-sample separation. By fitting my two-gap MAR theory to the I(V) data,
I obtain superconducting gaps of the tip and sample as a function of the tunnel
resistance Rn. I find the sample has nearly the full gap of bulk Nb (∆ ∼ 1.5 meV),
but the tip gap is only about 0.67 meV, and decreases for Rn ≤ 10 kΩ.
INVESTIGATION OF TUNNELING IN SUPERCONDUCTORS
USING A MILLIKELVIN SCANNING TUNNELING
MICROSCOPE
by
Wan-Ting Liao
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2019
Advisory Committee:
Professor Christopher J. Lobb, Chair/Advisor
Professor Frederick C. Wellstood, Co-Advisor
Dr. Michael Dreyer, Co-Advisor
Professor John Cumings
Dr. Kevin D. Osborn
©c Copyright by
Wan-Ting Liao
2019
Dedication
Dedicated to my father, my mother, my brothers, my aunts
and
my Fiancé Karl
ii
Acknowledgments
As a student at University of Maryland (UMD) and the Laboratory for Phys-
ical Sciences (LPS), the last few years have been the best time of my academic life.
None of the work in this thesis would have been possible without the help of many
people and I humbly thank them all.
I thank my advisors, Prof. Christopher Lobb, Prof. Fred Wellstood and Dr.
Michael Dreyer. In particular, I thank Dr. Lobb for being such a great mentor
and scientist. There were many times when I was working on the Multiple Andreev
Reflection (MAR) research where he often came to LPS to brainstorm the theoretical
problem with me and cheered me up (he always told me that the problem will be
solved by Monday). I thank Dr. Wellstood for being such a great scientist whom
I can always consult to discuss about physics. He can always figure out where I
am stuck in the problem and work our way on how to proceed next. His ability of
looking at a simple problem from different angles is amazing and this always inspires
me in my research. I thank Dr. Dreyer who is a real expert in scanning tunneling
microscopy (STM), low temperature techniques and the UHV system. I learned
most of the experimental skills that I needed during my graduate work from him.
There were times when I wanted to speed up the experiment and I skipped some
important steps. He warned me that, if something went wrong, then I would have
to spend even more time to fix the problem. There was also an incident where I was
totally stuck in my research. He just smiled and told me that one day you will run
out of the problems that you need to solve.
iii
I am grateful to Dr. Kevin Osborn for providing the TiN samples for the work I
describe in Chapter 6. He was always willing to discuss physics and the TiN project,
and agreed to be one of the committee members for my dissertation. I also want to
thank Prof. John cumings for agreeing to serve as the Dean’s representative for my
committee at the last minute.
I thank Dr. Butera, the STM group manager at LPS. He always gave me full
freedom to execute the projects and was very supportive when I needed help at
LPS or in my academic life. I am thankful to Dr. Sudeep Dutta, a great scientist
who is very detailed and could always spot any small flaws in my research that I
had ignored. At the same time he is a heart-warming person who helped me hang
on during dark times. I am grateful to my colleagues who worked on parts of this
project and shared their experiences with me, notably Drs. Anita Roychowdhury
and Rami Dana.
I thank the LPS cleanroom people, namely Toby Olver, Steve Brown, Jim
Fogleboch, Paul Hannah, Dan Hinkel, Doug Ketchum, Warren Berk, Debtanu Basu,
Curt Walsh, David Gutierrez, who were very knowledgeable and always helpful when
I had any issues related to the cleanroom. I thank the machinists in the LPS ma-
chine shop, namely Donald Crouse, Ruben Brun, John Sugrue, and Bill Donaldson,
who fabricated many different parts for my research and were always very fast and
responsive after I submitted my requests. I am grateful to the administrative staff
at LPS including Jonas Wood, Niquinn Fowler, Althia Kirlew, Cynthia Evans, Greg
Latini, Charla Pryor, Anthony Littlefield and Mark Thornton.
I am thankful for the support I received from Bruce Kane, Ben Palmer and
iv
Chris Richardson. I thank Ramesh Bhandari for being such a wonderful theoretical
physicist who gave me significant insight during my theoretical work on the MAR
problem described in Chapter 7. I thank Prof. Dmitri Averin, for guiding me through
generalizing the MAR problem and for providing the Fortran code that I modified
and then used in this thesis.
I am grateful to my friends from LPS, Luke Robertson, Joyce Coppock, Shavi
Premaratne, Neda Forouzani, Chih-Chiao Hung, Liuqi Yu, Tim Kohler, Joseph
Murray, Ashish Alexander, Chomani Gaspe, Margaret Samuels, Jennifer DeMell,
Kevin Dwyer, Siddharth Tyagi, Rui Zhang, Jen-Hao Yeh, Tamin Tai, Yizhou Huang,
and Philip Li for discussions of research, as well as trivial day-to-day greetings and
chatting with me every time we met in the LPS hallway.
I am thankful to the members of the running group at LPS, including Victor
Yun, Donghun Park, Alan I, Yuan Gu, Chenglin Yi. The daily jog at 11:30 AM
made me feel vibrant after half a day of work and this routine kept me both mentally
and physically healthy during my graduate study. I thank Charlie Kraft for inviting
me sailing several times during the summer, with his sister Amy and brother Hank,
who I enjoyed meeting.
Several friends who I met at UMD gave me support when I really needed
it, including Xiaoxiao Ge, I-Lin Liu, Michelle Lin, Yiran Li, Chih-Yuan Su, Ayan
Mallik and Sabyasachi Barik. I am grateful to my housemates Alice Rogers and
Sarah Cochrane. I would never forget those three wonderful years that we lived
together, shared our lives with each other and watched so many fun TV shows after
dinner. There are many other friends who I met outside of school who have become
v
my best friends. I thank them all and apologize for leaving them out here.
I would like to acknowledge financial support from the Laboratory for Physical
sciences and NSF DMR-1409925 for the work described in this thesis.
I am extremely grateful for my parents, my two brothers, and my aunts for
their love and support throughout graduate life. I especially thank my friend Yu-
Chun Wei, who shared many years of friendship and supported me during the first
two years at Maryland when I had a hard time adapting to life in the US.
Finally, I thank my Fiancé Karl, a wonderful person who I met in my graduate
life, who has gone through multiple ups and downs with me, who has traveled and
done many fun activities with me, and who I decided to spend the rest of my life
with.
vi
Table of Contents
Dedication ii
Acknowledgements iii
List of Tables x
List of Figures xi
List of Abbreviations xiv
Chapter 1 Introduction 1
Chapter 2 Scanning Tunneling Microscopy 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Theory of 1-D tunneling and principle of STM . . . . . . . . . . . . . 7
2.3 Topographic imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 Constant current mode . . . . . . . . . . . . . . . . . . . . . . 14
2.3.2 Constant height mode . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 N-I-N tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.2 N-I-S tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.3 S-I-S tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Cryogenic STM’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Chapter 3 Experimental Setup of the Millikelvin STM 33
3.1 Dilution refrigerator . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Overall experimental setup of the millikelvin STM . . . . . . . . . . . 39
3.3 Dual-tip STM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Wiring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5 Damage to the heat exchanger and partial repair . . . . . . . . . . . 52
3.6 Ultra high vacuum (UHV) system . . . . . . . . . . . . . . . . . . . . 55
3.7 Control electronics and measurement circuit . . . . . . . . . . . . . . 56
3.8 STM niobium and vanadium tip preparation . . . . . . . . . . . . . . 60
3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Chapter 4 Simultaneously Scanning Two Connected Tips in a
Dual-tip STM 62
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Room temperature system setup . . . . . . . . . . . . . . . . . . . . . 65
4.3 Scanning two tips independently . . . . . . . . . . . . . . . . . . . . . 67
vii
4.4 Simultaneously scanning two STM tips on test samples . . . . . . . . 75
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Chapter 5 Superconductivity and BTK Theory 87
5.1 Introduction to superconductivity . . . . . . . . . . . . . . . . . . . . 88
5.1.1 Density of states . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2 Andreev reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2.1 BTK theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2.2 Calculation of I-V Characteristics for S-I-N junction from the
BTK model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Chapter 6 Scanning Andreev Tunneling Microscopy of Titanium
Nitride Thin Films 116
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.2 TiN film growth and basic characteristics . . . . . . . . . . . . . . . . 118
6.3 STM measurements and Andreev effects . . . . . . . . . . . . . . . . 120
6.4 ∆, Z, T , h correlations in the 50 nm and 25 nm films . . . . . . . . . 128
6.5 Physical mechanism causing correlations between ∆, T , Z and h . . . 139
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Chapter 7 Theory of the I-V characteristics in S-I-S Junctions
with Multiple Andreev Reflection 146
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.2 MAR in symmetrical junction with transparency D = 1 . . . . . . . . 148
7.2.1 Averin-Bardas model . . . . . . . . . . . . . . . . . . . . . . . 148
7.2.2 Finding the expression for the current when the gaps are the
same . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.3 MAR in symmetrical junction with D 6= 1 . . . . . . . . . . . . . . . 172
7.4 MAR in Asymmetrical junctions . . . . . . . . . . . . . . . . . . . . . 176
7.5 The Josephson effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
7.6 Critical current in Asymmetrical junctions with MAR . . . . . . . . . 205
7.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Chapter 8 Ultra-small Josephson Junction Formed Using a Nb
STM Tip and Nb Sample 213
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
8.2 Sample preparation and experimental setup . . . . . . . . . . . . . . 214
8.3 I − V characteristics from a Nb tip and a Nb sample . . . . . . . . . 218
8.4 Dissipation analysis from the retrapping current . . . . . . . . . . . . 233
8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
Chapter 9 Conclusions 242
9.1 Summary of main results . . . . . . . . . . . . . . . . . . . . . . . . . 242
9.2 Some suggestions for further research . . . . . . . . . . . . . . . . . . 246
9.2.1 Low temperature requirement . . . . . . . . . . . . . . . . . . 246
viii
9.2.2 Smaller SQUID loop . . . . . . . . . . . . . . . . . . . . . . . 247
9.2.3 Better shielding . . . . . . . . . . . . . . . . . . . . . . . . . . 247
9.2.4 Intrinsic system constraint . . . . . . . . . . . . . . . . . . . . 248
Appendix A χ2 fit for the TiN film 250
Appendix B 50 nm TiN film showing S-I-S junction with a vana-
dium tip 253
Appendix C Fortran code for symmetrical S-I-S junction with MAR
effect 256
Appendix D Fortran code for Asymmetrical S-I-S junction with
MAR effect 262
Bibliography 271
ix
List of Tables
Table 5.1 BTK coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Table 6.1 Extracted parameters (∆, T, Z) from fitting conductance curves
to the BTK model . . . . . . . . . . . . . . . . . . . . . . . . . 126
Table 6.2 General properties of the 50 nm and 25 nm TiN films . . . . . . 130
Table 8.1 Extracted parameters Rn,∆tip,∆sample,∆sum, D, Z from fitting
the I−V characteristics in Fig. 8.3 (a)-(c) to the asymmetrical
MAR model at T = 0. . . . . . . . . . . . . . . . . . . . . . . . 221
x
List of Figures
Fig. 2.1 Particle incident on a barrier . . . . . . . . . . . . . . . . . . . 9
Fig. 2.2 Principle of STM . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Fig. 2.3 voltage-bias and current-bias circuits . . . . . . . . . . . . . . . 19
Fig. 2.4 N-I-N tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Fig. 2.5 N-I-S tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Fig. 2.6 Conductance and voltage plot of N-I-S tunneling . . . . . . . . 27
Fig. 2.7 S-I-S tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Fig. 2.8 I − V characteristic of S-I-S tunneling . . . . . . . . . . . . . . 31
Fig. 3.1 Phase diagram for 3He/4He mixture . . . . . . . . . . . . . . . 35
Fig. 3.2 Schematic diagram of dilution refrigerator . . . . . . . . . . . . 37
Fig. 3.3 Schematic of mK-STM setup . . . . . . . . . . . . . . . . . . . 40
Fig. 3.4 Schematic of the dual tip mK-STM . . . . . . . . . . . . . . . . 42
Fig. 3.5 Wiring and signal lines at 1 K pot stage . . . . . . . . . . . . . 44
Fig. 3.6 Wiring and signal lines at the still stage . . . . . . . . . . . . . 45
Fig. 3.7 Wiring and signal lines at the cold plate stage . . . . . . . . . . 46
Fig. 3.8 Wiring and signal lines at the mixing chamber stage . . . . . . 47
Fig. 3.9 Wiring and signal lines below the mixing chamber stage . . . . 48
Fig. 3.10 Wiring and signal lines at the STM stage . . . . . . . . . . . . 49
Fig. 3.11 Wiring of the drive lines at different stages . . . . . . . . . . . . 51
Fig. 3.12 Photographs of the broken heat exchanger . . . . . . . . . . . . 53
Fig. 3.13 Photographs of the re-sealed heat exchanger . . . . . . . . . . . 54
Fig. 3.14 Photograph of our mK-STM system . . . . . . . . . . . . . . . 57
Fig. 3.15 Schematic diagram of voltage-bias and current-bias modes . . . 59
Fig. 4.1 Bow-tie connectors . . . . . . . . . . . . . . . . . . . . . . . . . 66
Fig. 4.2 Two tips connected . . . . . . . . . . . . . . . . . . . . . . . . . 68
Fig. 4.3 Electronics setup for controlling two tips . . . . . . . . . . . . . 70
Fig. 4.4 I-z curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Fig. 4.5 |dI/dz|-z curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Fig. 4.6 Electronic circuit of safety box . . . . . . . . . . . . . . . . . . 76
Fig. 4.7 Topographic image of Au on Mica in current mode . . . . . . . 78
Fig. 4.8 Topographic image of Au on Mica in dI/dz mode . . . . . . . . 79
Fig. 4.9 Atomic image of HOPG in current mode . . . . . . . . . . . . . 80
Fig. 4.10 Atomic image of HOPG in dI/dz mode . . . . . . . . . . . . . 81
Fig. 4.11 Simultaneous scans of Au on Mica using two connected tips . . 83
xi
Fig. 4.12 Stability test on HOPG . . . . . . . . . . . . . . . . . . . . . . 84
Fig. 4.13 Stability test on HOPG showing dragging of the inner tip by
the outer tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Fig. 5.1 Superconducting gap as a function of temperature . . . . . . . . 98
Fig. 5.2 Density of states of a superconductor . . . . . . . . . . . . . . . 100
Fig. 5.3 Normal reflection and Andreev reflection amplitude vs. energy . 109
Fig. 5.4 Conductance as a function of energy for different Z at 75 mK . 112
Fig. 5.5 Conductance as a function of energy for different Z at 500 mK . 114
Fig. 5.6 Conductance as a function of energy for different Z at 500 and
75 mK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Fig. 6.1 Topographic image of the 50 nm TiN film and selected conduc-
tance curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Fig. 6.2 Topographic image of the 50 nm TiN film and fine scale view
of grains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Fig. 6.3 Fine scale topographic image maps of ∆, T , Z and line section
for the 50 nm TiN film . . . . . . . . . . . . . . . . . . . . . . . 127
Fig. 6.4 False color plot of conductance versus position and voltage
along the line cut . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Fig. 6.5 Large scale topographic image and line section of 50 nm and 25
nm TiN films . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Fig. 6.6 Large scale topographic image, maps of ∆, T , Z and line section
for the 50 nm TiN films . . . . . . . . . . . . . . . . . . . . . . 133
Fig. 6.7 Large scale topographic image, maps of ∆, T , Z and line section
for the 50 nm TiN films . . . . . . . . . . . . . . . . . . . . . . 134
Fig. 6.8 Large scale topographic image, maps of ∆, T , Z and line section
for the 25 nm TiN film . . . . . . . . . . . . . . . . . . . . . . . 135
Fig. 6.9 Large scale topographic image, maps of ∆, T , Z and line section
for the 25 nm TiN film . . . . . . . . . . . . . . . . . . . . . . . 136
Fig. 6.10 Gap histograms for 50 nm and 25 nm TiN films . . . . . . . . . 138
Fig. 6.11 Temperature dependent conductance curves for 25 nm TiN film
taken at a fixed location. . . . . . . . . . . . . . . . . . . . . . . 140
Fig. 6.12 Histograms of the gap and averaged height versus gap. . . . . . 141
Fig. 6.13 Correlations in the 50 nm and 25 nm TiN films . . . . . . . . . 143
Fig. 6.14 Thermal model describing ion milling process which leads to
higher barrier height and lower gap . . . . . . . . . . . . . . . . 145
Fig. 7.1 Schematic showing MAR reflection in S-I-S junction . . . . . . 149
Fig. 7.2 Electron-like quasiparticle incident from the left superconduct-
ing electrode and doing MAR. . . . . . . . . . . . . . . . . . . . 156
Fig. 7.3 Electron-like quasiparticle incident from the right supercon-
ducting electrode and doing MAR. . . . . . . . . . . . . . . . . 163
Fig. 7.4 I − V characteristics of a symmetrical S-I-S junction with MAR 177
xii
Fig. 7.5 Hole-like quasiparticle incident from the left superconducting
electrode and doing MAR. . . . . . . . . . . . . . . . . . . . . . 183
Fig. 7.6 Hole-like quasiparticle incident from the right superconducting
electrode and doing MAR. . . . . . . . . . . . . . . . . . . . . . 185
Fig. 7.7 I − V characteristics of an asymmetrical S-I-S junction with
MAR effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Fig. 7.8 Temperature effect on I − V characteristics . . . . . . . . . . . 198
Fig. 7.9 Temperature effect on I −V characteristics showing sensitivity
to the smaller superconducting gap . . . . . . . . . . . . . . . . 201
Fig. 7.10 AC current (first Fourier component) of the I−V characteristics.208
Fig. 7.11 Temperature dependence of the critical current . . . . . . . . . 211
Fig. 8.1 Relay box for voltage-biased and current-biased setup . . . . . . 216
Fig. 8.2 Nb surface topography . . . . . . . . . . . . . . . . . . . . . . . 217
Fig. 8.3 Superconducting Nb-Nb I −V characteristics at different junc-
tion resistances . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Fig. 8.4 Detailed view of the I − V curves from Fig. 8.3 . . . . . . . . . 220
Fig. 8.5 Normalized conductance G/G0 vs. distance z for different start-
ing normal resistance Rn . . . . . . . . . . . . . . . . . . . . . . 225
Fig. 8.6 Temperature effect on the sub-gap bump at V = ∆/e . . . . . . 226
Fig. 8.7 Detailed view of subgap regions in Fig. 8.6 . . . . . . . . . . . . 227
Fig. 8.8 Superconducting gap ∆ and transparency parameter D as a
function of junction resistance . . . . . . . . . . . . . . . . . . . 229
Fig. 8.9 Critical current, switching current and retrapping current as a
function of normal junction resistance Rn. . . . . . . . . . . . . 232
Fig. 8.10 Stewart-McCumber number βc and the capacitance C between
the tip and the sample as a function of normal resistance Rn . . 234
Fig. 8.11 Josephson energy EJ (black), charging energy EC (red), plasma
energy h̄ωp (blue) and MQT escape temperature vs normal re-
sistance Rn of the junction . . . . . . . . . . . . . . . . . . . . . 237
Fig. 8.12 Finite resistance R0 − Rc of the supercurrent branch versus
normal resistance Rn of the junction . . . . . . . . . . . . . . . 239
Fig. 9.1 Configuration for an STM-SQUID measurement . . . . . . . . . 245
Fig. A.1 Histograms of χ2 for the 50 nm thick and 25 nm thick TiN films.252
Fig. B.1 Topography, superconducting gap map, gap histogram and sam-
ple curve of the S-I-S data taken from the 50 nm thick TiN film. 255
xiii
List of Abbreviations
BTK Blonder, Tinkham and Klapwijk
BCS Bardeen, Cooper and Schrieffer
DOS Density of states
LDOS Local density of states
N-I-N Normal-Insulator-Normal
N-I-S Normal-Insulator-Superconductor
S-I-N Superconductor-Insulator-Normal
S-I-S Superconductor-Insulator-Superconductor
IVC Inner Vacuum Can
UHV Ultra-High Vacuum
HOPG Highly Ordered Pryolitic Graphite
TiN Titanium nitride
MAR Multiple Andreev Reflection
AB Averin and Bardas
STM Scanning Tunneling Microscopy
JSTM Josephson Scanning Tunneling Microscopy
STS Scanning Tunneling Spectroscopy
NSF National Science Foundation
xiv
CHAPTER 1
Introduction
High transition temperature superconductors [1–7] were discovered more than
30 years ago. Despite theoretical advances and experimental discoveries, many basic
questions remain unresolved; the most important being what is the mechanism
causing pairing [8–10]. Many interesting phenomena have been observed in high Tc
materials, including the existence of competing charge ordering phenomena [11, 12],
the possible existence of different superconducting order parameters [13–17] and the
occurrence of rapid variations in superconducting properties at atomic length scales
[18, 19]. Such phenomena have naturally been examined at the atomic scale using
low-temperature scanning tunneling microscopes (STM) [20]. Using normal metal
tips, STMs allow sensitive measurements of the superconducting gap and energy
level spectrum via quasiparticle tunneling [21–29], as I discuss in Chapter 2.
An STM with a superconducting tip and a superconducting sample can form
a Josephson junction. Josephson junctions allow current to flow from one electrode
to the other electrode with no voltage drop [30, 31]. Interestingly, STMs with
superconducting tips are not very common. Hamidian et al. [32] reported the use
1
of an STM with a superconducting tip to investigate variations in the Cooper pair
density around zinc impurities in Bi2Sr2CaCu2O8+x. Although in this case both
the tip and sample were superconducting, the relatively large tunneling resistance
(>1 MΩ) and the small tip-sample capacitance made it impossible to obtain a true
Josephson supercurrent due to phase diffusion [33]. Instead one finds a dissipative
current flow that peaks at a non-zero voltage [32–38] due to incoherent tunneling of
Cooper pairs [39].
Previously in our lab at LPS, Anita Roychowdhury et al. [39–41] built a dual-
tip mK STM as part of a project to operate an STM that showed a real Josephson
supercurrent [42]. They demonstrated that each Nb-STM tip could independently
scan the surface of a sample [40, 41] and observed photon-assisted tunneling of
incoherent Cooper pairs in a single Josephson junction [39].
In this thesis, I first report further progress on the developement of a mK
Josephson STM with two independent tips [43]. I describe the technique I used to
simultaneously scan two connected STM tips when only the total current through
both tips can be measured. To do this I had to connect the two tips together and
develop a novel feedback system.
The second project I discuss involved using our STM system to measure 50 nm
thick and 25 nm thick superconducting TiN films. I was surprised to observe features
in the superconductor-insulator-normal (S-I-N) tunneling characteristics that were
clearly due to Andreev reflection effects. This required me to analyze the data
by fitting it to the Blonder-Tinkham-Klapwijk (BTK) theory of Andreev tunneling
[44, 45].
2
TiN has been the subject of much research in recent years due to its po-
tential applications in superconducting quantum computation [46–50] and its high
kinetic inductance, which is potentially interesting for constructing superconducting
microwave kinetic inductance detectors for x-ray spectroscopy and sub-millimeter-
wave astronomy telescopes [51–55]. The final project I discuss involves observation
of multiple Andreev reflection effects in Nb-Nb S-I-S tunneling.
In Chapter 2, I discuss two fundamental elements which form the basis for the
rest of the thesis. I first discuss scanning tunneling microscopy (STM) and describe
the two imaging methods I used, constant current and constant height. I then discuss
scanning tunneling spectroscopy (STS), which I used for investigating the electronic
density of states of samples. In particular, I describe the two operating modes I
used, voltage biased mode and current biased mode, and why I needed to use the
current-biased mode for Josephson tunneling experiments. In addition I describe
the three basic tunneling situations I encountered: tunneling from a normal tip to
a normal sample (N-I-N), tunneling from a normal tip to a superconducting sample
(N-I-S), and tunneling from a superconducting tip to a superconducting sample
(S-I-S). Finally, I briefly discuss STM measurements at millikelvin temperatures.
In Chapter 3, I describe the mK STM setup that I used. I begin with a
discussion of the principles of operation of a dilution refrigerator and the operating
procedure. I move on to describe the dual-tip mK STM apparatus and the wiring
of the STM at each stage of the dilution refrigerator. I also discuss a difficulty I
encountered, when the heat exchanger exploded during mixture removal, and the
subsequent repairs to bring the system partially back into service. Finally, I discuss
3
the UHV chamber used for sample preparation, the control electronics for operating
the STM, and the superconducting STM tips that I used.
In Chapter 4, I describe an important step towards constructing a true Joseph-
son STM based on forming an asymmetric SQUID [42]. In particular, I first describe
how I connected the two STM tips in our dual-tip system with a short flexible link
made from thin Nb foil. I then discuss another challenge of this approach, which is
how to implement feedback to control both tips when they share the same current
lead. I present a new feedback technique that separates the current signal for each
tip and then show results from simultaneously scanning two connected tips at room
temperature. I finish by examining the electrical performance of the system and the
mechanical compliance of the link.
Chapters 5 and 6 cover the second main topic I worked on. In Chapter 5,
I discuss the Bardeen-Tinkham-Klapwijk (BTK) theory of Andreev reflection and
their expression for the I(V) characteristics of S-N junctions with different trans-
parencies. In Chapter 6, I discuss my STM measurements of 50 nm and 25 nm
thick superconducting TiN samples in the mK-STM. While at first sight most of
the I(V) characteristics appeared to show standard N-I-S tunneling, we eventually
realized we were seeing obvious Andreev reflection effects, including zero bias con-
ductance peaks at some locations. This is expected from Andreev reflection effects
for a highly transparent barrier. Using BTK theory, as opposed to a standard tun-
neling approach to fit the data, I extracted spatial maps of the superconducting
gap ∆, temperature T and barrier height Z which showed large spatial variations.
I found striking spatially-dependent correlations between these parameters, as well
4
as differences between the two films, and I conclude with a discussion of a simple
model which may explain this surprising aspect of the data.
Chapter 7 and Chapter 8 describe the final major topic that I worked on.
In Chapter 7, I extend Averin and Bardas’s theory of Multiple Andreev Reflection
(MAR) in S-I-S junctions to the case where the electrodes can have different gaps. I
use this theory to obtain current-voltage characteristics for a single superconducting
channel with arbitrary transparency at temperature T . Using this theory I also
work out the critical current of a Josephson junction for arbitrary transparency.
Although this chapter includes a lot of mathematical detail, it is based on the
simple and physically appealing scattering matrix approach of Averin and Bardas.
In Chapter 8 I present measurements at 50 mK and 1.5 K of the current-voltage
characteristics of S-I-S junctions formed by a Nb STM tip and a Nb(100) sample.
The characteristics show clear features due to MAR and I apply my two-gap MAR
theory from Chapter 7 to extract key parameters such as the superconducting gaps
of the sample and tip.
In Chapter 9, I conclude with a summary of my main findings and discuss
some possible future work to complete a dual-tip Josephson STM.
5
CHAPTER 2
Scanning Tunneling Microscopy
2.1 Introduction
The first scanning tunneling microscope was described by G. Binning and H.
Rohrer [56–58] in 1981. The ability to resolve individual atoms stimulated consid-
erable research world-wide and they were awarded the Nobel Prize in Physics in
1986. The invention of the STM opened up atomic scale imaging in real space and
allowed researchers to manipulate single atoms and molecules. Moreover, spectro-
scopic techniques were developed to reveal the electronic properties of samples, such
as the local density of states (LDOS). Unlike classical transport mechanisms such as
diffusion and drift, the underlying principle of operation of an STM is quantum me-
chanical tunneling of electrons through a vacuum barrier, which cannot be explained
using a classical picture.
In Section 2.2 I first discuss quantum mechanical tunneling in one dimension
and give a brief overview of scanning tunneling microscopy. Next, in Section 2.3 I
discuss two modes that are often used in imaging the surface of samples for STM,
6
2.2. THEORY OF 1-D TUNNELING AND PRINCIPLE OF STM
constant current mode and constant height mode. I then discuss a powerful spec-
troscopy tool for investigating the electronic states of the samples in Section 2.4. I
discuss voltage-biased spectroscopy in normal-insulator-normal (N-I-N) tunneling,
normal-insulator-superconductor (N-I-S) tunneling and superconductor-insulator-
superconductor (S-I-S) tunneling. I go on to describe current-biased spectroscopy
for use in Josephson STM tunneling. Finally, in Section 2.5 I briefly discuss a mK
cryogenic system is needed for conducting my STM experiments.
2.2 Theory of 1-D tunneling and principle of STM
Tunneling phenomena can be treated using a time-independent approach,
which involves finding the wavefunctions inside and outside of a barrier and match-
ing them at the boundaries, or using Fermi’s golden rule, which arises from first order
time-dependent perturbation theory [59–61]. Typical introductory discussions con-
sider simple one-dimensional barriers and this is a good approximation for planar
metal-oxide-metal junctions. On the other hand an STM tip and planer sample have
a 3D geometry and a more complicated analysis is required to accurately describe
the behavior.
Nevertheless, let me first review simple 1-D tunneling. To proceed, I consider
an electron with energy E and mass m that is in region A and is incident from
the left onto barrier B (see Fig. 2.1). The barrier in region B has a uniform height
u. Classically, the particle can only get over the barrier if its energy E is greater
7
2.2. THEORY OF 1-D TUNNELING AND PRINCIPLE OF STM
than the barrier height (E > u). In quantum mechanics, due to the wave nature of
matter, there is some probability to penetrate through the barrier even if E < u.
I can write Schrödinger’s equation and find the wave function in each of the three
regions (Fig. 2.1). In region A, Schrödinger’s equation is
h̄2 d2− Ψ1 = EΨ1 (2.1)
2m dx2
and the wavefunction is,
Ψ = eikx1 + Ae
−ikx (2.2)
where
2 2mEk = 2 . (2.3)h̄
For region B, Schrödinger’s equation is
2 2
− h̄ d Ψ2 + uΨ2 = EΨ2 (2.4)
2m dx2
and we can write
Ψ = Beχx + Ce−χx2 (2.5)
8
2.2. THEORY OF 1-D TUNNELING AND PRINCIPLE OF STM
Energy
A B C
u
E
x
0 s
Figure 2.1: A particle with energy E is incident from the left in region
A. In region B, the potential barrier u is larger than the energy E, but
there is some probability for the particle to penetrate through the barrier
into C.
9
2.2. THEORY OF 1-D TUNNELING AND PRINCIPLE OF STM
where
χ2
2m(u− E)
= 2 . (2.6)h̄
For region C, Schrödinger’s equation is
h̄2 2− d Ψ3 = EΨ3 (2.7)
2m dx2
and
Ψ = Deikx3 (2.8)
where
2 2mEk = 2 . (2.9)h̄
The rate at which electrons go through the barrier determines the tunnel cur-
rent. This is calculated from the ratio of the transmitted current density jt to the in-
cident current density ji; i.e. the transmission coefficient is T=jt/ji=(h̄k/m)|D|2/(h̄k/m)=|D|2.
By matching the wave function and the first derivative of the wave function at the
boundary x = 0 and x = s, one finds [62]
1
T = = |D|2. (2.10)
1 + (k2 + χ2)/(4k2χ2) sinh2(χs)
10
2.2. THEORY OF 1-D TUNNELING AND PRINCIPLE OF STM
Assuming a relatively high and wide barrier, so that the wave function in the barrier
is strongly attenuated,
≈ 16k
2χ2
T e−2χs (2.11)
(k2 + χ2)2
where from Eq. (2.6) χ is given by
√
2m(u− E)
χ = . (2.12)
h̄
The factor e−2χs in Eq. (2.11) is crucial for STM tunneling because it indicates
that the current depends exponentially on the barrier width s. Assuming a barrier
width of s = 5 Å and effective vacuum barrier height of u − E = 4 eV gives
T ∼ 10−5. For these parameters, changing the barrier width by 1 Å will change the
barrier transmission by an order of magnitude.
Figure 2.2 shows a schematic of the setup of an STM, including an atomically
sharp conducting tip, a vacuum tunnel barrier and the surface of a conducting
sample. To control the x, y and z motion of the tip, the tip is attached to a
piezoelectric scanner that has five electrode plates (x+, x−, y+, y−, z) insulated from
each other. Applying voltage to specific electrodes causes the tip to move in x, y or
z due to the piezo shrinking or expanding, depending on the voltage applied. When
the tip is brought very close to the sample and a voltage is applied between the
sample and the tip (typically mV to a few volts), a small current (typically in nA
range) tunnels through the vacuum gap between the tip and sample. The tunneling
11
2.2. THEORY OF 1-D TUNNELING AND PRINCIPLE OF STM
Figure 2.2: Schematic diagram of STM setup. The tip position is con-
trolled by applying voltages to the x+, x−, y+, y−, z electrodes on the
scan piezo. The tunneling current from the tip is detected when the
sample is voltage biased and the tip is close to the sample. For topo-
graphic imaging, the voltage applied to the z electrode is adjusted to
keep the tunneling current constant.
12
2.2. THEORY OF 1-D TUNNELING AND PRINCIPLE OF STM
current is amplified and converted to a voltage which is fed to the control electronics.
This voltage is compared to a set point and the difference is used to generate an error
signal that is fed back to the z-piezo to keep the tip–sample separation constant.
Because the tunnel current is extremely sensitive to the distance between the tip
and the sample, the feedback forces the vertical motion of the tip to precisely follow
the surface topography if the tip is moved laterally.
For a measureable tunneling current to be present, the tip has to be brought
very close to the sample. This is done using a coarse approach mechanism, in our
case a Pan-Style walker [63]. During the coarse approach, a dc bias voltage is applied
to the sample and the tip is effectively grounded. For each step of the walker the
tip typically moves forward a few 10s of nm. The z-piezo is then extended up to full
range to see if the tunneling current increases. If no current signal is found, the z
piezo is retracted, the walker moves one step closer toward the sample. This process
is repeated until the tip is close enough to the sample that a tunneling current is
detected.
Once the tip is in tunnel range, it is moved to an (x, y, z) position defined by
the user. To raster scan an area, the lateral (x, y) motion is controlled by applying
voltages to the x+, x−, y+ and y− electrodes. The vertical z-motion is controlled
by the z-piezo feedback control electronics. Recording the voltage applied to the
z-piezo as a function of the position (x, y) gives us the surface topography of the
scan area.
13
2.3. TOPOGRAPHIC IMAGING
2.3 Topographic imaging
As mentioned earlier, an STM tip has a 3D geometry and the 1D time-
independent Schrödinger equation does not accurately model the situation. Ter-
soff and Hamann [60, 61] used time-dependent perturbation theory to model the
3-dimensional tip structure and calculate the corresponding current and conduc-
tance. Their theory also produced an exponential dependence of the current on the
tip-sample separation, which is used in constant current mode of imaging. More-
over, they showed that the conductance is proportional to the density of states of
the sample. This is particularly useful for spectroscopy and understanding images
taken in the constant current mode. The other imaging mode I discuss in this sec-
tion is the constant height mode. This mode is not used as much as the constant
current mode, but it has some advantages in certain situations.
2.3.1 Constant current mode
In Section 2.2, I examined tunneling in one dimension and found the trans-
mission through the barrier depended exponentially on the barrier thickness. Thus
we expect that the tunneling current will obey
I ∝ e−2χs. (2.13)
14
2.3. TOPOGRAPHIC IMAGING
where χ is given roughly by Eq. (2.12) and s denotes the distance between the
sample surface and the front of the tip. The useful feature for imaging is that the
current is exponentially sensitive to the separation s.
A feedback loop is used to control the tip-sample separation so that the tunnel-
ing current between the tip and the sample remains constant. As such, the recorded
z signal can be interpreted as the topography as a function of the position (x, y) of
the sample.
2.3.2 Constant height mode
The main disadvantage of the constant current mode is the finite response time
of the feedback loop, which limits the maximum scanning speed. This drawback
also makes it harder to capture atomic scale features because the feedback responds
less to high-frequency components of the tunneling current due to small surface
features. An alternative is to use constant height mode, which involves turning off
the feedback and recording the tunneling current while scanning in x and y. A
significant drawback of constant height mode is that without feedback the tip is
prone to crashing into the sample if it is not atomically flat.
In practice, samples can be scanned using a mixture of constant height and
constant current modes. For this mixed mode, the system is set up for constant
current mode, and as usual when the tip scans over a sharp feature, the feedback
does not respond immediately and the current will not be constant. A map of the
tunnel current versus position will thus show some contrast from sharp topographic
15
2.4. SPECTROSCOPY
features. Thus by recording both the z-piezo voltage and the tunnel current, we can
obtain mixed-mode images.
2.4 Spectroscopy
Besides being able to generate topographic images, STMs can also perform
tunneling spectroscopy to measure electronic properties of a sample at the atomic
scale such as the local density of states (LDOS). Tunneling spectroscopy typically
involves moving to a fixed location, turning off the feedback, and then sweeping
the bias voltage V while recording the tunneling current I. The feedback is then
switched back on to prevent the tip from drifting into the sample.
In standard voltage-biased spectroscopy, voltage V is applied to the sam-
ple while the current I and the conductance dI/dV are acquired. The resulting
I − V or dI/dV − V characteristics depend on the materials used in the junction.
Next, I examine the standard cases of normal-insulator-normal (N-I-N) tunneling,
normal-insulator-superconductor (N-I-S) tunneling and superconductor-insulator-
superconductor (S-I-S) tunneling. In the standard approach [64] the tunnel barrier is
assumed to be relatively high so that Andreev reflection effects [44] can be neglected.
In Chapters 5 and 7 I present a more general discussion of I −V characteristics, in-
cluding effects from the Andreev processes in N-I-S and S-I-S junctions. In addition
to voltage-bias spectroscopy, I also briefly discuss current-bias spectroscopy, which
I used for investigating S-I-S junctions as described in Chapter 8.
16
2.4. SPECTROSCOPY
As Tersoff and Hamann [60, 61] emphasized, it is important to understand
that the conductance dI/dV can produce direct information about the local density
of states (LDOS) of a sample. In principle, one could measure the current at finely
spaced voltages V and then numerically take a derivative of the current with respect
to the voltage. However the result tends to be extremely noisy. Instead we add a
sinusoidal modulation to the DC bias voltage so that V (t) = V0 +A sinωt, and use
a lock-in amplifier to measure the resulting current Iac at the frequency ω. Using a
Taylor expansion of the current
∣
dI ∣ ( )
I(V ) ≈ I(V ) + ∣0 ∣ A cosωt+O A2 (2.14)dV V=V0
we see that the zeroth-order term is the non-modulated current I(V0). The ampli-
tude of the first-order term is proportional to the derivative of the current, i.e. the
conductance Gn = dI/dV . With a suitable choice of the drive frequency and lock-in
time constant, the I(V0) and dI/dV vs. V0 data can be obtained simultaneously.
Another useful technique is the current-bias mode. The I − V characteristics
of an S-I-S Josephson tunnel junction is often acquired by sweeping the current
while measuring the voltage across the junction. This is done because the voltage
will remain zero until the bias current exceeds the critical current, at which point
a voltage appears across the junction. Depending on the junction parameters, the
voltage may jump discontinuously from zero to a substantial value, typically on
the order of the gap. In this case one also typically sees hysteretic behavior when
the current sweeping direction is reversed, this hysteretic behavior is obscured by
17
2.4. SPECTROSCOPY
voltage-biased technique.
Naaman et al. [33], for example, used the voltage-biased method to measure the
I−V characteristics of a superconducting tip and a Pb film junction. Unfortunately,
the voltage-biased technique cannot tell the true behavior of the I(V) because of the
discontinuity of the voltage jump when the critical current is exceeded. In my work,
I modified the existing STM electronics to take both voltage-biased and current-
biased I-V characteristics for a given junction resistance.
Figure 2.3 shows schematics for the voltage-bias mode and the current-bias
mode. Figure 2.3 (a) shows a conventional setup for tunneling spectroscopy in
which voltage Vb is applied to the sample and the tunneling current I is recorded
as a function of Vb. Figure 2.3 (b) shows the current-bias mode which has a resistor
with a large resistance R connected in series with the STM junction to fix the
current I passing through the junction. A differential amplifier is used to measure
the voltage V across the junction as a function of the current I. If the resistance R
is much larger than the junction resistance, i.e. R  RJ , the current can be found
from the applied voltage Vb divided by R. Otherwise, I = Vb/(R + RJ) and the
junction resistance needs to also be taken into account to find the current flowing
through the junction.
2.4.1 N-I-N tunneling
Figure 2.4 shows a schematic diagram of electron energy levels for N-I-N tun-
neling. Figure 2.4 (a) shows the situation in thermal equilibrium with no applied
18
2.4. SPECTROSCOPY
(a) (b) V
Voltage bias Current Bias
R -
Sample
Sample
Tip Tip
I I
Vb A V Ab
Figure 2.3: (a) Schematic diagram of spectroscopy mode using voltage
bias Vb in which a voltage is applied to the sample and the current I
flowing through the junction is measured with a current preamplifier.
(b) In current-bias mode, a large resistor R is connected in series with
the junction to set the current I through the junction and the voltage
V across the junction is measured using a differential amplifier.
19
Bias voltage
Bias voltage
+
2.4. SPECTROSCOPY
voltage so that both Fermi levels are aligned (the shaded light blue areas indicate
filled states). Figure 2.4 (b) shows the situation when the right normal metal has
a voltage +V applied while the left electrode is grounded (0 V). As shown, this
effectively raises the Fermi level of the left electrode with respect to the Fermi level
of the right electrode and causes a net flow of electrons from the left to the right.
To calculate the net tunneling current, we need to find the difference between the
rightward and leftward currents. Following Tinkham [64] we can write the current
flowing from the left electrode to the right electrode as
∫
4πe ∞
IL→R = |M |2N1(− eV )f(− eV )N2()(1− f())d (2.15)
h̄ −∞
where N (E) is the density of states on the left, f() = 1/(e(E−µ)/kBT1 + 1) is the
Fermi distribution and M is the tunneling matrix element. Here N1( − eV )f()
represents the filled states in the left electrode and N2()(1 − f()) represents the
empty states in the right electrode. Similarly, the current flowing from the right
electrode to the left electrode is
∫
4πe ∞
IR→L = |M |2N2()f()N1(− eV )(1− f(− eV ))d. (2.16)
h̄ −∞
20
2.4. SPECTROSCOPY
The total current flowing from left to right is then
I = IL→R∫− IR→L
4πe ∞
= |M |2∫ N1(− eV )N2()[f(− eV )(1− f())− (1− f(− eV ))f()]dh̄ −∞
4πe ∞
= |M |2N1(− eV )N2()[f(− eV )− f()]d (2.17)
h̄ −∞
Let’s look at how Eq. (2.17) behaves at T = 0. Notice on the left side that
for  > eV , we have f( − eV ) = 0 (see region A in Fig. 2.4 (b)) while for  > eV
we have f() = 0 on the right. Therefore region A contributes no current. For
0 <  < eV (region B in Fig. 2.4 (b)), we have f(− eV ) = 1 for the left electrode
and f() = 0 for the right. Therefore [f(− eV )− f()] = 1 and Eq. (2.17) gives a
net current in this region. Finally for  < 0, we have f(− eV ) = f() = 1 in both
the left and right electrode (see region C in Fig. 2.4 (b)) and therefore this region
contributes no current.
Given the behavior of the Fermi function at T = 0, the integration range in
Eq. (2.17) can be cut off at zero and eV , and Eq. (2.17) becomes
∫
4πe eV
I = |M |2N1(− eV )N2()d (2.18)
h̄ 0
For typical metals of interest, we can take N1(− eV ) and N2() as constants N1(0)
and N2(0), the density of states at the Fermi level. Similarly, we will assume that
21
2.4. SPECTROSCOPY
(a) 
e
N1 N2
(b) 
A 
eV 
B 
0
C N1
N2
Figure 2.4: (a) Schematic of the density of states in N-I-N tunneling at
T = 0 for ideal normal metals when no voltage is applied. (b) The right
electrode is biased at voltage V above the left electrode, which effectively
raises the Fermi level of the left electrode, causing a net flow of electrons
from the left to the right electrode.
