Cooper L.N., Feldman D. (Eds.) BCS: 50 Years
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BCS: The Scientific “Love of My Life” 141
6. P. W. Anderson, Phys. Rev. Lett. 3, 325 (1959).
7. P. W. Anderson, Annales Henri Poincare 4, 1 (2003).
8. P. Morel, J. Phys. Chem. Solids 10, 277 (1959).
9. P. Morel and P. W. Anderson, Phys. Rev. 125, 1263 (1962).
10. A. A. Abrikosov, Phys. Rev. B 55, 11955 (1995).
11. N. N. Bogoliubov, Uzpekhi Fiz Nauk 67, 549 (1959).
12. G. M. Eliashberg, Soviet Physics — JETP 11, 696 (1960).
13. I. Giaevar, H. R. Hart and K. Megerle, Phys. Rev. 126, 941 (1962).
14. J. M. Rowell, P. W. Anderson and D. E. Thomas, Phys. Rev. Lett. 10, 334
(1963).
15. J. R. Schrieffer, D. J. Scalapino and J. W. Wilkins, Phys. Rev. Lett. 10, 336
(1963).
16. W. L. McMillan and J. M. Rowell, in Superconductivity, ed. R. D. Parks, p. 561
(Dekker, NY, 1969), for instance.
17. W. L. McMillan, Phys. Rev. 167, 331 (1968).
18. W. A. Little, Phys. Rev. 134, A1416 (1964). This paper is an example of
highly fertile scientific fields being stimulated by questionable ideas. There is an
enormous literature on organic superconductors and another on polymer met-
als, both fruitful systems but structurally neither of which much resemble the
Little scheme. Nonetheless they can be truthfully said to have been stimulated
by Little.
19. D. Allender, J. Bray and J. Bardeen, Phys. Rev. B 7, 1020 (1973).
20. M. L. Cohen and P. W. Anderson, AIP Conf. Proc. 4, 17 (1972).
21. P. W. Anderson, Theories of Fullerene T
c
’s Which Will not Work, unpublished
preprint 1992 (submitted to Nature).
22. M. Capone, M. Fabrizio, C. Castellani and E. Tosatti, Revs. Mod. Phys. 81,
943 (2009).
23. L. P. Pitaevskii, Sov. Phys. JETP 10, 1267 (1960).
24. D. Thouless, The Quantum Mechanics of Many-Body Systems (Academic Press,
NY 1961). David soon provided a fuller account: Ann. Phys. 10, 553 (1960)
but did not go very far into the properties of such a state.
25. R. Balian and N. R. Werthamer, Phys. Rev. 131, 1553 (1963).
26. F. Steglich, J. Aarts et al., Phys. Rev. Lett. 43, 1892 (1979).
27. G. G. Lonzarich, in Electron, ed. M. Springford, Ch. 6, p. 109 (1997).
28. D. Jerome, A. Mazaud, M. Ribault and K. Bechgaard, J. Phys. (Paris) Lett.
41, L95 (1980).
29. A. N. Bloch et al., Phys. Rev. Lett. 34, 1561 (1975).
30. G. Saitoh et al., Sol. St. Comm. 42, 557 (1982).
31. M. Y. Choi, P. M. Chaikin and R. L. Greene, Phys. Rev. B 34, 7727 (1983)
and subsequent work.
32. R. B. Laughlin, A. Bernevig et al., Phys. Rev. Lett. 91, 147003 (2003).
33. J. G. Bednorz and K. A. Muller, Z. Phys. B 64, 69 (1986); M. K. Wu et al.,
Phys. Rev. Lett. 52, 408 (1987).
34. P. W. Anderson, Science 235, 1196 (1987).
35. P. W. Anderson, Phys. Rev. 115, 2 (1959).
August 30, 2010 11:1 World Scientific Review Volume - 9.75in x 6.5in ch8
142 P. W. Anderson
36. T. M. Rice and K. Ueda, Phys. Rev. B 34, 3420 (1986); further described in
C. Gros, R. Joynt and T. M. Rice, Phys. Rev. B 36, 387 (1987).
37. J. E. Hirsch and D. J. Scalapino, Phys. Rev. Lett. 53, 706 (1984) et seq.
38. N. F. Berk and J. R. Schrieffer, Phys. Rev. Lett. 17, 433 (1966).
39. PWA, Science 316, 1705 (2007).
40. G. Kotliar and J. Liu, Phys. Rev. B 38, 5142 (1988); F. C. Zhang et al.,
Supercond. Sci. & Tech. 1, 36 (1988).