22
2.4. SPECTROSCOPY
|M |2 is independent of energy and this then gives
4πe2
I = |M |2N1(0)N2(0)V
h̄
= GNNV (2.19)
where the conductance of the N-I-N junction is
4πe2
GNN = |M2|N1(0)N2(0) (2.20)
h̄
For the same assumptions except that T > 0, one finds the same result [64].
Therefore we see that an N-I-N junction acts like an ideal, temperature-independent
ohmic resistor.
2.4.2 N-I-S tunneling
The situation becomes more interesting if one electrode is superconducting.
Tunneling experiments conducted by Giaever and Megerle [65] between a supercon-
ductor and a normal metal demonstrated that the superconducting gap could be
measured electrically and this helped to further establish the validity of the BCS
theory.
We can apply Eq. (2.17) to the N-I-S situation by writing [64]
∫
4πe ∞
INS = |M |2Nn1(− eV )Ns2()[f(− eV )− f()]d, (2.21)
h̄ −∞
23
2.4. SPECTROSCOPY
where Nn1 = Nn1(0) is the density of states of the normal metal on the left and
 |E|
 √ 2− 2Nn2(0) for |E| > ∆Ns2 =  E ∆ (2.22)0 for |E| < ∆
is the density of states of the superconductor on the right where Nn2(0) is the density
of states of the normal state of electrode 2. Figure 2.5 illustrates the situation for
T = 0. Note that if the voltage V applied to the superconducting electrode is
smaller than ∆/e, where ∆ is the superconducting gap, there are no states for
the electrons from the normal side to tunnel into the superconducting electrode.
Therefore there is no tunneling current unless eV > ∆. For T = 0, the Fermi
function and superconducting density of states cut off the integration to the range
∆ <  < eV . For T = 0 and V > ∆/e, Eq. (2.21) becomes
∫
4πe eV
INS = ∫ |M |
2Nn1(− eV )Ns2()d
h̄ ∆
4πe eV
= ∫ |M |
2Nn1(0)Ns2()d
h̄ ∆
4πe eV | Ns2()= M |2Nn1(0)Nn2(0) d
h̄ ∆ ∫ Nn2(0)
4πe eV
= |M |2 Nn1(0)Nn2(0) √ d
h̄ ∫ ∆ 2 −∆2
G eVnn √ = d
e √ 2 −∆2∆
Gnn (eV )2 −∆2
= . (2.23)
e
For eV < ∆, one finds I = 0, while Eq. (2.23) gives the current I as function of
24
2.4. SPECTROSCOPY
(a) 
e
Δ
Δ
N1 S2
(b) 
e
Δ eV 
Δ
N1 S2
Figure 2.5: (a) Schematic drawing of energy diagram for N-I-S tunneling
with no applied voltage. The left electrode (N1) is normal while the right
electrode (S2) is superconducting. (b) The superconducting side (S2) is
voltage biased by V , causing the electrons to flow from the left to the
right.
25
2.4. SPECTROSCOPY
voltage V for V > ∆/e. At T = 0, If we take the derivative of the current with
respect to the voltage we can show that
 √GnneVdIns  eV 2 −
2 for |eV | > ∆∆
Gns(V ) = = (2.24)
dV
0 for |eV | < ∆
Comparing Eq. (2.22) to Eq. (2.24) we see that the differential conductance
Gns(V ) has the same functional form as the superconducting density of state. This
is an example of how dI/dV can reveal the local density of states (LDOS), as Tersoff
argued.
For T > 0, thermal broadening becomes important and we write
∫
dI ∞ns Gnn  ∂f(− eV )
Gns = = √ d. (2.25)
dV e −∞ 2 −∆2 ∂V
The function ∂f/∂V in Eq. (2.25) is a peak of width kBT and effectively rounds off
the divergence in the density of states at the gap.
Figure 2.6 shows a plot of conductance vs. voltage for temperatures T= 50
mK (red), 300 mK (blue) and 1K (green) for ∆ = 0.5 meV. From the plot we can see
there is a sudden rise of conductance around the value of superconducting gap; these
are called coherence peaks. As Fig. 2.6 also shows, non-zero temperature smears
out the coherence peaks.
This is a good place to briefly discuss the Dynes formula [66]. This formula
has frequently been used to extract the superconducting gap from STM conductance
measurements. Dynes et al. [66] argued that the finite lifetime of quasiparticles
26
2.4. SPECTROSCOPY
5
4
3
2
1
0
- 2 - 1 0 1 2
V (mV)
Figure 2.6: Plot of normalized conductance 1 dIns vs. voltage V at
Gnn dV
different temperatures. The red curve is at 50 mK, the blue curve is at
300 mK, the green curve is at 1 K. In all three cases the superconducting
gap ∆ = 0.5 meV. The temperature mainly broadens the coherence
peaks, which occur at V = ±∆/e.
27
1 dIns
Gnn dV
2.4. SPECTROSCOPY
would cause broadening of the coherence peaks. To account for the lifetime they
modified the BCS density of states to :
Ns() [ − iΓ ]
= Re √ (2.26)
Nn(0) (− iΓ)2 −∆2
where Γ/h̄ is the quasiparticle scattering rate. The main effect of non-zero Γ is to
cause broadening of the coherence peaks. The advantage of using this form is that
it is simple to fit to conductance data because it is a simple analytical function. In
contrast, application of Eq. (2.25) requires a careful numerical integration. Although
the physical basis for Eq. (2.26) has been disputed, it captures the qualitative be-
havior and has found widespread application for extracting the superconducting gap
value.
2.4.3 S-I-S tunneling
For S-I-S tunneling, Eq. (2.17) becomes [64]:
∫
4πe ∞
ISS = ∫ |M |
2Ns1(− eV )Ns2()[f(− eV )− f()]d
h̄ −∞
G ∞nn Ns1(− eV ) Ns2()
= ∫ [f(− eV )− f()]de −∞ Nn1(0) Nn2(0)
G ∞nn |− eV | ||
= √ √ [f(− eV )− f()]d (2.27)
e −∞ (− eV )2 −∆2 2 −∆21 2
where Ns1(− eV ) is the density of states in the left electrode, Ns2() is the density
of states in the right electrode, and Gnn is the conductance of the junction when
28
2.5. CRYOGENIC STM’S
both electrodes are in the normal state.
Figure 2.7 shows the energy level diagram for S-I-S tunneling at T = 0. Notice
that when eV > ∆1 + ∆2, quasiparticles will flow from one electrode to the other
electrode, while for eV < ∆1 + ∆2, no current will flow. In general for T 6= 0
numerical integration is required to compute the I − V curves. Figure 2.8 shows
examples of I − V characteristics obtained by numerical integration of Eq. (2.27)
at temperatures of 300 mK (red), 3 K (blue) and 5 K (green). For these examples,
I used ∆1 = 1 meV and ∆2 = 0.5 meV. All three curves show a sharp rise at
V = (∆1 + ∆2)/e = 1.5 mV. When the temperature is large enough, one also
sees a sharp peak at |∆1 −∆2|/e = 0.5 mV and a region with negative differential
resistance. Although the numerical integration is relatively straightforward, I did
not see such peaks in my S-I-S tunneling data. Instead I saw step-like structures
inside for V < (∆1 + ∆2)/e. As I discuss in Chapter 7, this turned out to be due to
multiple Andreev reflection (MAR) in junctions with relatively high transparency.
2.5 Cryogenic STM’s
The superconductors that I investigated were TiN thin films, which had su-
perconducting transition temperatures of Tc ≈ 4 K, and a single niobium crystal
with Tc ' 9.2 K. Clearly I needed to cool below Tc to see superconducting behav-
ior. Furthermore, by using a much lower temperature, I could obtain less thermal
broadening, and get much better resolution spectroscopic data.
29
2.5. CRYOGENIC STM’S
(a) 
e
Δ e1 Δ2
Δ Δ1 2
S1 S2
(b) 
e
Δ e1 Δ1+Δ2= eV
Δ
Δ 21
S Δ1 S2 2
(c) 
e
Δ e1
Δ eV Δ1 2
S Δ1 S2 2
Figure 2.7: (a) Schematic drawing of electron energy levels in the left
and right electrode for S-I-S tunneling at T = 0 for V = 0. (b) Energy
levels for V = (∆1 + ∆2)/e and (c) V > (∆1 + ∆2)/e. When the bias
voltage V is greater than (∆1 + ∆2)/e, a current will flow.
30
2.5. CRYOGENIC STM’S
3
2
1
0
- 1
- 2
- 3
- 3 - 2 - 1 0 1 2 3
V (mV)
Figure 2.8: I−V characteristics from numerical evaluation of Eq. (2.27)
for S-I-S tunneling with ∆1 = 1 meV, and ∆2 = 0.5 meV. The red curve
is for temperature T = 300 mK, the blue curve is for T =3 K, and the
green curve is for T = 5 K. The current rises at V = (∆1 + ∆2)/e '1.5
mV. At the higher temperatures, we see a current peak at the difference
of the superconducting gaps V = (∆1 −∆2)/e = 0.5 mV.
31
I (arb.)
2.5. CRYOGENIC STM’S
Finally I note that a Josephson Scanning Tunneling Microscope (JSTM) must
be operated at mK temperatures in order to see a measureable critical current
[42, 67–70]. In a JSTM there are three important energy scales: the Josephson
energy EJ = h̄I0/2e, the charging energy EC = e
2/2C and the thermal energy kBT .
I0 is the critical current of the Josephson junction. In a standard STM setup, the
capacitance of the tip-sample junction is on the order of fF and in my setup the
critical current can be typically of order 1 nA. This leads to EJ/kB = 0.1 K and
EC/kB = 1 K. Therefore we need to cool our STM to T  EJ/kB or millikelvin
temperatures using a dilution refrigerator to prevent phase diffusion from thermal
excitation of the junction [34–36, 38, 71]. In addition, with EJ << EC charging
effects become important and this leads to suppression of the critical current [64, 72–
74] and even greater sensitivity to phase slips, as discussed in Chapter 7.
32
CHAPTER 3
Experimental Setup of the Millikelvin STM
In this chapter, I discuss details of the setup, construction and operation of our
millikelvin scanning tunneling microscope and associated systems. In Section 3.1 I
describe the working principle of a 3He/4He dilution refrigerator. Next in Sections
3.2, 3.3 and 3.4, I describe the design and operation of our dual-tip STM, including
the wiring and filtering used from room temperature to the mK temperature stage.
In Section 3.5 I discuss the accident in June 2016 which resulted in damage to the
dilution refrigerator and the subsequent partial repairs which allowed compromised
operation to about 0.5 K. I next describe the ultra high vacuum (UHV) sample
preparation system (Section 3.6), which allowed me to prepare and transfer samples
to the cold-stage while maintaining UHV conditions. Finally, I describe the opera-
tion and components of the control electronics (Section 3.7) and the preparation of
Nb and vanadium STM tips (Section 3.8).
33
3.1. DILUTION REFRIGERATOR
3.1 Dilution refrigerator
The working principle of a dilution refrigerator [75] is based on the behavior
of liquid mixtures of 3He and 4He (see Figure 3.1). Liquid 4He becomes superfluid
[76] at 2.177 K (zero viscosity) and can flow without loss of kinetic energy. 3He,
which is a fermion, does not become superfluid until 2.491 mK [76]. By diluting the
4He liquid with 3He, the temperature of the superfluid transition is lowered. The
boundary line is called the λ-line.
Above 67.5% atomic concentration of 3He in 4He , superfluidity does not exist.
Interestingly, below 0.867 K, 3He/4He mixtures separate into two phases. One phase
is rich in 3He (the concentrated phase), and the other phase is rich in 4He (the dilute
phase). In the limit where the temperature of the mixture is zero, the concentrated
phase becomes pure 3He. However, the concentration of 3He in the dilute phase
does not approach zero as T goes to zero, rather it reaches a limiting concentration
of 6.6% 3He [75]. This phase separation and finite solubility of 3He in 4He is crucial
to the operation of a 3He-4He dilution refrigerator.
Figure 3.2 shows a schematic diagram of a dilution refrigerator. For proper
operation, the dilute/concentrated phase boundary should be in the mixing chamber,
and the liquid level of the dilute phase should be in the still. Since 4He is denser
than 3He, the concentrated phase floats on top of the dilute phase. The cooling
power of the dilution refrigerator comes from the latent heat absorbed when 3He
atoms are transferred from the 3He rich side to the 3He poor side.
Examining Fig. 3.2, we see that the dilute phase extends from the still to the
34
3.1. DILUTION REFRIGERATOR
Figure 3.1: Phase diagram for 3He/4He mixtures from Ref. [75]. 4He
becomes superfluid at 2.177 K. Mixing 3He with 4He lowers the temper-
ature of superfluidity boundary. At 0.87 K, liquid helium separates into
two phases: a dilute phase with mostly 4He and a concentrated phase
with mostly 3He. Note that the dilute phase has 6.6 % 3He at 0 K.
35
3.1. DILUTION REFRIGERATOR
mixing chamber. A pumping line is attached to the top of the still. The still operates
at about 0.7 K, at which temperature the vapor pressure of 3He is much larger than
the vapor pressure of 4He, ensuring that mostly 3He evaporates. The difference
in 3He concentration between the still and the mixing chamber creates an osmotic
pressure gradient in the 3He, which acts to reduce the 3He concentration in the
dilute phase in the mixing chamber. In the mixing chamber 3He in the concentrated
phase then crosses the phase boundary to the dilute phase in order to maintain the
6.6 % 3He concentration of the dilute phase. This process is effectively 3He liquid
evaporating into a 4He vacuum and this absorbs latent heat. The latent heat is
provided from the mixing chamber and results in cooling of the mixing chamber.
In principle, there is no lower limit to the temperature that can be reached, but
in practice heat leaks, heating of the flowing liquids and inefficiency of the heat
exchangers limits the minimum temperature.
Most commercial refrigerators are now “dry” in that a separate 4K cryo-cooler
is used for the initial cool-down and for cooling the returning 3He gas. The additional
cryogenic systems create some vibrations and noise, which are serious problems for
the operation of a cryogenic STM. As a result, we use a “wet” refrigerator to cool
our millikelvin STM system. This involves maintaining 4 K liquid 4He around the
vacuum space (see Fig. 3.2). This bath needs to be refilled with liquid helium as
the liquid evaporates, but otherwise it is relatively free from vibration.
To cool the system from room temperature, I start by closing up the inner
vacuum can (IVC)(see Fig. 3.2) and raising the custom made super-insulated liquid-
helium cryostat [77]. I then fill this dewar with liquid nitrogen at 77 K and introduce
36
3.1. DILUTION REFRIGERATOR
LN2 
cold 
trap
IVC
20
Cu heat shield
Vacuum
Vacuum
Figure 3.2: Schematic diagram of our dilution refrigerator.
37
3.1. DILUTION REFRIGERATOR
a small amount of 3He exchange gas into the inner vacuum can to thermally couple
the dilution refrigerator unit to the 77 K bath (see Fig. 3.2). After about 1 day, the
system has settled to 77 K and the liquid nitrogen is then removed. The dewar is
then slowly filled with liquid helium to allow the refrigerator to cool to 4 K. Once
the refrigerator is at 4 K, the exchange gas is pumped out so that the refrigerator
stages are no longer thermally coupled to the bath.
With the system at 4 K and the exchange gas removed, we can begin to
circulate the 3He/4He mixture. A small amount of mixture is released from the
dump (a container that stores the mixture in gas form at room temperature) and
passed through a nitrogen trap kept at 77 K. This removes impurities such as N2
and O2, as well as oil from the pumps. The mixture next flows through a liquid He
cold trap which is inserted into the helium bath. This captures remaining impurities
that were not captured from the nitrogen trap, such as H2.
The mixture then flows into the condenser line and condenses at the 1 K pot
which is actually at about 1.5 K. The 1 K pot is a small copper chamber which
is connected to the 4He bath through a capillary tube. Pumping on the pot sucks
liquid helium from the bath and evaporation of the liquid in the pot causes cooling
to about 1.5 K, which is not cold enough for the mixture to reach phase separation,
but allows liquid to condense, cool and fill the heat exchangers, mixing chamber and
still. To achieve phase separation, we pump on the still, which cools below 1.2 K due
to evaporate of 3He. The condensed mixture from the 1 K pot passes through the
still heat exchanger and, as 3He is preferentially pumped from the still, the still cools
to the phase separation point. Additional pumping moves the phase boundary until
38
3.2. OVERALL EXPERIMENTAL SETUP OF THE MILLIKELVIN STM
eventually it reaches the mixing chamber, which then cools to a base temperature
of about 200 mK after several hours. With additional root pump running, it cools
down to 20 mK.
3.2 Overall experimental setup of the millikelvin STM
Figure 3.3 shows the overall experimental setup of the millikelvin STM system
[41]. A UHV sample preparation chamber and a UHV transfer chamber sit on top
of an optical table in an rf-shielded room. The 3He-4He dilution refrigerator is
mounted below the transfer chamber and the STM is bolted to the bottom of the
mixing chamber. This design allows us to clean the sample under UHV conditions
and transfer it into and out of the mK-STM without breaking vacuum, therefore
preserving the quality of the sample surface. We use a horizontal magnetic transfer
rod to bring samples from the preparation chamber to the transfer chamber. We
then use a vertical magnetic transfer rod [41], which runs through the center axis
of the dilution refrigerator to take the sample from the transfer chamber to the mK
stage of the STM.
Our dilution refrigerator is a customized Oxford Instruments Kelvinox with a
cooing power of 400 µW at 100 mK. The dilution refrigerator has an inner vacuum
can (IVC) which is pumped to high vacuum using a scroll pump and a turbo pump
prior to cooling down. The IVC and rest of the refrigerator insert sit in a custom
made super-insulated liquid-helium cryostat [77]. It serves as a liquid helium bath
39
3.2. OVERALL EXPERIMENTAL SETUP OF THE MILLIKELVIN STM
Figure 3.3: Illustration of mK-STM setup with UHV sample preparation
chamber and transfer rods.
40
3.3. DUAL-TIP STM
and enables our STM to thermally couple to the helium bath and cool to 4 K. The
base temperature of the refrigerator was 6-7 mK, but due to the heat load from
the large amount of wiring for the dual-tip STM [41], the actual operating base
temperature was around 30-35 mK.
3.3 Dual-tip STM
Our STM is a highly modified co-axial Pan-style design [63] which has two
independently controllable tips. The tips can be connected to operate in a SQUID
configuration [42] with a superconducting sample. A detailed description of the dual-
tip STM, including connection and characterization of the two electrically connected
STM tips, can be found in Chapter 4.
Figure 3.4 shows the overall layout of the STM. The main body is made out
of Macor. The sample is mounted on a sample stud (1) that is loaded from the
top and the two STM tips (2) approach from below. The two STM tips are spaced
about 1 mm apart with each tip attached to its own set of piezoelectric actuators
[41]. Each tip is controlled by a piezoelectric walker (5,6) and scanner (3,4) which
allow for coarse motion in z (vertical) and fine 3-D motion, respectively. The outer
tip assembly is moved vertically (z) by the large outer walker, which also carries the
smaller inner walker and inner tip assembly, thus moving the two tips together. The
inner walker moves the inner tip vertically (z) relative to the outer walker [40, 41].
To monitor the movement of the walkers and tips, a capacitance bridge [78]
41
3.3. DUAL-TIP STM
(a) (b)
1 2
5
3
4
7
6
Figure 3.4: (a) Cross sectional view and (b) photograph of the dual-tip
STM. Labels are (1) the sample stud, (2) two STM tips, (3) outer tip
scanner, (4) inner tip scanner, (5) outer tip walker, (6) inner tip walker
and (7) the capacitance sensor for the outer tip and the inner tip. Figure
from [40].
42
3.4. WIRING
was used to measure the capacitance (see Fig. 3.4) during the coarse approach of
each tip to the sample. The capacitance sensor reading for the outer walker varied
from 11 pF to 19 pF over the 24 mm z-range. For the inner walker, the capacitance
sensor reading varied from 1.8 pF to 6 pF over the 8 mm range. At cryogenic
temperatures, the maximum lateral scan range of the inner-tip piezo-tube scanner
was about 2 µm while the lateral scan range of the outer-tip piezo-tube scanner was
about 1 µm.
3.4 Wiring
We used two kinds of wires in our STM system to connect the room tempera-
ture electronics to the STM [41]. “Signal wires” were used for connecting to the two
STM tips, for applying sample bias and for reading out the two capacitance/monitor
sensors. “Drive wires” were used for supplying control/drive signals to move the two
walkers (6+6 lines), to drive the x-y-z motion of the tip piezos (5+5 lines) and for
the thermometry lines.
In order to transmit the very small voltages or currents carried by the signal
wires, the wired need to be well-shielded against rf pickup and microphonic noise.
We also need to minimize heat flow between different refrigerator stages. This was
done by using semi-rigid CuNi microcoax lines from 300 K to the mixing chamber
stage. Figs. 3.5-3.10 show detailed views of the signal lines at each stage. To heat
sink the wires as they went from 300 K to 30 mK, the coax was clamped to Cu
43
3.4. WIRING
8
1
3 cm
Figure 3.5: The 1 K pot (1) was custom built to 1.5 times the normal
size. (8) NbTi microcoax signal wires are bolted to the Copper heat
sinks.
44
3.4. WIRING
9
2
2 cm
Figure 3.6: Label (2) indicates the still. NbTi microcoax signal wires
are bolted to the Copper heat sinks (9).
45
3.4. WIRING
10 3
2 cm
Figure 3.7: Label (3) indicates the 3He/4He heat exchanger. NbTi mi-
crocoax signal wires are bolted to the Copper heat sinks (10).
46
3.4. WIRING
4
11
5
2 cm
Figure 3.8: Label (4) indicates a sintered silver heat exchanger and (5)
is the mixing chamber. (11) are NbTi microcoax signal wires, which are
bolted to the Copper heat sinks.
47
3.4. WIRING
6
2 cm
13 12
Figure 3.9: Label (6) is the bronze powder filter. The NbTi microcoax
wires (12) entered the bronze powder filter with output on Cu coaxial
wires (13) at the mixing chamber stage.
48
3.4. WIRING
14
7
6 cm
Figure 3.10: Label (7) is the can inside of which the STM is mounted.
All of the Cu coaxial wires are heat sunk by clamps (14) that press the
wires onto the Au-plated Cu extension.
49
3.4. WIRING
posts at the 1K pot stage (see Fig. 3.5), the 600 mK still stage (Fig. 3.6), and
7.5 cm long posts at the 150 mK stage (Fig. 3.7). At the mixing chamber stage
(Fig. 3.8), these microcoaxe lines were fed into bronze powder filters (Fig. 3.9) [79]
with SSMC connectors. From the mK bronze powder filters to the STM (Fig. 3.10),
the signal lines were made of semi-rigid coax with Cu shielding and an Ag-plated
Cu inner conductor. The Cu coax lines helped to carry away heat from the STM
tip and the sample. This was critical to cool the STM, since the main body of the
STM were made from macor, which is very thermally insulating.
The drive wires were made from four Oxford Instruments wiring looms [80],
each with 12 twisted pairs of 100 µm diameter wire. From 300 K to 1.4 K, two
Constantan wiring looms were used as they provided a low heat load and resistances
that varied only slightly with temperature. From 1.4 K to the mixing chamber, we
used two CuNi-clad NbTi looms that were superconducting below 9 K. To ensure
adequate thermal anchoring, the looms are wrapped around Cu heat sinks at each
stage on the refrigerator. For the last section from the mixing chamber to the STM,
semi-rigid Cu coax wires which are the same type used for the signal wires are used.
Figure 3.11 shows detailed views of the looms and copper heat sinks at each stage.
Figure 3.11 (a) shows the 1K pot. Figure 3.11 (b) shows the still and the heat
exchanger at the still and the cold plate stage. Fig. 3.11 (c) shows the filters and
wiring at the mixing chamber stage, while Fig. 3.11 (d) shows the filter and wiring
section from the mixing chamber to the STM stage.
50
3.4. WIRING
(a) (b)
2
1 64 3 5
5 cm 5 cm
(c) (d)
7
8 10
9
4 cm 10 cm
11
Figure 3.11: (a) Cylindrical heat sinks for the loom wires ((4),(5),(7)
and (9)) are tightly bolted onto the 1K stage, (b) the still, and (c) the
mixing chamber plate. (d) At the mixing chamber stage, the loom wires
are soldered into a plug and connected to Cu coax lines using thin Cu
wires. The signal wires and all Cu coaxial wires are heat sunk by clamps
(10) that press the wires onto the Au-plated Cu extension. (11) is the
STM.
51
3.5. DAMAGE TO THE HEAT EXCHANGER AND PARTIAL REPAIR
3.5 Damage to the heat exchanger and partial repair
While the mixture was being removed early on June 29, 2016, a diaphragm
valve between the still and the 3He circulation pump broke. Air entered the system
and created a complete blockage in the 3He/4He line. Since the 1K pot was not
running at that time, the mixture slowly warmed up and eventually the remaining
liquid in the refrigerator turned into gas. This caused one of the sintered silver
heat exchangers to fail due to severe over-pressure and we ended up losing all the
mixture. Figure 3.12 (a) shows a photograph of the broken sintered silver heat
exchanger. The white objects are sintered silver plates from the inside of the heat
exchanger, which increase the contact surface area with the mixture.
Due to limited funding we were not able to immediately replace the unit and
instead, we applied a temporary fix to get the system back in operation. Figure 3.12
(b) shows the mixing chamber stage, detached from the rest of system. Figure 3.12
(c) shows the brass flange for the input line at the cold plate stage (2) and the bigger
brass flange (3) for the still line.
To reseal the heat exchanger, I tested two solders. I first tried EutecRod 157
[81], which is a lead-free soft solder that can be used for joining capillary tubes. The
second solder I tried was Wood’s metal. The Wood’s metal did not stick well to the
surface of the heat exchanger, so I used the EuteRod 157 for the repair.
Figure 3.13 (a) shows the mixing chamber stage detached from the rest of the
system and clamped onto a work table, ready for soldering. The soldering process
was done at 450 ◦C using EutecRod 157 [81]. Once the soldering was done, a 4He
52
3.5. DAMAGE TO THE HEAT EXCHANGER AND PARTIAL REPAIR
(a) (b)
1
(c)
2 3
Figure 3.12: (a) and (b) show photos of the damaged second sintered
silver heat exchanger. Label (1) indicates the second plate of the heat
exchanger. (c) Photos of the brass flange connector (2) for the incoming
line for the mixture, and the larger brass flange (3) for the return line
that goes to the still.
53
3.5. DAMAGE TO THE HEAT EXCHANGER AND PARTIAL REPAIR
(a) (b)
Figure 3.13: Photos of the re-sealed sintered silver heat exchanger. (a)
The system is clamped onto a work table for soldering. (b) The repaired
heat exchanger has been reattached to the dilution refrigerator. The
color of the brass flange changed due to the soldering.
54
3.6. ULTRA HIGH VACUUM (UHV) SYSTEM
leak test was performed to verify the seal was leak tight at room temperature. After
reassembling the dilution refrigerator system, we performed a leak test again with
the system at 4 K. Figure 3.13 (b) shows the final result after the dilution refrigerator
was reassembled.
After we replenished the mixture, the refrigerator was able to cool to a min-
imum temperature of only about 300 mK. However the operation was not stable
and the phase boundary did not seem to be in the mixing chamber. We also found
out that there was a cross leak between the input and output lines between the 1K
pot and the still plate; presumably this leak was created during the accident. This
was likely the main reason the system would not cool below 0.3 K. Nevertheless,
the system was able to maintain a stable temperature at 500 mK and this was the
temperature of the titanium nitride (TiN) data presented in Chapter 6.
3.6 Ultra high vacuum (UHV) system
Our STM system has a UHV preparation chamber and a transfer chamber
that are separated by a gate valve (see Fig. 3.14). Each chamber has its own ion
getter pump [82], titanium sublimation pump [83], and ion gauge [84]. To prepare
the UHV environment for both chambers, we first pump it out using a turbo pump
[85] backed by an oil-free scroll pump [86]. The chambers are then baked at 150 ◦C
for 24 hours to remove water and allow outgassing.
The sample preparation chamber has a residual gas analyzer [87], two electron
55
3.7. CONTROL ELECTRONICS AND MEASUREMENT CIRCUIT
beam evaporators [88], and an argon ion sputter gun [89] (see Fig. 3.15). The sample
stage is attached to an XYZ manipulator [90]. Samples can be heated to 600 ◦C by
a resistive heater or by a direct current heater.
Samples are mounted on copper, stainless steel, or molybdenum sample studs
that fit into the STM. To start, we epoxy a sample to a sample stud, put it into
the load lock, evacuate the load-lock, and transfer the sample into the preparation
chamber. After the sample is treated (cleaned, milled, annealed, etc.) in the cham-
ber, it is moved to the transfer chamber. To move samples, the sample stud sits
inside a sample transfer plate and we use a magnetic transfer rod to transfer the
sample plate [40]. To move a sample from the transfer chamber to the cold STM,
we use a top loading system with a 3 cm diameter access shaft in the dilution re-
frigerator. A detailed description of the transfer mechanism can be found in Ref.
[40].
3.7 Control electronics and measurement circuit
To control the two tips, each tip has its own electronics to allow it to approach
and scan a sample. One of the tips is controlled by an RHK Technology STM control
system [91]. It has a maximum piezo voltage of 220 V and uses analog feedback.
This system uses an IVP-300 RHK trans-impedance amplifier with a conversion
factor of 1 nA/V to maintain the tunnel current. The other tip is controlled by an
RHK-R9 system [92]. It has a maximum piezo voltage of 220 V and uses digital
56
3.7. CONTROL ELECTRONICS AND MEASUREMENT CIRCUIT
5
1 4
3
2
1
2
4
6
7
Figure 3.14: The UHV preparation chamber sits on top of an optical
bench. The transfer chamber sits on top of the optical table on the left
side. The dilution refrigerator is underneath the transfer chamber and
the STM unit is bolted to the mixing chamber plate from below. Label
(1) indicate magnetic transfer rods, (2) are the XYZ manipulators, (3)
is the argon ion sputter gun, (4) are the electron beam evaporators, (5)
is the gate valve, (6) is the dilution refrigerator and (7) is the STM.
57
3.7. CONTROL ELECTRONICS AND MEASUREMENT CIRCUIT
feedback. The current amplifier [93] for this tip can run from 10−5 to 10−11 A/V.
In standard STM operation, the STM is run in a voltage-biased mode: a
voltage bias V is applied to the sample and the tunneling current I is detected
from the tip. The feedback mechanism adjusts the z-piezo to maintain a constant
tunneling current as the tip scans in x-y across the sample, allowing us to image the
surface topography. I−V spectroscopy can be obtained by turning off the feedback
and then sweeping the voltage bias V while recording the tunnel current I. A circuit
diagram of the voltage-biased mode is shown in Figure 3.15 (red). In our two-tip
system, the other tip can be kept away from the sample so that no current flows
through that branch or it can be brought close to the sample to allow simultaneous
tunneling.
To facilitate measurement of hysteretic I − V characteristics we also used a
relay box that let us switch from the voltage-biased mode to the current-biased
mode. In current-biased mode, we sweep the current I applied to the junction and
measure the voltage V across the junction. To measure I − V curves in current-
biased mode during a scan of a sample, we first set the junction resistance Rn while
biased at a certain voltage (Rn defined as the voltage across the junction divided
by the tunneling current), turn off the feedback, switch the system to the current-
biased mode using the relay box, take I − V curves, switch back to voltage-biased
mode, and then turn the feedback loop back on, and move to the next location.
58
3.7. CONTROL ELECTRONICS AND MEASUREMENT CIRCUIT
Current Bias 300 K
-
RW
CPF Copper
RW
CPF
A
RW Tip Tip
CPF
Sample
RW R SuperconductorCPF C
50 mK-1.5K
Figure 3.15: Schematic diagram of setup for interchangeable voltage-
biasing mode (red) and current-biasing mode (green) using a home built
relay box. We utilized the two measurement configurations in measure-
ments of hysteretic Nb-Nb S-I-S tunneling characteristics (see Chap-
ter 8).
59
R1
+
3.8. STM NIOBIUM AND VANADIUM TIP PREPARATION
3.8 STM niobium and vanadium tip preparation
Much of my research was aimed at realizing a true Josephson STM. For it to
work, it is necessary to have an S-I-S junction with a non-negligible super current
between a superconducting STM tip and a superconducting sample. As discussed
in Chapter 8, I used a Nb crystal and a Nb superconducting tip for these tests.
The process I used for making a Nb STM tip is briefly described below. The
recipe was developed by Anita Roychowdhury and additional details may be found
in her thesis [40] and Ref. [94]. To make tips, I first cut 50 mm-long sections of
250 µm diameter Nb wire and place them, sticking up, in a 12 cm × 12 cm × 2 cm
aluminum block. I then used a drop of Fujifilm 906-10 photoresist around each tip
to glue them in place and then baked the aluminum block at 150 ◦C for 10 minutes.
I next placed the block with its Nb tips in a plasma reactive ion etcher (RIE) [95]
and etched the tips for 85 minutes. The recipe used an SF6 gas with flow rate of 10
SCCM, a pressure of 100 mTorr, and an applied rf power of 150 W. No O2 was used
Once the tips were etched, I chose the tip that looked sharpest and placed
it into the STM. Newly made Nb tips usually lasted for about 12 hours in air at
300 K before they became too oxidized to use. This allowed enough time for us to
mount the STM back onto the system, test it at room temperature, make sure it
was working and then close the system and cool it to cryogenic temperatures. Once
the STM reached base temperature, we usually kept it cold for a few months.
Finally, I also found out that I could use a vanadium tip as a replacement
for the Nb tip. The main advantage was that it could be easily made at room
60
3.9. CONCLUSION
temperature and put into the tip holder right away. This process involves using
pliers to rip off the end of the vanadium wire. This not only saved me time, but
also meant I did not need access to the clean room equipment. This was important
because there were a few months when the clean room was not running. Although
vanadium tips seemed to work pretty well, as I discuss in Chapter 6 I did not observe
a superconducting gap for the vanadium tips when the tip was scanning my TiN
samples. This may have been due to contamination on the tip or some other issue
with V. Another drawback of using the vanadium tip is that the critical temperature
of vanadium is only 5.6 K. I also found that it was harder to prepare vanadium so
that it had a nice sharp tip, in contrast to Pt-Ir, which are very easy to prepare so
that the tip is sharp.
3.9 Conclusion
In this chapter, I described the overall layout of our millikelvin STM sys-
tem including the dilution refrigerator, dual-tip STM, wiring and the UHV sample
preparation system. Additional details about the construction and operation of the
dual-tip STM can be found in Chapter 4.
61
CHAPTER 4
Simultaneously Scanning Two Connected Tips in
a Dual-tip STM
4.1 Introduction
Josephson STMs that are sensitive to coherent tunneling of Cooper pairs have
recently been proposed [42, 67–70]. Unlike existing STMs, in principle such systems
could be used to measure variations in both the magnitude and phase of the order
parameter of the superconducting state on the atomic scale.
The approach I describe here involves obtaining a phase coherent signal from
an atomic size junction (J1) by connecting it to a larger superconducting junction
(J2) using an inductive shunt L to form an asymmetric SQUID [96]. The total
switching current Is of the SQUID will be dominated by J2 and can be deter-
mined accurately using current switching measurements [42, 97]. For I01  I02,
the SQUID’s switching current Is achieves a maximum value of I02 + I01, when
the current through J1 and J2 are in phase. A minimum switching current of I02
- I01 results when the junctions are 180
◦ out of phase. I01 and I02 are the critical
62
4.1. INTRODUCTION
currents of the small and large junctions, respectively and I have assumed I01  I02
 Φ0/2L.
By coupling J1 to the larger junction J2 we can lock the phase difference across
J1 with respect to J2 and reduce its uncertainty. The relative phase difference across
the two junctions is determined by the effective flux, which is the flux applied to
the SQUID loop plus contributions due to the pairing symmetry of the sample at
the location of the tips. By measuring the SQUID’s switching current Is versus
applied flux Φa, one can extract the critical current I01 of the small junction and
the effective flux for a given tip location. For example, if one junction moves from
a location where it is tunneling into a positive phase in a d-wave superconductor to
a location where it is tunneling into a negative phase, we would see an equivalent
shift of Φ0/2 in the Is(Φa) characteristic [98].
For this approach to work the tips must be independently controllable, and
the connection between the tips must be superconducting and have a sufficiently
low inductance. The connection must be mechanically flexible enough that the tips
can be moved independently. The total loop inductance L of the SQUID should be
small because the minimum uncertainty σγ1 in the phase difference across the small
junction is [97] √ ( )
2e2
−1/4
' C1 C1σγ1 + , (4.1)
h̄ LJ1 L
where LJ1 = Φ0/2πI01 is the Josephson inductance and C1 the capacitance of the
sm√all junction. For L  LJ1 the minimum phase uncertainty will be small if Zs
= L/C1  h̄/2e2 = Rq = 2.035 kΩ. For a 1 nA STM junction, a 1 µA fixed
63
4.1. INTRODUCTION
junction, a loop inductance of 1 nH, and an STM junction with total capacitance
6 fF, one finds Zs = 234 Ω and σγ ≈ 0.44 radians, which is small enough to obtain
a measurable critical current [42], provided thermal fluctuations and external noise
are sufficiently small.
Sullivan et al. [42] completed a proof-of-principle experiment that demon-
strates the phase across a small junction can be significantly stabilized by coupling
to a large junction via an inductance. A small Al/Al2O3/Al junction was first pre-
pared (C ∼ 6 fF), that exhibited a highly suppressed switching current at 50 mK
due to phase diffusion. An identically prepared small junction was then fabricated
as part of a highly asymmetric, hysteretic SQUID with the junctions having capac-
itance of 100 fF and 6 fF. The switching current of the SQUID was measured, and
the I − Φ curve displayed a periodic modulation of ∼ 1 nA on top of the ∼ 800
nA total switching current. This result strongly suggests that a superconducting
dual-tip STM with comparable parameters will be capable of imaging the gauge
invariant phase of superconductors on the atomic scale.
To construct an STM with two connected tips, we started from an existing
dual-tip STM [41] that can operate at room temperature or in a dilution refrigera-
tor with a base temperature of 30 mK. Previously, the feasibility of scanning both
tips independently at mK temperatures was demonstrated using plasma-etched su-
perconducting Nb tips [94]. In Section 4.2, I describe how I modified this system
to enable operation with two tips that are electrically connected via a flexible link
made from thin Nb foil. In Section 4.3, I next discuss one of the main challenges of
operating two connected tips independently: only the total tunnel current through
64
4.2. ROOM TEMPERATURE SYSTEM SETUP
both tips can be measured, so that a new technique for controlling both tips us-
ing one current needed to be devised. I describe a technique for keeping each tip
in feedback and then demonstrate simultaneous imaging with both tips. Finally, in
Section 4.4, I describe images obtained by simultaneously scanning two tips at room
temperature and examine the electrical performance and mechanical characteristics
of the system.
4.2 Room temperature system setup
To form a SQUID, the two tips, connecting link and sample must be super-
conducting. We chose to use niobium for the tips, tip holders and connecting link
due to its toughness, high critical temperature, ability to make sharp tips and slow
oxidation rate [94]. (Alternatively, vanadium could also be used. Vanadium wire
is not as brittle as un-annealed niobium wire and it can be cut using wire cutters
to produce sharp points.) By attaching the connecting link to tip holders, instead
of directly to the tips, we get a stronger connection and can exchange tips without
having to remake the connection. Each tip slides into a small hole in its holder and
is held by friction. A disadvantage of connecting the holders is that this makes the
connector longer, which increases the SQUID loop inductance L and pickup area,
which would increase the phase diffusion in J1 and make the device more sensitive
to magnetic noise.
To make the link I used bow-tie shaped connectors that were cut from 25 µm
65
4.2. ROOM TEMPERATURE SYSTEM SETUP
Figure 4.1: Two laser-cut Nb foil connectors shown on a ruler with 1 mm marks.
66
4.3. SCANNING TWO TIPS INDEPENDENTLY
thick Nb foil (see Fig. 4.1). The cutting was done using an 800 nm wavelength laser
with a 100 fs pulse length, an average power of 1 W and spindle speed of 100 rpm.
The pads of the bow-tie allow a larger contact welding area to the tip holders and
the neck in the bow-tie gives flexibility so that each tip does not pull too strongly
on the other. The Nb connector was spot welded [99] to each tip holder (see Fig.
4.2) using a hand-held Cu spot-welding clamp.
4.3 Scanning two tips independently
With the tips electrically connected together by the Nb bowtie, the controller
can only measure the sum of the tunneling current from both tips. Since an inde-
pendent current signal from each tip is not available, a conventional STM constant-
current feedback mode (Chapter 2) cannot be used to control the tip-sample sep-
aration of both tips simultaneously. Instead, we devised a technique that involves
modulating each tip’s z-piezo voltage at a different frequency and then extracting
from the total current a separate feedback signal for each tip.
To understand this feedback technique, note that for a single STM tip at
a distance z above a surface, the tunneling current can be written as [61] (see
67
4.3. SCANNING TWO TIPS INDEPENDENTLY
Outer tip
Inner tip
Tip holder
Nb connector
1mm
Figure 4.2: Nb tip holders connected by spot welded Nb foil.
68
4.3. SCANNING TWO TIPS INDEPENDENTLY
Chapter 2)
∫∞
I ∝ |M |2Ntip(− eV ) ·Nsample()[f(− eV, T )− f(, T )]d
−∞
∝ e−κz (4.2)
where T is the temperature, 1/κ is the effective tunneling length scale,  is the
electron energy (with respect to the Fermi level of the sample), Ntip is the density
of states of the tip, Nsample is the density of states of the sample, V is the voltage
applied to the sample with respect to the tip and |M |2 is the tunneling matrix
element, which decreases exponentially with distance from the sample.
From Eq. (4.2), the derivative of the tunneling current with respect to distance
z gives
dI
= −κI. (4.3)
dz
Thus the dI/dz signal from a tip is proportional to the current I through the tip,
with a constant factor of simply -κ.
Figure 4.3 shows our dual-tip electronics setup. One of the tips is controlled
by an RHK Technology SPM 1000 Scanning Probe Microscope Control System [91].
It has a maximum output piezo voltage of 220 V and uses analog feedback. The
other tip uses a Topometrix SPM control system [100] [obsolete] with a maximum
output piezo voltage of 220 V and digital feedback.
Figure 4.4 shows an STM measurement of I versus z and Fig. 4.5 shows a
measurement of dI(z)/dz versus z, where I used a Pt-Ir tip and a Au/mica sample
69
4.3. SCANNING TWO TIPS INDEPENDENTLY
Topometrix Out1 Safety Out2
I In2 RHKIn1
In In
Out Out
    Lock-in #1     Lock-in #2 
Mod In In Mod
z f f Mod zx/y
1 z x/y2 Bias
In Mod
Mod z Out z z
Tip 1 x y x y Tip 2
Sample
Figure 4.3: Experimental configuration for using dI/dz signals to gener-
ate feedback for two electrically connected STM tips with two indepen-
dent position controllers. The RHK electronic controller has a built-in
modulation mode for the z–piezo, whereas the Topometrix system re-
quires an external modulation box. The safety box checks the total
output current and retracts both tips if the total current exceeds a set
value, preventing the tips from crashing into the sample.
70
4.3. SCANNING TWO TIPS INDEPENDENTLY
at room temperature. For this data, I parked the tip close enough to the sample
that there was an appreciable tunneling current (nA), and turned off the feedback.
I then retracted the tip a few Å from the surface while recording the tunnel current
and the dI/dz signal while the tip z position was modulated at an amplitude of 12
pm. As expected, dI/dz and I both decreased exponentially with the tip-sample
distance z. Thus the plots of I vs. z and dI/dz vs. z are approximately straight lines
on a semilog plot. Fitting Eq. (4.2) to the I(z) data, we find 1/κ ' 0.30 nm. Since
dI/dz is proportional to I (see Fig. 4.5), we can use the dI/dz from a z-modulated
tip as a feedback error signal when operating the STM.
To distinguish the currents from each tip, we add a small modulation of a
few mV at different frequencies (between 1 and 10 kHz) to the z-piezo of each
tip. I then used two lock-in amplifiers to monitor the total current and detect
dI/dz at each driving frequency. In practice we found that the total current has
an in-phase (resistive) component and a 90◦ out of phase (capacitive) component
at each frequency. The amplitude of the capacitive contribution to the current is
I ∼c = 2πfV Cd/z, where z is the distance between the sample and the tip, C is the
effective tip-sample capacitance, d is the amplitude of the tip z-modulation, and the
sample is held at voltage V . For example, for C = 10 fF, frequency f = 5 kHz, bias
voltage V = 1 V, z = 0.5 nm and d = 6 pm, the capacitive current amplitude is Ic
= 10 pA which is not small given a typical tunneling current range of pA to nA.
Fortunately, since the two components are out of phase by 90◦, we can easily
measure and separate out the unwanted capacitive part; only the in-phase resistive
component will obey Eq. (4.3) and be useful for generating the feedback signal. To
71
4.3. SCANNING TWO TIPS INDEPENDENTLY
100
50
10
5
1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
z (nm)
Figure 4.4: Semi-log plot of measured tunneling current I (blue points)
versus distance z between a Pt-Ir tip and a Au/mica surface. The red
line is a fit to an exponential decay with 1/κ = 0.30 nm for z < 0.3 nm.
72
I (pA)
4.3. SCANNING TWO TIPS INDEPENDENTLY
1
0.50
0.10
0.05
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
z (nm)
Figure 4.5: Semi-log plot of measured |dI/dz| (blue points) versus dis-
tance z acquired simultaneously with Fig. 4.4. The red line is a fit to
data for z < 0.3 nm.
73
dI/dz (nA/nm)
| |
4.3. SCANNING TWO TIPS INDEPENDENTLY
find the phase ϕc of the capacitive current, we measure the current while the tip is
somewhat beyond the z-range where there is a detectable tunneling current. Apply-
ing an ac voltage to the scanner, we find a capacitive signal and determine the phase
ϕc at which the output from the lock-in is a maximum at the applied frequency;
this is the phase ϕc of the capacitive component. With the tip in tunneling range,
we then extract the component at ϕc ± 90◦ using the lock-in amplifier and use this
signal to generate the z-feedback signal for the tip, note this also gives a way to
measure C, which is an important parameter for the STM SQUID.
There are several factors to consider when choosing the z-modulation frequency
for our dI/dz feedback technique. A high frequency is desirable because this allows
a large feedback bandwidth and thus a high scan rate. If, however, the modulation
frequency is much higher than the piezotube’s resonant frequency, the tip will stop
responding to the applied modulation voltage. Our outer scanner has a resonance
at a few kHz and cuts off at 4.6 kHz, whereas for the inner scanner, the cut off is
around 11 kHz. The large difference in the scanner resonance frequencies allows us
to drive the two piezos with little crosstalk or interference using two well-separated
frequencies.
Our use of two modulation frequencies requires some additional modifications
to the standard feedback control. In standard constant current imaging mode (see
Chapter 2), the feedback control system naturally prevents the tip from crashing
into the sample. The tunnel current is compared to a set point and the feedback
system tries to maintain this value by moving the tip toward or away from the
sample. Even if the current amplifier saturates when the tip gets too close to the
74
4.4. SIMULTANEOUSLY SCANNING TWO STM TIPS ON TEST SAMPLES
sample, the error signal will be large and the feedback control will pull back the tip.
However, for our dI/dz technique we are only monitoring dI/dz. Saturating the
current amplifier will cause the dI/dz feedback signal to drop to zero, which is less
than the set point, and the feedback control will push the tip forward, causing the
tip to crash into the sample.
To prevent this, I incorporated a safety box into the control system (see
Fig. 4.3). The inputs to the safety box (see Fig. 4.6) are the two dI/dz signals
from both lock-in amplifiers and the total tunnel current. If the total current is
larger than a threshold value (typically 7 nA), the safety box outputs 10 V to the
tip feedback system to force both tips to be pulled back. Otherwise the safety box
just outputs the detected dI/dz signal for each tip. A downside of this method is
that both tips are pulled back, since I do not determine which tip caused the safety
box to trigger. In principle, this could be determined by examining the dI/dz
signals. In practice, I set the threshold current close to the maximum output of
the current amplifier and adjust the scan parameter so that the safety box is not
triggered too often during a scan.
4.4 Simultaneously scanning two STM tips on test samples
To test the two-tip feedback system with both tips connected together, I used
the STM to image HOPG and Au/mica samples with Pt-Ir tips at room temperature
in air. In the first set of tests I brought one tip into tunnel range and scanned the
75
4.4. SIMULTANEOUSLY SCANNING TWO STM TIPS ON TEST SAMPLES
dI/dz
I
dI/dz
Figure 4.6: Simplified circuit for the safety box. If the combined tunnel-
ing current signal sent to Input I is higher than the set point, the output
of the comparator switches from 0 V to -10 V, which makes outputs 1
and 2 switch to 10 V, causing both tips to retract. Otherwise the output
of the comparator is 0 V and feedback outputs 1 and 2 are determined
by the corresponding dI/dz signal.
sample while the other tip was out of tunnel range and vice versa. I used the RHK
controller as the feedback system and set the lock-in phase so that the capacitive
contribution to the lock-in output was minimized.
Figure 4.7 shows a topographic image of the Au/mica surface that was scanned
with the inner tip using its tunneling current as the feedback signal in the standard
way. Single atomic layer steps are clearly seen. After the image was taken, I added
a small modulation of ∼6 pm at a frequency of 9.103 kHz to the inner tip z–
piezo and recorded the modulation of the current signal at that frequency using a
lock-in amplifier with an averaging time constant of 0.3 ms. Figure 4.8 shows the
corresponding image of the same region taken using the inner tip in dI/dz mode.
Before switching to the dI/dz mode, I moved the tip back ∼ 20 nm out of tunnel
76
4.4. SIMULTANEOUSLY SCANNING TWO STM TIPS ON TEST SAMPLES
range. I then switched the feedback input to dI/dz, brought the tip back into tunnel
range, and resumed taking images. Comparing the images, the steps in Fig. 4.7 are
not as sharp as in Fig. 4.8, but overall the two images are remarkably similar, as
expected.
Figure 4.9 shows an image with atomic resolution on an HOPG sample taken
with the outer tip using current feedback. The corresponding image in dI/dz mode
is shown in Fig. 4.10 and it is quite similar, although some distortion is visible in
Fig. 4.10 due to the feedback reacting slower during the scan. With the inner tip, we
were not able to achieve atomically resolved images on HOPG. Typically, the inner
tip was less stable than the outer tip because the inner scanner is located above
the inner walker [41], so the overall lever arm is longer, making the inner tip more
susceptible to vibration.
After demonstrating the feasibility of using the dI/dz mode for imaging, I
scanned both tips simultaneously. To do this, I used the Topometrix SPM control
system for the inner tip and the RHK SPM100 for the outer tip (see Fig. 4.3).
The modulation frequency was 4.517 kHz for the outer tip and 10.007 kHz for the
inner tip. The threshold of the safety box was set to 6 nA and the tips were raster
scanned at different rates. Figure 4.11 (a) and (b) were taken with the outer tip
while Fig. 4.11 (c) and (d) were taken with the inner tip. Figure 4.11 (a) and (c)
were taken with one tip scanning using dI/dz as the feedback signal while the other
tip was retracted. Figure 4.11 (b) and (d) were taken with both tips scanning at
the same time and the resulting images can be compared directly to Fig. 4.11 (a)
and (c). The two images taken with the outer tip are very similar. White spots in
77
4.4. SIMULTANEOUSLY SCANNING TWO STM TIPS ON TEST SAMPLES
50 nm
Figure 4.7: Topographic STM image of Au/mica taken with the inner
Pt-Ir tip using current for generating the feedback signal for the z-piezo.
The sample bias is 1 V and the tunneling current is 80 pA. The z color
scale range represents the height difference of 8 nm.
78
4.4. SIMULTANEOUSLY SCANNING TWO STM TIPS ON TEST SAMPLES
50 nm
Figure 4.8: Topographic STM image of Au/mica taken with the inner
Pt-Ir tip using dI/dz for generating the feedback signal for the z-piezo.
The same region as in Fig. 4.7 has been imaged.
79
4.4. SIMULTANEOUSLY SCANNING TWO STM TIPS ON TEST SAMPLES
(a)
0.5 Å
(b) 0 Å
x
Figure 4.9: (a) Topographic STM image of HOPG showing atomic reso-
lution taken with the outer Pt-Ir tip using current as the feedback signal.
(b) Line section of the topographic signal z vs. x along the blue line in
(a).
80
z
4.4. SIMULTANEOUSLY SCANNING TWO STM TIPS ON TEST SAMPLES
(a)
0.5 Å
(b) 0 Å
x
Figure 4.10: (a) Topographic STM image of HOPG showing atomic
resolution taken with the outer Pt-Ir tip using dI/dz as the feedback
signal. (b) Line section of the topography z vs. x along the blue line in
(a).
81
z
4.4. SIMULTANEOUSLY SCANNING TWO STM TIPS ON TEST SAMPLES
Fig. 4.11 (d) are pixels at which the outer tip scanned across a step edge and caused
the safety box to trigger and retract both tips. Other than that, the tips seem not
to affect each other significantly.
Although the images in Fig. 4.11 do not show obvious effects from mechanical
coupling between the tips which were connected together with the Nb bow-tie, note
that the inner tip assembly, due to its long lever arm, will be pulled by the outer
tip if the outer tip moves a significant distance. Figure 4.12 shows an image of the
HOPG sample taken with the inner tip while the outer tip was scanning at the same
time with a scan range of 16 nm × 16 nm. The appearance of the steps was not
affected significantly while the outer tip was scanning over this small range. For
comparison, Fig. 4.13 shows an image of the same region taken using the inner tip,
but this time while the outer tip was moved around a 2 µm× 2 µm region. Initially
the outer tip was at the bottom right corner of the total scan area. After scanning a
few lines with the inner tip, we moved the outer tip to the bottom left corner of its
scan area and waited for a few lines while the inner tip kept scanning. We repeated
the same procedure, moving the outer tip to the other corners before returning the
outer tip back to the starting corner and repeating the entire cycle.
Comparison of Fig. 4.12 and Fig. 4.13 shows that while Fig. 4.13 was being
acquired the inner tip moved significantly each time the position of the outer tip
changed. The white arrows and labels 1 to 4 indicate the image sections for which
the outer tip was at the extreme corner of its scan range (see inset). Analysis of
the images shows that the inner tip moved by about 40 nm in Fig. 4.13 when the
outer tip changed from position 1 to position 2 (indicated by a green line at 45◦ with
82
4.4. SIMULTANEOUSLY SCANNING TWO STM TIPS ON TEST SAMPLES
(a) (b)
10 nm 10 nm
(c) (d)
10 nm 10 nm
Figure 4.11: Topographic STM images of Au on mica. The top row
images (a) and (b) were taken using the outer tip with a z-modulation
frequency of 4.517 kHz. Bottom row images (c) and (d) were taken using
the inner tip and a z-modulation frequency of 10.007 kHz. (a) and (c)
used dI/dz as feedback input while the other tip was out of tunneling
range. Images (b) and (d) were taken simultaneously, i.e. with both
tips scanning at the same time. White spots in (d) are pixels where the
safety box retracted both tips when the outer tip scanned over a large
step edge.
83
4.4. SIMULTANEOUSLY SCANNING TWO STM TIPS ON TEST SAMPLES
50 nm
Figure 4.12: Topographic images taken with the inner STM tip on the
surface of HOPG while at the same time, the outer tip was scanning a
16 nm × 16 nm area.
84
4.4. SIMULTANEOUSLY SCANNING TWO STM TIPS ON TEST SAMPLES
50 nm 1
4
3
2
1
4
3 4 3
2
2 1 1
Figure 4.13: Topographic images taken with the inner STM tip on the
surface of HOPG while, at the same time, the outer tip was moved
around its maximum 2 µm × 2 µm scan area. Step edges are shifted
due to the outer tip pulling the inner tip as the outer tip is moved to
a new location. Labels 1, 2, 3, 4 indicate the corner at which the outer
tip was located when the inner tip was scanning the corresponding lines.
The green line shows the inner tip displacement vector change when the
outer tip moved from corner 1 to corner 2.
85
4.5. CONCLUSIONS
respect to the horizontal axis). This corresponds to a reduction by about a factor
of 50 in the relative motion of the inner tip due to the outer tip, indicating a very
flexible connection. We note that motion of the inner tip has a much smaller effect
on the outer tip due to the greater rigidity of the outer tip walker. This behavior
suggests it would be best to use the outer tip as a large fixed junction (reference
junction) and use the inner tip as the small junction for scanning across the sample
to image changes in superconducting phase.
4.5 Conclusions
In this chapter I described how I electrically connected two tips together in a
dual-tip STM using a bowtie shaped Nb foil connector. I also described images of
HOPG and Au/mica samples obtained with the system at room temperature. Both
tips showed good images, with the outer tip obtaining atomically resolved images on
HOPG. Using dI/dz feedback, I obtained topographic images, successfully scanned
both tips at the same time and measured the mechanical coupling between the tips.
These results are a significant step towards using our dual-tip STM to measure the
gauge-invariant phase on a superconducting sample.
86
CHAPTER 5
Superconductivity and BTK Theory
In Section 5.1 I start by describing the basic principles of superconductivity
and the Bardeen-Cooper-Schrieffer (BCS) theory. I introduce a few important con-
cepts which will be used throughout the thesis, namely the superconducting gap ∆,
critical temperature Tc, and density of states D(E). In Section 5.2 I then discuss the
theory of Andreev reflection and subgap structure observed in the I − V character-
istics of N-I-S junctions. When an electron-like quasiparticle is incident on an S-N
interface from the normal side, a hole-like quasiparticle is reflected at the interface
due to the Andreev reflection, thus creating a Cooper pair on the superconducting
side. In the discussion for the BTK theory, I show that the theory describes an S-N
junction for arbitrary transparency. This is different from the behavior of a standard
N-I-S junction in the low transparency limit (tunneling limit) that I described in
the Chapter 2.
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5.1. INTRODUCTION TO SUPERCONDUCTIVITY
5.1 Introduction to superconductivity
The Bardeen, Cooper, Schrieffer (BCS) theory [101, 102] provides a micro-
scopic quantum description of superconducting phenomena. A key feature of the
theory is that two electrons with opposite momenta can bind together to form a
Cooper pair [103]. The attractive force is due to the first electron polarizing the
medium (lattice) by attracting positive ions. The ions in turn attract the second
electron. The effective attractive interaction between the electrons can overcome
the Coulomb repulsion between electrons, which is reduced by screening.
At sufficiently low temperature, the formation of Cooper pairs creates an in-
stability in the Fermi sea of electrons, and the Cooper pairs condense into a single
ground state. The BCS ground state wavefunction of the electrons can be written
as [101, 102] ∏
|ΨBCS〉 = (uk + νkc†k,↑c
†
−k,↓) |Φ0〉 (5.1)
k=k1,k2....kn
where c† †k,↑c−k,↓ is the pair creation operator with zero total momentum, |ν 2k| is the
probability of the pair being occupied, |u |2k =1 − |v |2k is the probability that the
pair is unoccupied and |Φ0〉 is the vacuum sate with no particles present.
The BCS Hamiltonian of a superconducting system can be written as [101,
102]: ∑ ∑
Ĥ = ( − µ)c† c + V c† c†k k,σ k,σ kl k,↑ −k,↓c−l,↓cl,↑ (5.2)
k,σ k,l
The first term is the kinetic energy of all the free electrons with respect to the
chemical potential µ of the system. The second term is the attractive interaction
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5.1. INTRODUCTION TO SUPERCONDUCTIVITY
between electrons with interaction coupling strength Vkl.
Given Eq. (5.1) and Eq. (5.2) the goal is to find the parameters uk and νk for the
ground state such that the energy E=〈ΨBCS| Ĥ |ΨBCS〉 is minimized. The easiest
way to do this is to use a variational method [101, 102]. However the technique is
somewhat clumsy when dealing with excited states that occur for T > 0. Instead,
I will use mean-field theory [104, 105] to handle both the ground state and excited
states. I note that this approach only works well in standard BCS theory in the
weak-coupling limit [106, 107].
The ground state of a BCS superconductor is a many-body state composed of
a phase-coherent superposition of pairs of electrons occupying states (k ↑, −k ↓).
Due to the coherence, operators c−k↓ck↑ can have non zero expectation values and
we can write c−k↓ck↑ = 〈c−k↓ck↑〉 + (c−k↓ck↑−〈c−k↓ck↑〉), where the second term can
be small. Substituting this expression into the interaction term in the Hamiltonian
Eq. (5.2), one finds:
∑ ∑
Ĥ = (k − µ)c† † †k,σck,σ + Vkl(〈ck↑c−k↓〉 〈c c 〉+ 〈c
† c†−l↓ l↑ k↑ −k↓〉(c−l↓cl↑
k,σ k,l
∑ − 〈c−l↓cl↑〉) + (∑c
†
k↑c
†
−k↓ − 〈c
† c†k↑ −k↓〉) 〈c−l↓cl↑〉)
= (k − µ)c†k,σck,σ + Vkl(〈c
† c† † †k↑ −k↓〉c−l↓cl↑ + ck↑c−k↓ 〈c−l↓cl↑〉
k,σ k,l
− 〈c† †k↑c−k↓〉 〈c−l↓cl↑〉) (5.3)
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5.1. INTRODUCTION TO SUPERCONDUCTIVITY
∑
Defining the order parameter ∆k = − Vkl〈c−l↓cl↑〉, the Hamiltonian then becomes
l
∑ ∑
Ĥ = (k − µ)c†k,σck,σ − (∆
†
kc−k↓ck↑ + ∆kc
†
k↑c
† † †
−k↓ −∆k〈ck↑c−k↓〉) (5.4)
k,σ k
The last sum in Eq. (5.4) is just a constant term, which we will retain because of
the condensation energy of the superconducting state.
To diagonize the Hamiltonian, we introduce the quasiparticle operators γ , γ†k0 k1
via the Bogoliubov-Valatin transformation [105]
c ∗k↑ = ukγ + ν γ
†
k0 k k1
c† ∗−k↓ = −νkγk0 + ukγ
†
k1 (5.5)
I now rewrite Eq. (5.4) in the matrix form
∑[( )  
Ĥ = c† c  ξk −∆k ck↑  ]†− ↓ + ξk + ∆kbk (5.6)k↑ k
k −∆† †k −ξk c−k↓
where ξk = k − µ and bk = 〈c−k↓ck↑〉. Using Eq. (5.5), we can now write
∑[( )    u −ν  ξ −∆  u∗k k k k k νkĤ = γ†k0 γk1 γk0
k ] ν∗ u∗ −∆+ −ξ −ν∗ u γ†k k (5.7)k k k k k1
+ξ + ∆ b†k k k
To diagonize the Hamiltonian Eq. (5.7) in the γ basis, we expand the middle
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5.1. INTRODUCTION TO SUPERCONDUCTIVITY
three matrices and choose the uk and νk so that the coefficients of the off-diagonal
terms γ†k0γ
†
k1 and γk1γk0 vanish. Multiplying out the three middle matrices from
Eq. (5.7) gives:
 (|u 2k| − |ν |2k )ξ + ∆ u ν∗k k k k + ∆∗ku∗kνk 2ukνkξ − u2 2 ∗k k∆k + νk∆k 
2u∗ν∗ξ − u∗2∆∗ + ν∗2k k k k k k ∆k −((|u 2 2 ∗k| − |νk| )ξk + ∆kukνk + ∆∗ ∗kukνk)
(5.8)
The coefficients of the unwanted (off-diagonal) terms will vanish if
2ξkukνk + ∆
∗
kν
2
k −∆ku2k = 0 (5.9)
Multiplying Eq. (5.9) by ∆∗ 2k/uk gives:
ξ u ν ∆∗ ∆∗ν2∆∗k k k
2 k + k k k −∆ ∆∗
u2 u2
k k = 0 (5.10)
k k
Rearranging gives:
∆∗ ∗kνk ∆ νk( )2 + 2ξk(
k )− |∆ |2k = 0 (5.11)
uk uk
and solving the resulting quadratic equation gives:
∆∗
√
kνk = ξ2k + |∆k|2 − ξk = Ek − ξk (5.12)uk
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5.1. INTRODUCTION TO SUPERCONDUCTIVITY
where
√
Ek = ξ2k + |∆ 2k| (5.13)
and Ek is the quasiparticle energy. From Eq. (5.13) the minimum energy to excite
a single quasiparticle is the energy gap |∆k|. The quantity ∆k is also the order
parameter [64, 108, 109] for the superconducting state and vanishes for T > Tc.
Given the normalization requirement |u 2k| + |vk|2 = 1 and Eq. (5.12), we can
write:
|νk|2 = 1− |uk|2
1 − ξk= (1 ) (5.14)
2 Ek
The remaining diagonal terms in Hamiltonian Eq. (5.7) can be evaluated and the
Hamiltonian becomes
∑ ∑
Ĥ = Ek(γ
† † †
k0γk0 + γk1γk1) + (ξk + ∆kbk − Ek) (5.15)
k k
The last term is the energy difference between the superconducting state and normal
state at T = 0, and is called the condensation energy. Equation (5.15) gives the
excitation energy from the ground state in terms of the number operators γ†kγk.
Therefore the γk describe the excitations of the systems. They are called ’Bogoliubov
qausiparticles’ or ’Bogoliubovons’.
To see what the γk0 and γk1 mean physically, I invert the transformation
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5.1. INTRODUCTION TO SUPERCONDUCTIVITY
Eq. (5.5), and get
γ† = u∗k0 kc
† ∗
k↑ − νkc−k↓
(5.16)
γ† ∗ †k1 = ukc−k↓ + ν
∗
kck↑
γ†k0 effectively creates Bogoliubovon of momentum k and spin ↑, since c−k↓ subtracts
a particle with (−k ↓) from the system, which is the same as adding a particle (k ↑).
Similarly, γ†k1 creates a Bogoliubovon with wave vector −k and spin ↓. ∑
Inserting Eq. (5.5) into the equation for the order parameter ∆k = − Vkl〈cl↓cl↑〉,
l
one finds: ∑
∆ = − V u∗k kl l ν (1− 〈γ
†
l l0γl0〉 − 〈γ
†
l1γl1〉). (5.17)
l
At T = 0, no quasiparticles are excited and Eq. (5.17) becomes
∑ 1 ∑ |∆l|
∆k = − Vklu∗l νl = − Vkl (5.18)2 El
l l
where the BCS interaction is,
 −V if |ξk| < h̄ωVkl =  (5.19)0 otherwise.
Here ω is the Debye cutoff frequency of the ions in the lattice. This means that
the electron-phonon interaction happens within a thin shell near the Fermi surface.
In the BCS model of the elctron-phonon interaction [101, 102], the gap is constant
and does not vary with the direction of k, i.e. it is isotropic or s-wave symmetric.
This model can be extended to describe unconventional superconductivity where
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5.1. INTRODUCTION TO SUPERCONDUCTIVITY
the order parameter (energy gap ∆) has other symmetries. For example, p-wave
superconductors [110] or d-wave superconductors [111].
Inserting Vkl from Eq. (5.19) into Eq. (5.18) one obtains a self-consistency
equation
 ∑V ∆ if |ξ | < h̄ω2
 E
k
∆ = 
k
k (5.20)
0 otherwise.
Note in Eq. (5.20) that the sum is over a thin shell around the Fermi energy with
|ξk| = |k − µ| < h̄ω. The sum over k can be replaced by an integration over energy
ξ from −h̄ω to h̄ω. Canceling the gap ∆ from both sides, Eq. (5.20) becomes
∫ h̄ω 1
1 = V √ D(ξ)dξ (5.21)
0 ξ2 + ∆2
The factor of 2 in Eq. (5.20) disappears due to the symmetry of the integration
over ξ. To proceed, I substitute ξ = x∆ and note that the normal metal density of
states D(ξ) can be taken as constant D0 over the small integration range provided
h̄ω  EF . Equation (5.21) becomes
∫ h̄ω/∆ ∫∆dx h̄ω/∆√ √ dx1 = V D0 ∫ = V D0∆ 1 + x20 0tanh−1(h̄ω/∆) 2 ∫ 1 + x2sec θdθ tanh−1(h̄ω/∆)
= V D0 = V D0 sec θdθ
0 sec θ 0
√
tanh−1(h̄ω/∆) h̄ω/∆
= V D0 ln | sec θ + tan θ|| 20 = V D0 ln | 1 + x + x||0
−1 |h̄ω/∆ h̄ω= V D0 sinh x 0 = V D0 sinh−1( ) (5.22)∆
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5.1. INTRODUCTION TO SUPERCONDUCTIVITY
The gap equation then becomes
h̄ω
∆ = ≈ 2h̄ωe−1/D0V (5.23)
sinh(1/V D0)
The last approximation is valid in the limit V D0  1.
For finite temperatures, the presence of quasiparticles needs to be taken into
account. The probability of having a quasiparticle excitation with energy Ek is given
by the Fermi function
1
f(Ek) = (5.24)
eEk/kBT + 1
Equation (5.17) then becomes [64, 101, 102]:
∑ ∑ ( )
∆k = − V ∗
∆ El
klul νl[1− 2(f(El))] = − Vkl tanh (5.25)2El 2kBT
l l
Again using Eq. (5.19) and assuming that ∆ is independent of the k direction, I
obtain an implicit equation for the gap as a function of temperature:
1 1 ∑ tanh(Ek/2kBT )
= (5.26)
V 2 Ek
k
Similar to the T = 0 case, I get
∫ (√ )2 2
1 h̄ω tanh √ξ + ∆ /2kBT= D(ξ)dξ (5.27)
V 0 ξ2 + ∆2
Using equation Eq. (5.27), one can find the superconducting gap as a function
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5.1. INTRODUCTION TO SUPERCONDUCTIVITY
of temperature T and the critical temperature where the gap disappears and the
material becomes a normal metal. Substituting ∆ = 0 at T = Tc, we get:
∫ h̄ω tanh(ξ/2kBTc)
1 = V D0 dξ. (5.28)
0 ξ
With ξ/2kBTc = x, Eq. (5.28) becomes
∫ h̄ω/2kBT ∫tanh(x)dx h̄ω/2kBTh̄ω/2k T
1 = V D0 = V D0(lnx tanhx| B0 − lnx sec2 xdx)
0 x 0
(5.29)
Assuming h̄ω  2kBT , one finds:
1
= (ln (h̄ω/2kBT )− ln 0.44) (5.30)
V D
and thus
k T = 1.136h̄ωe−1/V D0B c (5.31)
If we compare Eq. (5.23) and Eq. (5.31), we obtain a relationship between the
superconducting gap ∆(0) at zero temperature T = 0 and the critical temperature
Tc,
∆(0) 2
= = 1.764 (5.32)
kBTc 1.13
Thus there is a direct relationship between the critical temperature Tc and the
superconducting gap ∆(0) at T = 0 in conventional BCS theory.
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5.1. INTRODUCTION TO SUPERCONDUCTIVITY
Finally to calculate the superconducting gap as a function of temperature
[64], we have to use numerical integration, as there is no direct way of simplifying
Eq. (5.27). In Fig. 5.1, I plot the superconducting gap ∆ versus the temperature T
from the BCS theory. The temperature T is normalized to the critical temperature
Tc and the superconducting gap ∆ is normalized to the superconducting gap ∆(0) at
zero temperature. From the plot, we can see that the superconducting gap does not
vary much until T ≥ 0.4Tc of the critical temperature of the material. Other than
Tc and ∆(0), this behavior does not depend on the material, provided the material
has s-wave pairing and a weak electron-phonon coupling constant.
5.1.1 Density of states
Here I derive the density of the excited states in a superconducting material.
In a normal metal, states above the Fermi energy can be occupied by exciting
an electron, while an empty electron state below the Fermi level can be treated as
a hole excitation. The density N(E) of excited states in a superconductor obeys:
N(E)dE = N(ξ)dξ (5.33)
For a metal made which contains a non-interacting Fermi gas, dξ/dE = 1, and
Nn(E) = Nn(ξ).
In the superconducting case, there are no excited states with |E| < ∆, but we
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5.1. INTRODUCTION TO SUPERCONDUCTIVITY
1.0
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4 / 0.6 0.8 1.0T TC
Figure 5.1: Normalized BCS superconducting gap ∆/∆0 as a function
of normalized temperature T/Tc.
98
Δ/Δ(0)
5.2. ANDREEV REFLECTION
still have
Ns(E)dE = Nn(ξ)dξ (5.34)
since we are only interested in energies that are within a few meV from the Fermi
energy (the typical energy scale of ∆), we can take Nn(ξ) = Nn(0) as constant.
From Eq. (5.34) and Eq. (5.13), we then find
 E
Ns(E) dξ
 √
− (|E| > ∆)2 2= = E ∆ (5.35)
Nn(0) dE  0 (|E| < ∆)
Figure 5.2 shows a plot of Ns(E)/Nn(0) as a function of E. Note in particular
that the density of states diverges as E approaches ∆ from above, while for |E| < ∆,
there is zero density of states.
5.2 Andreev reflection
In Chapter 2, I discussed the current-voltage characteristics of tunnel junctions
made with a normal metal electrode and a superconducting electrode. For S-I-N
tunneling the main takeaway was that since there are no states in the superconductor
with energy |E| < ∆, there was no tunneling current for |eV | < ∆ at T = 0. However
experimentally when the tunnel barrier is low enough, step-like increments of the
current occurs for |E| < ∆. This phenomenon is explained by Andreev reflection
[44]. Andreev first considered the situation where there is no tunnel barrier. For a
99
5.2. ANDREEV REFLECTION
4
3
2
1
0
0 1 2 3 4
E/Δ
Figure 5.2: Normalized plot of density of states of a superconductor as
a function of energy.
100
Ns(E)/Nn(0)
5.2. ANDREEV REFLECTION
quasiparticle with momentum h̄kF and a plane wave as a trial function, he solved the
superconducting Bogoliubov-de-Gennes equation [64, 104, 105, 112]. After ignoring
higher order terms in the Hamiltonian, he found a wavefunction with one component
representing an electron traveling to the right in the normal metal and another
component representing a hole traveling to the left in the normal metal. When the
electrons are incident on the S-N interface from the normal metal at energies E < ∆,
they cannot enter the superconductor because there are no available quasiparticle
states with E < ∆. Instead, Andreev’s solution reveals that a hole is reflected back
and a Cooper pair with charge 2e is added to the superconductor. This process is
called Andreev reflection.
Blonder, Tinkham, and Klapwijk (BTK) [45] generalized Andreev’s picture to
obtain the I−V curve of S-N interfaces with different transparency [44]. To do this
they introduced a δ-function potential barrier of strength Z at the S-N interface
and solved the Bogoliubov equations [45] to find the probability of tunneling as a
function of Z. When the barrier is high, Z  1, one recovers the tunnel limit
results described in Chapter 2. This is the high-Z limit where Andreev effects are
suppressed. In contrast, when Z  1, the barrier height is low, the interface is
very transparent, most of the electrons are Andreev reflected, and Andreev effects
dominate the conductance.
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5.2. ANDREEV REFLECTION
5.2.1 BTK theory
To find the I − V characteristics of a normal–superconductor junction, the
key idea of the BTK theory is to use the Bogoliubov Hamiltonian Eq. (5.6) to
find the wavefunctions of electrons and quasiparticles in the normal electrode and
superconducting electrode. To deal with the interface, they included a δ-function
potential. Matching the wavefunctions at the interface yields the transmission and
reflection coefficients.
For the superconductor, one can start from Eq. (5.6) and write the one-
dimensional Bogoliubov equation [45, 64]:
    − h̄
2 ∂2 − µ+H δ(x) ∆(x) u(x)
2m ∂x2 0   u(x)= E  (5.36)
2 2
∆∗(x) h̄ ∂ 2 + µ−H0δ(x) ν(x) ν(x)2m ∂x
On the normal side there is no superconducting gap and Eq. (5.36) reduces to
    − h̄
2 ∂2
2 − µ 0 u(x) u(x)2m ∂x  = E  (5.37)
2
0 h̄ ∂
2
+ µ ν(x) ν(x)
2m ∂x2
√
2 2
Solving this equation for the eigenvalues E, we obtain E = + ( h̄ k − µ)2 where
2m
h̄2k2
µ = F . One solution corresponds to an electron with wavevector ke such that2m
2 2
ke > kF and energy E =
h̄ ke − µ, the other solution corresponds to a hole with
2m
h̄2k2
wavevector k hh such that |kh| < kF and energy E = − +µ. The solutions for the2m
102
5.2. ANDREEV REFLECTION
wave function can then be written as
  √1 2m
Ψe(x) =   e±ikex, where ke = 2 (E + µ) (5.38a)

0 h̄
0
Ψ (x) =  
√
 2me±ikhxh , where kh = 2 (µ− E) (5.38b)h̄
1
√
2 2 h̄2k2
On the superconducting side, we obtain E = + ( h̄ q − µ)2 + ∆2 where µ = F .
2m 2m
√ 2 2
Again there are two solution for the energy, E2 −∆2 = h̄ qe − µ for the electron
2m
√ h̄2q2
case with q > k , and E2 −∆2 = − he F + µ for the hole case with qh < kF . The2m
wavefunction on the superconducting side can be written as
u0 √2m √
Ψe(x) =   e±iqex, where q = 2 2e 2 ( E −∆ + µ) (5.39a)
 ν0 
h̄
√
ν0 2m √Ψh(x) =  e±iqhx, where qh = 2 (µ− E2 −∆2) (5.39b)h̄
u0
If we plug in√the energy E and wavefunc√tion defined in Eq. (5.39) into Eq. (5.36),√ √
we get u = 1
2 2
(1 + E −∆ ) and ν = 1
2 2
0 0 (1− E −∆ ). The form for u and ν2 E 2 E 0 0
are the same as in Eq. (5.14).
Given Eq. (5.38) for the wavefunction in the normal (left) electrode and
Eq. (5.39) for the wavefunction for the superconductor (right) side, we now as-
sume that the wavefunction for the left side has incident and reflected waves. The
incident electron wave function, incoming from the normal left electrode towards
103
5.2. ANDREEV REFLECTION
the interface S-N, is
 1 1
Ψ (x) = √   eikexin (5.40)ve
0
This wave is reflected back into the normal left region and consists of a left-moving
electron and a a left-moving hole. Note that the momentum k of a left moving hole
will be negative [113] and thus
   
r 1 0ee   rhe  
Ψreflect(x) = √   e−ikex + √   e+ikhx. (5.41)ve vh
0 1
Here ree represents the amplitude of the reflection coefficient of the electron,
rhe represents the amplitude of the Andreev reflection coefficient of the holes from
incident electrons and ve ' vh ' vF is the Fermi velocity on the left side.
On the right side (superconductor), the transmitted wave contains right-moving
electron-like particles and right-moving hole-like particles
tee 
  
u0 the 

ν0
Ψ (x) = √ e+iqex + √  e−iqhxtrans (5.42)we wh
ν0 u0
where tee is the amplitude of the transmission coefficient of the electron and the is the
amplitude of the transmission coefficient of the holes. I note that the wavefunctions
are normalized by the different velocities w and v, on the respective sides of the
barrier, due to conservation of probability. To determine the velocity of the electrons
104
5.2. ANDREEV REFLECTION
and holes in t∣he ∣normal electrode and superconductor electrode, we can use the
relation v = 1 ∣dE ∣ to obtain:
h̄ dk
h̄ke
ve = '
h̄kh
vF vh = ' vF
√m √ m
E2 −∆2 h̄q 2 2e E −∆ h̄qh
we = wh = (5.43)
E m E m
Finally, we need to match the two wavefunctions at the S-N interface. From
Eq. (5.36), we have
− h̄
2 ∂2
( − µ)u(x) +H0δ(x)u(x) + ∆(x)ν(x) = Eu(x) (5.44)
2m∂x2
Integrating the equation over an infinitesimal section spanning the boundary at
x = 0, we obtain
∣
2
− h̄ ∂u ∣∣∣0+ +H0u(0) = 0 (5.45a)2m ∂x 0−
∂u(0+) ∂u(0−− ) 2mH0= 2 u(0) (5.45b)∂x ∂x h̄
This is also true for the ν(x) case. Thus we obtain four equations that need to be
105
5.2. ANDREEV REFLECTION
satisfied at the boundary:
u(0+) = u(0−) (5.46a)
ν(0+) = ν(0−) (5.46b)
∂u(0+) − ∂u(0
−) 2mH0
= 2 u(0) (5.46c)∂x ∂x h̄
∂ν(0+) ∂ν(0−− ) 2mH0= 2 ν(0) (5.46d)∂x ∂x h̄
We now plug Eq. (5.40), Eq. (5.41) and Eq. (5.42), into Eq. (5.46) and arrive at
four equations for the parameters tee, the, ree, rhe:
1 r
√ √ee
tee t√ √he+ = u0 + ν0 (5.47a)
ve ve we wh
r
√he
t
√ee
the
= ν0 + √ u0 (5.47b)
vh we wh
t t 1 r 2mH 1 r
iqe√ee he ee 0 eeu0 − iqh√ ν0 − ike√ + ike√ = (√ + √ ) (5.47c)
we wh ve v
2
e h̄ ve ve
t
√eeiqe ν0 −
t r 2mH r
iqh√he u0 − ikh√he 0 he= √ (5.47d)
we wh v
2
e h̄ ve
Blonder et al. [45] solved these four equations Eq. (5.47) by using a semi-
classical approximation. This approximation is based on the fact that E and ∆
are relatively small compared to the Fermi energy µ. To proceed, in Eq. (5.38√) and
Eq. (5.39), we retain just the lowest order in k, with ke = kh = q
2mµ
e = qh = kf = h̄2
106
5.2. ANDREEV REFLECTION
so that Eq. (5.43) then becomes
h̄kf
ve = vh = (5.48a)
√m
E2 −∆2 h̄kf
we = wh = (5.48b)
E m
Plugging Eq. (5.48) into Eq. (5.47) and solving the four equations for the four
parameters, we obtain:
u0ν0
rhe = (5.49a)
u2 + Z2(u2 − ν20 0 0)
(Z2 + iZ)(ν2 − u2)
ree =
0 0 (5.49b)
u2 + Z2(√u20 0 − ν20)
(1− iZ)u0 u2 − ν2
t = √ 0 0ee (5.49c)u20 + Z2(u20 − ν20)
iZν u2 − ν20
t = 0 0he (5.49d)
u2 + Z20 (u
2 2
0 − ν0)
where the barrier height parameter is Z = H m/h̄20 kf = H0/h̄vf .
The transmission and reflection coefficients can be obtained from Eq. (5.49).
Here we preserve the convention in Ref. [45]. A = |r 2he| gives the probability
of Andreev reflection. B = |r |2ee gives the ordinary reflection probability of elec-
trons, C = |tee|2 is the transmission amplitude of electrons without branch cross-
ing while D = |t |2he is the transmission of the holes with bran√ch crossing. Ta-√
2 2
ble 5.1√gives A,B,C,D for both E > ∆ and E < ∆, where u0 = 1(1 + E −∆ ),2 E√
1 − E2 2ν = (1 −∆0 ), for the E > ∆ case. For E < ∆, we have to use the defi-2 E
107
5.2. ANDREEV REFLECTION
Table 5.1: Transmission and reflection coefficients in the BTK theory. A is the
probability of Andreev reflection. B is the probability of ordinary reflection of
electrons. C is the probability of transmission of the electrons. D is the probability
of transmission of the holes.
A B C D
u2ν2 (u2−ν2)Z2(1+Z2) u2(u2−ν2)(1+Z2) ν2(u2−ν2E > ∆ 0 0 0 0 0 0 0 0 0 0 )
[u20+Z
2(u20−ν2 20 )] [u2+Z2(u2−ν2)]2 [u2+Z2(u20 0 0 0 0−ν2 20 )] [u2+Z2(u2−ν20 0 0 )]2
E < ∆ ∆2 1− A 0 0
E2+(∆2−E2)(1+2Z2)2
√ √ √ √
2
nition u = 1(1 + i ∆ −E
2 2 2
0 ), ν0 =
1(1− i ∆ −E ) and plug into Eq. (5.49). I
2 E 2 E
note that in this case C = D = 0.
In Fig. 5.3, I plot the coefficients A (blue) and B (red) as a function of energy
for different values of Z. As we can see in Fig. 5.3(a), Z = 0 is the ideal transparent
limit. In this case A = 1 inside the gap, meaning all incident electrons in the N-
region are Andreev reflected as holes. For energies above the gap, we see that an
electron has a finite probability to be transmitted as an electron. On the other
hand, for E < ∆ and Z  1 (see Fig. 5.3(d)), the electrons are almost completely
reflected as electrons, and there is barely any Andreev reflection. This is also the
reason why in the tunnel limit, no current flows across the junction for V < ∆/e.
Note also that for E < ∆: A = 1 − B, i.e. the probability of transmission plus
reflection must be equal to 1. So an increased probability of electron reflection will
lead to a decrease of Andreev reflection.
108
5.2. ANDREEV REFLECTION
(a) (b)
1.0 1.0
0.8 Z=0 0.8 Z=0.2
A 0.6 A 0.6
or 0.4 or 0.4
B 0.2 B 0.2
0.0 0.0
0 1 2 3 4 0 1 2 3 4
E/Δ E/Δ
(c) (d)
1.0 1.0
0.8 Z=1 0.8 Z=3
A 0.6 A 0.6
or 0.4 or 0.4
B 0.2 B 0.2
0.0 0.0
0 1 2 3 4 0 1 2 3 4
E/Δ E/Δ
Figure 5.3: (a)-(d) plot of Andreev reflection probability A (blue) and
normal electron reflection amplitude B (red) as a function of normalized
energy E/∆ at different values of the barrier height Z. For Z = 0 and
E < ∆, all electrons are Andreev reflected at the interface. For E > ∆
electrons are partially reflected. For Z  1, the electrons are completely
reflected for E < ∆ and partially transmitted for E > ∆.
109
5.2. ANDREEV REFLECTION
5.2.2 Calculation of I-V Characteristics for S-I-N junction from the
BTK model
In this section I discuss the BTK model for how the current-voltage charac-
teriscis behave as a function of the barrier height Z. The derivation is somewhat
complicated and a very nice and detailed derivation can be found in lecture notes
from Kopnin [114].
Essentially there are four processes that contribute to current flowing across
an S-I-N interface. The first process is due to electrons that are incident on the
S/N interface from the normal side. Such electrons have a probability to be An-
dreev reflected as holes and normally reflected as electrons, while the rest will be
transmitted as quasiparticles into the superconducting side. The second process is
due to holes incident on the S/N interface from the normal side. The third process
involves electron-like quasiparticles starting out on the superconducting side and
transferring electrons or holes to the normal side. The last process involves hole-like
quasiparticles incident on the S/N interface from the superconductor. We need to
sum up the current from these processes to get the total current. One finds a fairly
simple expression by considering the resulting net current on the normal side [45]:
∫ ∞
I = 2N(0)evfσ [f0(E − eV )− f0(E)](1 + A(E)−B(E))dE. (5.50)
−∞
Here N(0) is the density of states of single spins at the Fermi energy µ, σ is the
effective cross sectional area of the junction, and vf is the Fermi velocity of the
110
5.2. ANDREEV REFLECTION
electrons in the normal metal. The normal junction resistance Rn = 1/2N(0)e
2vfσ.