41. T. A. Maier, D. Poilblanc and D. J. Scalapino, Phys. Rev. Lett. 100, 237001
(2008); J. Hwang, T. Timusk et al., Phys. Rev. B 75, 155708 (2007).
42. Y. Kamihara, H. Hosono et al., J. Am. Chem. Soc. 128, 10012 (2006).
August 25, 2010 14:12 World Scientific Review Volume - 9.75in x 6.5in divider
II. Fluctuations, Tunneling
and Disorder
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September 8, 2010 10:53 World Scientific Review Volume - 9.75in x 6.5in ch9
SQUIDs: THEN AND NOW
John Clarke
Department of Physics, University of California
and
Materials Sciences Division,
Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA
jclarke@berkeley.edu
This chapter is dedicated to the memory of Brian Pippard.
Following Brian Josephson’s prediction in 1962, Anderson and Rowell
observed Josephson tunneling in 1963. The following year, Jaklevic,
Lambe, Silver and Mercereau demonstrated quantum interference in a
superconducting ring containing two Josephson tunnel junctions. Subse-
quently, the first practical devices emerged, including the point-contact
dc and rf SQUIDs (Superconducting QUantum Interference Devices) of
Zimmerman and Silver and Clarke’s SLUG (Superconducting Low-
inductance Undulatory Galvanometer) — a blob of solder frozen around
a length of niobium wire. The return to the tunnel junction as the Joseph-
son element was heralded by the cylindrical SQUID in 1976. The square
washer dc SQUID developed by Ketchen and Jaycox in 1982 remains the
workhorse design for most applications. Theories for the dc and rf SQUIDs
were worked out in the 1970s. Today, SQUIDs (mostly dc) are used in
a variety of configurations — for example, as magnetometers, gradiome-
ters, cryogenic current comporators, low-frequency and microwave ampli-
fiers, and susceptometers — in applications including magnetoencephalog-
raphy, magnetocardiography, geophysics, nondestructive evaluation, preci-
sion gyroscopes, standards, cosmology, nuclear magnetic resonance, reading
out superconducting quantum bits, and a myriad of one-of-a-kind exper-
iments in basic science. Experiments are described to search for galaxy
clusters, hunt for the axion, and perform magnetic resonance imaging in
microtesla fields.
145
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146 J. Clarke
1. Introduction
The pairing of electrons, predicted by Leon Cooper
1
in 1956, is the corner-
stone of the BCS (Bardeen, Cooper, Schrieffer) theory of superconductivity
2
that appeared in 1957. Among the myriad consequences of Cooper pairs
are two that underlie the physics of the SQUID (Superconducting QUan-
tum Interference Device): flux quantization and Josephson tunneling. The
first implies that the magnetic flux embraced by a superconducting loop is
quantized in units of the flux quantum Φ
0
≡ h/2e ≈ 2.07 × 10
−15
Tm
2
;
here, h ≡ 2π~ is the Planck constant and e is the electronic charge. The
concept of the flux quantum was in fact introduced much earlier by Fritz
London,
3
who, obviously not knowing about Cooper pairs, predicted that
its value should be h/e. The experimental observation of the quantization
of flux in 1961 by Deaver and Fairbank
4
and Doll and N¨abauer
5
— needless
to say, in units of h/2e — was a striking confirmation of pairing and of the
existence of a macroscopic wave function in the superconducting state. The
wave function
Ψ(r, t) = |Ψ(r, t)|exp[iφ(r, t)] , (1)
which depends on position r and time t, describes the entire condensate
of Cooper pairs in a given superconductor; φ(r, t) is a phase factor. Flux
quantization arises from the fact that Ψ(r, t) must be single-valued in going
once around a superconducting loop. In the absence of any applied cur-
rents or magnetic fields, the phase φ(r, t) has the same value throughout the
superconductor. When the loop is threaded by a magnetic flux, however, the
phase around the loop changes by 2πn, where the integer n is the number
of enclosed flux quanta.
In the year preceding the observation of flux quantization, Ivar Giaever
6
demonstrated the tunneling of single electrons between a superconductor
(S) and a normal metal (N) separated by a thin insulating barrier (I).
Subsequently, he observed single-electron tunneling through SIS junctions.
7
Morel Cohen, Leo Falicov and Jim Phillips
8
explained these effects in terms
of a tunneling Hamiltonian. Taking this calculation to higher order, in
1962 Brian Josephson predicted the tunneling of Cooper pairs through a
barrier between two superconductors.