Another way of looking at this expression is that incoming electrons with unity
probability are Andreev reflected as holes with probability A(E), which increases
the electron flow. The probability B(E) describes normally reflected electrons which
decrease the net flow of electrons through the interface. Taking the derivative of I
with respect to the voltage V , we get the conductance:
∫
dI 1 ∞ ∂f0(E − eV )
Gns(V ) = = (1 + A(E)−B(E))dE (5.51)
dV eRn −∞ ∂V
Scanning tunneling microscopy (STM) enables us to measure the conductance G
versus V , which can be compared directly to Eq. (5.51). The resulting dI/dV curves
depend strongly on the transparency parameter Z, provided Z is not too large
compared to unity.
Figure 5.4 shows an example of plots of the conductance versus energy for
Z = 0 (red), 0.5 (blue), 1.5 (green), and 5 (purple) at 75 mK. As we can see, when
the junction is very transparent (Z = 0) and eV < ∆, the conductance is doubled
in value compared to the conductance at eV  ∆. Conversely, in the tunnel limit
Z  1, there is zero conductance for eV < ∆, recovering the traditional S-I-N
tunnel junction behavior discussed in Fig. 2.6.
For comparison, Fig. 5.5 shows an example of plots of the conductance versus
energy for Z = 0 (orange), 0.5 (cyan), 1.5 (yellow), and Z = 5 (magenta) at
T = 500 mK. As we can see, thermal broadening rounds off conductance peaks,
and the effect shows up more in the tunnel limit (Z  0) than in the low Z limit.
111
5.2. ANDREEV REFLECTION
5
4
3
2
1
0- 2 - 1 0 1 2
eV/Δ
Figure 5.4: Normalized BTK S-I-N conductance RndI/dV as a function
of normalized voltage eV/∆ at a temperature of T=75 mK for supercon-
ducting gap ∆ =0.5 meV. Red curve is for Z = 0, blue for Z = 0.5, green
for Z = 1.5 and purple for Z = 5. Notice that in the very transparent
limit (Z ≈ 0), the conductance becomes twice that of the normal state.
In the tunnel limit Z  0, the curve recovers the classic tunneling limit.
112
R dIndV
5.2. ANDREEV REFLECTION
When the junction is very transparent (Z = 0) the conductance at edge eV ∼ ∆ is
rounded but this is less obvious than rounding of the coherence peak in the high Z
limit.
Figure 5.6 shows all the curves from Fig. 5.4 and Fig. 5.5. From the plot we
see that the coherence peaks in the limit Z → ∞ is effected strongly due to the
thermal broadening. In contrast, for the low-barrier case Z ≈ 0, the curves appear
to be only slightly rounded around eV ∼ ∆.
113
5.2. ANDREEV REFLECTION
5
4
3
2
1
0- 2 - 1 0 1 2
eV/Δ
Figure 5.5: Normalized BTK S-I-N conductance RndI/dV as a function
of normalized voltage eV/∆ at a temperature of T=500 mK for super-
conducting gap ∆ =0.5 meV. Orange curve is for Z = 0, cyan is for
Z = 0.5, yellow is for Z = 1.5 and magenta is for Z = 5. Comparing
Fig. 5.4 to Fig. 5.5, we see that thermal broadening affects the curves
more for Z  0 than for (Z  1).
114
R dIndV
5.2. ANDREEV REFLECTION
5
4
3
2
1
0- 2 - 1 0 1 2
eV/Δ
Figure 5.6: Overlapping of all curves shown in Fig. 5.4 and Fig. 5.5 to
get a better comparison of the thermal broadening effect for different
barrier height Z.
115
dI
RndV
CHAPTER 6
Scanning Andreev Tunneling Microscopy of
Titanium Nitride Thin Films
6.1 Introduction
Thin-film superconducting titanium nitride (TiN) has attracted recent interest
because it can have very low loss and this makes it attractive for building microwave
resonators and possibly other devices for applications in quantum computation [46–
50]. TiN can also have a high kinetic inductance and this also makes it potentially
interesting for constructing superconducting microwave kinetic inductance detec-
tors for x-ray spectroscopy and sub-millimeter-wave astronomy telescopes [51–55].
In practice, good superconducting films of TiN are not so easy to grow repro-
ducibly. In particular the loss, the superconducting critical temperature Tc, the
micro-structure, grain size and stress all depend on the growth conditions. Al-
though kinetic inductance measurements suggest a well-defined gap [53], atomic
scale scanning tunneling microscopy (STM) [20] of typical films show very rough
surfaces and small grains. Furthermore measurements of thin TiN films have re-
116
6.1. INTRODUCTION
vealed a superconductor-insulator transition and inhomogeneous superconducting
gap [115–117].
In this chapter, I describe my results on scanning tunneling microscopy [41] of
superconducting 50 nm and 25 nm thick TiN films taken at 500 mK. In Section 6.2,
I discuss some basic properties of the films. In Section 6.3, I show fine scale maps
of the 50 nm film which show significant spatial variations in the superconducting
gap that are correlated with the topography. In addition I observed a distinct zero-
bias conductance peak at some locations. Such peaks are characteristic of Andreev
S/N tunneling [44] through a high transparency barrier, as opposed to conventional
superconductor-normal quasiparticle tunneling [66, 118] through a low transparency
barrier. Using an analysis based on the Blonder-Tinkham-Klapwijk (BTK) model of
Andreev tunneling [45] (see Chapter 5), I extract maps of the superconducting gap
∆, temperature T and barrier height Z and examine some interesting features in
the maps. In Section 6.4, I then show the topography and maps of ∆, Z and T on a
larger scale. I find that both the 50 nm thick and 25 nm thick films show significant
variations in ∆. Much of the variation in ∆ in the thicker film (50 nm) appears to be
due to variations in the local temperature, while in the thinner film (25 nm) the ∆
variations are correlated with variations in the barrier height. Finally, in Section 6.5,
I discuss some possible explanations of the T variations and the correlations between
∆, Z and T .
117
6.2. TIN FILM GROWTH AND BASIC CHARACTERISTICS
6.2 TiN film growth and basic characteristics
Our TiN films were deposited by Kevin Osborn’s group [119] on a high re-
sistivity (100) silicon wafer (ρ > 20000 µΩ · cm, float-zone). The substrate was
pretreated in an O2 plasma for 30 s before being put into the sputtering system.
After evacuating the sputter chamber, the substrate was heated to 350◦C. During
DC magnetron sputtering 400 W of rf power was applied to a 7.6 cm diameter Ti
target while maintaining a pressure of 3.5 mTorr with Ar and N2 gas flowing at 15
SCCM and 10 SCCM, respectively.
The Tc of the 50 nm thick film was found to be 4.7 K using an inductive
measurement while the 25 nm TiN film had Tc=3.6 K. From Tc, the BCS coherence
length was estimated to be ξ = 18 nm for the 50 nm thick film and ξ = 15 nm for
the 25 nm thick film. (Measurements were done by Serena Merteen from Leonardo
Civale’s group at Los Alamos National Laboratory.) Resonators made from the 50
nm film showed an internal quality factor of Q 5i = 2×10 , whether or not an O2
pretreatment was used, while the 25 nm thick film had Qi= 1.5×104 with an O2
pretreatment, compared to Q = 2×105i without O2 pretreatmeant. (Quality factor
measurements were done from Kevin Osborn’s group at Laboratory for Physical
Sciences.) For the 50 nm film k 2 2/3 2 1/3Fl=h̄(3π n) /e ρn ≈ 24, while kFl ≈ 16 for the
25 nm film, where n is the carrier density obtained from Hall measurements and ρ
is the resistivity. Based on these values neither films violates the Ioffe-Regel limit
[120] and both films are weakly disordered.
Prior to imaging with the STM, each film was mounted on an STM sample
118
6.2. TIN FILM GROWTH AND BASIC CHARACTERISTICS
stud [41], and then sonicated in isopropanol for five minutes to remove dust particles.
The sample was then transferred to the custom UHV setup [41] and the surface was
cleaned by sputtering using 750 eV Ar ions while gently heating the sample at
T ∼ 300 ◦C for 60 minutes. We note that heating the TiN samples above 400
◦C caused Tc to decrease substantially. The sample was then transferred without
breaking vacuum to the cold-stage of a mK-STM [41] for imaging with a vanadium
tip or a Nb tip. The vanadium tip was used to image the 50 nm thick TiN film
while the Nb tip was used to image the 25 nm thick TiN film.
Since niobium has a critical temperature Tc of 9.2 K and vanadium has a
critical temperature Tc of 5.03 K, one may ask why we see an S-I-N junction. The
possible reason is that since the TiN film is very granular and the surface is not
flat, the tip easily picks up debris from the surface which makes it an effectively
normal tip. Another possible reason is that for spectroscopy measurements, I needed
to set the sample bias as low as 5 meV. This leads to a junction resistance of
over 50 MΩ and places the tip close to the sample. Normally in an STM, the
sample bias is around 1 V when STM performs surface imaging, and the junction
resistance is around 1 GΩ. Due to how close the tip is to the sample when performing
spectroscopy measurements, the tip can easily pick up debris and become normal.
I note that I did find some data sets that showed S-I-S behavior, however the tip
changed after acquiring just part of the spectroscopic map (see Appendix B).
119
6.3. STM MEASUREMENTS AND ANDREEV EFFECTS
6.3 STM measurements and Andreev effects
Figure 6.1 (a) shows an STM topographic map of a 42 nm × 42 nm region on
the 50 nm thick TiN film. This image was acquired at 0.5 K using a vanadium tip
and there are 256 × 256 pixels. I used a voltage bias V = 4 mV and a tunneling
current I = 100 pA. In this false color topographic image, blue and purple are higher
grains while red and yellow are lower. The surface is very granular.
As the topography was being measured for Fig. 6.1 (a), I-V curves were also
acquired at 128 × 128 pixels by sweeping the bias voltage V from 4 mV to -4 mV
while recording the tunnel current I. Conductance dI/dV versus V curves were also
obtained simultaneously by adding a small ac modulation voltage (140 µV) to the
voltage bias and using a lock-in amplifier to extract the ac signal from the tunnel
current.
Figure 6.1 (b)-(f) show dI/dV vs. V curves at five selected locations in Fig. 6.1
(a) taken at 0.5 K using a vanadium tip. The range of behavior is striking. Quali-
tatively Figs. 6.1 (b) and (c) show a clearly recognizable superconducitng gap, as ex-
pected for superconducting-normal tunneling through a convenional low-trasnparency
barrier [64], although there appears to be much more leakage current at low voltages
than expected. On the other hand, Fig. 6.1 (d) seems to show a gap and coherence
peaks, but the conductance at zero volts is almost the same as at high voltage, which
is not what is expected for a conventional S/N tunnel barrier. Finally, Figs. 6.1 (e)
and (f) show a distinct peak in the conductance at zero bias, reaching about two or
three times the conductance at high voltage, respectively. This is what is expected
120
6.3. STM MEASUREMENTS AND ANDREEV EFFECTS
(a) h (nm)40 (b) 80
2.5
30 602.0
20 401.5 20
10 1.0 0
0 0.5 -3 -2 -1 0 1 2 3
(c) 0 10 20 30 4080 (d)
V (mV)
80
60 60
40 40
20 20
0 0
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
(e) V (mV) V (mV)80 (f)160
140
60 120
40 10080
20 6040
0 20
-3 -2 -1V (m0 V)1 2 3 -3 -2 -1V (0mV)1 2 3
Figure 6.1: (a) Fine scale (42 nm × 42 nm) topographic map of 50
nm thick TiN film at a temperature of 0.5 K using a vanadium tip.
(b) Measured conductance dI/dV versus voltage V (black points) at
(x,y)=(36.42 nm, 9.19 nm) as well as fit to BTK model (blue) for Z =
3.15 and ∆ = 0.35 meV and Dynes model (red) for ∆ = 0.395 and Γ =
0.235 meV. (c) Measured dI/dV vs. V (black points) at (x,y)=(26.91
nm, 13.78 nm) as well as fit to BTK model (blue) for Z = 1.44 and
∆ = 1.39 meV and Dynes model (red) for ∆ = 1.274 and Γ = 0.34 meV.
(d)-(f) dI/dV vs. V at (2.63 nm, 5.25 nm), (8.20 nm, 36.09 nm) and
(34.78 nm, 13.45 nm), respectively. Blue curves are fit to BTK model
for Z =0.56, 0.00294 and 0.00217, respectively, and ∆ =0.66, 0.44, 0.25,
respectively.
121
dI/dV (nS) dI/dV (nS)
dI/dV (nS) dI/dV (nS) dI/dV (nS)
6.3. STM MEASUREMENTS AND ANDREEV EFFECTS
from S/N tunneling through a highly transparent barrier when Andreev reflection
[44, 45] is taken into account. Reexamining the curve in Fig. 6.1 (d), it appears
consistent with Andreev tunneling through a barrier that is of intermediate barrier
height.
To analyze the conductance vs. V data, I fit it to the conductance from the
BTK [45] model of Andreev reflection at a superconductor/normal interface. The
key model parameter is the barrier height parameter Z of the interface. For Z much
less than 1, the interface is very transparent, i.e. D = 1/(1 + Z2) ≈ 1 while for Z
much greater than 1, the barrier has a low transparency, i.e. D = 1/(1+Z2) ≈ 1 and
one recovers the conventional S/N tunneling limit [118]. As discussed in Chapter 5,
the BTK model conductance can be expressed as [45]:
∫ ∞
× ∂f0(E − eV )G(V ) = A0 (1 + A(E)−B(E))dE. (6.1)
−∞ ∂V
Here f0(E) is the Fermi function, which is dependent on the temperature T , A(E) is
the amplitude of the Andreev reflection at the S/N interface, B(E) is the amplitude
for the reflection of normal electrons of energy E and A0 = (1 + Z
2)/(eRN). A(E)
and B(E) depend on the barrier height Z of the barrier and the superconducting
gap ∆ [45]:
 ∆2
 2 2− 2 2 2 , for E < ∆[E +(∆ E )(1+2Z ) ]A(E) = (6.2)√ ∆2 , for E > ∆
[E+ E2−∆2(1+2Z2)]2
122
6.3. STM MEASUREMENTS AND ANDREEV EFFECTS
 ∆2
 1− [E2+(∆2−E2)(1+2Z2)2 , for E < ∆]B(E) = (6.3)4Z2√(1+Z2)(E2−∆2) , for E > ∆.
[E+ E2−∆2(1+2Z2)]2
I evaluated Eqs.(6.1) to (6.3) numerically and varied ∆, T and Z to obtain the
best fit to the measured dI/dV vs. V curve at each location. As the blue curves in
Figs. 6.1 (b)-(f) show, we find good qualitative agreement except for the curve in
Fig. 6.1 (f).
For comparison, the red curves in Figs. 6.1 (b)-(f) show fits to the conductance
in a junction with a conventional tunnel junction characteristic, i.e. through a low
transparency barrier, with a Dynes gap function [66]. In this case we can write:
∫
1 ∞ ∂f0(E − eV )√ E − iΓG(V = dE (6.4)
eRn −∞ ∂V (E − iΓ )2 −∆2
here Γ is the broadening parameter, f0(E) is the Fermi function and ∆ is the gap.
Comparing the measurements (blue) to the Dynes fits (red) in Figs. 6.1 (b)-(f), we
see reasonable fits for Figs. 6.1 (b) and (c), but quantitatively and qualitatively poor
fits in Figs. 6.1 (d)-(f) where the best fits yield ∆ ' 0.
Table 6.1 summarizes the fit parameters (∆, Z, T ) obtained from the BTK fits
shown in Fig. 6.1 (blue). The curves have significantly different gaps, temperatures,
and Z values. For example, for Fig. 6.1 (b) I found Z = 3.15, which is somewhat
large compared to 1, indicating the junction is not far from being in the conventional
S/N tunneling limit [64]. For comparison, fitting to the curve in Fig. 6.1 (d) yield
Z = 0.56, which indicates a fairly transparent barrier. BTK fits to Figs. 6.1 (e)
123
6.3. STM MEASUREMENTS AND ANDREEV EFFECTS
and (f) yielded Z ∼= 0.003 and Z ∼= 0.002, respectively, which indicates a highly
transparent barrier. I note in particular the BTK fit in Figs. 6.1 (e) is reasonable and
the behavior is just what one expects for Andreev tunneling in a highly transparent
S/N junction.
I note that this application of the BTK model only assumed a single channel
with well-defined Z, ∆ and T . Better fits could be obtained by allowing for two or
more channels with different Z. Also, the BTK fit in Fig. 6.1 (e) is not good around
the zero bias voltage and it might suggest that mid-gap states are present [26].
Although only a few sample curves are shown in Fig. 6.1, the range of behavior
is just what one expects from S/N Andreev tunneling through barriers that range
from highly transparent (Fig. 6.1 (e) and (f)) to low transparency (Fig. 6.1 (b)
and (c)). Although the fits to the BTK model are not perfect, they much more
faithfully represent the data than the fit to a Dynes model, Eq. (6.4). Thus I
conclude that Andreev effects are important in interpreting the STM data from
our TiN samples and that using a Dynes model here would potentially result in a
mistaken interpretation.
To better understand the sample, I took additional images of the region (see
Fig. 6.2). Figure 6.2 (a) shows the same topographic STM image as in Fig. 6.1 (a),
with a different color scale. Figure 6.2 (b) shows a topographic image of the same
region, scanned just before taking the spectroscopy. It shows the grains more clearly
and there is no apparent drift (white boxes in (a) and (b) represent the same area).
Figure 6.2 (c) shows a topographic image of a larger region on the same 50 nm film.
There seems to be some features on the grains that line up in different directions on
124
6.3. STM MEASUREMENTS AND ANDREEV EFFECTS
(a) (b)
10 nm 10 nm
(c) (d)
20 nm 5 nm
Figure 6.2: (a) Topographic image showing the same data as Fig. 6.1 (a)
but with a different color palette. (b) This image was taken before the
spectroscopy data and has better resolution and less drift. (c) Another
nice topographic image of the sample and (d) high resolution image
showing fine texture on top of individual grains.
125
6.3. STM MEASUREMENTS AND ANDREEV EFFECTS
Table 6.1: Extracted parameters (∆, T, Z) from fitting the conductance curves in
Figs. 6.1 (b)-(f) to the BTK model Eq. (6.1).
No. ∆ (meV) T (K) Z
(b) 0.35 1.95 3.15
(c) 1.39 2.43 1.44
(d) 0.66 0.94 0.56
(e) 0.44 2.60 0.003
(f) 0.25 0.42 0.002
different grains, with no clear step structure. Figure 6.2 (d) shows a fine scale image
with many small grains decorated with particles that have different alignment on
each grain. The surface is clearly poly-crystalline, with a grain size of order 10 nm
in this small region.
Figure 6.3 (a) again shows the topography of the 50 nm TiN film, while
Figs. 6.3(b)-(d) show the corresponding maps of ∆, T and Z obtained from fit-
ting the conductance curves at each point to the BTK model. Again, all the data
was taken at 0.5 K. First, note that there are distinct, spatially correlated variations
in all four maps, but they are most obvious in the topography and gap, i.e. Figs. 6.3
(a) and (b). If we compare the topography in Fig. 6.3 (a) to the ∆ map in Fig. 6.3
(b) one sees a distinct correspondence between the colors. Thus for example, topo-
graphically high points (blue in Fig. 6.3 (a)) correspond to small values of ∆ (blue
in Fig. 6.3 (b)). Similar but less pronounced variations in T are evident in Fig. 6.3
126
6.3. STM MEASUREMENTS AND ANDREEV EFFECTS
(a) (c)
 h (nm) T (K)
3.5
2.5 3.0
2.0 2.5
1.5 2.0
1.5
1.0 1.0
0.5 0.5
10 nm
(b) (d)
 
  (meV) Z
1.1 3.5
1.0 3.0
0.9 2.5
0.8
0.7 2.0
0.6 1.5
0.5 1.0
0.4 0.5
0.3 0.0
Figure 6.3: (a) Topographic and (b) false color map of superconducting
gap ∆ for 50 nm thick TiN film. (c) Maps of temperature T and (d)
barrier height Z, all from fitting the conductance curves to the BTK
model. The dark blue regions indicate locations with a zero bias con-
ductance peak and small Z. All of these data was taken at 0.5 K using
a vanadium tip.
127
6.4. ∆, Z, T , H CORRELATIONS IN THE 50 NM AND 25 NM FILMS
(c), with higher points tending to give higher temperature. Such topography corre-
lated variations are also present in Z, but they are much less obvious due to larger
point-to-point random variations.
To verify that the large spatially correlated variations in the gap are directly
visible in the raw data, Fig. 6.4 shows a false color plot of dI/dV vs. V along a
line cut (indicated by arrows in Fig. 6.3 (b)). Distinct regions are clearly visible
where the superconducting gap differs by a factor of 2. The transition appears to
be abrupt on the nm scale, perhaps suggesting grain-to-grain variations from one
disconnected grain to another. One also sees a section with a prominent zero-bias
conductance peak (red), consistent with Andreev effects in an S/N junction with a
high transparency. The fact that TiN shows large local variations in ∆ is perhaps not
surprising given the granular nature of the films. Also, variations in barrier height
Z are not surprising because they are likely due to surface contamination or local
damage created while preparing the film. What is surprising is to see correlations
between the topography and T or Z. To understand what is going on, I acquired
additional data, including on the 25 nm TiN film, as discussed in the next section.
6.4 ∆, Z, T , h correlations in the 50 nm and 25 nm films
Figure 6.5 shows larger scale topographic images of the 50 nm and 25 nm
TiN films. Both of the films were very granular. From analysis of the images, the
maximum grain size of the 50 nm thick film was 20 nm while for 25 nm thick film
128
6.4. ∆, Z, T , H CORRELATIONS IN THE 50 NM AND 25 NM FILMS
V (mV)  dI/dV (nS)
2 100
1
0
-1 50
-2 0
0 10 20 30 40
x (nm)
Figure 6.4: False color plot of conductance dI/dV versus position x and
voltage V along the line cut indicated by black arrows in Fig. 6.3 (a).
129
6.4. ∆, Z, T , H CORRELATIONS IN THE 50 NM AND 25 NM FILMS
Table 6.2: General properties of the 50 nm and 25 nm thick TiN film.
Film thickness Tc (K) ξ (K) Qi roughness (nm) grain sizes (nm)
50 nm 4.7 18 2×105 4 20
25 nm 3.6 15 1.5×104 2 10
the average grain size was about 10 nm. The surface roughness is 4 nm for the 50
nm thick film and 2 nm for the 25 nm thick film. Overall we see that the thicker
(50 nm) film had a rougher surface and a larger grain size. Table 6.2 summarizes a
few properties of the two films.
Figure 6.6 (a) shows a 350 × 160 nm2 topographic image of the 50 nm film
acquired at 500 mK taken with a vanadium tip. For this image, we used a voltage
bias V = 4 mV and a tunneling current I = 100 pA. The resulting topographic
map had 256 × 113 points, and I acquired spectra at each location. Figure 6.6
(b) shows the corresponding map of the superconducting gap ∆ obtained by fitting
the conductance dI/dV vs. V data at each location to the BTK model, Eq. (6.1).
Gray points in the map indicate curves where we did not obtain a good fit (see
Appendix A). Figure 6.6 (c) shows a line section through the topographic image
(gray) and the gap map (blue). Examining the plot, we see that the superconducting
gap is strongly anti-correlated with surface height (note the reversed scale for ∆).
To verify that this is not an artifact from fitting to the BTK model, I also show in
red the gap extracted from a fit to the Dynes model, Eq. (6.4). Although Eq. (6.4)
does not fit the conductance data as well as Eq. (6.1), the resulting line section
130
6.4. ∆, Z, T , H CORRELATIONS IN THE 50 NM AND 25 NM FILMS
(a) (b)
50 nm 50 nm
(c) (d)
6 6
5 5
4 4
3 3
2 2
1 1
0 0
0 100 200 300 0 50 100 150 200 250
x (nm) x (nm)
Figure 6.5: (a) Topographic image of 50 nm thick TiN film and (b) the
25 nm TiN film. (c) Line section along white line in (a), and (d) along
white line in (b), showing surface height versus distance x along line.
131
h (nm)
h (nm)
6.4. ∆, Z, T , H CORRELATIONS IN THE 50 NM AND 25 NM FILMS
plot nevertheless shows the same qualitative anti-correlation between ∆ and the
topography.
Figure 6.7 (b) and (c) show the corresponding map of the temperature T
and the barrier height Z from fitting the conductance data to the BTK model.
Comparing the topography Fig. 6.7 (a) to the gap, temperature and barrier height
maps, we again see some remarkable correlations: the higher the grain, the smaller
the superconducting gap and the higher the temperature. The barrier height map
appears featureless.
Similar behavior shows up in the 25 nm thick TiN film. Figure 6.8 (a) shows a
280 nm× 112 nm topographic image of the 25 nm thick film taken at T = 0.5 K with
a Nb tip. The grains in the image are somewhat distorted by a tip artifact; the actual
grain size is somewhat smaller (Fig. 6.5 (b)). Spectroscopy data was taken at 256
× 101 = 25856 locations; I vs. V and dI/dV vs. V were recorded simultaneously.
Figure 6.8 (b) shows the corresponding map of ∆, extracted by fitting conductance
curves at each point to the BTK model. Examining the topographic image (Fig. 6.8
(a)) and the gap (Fig. 6.8 (b)), we again see a distinct anti-correlation, with higher
regions having smaller superconducting gap. To see more clearly the correlation
between the topography and the gap, Fig. 6.8 (c) shows a line cut through the
surface topography (gray) and the BTK gap map ∆ (blue). For comparison we also
show ∆ extracted from fitting to the Dynes gap model (red). Again note the reverse
scale for ∆. As with the 50 nm thick film, the 25 nm film shows a remarkable
anti-correlation between surface height and gap.
Figure 6.9 (b)-(c) show the corresponding maps of T and Z, extracted by
132
6.4. ∆, Z, T , H CORRELATIONS IN THE 50 NM AND 25 NM FILMS
(a) h (nm)
6
5
4
3
2
1
0
(b) 50 nm Δ (meV)
(c)
6 0.3
5 Dynes ∆BTK ∆ 0.4
4 0.5
3
2 0.6
1 0.7
0 0.8
0 50 100 150 200 250 300 350
x (nm)
Figure 6.6: (a) Topography of 50 nm thick TiN film taken at 100 pA
and 4 mV bias showing rough surface. (b) Corresponding maps of the
gap ∆. (c) Line section through (a) and (b) showing anti-correlation
between the gap and topography (gray). Note reversed scale for ∆, i.e.
large gaps correspond to small h. Red curve shows ∆ vs. x from fits to
the Dynes model. All data taken at T = 0.5 K using a vanadium tip.
133
hz ((nnmm))
Δ (meV)
6.4. ∆, Z, T , H CORRELATIONS IN THE 50 NM AND 25 NM FILMS
(a) h (nm)
6
5
4
3
2
12.5
02.
12.5
(b) 50 nm T (K)12.
2.501.5
2.01.
1.50.5
1.0
0.5
0
(c) Z
3
2
1
0
Figure 6.7: (a) Topography of 50 nm thick TiN film taken at 100 pA
and 4 mV bias showing rough surface. Corresponding maps of the (b)
temperature T and (c) barrier height parameter Z from fits to the BTK
model. All data taken at T = 0.5 K using a vanadium tip.
134
6.4. ∆, Z, T , H CORRELATIONS IN THE 50 NM AND 25 NM FILMS
(a) h (nm)
3.0
2.5
2.0
1.5
1.0
0.5
0.0
(b) 50 nm Δ (meV)
(c)
4 0.2
Dynes ∆
3 BTK ∆ 0.3
2
0.4
1
0.5
0
0 50 100 150 200 250
x (nm)
Figure 6.8: (a) Topography of 25 nm thick TiN film taken at I=100 pA
and V=3 mV. There is some distortion from the rough surface imaging
the tip shape, instead of the tip imaging the surface. (b) Corresponding
map of gap ∆. (c) Line sections through (a) and (b) showing anti-
correlation between the gap ∆ (blue) and the topography h (gray). Note
reversed scale for ∆, i.e. large gaps correspond to small h. Red curve
shows ∆ vs. x from fits to the Dynes model. All data taken at T = 0.5
K using a Nb tip.
135
zh ((nnmm))
Δ (meV)
6.4. ∆, Z, T , H CORRELATIONS IN THE 50 NM AND 25 NM FILMS
(a) h (nm)
3.0
2.5
2.0
1.5
12.0.5
02.5.
0.0
12.5
50 nm
(b) T (K)12.2.501.5
2.01.
1.50.5
1.0
0.5
0
(c) Z
3
2
1
0
Figure 6.9: (a) Topography of 25 nm thick TiN film taken at I=100 pA
and V=3 mV. There is some distortion from the rough surface imaging
the tip shape, instead of the tip imaging the surface. Corresponding
maps of (b) temperature T and (c) barrier parameter Z extracted from
fits to the BTK model. All data taken at T = 0.5 K using a Nb tip.
136
6.4. ∆, Z, T , H CORRELATIONS IN THE 50 NM AND 25 NM FILMS
fitting conductance curves at each point to the BTK model. Unlike what I found
in the 50 nm film, the temperature map for the 25 nm film (see Fig. 6.9 (b)) shows
little obvious correlation with the topography. On the other hand, the Z map (see
Fig. 6.9 (c)) reveals that high Z regions tend to have lower superconducting gap
(see Fig. 6.8 (b)), which was not obvious in the 50 nm thick film in Fig. 6.7 (c).
To better understand the range over which the gap is varying, the blue curve
in Fig. 6.10 (a) shows the histogram of gap values obtained from Fig. 6.6 (b) on the
50 nm thick film. The red curve shows the results from the Dynes fits. In general,
the gap distribution from the BTK fits is more symmetric than that of the Dynes fit,
which as we noted above does not do a good job of capturing the observed behavior.
In addition, the Dynes fit gives counts at ∆=0 meV, because of its inability to fit
data with a zero bias conductance peak, while no such points occur in the histogram
of ∆ from the BTK fits. Fig. 6.10 (b) shows the corresponding histograms for the
25 nm film for the data set shown in Fig. 6.8 (b). Note that the histogram for the 50
nm film has a peak at about 0.6 meV, while the histogram for the 25 nm has a peak
at about 0.4 meV. This is only qualitatively consistent with the Tc measurements,
which gave Tc = 4.7 K for the 50 nm thick film and Tc = 3.6 K for the 25 nm thick
film; for a BCS superconductor, these transition temperature would imply a gap of
about 0.72 meV for the 25 nm film and about 0.55 meV for the 50 nm thick film.
137
6.4. ∆, Z, T , H CORRELATIONS IN THE 50 NM AND 25 NM FILMS
(a) (b)
1400 1400
Dynes Dynes
1200 BTK 1200 BTK
1000 1000
800 800
600 600
400 400
200 200
0 0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8
Δ (meV) Δ (meV)
Figure 6.10: The blue curves show histograms of the superconducting
gap from (a) the gap map in Fig. 6.6 (b) for the 50 nm thick film and
(b) the gap map in Fig. 6.8 (b) for the 25 nm thick film. Red curves
show histograms of the superconducting gap using the Dynes fit. Black
arrows indicate ∆ from measured Tc of the film.
138
Counts
Counts
6.5. PHYSICAL MECHANISM CAUSING CORRELATIONS BETWEEN ∆, T ,
Z AND H
6.5 Physical mechanism causing correlations between ∆, T ,
Z and h
To check whether my local measurements of the gap were consistent with weak-
coupling BCS theory, I measured the temperature dependence of the gap at a fixed
location. Figure 6.11 shows selected, averaged conductance curves as a function of
temperature from the 25 nm TiN film. These were obtained by increasing T , and
using a vanadium tip at a fixed location. Above 2.65 K, there was no hint of the
superconducting gap and we see a featureless background curve from the normal
state. Below 2.5 K, we see a clear gap and coherence peaks that both increase as T
decreases.
To understand what might be producing the correlations between h vs. ∆, I
did some further analysis of the maps shown in Fig. 6.6 and Fig. 6.8. Figure 6.12
shows the correlation between ∆ and h from the smaller scale 50 nm TiN film (42
× 42 nm2), the large scale 50 nm TiN film (350 × 160 nm2) and the large scale 25
nm TiN film (280 × 112 nm2). Here the binned height is plotted (red, left axis) vs.
the binned superconducting gap. For reference, I also show the number of counts
within each superconducting gap bin size of 0.01 meV (black, right axis). In each
case, we clearly see that the height h of the surface and ∆ are anti-correlated.
For comparison, Fig. 6.13 shows the 3D relation between ∆, Z and T for the
small scale 50 nm, large scale 50 nm and 25 nm film. For the 50 nm thick TiN
film, the superconducting gap is strongly anti-correlated with T (see red projected
139
6.5. PHYSICAL MECHANISM CAUSING CORRELATIONS BETWEEN ∆, T ,
Z AND H
30
25
20  1.59 K 1.71 K
 1.83 K
 1.94 K
15  2.06 K 2.18 K
 2.29 K
10  2.41 K 2.52 K 2.65 K
-4 -2 0 2 4
V (mV)
Figure 6.11: Temperature dependence of the conductance dI/dV curves
vs. V at a fixed location on the 25 nm TiN film. For T > 2.5 K, no
superconducting gap was observed.
140
dI/dV (nS)
6.5. PHYSICAL MECHANISM CAUSING CORRELATIONS BETWEEN ∆, T ,
Z AND H
(a) 0.6 1000
0.5 800
0.4
600
0.3
400
0.2
0.1 200
0.0 0
0.3 0.4 0.5 0.6 0.7
Δ �meV)
(b)
1400
1.5 1200
1000
1.0 800
600
0.5 400
200
0.0 0
0.3 0.4 0.5 0.6 0.7
Δ �meV)
(c)
0.8 1000
0.7
0.6 800
0.5 600
0.4
0.3 400
0.2 200
0.1
0.0 0
0.2 0.3 0.4 0.5 0.6
Δ �meV)
Figure 6.12: (a) Black points show the histogram of the superconducting
gap ∆ with a bin size of 0.01 meV and red points show the corresponding
average height h versus the gap of points corresponding to this height
for the 42 × 42 nm2 image of the 50 nm TiN film. (b) Corresponding
plots for the 350 × 116 nm2 image of the 50 nm film and for the (c) 280
× 112 nm2 image of the 25 nm film.
141
h (nm) h (nm) h (nm)
Count Count Count
6.5. PHYSICAL MECHANISM CAUSING CORRELATIONS BETWEEN ∆, T ,
Z AND H
curve in Fig. 6.13 (a)); the higher the temperature T , the lower the gap ∆. This
behavior suggests that the gap is being reduced at some locations due to heating.
In contrast, for the 25 nm film, ∆ appears to be well correlated with Z, and less
well correlated with T , which varies over a small range. This behavior suggests that
there are thermally disconnected regions in the thick film, but not in the thin film.
Given that TiN films can have large stress [121], this suggests that the thicker films
are cracking and producing grains which may be poorly connected to the rest of the
film.
We know that surface roughness is higher in the 50 nm film, indicating that
the grains stick out further from the surface. During tunneling spectroscopy mea-
surements, heat is generated due to the tunneling current (Joule heating) and it will
have a harder time dissipating from disconnected higher grains. Therefore those
regions should show a higher temperature. This argument works for the 50 nm film
as a possible explanation for why there is a smaller gap on higher grains. However,
for the 25 nm film the temperature map does not show a similar correlation with h.
Instead we see a clear correlation between Z and h. Therefore the lower supercon-
ducting gap in parts of the 25 nm film does not appear to be due to an increase in
temperature, but something such as surface damage or contamination that causes
an increase in the barrier height Z while suppressing ∆.
Figure 6.14 illustrates a possible mechanism which would generate correlations
in ∆, T, Z and h. During Ar ion cleaning isolated grains will charge up, leaving
disconnected regions that stick up and are less clean. This leads to a higher barrier
height in regions with a smaller superconducting gap and these are also regions that
142
6.5. PHYSICAL MECHANISM CAUSING CORRELATIONS BETWEEN ∆, T ,
Z AND H
(a)
2.0 (d)
1.8
1.6
1.4
1.2
2.
1.0
1
2.0
.30 10 5 .90.3 1.80.40 0.45 0 1.0.5
7
Δ 0.55
1.6
(meV) 0.6
0
0.65
1.5
0.70 1.4
(b)
3.2
3.0
2.8
2.6
2.4
2.2
2.0
1.8 2.2
2.1
1.6 2.01.9
1
1.4 0.3 .8
0.4
1.7
.5 10 .61
Δ .5 (meV 1.) 41.3
(c)
2.2
2.0
1.8
1.6
1.4
1.2 1.5
.2 1.0 4
0.3
1.3
1 )Δ 0.4 .( 2 T (KmeV) 0.5 1.1
0.6 1.0
Figure 6.13: Three-dimensional scatter plots show correlation between
the superconducting gap ∆, temperature T and barrier height Z for the
(a) 42 × 42 nm2 image of the 50 nm film, (b) 350 × 116 nm2 image of
the 50 nm film and (c) 280 × 112 nm2 image of the 25 nm film.
143
0.6
0.7
0.8
T
T ( K(K ))
Z Z Z
6.6. CONCLUSIONS
tend to stick up. Since isolated grains may also be thermally disconnected, they will
also correspond to higher T .
6.6 Conclusions
In conclusion, I used an STM to image 50 nm and 25 nm thick TiN films
at 500 mK. The films show rough granular surfaces, and a large variations in the
superconducting gap that are anti-correlated with the topography. My BTK analysis
of conductance provided a better fit to the data than fitting to the Dynes model.
It also allowed me to extract temperature and transparency maps, and accounted
for distinct Andreev features in the spectroscopy that occur for highly transparent
tunnel barriers. I found a puzzling correlation between topography, ∆, T and Z,
with reduced gap and higher barriers and higher temperature on higher surfaces.
This may be due in part to increased heating from poor thermal conduction in
disconnected grains, which also leads to higher barriers on top of the grains because
the grains are not cleaned well during ion milling because they charge up.
144
6.6. CONCLUSIONS
Ar+
cleaning
after cleaning
Figure 6.14: Illustration showing isolated grains will charge up during
the Ar ion cleaning, leaving disconnected regions that stick up and are
less clean, which gives a higher barrier height Z and reduced gap, and
will heat up more because they are disconnected.
145
+ + + + +
CHAPTER 7
Theory of the I-V characteristics in S-I-S
Junctions with Multiple Andreev Reflection
7.1 Introduction
In Chapter 5 and Chapter 6, I described the physics of S-I-N junctions, in-
cluding the phenomenon of Andreev reflection. In this chapter, I examine S-I-S
junctions. When two superconductors with gap ∆ are connected by a weak link, the
current rises when the voltage V = 2∆/e. In addition, Andreev effects give current
steps at subgap voltages V = ∆/en, where n is a positive integer. The theory used
to explain this phenomenon involves Multiple Andreev Reflections (MAR) and the
Josephson effect [30, 31]. In the Josephson effect two superconductors are coupled
by a weak link and current can flow through the junction without any voltage ap-
plied. This current is called supercurrent and the junction is called a Josephson
junction.
In Section 7.2 and Section 7.3, I discuss the approach developed by Averin and
Bardas [122] to describe MAR behavior in symmetrical S-I-S junctions with trans-
146
7.1. INTRODUCTION
parency D = 1 and D 6= 1. This theory describes the current-voltage charasteristics
of a single S-I-S superconducting channel with arbitrary transmission [122–125].
While there have been many theoretical descriptions of MAR, many are not easy to
follow. Moreover, most of the calculations, including Averin and Bardas’s, assumed
the two superconductors had the same superconducting gap. In my STM, the S-I-S
junction formed by the tip and the sample were usually asymmetrical, i.e. the tip
and sample had different gaps. I then need a theory that can describe asymmetrical
junctions.
In Section 7.4 I generalize the scattering approach developed by Averin et al.
[122] to the asymmetrical case, and demonstrate that this yields results that are in
agreement with Wu et al. [126], who used a Green’s function approach to describe
MAR in asymmetrical Josephson junctions. I make some general comments about
asymmetrical junctions and describe results for the supercurrent and subgap current
from this model.
In Section 7.5, I discuss the dc and ac Josephson effects. In Section 7.6, I
examine expressions for the critical current derived previously and my derivation of
MAR effects in asymmetrical junctions. Finally, in Section 7.7, I conclude with a
summary of my main results from the analysis.
147
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
7.2 MAR in symmetrical junction with transparency D = 1
7.2.1 Averin-Bardas model
In 1995, Averin and Bardas (AB) presented an accesible description of An-
dreev and ac Josephson effects in an S-I-S junction with a single quantum channel
[122]. They considered a model of a junction as a short constriction between two
superconductors and obtained a quantitative description of the current-voltage char-
acteristics and how it depends on the temperature and barrier transparency. The
single quantum channel can be thought of as a normal channel (∆=0) and, de-
pending on the transparency, the junction can be used to describe S-N-S or S-I-S
tunneling.
Figure 7.1 shows an illustration of the AB model of an S-I-S junction. The
left lead is a superconductor with gap ∆1 and the right lead is an superconductor
with gap ∆2. The center is the weak link, which acts as a scattering region for
quasiparticles. If there is no scattering, the link is completely transparent, and
the channel has transparency D = 1. On the other hand, if the scattering is not
zero the quantum channel will not be completely transparent, and the transparency
parameter D will be between 0 and 1. Here the transparency D is different from the
barrier height Z defined in Chapter 6. However they are related as D = 1/(1 +Z2).
Andreev reflections [44, 45] happen at each S/N interface and quasiparticles also
scatter within the link region. This leads to the possibility of multiple Andreev
reflections and multiple scattering, which greatly complicates the analysis. Averin
148
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
S N barrier1 1 N2 S2
Incoming  electron: Bn Incoming  electron: Cn
Incoming  hole: An Incoming  hole: Dn
Δ1 Δ2
Andreev reflected hole: a2nBn Andreev reflected hole: a2n+1Cn
Andreev reflected electron Andreev reflected electron: a D
+Source term J incident 2n+1 n
from the left: a2nAn+Jδn0
Figure 7.1: Illustrate of Andreev reflections off of two S/N interfaces
and Bardas proceeded by defining a scattering matrix and determining separate
relations for Andreev reflections at each S/N boundary.
As I discussed in Chapter 5, Andreev reflection at an NS interface can be
characterized by a reflection amplitude a that depends on quasiparticle energy ε
[122]:
 ε− sgn(ε)(ε21  −∆
2)1/2, for |ε| > ∆
a(ε) = × (7.1)∆ ε− i(∆2 − ε2)1/2, for |ε| < ∆
149
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
AB then defined a scattering matrix to account for the weak link [122]:
 
S = r t el . (7.2)
t −r∗t/t∗
where r is the probability of the quasiparticle being reflected in the normal region,
and t is the probability of the quasiparticle being transmitted in the normal region.
Here the transparency is D=|t|2 = 1/(1 + Z2). They then wrote the wavefunctions
in region 1 in the channel (see Fig. 7.1) generated by a quasiparticle incident from
the left superconductor as [122]:
∑
ψ = [(a A + Jδ )eikx +B e−ikx]e−i(ε+2neV )t/h̄el 2n n n0 n (7.3a)
∑n
ψ = [A eikxh n + a
−ikx
2nBne ]e
−i(ε+2neV )t/h̄ (7.3b)
n
Here k and ε are the wave-vector and energy of the incident quasiparticle, and
am = a(ε + meV ). The second term in Eq. (7.3a) corresponds to a quasiparticle
incident from the superconductor, which produces an electron in the normal region
with effective source amplitude J(ε) = [1−|a(ε)|2]1/2. For this case the wave function
in region 2 of the channel (see Fig. 7.1) can be written as,
∑
ψ = [C eikx + a D e−ikx]e−i(ε+(2n+1)eV )t/h̄el n 2n+1 n (7.4a)
∑n
ψh = [a
ikx
2n+1Cne +D e
−ikx]e−i(ε+(2n+1)eV )t/h̄n (7.4b)
n
150
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
where the sum over n represents contributions from multiple reflections and is taken
over all integers from -∞ to ∞.
From the basic properties of waves, we can relate the incoming electrons to
the outgoing electrons in regions 1 and 2 using a scattering matrix. Similarly there
will be a scattering matrix connecting the holes in the two regions. AB wrote this
as:
Bn

 
     
a2nAn + Jδn0= Sel  ,  An   a2nBn = S h (7.5)
Cn a2n+1Dn Dn−1 a2n−1Cn−1
where  ∗ r t Sh = Sel, Sel =   . (7.6)
−r∗t tt∗
At this point in this paper [122], AB present a recursion relation for finding
the An, Bn, Cn and Dn. The authors omit detailed steps and only consider the case
where the left and right superconductors have the same gap. This is clearly not true
in our case, and it is important to understand how to find the recursion relation
before trying to generalize it to our situation where the superconducting gaps are
different on the left and right.
If we stare at the Eq. (7.5), we see that Bn is a function of An and Dn. Moving
the equation around, we get Dn as a function of An and Bn. Since Cn is a function
of An and Dn, we can now use Dn(An, Bn) to get Cn(An, Bn). From Eq. (7.5) I find:
151
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
Bn = ra2nAn + rJδn0 + ta2n+1Dn (7.7a)
r∗t
Cn = ta2nAn + tJδn0 − ∗ a2n+1Dn (7.7b)t
To proceed, I rearrange Eq. (7.7a) and plug it into Eq. (7.7b) to get Cn and
Dn as a function of An and Bn:
1 ra2n
Dn = Bn − An −
rJδn0
(7.8)
ta2n+1 ta2n+1 ta2n+1
|r|2 − r
∗ |r|2
Cn = ( ∗ a2n + ta2n)An ∗Bn + ( ∗ + t)Jδn0 (7.9)t t t
Similarly for the hole equation, I find:
An = r
∗a ∗2nBn + t a2n−1Cn−1 (7.10a)
rt∗
D ∗n−1 = t a2nBn − a2n−1Cn−1 (7.10b)
t
From Eq. (7.10a), we can write:
An+1 = r
∗a ∗2n+2Bn+1 + t a2n+1Cn (7.11)
152
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
I now substitute Cn from Eq. (7.9) into Eq. (7.11) to get:
|r|2 r∗ |r|2
An+1 = r
∗a ∗2n+2Bn+1 + t a2n+1[(
t∗
a2n + ta2n)An − ∗Bn + ( + t)Jδn0]t t∗
= r∗a2n+2Bn+1 +Ra2na2n+1An +Da2na
∗
2n+1An − r a2n+1Bn
+Ra2n+1Jδn0 +Da2n+1 (7.12)
and thus:
√
An+1 − a2n+1a2nAn = R(a2n+2Bn+1 − a2n+1Bn) + Ja1δn0 (7.13)
where R=|r|2. Equation (7.13) is exactly the same as the second recursion relation
Eq. (5) from the AB paper [122].
To derive the recursion relation for Bn, I proceed as follows. From Eq. (7.10b),
I can rewrite :
∗
D = t∗
rt
n a2n+2Bn+1 − a2n+1Cn (7.14)
t
153
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
Plugging in Dn and Cn from Eq. (7.8) and Eq. (7.9) then gives
1 Bn − ra2n A − rJδn0ta2n+1 ta n2n+1 ta2n+1
∗ − rt∗ |r|
2
− r∗ |r|
2
= t a2n+2Bn+1 a2n+1[( ∗ a2n + ta2n)An ∗Bn + (t t t t∗ + t)Jδn0]
a22n+1ra2nAn − ra2nAn
= Da2n+1a2n+2Bn+1 + [a
2
2n+1R− 1]Bn + rJδ 2n0 − a2n+1rJδn0
a2 R−1
ra2nA
a2n+1a2n+2 2n+1
n = D Ba2 n+1 + 2 Bn − rJδn02n+1−1 a2n+1−1
(7.15)
Eq. (7.15) is important because it shows that An is related to Bn+1 and Bn. Plugging
Eq. (7.15) into Eq. (7.13) gives us a recursion relation for Bn+1, Bn, Bn−1 which
eventually gives me the first recursion relation Eq. (5) from the AB paper [122].
To proceed, I rewrite the index from Eq. (7.13) so that
A − a a A = R1/2n 2n−1 2n−2 n−1 (Bna2n −Bn−1a2n−1) + Ja1δn−1,0 (7.16)
Plugging An from Eq. (7.15) into Eq. (7.16), I get:
D a 22n+1a2n+2 1 a2n+1R− 1 − JBn+1 + Bn δn0
ra a22n 2n+1 − 1 ra 22n a2n+1 − 1 a2n
Da22n−1a2n a2n−1 a
2
− B − ( 2n−1
R− 1
n )B2 − 2 − n−1
+ Ja2n−1δn−1,0
r(a2n−1 1) r a2n−1 1
√
= R(a2nBn − a2n−1Bn−1) + Ja1δn−1,0 (7.17)
After rearranging terms, I arrive at the first recursion relation Eq. (5) in the AB
154
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
paper [122]:
Da 2 2 22n+1a2n+2 a
B − ( 2n+1
R− 1 Da2n−1a2n
− 2 n+1
−
1 a a2 22n+1 2n+1 − 1 a2n−1 − 1
2 a
2
− 2n−1
a2n(a R− 1)
Ra2n)Bn + (
2n−1 −Ra
2 − 2n
a2n−1)Bn−1
a2n−1 1
= −rJδn,0 − rJa2a1δn−1,0 + rJa2a1δn−1,0 (7.18)
This recursion relation is important because it gives us the wavefunction am-
plitudes An and Bn for different transparency D. From the wavefunction, we can
determine an expression for the current flowing through the S-I-S junction, as I
discuss in the next section. After that, I discuss how to generalize to the case of
different gaps for the left and right superconductors.
7.2.2 Finding the expression for the current when the gaps are the
same
In this section, I work out Eq. (6) from Averin-Bardas. This gives the current
through an S-I-S junction when the superconductors have the same gap and there is
Multiple Andreev Reflection (MAR). This will help us understand the physics and
how to generalize to the case where the left and right superconductors have different
gaps. To simplify the discussion, I examine the situation for D=1. This is the case
of no scattering of the electrons or holes, i.e. a completely transparent weak link
with Z = 0.
For an electron-like quasiparticle with energy ε that is incident from the left
155
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
S1 0 N V S2
J(ε)a1(ε+eV)a2(ε+2eV)a3(ε+3eV)
               a4(ε+4eV)=a4A2
J(ε)a1(ε+eV)a2(ε+2eV)
      =a2A1 ε+5eV
incident electron-like J(ε) ε+3eV
quasiparticle
EF ε+eV
eV
EF
ε+2eV
ε+4eV
J(ε)a1(ε+eV)=A1
J(ε)a1(ε+eV)a2(ε+2eV)a3(ε+3eV)
      =A2
J(ε)a1(ε+eV)a2(ε+2eV)a3(ε+3eV)
    a4(ε+4eV)a5(ε+5eV)=A3
Figure 7.2: Configuration of the multiple Andreev reflection at trans-
parency D = 1. An electron-like quasiparticle (filled circle) is incident
from the left with energy ε and is Andreev reflected with amplitude A
as a hole (open diamond) from the right S/N interface. Note that a hole
below the Fermi level has positive energy.
156
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
superconducting electrode onto the S/N interface, the p√robability amplitude of an
electron being created in the normal region is [122]: J = 1− |a(ε)|2. I now assume
voltage V is applied to the right electrode and the left electrode is at 0 V (Fig. 7.2).
Again for simplicity, I consider D = 1, so there is no scattering, and nothing is
reflected back while traveling in the normal region, so that all particles reach the
right electrode. Since the right electrode is at voltage V (V > 0), the quasiparticle
arrives at the right electrode with E=ε + eV . The particle will then be Andreev
reflected back [44, 122] as a hole with amplitude a1(ε+ eV ). This Andreev reflected
hole will travel back to the left, and since it is a hole, it will pick up additional
energy eV and reach the left electrode with energy ε+2eV . It will then be Andreev
reflected as an electron with amplitude a2(ε+2eV ), etc. (see Fig. 7.2). Keep in mind
that, if the energy E = |ε+neV | after reflection is smaller than the superconducting
gap ∆ of the electrode at the interface, then the electron will be Andreev reflected
as a hole.
Since D = 1, we have R = 0 in the scattering matrix Eq. (7.6), and the
scattering matrices become
 0 1
Sh = S
∗
el and Sel =   . (7.19)
1 0
Plugging this into Eq. (7.5), I get four relations:
Bn = a2n+1Dn (7.20a)
157
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
Cn = a2nAn + Jδn0 (7.20b)
An = a2n−1Cn−1 (7.20c)
Dn−1 = a2nBn (7.20d)
In this case, with D = 1, A0 = 0, there is no scattering so Bn = 0 and thus Dn = 0.
From Eq. (7.20), I can now construct An and Cn recursively. For example:
C0 = J,A1 = a1C0 = a1J,C1 = a2A1, A2 = a3C1 = a3a2a1J , etc. We can also see
that the An are only non-zero for positive n and for D = 1, I find
A−3 = A−2 = A−1 = A0 = 0 (7.21a)
A1(ε, V ) = J(ε)a1(ε+ eV ) (7.21b)
A2(ε, V ) = J(ε)a1(ε+ eV )a2(ε+ 2eV )a3(ε+ 3eV ) (7.21c)
A3(ε, V ) = J(ε)a1(ε+ eV )a2(ε+ 2eV )a3(ε+ 3eV )a4(ε+ 4eV )a5(ε+ 5eV )
(7.21d)
... (7.21e)
Thus, the An(ε, V ) are functions of the incident quasiparticle amplitude J at energy
ε and the quasiparticle picks up energy eV each time it transits the normal region.
158
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
From Eq. (7.18), Eq. (7.20) and Eq. (7.3) the wavefunction can now be written as:
∑
ψ = [(a A + Jδ )eikx]e−i(ε+2neV )t/h̄el 2n n n0 (7.22a)
∑n
ψ = [A eikxh n ]e
−i(ε+2neV )t/h̄ (7.22b)
n
where the n’s are positive integers. Notice that the electron wavefunction has com-
ponents traveling to the right, represented as eikx, while the hole wavefunction has
components traveling to the left, represented as eikx; note that the hole travels in
the opposite direction of the k vector [113].
From the wavefunction, we can calculate the electron and the hole probability
current density using the well-known quantum expression:
h̄
j = (ψ∗∇ψ − ψ∇ψ∗) (7.23)
2mi
For the electron probability current density, I find:
ψ = eikx(Je−iεt/h̄ + a A e−i(ε+2eV )t/h̄ + a A e−i(ε+4eV )t/h̄el 2 1 4 2 + a
−i(ε+6eV )t/h̄
6A3e + ...)
(7.24a)
ψ∗ = e−ikx(Jeiεt/h̄ + a∗A∗ei(ε+2eV )t/h̄ + a∗ ∗ i(ε+4eV )t/h̄ ∗ ∗ i(ε+6eV )t/h̄el 2 1 4A2e + a6A3e + ...)
(7.24b)
159
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
and thus:
(
h̄
jel = ∗ 2ik [|J |2 + |a 2 22| |A1| + |a 2 24| |A2| + |a6|2|A |23 + ...]
2mi
+ [Ja∗A∗2 1 + a2A1a
∗A∗4 2 + a4A2a
∗
6A
∗ + ...]ei2eV t/h̄3
+ [Ja∗ ∗4A2 + a2A a
∗A∗ + a A a∗A∗ + ...]ei4eV t/h̄ + [...]ei6eV t/h̄1 6 3 4 2 8 4 + ...
+ [J∗a2A1 + a
∗
2A
∗
1a
∗ ∗
4A2 + a4A2a A + ...]e
−i2eV t/h̄
6 3 )
+ [J∗a A + a∗A∗a A ∗ ∗ −i4eV t/h̄ −i6eV t/h̄4 2 2 1 6 3 + a4A2a8A4 + ...]e + [...]e + ... (7.25)
For the probability current density from the holes that are generated by
electron-like quasiparticles incident from the left, I find:
ψh = e
ikx(A e−i(ε+2eV )t/h̄ + A e−i(ε+4eV )t/h̄ + A e−i(ε+6eV )t/h̄1 2 3 + ...) (7.26a)
ψ∗ = e−ikx(A∗ei(ε+2eV )t/h̄ + A∗ei(ε+4eV )t/h̄ + A∗ei(ε+6eV )t/h̄h 1 2 3 + ...) (7.26b)
(
h̄
jh = ∗ 2ik [|A1|2 + |A2|2 + |A 23| + ...]
2mi
+ [A∗1A2 + A
∗
2A + ..]e
−i2eV t/h̄
3 + [A
∗
1A3 + A
∗
2A4 + ..]e
−i4eV t/h̄ + [..]e−i6eV t/h̄ +)...
+ [A∗A + A∗2 1 3A2 + ..]e
i2eV t/h̄ + [A∗3A + A
∗
1 4A2 + ..]e
i4eV t/h̄ + [...]ei6eV t/h̄ + ...
(7.27)
The total probability current density is then j = jel + jh, where the direction of
the charge flow has been taking care of from the definition of the wavefunction
160
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
Eq. (7.22). I find
(
h̄k
j(ε, t) = [|J |2 + |A |21 (1 + |a 22| ) + |A 22| (1 + |a 24| ) + |A 23| (1 + |a6|2) + ...]
m
+ [Ja∗A∗ + A A∗(1 + a a∗) + A A∗(1 + a a∗) + ...]ei2eV t/h̄2 1 1 2 2 4 2 3 4 6
+ [Ja∗A∗ + A A∗(1 + a a∗) + A A∗(1 + a a∗) + ...]ei4eV t/h̄ + [...]ei6eV t/h̄4 2 1 3 2 6 2 4 4 8 + ...
+ [J∗a A + A A∗(1 + a a∗) + A A∗(1 + a a∗) + ...]e−i2eV t/h̄2 1 2 1 4 2 3 2 6 4
+ [J)∗a4A2 + A3A∗1(1 + a6a∗2) + A4A∗2(1 + a a∗8 4) + ...]e−i4eV t/h̄ + [...]e−i6eV t/h̄
+ ... (7.28)
Equation (7.28) gives the probability current density due to an electron-like
quasiparticle with energy ε incident from the left superconducting electrode. The
probability current density can be written as a sum of Fourier components:
h̄k ∑
j(ε, t) = j (ε, V )ei2KeV t/h̄K (7.29)
m
K=0,±1,±2,..
where K are integers. Summing up the probability current density from all energies,
weighted by the Fermi distribution, I obtain the total probability current density
coming from the left electrode.
Since this is a 1-channel problem, the total electrical or charge current is
directly proportional to the probability current density, and I can write:
∫
e m ∞
IL(t) = dεj(ε, t)f(ε) (7.30)
2πh̄ h̄k −∞
161
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
where f(ε) is the Fermi distribution for the incident quasiparticle at energy ε. For
the left electrode, this gives
∫ (
e
IL(ε, t) = dε [|J |2 + |A1|2(1 + |a2|2) + |A |22 (1 + |a 24| ) + |A3|2(1 + |a6|2)
2πh̄
+ ...] + [Ja∗2A
∗
1 + A1A
∗ ∗
2(1 + a2a4) + A A
∗(1 + a a∗) + ...]ei2eV t/h̄2 3 4 6
+ [Ja∗A∗4 2 + A A
∗ ∗ ∗ ∗ i4eV t/h̄
1 3(1 + a2a6) + A2A4(1 + a4a8) + ...]e + [...]e
i6eV t/h̄ + ...
+ [J∗a2A1 + A A
∗
2 1(1 + a4a
∗
2) + A3A
∗
2(1 + a
∗
6a4) + ...]e
−i2eV t/h̄
+ [J)∗a4A2 + A A∗(1 + a a∗) + A A∗(1 + a a∗) + ...]e−i4eV t/h̄ + [...]e−i6eV t/h̄3 1 ∑6 2 4 2 8 4∞1 (K)
+ ... × = IL (ε)e
i2KeV t/h̄ (7.31)
eε/kBT + 1
K=−∞
where the last expression is written as a sum over Fourier components.
This completes half of the story. For the net current, we also have to consider
the current due to electron-like quasiparticles incident from the right electrode.
This is similar to the case discussed above of quasiparticles incident from the left
electrode, however an electron-like quasiparticle traveling from right to left picks up
energy −eV rather than eV as it traverses the normal region.
To proceed, note again that for D = 1, I will have Bn = 0 and Dn = 0, i.e.
no scattered wave. Since the right electrode has applied voltage V , it effectively
shifts the Fermi level down. To simplify the equations, I consider an electron-like
quasiparticle incident from the right with energy ε+ eV and source term J(ε+ eV ).
Examining Fig. 7.3, one can see that when an electron travels from right to left,
it loses energy eV , and thus then Andreev reflection amplitude at the left S/N
162
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
S1 0 N V S2
J(ε+eV)a1(ε)a2(ε-eV)a3(ε-2eV)
    a4(ε-3eV)a5(ε-4eV)=A3
J(ε+eV)a1(ε)a2(ε-eV)a3(ε-2eV)
J(ε+eV) incident electron-like
      =A2 quasiparticle
EF ε ε-3eV
ε-2eV eVJ(ε+eV)a (ε)=A1 ε-eV1 EF
J(ε+eV)a1(ε)a2(ε-eV)
ε-4eV
      =a2A1
J(ε+eV)a1(ε)a2(ε-eV)a3(ε-2eV)
               a4(ε-3eV)=a4A2
Figure 7.3: Illustration of multiple Andreev reflection of an electron-like
quasiparticle that is incident at ε from the right electrode. Since the
right electrode has voltage V , it effectively brings down the Fermi level
of the right side.
163
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
interface will be a1(ε+ eV − eV ) = a1(ε). Thus the energy of the electron that will
be Andreev reflected as an hole is ε. The hole will travel to the right, and it loses
energy eV , which gives a reflection amplitude of a2(ε − eV ) at the right electrode.
This process continues and each time the particle’s energy decreases by eV when
traveling across the normal region. Thus I obtain:
A−3 = A−2 = A−1 = A0 = 0 (7.32a)
A1(ε+ eV,−V ) = J(ε+ eV )a1(ε) (7.32b)
A2(ε+ eV,−V ) = J(ε+ eV )a1(ε)a2(ε− eV )a3(ε− 2eV ) (7.32c)
A3(ε+ eV,−V ) = J(ε+ eV )a1(ε)a2(ε− eV )a3(ε− 2eV )a4(ε− 3eV )a5(ε− 4eV )
(7.32d)
... (7.32e)
The wave function due to an electron-like quasiparticle incident from the right side
is thus, for D = 1:
∑
ψ = [(a A −ikx −i(ε−2neV )t/h̄el 2n n + Jδn0)e ]e (7.33a)
∑n
ψ = [A e−ikx]e−i(ε−2neV )t/h̄h n (7.33b)
n
The corresponding current due to electron-like quasiparticles incident from the right
164
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
is then:
∫ ∞ (
IR(ε, t) = −
e
dε [|J |2 + |A 2 21| (1 + |a2| ) + |A 2 22| (1 + |a4| ) + |A |23 (1 + |a 26|
2πh̄ −∞
+ ...] + [Ja∗A∗ + A A∗(1 + a a∗) + A A∗2 1 1 2 2 4 2 3(1 + a a
∗
4 6) + ...]e
−i2eV t/h̄
+ [Ja∗A∗4 2 + A
∗
1A3(1 + a2a
∗ ∗ ∗
6) + A2A4(1 + a4a8) + ...]e
−i4eV t/h̄ + [..]e−i6eV t/h̄ + ...
+ [J∗a2A1 + A A
∗
2 1(1 + a4a
∗
2) + A A
∗
3 2(1 + a6a
∗
4) + ...]e
i2eV t/h̄ )
+ [J∗a4A + A A
∗
2 3 1(1 + a a
∗
6 2) + A A
∗(1 + a a∗) + ...]ei4eV t/h̄ + [...]ei6eV t/h̄4 2 8 4 + ...∑∞
× 1 (K)= I (ε)ei2KeV t/h̄ (7.34)
e(ε−µR)/kBT + 1 R
K=−∞
where the last expression is written as a sum over Fourier components. Also note
in Eq. (7.34) that the chemical potential µR of the right electrode is -eV .
I can now find the total current by combining the current from the left and
the right side. For simplicity, let’s look at the DC case. Taking the zeroth Fourier
component K = 0 in Eq. (7.31) and Eq. (7.34), I find:
∫ ∞
I 0 0DC = [d∫ε(IL(ε) + IR(ε))−∞
e ∞
= dε[|J(ε)|2 + |A1(ε, V )|2(1 + |a2(ε+ 2eV )|2) + |A2(ε, V )|2(1
2πh̄ −∞
1
+ |∫a4(ε+ 4eV )|
2) + |A3(ε, V )|2(1 + |a6(ε, V )|2) + ...]
eε/kBT + 1
∞
− dε[|J(ε+ eV )|2 + |A 2 21(ε+ eV,−V )| (1 + |a2(ε+ eV − 2eV )| )
−∞
+ |A2(ε+ eV,−V )|2(1 + |a4(ε+ eV − 4eV] )|2) + |A3(ε+ eV,−V )|2(1+
|a (ε+ eV − 6eV )|2 16 ) + ...] (7.35)
e(ε+eV )/kBT + 1
165
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
Substituting for the An from Eq. (7.21) and Eq. (7.32), I can write
[ ∫
e ∞
IDC = dε[|J(ε)|2 + |J(ε)|2|a1(ε+ eV )|2 + |J(ε)|2|a 21(ε+ eV )|
2πh̄ −∞
|a (ε+ 2eV )|2 + |J(ε)|2| 12∫ a1(ε+ eV )|
2|a2(ε+ 2eV )|2|a3(ε+ 3eV )|2...]
eε/kBT + 1
∞
− dε[|J(ε+ eV )|2 + |J(ε+ eV )|2|a (ε)|2 + |J(ε+ eV )|21 |a1(ε)|2]|a2(ε− eV )|
2
−∞
+ |J(ε+ eV )|2|a (ε)|2|a (ε− eV )|2 11 2 |a3(ε− 2eV )|2...] (7.36)
e(ε+eV )/kBT + 1
Let’s look at the |J(ε)|2 terms first. I find:
∫ ∞ ( )1
dε |J(ε)|2 − |J(ε+ eV )|2 1∫ ( eε/kBT + 1 e(ε+eV )/k T−∞ B + 1∞ )
= ∫ dε[ (1− |a(ε)|
2 1 1) − (1− |a(ε+ eV )|2)
eε/kBT−∞ + 1 e
(ε+eV )/kBT + 1
∞ 1 1 1
= dε ( − )− (|a(ε)|2
eε/kBT + 1 e(ε+eV])/kBT + 1 eε/k−∞ BT + 1
− |a(ε+ eV )|2 1 ) = eV (7.37)
e(ε+eV )/kBT + 1
This eV is the first term in Eq. (6) in Averin and Bardas [122].
For the rest of the terms, I change variables for the incoming current from the
right. Defining ε′ = ε+ eV , I can now write Eq. (7.36) as
[
2 ∫e V e ∞
I = + dε[|J(ε)|2 2 2DC |a1(ε+ eV )| + |J(ε)| |a1(ε+ eV )|2|a2(ε+ 2eV )|2
2πh̄ 2πh̄ −∞
+ |∫J(ε)|
2| 1a1(ε+ eV )|2|a2(ε+ 2eV )||a3(ε+ 3eV )|2...]
eε/kBT + 1
∞+eV
− ′ | 2 2 2 2 2dε [ J(ε′)| |a (ε′1 − eV )| + |J(ε′)| |a1(ε′ − eV )| |a2(ε′ − 2eV )|
−∞+eV
166
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
]
+ |J(ε′)|2|a1(ε′ − 2 2 2
1
eV )| |a ′2(ε − 2eV )| |a ′3(ε − 3eV )| ...] ′ (7.38)eε /kBT + 1
We see there is√some kind of the symmetry for the case of equal gaps. From the
definition of J(ε) = (1− |a(ε)|2), this is an even function: J(ε)=J(−ε). In terms
of a(ε), by inspecting the definition from Eq. (7.1), we get the following relation:
 a(−E) = −a(E) = −a
∗(E), for |E| > ∆
 (7.39)a(−E) = −a∗(E), for |E| < ∆
I use E to represent the total energy of the particle to avoid confusion in the notation.
ε is the original energy that the incident particle carries. V is the voltage that the
incident quasiparticle sees. Thus:
an(−ε,−V ) = a(−ε− neV ) = a(−E)
= −a∗(E) = −a∗(ε+ neV ) = −a∗n(ε, V ) (7.40)
Using Eq. (7.40), I can show that there is a relation between A1(ε, V ) =
J(ε)a1(ε+ eV ) and A1(−ε,−V ) = J(−ε)a1(−ε− eV ), where I find:
A1(−ε,−V ) = −A∗1(ε, V ) (7.41)
This relation holds for A1, A2, A3..., as each An has an odd number of factors of a(ε),
167
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
therefore it always picks up a minus sign. Thus the general relation for A(ε, V ) is:
An(−ε,−V ) = −A∗n(ε, V ) (7.42)
as stated by AB [122].
Using Eq. (7.41) and Eq. (7.42), the negative ε′ terms in Eq. (7.38) can combine
with the positive ε terms, and I find:
[ ∫
e ∞
IDC = eV + dε[|J(ε)|2|a1(ε+ eV )|2 + |J(ε)|2|a1(ε+ eV )|2|a2(ε+ 2eV )|2
2πh̄ −∞
+ |∫J(ε)|
2|a (ε+ eV )|2| 11 a2(ε+ 2eV )|2|a3(ε+ 3eV )|2...]
eε/kBT + 1
∞−eV
− dε[|J(−ε)|2|a1(−ε− eV )|2 + |J(−ε)|2|a1(−ε− eV )|2
−∞−eV
|a2(−ε− 2eV )|2 + |J(−]ε)|2|a1(−ε− eV )|2|a2(−ε− 2eV )|2|a3(−ε− 3eV )|2
1
[ .∫..]e−ε/kBT + 1
e ∞
= eV + dε[|J(ε)|2|a (ε+ eV )|2 + |J(ε)|21 |a1(ε+ eV )|2|a2(ε+ 2eV )|2
2πh̄ −∞
+ |J(ε)|2|a (ε+ eV )|2|a (ε+ 2eV )|21 2 |a3(ε+ 3eV )|2
1
...](∫ −
1
− )eε/kBT + 1 e ε/kBT + 1
−∞
− dε[|J(−ε)|2|a1(−ε− eV )|2 + |J(−ε)|2|a1(−ε− eV )|2|a2(−ε− 2eV )|2
−∞−eV
+ ∫|J(−ε)|
2| 1a1(−ε− eV )|2|a2(−ε− 2eV )|2|a3(−ε− 3eV )|2...]
e−ε/kBT + 1
∞
+ dε[|J(−ε)|2|a1(−ε− eV )|2 + |J(−ε)|2|a1(−ε− eV )|2|a2(−ε− 2]eV )|
2
∞−eV
+ [|J(− |
2 1ε)∫|a (−ε− eV )|
2
1 |a2(−ε− 2eV )|2|a3(−ε− 3eV )|2...]
e−ε/kBT + 1
e ∞
( )(
= eV − εdε tanh |J(ε)|2|a1(ε+ eV )|2 + |J(ε)|2|a1(ε+ eV )|2
2πh̄ −∞ 2kBT
168
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
)
|a2(∫ε+ 2eV )|
2
(+ |J(ε)|
2|a (ε+ eV )|21 |a2(ε+ 2eV )|2|a3(ε+ 3eV )|2...
∞+eV
− dε |J(ε)|2|a1(ε− eV )|2 + |J(ε)|2|a1(ε− eV))|
2|a (ε− 2eV )|22
∞
+ |J(ε)|2∫ |a1(ε(− eV )|
2| 1a2(ε− 2eV )|2|a3(ε− 3eV )|2...
eε/kBT + 1
−∞+eV
+ dε |J(ε)|2|a1(ε− eV )|2 + |J(ε)|2|a1(ε− e)V )|
2|a 22(ε−]2eV )|−∞
+ [|J(ε)|
2| 2∫a1(ε− eV )|(|a2(ε−)2(eV )|
2|a3(ε− 3eV )|2
1
...
eε/kBT + 1
e ∞− ε= eV dε tanh |J(ε)|2|a1(ε+ eV )|2 + |J(ε)|2|a1(ε+ eV )|2
2πh̄ −∞ 2kBT )]
|a2(ε+ 2eV )|2 + |J(ε)|2|a (ε+ eV )|2|a (ε+ 2eV )|2 21 2 |a3(ε+ 3eV )| ...
(7.43)
The second integral vanishes because the Fermi function vanishes as ε → ∞. The
last integral also vanishes because a(ε) goes to zero as ε → −∞, as can be seen
from Eq. (7.1). Later on, when I consider the case where D is not equal to 1, the
last term will need to be reexamined.
A similar analysis can be done with the other Fourier components in Eq. (7.31)
and Eq. (7.34). In particular, I find that for the first Fourier component K = 1:
IAC = I
1
L(t) + I
1
R(t∫) ∞
= ei2eV t/h̄
e
dε[J(ε)a∗(ε, V )A∗(ε, V ) + A (ε, V )A∗2 1 1 2(ε, V )(1 + a2(ε, V )2πh̄ −∞
a∗
1
4(ε, V )) +∫A2(ε, V )A∗3(ε, V )(1 + a4(ε, V )a∗6(ε, V )) + ...]eε/kBT + 1∞+eV
− ei2eV t/h̄ e dε[J∗(ε)a2(ε,−V )A ∗1(ε,−V ) + A1(ε,−V )A2(ε,−V )(12πh̄ −∞+eV
+ a∗2(ε,−V )a4(ε,−V )) + A∗2(ε,−V )A3(ε,−V )(1 + a∗4(ε,−V )a6(ε,−V ))
169
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
1
+ ...] (7.44)
eε/kBT + 1
This can be rearranged to give
∫ ∞
I = ei2eV t/h̄
e
AC dε[J(ε)a
∗(ε, V )A∗2 1(ε, V ) + A (ε, V )A
∗
1 2(ε, V )(1 + a2(ε, V )2πh̄ −∞
a∗
1
4(ε, V )) +∫A2(ε, V )A∗3(ε, V )(1 + a4(ε, V )a∗6(ε, V )) + ...]eε/kBT + 1∞−eV
− ei2eV t/h̄ e dε[J∗(−ε)a2(−ε,−V )A1(−ε,−V ) + A∗1(−ε,−V )2πh̄ −∞−eV
A2(−ε,−V )(1 + a∗2(−ε,−V )a ∗4(−ε,−V )) + A2(−ε,−V )A3(−ε,−V )
(1 + a∗4(−ε,−∫V )a6(− −
1
ε, (V )) + .)..]e−ε/kBT + 1∞
= −ei2eV t/h̄ e εdε tanh [J(ε)a∗2(ε, V )A∗1(ε, V )2πh̄ −∞ 2kBT
+ A ∗1(ε, V )A2(ε, V )(1 + a2(ε, V )a
∗
4(ε, V )) + A2(ε, V )A
∗
3(ε, V )
(1 + a4(ε, V∫ )a∗6(ε, V )) + ...]
− i2eV t/h̄ e
∞+eV
e dε[J∗(ε)a2(ε,−V )A1(ε,−V ) + A∗1(ε,−V )A2(ε,−V )2πh̄ ∞
(1 + a∗2(ε,−V )a4(ε,−V )) + A∗2(ε,−V )A3(ε,−V )(1 + a∗4(ε,−V )a6(ε,−V ))
1
+ .∫..]eε/kBT + 1−∞+eV
+ ei2eV t/h̄
e
dε[J∗(ε)a2(ε,−V )A ∗1(ε,−V ) + A1(ε,−V )A2(ε,−V )2πh̄ −∞
(1 + a∗2(ε,−V )a4(ε,−V )) + A∗2(ε,−V )A ∗3(ε,−V )(1 + a4(ε,−V )a6(ε,−V ))
1
+ ...] (7.45)
eε/kBT + 1
170
7.2. MAR IN SYMMETRICAL JUNCTION WITH TRANSPARENCY D = 1
Finally I find:
∫ ∞ ( )
I = −ei2eV t/h̄ e εAC dε tanh [J(ε)a∗2(ε, V )A∗1(ε, V ) + A ∗1(ε, V )A (ε, V )2πh̄ 2−∞ 2kBT
(1 + a2(ε, V )a
∗
4(ε, V )) + A2(ε, V )A
∗
3(ε, V )(1 + a (ε, V )a
∗
4 6(ε, V )) + ...]
(7.46)
If we do the analysis for all the Fourier components, I find:
∑
I(t) = I ei2keV t/h̄k (7.47)
k
where
∫
e ∞
( )
ε (
Ik = 2× [eV δk0 − dε tanh J(ε)(a∗ A∗2k k + a−2kA−k)2πh̄ ∑ −∞ 2kBT∞ )
+ (1 + a ∗ ∗2na2(n+k))(AnAn+k) ] (7.48)
n=−∞
I note that this expression has Ak=0 for k < 0 and it holds only for the very special
case where there is no scattering, i.e. D = 1.
Examining Eq. (7.48), we see that the current component Ik has a leading
factor of 2. This accounts for a hole-like quasiparticle incident from the left and
a hole-like quasiparticle incident from the right, where the sum of both terms con-
tributes exactly the same as an electron-like quasiparticle incident from the right
and an electron-like quasiparticle incident from the left. However this situation only
applies when the left and right superconductors have the same gap ∆. In a later
171
7.3. MAR IN SYMMETRICAL JUNCTION WITH D 6= 1
section, we will see that when the electrons have different gaps, one needs to consider
four different scenarios (electron-like quasiparticle incident from the left, electron-
like quasiparticle incident from the right, hole-like quasiparticle incident from the
left, hole-like quasiparticle incident from the right) in order to obtain the correct
current expression.
7.3 MAR in symmetrical junction with D 6= 1
To find the current where the barrier transparency D 6= 1, we need to account
for quasiparticles being reflected at the barrier in the weak link (see Fig. 7.1). For
D 6= 1, when a quasiparticle is incident from the left the wave function is
∑
ψ = [(a A + Jδ )eikx +B e−ikx]e−i(ε+2neV )t/h̄el 2n n n0 n (7.49a)
∑n
ψ = [A eikxh n + a
−ikx
2nBne ]e
−i(ε+2neV )t/h̄ (7.49b)
n
From this wavefunction, we can get the total probability current j = jel + jh using
Eq. (7.23). As we see above the wavefunction of the electron and the hole have
taken care of the current flow.
For the electron wave function:
ψel = e
ikx(a A e−i(ε−4eV )t/h̄−4 −2 + a A
−i(ε−2eV )t/h̄ −iεt/h̄
−2 −1e + (J + a0A0)e
+ a A e−i(ε+2eV )t/h̄ + a A e−i(ε+4eV )t/h̄ + a A e−i(ε+6eV )t/h̄2 1 4 2 6 3 + ...)
172
7.3. MAR IN SYMMETRICAL JUNCTION WITH D =6 1
+ e−ikx(B e−i(ε−4eV )t/h̄ +B e−i(ε−2eV )t/h̄−2 −1 +B0e
−iεt/h̄ +B e−i(ε+2eV )t/h̄1
+B e−i(ε+4eV )t/h̄2 + ...) (7.50)
and thus
ψ∗ = e−ikx(a∗ A∗ ei(ε−4eV )t/h̄ + a∗ A∗ ei(ε−2eV )t/h̄el −4 −2 −2 −1 + (J + a
∗A∗ iεt/h̄0 0)e
+ a∗A∗ei(ε+2eV )t/h̄ + a∗A∗ei(ε+4eV )t/h̄ + a∗A∗ei(ε+6eV )t/h̄2 1 4 2 6 3 + ...)
+ eikx(B∗ ei(ε−4eV )t/h̄ +B∗ ei(ε−2eV )t/h̄ +B∗eiεt/h̄−2 −1 0
+B∗ei(ε+2eV )t/h̄ +B∗ei(ε+4eV )t/h̄1 2 + ...). (7.51)
I then find the probability current density:
{(
h̄
jel = ∗ 2ik [...+ |a 2−4| |A−2|2 + |a 2 2−2| |A−1| + |J |2 + J(a0A0 + a∗ ∗
2mi 0
A0)
+ |a 20| |A |2 + |a |2 + |A |2 +)|a |(20 2 1 4 |A 22| + |a |26 |A 23| + ...]− [...+ |B−2|2 + |B 2−1|
+ |B |20 + |B1|2 + |B |22 + ...] + [...a A a∗ ∗ ∗ ∗−4 −2 −2A−1 + a−2A−1(J + a0A0)
+ (J + a A )a∗0 0 2A
∗
1 + a)A a∗A∗2 1 4 2 + a4A2a∗A∗6 3 + ...]− [...+B ∗ ∗−2B−1 +B−1B0
+B B∗(0 1 +B B
∗ + ...] ei2eV t/h̄ + [...]ei4eV t/h̄ + [...]ei6eV t/h̄1 2 + ...
+ [...+ a A a∗ ∗−2 −1 −4A−2 + a
∗ ∗
−2A−1(J + a0A0) + (J + a
∗A∗0 0)a2A1 +)a
∗ ∗
2A1a4A2
+ a∗A∗a A + ...]− [...+B B∗4 2 6 3 −1 −2}+B0B
∗ +B ∗ ∗−1 1B0 +B2B1 + ...] e
−i2eV t/h̄
+ [...]e−i4eV t/h̄ + [...]e−i6eV t/h̄ + ... (7.52)
173
7.3. MAR IN SYMMETRICAL JUNCTION WITH D =6 1
For the hole wave function:
ψ = eikx(A e−i(ε−4eV )t/h̄ + A e−i(ε−2eV )t/h̄ + A e−iεt/h̄ + A e−i(ε+2eV )t/h̄h −2 −1 0 1
+ A2e
−i(ε+4eV )t/h̄ + ...) + e−ikx(a−4B e
−i(ε−4eV )t/h̄ + a B e−i(ε−2eV )t/h̄−2 −2 −1
+ a B e−iεt/h̄ + a B e−i(ε+2eV )t/h̄ + a B e−i(ε+4eV )t/h̄0 0 2 1 4 2 + ...) (7.53)
and thus
ψ∗ = e−ikx(A∗ ei(ε−4eV )t/h̄ + A∗ ei(ε−2eV )t/h̄ + A∗eiεt/h̄ + A∗ei(ε+2eV )t/h̄h −2 −1 0 1
+ A∗2e
i(ε+4eV )t/h̄ + ...) + eikx(a∗ B∗ i(ε−4eV )t/h̄−4 −2e + a
∗ B∗ ei(ε−2eV )t/h̄−2 −1
+ a∗B∗eiεt/h̄ + a∗B∗ei(ε+2eV )t/h̄ + a∗B∗ei(ε+4eV )t/h̄0 0 2 1 4 2 + ...) (7.54)
I then find the probability current density:
{(
h̄
jh = ∗ 2ik − [...+ |A−2|2 + |A 2 2 2 2−1| + |A0| + |A1| + |A2| + ...]
2mi )
+ [(|a
2 2 2
−4| |B−2| + |a−2| |B−1|2 + |a0|2|B0|2 + |a 2 22| |B1| + |a4|2|B |22 + ...]
+ − [A ∗ ∗ ∗ ∗ ∗1A2 + A0A1 + A−1A0 +)A−2A−1...] + [a2B1a4B
∗
2 + a
∗
0B0a2B
∗
1
+ a B a∗B∗ + a B a∗ B∗ ] ei2eV t/h̄ + [...]ei4eV t/h̄(−2 −1 0 0 −4 −2 −2 −1 + [...]e
i6eV t/h̄ + ...
+ − [A∗A + A∗A + A∗ A + A∗ ∗ ∗ ∗ ∗1 2 0 1 −1 0 ) −2A−1...] + [a2B1a4B2 + a0B0a2B1 }
+ a∗ ∗ ∗ ∗−2B−1a0B0 + a−4B−2a−2B−1] e
−i2eV t/h̄ + [...]e−i4eV t/h̄ + [...]e−i6eV t/h̄ + ...
(7.55)
174
7.3. MAR IN SYMMETRICAL JUNCTION WITH D 6= 1
From Eq. (7.52) and Eq. (7.55), I can now get a version of Averin and Bardas’s
Eq. (6) in [122] that is valid for all D:
∑
I(t) = I ei2keV t/h̄k (7.56)
k
where
{ ∫
e ∞
( )
× − ε
[
Ik = 2 eV δk0 dε tanh J(ε)(a
∗ ∗
2]k
Ak∫+ a−2kA−k)2πh̄∑ −∞ 2kBT∞ −∞+eV [
+ (1 + a ∗ ∗2na2(n+k))(AnAn+k −BnB∗n+k) + dε J(ε)(a∗2kA∗} kn=−∞ ∑ −∞∞ ] 1
+ a−2kA−k) + (1 + a a
∗ ∗ ∗
{ ∫ ( 2n 2(n)+k)
)(AnAn+k −BnBn+k) eε/kBT + 1
n=−∞
e ∞− ε
[
= eV Dδk0 dε tanh J(ε)(a
∗ A∗ + a−2kA−k)
πh̄ ∑ −∞ 2kBT 2k∞ ]k}
+ (1 + a a∗ ∗ ∗2n 2(n+k))(AnAn+k −BnBn+k) (7.57)
n=−∞
I note that the eV D δk0 term in the final expression is not equal to eV in this case
because the last term does not disappear when D 6= 1. One finds that the B0B∗0
terms in the sum yield eV R, which then combine with the first term eV − eV R =
eV D. Note that in Eq. (7.57) the A and B terms are in general non-zero for all k,
which are the integers 0,±1,±2,±3, ....
To calculate the I − V characteristics from Eq. (7.57), one needs to use the
definition of a(ε) from Eq. (7.1) to plug into the recursion relation Eq. (7.18) and
Eq. (7.13) to get An, Bn. Once I have the An and Bn terms, I plug them into
Eq. (7.57) and numerically integrate the equation to get the current at coltage V
175
7.4. MAR IN ASYMMETRICAL JUNCTIONS
(see Appendix C).
Figure 7.4 shows the calculated dc I − V characteristics of an S-I-S junction
that I obtained from Eq. (7.57) for k=0. The different curves are for different trans-
parency D. For example, for D = 0.01 and T = 0, the barrier has low transparency,
and the junction I(V) curve goes to the conventional S-I-S tunneling limit [64].
There is a clear rise in the current at the sum of the superconducting gaps, but no
obvious subgap behavior on this scale. In contrast, the other curves with D ≥ 0.1
are for more transparent junctions, and these show current steps at eV = 2∆/n,
where n = 1, 2, 3...is an integer.