9
Brian showed that the phenomenon
of Josephson tunneling can be summarized by two succinct equations. First,
the supercurrent I flowing through the junction can be expressed as
I = I
0
sin δ . (2)
September 8, 2010 10:53 World Scientific Review Volume - 9.75in x 6.5in ch9
SQUIDs: Then and Now 147
0
0
2
4
6
8
1 2 3
VOLTAGE IN mV
CURRENT IN mA
(a)
(b)
Fig. 1. Current-voltage characteristic for a tin-tin oxide-lead tunnel junction at
about 1.5 K. (Reproduced with permission from Ref. 10.)
Here, δ(t) = φ
1
(t) − φ
2
(t) is the difference between the phases φ
1
and φ
2
of the condensates in the superconducting electrodes 1 and 2, and I
0
is
the critical current — the maximum supercurrent the junction can sustain.
Second, in the presence of a voltage V across the junction, δ(t) evolves with
time as
dδ/dt = 2eV /~ = 2πV/Φ
0
. (3)
Equations (2) and (3) describe the “dc” and “ac” properties of a Josephson
junction. In the presence of an external current through the junction, the
flow of Cooper pairs constitutes a supercurrent and, in the absence of fluctu-
ations, the voltage across the junction remains zero until the current exceeds
I
0
. At higher currents, there is a voltage across the junction and δ(t) evolves
according to Eq. (3). Subsequently, it became clear that this behavior is
generally valid for all weakly-coupled superconductors, although there may
be significant departures from the sinsusoidal current-phase relation.
One year later, Phil Anderson and John Rowell
10
observed the dc Joseph-
son effect in a thin-film, Sn-SnOx-Pb junction cooled to 1.5 K. The I–V
characteristic is reproduced in Fig. 1. Subsequently, Rowell
11
showed that
September 8, 2010 10:53 World Scientific Review Volume - 9.75in x 6.5in ch9
148 J. Clarke
−1000 −500 0
JOSEPHSON CURRENT
(ARBITRARY UNITS)
B
A
(1)
(2)
c
b
a
d
MAGNETIC FIELD, B in milligauss
500 1000
A
Fig. 2. The first dc SQUID. Josephson critical current versus magnetic field for
two Sn-SnOx-Sn junctions (1) and (2) in parallel (inset) showing interference effects.
Magnetic field applied normal to the area between junctions. Curve (A) shows
interference maxima spaced at ∆B = 8.7 × 10
−3
G, curve (B) spacing ∆B =
4.8 × 10
−3
G. Maximum Josephson current is approximately 1 mA. (Reproduced
with permission from Ref. 12.)
a magnetic field B, applied in the plane of the films, caused the critical
current modulation
I
0
(Φ) = I
0
(0)|sin(πΦ/Φ
0
)/(πΦ/Φ
0
)|. (4)
Here, Φ = Bw(d + λ
1
+ λ
2
); w is the width of the junction perpendicular
to B, d is the barrier thickness and λ
1
and λ
2
are the penetration depths of
the two superconductors. The observation of this Fraunhofer-like result —
analogous to the diffraction of coherent, monochromatic light by a slit — is
a dramatic demonstration of the validity of the current-phase relation.
A few months later, Bob Jaklevic, John Lambe, Arnold Silver and
Jim Mercereau
12
at the Scientific Laboratory of Ford Motor Company,
Dearborn, Michigan demonstrated quantum interference between two thin-
film Josephson junctions connected in parallel on a superconducting loop.
The dependence of the critical current on the applied magnetic field is shown
in Fig. 2. The slowly varying modulation arises from the Fraunhofer-like
diffraction of the two junctions [Eq. (4)]. The rapid oscillations are due to
quantum interference between the two junctions, and their period is given
by the field required to generate one flux quantum in the loop: thus, critical
current maxima occur at Φ/Φ
0
= 0, ±1, ±2, . . . . The observation of these
oscillations — analogous to two-slit interference in optics — was the birth
of the dc SQUID.
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SQUIDs: Then and Now 149
2. Early dc SQUIDs
The lead and tin thin-film Josephson junctions used in the pioneering exper-
iments proved to be fragile and subject to failure after even a single thermal
cycling. Later in 1964, Jim Zimmerman and Silver
13
came up with a simple
alternative technology consisting of a narrow sheet of niobium with a thin
niobium wire bent over the top of it to form two junctions. The nature of
the junctions was not entirely clear. Either the native oxide on the surface
of the niobium formed a tunnel barrier or there was a very narrow metallic
constriction between the pieces of niobium. Nonetheless, the device behaved
as a SQUID.