7.4 MAR in Asymmetrical junctions
In the previous section I showed how to obtain Averin and Bardas’s expres-
sion for the I(V) characteristics of an S-I-S junction when both electrodes have the
same gap. Here I generalize their approach to the case where the electrodes have
different superconducting gaps. The key thing to note is that the Andreev reflection
amplitude will now depend on which interface is reflecting the particle. To simplify
the discussion, let’s again start by looking at the case D = 1. Consider an electron-
like quasiparticle incident from the left with energy ε and a voltage V applied to
the right electrode. Now when a transmitted electron goes from the left to the
right interface, its energy becomes ε+ eV and the Andreev reflection amplitude will
be A1(ε, V ) = J
L(ε)aR1 (ε + eV ). An Andreev reflected hole will then travel from
176
7.4. MAR IN ASYMMETRICAL JUNCTIONS
12
10
D
8 0.01
0.1
0.4
6
0.7
0.9
4 0.99
1
2
0
0 0.5 1 1.5 2 2.5 3
eV/Δ
Figure 7.4: Calculated I−V characteristics of a symmetrical S-I-S junc-
tion with MAR. The superconducting gap is the same for the two super-
conductors and the different curves correspond to D values of 1, 0.99,
0.9, 0.7, 0.4, 0.1 and 0.01 from top to bottom. GT is the normal state
conductance of the channel G 2T = e D/πh̄.
177
I/(ΔGT /2e)
7.4. MAR IN ASYMMETRICAL JUNCTIONS
right to left and pick up extra energy eV . This hole will be Andreev reflected at
the left interface, producing a rightward traveling electron with amplitude aL2 (ε)A1
=JL(ε)aR1 (ε+eV )a
L
2 (ε+2eV ). When the electron again travels to the right, it picks
up another energy increment of eV and at the right interface yields an Andreev
reflected hole with amplitude JL(ε)aR1 (ε+ eV )a
L
2 (ε+ 2eV )a
R
3 (ε+ 3eV ). This is the
same process shown in Fig. 7.2, it is just that we need to keep in mind that the
left and right sides produce different reflection amplitudes because they may have
different gaps.
As we can see from this description, all the odd indices for an come from the
right electrode, and all the even indices for an are from the left electrode. To get
A1, A2, A3..., we have to calculate the amplitude using corresponding a
L R
even and aodd.
To help keep things straight, we emphasize that here we first consider quasiparticles
incident from the left electrode at energy ε, and now we relabel the A1, A2,.. as
A1L(ε, V ), A2L(ε, V ), A3L(ε, V )... We will go on to consider quasiparticles incident
from the right.
Previously for the case of electrodes with the same superconducting gap, I
obtained Eq. (7.39) and Eq. (7.42). These relations are important because they
allowed me to simplify the calculation of the current. For the case of different gaps,
I need to re-examine these relations to see if they still hold.
To see what happens to Eq. (7.42), let’s examine the situation when an
electron-like quasiparticle is coming from the right electrode with energy −ε and
the left electrode is at the potential of 0 V while the right electrode is at the poten-
tial of V . When it goes from the right to the left interface, its energy decreases by
178
7.4. MAR IN ASYMMETRICAL JUNCTIONS
eV with respect to the Fermi energy on the left side. An Andreev reflected hole with
amplitude A1(−ε,−V ) = JR(−ε)aL1 (−ε − eV ) will then travel from the left to the
right and its energy decreases by eV on reaching the right side. An Andreev reflected
electron then propagates from the right to the left with amplitude aR2 (−ε− 2eV )A1
=JR(−ε)aL1 (−ε − eV )aR2 (−ε − 2eV ). At the left interface the electron energy will
again decrease by eV , and an Andreev reflected hole will be generated with ampli-
tude A R2 = J (−ε)aL1 (−ε− eV )aR2 (−ε− 2eV )aL3 (−ε− 3eV )....etc.
Clearly A1(ε, V ) = J
L(ε)aR1 (ε + eV ) and A1(−ε,−V ) = JR(−ε)aL1 (−ε − eV )
if the gaps are not equal, i.e.
A1(ε, V ) 6= −A∗1(−ε,−V ) (7.58)
because aR1 (ε+ eV ) is different than a
L
1 (−ε− eV ) because of the different gaps.
Therefore, to calculate the total current, we have to first calculate the current
IL from quasiparticles incident from the left by summing over all possible energies
and weighting them by the Fermi distribution. I then need to find the current
IR due to quasiparticles incident from the right superconductor by summing over
all possible energies and weighting with the Fermi distribution. I can then find
I = IL − IR, which is the net current that flows through the junction.
Let’s again start by finding the dc part. Starting from Eq. (7.43) and keeping
track of the left and right, I can write,
IDC = IL + IR
179
7.4. MAR IN ASYMMETRICAL JUNCTIONS
[ ∫
e ∞ ∣∣ ∣∣2 ∣∣ ∣∣2∣∣ ∣ ∣ ∣ ∣ ∣= dε[ JL(ε) + JL(ε) aR 2 2(ε+ eV )∣ + ∣JL(ε)∣ ∣aR(ε+ eV )∣2
2πh̄ 1 1∣∣ −∞ ∣2 ∣ ∣L 2∣ ∣2∣ ∣2
∣a2 (ε+ 2eV )
∣ + ∣JL∣ (ε)
∣ ∣aR(ε+ eV )∣ ∣aL1 2 (ε+ 2eV )∣
∣ R 2 1∫ a3 (ε+ 3eV )∣ ...]eε/kBT + 1∞ ∣ ∣2 ∣ ∣2∣ ∣2 ∣ ∣2∣ ∣− 2dε[∣JR(ε+ eV )∣ + ∣JR(ε+ eV )∣ ∣aL1 (ε)∣ + ∣JR(ε+ eV )∣ ∣aL1 (ε)∣
−∞∣∣ ∣R − ∣2 ∣∣ ∣R ∣2∣∣ ∣L] ∣2
∣ ∣2
∣a2 (ε eV ) ∣+ J (ε+ eV ) a1 (ε)
∣aR ∣2 (ε− eV )
∣ 2 1[a
L ∣
3∫(ε− 2eV ) ...]∣ ∣ e(ε+∣eV )/kBT∣ + 1e ∞ ∣ ∣2 ∣ 2∣ ∣2 ∣ ∣2∣ ∣2= dε[ JL(ε) + JL(ε)∣ ∣aR(ε+ eV )∣ + ∣JL(ε)∣ ∣aR(ε+ eV )∣
2πh̄ 1 1∣∣ −∞aL(ε+ 2eV )∣∣2 ∣∣ ∣2∣ ∣2∣ ∣2∣ 2 ∣ + J
L(ε)∣ ∣aR(ε+ eV )∣ ∣1 aL2 (ε+ 2eV )∣
∫ ∣ R ∣
2 1
a3 (ε+ 3eV ) ...]
∞+eV ∣ ∣ e∣ε/kBT∣ +∣ ∣1 ∣ ∣ ∣ ∣ ∣− dε[ JR 2(ε)∣ + ∣JR 2 2 2 2(ε)∣ ∣aL1 (ε− eV )∣ + ∣JR(ε)∣ ∣aL ∣1 (ε− eV )
−∞∣∣+eV ∣ ∣ ∣ ∣ ∣ ∣ ∣aR 2 2 2 2∣ 2 (ε− 2eV )
∣ + ∣JR(ε)∣ ∣aL(ε− eV )∣ ∣aR ∣1 2 (ε− 2eV )
∣ ∣ ]aL3 (ε− 2 13eV )∣ ...] (7.59)eε/kBT + 1
To proceed I now arrange things to make the integration limits look much
nicer. This will result in an expression with the same “format” as Averin found in
a paper on multi-channel MAR for the case of identical gaps [127]. Working with
the limits of integration, I can write:
[ ∫
e ∞
I = dε[∣∣ ∣L ∣2 ∣∣ ∣L ∣2∣ ∣2 ∣ ∣2∣ ∣2DC J (ε) + J (ε) ∣aR(ε+ eV )∣ + ∣JL(ε)∣ ∣aR(ε+ eV )∣
2πh̄ 1 1−∣∣∞−eV ∣2 ∣aL(ε+ 2eV )∣ + ∣JL(ε)∣∣2∣∣ ∣∣2∣R ∣ ∣∣L 22 a1 (ε+ eV ) a2 (ε+ 2eV )
180
7.4. MAR IN ASYMMETRICAL JUNCTIONS
∣∣ ∣R 2 1∫ a3 (ε+ 3eV )∣ ...] ε∞ ∣∣ ∣∣ e2 ∣
/kBT
∣ + 1 ∣∣2∣∣ ∣∣2 ∣∣ ∣∣2∣∣ ∣− 2dε[ JR(ε+ eV ) + JR(ε+ eV ) aL1 (ε) + JR(ε+ eV ) aL1 (ε)∣
−∞−eV ∣∣ ∣ ∣ ∣ ∣ ∣R − ∣2 ∣ R ∣2∣ L] ∣2a ∣
∣ ∣
R 2
∣ 2 (ε eV ) ∣+ J (ε+ eV ) a1 (ε) a2 (ε− eV )
∣
L
[ ∫ ∣a (ε− 2eV )∣
2 1
3 ...]
∣ ∣ e(ε+eV )/kBe ∞ 2 ∣ ∣
T∣+ 12
= dε[∣JL(ε)∣ + ∣JL(ε)∣ ∣aR(ε+ eV )∣∣2 ∣∣ ∣L 2∣ ∣+ J (ε)∣ ∣aR(ε+ eV )∣2
2πh̄ 1 1−∣∣∞−eV ∣2 ∣ ∣2∣ ∣2∣ ∣2
∣a
L
2 (ε+ 2eV )∣ ∣ L ∣ ∣∣ + J (ε) a
R ∣ ∣ L ∣
1 (ε+ eV ) a2 (ε+ 2eV )
∣ R ∣2 1∫ a3 (ε+ 3eV ) ...]∣ ε/kBT∞+eV
− dε[∣JR(ε)∣∣2 ∣∣
e ∣∣2∣∣ + 1 ∣R L 2 ∣+ J (ε) a (ε− eV )∣ + ∣JR(ε)∣∣2∣∣ ∣21 aL1 (ε− eV )∣
−∞ ∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣aR − ∣2 2 2 2∣ 2 (ε 2eV )∣ +
∣JR(ε)∣ ∣a]L(ε− eV )∣ ∣ R1 a2 (ε− 2eV )∣
2 1
{∫ ∣a
L
3 (ε− 3[eV )∣ ...]eε/kBT + 1
e ∞ ∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣L ∣2 ∣ L ∣2∣ R ∣2 ∣ L ∣2∣ R ∣2= dε J (ε) + J (ε) a1 (ε+ eV ) + J (ε) a1 (ε+ eV )2πh̄ −∣∣∞−eV ∣ ∣ ∣ ∣ ∣ ∣ ∣aL 2 2 2 2∣ 2 (ε+ 2eV )
∣
∣ +]
∣JL(ε)∣ ∣aR(ε+ eV )∣ ∣aL1 2 (ε+ 2eV )∣
R 2 1
∫ ∣a[3 (ε+ 3eV )∣ ... eε/kBT + 1∞ ∣
− dε ∣JR(−ε)∣∣2 ∣∣ ∣∣R − 2∣ ∣ ∣ ∣+ J ( ε) ∣aL 2 21 (−ε− eV )∣ + ∣JR(−ε)∣
−∞−eV ∣∣aL(−ε− eV )∣∣2∣∣ ∣ ∣ ∣ ∣ ∣R∣ 1 ∣ a∣2 (−ε−
2 2 2
2eV )∣ + ∣]JR(−ε)∣ ∣∣ a
L
1 (}−ε− eV )∣∣aR2 (− − 2 2 1ε 2eV )∣ ∣aL(−ε− 3eV )∣3 ... − (7.60)e ε/kBT + 1
Note that when the gap of the left and right electrodes are the same, the term in
square brackets in each integral is the same and the two integrals can be combined
such that the two Fermi functions produce an overall leading factor of 1
eε/k
−
BT+1
181
7.4. MAR IN ASYMMETRICAL JUNCTIONS
( )
1 ε
−ε/k T = − tanh . This yields an equation for the current that has thee B +1 2kBT
same overall form as Eq. (4) in Averin’s multi channel paper [127].
Unfortunately, when the electrodes have different gaps, a quasiparticle will
have different reflection amplitudes at the left and right S/N interfaces. In this
case I realized that it was necessary to explicitly consider the hole-like quasiparticle
current incident from the left and right, as well as the electron-like quasiparticle
incident from the left and right, a total of four separate processes.
Let’s now consider the situation when a hole-like quasiparticle is incident from
left superconducting electrode onto the right S/N interface and voltage V is applied
to the right electrode. Figure 7.5 illustrates this situation for a hole-like quasiparticle
incident above the Fermi energy with energy ε. In contrast to the case of an electron,
the Fermi level for the holes on the right electrode is shifted upwards; i.e. the
hole loses kinetic energy as it moves from left to right across the normal region.
Considering the D = 1 case, An can then be written as:
A−3 = A−2 = A−1 = A0 = 0 (7.61)
A1h(ε,−V ) = JL(ε)aR1 (ε− eV ) (7.62)
A L R2h(ε,−V ) = J (ε)a1 (ε− eV )aL2 (ε− 2eV )aR3 (ε− 3eV ) (7.63)
A3h(ε,−V ) = JL(ε)aR1 (ε− eV )aL2 (ε− 2eV )aR3 (ε− 3eV )aL4 (ε− 4eV )aR5 (ε− 5eV )
(7.64)
The wavefunction produced when a hole-like quasiparticle is incident from the left
can then be written as
182
7.4. MAR IN ASYMMETRICAL JUNCTIONS
S1 0 N V S2
J(ε)a1(ε-eV)a2(ε-2eV)a3(ε-3eV)
    a4(ε-4eV)a5(ε-5eV)=A3
J(ε)a1(ε-eV)a2(ε-2eV)a3(ε-3eV)
incident hole-like J(ε)       =A2
quasiparticle
|ε-eV|
|ε-4eV| ε EeV F
E |ε-3eV|F |ε-2eV| J(ε)a1(ε-eV)=A1
J(ε)a1(ε-eV)a2(ε-2eV) |ε-5eV|
      =a2A1
J(ε)a1(ε-eV)a2(ε-2eV)a3(ε-3eV)
               a4(ε-4eV)=a4A2
Figure 7.5: Illustration of multiple Andreev reflection for transparency
D = 1 for a hole-like quasiparticle incident from the left with energy ε
and voltage V applied to the right electrode while the left electrode is
grounded. The hole experiences repulsive force from the right electrode
and decreases its kinetic energy. This effectively shifts the Fermi energy
of the holes on the right electrode up by eV .
183
7.4. MAR IN ASYMMETRICAL JUNCTIONS
∑
ψ = [(a A + Jδ )e−ikx]e−i(ε−2neV )t/h̄h 2n n n0 (7.65)
n
∑
ψ −ikx −i(ε−2neV )t/h̄el = [Ane ]e (7.66)
n
The electrical current that is then produced by an incoming hole from the left, can
then be written as:
∫ (
e
ILh(ε, t) = − dε [|J |2 + |A |2(1 + |a |2) + |A |2(1 + |a |2) + |A |2(1 + |a |21 2 2 4 3 6 )
2πh̄
+ ...] + [Ja∗A∗ + A A∗(1 + a a∗) + A A∗(1 + a a∗) + ...]e−i2eV t/h̄2 1 1 2 2 4 2 3 4 6
+ [Ja∗A∗ + A A∗(1 + a a∗) + A A∗(1 + a a∗) + ...]e−i4eV t/h̄4 2 1 3 2 6 2 4 4 8 + [..]e
−i6eV t/h̄
+ ...+ [J∗a2A1 + A2A
∗
1(1 + a4a
∗) + A A∗ ∗ i2eV t/h̄2 3 2(1 + a6a4) + ...]e
+ [J)∗a A + A A∗4 2 3 1(1 + a ∗6a2) + A A∗4 2(1 + a a∗8 4) + ...]ei4eV t/h̄ + [...]ei6eV t/h̄
× 1+ ... (7.67)
eε/kBT + 1
I also need to consider the situation when there is a hole-like quasiparticle
incident from the right with energy ε− eV and voltage V is applied to the right. As
shown in Fig. 7.6, this time the hole increases its kinetic energy each time it travels
from the right to the left. I can then write for this process:
A−3 = A−2 = A−1 = A0 = 0 (7.68)
A1h(ε− eV, V ) = JR(ε− eV )aL1 (ε) (7.69)
184
7.4. MAR IN ASYMMETRICAL JUNCTIONS
S1 0 N V S2
J(ε-eV)a1(ε)a2(ε+eV)a3(ε+2eV)
               a4(ε+3eV)=a4A2
J(ε-eV)a1(ε)a2(ε+eV)
|ε+4eV|       =a2A1
|ε+2eV| J(ε-eV) incident hole-like
ε-eV quasiparticle
ε EeV F
EF |ε+eV|
|ε+3eV|
J(ε-eV)a1(ε)=A1 |ε+5eV|
J(ε-eV)a1(ε)a2(ε+eV)a3(ε+2eV)
      =A2
J(ε-eV)a1(ε)a2(ε+eV)a3(ε+2eV)
    a4(ε+3eV)a5(ε+4eV)=A3
Figure 7.6: Illustration of multiple Andreev reflection for a hole-like
quasiparticle incident with energy ε−eV from the right electrode, which
has voltage V applied. The hole increases its kinetic energy when it
moves from right to left.
185
7.4. MAR IN ASYMMETRICAL JUNCTIONS
A2h(ε− eV, V ) = JR(ε− eV )aL1 (ε)aR2 (ε+ eV )aL3 (ε+ 2eV ) (7.70)
A3h(ε− eV, V ) = JR(ε− eV )aL1 (ε)aR2 (ε+ eV )aL R L3 (ε+ 2eV )a4 (ε+ 3eV )a5 (ε+ 4eV )
(7.71)
The resulting wavefunction for the case of a hole incident from the right electrode,
then can be written as:
∑
ψh = [(a
ikx
2nAn + Jδn0)e ]e
−i(ε+2neV )t/h̄ (7.72)
n
∑
ψ = [A eikx]e−i(ε+2neV )t/h̄el n (7.73)
n
The electrical current produced by the hole incident from the right side is then:
∫ (
e
I 2Rh(ε, t) = dε [|J | + |A1|2(1 + |a |2) + |A |2(1 + |a |2) + |A |22 2 4 3 (1 + |a 26| )
2πh̄
+ ...] + [Ja∗A∗2 1 + A
∗
1A2(1 + a a
∗) + A A∗(1 + a a∗) + ...]ei2eV t/h̄2 4 2 3 4 6
+ [Ja∗A∗4 2 + A
∗
1A3(1 + a a
∗
2 6) + A2A
∗
4(1 + a4a
∗
8) + ...]e
i4eV t/h̄ + [...]ei6eV t/h̄ + ...
+ [J∗a2A1 + A2A
∗
1(1 + a4a
∗ ∗
2) + A3A2(1 + a6a
∗
4) + ...]e
−i2eV t/h̄
+ [J∗a4A2 + A A
∗
3 1()1 + a ∗ ∗ ∗ −i4eV t/h̄6a2) + A4A2(1 + a8a4) + ...]e
+ [...]e−i6eV t/h̄
1
+ ... × (7.74)
e(ε−µR)/kBT + 1
where the chemical potential µR of the right electrode is −eV .
The total dc current due to hole-like quasiparticles incoming from the left and
right can now be expressed as:
[ ∫
e ∞ ∣ ∣2 ∣ ∣− 2∣ ∣2 ∣ ∣2∣ ∣2IDCh = dε[∣JL(ε)∣ + ∣JL(ε)∣ ∣aR(ε− eV )∣ + ∣JL(ε)∣ ∣aR1 1 (ε− eV )∣2πh̄ −∞
186
7.4. MAR IN ASYMMETRICAL JUNCTIONS
∣∣ ∣2 ∣aL(ε− 2eV )∣ + ∣JL(ε)∣∣2∣∣ ∣R − ∣2∣ ∣a (ε eV ) ∣aL∣ 2 ∣ 1 2 (ε−
2
2eV )∣
2 1
∫ ∣aR3 (ε− 3eV )∣ ...]∞ ∣∣ ∣∣2 ∣
ε/kBT
+ dε[ JR(ε− eV ) + ∣
e + 1
JR(ε− eV )∣∣2∣∣ ∣ ∣ ∣ ∣ ∣aL 2 2 21 (ε)∣ + ∣JR(ε− eV )∣ ∣aL1 (ε)∣
−∞ ∣∣ ∣2 ∣ ∣2∣ ∣2∣ ∣2
∣a
R
2 (ε+ eV )∣ + ∣JR(ε− eV )∣ ∣a]L∣ 1 (ε)
∣ ∣aR2 (ε+ eV )∣
L
[ ∣∫a (ε+ 2eV )∣
2 1
3 ...]
∣ ∣ e(ε−eV )/kBT + 1e ∞
= − dε[∣JL(ε)∣2 ∣∣ ∣∣L 2∣ ∣ ∣+ J (ε) ∣aR 2(ε− eV )∣ + ∣JL(ε)∣∣2∣∣ ∣R 21 a1 (ε− eV )∣2πh̄
∣∣ −∞aL(ε− 2eV )∣∣2 ∣∣ ∣∣2∣ ∣L ∣ R ∣2∣∣ ∣L 2∣ 2 ∣ + J (ε) a1 (ε− eV ) a2 (ε− 2eV )
∣
∫ ∣
2 1
aR3 (ε− 3eV )∣ ...]∣ ∣ ∣ eε/kBT + 1∞−eV 2 ∣2∣ ∣2 ∣ ∣2∣ ∣2
+ dε[∣JR(ε)∣ + ∣JR(ε)∣ ∣aL(ε+ eV )∣ + ∣JR(ε)∣ ∣ L1 a1 (ε+ eV )∣
−∞−eV ∣∣ ∣2 ∣ ∣2∣ ∣2∣ ∣2
∣a
R
2 (ε+ 2eV )∣ ∣∣ + J
R(ε)∣ ∣a]L1 (ε+ eV )∣ ∣aR2 (ε+ 2eV )∣∣ L ∣2 1a3 (ε+ 3eV ) ...] (7.75)eε/kBT + 1
To summarize where we are in the calculation, I now have Eq. (7.59) for the
current due to electron-like quasiparticles incident from the left and right. I also
have the current due to hole-like quasiparticles incident from the left and right
superconductors Eq. (7.75). Adding the electron-like quasiparticle current and the
hole-like quasiparticle current, I get the total dc current.
IDC = IDCe[+∫IDCh
e ∞
= dε[∣∣ ∣2 ∣ ∣2∣ ∣2 ∣ ∣2∣ ∣2JL(ε)∣ + ∣JL(ε)∣ ∣aR(ε+ eV )∣ + ∣JL(ε)∣ ∣ R ∣
2πh̄ 1
a1 (ε+ eV )
−∞
187
7.4. MAR IN ASYMMETRICAL JUNCTIONS
∣∣ ∣aL 2(ε+ 2eV )∣ + ∣∣ ∣∣2∣∣ ∣∣2∣ ∣JL R ∣ L 2∣ 2 ∣ (ε) a1 (ε+ eV ) a2 (ε+ 2eV )
∣
∫ ∣
2 1
aR3 (ε+ 3eV )∣ ...]∣ ∣ eε/kBT∞+eV
− 2dε[∣JR(ε)∣ + ∣∣
+
JR(ε)∣∣2∣∣
1 ∣2 ∣ ∣2∣ ∣2
aL1 (ε− eV )∣ + ∣JR(ε)∣ ∣aL1 (ε− eV )∣
−∞∣∣+eV ∣∣2 ∣∣ ∣∣R − R 2∣∣ ∣ ∣ ∣∣a2 (ε 2eV )∣ + J (ε) a
L
1 (ε−
2
eV )∣ ∣aR 22 (ε− 2eV )∣
∫ ∣aL(ε− 3eV )∣
2 1
3 ...]∣ ∣ ∣ eε/k∞ 2 ∣BT + 12∣ ∣2 ∣ ∣2∣ ∣− 2dε[∣JL(ε)∣ + ∣JL(ε)∣ ∣aR1 (ε− eV )∣ + ∣JL(ε)∣ ∣aR1 (ε− eV )∣
−∞∣∣ ∣2 ∣ ∣2∣ ∣2∣ ∣2
∣a
L
2 (ε− 2eV )∣ + ∣ L∣ J (ε)
∣ ∣aR1 (ε− eV )∣ ∣aL2 (ε− 2eV )∣
2 1
∫ ∣aR3 (ε− 3eV )∣ ...] ε/kBT∞−eV
+ dε[∣∣ ∣JR(ε)∣
e
2 ∣∣ +∣2∣1 ∣2 ∣ ∣2∣ ∣2+ JR(ε)∣ ∣aL1 (ε+ eV )∣ + ∣JR(ε)∣ ∣aL1 (ε+ eV )∣
−∞∣∣−eV ∣∣2 ∣∣ ∣R R ∣2a (ε+ 2eV ) + J (ε) ∣∣ ∣2∣ ∣a]L(ε+ eV )∣ ∣aR(ε+ 2eV )∣2∣ 2 ∣ 1 2∣ L ∣2 1a3 (ε+ 3eV ) ...] (7.76)eε/kBT + 1
Regrouping terms and changing variables, I find:
[ ∫
e ∞ ∣∣ ∣∣2 ∣∣ ∣∣2∣ ∣2 ∣ ∣2∣ ∣2I = dε[ JL(ε) + JL(ε) ∣aR(ε+ eV )∣ + ∣DC 1 JL(ε)∣ ∣aR(ε+ eV )∣2πh̄ 1∣∣ −∞ ∣2 ∣ ∣2∣ ∣2∣ ∣2
∣a
L
2 (ε+ 2eV )∣ + ∣JL(ε)∣ ∣aR1 (ε+ eV )∣ ∣aL2 (ε+ 2eV )∣
∫ ∣
∣
R 2 1a3 (ε+ 3eV )∣ ...]
∣∣ ∣∣ ∣ e
ε/kBT
∣ +∞ 2 ∣∣2∣1 ∣ ∣ ∣ ∣ ∣− dε[ JL(−ε) + JL(−ε) ∣aR ∣2 ∣ L 2 21 (−ε− eV ) + J (−ε)∣ ∣aR(−ε− eV )∣1
−∞∣∣aL(−ε− 2eV )∣∣2 ∣ ∣+ ∣JL(− 2ε)∣ ∣∣ ∣ ∣ ∣R∣ 2 ∣ a1 (−ε−
2 2
eV )∣ ∣aL(−ε− 2eV )∣2
∣ R − − 2 1a3 ( ε 3eV )∣ ...]e−ε/kBT + 1
188
7.4. MAR IN ASYMMETRICAL JUNCTIONS
∫ ∞+eV ∣∣ ∣∣2 ∣∣ ∣∣2∣∣ ∣2 ∣ ∣2∣ ∣− 2dε[ JR(ε) + JR(ε) aL1 (ε− eV )∣ + ∣JR(ε)∣ ∣aL1 (ε− eV )∣
−∞∣∣+eV ∣∣2 ∣∣ ∣∣2∣∣ ∣2∣ ∣R R L 2
∣a2 (ε− 2eV )∣ + J (ε) a1 (ε− eV )
∣ ∣aR2 (ε− 2eV )∣
∣ L − ∣2 1∫ a3 (ε 3eV ) ...]eε/kBT + 1∞+eV ∣ ∣2 ∣ ∣2∣ ∣2 ∣ ∣2
+ dε[∣JR(−ε)∣ + ∣JR(−ε)∣ ∣aL1 (−ε+ eV )∣ + ∣JR(−ε)∣
−∞∣∣+eV ∣2∣ ∣2 ∣ ∣2∣ ∣2
∣a
L
1 (−ε+ eV )∣ ∣aR∣ ∣2 (−ε+ 2eV )
∣ + ∣JR(−ε)∣ ∣ L∣ a1](−ε+ eV )
∣
∣ R − ∣2 2 1a2 ( ε+ 2eV ) ∣aL3 (−ε+ 3eV )∣ ...]e− (7.77)ε/kBT + 1
Additional simplification gives:
[ ∫
e ∞ ∣
I = dε[∣ ∣2 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣JL(ε)∣ + ∣JL ∣2∣ R 2 2(ε) a (ε+ eV )∣ + ∣JL(ε)∣ ∣aR ∣2DC 1 1 (ε+ eV )2πh̄∣∣ −∞ ∣aL 2(ε+ 2eV )∣ + ∣∣ ∣∣L 2∣∣ ∣2∣ ∣2∣ 2 ∣ J (ε) a
R
1 (ε+ eV )∣ ∣aL2 (ε+ 2eV )∣
∣ R ∣2 1 1∫ a3 (ε+ 3eV ) ...]( − )∣ ∣ ε/kBT −ε/kBT∞+eV
− 2dε[∣JR(ε)∣ + ∣∣
e ∣∣+2∣1 e ∣ + 1∣ ∣ ∣ ∣JR(ε) ∣aL(ε− eV )∣2 ∣ R 2 21 + J (ε)∣ ∣aL(ε− eV )∣1
−∞∣∣+eV ∣2 ∣ ∣2∣ ∣2∣ ∣2
∣a
R
2 (ε− 2eV )∣ + ∣JR(ε)∣ ∣aL(ε− eV )∣ ∣aR1 2 (ε− 2eV )∣
∣ ∣L − ∣2 1 1[a3∫(ε 3eV ) ...]( − )∣ ∣ eε/∣kBT + 1 e−ε/kBT + 1e ∞ 2 ∣2∣ ∣2 ∣ ∣2∣ ∣2
= dε[∣JL(ε)∣ + ∣JL(ε)∣ ∣aR1 (ε+ eV )∣ + ∣JL(ε)∣ ∣aR1 (ε+ eV )∣2πh̄
∣∣ −∞ ∣∣2 ∣∣ ∣∣2∣∣ ∣∣L L R 2∣ ∣a (ε+ 2eV ) + J (ε) a (ε+ eV ) ∣aL ∣2∣ 2∣ ∣
1 2 (ε+ 2eV )
2 1 1
∫ aR3 (ε+ 3eV )∣ ...]( − )∣ ∣ ∣ eε/∣kB∣T + 1 e−ε∞ ∣/kBT ∣+ 1 ∣ ∣ ∣
− dε[∣ 2JR(ε)∣ + ∣JR(ε)∣2∣aL(ε− eV )∣2 + ∣JR 2 21 (ε)∣ ∣aL1 (ε− eV )∣
−∞∣∣aR(ε− 2eV )∣∣2 ∣ ∣2∣ ∣2∣ ∣22 + ∣JR(ε)∣ ∣aL ∣ ∣ R ∣1 (ε− eV ) a2 (ε− 2eV )
189
7.4. MAR IN ASYMMETRICAL JUNCTIONS
∣
∫ ∣
∣
aL(ε− 3eV )∣2 1 13 ...]( − )∣∣ ∣∣ e
ε/kBT + 1 e−ε/kBT + 1
∞+eV
− R 2 1∫dε J (ε) ( −
1
)
eε/kBT + 1 e−ε/k T∞ B + 1
−∞+eV ∣∣ ∣R ∣ ]2 1 1+ dε J (ε) ( − ) (7.78)
eε/kBT + 1 e−ε/k T−∞ B + 1
√
Since JR(ε) = 1 ∫− | 2aR0 (ε)| and |aR| → 0 as ε → ∞, the second to lastL+eV 1
integral reduces to − lim dε(− − ) = eV . Similarly the last integralL→∞ e ε/kBTL + 1
also reduces to eV . Therefore the dc current for the case of D = 1 and different
gaps case can be written as:
[ ∫ ∞ ( )e ε ∣ ∣2 ∣ ∣2∣ ∣2
IDC = 2eV − dε tanh [∣JL(ε)∣ + ∣JL(ε)∣ ∣aR1 (ε+ eV )∣2πh̄ 2kBT
∣∣ ∣ ∣ −∞2 ∣2∣ ∣2 ∣ ∣2∣ ∣2+ ∣J
L(ε)∣ ∣aR(ε+ eV )∣ ∣aL(ε+ 2eV )∣ + ∣JL(ε)∣ ∣ R ∣1 ∣ ∣ 22 ∣
a1 (ε+ eV )
2
+ ∣aL ∣ ∣ R ∣2∫(ε+ 2eV ) a3 (ε+ 3eV ) ...]∞ ( )ε ∣ ∣2 ∣ ∣2∣ ∣2 ∣ ∣2
+ dε tanh [∣JR(ε)∣ + ∣JR(ε)∣ ∣aL1 (ε− eV )∣ + ∣JR(ε)∣
∣∣ −∞ ∣
2kBT
L − ∣2∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣R ] − ∣2 ∣ R ∣2∣ L − ∣2∣ R − ∣2∣a1 (ε eV ) ∣ a2 (ε 2eV ) + J (ε) a1 (ε eV ) a2 (ε 2eV )∣aL3 (ε− 23eV )∣ ...] (7.79)
To check this expression, if ∆1 = ∆2 then Eq. (7.79) becomes
[ ∫
e ∞
( )
I = 2eV − εdε tanh [|J(ε)|2DC + |J(ε)|2|a1(ε+ eV )|2 + |J(ε)|2
2πh̄ −∞ 2kBT
|a1(ε+ eV )|2|a2(ε+ 2eV )|2 + |J(ε)|2|a1(ε+ eV )|2
|a2(ε+ 2eV )|2|a3(ε+ 3eV )|2...]
190
7.4. MAR IN ASYMMETRICAL JUNCTIONS
∫ ∞ ( )ε
+ dε tanh [|J(ε)|2 + |J(ε)|2|a1(ε− eV )|2 + |J(ε)|2
−∞ 2kBT
|a1(ε− eV )|2|a2(ε]− 2eV )|2 + |J(ε)|2|a1(ε− eV )|2|a2(ε− 2eV )|2
|[a3(ε− 3∫eV )|
2...]
∞ ( )e ε
= 2eV − dε tanh [|J(ε)|2|a (ε+ eV )|21 + |J(ε)|2
2πh̄ −∞ 2kBT
|a (ε+ eV )|2|a (ε+ 2eV )|21 2 + |J(ε)|2|a1(ε+ eV )|2
|∫a2(ε+ 2eV )(|2|a (ε+ 3)eV )|23 ...]∞ ε
+ dε tanh − [|J(−ε)|2|a1(−ε− eV )|2 + |J(−ε)|2
−∞ 2kBT
|a1(−ε− eV )|2|a2(−ε− 2eV )|2 + |J](−ε)|2|a1(−ε− eV )|2
|[a2(−ε− 2eV )|
2
∫ |a3(−ε(− 3eV )
2
)| ...]
e ∞− ε= 2eV 2 dε tanh [|J(ε)|2|a1(ε+ eV )|2 + |J(ε)|2
2πh̄ −∞ 2kBT
|a (ε+ eV )|2|a (ε+ 2eV )|2 + |J](ε)|21 2 |a1(ε+ eV )|2
+[|a2(ε+∫ 2eV )|
2|a 23(ε(+ 3eV))| ...]
e ∞ ε
= eV − dε tanh [|J(ε)|2|a1(ε+ eV )|2 + |J(ε)|2
πh̄ −∞ 2kBT
|a1(ε+ eV )|2|a2(ε+ 2eV )|2 + |J](ε)|2|a 21(ε+ eV )|
+ |a2(ε+ 2eV )|2|a3(ε+ 3eV )|2...] (7.80)
which is the same as Eq. (7.43) for the case D = 1 except Eq. (7.80) is twice as
large because we included both hole-like quasiparticles incident from the left and
right and electron-like quasiparticle incident from the left and right.
I can now generalize to the case of two different gaps and transparency D not
191
7.4. MAR IN ASYMMETRICAL JUNCTIONS
restricted to one. Consider for example the k = 1 Fourier component which gives
the ei2eV t/h̄ terms. To make the source of each contribution clearer, I use subscripts
R and L to label contributions that are from the right (R) and left (L) interfaces or
electrodes. Thus I find from:
∫ ∞
I i2eV t/h̄
e ∗ ∗ ∗
AC = e dε[JL(ε)a2L(ε, V )A1L(ε, V ) + A1L(ε, V )A2L(ε, V )2πh̄ −∞
(1 + a ∗ ∗2L(ε, V )a4L(ε, V )) + A2L(ε, V )A3L(ε, V )
1
(1 + a4L∫(ε, V )a∗6L(ε, V )) + ...]eε/kBT + 1
− e
∞
ei2eV t/h̄ dε[J∗R(ε)a2R(ε,−V )A1R(ε,−V ) + A∗2πh̄ 1R
(ε,−V )A2R(ε,−V )
−∞
(1 + a∗2R(ε,−V )a4R(ε,−V )) + A∗2R(ε,−V )A3R(ε,−V )
(1 + a∗
1
4R∫(ε,−V )a6R(ε,−V ) + ...]eε/kBT + 1
e ∞− ei2eV t/h̄ dε[[J∗L(ε)a2L(ε,−V )A1L(ε,−V ) + A2L(ε,−V )A∗1L(ε,−V )2πh̄ −∞
(1 + a ∗4L(ε,−V )a2L(ε,−V )) + A (ε,−V )A∗3L 2L(ε,−V )
1
(1 + a6L∫(ε,−V )a∗4L(ε,−V )) + ...]eε/kBT + 1
e ∞
+ ei2eV t/h̄ dε[J (ε)a∗ (ε, V )A∗R 2R 1R(ε, V ) + A
∗
1R(ε, V )A2R(ε, V )2πh̄ −∞
(1 + a2R(ε, V )a
∗
4R(ε, V )) + A2R(ε, V )A
∗
3R(ε, V )
1
(1 + a4R(ε, V )a
∗
6R(ε, V )) + ...] (7.81)eε/kBT + 1
The first term in Eq. (7.81) comes from electron-like quasiparticles that are incident
from the left electrode. The second term is due to electron-like quasiparticles that
are incident from the right electrode. The third term is due to hole-like quasipar-
192
7.4. MAR IN ASYMMETRICAL JUNCTIONS
ticles that are incident from the left electrode and the last term is due to hole-like
quasiparticles that are incident from the right electrode.
Examining the first term and the third term, we will show that they can
be combined. Similarly, we will see that the second and the fourth terms can be
combined. To see this, I change the integration variable from ε to -ε in the third
and the fourth terms, and after some rearranging so that the two L terms are first
and the R terms second:
[ ∫ ∞
I = ei2eV t/h̄
e ∗
AC dε[JL(ε)a2L(ε, V )A
∗ ∗
1L(ε, V ) + A1L(ε, V )A (ε, V )2πh̄ 2L−∞
(1 + a2L(ε, V )a
∗
4L(ε, V )) + A2L(ε, V )A
∗
3L(ε, V )
1
∫ (1 + a4L(ε, V )a∗6L(ε, V )) + ...]eε/kBT + 1∞
− dε[[J∗L(−ε)a2L(−ε,−V )A1L(−ε,−V ) + A2L(−ε,−V )A∗1L(−ε,−V )
−∞
(1 + a4L(−ε,−V )a∗2L(−ε,−V )) + A3L(−ε,−V )A∗2L(−ε,−V )
1
∫ (1 + a6L(−ε,−V )a∗4L(−ε,−V )) + ...]e−ε/kBT + 1∞
− dε[J∗R(ε)a2R(ε,−V )A1R(ε,−V ) + A∗1R(ε,−V )A2R(ε,−V )
−∞
(1 + a∗2R(ε,−V )a4R(ε,−V )) + A∗2R(ε,−V )A3R(ε,−V )
1
∫ (1 + a∗4R(ε,−V )a6R(ε,−V )) + ...]eε/kBT + 1∞
+ dε[JR(−ε)a∗2R(−ε, V )A∗1R(−ε, V ) + A1R(−ε, V )A∗2R(−ε, V )
−∞
(1 + a2R(−ε, V )a∗4R(−ε, V )) + A2R(−ε, V )A∗3R](−ε, V )
(1 + a4R(−ε, V )a∗6R(−
1
ε, V )) + ...] (7.82)
e−ε/kBT + 1
193
7.4. MAR IN ASYMMETRICAL JUNCTIONS
I can now use Eq. (7.39) and Eq. (7.42) to simplify Eq. (7.82) and arrive at
[ ∫
i2eV t/h̄ e
∞
IAC = e − dε[JL(ε)a∗ ∗ ∗2L(ε, V )A1L(ε, V ) + A1L(ε, V )A2L(ε, V )2πh̄ −∞
(1 + a2L(ε, V )a
∗
4L(ε, V )) + A2L(ε, V )A
∗
3L(ε, V )
ε
∫ (1 + a (ε, V )a
∗
4L 6L(ε, V )) + ...] tanh ( )2kBT
∞
+ dε[J∗R(ε)a2R(ε,−V )A1R(ε,−V ) + A∗1R(ε,−V )A2R(ε,−V )
−∞
(1 + a∗2R(ε,−V )a4R(ε,−V )) + A∗2R(ε,−V )A3R(]ε,−V )
ε
(1 + a∗4R(ε,−V )a6R(ε,−V )) + ...] tanh ( ) (7.83)2kBT
Finally take all the Fourier components and combine them with the dc term, we get
the final current expression for the case of different gaps:
∑
I(t) = I ei2keV t/h̄k (7.84)
k
where the k-th Fourier component is:
[ ∫ ∞ ( )e ε (
I = 2eV Dδ − dε tanh JL ∗(L) ∗(L) (L) (L)k k0 (ε)(a
2πh̄ ∑ 2k T 2k
Ak + a−2kA−k )
−∞ B
(L) ∗(L) )
+ (1 + a a )(A(L)
∗(L)
A −B(L) ∗(L)
∫ ( )2n 2(n+k) n n+k n
Bn+k )
n
∞ ε ( R (R) (R) ∗(R) ∗(R)+ dε tanh J (ε)(a2k Ak + a∑ 2k T −2k
A−k )
−∞ B ]
∗(R) (R) ∗(R) (R) ∗(R) (R)+ (1 + a2n a2(n+k))(An An+k −Bn Bn+k) . (7.85)
n
(L) (L)
In this expression a Right Leftm−>odd = a (ε+meV ) and am−>even = a (ε+meV ).
194
7.4. MAR IN ASYMMETRICAL JUNCTIONS
Note that for a quasiparticle incident from the left, the Andreev reflection amplitude
depends on the interface it hits. If it is an odd number subscript, then the Andreev
reflection amplitude is for the right superconducting gap (∆R). On the contrary, for
an even number, the Andreev reflection amplitude is for the left superconducting gap
(R)
(∆L). Similarly for a quasiparticle incident from the right, a
Left
m−>odd = a (ε−meV )
(R)
and a = aRightm−>even (ε −meV ); the odd indices are for the left superconducting
gap (∆L) while the even indices use the right superconducting gap (∆R). Thus:

1 

 ε− sgn(ε)(ε
2 −∆2 )1/2L , |ε| > ∆L
aLeft(ε) = × (7.86)∆L ε− i(∆2 − ε2)1/2L , |ε| < ∆L
while 
1  ε− sgn(ε)(ε
2 −∆2 )1/2R , |ε| > ∆R
aRight(ε) = × (7.87)∆R ε− i(∆2 2 1/2R − ε ) , |ε| < ∆R.
To calculate the I − V characteristics from Eq. (7.57) for the asymmetrical
junction with the MAR effect, one needs to use the definition of a(ε) from Eq. (7.86)
and Eq. (7.87) to plug into the recursion relation Eq. (7.18) and Eq. (7.13) to get
(R) (L) (R) (L)
An , An , Bn and Bn . However keep in mind that the two electrodes have
different superconducting gap, thus we need to be careful of which Andreev reflection
amplitude to use. For example, the recursion relation for an incoming electron
incident from the left can be expressed as:
√
(L)
A − aR L (L) L (L) R (L) Rn+1 2n+1a2nAn = R(a2n+2Bn+1 − a2n+1Bn ) + Ja1 δn0 (7.88)
195
7.4. MAR IN ASYMMETRICAL JUNCTIONS
and
[ ( ) ]
aR aL (aR 2 L 2
D 2n+1 2n+2
(L)
B − D 2n+1
) (a )
+ 2n + 1− (aL )2 B(L)
1− (aL )2 n+1 1− (aR )2 1− (aR )2 2n n2n+1 2n+1 2n−1
aL R √
+D 2n
a2n−1 (L)Bn−1 = − RJδn,0 (7.89)1− aR2n−1
(L) (L) (R) (R)
we then obtain An and Bn terms. Similarly, I can also get An and Bn . I
plug them into Eq. (7.85) and numerically integrate the equation to get the cur-
rent. For this calculation, I modified the Fortran code provided from Averin for the
symmetrical MAR effect case (see Appendix D).
Figure 7.7 shows an example of the dc I−V characteristics of an asymmetrical
S-I-S junction calculated using Eq. (7.85) for different values of D and for T = 0.
Here I set the ratio of the two superconducting gaps to ∆R/∆L = 2 so that the
sum of the gaps is ∆L + ∆R = 1.5∆R. As expected, the I − V shows a sudden rise
at V = 1.5∆R/e. For D = 0.4, for example, we see clear subgap current steps at
V = ∆R/e and V = ∆L/e.
I also used Eq. (7.85) to examine what happens to the I(V) characteristics
when the temperature is not at T = 0 K. Figure 7.8 shows examples of the dc I−V
characteristics calculated using Eq. (7.85) at different temperatures. In this case,
I set the transparency parameter to D=0.25 and used a superconducting gap ratio
∆R/∆L=1/0.47. Notice that when the temperature is higher, the current increases
in the subgap region. This happens because there are more quasiparticles present
to generate MAR. Examining Fig. 7.8, one also sees that the subgap steps become
more rounded for higher temperatures.
196
7.4. MAR IN ASYMMETRICAL JUNCTIONS
10
8 D
0.01
6 0.1
0.4
4 0.7
0.9
0.99
2 1
0
0 0.5 1 1.5 2 2.5 3
eV/ΔR
Figure 7.7: Calculated I − V characteristics of an asymmetrical S-I-S
junction with MAR at T = 0. The ratio of the two superconducting
gaps is ∆R/∆L = 2 and the different curves correspond to D value of 1,
0.99, 0.9, 0.7, 0.4, 0.1. and 0.01 from top to bottom.
197
I/(ΔRGT /2e)
7.4. MAR IN ASYMMETRICAL JUNCTIONS
6
5 kBT/ΔR
4 0.005
0.1
3
0.2
2 0.3
1 1
0
0 0.5 1 1.5 2 2.5 3
eV/ΔR
Figure 7.8: Calculated I − V characteristics for transparency D = 0.25
and ∆R/∆L = 1/0.47. The different curves correspond to kBT/∆R val-
ues of 1, 0.3, 0.2, 0.1 and 0.005 from top to bottom.
198
I/(ΔRGT /2e)
7.4. MAR IN ASYMMETRICAL JUNCTIONS
Another interesting effect that we can examine by a trivial modification of
Eq. (7.85) is what happens to the I(V) curves when the big gap superconductor
and the small gap superconductor have different temperatures. Asymmetric junc-
tions have been used as electronic refrigerators [128–130] and thermal transport and
heating effects can be important in small junctions. For example, in the supercon-
ducting STM, since the STM tip sits on the end of a long thin superconducting wire,
it may have more difficulty dissipating heat and reach a higher temperature than a
superconducting sample when the junction is biased at a non-zero voltage. I also
show in Chapter 5 how local heating effects seamed to be important.
Figure 7.9 shows four I − V characteristics for four different scenarios. For all
the curves I set the superconducting gap ratio to ∆R/∆L=1/0.47, the transparency
to D = 0.1 and ∆R = 1 meV. The red points are for TL = TR = 0.06 K. The black
points are TL = 0.06 K and TR = 2.3 K. The blue points are for TL = TR = 2.3 K
and green diamonds are for TL = 2.3 K and TR = 0.06 K. Comparing the red and
the black curves, we see that the characteristics seem not to be affected. This is not
surprising because ∆R  kBT and ∆L  kBT . In contrast, comparing the green
curve to red or the black curve in Fig. 7.9, we see the current is somewhat higher
in the subgap region for the green curve, causing the I − V characteristic curve to
be shifted upwards. This suggests that the characteristics are more sensitive to the
temperature of the electrode with a smaller superconducting gap, which is what we
would have expected since more quasiparticles will be excited in this case. Finally,
comparing the blue and green curves, we see the I−V characteristics are essentially
the same, even though the temperature of the high gap side is different. Again, this
199
7.5. THE JOSEPHSON EFFECT
is consistent with the characteristics being less sensitive to the temperature of the
electrode with the larger superconducting gap. In principle this behavior can be
used to determine the temperature of each electrode in the junction.
7.5 The Josephson effect
Electrons in a superconductor form Cooper pairs with charge 2e below the
transition temperature Tc. The ground state of a superconductor system can be
√
described by a single macroscopic wave function Φ = | N | exp(iφ) where the phase
φ is well defined and N is the number density of pairs. A Josephson junction is
formed when two superconducting pieces are separated by a thin insulating layer.
Ignoring quasiparticles and assuming that the barrier is sufficiently high so that
we can ignore MAR, many of the electrical properties of such a junction can be
understood by using simple quantum mechanics [131].
Following a discussion in Feynman’s Lecture on Physics, we can write the
wavefunction of the pair as ∑2
Ψ = Cαψα (7.90)
α=1
where ψ1 is non-zero only in superconductor 1 and ψ2 is non-zero only in supercon-
ductor 2. If the two superconductors are uncoupled, then we can write Schrödinger’s
equation for each superconductor as:
∂ψα
ih̄ = Eαψα (7.91)
∂t
200
7.5. THE JOSEPHSON EFFECT
4 1.0
0.8
3 0.6
0.4
0.2
2 0.0
0.5 eV/ 1.0
 R
1
0
0.0 0.5 1.0 1.5 2.0
eV/
 R
Figure 7.9: Calculated I − V characteristics for different electrode tem-
peratures with transparency D =0.1, ∆R = 1 meV and ∆L = 0.47 meV.
The open red triangles have TL = TR = 0.06 K. The solid black squares
lie on the red points but have TL = 0.06 K and TR = 2.3 K. The open
green diamonds have TR = 0.06 K and TL = 2.3 K. The blue points have
TL = TR = 2.3 K and lie on top of the green diamonds.
201
I /(
 RGT /2e)
I /(
 RGT /2e)
7.5. THE JOSEPHSON EFFECT
Here Eα (for α = 1, 2) are the energies of the uncoupled superconductor 1 and 2.
If the superconductors are now joined by a thin insulating region and the
insulator thickness is small enough, the wavefunctions from the two superconductors
can overlap with each other, and the wavefunction of the coupled superconductors
should satisfy the Schrödinger equation
∂Ψα
ih̄ = ĤΨα (7.92)
∂t
where
 E1 + e∗V/2 −K 
H =   (7.93)
−K E2 − e∗V/2
is the total Hamiltonian. The diagonal elements H11 = E1 + e
∗V/2 and H2 =
E − e∗2 V/2 correspond to the energies of state 1 and 2, where e∗ = −2e is the
charge of a Cooper pair. The off-diagonal elements H12 = H21 = −K describe
couplings between the two superconductors. Substituting Eq. (7.90) and Eq. (7.93)
into Eq. (7.92), we have
 ∂C1ih̄ = eV C1(t)−KC2(t) ∂t (7.94)∂C2ih̄ = −KC1(t)− eV C2(t)
∂t
√ √
I now let C1 = N1e
iφ1 , C2 = N2e
iφ2 where N1 and N2 are the number density of
Cooper pairs in superconductor 1 and 2. Separating the real and imaginary parts,
we find equations for the N1 part,
202
7.5. THE JOSEPHSON EFFECT
dN √1
h̄ = −2K N1N2 sin(φ2 − φ1) (7.95)
dt
or:
dφ √1
h̄N1 = −eV N1 +K N1N2 cos(φ2 − φ1). (7.96)
dt
Similarly we have for the N2 part,
dN √2
h̄ = 2K N1N2 sin(φ2 − φ1) (7.97)
dt
or:
dφ √2
h̄N2 = eV N2 +K N1N2 cos(φ2 − φ1) (7.98)
dt
Adding Eq. (7.95) and Eq. (7.97) we find d(N1 +N2)/dt=0 and N1 +N2 must
be constant, as required by charge conservation. If we multiply Eq. (7.95) and
Eq. (7.97) by the charge −2e, I find
− dN1 K
√
2e = 4e N1N2 sin(φ2 − φ1) (7.99)
dt h̄
dN √2 K
2e = 4e N1N2 sin(φ2 − φ1) (7.100)
dt h̄
Examining Eq. (7.99) and Eq. (7.100) we see that they have the same form as the
dc Josephson equation [30, 31],
Is = Ic sinφ (7.101)
where φ = φ1−φ2 is the phase difference between the two electrodes, Is = dQ/dt =
203
7.5. THE JOSEPHSON EFFECT
2e(dN2/dt) = −2e(dN1/dt) is the supercurrent that flows between the two super-
√
conductors and the critical current of the junction Ic = 4eK N1N2/h̄.
Next consider Eq. (7.96) and Eq. (7.98). Since the individual phases should
play no physical role, we can set φ2 = φ/2. φ1 = −φ/2. Equation (7.96) and
Eq. (7.98) becomes,
 dφ √h̄N1 = eV N1 −K√N1N2 cos(φ2 − φ1) dt (7.102)dφh̄N2 = eV N2 +K N1N2 cos(φ2 − φ1).
dt
Adding these two equations, we get the ac Josephson equation [30, 31],
h̄(N1 +N2) dφ
= eV (N1 +N2), (7.103)
2 dt
which reduces to
dφ 2eV
= (7.104)
dt h̄
Thus the voltage drop V across the junction is related to the rate at which the phase
difference φ changes.
204
7.6. CRITICAL CURRENT IN ASYMMETRICAL JUNCTIONS WITH MAR
7.6 Critical current in Asymmetrical junctions with MAR
In the previous section, I presented a simple discussion of the critical current
of a Josephson junction in the limit of small barrier transparency. However, it is
not obvious from that discussion exactly how the critical current is related to the
superconducting gap. In the next chapter I discuss my STM setup with a supercon-
ducting Nb tip and a superconducting Nb sample, which formed a Josephson tunnel
junction. In most cases, the superconducting tip did not show a full gap and thus
we need to understand how this affects the critical current.
Van Duzer in ”Principles of Superconducting, Devices and Circuits” [132]
pointed out that when one electrode has gap ∆1 and the other gap ∆2, the quasi-
particle current rise Iss at voltage (∆1+∆2)/e is given by:
√
Gnπ ∆1∆2
Iss = (7.105)
2e
In the experiment described in the next chapter, I show that ∆1 and ∆2 can be ex-
tracted from the voltage at which Andreev steps are observed. We also can measure
the point where the quasiparticle current rises and the resistance Rn = 1/Gn of the
I(V) characteristics for V  (∆1 + ∆2)/e. Thus we can compare the theoretical Iss
value to the measured value Iqp. Of course these results are not for the junction
critical current but are for the quasiparticle current in a low-transparency barrier.
I can also measure the critical current of a junction and compare it to the
theoretical prediction of Ambegaokar and Baratoff [133, 134]. In this paper they
205
7.6. CRITICAL CURRENT IN ASYMMETRICAL JUNCTIONS WITH MAR
used thermodynamic Green’s functions to find an expression for the critical current
of a junction with arbitrarily low transparency (i.e. MAR effects are not included).
They found:
π ∑ −1/2
I0 = R
−1∆ (T )∆ (T ) [(ω2 + ∆2n 1 2 l 1(T ))(ω
2 2
βe l
+ ∆2(T ))] (7.106)
l=0,±1,±2...
where ωl=π(2l+ 1)/β, β = 1/kBT . In the limit T → 0, the sum can be replaced by
an integral and they found:
1
I = R−1∆ (T )K([1−∆2(T )/∆2(T )]1/20 n 1 1 2 ) (7.107)e
where ∆1 is the smaller of the two energy gaps and K is the modified Bessel function
[135]. For a symmetrical junction Eq. (7.107) can be analytically simplified to:
π (1 )
I0 = R
−1
n ∆(T )tanh β∆(T ) (7.108)2e 2
which agrees with the well-known Ambegaokar-Baratoff formula [133, 134] for the
critical current of a Josephson junction.
Of interest here is that the critical current can also be obtained from the MAR
theory discussed in the previous parts of this chapter. In particular, Averin’s [122]
calculation of the ac current terms can be used to extract the critical current of a
junction with a single channel at different transparency parameters D, while my
analysis of the case of different gaps allows me to obtain I0 for different gaps at
arbitrary transparency. In particular, the critical current is simply related to the
206
7.6. CRITICAL CURRENT IN ASYMMETRICAL JUNCTIONS WITH MAR
first Fourier component.
Figure 7.10 (a) shows examples of the real part and the imaginary part of the
first Fourier component of the current I1(t)=I e
2ieV t/h̄
1 from Eq. (7.85) with k = 1.
Note Re(I1)=0 when the transparency D < 1. Therefore the only contribution to
the critical current is from Im(I1). Notice in Fig. 7.10 (b), when the voltage V
approaches zero, the current approaches a finite value. This “a.c. current at zero
volts” is due to the critical current of the junction.
To find an approximate expression for the critical current, we know from
Eq. (7.85) with k = 1 that we can write:
π 2ieV t
I1 = | Im(I −i1)|e 2 e h̄ (7.109)
π 2ieV t
I−1 = | Im(I−1)|ei e−2 h̄ (7.110)
where | Im(I−1)| = | Im(I1)|. We will assume the critical current can be approxi-
mated as the sum of the first Fourier component term (i.e. I2, I3...are small) and
write:
π 2eV t π 2eV t
I0 = I1 + I−1 = | I(m(I )|(e−i( )− ) + ei( − )1 2 h̄ 2 h̄ () )
π 2eV t 2eV t
= 2| Im(I1)| cos − = 2| Im(I1)| sin (7.111)
2 h̄ h̄
Equation (7.111) is the same form as expected for the Josephson supercurrent.
207
7.6. CRITICAL CURRENT IN ASYMMETRICAL JUNCTIONS WITH MAR
(a) eV/ΔR
0 0.5 1 1.5 2 2.5 3
0 D
0.01
-0.2
0.1
-0.4 0.4
0.7
-0.6 0.9
0.99
-0.8 1
-1
-1.2
(b)
3
2.5 D
2 0.01
0.1
1.5 0.4
0.7
1 0.9
0.99
0.5 1
0
0 0.5 1 1.5 2 2.5 3
eV/ΔR
Figure 7.10: (a) Normalized real part and (b) imaginary part of the AC
current (first Fourier component) versus normalized voltage for different
transparency D and for ∆R/∆L=1/0.5 at T=0.
208
-Im(I1)/(ΔRGT /2e) Re(I1)/(ΔRGT /2e)
7.6. CRITICAL CURRENT IN ASYMMETRICAL JUNCTIONS WITH MAR
To see this, note that Eq. (7.101) and Eq. (7.104) can be combined to yield
( )
2eV t
Is = Ic sin (7.112)
h̄
Comparing equations Eq. (7.111) and Eq. (7.112), we find the critical current of the
asymmetrical junction is:
Ic = 2| Im(I1)| (7.113)
where I1 can be obtained numerically from Eq. (7.85) for arbitrary D,∆1,∆2 and
T .
As mentioned in Ref. [136], Haberkorn et al. obtained results for the critical
current for a junction with arbitrary transparency and electrodes with the same gap.
In fact, Averin et al. note in their paper that they obtained the same form as [136]
by using the ac current expression [122]. However the expression Haberkorn et al.
[136] found was only for a symmetrical junction. Here I have obtained results for
the critical current of an arbitrary transparency asymmetrical junction. In practice,
Eq. (7.85), Eq. (7.86) and Eq. (7.87) can be evaluated numerically to calculate the
value I0 = 2|Im(I1)| in the limit of V = 0.
Finally, in Fig. 7.11 I plot the normalized critical current as a function of
the temperature T by evaluating Eq. (7.85) for ∆R = 1 meV, k = 1, ∆R/∆L=2
and D=0.5. Note that the critical current reaches a maximum value of about
0.62(2∆/eRn) at T = 0 and decreases rapidly above about 2 K. For the blue curve,
209
7.7. CONCLUSION
I assumed that ∆R and ∆L are independent of temperature, which is why the plot
still shows a critical current at relatively high temperature. For the orange curve
I assumed a BCS temperature dependence for the gaps with ∆R(0) = 1 meV and
∆L(0) = 0.5 meV. I then used BCS theory to calculate the corresponding gap at
specific temperatures and with these gap values, obtained the critical current by
evaluating Eq. (7.85). Note that this critical current decreases much more rapidly
above 2 K and goes to zero at 3.5 K, when the left electrode goes normal.
7.7 Conclusion
In conclusion, I first discussed Averin and Bardas’s derivation of a theoretical
model of the I(V) of symmetrical S-I-S junctions with MAR [122]. Once I under-
stood their Eq. (6) for the current [122], I moved on to generalize the theory to the
asymmetrical S-I-S junction with MAR. I showed that this generalization essentially
yields the same result for the I(V) as in Ref. [126], which used a Green’s function
approach.
Using the generalized theory, I also obtained an expression for the critical
current of an asymmetrical S-I-S junction and its dependence on the transparency
D. In Chapter 8, I will show the experimental results from STM of an asymmetrical
S-I-S junction with a Nb tip and a Nb sample. Although both the tip and the
sample are nominally the same material, the tip does not show a full Nb gap, hence
our junction is essentially an asymmetrical junction with different superconducting
210
7.7. CONCLUSION
3
2.5
2
1.5
1
0.5
0
0.1 1 10
T (K)
Figure 7.11: Semi-log plot of normalized critical current Ic/(∆RGT/2e)
vs. logarithmic scale of temperature T calculated from Eq. (7.85),
Eq. (7.86) and Eq. (7.87) for transparency D = 0.5. The blue curve uses
fixed ∆R = 1 meV, ∆R/∆L=2 while the orange curve uses temperature
dependent gap from BCS theory ∆R(T ) and ∆L(T ) with ∆R(0) = 1 meV
and ∆L(0) = 0.5 meV.
211
Ic/(ΔRGT /2e)
7.7. CONCLUSION
gaps. I also discuss the critical current of our STM formed Josephson junction using
the MAR theory of the critical current that I described in Section 7.6. Finally, I
note that a critical current is a signature of a phase coherent response of the tunnel
junction and it is essential for constructing a SQUID STM.
212
CHAPTER 8
Ultra-small Josephson Junction Formed Using a
Nb STM Tip and Nb Sample
8.1 Introduction
Multiple Andreev Reflection (MAR) effects [137–141] become important when
two superconductors are connected through a weak link that has a non-negligible
transparency. As I discussed in Chapter 7, MAR causes current to flow at subgap
voltage V = ∆/ne, where n is a positive integer. The current-voltage charasteristics
for a single superconducting channel with MAR and arbitrary transmission has been
calculated theoretically [122–125] and my own analysis for the case of different gaps
is presented in Chapter 7. Numerous previous experiments have demonstrated MAR
effects in mechanically controllable break-junctions [137, 138, 141, 142] and scanning
tunneling microscopy (STM) [71, 139, 140, 143]. However, investigations of MAR
effects in Josephson junctions that have electrodes with different superconducting
gaps is apparently relatively rare, not only experimentally but also theoretically
[71, 126]. Recent interest in Josephson STMs [32, 36, 144–146] has been driven
213
8.2. SAMPLE PREPARATION AND EXPERIMENTAL SETUP
by the need for better understanding of the physics of small Josephson junctions
[147–154] and the behavior of high Tc superconductors.
In this chapter, I describe my use of a scanning tunneling microscope (STM)
[20] to look at atomic-scale superconducting tunnel junctions formed between a Nb
tip and a Nb sample. In Section 8.2, I discuss the sample preparation and the
experimental setup. In Section 8.3, I show the I − V characteristics obtained at 50
mK and 1.5 K for junction tunnel resistance Rn from 10 MΩ to 300 Ω. I analyzed
the I − V characteristics by fitting to the MAR theory of Chapter 7. Depending on
the temperature T and distance z between the tip and the sample, which determines
the tunneling resistance Rn and the junction capacitance C, the junction can be in
the phase-diffusion regime [34–36, 38, 71], the underdamped small junction limit
[64] or the point contact regime [143, 155]. Next, in Section 8.4 I discuss some
important parameters extracted from the fits and some other phenomena, including
early switching current and finite resistance on the supercurrent branch. Finally, I
conclude in Section 8.5 with a summary and discussion of some implications.
8.2 Sample preparation and experimental setup
Each tip in the millikelvin dual-tip STM [41, 42] (see Fig. 8.1) was made from
250 µm diameter Nb wire etched in an SF6 reactive ion etcher for about 90 minutes
[94]. Each tip was mounted on the STM and was usually cleaned by performing
high voltage field emission on an Au(100) or Au(111) single crystal under vacuum
214
8.2. SAMPLE PREPARATION AND EXPERIMENTAL SETUP
conditions. To perform field emission on a tip, I turn off the z-feedback, pull the tip
a few nanometers away from the sample, and then ramp the sample bias to 80-100
V. The current is monitored during the ramp until we see a sudden drop of the
current to zero, indicating that the front part of the tip has been ripped of. After
cleaning, the quality of the tip is checked by taking I − V spectroscopy on the Au
sample to see if the tip is metallic (or superconducting) and by taking images to see
if the tip is atomically sharp. Once the quality of the tip is good, I use the sample
transfer system to take out the Au sample and put in the sample that I would like
to examine next.
I used a 4.5 mm x 4.5 mm x 1 mm Nb (100) single crystal as our sample [156].
The sample was prepared by Ar ion sputtering with a kinetic energy of 2 keV at
an Ar pressure of 2.0 × 10−5 mbar for 3-4 hours followed by annealing to 900◦C in
UHV for 20 minutes. This cycle was repeated two or three times and the sample
was then transferred to the STM mounted at the mixing chamber without breaking
vacuum [41]. Figure 8.2 shows topographic images taken from the Nb(100) surface
at 1.5 K and 50 mK. Atomically flat terraces are clearly visible.
To measure I(V) characteristics I used a relay box to switch from a standard
voltage-biased mode to a current-biased mode (See Fig. 8.1). In voltage-biased
mode, I acquired topographic images, I(V) and dI/dV − V curves. In the current-
biased mode, I used either a 1 MΩ, 10 MΩ or 100 MΩ resistor in series with the
voltage source to set the current I and then measured the voltage V across the
junction using an SR560 voltage amplifier. See Section 2.4 for a detailed description
of the current-bias mode.
215
8.2. SAMPLE PREPARATION AND EXPERIMENTAL SETUP
Current Bias 300 K
-
RW
CPF Copper
RW
CPF
A
RW Tip Tip
Rb CPF Sample
RW
R SuperconductorCPF C
V 50 mK-1.5Kb
Figure 8.1: Relay box used to switch from STM topographic mode to
current bias mode. Rb is set to 1 MΩ, 10 MΩ or 100 MΩ, depending on
the tunnel junction resistance. Rw ≈ 220 Ω is the resistance of each wire
and Rc is the contact resistance between the sample mounting stud and
the measurement wires.
216
+
8.2. SAMPLE PREPARATION AND EXPERIMENTAL SETUP
(a) (b)
(c) 50 nm (d) 50 nm
50 nm 50 nm
(e) (f)
50 nm 10 nm
Figure 8.2: Topography of different regions on the Nb crystal surface
obtained using the STM at (a)-(d) 1.5 K and (e)-(f) 50 mK.
217
8.3. I − V CHARACTERISTICS FROM A NB TIP AND A NB SAMPLE
8.3 I − V characteristics from a Nb tip and a Nb sample
Figure 8.3 shows three I − V characteristics for junction tunnel resistances of
Rn =72 kΩ, 7.9 kΩ and 460 Ω. This data was acquired using the current-biased
mode at 1.5 K. The blue points are from the backward sweeps (from positive current
to negative current) and the black points are from the forward sweeps (from negative
current to positive current). Before plotting, the data was processed by subtracting
the effect of a contact resistance Rc ≈ 173 Ω between the sample mounting stud
and the measurement wires (see Fig. 8.1).
Examining Fig. 8.3 (a), which shows I(V) when Rn = 72 kΩ, we see that the
curve looks somewhat like a standard S-I-S tunneling junction with a well defined
current rise at voltage V = (∆tip + ∆sample)/e ≈ 2.1 mV. The quasiparticle current
rise at V = 2.1 mV is Iqp = 23 nA. The main discrepancies between this I(V) and
standard S-I-S tunneling characteristics are the apparent absence of a supercurrent
and the significant subgap current steps at about 0.67 mV and 1.4 mV. The current
steps are more easily seen in the detailed plot of the sub-gap current shown in
Fig. 8.4 (a), which also reveals a supercurrent of about 450 pA. The current steps
are what one expects from MAR and the voltages at which they occur provide a
direct measure of the gap in the electrodes. Moreover, Fig. 8.3 (b) and (c) show
that the steps and supercurrent increase rapidly as the tunnel junction resistance
decreases.
The red curves in Fig. 8.3 and Fig. 8.4 show my fits to the MAR theory, with
T = 0 and the other parameters given in Table 8.1. The theory replicates the clear
218
8.3. I − V CHARACTERISTICS FROM A NB TIP AND A NB SAMPLE
(a) 50
40
30
20
10
0
(b) 0 1 V (2mV) 3 4500  
400
300
200
100
0
0 1 2 3 4
(c) 7 V (mV) 
6
5
4
3
2
1
0
0.0 0.5 1V.0 (m1V.5) 2.0 2.5 
Figure 8.3: I − V characteristics for junction tunneling resistance mea-
sured at T =1.5 K of (a) Rn = 72 kΩ, (b) 7.9 kΩ and (c) 0.46 kΩ. In
each, the blue curve is a forward sweep, the black curve is a backward
sweep, and the red curve is a fit to the asymmetrical MAR theory of
Chapter 7. Note the red line at zero voltage is the ideal critical cur-
rent I0 predicted from the MAR theory at 1.5 K. See Table 8.1 for fit
parameters.
219
I (µA) I (nA) I (nA)
8.3. I − V CHARACTERISTICS FROM A NB TIP AND A NB SAMPLE
(a)100
10 Is 1
0.1
0.01
(b) 0 1 V (2mV) 3 450  
40 Is 30
20
10 Ir 
0
0.0 0.5
(c) 3 V (mV
1).0 1.5 
2
1
0
0.0 V 0(.m5 V) 1.0 
Figure 8.4: (a) Semi-log plot of I−V characteristics from Fig. 8.3 (a). (b)
and (c) show detailed view of low voltage section of I−V characteristics
from Fig. 8.3 (b) and (c). The ideal critical current is off-scale and not
shown.
220
I (µA) I (nA) I (nA)
8.3. I − V CHARACTERISTICS FROM A NB TIP AND A NB SAMPLE
Table 8.1: Extracted parameters Rn,∆tip,∆sample,∆sum, D, Z from fitting the I −V
characteristics in Fig. 8.3 (a)-(c) to the asymmetrical MAR model at T = 0. Z is
the barrier height parameter and D is the transparency parameter.
∆sample ∆tip ∆sum
Rn (kΩ) D Z
(meV) (meV) (meV)
Fig. 8.3 (a) 72 kΩ 1.43 0.668 2.098 0.035 4.25
Fig. 8.3 (b) 7.9 kΩ 1.48 0.592 2.072 0.27 1.644
Fig. 8.3 (c) 0.46 kΩ 1.4 0.56 1.96 0.6 0.816
sudden rise at the sum of the superconducting gaps, and the subgap steps. The
subgap structure is very sensitive to the transparency value D, with a best fit value
of 0.035 for the Rn = 72 kΩ curve in Fig. 8.3 (a) and Fig. 8.4 (a). Fitting this curve
also gave two gaps with ∆sample = 1.43 meV and ∆tip = 0.67 meV.
Figure 8.3 (b) shows the corresponding I − V characteristics for Rn = 7.9 kΩ,
again with the red curve being the fit to the MAR theory. At V = ∆sum/e ≈ 2.1 mV,
the quasiparticle current Iqp ≈ 250 nA. Fitting this curve to the MAR theory with
T = 0 gives D = 0.27. Thus the junction characteristics in Fig. 8.3 (b) correspond
to a much more transparent tunnel barrier than that of Fig. 8.3 (a). The subgap
currents are also much larger in Fig. 8.3 (b) than Fig. 8.3 (a). Moreover, in Fig. 8.4
221
8.3. I − V CHARACTERISTICS FROM A NB TIP AND A NB SAMPLE
(b), the switching current is Is ≈ 32 nA, which is about 80 times larger than the
switching current in Fig. 8.4 (a), even though Rn differs by less than a factor of 10.
Note however that the switching current is not a true super-current because this
part of the I(V) curve has a non-zero resistance R0 ≈ 373 Ω. Fig. 8.4 (b) also shows
hysteretic behavior with a retrapping current Ir (the current when switching from
a finite voltage state to the zero voltage state) of 4.6 nA.
If we keep lowering the junction resistance, the I(V) curve continues to evolve.
Figure 8.3 (c) shows the situation for Rn = 460 Ω. The characteristics are very
rounded and very unlike a traditional S-I-S tunneling characteristic. Note that Iqp
has increased to nearly 5.1 µA while the switching current has increased to nearly
2 µA. In this regime, unlike the hysteretic behavior in Fig. 8.3 (b), the switching
current Is is nearly equal to the retrapping current Ir. Fitting the MAR theory to
this data at T = 0, I found D = 0.6 and gaps of 0.56 meV and 1.4 meV.
Here is a good place to comment on the identification of the gap of the sample
and tip. I typically saw that one of the gaps was close to the value expected for
bulk Nb (1.52 meV) while the second gap was typically smaller and varied with the
tip cleaning. For this reason, I identified the larger gap as being due to the sample
and the smaller gap as being due to the tip. As Table 8.1 shows, ∆tip is significantly
smaller at Rn = 460 Ω compared its value at Rn = 72 kΩ. I also note that the red
fit curve in Fig. 8.3 (c) disagrees noticeably with the data below 1.5 mV, but overall
the fit does a remarkably good job of fitting the data. One possible explanation is
that our single channel MAR theory does not apply to the situation of low junction
resistance, and one needs to consider multiple channels with different transparencies
222
8.3. I − V CHARACTERISTICS FROM A NB TIP AND A NB SAMPLE
[137–141].
Since we expect a single conductance channel can at most contribute a conduc-
tance of 2 × 2e2/h ' (6.5 kΩ)−1, it is not surprising that the single channel MAR
theory may fail for Rn ≤ Rq = 12.9 kΩ. To better understand this limit I measured
how the tunneling current at fixed voltage varied with the distance from the surface.
For these measurements, I set the sample voltage at 3 meV, which is above the gap,
and adjusted z to set Rn. I then recorded the current I while retracting the tip
a distance ∆z = 1.5 nm. Figure 8.5 shows the conductance G = I/V , normalized
by the quantum conductance (2e2/h) vs. the retraction distance ∆z. The different
colors represent curves with different starting resistances Rn. This also means that
the initial point of each curve has a different distance between the tip and the sam-
ple. I note that at V  (∆1 + ∆2)/e, the junction resistance in the STM is set by
the bias voltage V and the set current I, with Rn ≈ V/I. Here I actually set the
current, and since I ∝ e−αz this requires a smaller tip-sample separation for a lower
junction resistance. The curves shown in the plot are not in chronological order.
Examining Fig. 8.5, one sees two orange curves starting at Rn = 7.5 kΩ. One
curve shows a fairly smooth exponential-like dependence while the other curve just
shows some small sudden jumps in conductance. This second orange curve was taken
after a scan in which the tip had made contact with the sample surface. Evidently
a slight deformation of the tip or sample occurred during the previous scan. The
red curves show the behavior when I set the initial resistance to 3 kΩ. Many large
random jumps appear as the tip is withdrawn. We can safely conclude that at 3
kΩ, the tip is making physical contact with the sample, causing deformation of the
223
8.3. I − V CHARACTERISTICS FROM A NB TIP AND A NB SAMPLE
tip or sample, or causing the transfer of material between the tip and sample. As
the tip moves away from the sample, different parts of the tip leave the sample at
random times causing this irregular behavior in the I-∆z curve.
As observed in break junctions, the typical variations in conductance are of
the order of the quantum conductance [137, 138, 140, 141]. Since the tip makes
solid contact with the sample for R ≤ 3.9 kΩ, and a single channel would have a
conductance of 6.5 kΩ (considering quasiparticles with both up and down spin), at
least two channels must be contributing. If these channels have different barrier
transparency, then my single channel asymmetrical MAR theory will not fit.
Another small but significant discrepancy between the MAR theory and the
measured characteristics is visible in Fig. 8.3 (a) and Fig. 8.3 (b) at V = 2.1 mV;
the theory systematically predicts less current or, we could say, the current rises
above the theory at the sum of the gaps.
To illustrate what might be going on, Fig. 8.6 shows the measured I − V
characteristics (blue) for a junction resistance of Rn = 13.5 kΩ. A detailed view
of the subgap region of this curve is shown in Fig. 8.7. One sees a clear hysteretic
behavior with a switching current Is = 12 nA and a retrapping current Ir = 1
nA. The green, pink, purple and red curves in Fig. 8.6 were calculated using MAR
theory for temperatures of 13.3 K, 8.3 K, 5 K and 0 K, respectively. Notice that
the 13.3 K curve (green) has more current at all voltages and in particular it follows
the data quite nicely for V > 2.1 mV. Of course T < 13.3 K and the curve does not
follow the data for V < 2.1 mV. Nevertheless this suggests that the junction may
be self-heating to a rather high temperature when biased at voltages above the sum
224
8.3. I − V CHARACTERISTICS FROM A NB TIP AND A NB SAMPLE
Δz (nm)
Figure 8.5: Normalized conductance (G/G0) vs distance ∆z for different
starting normal resistance Rn, where G0 = 2e
2/h is the conductance
quantum and ∆z is the distance from the original tip position where we
set Rn. All measurements were made at V = 3 mV and T = 1.5 K.
Around G = G0, the curves have occasional sudden jumps, suggesting
the tip is forming a point contact with the surface. For G > 3 G0, the
tip probably makes solid contact with the sample and the large variation
in conductance is probably due to different parts of the tip leaving the
surface at slightly different times.
225
G/G0
8.3. I − V CHARACTERISTICS FROM A NB TIP AND A NB SAMPLE
200
150
100
50
00.0 0.5 1.0 1.5 2.0 2.5 3.0
V (mV) 
Figure 8.6: Blue and black points show measured I − V characteristics
at T = 1.5 K for Rn = 13.5 kΩ. Green, pink, purple and red curves
correspond to calculated MAR theory curves for T =13.3 K, 8.3 K, 5 K,
0 K, respectively. Here I used ∆sample = 1.43, ∆tip = 0.667 and D = 0.15
for all curves.
226
I (nA)
8.3. I − V CHARACTERISTICS FROM A NB TIP AND A NB SAMPLE
20
15
10
5
0
0.0 0.5 1.0 1.5
V (mV) 
Figure 8.7: Detailed view of subgap current behavior shown in Fig. 8.6.
The blue points show the measured forward sweep (large current to small
current), the black points show the measured backward sweep and the
red curve is a simulation at T = 0 K. Note that the zero voltage supercur-
rent branch has a non-zero resistance. For this simulation ∆sample = 1.43,
∆tip = 0.667, D = 0.15, T = 0 K and Rn = 13.5 kΩ.
227
I (nA)
8.3. I − V CHARACTERISTICS FROM A NB TIP AND A NB SAMPLE
of the gaps. Note that this heating effect cannot be as significant in the sub-gap
region, but of course the dissipated power I × V is also much less in the subgap
region.
Figure 8.8 (a) shows ∆tip + ∆sample (black squares), ∆tip (red circles), and
∆sample (blue triangles) versus Rn, extracted from fitting the MAR theory to the
measured I(V) curves. Note that ∆tip + ∆sample is constant at about 2.2 mV for
Rn > 10 kΩ, but for Rn < 10 kΩ, it drops steadily. The red dots show ∆sample ≈ 1.48
meV which are nearly constant, this is close to the full superconducting gap of bulk
Nb sample (1.5 mV). The blue triangles show the smaller gap value ∆tip, which is
only ∼ 0.7 meV: since the Nb tip was exposed to air and Nb with O dissolved in it
has a lower Tc, it is perhaps not too surprising to see a reduced gap value. A gap of
∆tip = 0.7 meV; suggests Tc ∼ 4.5 K, which is consistent with 5 % oxygen in Nb
[157]. This is below the solubility limit of 9% [158]. It may also be note that the
end of the tip was contaminated in some other way, producing a superconducting
gap that is less than the bulk Nb gap.
From Figure 8.8 (a), we see that the gap of the Nb sample ∆sample stays almost
constant over the full range of Rn, actually there is a drop of about 0.03 meV at
small Rn values, which is barely visible on the plot. In contrast, ∆tip decreases as
Rn decreases below 5 kΩ. I note that this decrease is consistent with the decreasing
in the sum of the superconducting gaps, which was measured separately from the
I(V) characteristics. One possible explanation is that when the junction resistance
is below 5 kΩ, the tip starts to make contact with the sample. This applies pressure
to the material at the tip, reducing its Tc. In contrast, the sample Tc decreases
228
8.3. I − V CHARACTERISTICS FROM A NB TIP AND A NB SAMPLE
(a)
2.0
1.5
1.0
0.5
0.0
10-1 100 101 102 3 4
(b) R (k )
10 10
n Ω
1.0
0.8
0.6
0.4
0.2
0.0
10-1 100 101 102 103 104
Rn (kΩ)
Figure 8.8: (a) Superconducting gap of the Nb sample (red), Nb tip
(blue) and the sum of the gaps of the Nb sample and the tip (black) as
a function of tunneling resistance Rn of the junction. (b) Transparency
parameter D as a function of Rn.
229
D 
  (meV)
8.3. I − V CHARACTERISTICS FROM A NB TIP AND A NB SAMPLE
relatively little, perhaps because pure Nb is less sensitive to pressure than oxygen-
doped Nb, or perhaps because the spindly tip compresses more than the sample.
Another possible explanation is that the large current running through the junction
causes heating. The small tip not only has a harder time dissipating heat than the
sample, but it has a smaller Tc due to contaminants compared to the sample, leading
to a larger decrease in the superconducting gap.
Figure 8.8 (b) shows the corresponding plot of transparency D vs. Rn from
fitting the asymmetrical MAR theory to the I(V) data at 1.5 K. As expected, D  1
when the junction resistance Rn is high, indicating very low transparency and the
junction is in the standard S-I-S tunneling limit [64]. As the tip and sample get
closer, Rn decreases and the transparency D becomes larger. Interestingly, D did
not reach 1 as Rn → 0 but saturated at around 0.6.
Figure 8.9 shows plots of Iqp, Is and Ir vs. the junction tunneling resistance Rn.
Iqp is the largest current value around the voltage of the sum of the superconducting
gap where it has the sudden rise in current. Is is the current where it switches out to
the finite voltage state from the supercurrent branch, while Ir is the current where
the junction is retrapped back to the supercurrent branch from the finite voltage
state. The filled points (squares, circles, triangles) were taken at 1.5 K and the
open points (squares, circles, triangles) were taken at 50 mK. The cyan stars are the
theoretical prediction of the critical current I0 from the asymmetric MAR theory
at 1.5 K and the red hexagons are the theoretical prediction of the quasiparticle
current Iqp from the asymmetrical MAR theory. The behavior can be divided into
three regimes. For 10 kΩ < Rn > 100 kΩ, no hysteresis and no switching current
230
8.3. I − V CHARACTERISTICS FROM A NB TIP AND A NB SAMPLE
is observed, consistent with thermal fluctuations destroying phase coherence [34–
36, 38, 71]. For Rn < 100 kΩ, a small supercurrent branch and a switching current
Is are observed (triangles). In this regime, the switching current Is is only 1 % to
10 % of the ideal critical current I0 (cyan stars). The retrapping current Ir (green
circles) is relatively large, indicating significant damping in the junction. As Rn
decreases, the retrapping current Ir increases much more rapidly than the switching
current Is. For Rn < 3 kΩ, where the tip and the sample make contact, Ir ≈ Is.
I note that although there was a strong dependence of the junction behavior
on Rn, there was very little difference between the data at 1.5 K (filled symbols)
and 50 mK (open symbols). Although the thermal energy of the junction at 50 mK
was nominally 30 times smaller than at 1.5 K, Is was not significantly larger at 50
mK, nor was Ir substantially smaller at 50 mK. This suggests that the value of the
switching current in our STM junction is not being limited to the thermal energy
of the bath. One alternative is macroscopic quantum tunneling [64, 159, 160] of
the phase, which becomes important because of the small critical current and small
capacitance between the sample and the tip. Another possibility is external noise
driving the system to switch into the non-zero voltage state. A third possibility
is that with an applied bias current, heating in the leads is creating temperatures
well-above the bath temperature. This last two alternatives are unlikely, as principle
measurements in our system have shown relatively low noise [40].
231
8.3. I − V CHARACTERISTICS FROM A NB TIP AND A NB SAMPLE
104
103
102
101
100
10-1
10-2
10-310-1 100 101 102 3R (k ) 10 10
4
n Ω
Figure 8.9: Nb-Nb junction parameters including extracted quasiparticle
current Iqp (black squares), switching current Is (blue triangles), and
retrapping current Ir (purple circles) plotted vs. Rn. The filled symbols
were taken at 1.5 K whereas the open points were taken at 50 mK. The
cyan stars are the critical current values obtained by fitting the I(V) data
to the asymmetric MAR theory of Chapter 7 at 1.5 K. Red hexagons
are the quasiparticle current values obtained by fitting to the asymmetric
MAR theory at 1.5 K. Above 0.1 MΩ, there was no observed supercurrent
branch and therefore Is and Ir were zero. Below 3 kΩ, the STM tip makes
contact with the sample and hysteretic behavior disappeared. For Rn
between 0.1 MΩ and 3 kΩ, the junction showed hysteretic behavior.
232
I (nA)
8.4. DISSIPATION ANALYSIS FROM THE RETRAPPING CURRENT
8.4 Dissipation analysis from the retrapping current
So far, I have shown that I could obtain reasonable fitting of the MAR I −
V characteristics to my data. However, my observation of a relatively and large
retrapping current indicate that the junction is subject to significant dissipation. In
this section, I compare my results to the standard RCSJ model [64] to extract some
additional important junction parameters.
In the RCSJ model, the Stewart-McCumber [161, 162] hysteresis parameter
βc can be extracted from a hysteretic I(V) characteristics by using [64, 163]:
βc = (Iqp/Ir)
2 (8.1)
where Ir is the retrapping current and Iqp is the quasiparticle current at V = (∆1 +
∆2)/e. For βc > 1, the system is underdamped, while for βc < 1, the system is
overdamped. For small√disturbance of the phase, the phase will oscillate at the√
plasma frequency ωp = 2πI0/Φ0C with a quality factor Q = βc.
Figure 8.10 shows extracted values of βc as a function of Rn (black squares).