This was the scene when I became a research student in the Royal Soci-
ety Mond Laboratory — part of the Cavendish Laboratory — at the Uni-
versity of Cambridge in October 1964. The Mond was presided over —
with great kindness — by David Shoenberg. My thesis supervisor was Brian
Pippard. He gave me a project to investigate the electrical resistance of
the superconductor-normal metal (SN) interface — a topic that later be-
came of considerable interest.
14
Brian pointed out that the way to measure
this resistance was to make SNS sandwiches in which the normal metal
was too thick to sustain a supercurrent — and that one needed to measure
tiny voltages, on the order of 1 pV. Brian asked me to look into improv-
ing the performance of the cryogenic galvanometer he and George Pullan
had previously developed.
15
This device involved a mirror and magnet sus-
pended by a quartz fiber inside a vacuum can, surrounded by liquid helium,
at the center of a Helmholtz coil through which one passed the current to
be measured.
My thinking about ways to upgrade the galvanometer was brought to an
abrupt end, however, by a Mond seminar given by Brian Josephson. In his
talk, Brian described the experiments demonstrating flux quantization,
4,5
his
own theory of pair tunneling,
9
and the observation of quantum interference
12
earlier that year. To me, as a new research student, it was utterly fascinating,
and I remember thinking how I would like to work on these new ideas. My
opportunity came much sooner than I could possibly have expected! The
very next day, Brian Pippard bounced into my laboratory with a big smile on
his face. “John, how would you like a voltmeter with a resolution of 2×10
−15
volt in 1 second?” he asked. Not quite knowing how to answer, I somewhat
cautiously admitted that this did seem like an interesting possibility. Brian
— with a good deal of excitement — explained that he had dreamed up
his new concept of a voltmeter the previous evening after hearing Brian
Josephson’s seminar. He sketched on the blackboard a voltage source υ in
September 8, 2010 10:53 World Scientific Review Volume - 9.75in x 6.5in ch9
150 J. Clarke
series with a resistance R and an inductance L that was perfectly coupled
to a SQUID, also with inductance L. The current through the resistor and
inductor required to generate the flux quantum in the SQUID was simply
Φ
0
/L, and the corresponding static voltage υ was thus (Φ
0
/L)R. Setting
the time constant τ = L/R = 1 s, Brian pointed out that the resolution was
2 ×10
−15
V. It is interesting to note that Brian’s initial idea was to make a
digital voltmeter, each digit being a flux quantum.
Needless to say, I was thrilled — not least because my enthusiasm for
dangling a tiny magnet and mirror from a quartz fiber inside a vacuum can
had already begun to wane. I talked to John Adkins, who had previously
made thin-film Josephson junctions in the Mond. He was not encouraging
about the longevity of such devices, and I looked around for better alter-
natives. I came across the Zimmerman-Silver
13
article, which seemed like a
good place to start. I made a similar device with the additional twist that I
could adjust the vertical force on the bent wire from the top of the cryostat.
I remember rolling out a piece of niobium wire to make the narrow piece
of sheet. In those days, one took one’s glass cryostat to the helium liquifier
for Paul Booth to fill it — portable storage dewars for liquid helium seemed
not to exist in 1964. After returning my dewar safely to my lab, I connected
up the simple current-voltage measurement system I had built — and my
SQUID worked! I spent much of that day adjusting the force on the wire,
and discovered that I could vary the critical current quite readily. Sometimes
I observed oscillations in critical current when I varied the applied fields and
sometimes I did not. I made several variants of this design, and they all
more or less worked, but I was not entirely convinced that the device was
sufficiently stable to use routinely as a voltmeter.
An important part of one’s life in the Cavendish was — and still is —
coffee at 11 am and tea at 4 pm. The Mond students and staff always sat at
the same table. One tea-time, I discussed how I was looking for a Josephson
junction technology that did not require thin films yet was mechanically
stable. Paul Wraight — with whom I shared my lab — suddenly looked at
me and said something like “How about a blob of solder on a piece of niobium
wire? Solder is a superconductor and you keep telling me that niobium has
a surface oxide layer.” We rushed back to our lab, where fortunately I still
had some liquid helium left in my dewar from the experiment I had been
running earlier in the day. I made two devices consisting of a blob of lead-tin
solder melted onto a short length of niobium wire, attached some leads and
lowered the devices into the helium bath. They both worked! The critical
currents were roughly 1 mA. Paul and I were thrilled. The next morning,