Note that for all Rn values I found βc > 1. If we look carefully at Fig. 8.3 (c),
this I(V) curve at Rn = 490 Ω and βc ≈ 10 does not look like a conventional S-I-S
overdamped I(V) curve because of MAR effects. Nevertheless, we see that hysteresis
is present, Ir is about 90 % of the switching current I
1
s, and Is ∼ I .3 qp
233
8.4. DISSIPATION ANALYSIS FROM THE RETRAPPING CURRENT
109
108
107
106
105 100
104
103
102
101 10-1
10010-1 100 101 102
Rn (kΩ)
Figure 8.10: Extracted values of Stewart-McCumber parameter βc
(black) and total effective capacitance C (red) of the junction versus
tunneling resistance Rn. βc  1 indicates the junction is strongly un-
derdamped. The capacitance is roughly consistent with expected STM
junction capacitance in the range of fF to aF. Filled symbols represent
data taken at 1.5 K while open symbols represent data taken at 50 mK.
234

 c
C (f F)
8.4. DISSIPATION ANALYSIS FROM THE RETRAPPING CURRENT
The Stewart-McCumber parameter [161, 162] can also be written as:
βc = 2πI0R
2C/Φ0 (8.2)
where I0 is the true critical current of the junction (as opposed to the measured
switching current Is), R is the total effective resistance shunting the Josephson
junction and C is the total effective capacitance across the junction. Using the
extracted values of βc and assuming I0 = πIqp/4, I can use Eq. (8.2) to obtain:
C = 4βch̄/2eIqpR
2π. (8.3)
where the effective resistance at the retrapping voltage is simply:
R = Vr/Ir (8.4)
and Vr is the corresponding voltage at the retrapping current Ir.
Figure 8.10 shows a plot of the resulting capacitance C as a function of Rn
(red points) at 50 mK (open circles) and 1.5 K (filled circles). This method yields
a capacitance value that appears to be in a reasonable agreement with one would
expect for an STM tip that is near to a conducting surface [164–166], which is on
the order of fF to aF. As expected, C increases as Rn becomes smaller, because
the distance between the tip and the sample becomes smaller, resulting in a larger
capacitance value. For Rn ≥ 100 kΩ, I could not use this approach to determine C
because the retrapping current was too small to measure.
235
8.4. DISSIPATION ANALYSIS FROM THE RETRAPPING CURRENT
Although I used the RCSJ model to estimate βc and C, the I−V characteristics
show some features that cannot be explained with just this simple model, including
MAR, early switching out of the supercurrent state and finite resistance of the
supercurrent branch. Although MAR explains much of the behavior for a non-
zero voltage, small Josephson junction physics is needed to explain the behavior
near V = 0. This behavior of Josephson junction [64] is governed by the level of
dissipation and the relative size of four different energies: the Coulomb charging
energy EC = e
2/2C, the Josephson energy EJ = I0Φ0/2π, the thermal energy kBT
and the plasma energy ωp × h̄, which is approximately the energy required to excite
the junction out of its lowest energy quantum state. Since βc > 1 for all Rn (see
Fig. 8.10), the junctions are in the relatively low dissipation limit. Figure 8.11 shows
Josephson energy, Coulomb energy and plasma energy as a function of the tunneling
resistance Rn. As Fig. 8.11 shows, EJ > h̄ωp > Ec > kBT for Rn < 10 kΩ. This
is the phase qubit limit, where phase is well-defined but quantum tunneling effects
[64, 159, 160] can be important and produce early switching of the super-current.
Also shown in Fig. 8.11 is the escape temperature TESC and the thermal energy
kBT . From the plot, TESC ≈ 1 K which suggests macroscopic quantum tunneling is
important. For Rn greater than about 100 kΩ, Ec dominates. In this limit charging
effects, phase diffusion and the impedance of the bias leads become important in
determining the I(V) characteristics, including in particular the size of the observed
phase-coherent supercurrent and the resistance of the supercurrent branch. As far
as we know, a full theory, including MAR, charging effects, and circuit impedance
has yet to be developed.
236
8.4. DISSIPATION ANALYSIS FROM THE RETRAPPING CURRENT
102 EJ /kB Ecℏ/kB101 ω /k
1.5 K p B
100 TESC
10-1 50 mK
10-2
10-3
10-1 100 101 102 103 104
Rn (kΩ)
Figure 8.11: Josephson energy EJ (black), charging energy EC (red)
and plasma energy h̄ωp (blue circles) of the Nb-Nb STM junction ver-
sus normal resistance Rn of the junction. Dashed dark gray lines show
temperature of 1.5 K and 50 mK. Green circles show the MQT escape
temperature vs. Rn. Note that TEQC ∼ 1K for a wide range of Rn.
237
E/kB (K)
8.4. DISSIPATION ANALYSIS FROM THE RETRAPPING CURRENT
An important part of the small Josephson junction theory is und√erstanding
effects produced by the lead impedance [73]. The plasma frequency ωp = 2eI0/h̄C
of a small junction is typically around 1010 Hz to 1011 Hz and at such frequencies the
shunting impedance Z1 of the bias leads has non-zero real and imaginary parts that
typically are of order 50 Ω. Thus Z1 is typically orders of magnitude smaller than
the quasiparticle resistance. Our output leads act like a mismatched transmission
line and give high-frequency damping which produces the retrapping current and
a measurable non-zero resistance on the supercurrent branch due to the inelastic
tunneling of Cooper pairs [64, 159, 160, 167].
The high frequency damping effect also produces noise which causes phase
diffusion [74, 152, 153] and an average voltage in the supercurrent branch. For
thermally generated noise and EJ  kBT , one finds a resistance of [64, 74]:
R0 = 2Z1(kBT/E )
2
J (8.5)
In our system there can also be a contact resistance Rc between the Nb sample
output leads and the sample bias lines (see Fig. 8.1). Equation (8.5) can then be
written as: [ ]
2 24e kBTRn
R0 = 2Z1 +Rc(T ) (8.6)
h̄π∆
Thus we expect that R0−Rc is proportional to the square of the normal resistance,
at least in the limit EJ  kBT .
Figure 8.12 shows a plot of R0 − Rc as a function of Rn. The black squares
238
8.4. DISSIPATION ANALYSIS FROM THE RETRAPPING CURRENT
105
104
103
102
101
1 10 100
Rn(kΩ)
Figure 8.12: Plot of resistance R0 − Rc of the supercurrent branch ver-
sus normal tunnel resistance Rn of the junction. Blue circles are data
obtained at 50 mK, black squares show data obtained at 1.5 K and red
dashed line is a guide line showing R 20 −Rc is proportional to Rn.
239
R0-Rc (Ω)
8.5. CONCLUSION
were taken at 1.5 K while the blue circles were taken at 50 mK. The red dashed
line is a guide line showing R0−Rc scales as R2n. I estimated the contact resistance
Rc = 173 Ω at 1.5 K and Rc = 15 Ω at 50 mK. Although the data does not scale
exactly as R2n, it is remarkably close, suggesting that the critical current is being
suppressed by phase diffusion, as expected.
8.5 Conclusion
In conclusion, by varying the distance between a Nb STM tip and a Nb sample
at 1.5 K, I varied the normal junction resistance Rn and observed I − V character-
istics that ranged from the low-transparency S-I-S tunnel limit to the high trans-
parency S-I-S limit with prominent MAR effects. The junction resistance varied
from Rn = 5 MΩ to 500 Ω, with standard S-I-S low-transparency behavior above
100 kΩ. For Rn between 3 kΩ and 100 kΩ, the junction behaved like a hysteretic
small Josephson junction with an early (premature) switching current Is and a small
but non-zero resistance R0 on the “supercurrent” branch. Below 3 kΩ, the STM
junction acts like a point contact and multiple channel MAR theory needs to be
taken into consideration. I also found that the superconducting gap of the tip was
significantly lower (70%) in this limit.
Using the retrapping current, I extracted estimates for the Stewart-McCumber
hysteresis parameter βc and the capacitance C of the junction. I also observed a
small, non-zero resistance of the supercurrent branch and found that this scaled as
240
8.5. CONCLUSION
R2n, as expected for phase diffusion. I note that early switching of the supercurrent
could be suppressed and the phase could be stabalized by using a SQUID-STM
configuration [39, 42, 43] in which two superconducting junctions can form a SQUID.
One tip could be pushed into the sample to achieve a relatively latge and stable
critical current while the other tip is scanned. One then picks up the coherent phase
signal detected by applying a small magnetic flux to the loop and then measuring
the switching current [39, 42, 43, 168, 169].
241
CHAPTER 9
Conclusions
9.1 Summary of main results
In this thesis, I described results from my research on three main topics: (i)
work towards developing a dual-tip STM that can image the gauge-invariant phase
difference on a superconducting surface, (ii) STM imaging of Andreev reflection
effects on the surface of TiN which allowed mapping of local variations in the gap,
transparency and temperature, and (iii) the development of a theory of multiple
Andreev reflection and its application to the analysis of STM Nb-Nb S-I-S tunneling
characteristics. After an overview of the thesis in Chapter 1 and a brief review of
tunneling in Chapter 2, I described the millikelvin dual-tip STM system that I used
for all my research in Chapter 3.
In Chapter 4, I described how I modified the dual-tip STM to connect two
tips using a short flexible link made of Nb foil. I then described a new technique to
control both tips using dI/dzand showed results from simultaneous imaging with
both tips at room temperature and discussed the potential application of the system
242
9.1. SUMMARY OF MAIN RESULTS
to imaging superconducting phase differences.
In Chapter 5, I reviewed the BTK theory of Andreev reflection effects in S-N
tunneling. In Chapter 6, I then showed STM images and tunneling spectroscopy
on 50 nm and 25 nm thick TiN film at 500 mK. For the 50 nm thick TiN film I
used a vanadium tip while for the 25 nm thick TiN film I used a Nb tip. I observed
prominent Andreev effects in the tunneling characteristics and used the BTK theory
to analyze the I(V) and dI/dV (V) data that I acquired. From the BTK analysis I
was able to extract maps of the gap, temperature and barrier transparency, which
showed large, correlated local variations. I presented a model that may explain the
correlated variation as well as differences between the thick and thin films.
In Chapter 7, I generalized the Averin and Bardas theory of MAR to describe
an S-I-S junction with electrodes with different superconducting gaps. Using this
theory I obtained an expression for the I(V) characteristics and a prediction of the
critical current as a function of the barrier transparency, gaps, and temperature. In
Chapter 8, I described my measurement of I(V) characteristics obtained at 50 mK
and 1.5 K using an STM with a Nb tip and a Nb sample. I then applied the theory
developed in Chapter 7 to analyze the I(V) characteristics and concluded that the
asymmetrical MAR theory worked remarkably well at describing the characteristics
over the full range of Rn values. The main discrepancies occurred for Rn < 5
kΩ, where the tip began to make contact with the sample. I also examined the
switching current and retrapping current and identified three regimes: the phase
diffusion regime (for Rn > 100 kΩ), the underdamped regime (for 3 kΩ < Rn < 100
kΩ) and the point contact regime (for Rn < 3 kΩ).
243
9.1. SUMMARY OF MAIN RESULTS
One may well ask, since you succeeded in connecting two Nb STM tips together
and simultaneously scanning both tips, you should have been able to form an STM-
SQUID, measure flux sensitivity and image the gauge invariant phase difference. In
addition, was the observed supercurrent from the single S-I-S Josephson junctions
discussed in Chapter 8 phase coherent or incoherent tunneling of Cooper pairs?
Figure 9.1 shows the configuration of the STM controllers and read out elec-
tronics for running the SQUID setup. To operate the dual-point system as a SQUID,
I first stabilize the two STM tips at the desired junction tunneling resistance Rn,
turn the z feedback off and switch the relay to the current bias mode. I then use the
function generator to output the current to the junctions and start the counter. The
counter waits for a signal from the relay box; once the voltage across the SQUID
reaches the threshold (because the SQUID switching out from the supercurrent
branch), the counter stops. I repeat this switching measurement many times and
for different flux applied to the loop. I can plot the applied flux Φ vs. the switching
current Is (converted from the time for the supercurrent state to switch out to the
voltage state). Once all the measurements are done, the relay box switched the
STM electronics back to the voltage bias mode, and the STM feedback is turned
back on.
I actually spent a total of about 6 months attempting to run the STM-SQUID
at 30 mK before the refrigerator was damaged and then at 1.5 K after the refriger-
ator was temporarily fixed. However, I did not observe an obvious critical current
modulation when the flux was applied to the SQUID loop [42]. Below I discuss
several possible reasons and suggest some possible improvements for the future.
244
9.1. SUMMARY OF MAIN RESULTS
1-10 MΩ
FG Output
Burst
Counter StartStop
Data
Arm Current bias STM
Arm
PC Data Relay
Burst
Relay
Start
R9 Start
Current
Bias
Piezo
Figure 9.1: Configuration of the STM controllers and read out electronics
for running the SQUID setup.
245
-
Voltage bias
+
9.2. SOME SUGGESTIONS FOR FURTHER RESEARCH
9.2 Some suggestions for further research
9.2.1 Low temperature requirement
Previously most of the reports on Josephson STMs were conducted in the
phase diffusive regime [34–36, 38, 71] where the critical current is highly suppressed.
For a symmetric superconducting junctions (with electrodes that have the same
superconducting gap), the Ambegaokar-Baratoff formula [133, 134] gives a critical
current
( )
π∆ ∆
I0 = tanh (9.1)
2eRN 2kBT
where RN is the tunneling resistance of the junction in the normal state [64]. For the
critical current I0 to be close to the measured switching current Is, the Josephson
coupling energy E = h̄I0J must be much greater than the thermal energy E2e T =
kBT . As we saw in Fig. 8.11, the Josephson energy EJ (black), charging energy
EC (red), and thermal energy kBT (blue) can be relatively close to each other, and
for phase coherent operation for typical parameters we need to use a mK setup.
Unfortunately, as I described in Section 3.5, after the accident which damaged the
step heat exchanger, we found the refrigerator was stable down to about 0.5 K and
could not cool below 0.3 K. I tried to run the SQUID measurement at 0.5 K but
it was quite possible that the thermal energy was too big and the phase coherence
was not present for the range of junction parameters I tried. Clearly it would be
246
9.2. SOME SUGGESTIONS FOR FURTHER RESEARCH
interesting to replace the refrigerator with a working mK unit and try again at 50
mK.
9.2.2 Smaller SQUID loop
As Roychowdhury et al. showed theoretically [40], it is better to have a smaller
inductance SQUID loop to observe a phase coherent response if one of the junctions
has a small critical current. Unfortunately, the STM tip length was relatively large
(∼ 2 mm) (see Fig. 4.2). This also meant that the SQUID had a large pickup area,
which makes it very sensitive to magnetic field noise. A total RMS field noise as
small as 1 nT would have destroyed the SQUID modulation with applied flux. It
clearly would be interesting to try again with a smaller loop formed from shorter
STM tips and holders. One could also bend the tips toward each other so that when
the tips approached the sample, they would form a smaller SQUID loop.
9.2.3 Better shielding
Currently the IVC can is enclosed in an Amuneal Manufacturing Amumetal
4K shield [170] to reduce the strength of low-frequency magnetic fields and electrical
noise in the system. I note that this level of shielding may not be adequate given the
relatively large pickup area of the SQUID. With a pickup area of ∼ 2 mm2, a field
change of 1 nT would produce a flux change of 1 Φ0 in the SQUID. Fluctuations
of this size would completely obscure measurements of the SQUID critical current.
247
9.2. SOME SUGGESTIONS FOR FURTHER RESEARCH
Since background 60 Hz fields alone are typically 100 nT, we should shield by at
least a factor of 1000. In the future, adding one or two more Amumetal shields
or a superconducting shield would be a priority. It would also be advisable to add
additional radiation shields at the plate or still to reduce black-body radiation at
the STM which can produce excess quasiparticle at mK temperature.
9.2.4 Intrinsic system constraint
Dan Sullivan’s proof of principle experiment [42] demonstrated the feasibility
of using a highly asymmetric SQUID to make phase sensitive measurements. He
used e-beam lithography to pattern thin film junctions that mimicked actual STM
conditions. In particular, the SQUID had ∼ 1 nA of critical current modulation on
top of ∼ 1 µA critical current. To satisfy this requirement in a dual-point STM,
that means one tip would have ∼ 1 µA critical current, which is in the point contact
regime. The other tip can be in the nA range, which is in the underdamped regime,
just as Dan’s experiment.
During my data acquisition of the I−V characteristics, I often noticed that it
was hard to stabilize the tip with a fixed switching current in the nA range. When
I turned off the feedback to make SQUID measurements, the tip typically ended
up drifting and within a short time the junction’s switching current would change
significantly. Although I did not try to make the system run with a large and a small
junction, I did try to operate at 0.5 K with both tips having a large switching current.
It was stable enough to run the SQUID measurements. Unfortunately, I did not see
248
9.2. SOME SUGGESTIONS FOR FURTHER RESEARCH
any modulation of the device critical current when I varied the flux. Although this
was in the point contact regime, I still should have observed a modulation. It is
possible that there was too much flux noise, or something was wrong with the coil I
was using to apply flux. In the future, it would be interesting to check these possible
problems by testing a conventional thin-film lithographically defined SQUID in the
apparatus. Another critical test of both the SQUID configuration and the single
STM point is to check the response to applied microwaves. As Roychowdhury et
al. showed, this can be used to distinguish phase coherent tunneling of pairs from
incoherent tunneling of pairs [40].
249
APPENDIX A
χ2 fit for the TiN film
In Chapter 6 I showed several maps of the gap in TiN films. These supercon-
ducting gap maps had gray points which indicated points where I got a bad fit to
the BTK theory. Each conductance curve had 115 points and the uncertainty of
each measurement of conductivity was estimated to be 5 nS for V > ∆/e and 1 nS
for V < ∆/e. I used χ2 ≥ 5000 as the threshold for a truly bad fit in the image, in
Fig. 6.6 (b) and Fig. 6.8 (b), which had 28928 and 25856 pixels. Note that a good
fit should yield χ2 ≈ 115. Points with χ2 > 5000 were colored gray in the maps in
Chapter 6. For Fig. 6.6 of the 50 nm film, there were 6617 points that gave bad fits
to the Dynes model in Fig. A.1 (a) (22.87 %), while 6748 points had bad fits to the
BTK model in Fig. A.1 (b) (23.32 %). For the 25 nm film, there are 6258 points
with bad fits to the Dynes model in Fig. A.1 (c) (24.20 %), while 4939 points had
bad fits to the BTK model in Fig. A.1 (d) (19.1 %). In the 50 nm film, the quality
of the fits to the BTK model and to the Dynes model are not so different. In the
25 nm film, the BTK model gave significantly better fits.
Figure A.1 shows gap maps without using gray to indicate poor fits. Most of
250
the formally gray points are now red, which suggests that the gap is high. However,
examination of these fits shows that this is misleading, as these fit curves have a
very poor correspondence with the data. Figure A.1 also shows histograms of the
corresponding χ2 values, which display a long tail.
251
(a) (e)
500
400
300
200
100
0
0 1000 2000 3000 4000 5000
χ2
(b) (f)
500
400
300
200
100
0
0 1000 2000 3000 4000 5000
χ2
(c) (g)
400
300
200
100
0
0 1000 2000 3000 4000 5000
2
(d) (h) χ
700
600
500
400
300
200
100
0
0 1000 2000 3000 4000 5000
χ2
Figure A.1: (a) Gap ∆ map of the 50 nm thick TiN film from fitting to
the Dynes model and (b) the BTK model. (c) ∆ map of the 25 nm thick
TiN film from fitting to the Dynes model and (d) the BTK model. (e)
χ2 histograms of the 50 nm thick TiN film by fitting to the Dynes model
and (f) the BTK model. (g) χ2 histograms of the 25 nm thick TiN film
by fitting to the Dynes model (h) and the BTK model.
252
Count Count Count Count
APPENDIX B
50 nm TiN film showing S-I-S junction with a
vanadium tip
In the thesis, I mentioned that the superconducting tip did not show super-
conducting behavior due to it easily picking up debris from the surface. Here I show
that I successfully acquired one data set which shows the S-I-S junction on the TiN
surface.
Figure B.1 (a) shows a topographic image of a 70 nm × 33 nm region on
the 50 nm thick TiN film. The image was acquired at 0.5 K using a vanadium
tip and there are 256 × 122 pixels. The sample bias was set at 5 mV while the
tunneling current was set at 100 pA. Figure B.1 (b) shows the corresponding map of
∆ obtained from fitting the conductance curves at each point to the Dynes model.
Although in this case I did not use the BTK model to do the fitting (it is very
time consuming), from the map we can see that in general the values of the gap are
larger than the S-I-N data shown in Chapter 6. Figure B.1 (c) shows a histogram
of the superconducting gap using the Dynes fit. Compared to Fig. 6.10 where the
maximum of the superconducting gap value lies around 0.6 meV for the 50 nm film,
253
in this case the peak of the superconducting gap is around 1.5 meV. This suggests
that the junction is S-I-S like, instead of S-I-N like.
Figure B.1 (d) shows a dI/dV vs. V curve taken from one location in Fig. B.1
(a). Qualitatively we see that the coherence peaks are much sharper and the spacing
of the coherence peaks is much wider, corresponding to the sum of gaps, 2(∆tip +
∆sample). Moreover, the conductance curve bottoms out at 0 for V < ∆tip + ∆sample.
This suggest that there is no current flow inside the superconducting gap, which is
expected for a standard S-I-S curve. The red curve in Fig. B.1 (d) shows the fit
to the Dynes model. As we can see here, the fit is not very good, and it tends to
underestimate the gap. In this case the fit yields the superconducting gap of ∆tip +
∆sample = 1.44 meV. In principle, a better fit could be obtained from the Tinkham
formula which involves the numerical integration of the expression Eq. (2.27).
254
(a) h (nm)
3.0
2.5
2.0
1.5
1.0
0.5
0.0
(b) 10 nm Δ (meV)
1.6
1.4
1.2
1.0
0.8
(c) (d)
1200 40
1000
30
800
600 20
400
10
200
0 0
0.0 0.5 1.0 1.5 2.0 2.5 - 4 - 2 0 2 4
Δ (meV) V (mV)
Figure B.1: (a) Topography of 50 nm thick TiN film taken at 100 pA
and 5 mV bias.(b) Superconducting gap ∆ map of the 50 nm thick TiN
film from fitting to the Dynes model. (c) Histogram of superconducting
gap from gap map in Fig. B.1 (b) using the Dynes fit. (d) Measured
conductance dI/dV vs. V (blue curve) as well as fit to the Dynes model
(red curve).
255
Counts
dI/dV (nS)
APPENDIX C
Fortran code for symmetrical S-I-S junction with
MAR effect
In the symmetrical S-I-S junction code with MAR effect that I present below,
t is the transparency which is D in the Chapter 7, r is the reflection probability
which is R in the Chapter 7 and th is the temperature in kelvin. ne is the largest
integration upper limit ε0, e is the integration variable ε, v is the voltage V while
ai is the current I.
I use function scat(t,r,n,e,v) to calculate An and Bn defined in Chapter 7, and
it yields a(k) and b(k), respectively. The Andreev reflection amplitude is defined as
an(x)(in the c)ode. The Source J(ε) in this code is used as so(x). ft(x,th) gives the
tanh ε/2kBT function.
Finally, I would like to thank Prof. Averin again for providing the Fortran code
for the symmetrical S-I-S junction case. I then modified the code and generalized it
to the asymmetrical S-I-S junction case (see Appendix C).
256
1 program main
2 double complex a( -1000:1000) ,b( -1000:1000) ,w( -2000:2000)
3 double complex y,yy ,sa,ai,an
4 double precision e,ft,so
5
6 common/amp/a,b,w
7
8 open(1,file=’tmp1.dat’)
9 open(2,file=’tmp2.dat’)
10 open(3,file=’tmp3.dat’)
11 open(4,file=’tmp4.dat’)
12
13 !! This is the transparency parameter defined as D in the theory
14 write (*,*)’input barrier transparency t’
15 read (*,*)t
16
17 !! This is the temperature , normalized
18 th =0.005
19
20
21 !! R+T=1
22 r=1.0-t
23
24 !! jj is the Fourier component of the current term
25 jj=0.
26
27 !! v is the voltage
28 do 2 kv=1,150
29 v=0.02* kv
30
31 !! h is the integration interval
32 h=.001
33
34
35 !! n is how many terms of a(k) and b(k) have
36 n=int (2./v)
37 if(n.LT.20) n=20
38
39 !! ne is the maximum energy where the integration of the energy
goes up to
40 ne=int (20./h)
41 if(v.GT.1)ne=int ((18.+2.*v)/h)
42
43
44 !! je is just to alternate the integration terms , one term is 4
one terms is 2, in the end it divides by 3
45 !! "do 3" command sums up the current sa at each energy ke
46 sa=0.0
47 je=-1
48 do 3 ke=0,ne
49 je=-je
50
51 do 33 jth=1,2
52
53 !! both positive energy side and negative energy side have to be
summed up
54 e=((-1)**jth)*(h*ke +1.0005)
257
55
56 !! y is to calculate the current at certain energy e
57 call scat(t,r,n,e,v)
58 y=(w(2*jj)*a(jj)+conjg(w(-2*jj)*a(-jj)))*so(e)
59 do 4 k=-n,n
60 kk=2*k
61 kj=k-jj
62 yy=1.+w(kk)*conjg(w(kk -2*jj))
63 y=y+yy*(a(k)*conjg(a(kj))-b(k)*conjg(b(kj)))
64 4 continue
65
66 !! sa is the summed current at specefic voltage v with fermi
distribution ft
67 sa=sa +(3.+je)*y*ft(e,th)
68 33 continue
69
70 3 continue
71
72 !! the final current needs to multiply the interval of the
integration energy
73 ai=sa*h/(3.*t)
74
75 write (*,*)v,ai
76 write (3,*)v,real(ai)
77 write (4,*)v,-imag(ai)
78
79 2 continue
80
81 stop
82 end
83
84
85
86 subroutine scat(t,r,n,e,v)
87 double complex a( -1000:1000) ,b( -1000:1000) ,w( -2000:2000)
88 double complex an,s,x,q,u,p,f
89 double precision e,so
90
91 common/amp/a,b,w
92
93 rr=sqrt(r)
94
95 do 1 k=-n-2,n+2
96 a(k)=cmplx (0. ,0.)
97 1 b(k)=cmplx (0. ,0.)
98
99 do 2 k=-2*n-4,2*n+4
100 w(k)=an(e+k*v)
101 2 continue
102
103 s=(0. ,0.)
104 do 3 k=n,1,-1
105 j=2*k
106 j1=j-1
107 i1=j+1
108 u=t*w(j)/((1. ,0)-w(j1)**2)
109 f=t*w(i1)/((1. ,0)-w(i1)**2)
258
110 q=f*w(j+2)
111 x=u*w(j1)
112 p=u*w(j)+f*w(i1)+(1. ,0.)-w(j)**2
113 s=x/(p-q*s)
114 3 b(k)=s
115
116
117 s=(0. ,0.)
118 do 4 k=-n,-1
119 j=2*k
120 j1=j-1
121 i1=j+1
122 u=t*w(j)/((1. ,0.)-w(j1)**2)
123 f=t*w(i1)/((1. ,0.)-w(i1)**2)
124 q=f*w(j+2)
125 x=u*w(j1)
126 p=u*w(j)+f*w(i1)+(1. ,0.)-w(j)**2
127 s=q/(p-x*s)
128 4 b(k)=s
129
130
131 u=t*w(0) /((1. ,0.)-w(-1)**2)
132 f=t*w(1) /((1. ,0.)-w(1) **2)
133 q=f*w(2)
134 x=u*w(-1)
135 p=u*w(0)+f*w(1) +(1. ,0.)-w(0) **2
136 b(0)=rr*so(e)/(p-b(1)*q-b(-1)*x)
137
138
139
140 !! calculation of b(k)
141 b(1)=b(0)*b(1)
142 b(-1)=b(0)*b(-1)
143 do 5 k=2,n
144 b(k)=b(k-1)*b(k)
145 b(-k)=b(-k+1)*b(-k)
146 5 continue
147
148
149 !! calculation of a(k)
150 do 6 k=-n,-1
151 j=2*k
152 j1=j+1
153 k1=k+1
154 u=a(k)*w(j)*w(j1)
155 f=rr*(b(k1)*w(j+2)-b(k)*w(j1))
156 a(k1)=u+f
157 6 continue
158
159 u=a(0)*w(0)*w(1)
160 f=rr*(b(1)*w(2)-b(0)*w(1))
161 a(1)=u+f+w(1)*so(e)
162
163
164
165 do 7 k=1,n
166 j=2*k
259
167 j1=j+1
168 k1=k+1
169 u=a(k)*w(j)*w(j1)
170 f=rr*(b(k1)*w(j+2)-b(k)*w(j1))
171 a(k1)=u+f
172 7 continue
173
174
175
176 return
177 end
178
179 !! calculation of Andreev reflection amplitude
180 double complex function an(x)
181 double precision x,g
182 g=x**2
183 if(g.GE.1.) goto 2
184 an=cmplx(x,dsqrt(1.-g))
185 goto 1
186 2 t=x*(1.- dsqrt(g-1.)/abs(x))
187 an=cmplx(t,0.)
188 1 continue
189 return
190 end
191
192 !! calculation of incoming current source J(e)
193 double precision function so(x)
194 double precision x,z
195 double complex an
196 z=Real(an(x))
197 so=dsqrt (1.-z**2)
198 return
199 end
200
201 function sig(x)
202 double precision x
203 sig=x/abs(x)
204 return
205 end
206
207 !! ft corresponds to -tanh(e/2T)
208 double precision function ft(x,th)
209 double precision x,xx
210 xx=x/th
211 if(xx.LT.0.) goto 1
212 if(xx.GT.40) goto 2
213 z=dexp(-xx)
214 ft=(z-1.) /(1.+z)
215 goto 3
216 2 ft=-1.
217 goto 3
218 1 if(xx.LT.-40.) goto 4
219 z=dexp(xx)
220 ft=(1.-z)/(z+1.)
221 goto 3
222 4 ft=1.
223 3 continue
260
224 return
225 end
261
APPENDIX D
Fortran code for Asymmetrical S-I-S junction
with MAR effect
In this Fortran code, I calculated the ac part of the tunneling current, i.e.
jj=1, which is different from the dc part calculated in the previous Appendix C. As
I mentioned in Chapter 7, the main difference between the symmetrical junction and
the asymmetrical junction is that one has to be careful whether the quasiparticle is
reflected off the left electrode or the right electrode. For the case where the current
IL is produced from the left incident quasiparticle LAC, the left electrode Andreev
reflection amplitude is an(x), while for the right electrode the Andreev reflection
amplitude is bn(x). In contrast, for the case where the current IR is produced from
the right incident quasiparticle RAC, the left electrode is defined as bn(x) in the
code while an(x) is for the right electrode.
Finally, the total current from the asymmetrical S-I-S junction is calculated
using I = IL − IR. In my calculation here, the ratio of the gap of the left electrode
to the gap of the right electrode is ∆L/∆R = 1/0.5 = 2.
262
LAC :
1 program main
2 double complex a( -1000:1000) ,b( -1000:1000) ,w( -2000:2000)
3 double complex y,yy ,sa,ai,an ,bn
4 double precision e,ft,so
5
6 common a,b,w
7
8 open(3,file=’tmp3.dat’)
9 open(4,file=’tmp4.dat’)
10
11
12 jj=1.
13
14 do 2 kt=1,50
15 th =0.05* kt
16 t=0.5
17 v=0.01
18
19 h=.001
20 r=1.-t
21
22
23 n=int (2./v)
24 if(n.LT.20) n=20
25
26 ne=int (20./h)
27 if(v.GT.1)ne=int ((18.+2.*v)/h)
28
29 sa=0.0
30 je=-1
31 do 3 ke=0,ne
32 je=-je
33
34 do 33 jth=1,2
35
36 e=((-1)**jth)*(h*ke +1.0005)
37
38
39 call scat(t,r,n,e,v)
40 y=(w(2*jj)*a(jj)+conjg(w(-2*jj)*a(-jj)))*so(e)
41 do 4 k=-n,n
42 kk=2*k
43 kj=k-jj
44 yy=1.+w(kk)*conjg(w(kk -2*jj))
45 y=y+yy*(a(k)*conjg(a(kj))-b(k)*conjg(b(kj)))
46 4 continue
47
48 sa=sa +(3.+je)*y*ft(e,th)
49 33 continue
50 3 continue
51
52 ai=sa*h/(3.*t)
53
54 write (*,*)th,ai
263
55 write (3,*)th,real(ai)
56 write (4,*)th,-imag(ai)
57
58
59 2 continue
60
61 stop
62 end
63
64 subroutine scat(t,r,n,e,v)
65 double complex a( -1000:1000) ,b( -1000:1000) ,w( -2000:2000)
66 double complex an,bn,s,x,q,u,p,f
67 double precision e,so
68
69 common a,b,w
70
71 rr=sqrt(r)
72
73 do 1 k=-n-2,n+2
74 a(k)=cmplx (0. ,0.)
75 1 b(k)=cmplx (0. ,0.)
76
77 do 2 k=-2*n-4,2*n+4,2
78 w(k)=an(e+k*v)
79 2 continue
80
81 do 3 k=-2*n-3,2*n+3,2
82 w(k)=bn(e+k*v)
83 3 continue
84
85 s=(0. ,0.)
86 do 4 k=n,1,-1
87 j=2*k
88 j1=j-1
89 i1=j+1
90 u=t*w(j)/((1. ,0)-w(j1)**2)
91 f=t*w(i1)/((1. ,0)-w(i1)**2)
92 q=f*w(j+2)
93 x=u*w(j1)
94 p=u*w(j)+f*w(i1)+(1. ,0.)-w(j)**2
95 s=x/(p-q*s)
96 4 b(k)=s
97
98
99 s=(0. ,0.)
100 do 5 k=-n,-1
101 j=2*k
102 j1=j-1
103 i1=j+1
104 u=t*w(j)/((1. ,0.)-w(j1)**2)
105 f=t*w(i1)/((1. ,0.)-w(i1)**2)
106 q=f*w(j+2)
107 x=u*w(j1)
108 p=u*w(j)+f*w(i1)+(1. ,0.)-w(j)**2
109 s=q/(p-x*s)
110 5 b(k)=s
111
264
112
113 u=t*w(0) /((1. ,0.)-w(-1)**2)
114 f=t*w(1) /((1. ,0.)-w(1) **2)
115 q=f*w(2)
116 x=u*w(-1)
117 p=u*w(0)+f*w(1) +(1. ,0.)-w(0) **2
118 b(0)=rr*so(e)/(p-b(1)*q-b(-1)*x)
119
120
121 b(1)=b(0)*b(1)
122 b(-1)=b(0)*b(-1)
123 do 6 k=2,n
124 b(k)=b(k-1)*b(k)
125 b(-k)=b(-k+1)*b(-k)
126 6 continue
127
128
129 do 7 k=-n,-1
130 j=2*k
131 j1=j+1
132 k1=k+1
133 u=a(k)*w(j)*w(j1)
134 f=rr*(b(k1)*w(j+2)-b(k)*w(j1))
135 a(k1)=u+f
136 7 continue
137
138 u=a(0)*w(0)*w(1)
139 f=rr*(b(1)*w(2)-b(0)*w(1))
140 a(1)=u+f+w(1)*so(e)
141
142
143 do 8 k=1,n
144 j=2*k
145 j1=j+1
146 k1=k+1
147 u=a(k)*w(j)*w(j1)
148 f=rr*(b(k1)*w(j+2)-b(k)*w(j1))
149 a(k1)=u+f
150 8 continue
151
152
153 return
154 end
155
156
157 double complex function an(x)
158 double precision x,g
159 g=x**2
160 if(g.GE.1.) goto 2
161 an=cmplx(x,dsqrt(1.-g))
162 goto 1
163 2 t=x*(1.- dsqrt(g-1.)/abs(x))
164 an=cmplx(t,0.)
165 1 continue
166 return
167 end
168
265
169 double complex function bn(x)
170 double precision x,g
171 g=x**2
172 if(g.GE..25) goto 3
173 bn=cmplx (2.*x,dsqrt (1. -4.*g))
174 goto 4
175 3 t=x*(2.- dsqrt (4.*g-1.)/abs(x))
176 bn=cmplx(t,0.)
177 4 continue
178 return
179 end
180
181
182 double precision function so(x)
183 double precision x,z
184 double complex an
185 g=x**2
186 if(g.LT.1.) goto 2
187 z=Real(an(x))
188 so=dsqrt (1.-z**2)
189 goto 1
190 2 so=0.0
191 1 continue
192 return
193 end
194
195
196 double precision function ft(x,th)
197 double precision x,xx
198 xx=x/th
199 if(xx.LT.0.) goto 1
200 if(xx.GT.40) goto 2
201 z=dexp(-xx)
202 ft=(z-1.) /(1.+z)
203 goto 3
204 2 ft=-1.
205 goto 3
206 1 if(xx.LT.-40.) goto 4
207 z=dexp(xx)
208 ft=(1.-z)/(z+1.)
209 goto 3
210 4 ft=1.
211 3 continue
212 return
213 end
266
RAC :
1 program main
2 double complex a( -1000:1000) ,b( -1000:1000) ,w( -2000:2000)
3 double complex y,yy ,sa,ai,an ,bn
4 double precision e,ft,so
5
6 common a,b,w
7
8 open(3,file=’tmp3.dat’)
9 open(4,file=’tmp4.dat’)
10
11 !! calculate the AC part first Fourier component
12 jj=1.
13
14 do 2 kt=1,50
15 th =0.05* kt
16 t=0.5
17 v=0.01
18 h=.001
19
20
21 r=1.-t
22 n=int (2./v)
23 if(n.LT.20) n=20
24
25
26 !! now the ne goes a little bit further due to e only starts from
0.5005 instead of 1.005
27 ne=int (20.5/h)
28 if(v.GT.1)ne=int ((18.5+2.*v)/h)
29
30 sa=0.0
31 je=-1
32 do 3 ke=0,ne
33 je=-je
34
35 do 33 jth=1,2
36
37 !! the energy e where it starts to contribute some current is
outside the superconducting gap which is 0.5
38 e=((-1)**jth)*(h*ke +0.5005)
39
40
41 call scat(t,r,n,e,v)
42 y=(w(-2*jj)*a(-jj)+conjg(w(2*jj)*a(jj)))*so(e)
43 do 4 k=-n,n
44 kk=2*k
45 kj=k-jj
46 yy=1.+ conjg(w(kk))*w(kk -2*jj)
47 y=y+yy*(conjg(a(k))*a(kj)-conjg(b(k))*b(kj))
48 4 continue
49
50 sa=sa +(3.+je)*y*ft(e,th)
51 33 continue
52 3 continue
267
53
54 ai=sa*h/(3.*t)
55
56 write (*,*)th,ai
57 write (3,*)th,real(ai)
58 write (4,*)th,-imag(ai)
59
60
61 2 continue
62
63 stop
64 end
65
66
67 subroutine scat(t,r,n,e,v)
68 double complex a( -1000:1000) ,b( -1000:1000) ,w( -2000:2000)
69 double complex bn,an,s,x,q,u,p,f
70 double precision e,so
71
72 common a,b,w
73
74 rr=sqrt(r)
75
76 do 1 k=-n-2,n+2
77 a(k)=cmplx (0. ,0.)
78 1 b(k)=cmplx (0. ,0.)
79
80
81 !! for the odd term it sees bn(x) function , even term it sees an(x
) function
82 do 2 k=-2*n-4,2*n+4,2
83 w(k)=an(e-k*v)
84 2 continue
85
86 do 3 k=-2*n-3,2*n+3,2
87 w(k)=bn(e-k*v)
88 3 continue
89
90 s=(0. ,0.)
91 do 4 k=n,1,-1
92 j=2*k
93 j1=j-1
94 i1=j+1
95 u=t*w(j)/((1. ,0)-w(j1)**2)
96 f=t*w(i1)/((1. ,0)-w(i1)**2)
97 q=f*w(j+2)
98 x=u*w(j1)
99 p=u*w(j)+f*w(i1)+(1. ,0.)-w(j)**2
100 s=x/(p-q*s)
101 4 b(k)=s
102
103
104 s=(0. ,0.)
105 do 5 k=-n,-1
106 j=2*k
107 j1=j-1
108 i1=j+1
268
109 u=t*w(j)/((1. ,0.)-w(j1)**2)
110 f=t*w(i1)/((1. ,0.)-w(i1)**2)
111 q=f*w(j+2)
112 x=u*w(j1)
113 p=u*w(j)+f*w(i1)+(1. ,0.)-w(j)**2
114 s=q/(p-x*s)
115 5 b(k)=s
116
117
118 u=t*w(0) /((1. ,0.)-w(-1)**2)
119 f=t*w(1) /((1. ,0.)-w(1) **2)
120 q=f*w(2)
121 x=u*w(-1)
122 p=u*w(0)+f*w(1) +(1. ,0.)-w(0) **2
123 b(0)=rr*so(e)/(p-b(1)*q-b(-1)*x)
124
125
126 b(1)=b(0)*b(1)
127 b(-1)=b(0)*b(-1)
128 do 6 k=2,n
129 b(k)=b(k-1)*b(k)
130 b(-k)=b(-k+1)*b(-k)
131 6 continue
132
133
134 do 7 k=-n,-1
135 j=2*k
136 j1=j+1
137 k1=k+1
138 u=a(k)*w(j)*w(j1)
139 f=rr*(b(k1)*w(j+2)-b(k)*w(j1))
140 a(k1)=u+f
141 7 continue
142
143 u=a(0)*w(0)*w(1)
144 f=rr*(b(1)*w(2)-b(0)*w(1))
145 a(1)=u+f+w(1)*so(e)
146
147
148 do 8 k=1,n
149 j=2*k
150 j1=j+1
151 k1=k+1
152 u=a(k)*w(j)*w(j1)
153 f=rr*(b(k1)*w(j+2)-b(k)*w(j1))
154 a(k1)=u+f
155 8 continue
156
157
158 return
159 end
160
161 !! bn(x) function represents the full superconducting gap
162 double complex function bn(x)
163 double precision x,g
164 g=x**2
165 if(g.GE.1.) goto 2
269
166 bn=cmplx(x,dsqrt(1.-g))
167 goto 1
168 2 t=x*(1.- dsqrt(g-1.)/abs(x))
169 bn=cmplx(t,0.)
170 1 continue
171 return
172 end
173
174 !! an(x) function represents the half superconducting gap (0.5)
175 double complex function an(x)
176 double precision x,g
177 g=x**2
178 if(g.GE..25) goto 3
179 an=cmplx (2.*x,dsqrt (1. -4.*g))
180 goto 4
181 3 t=x*(2.- dsqrt (4.*g-1.)/abs(x))
182 an=cmplx(t,0.)
183 4 continue
184 return
185 end
186
187 !! so(x) function function starts from gap 0f 0.5 (see codition g.
LT..25)
188 double precision function so(x)
189 double precision x,z
190 double complex an
191 g=x**2
192 if(g.LT..25) goto 2
193 z=Real(an(x))
194 so=dsqrt (1.-z**2)
195 goto 1
196 2 so=0.0
197 1 continue
198 return
199 end
200
201
202 double precision function ft(x,th)
203 double precision x,xx
204 xx=x/th
205 if(xx.LT.0.) goto 1
206 if(xx.GT.40) goto 2
207 z=dexp(-xx)
208 ft=(z-1.) /(1.+z)
209 goto 3
210 2 ft=-1.
211 goto 3
212 1 if(xx.LT.-40.) goto 4
213 z=dexp(xx)
214 ft=(1.-z)/(z+1.)
215 goto 3
216 4 ft=1.
217 3 continue
218 return
219 end
270
Bibliography
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and R. J. Matson, “Superconductivity at 90 K in the Tl-Ba-Cu-O system,”
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