The current-phase relation in Josephson junctions
A. A. Golubov
*
Faculty of Science and Technology, University of Twente,
P. O. Box 217, 7500 AE Enschede, The Netherlands
M. Yu. Kupriyanov
†
Nuclear Physics Institute, Moscow State University, 119992 Moscow, Russia
E. Il’ichev
‡
Department of Cryoelectronics, Institute of Physical High Technology,
D-07702 Jena, Germany
(Published 26 April 2004)
This review provides a theoretical basis for understanding the current-phase relation (C⌽R) for the
stationary (dc) Josephson effect in various types of superconducting junctions. The authors summarize
recent theoretical developments with an emphasis on the fundamental physical mechanisms of the
deviations of the C⌽R from the standard sinusoidal form. A new experimental tool for measuring the
C⌽R is described and its practical applications are discussed. The method allows one to measure the
electrical currents in Josephson junctions with a small coupling energy as compared to the thermal
energy. A number of examples illustrate the importance of the C⌽R measurements for both
fundamental physics and applications.
CONTENTS
I. Introduction 412
II. dc Josephson Effect 412
A. General properties of the Josephson current-
phase relation 414
B. Basic Josephson structures 415
III. Josephson Effect in Point Contacts 417
A. Aslamazov-Larkin model 417
B. Kulik-Omelyanchuk model, dirty limit (KO-1) 417
C. Kulik-Omelyanchuk model, clean limit (KO-2) 418
D. Point contact, the general case 418
IV. Josephson Junctions with Tunnel-Type Conductivity 420
A. Ideal tunnel junctions 420
B. Tunnel junctions with extended barriers 421
C. Tunneling via localized states 421
D. SIS structures with high transparency: Depairing
by current 423
V. SNS Junctions 424
A. The limit of high temperature, T⬇T
c
424
1. Weak depairing in the electrodes (rigid
boundary conditions) 425
2. Depairing in superconducting electrodes by
the proximity effect 425
3. Depairing in superconducting electrodes by
supercurrent 425
4. Nonlinear phase shift in the electrodes 426
5. Influence of the critical temperature of
weak-link material on I
C
R
N
427
B. Dirty limit, arbitrary temperature 427
1. Rigid boundary conditions 428
a. Structures made from one material (T
c
⫽ T
c
⬘
) 428
b. SS
⬘
S junctions (T
c
⫽T
c
⬘
) 429
2. SNS junction at arbitrary T: Depairing due
to the proximity effect 430
3. SS
⬘
S junction at arbitrary T: Depairing by
the supercurrent 431
C. Clean SNS junctions 431
VI. Double-Barrier SINIS Junctions 433
A. SINIS junctions, clean limit 434
1. The general case 434
2. Coherent regime (broad resonances) 435
3. Incoherent regime (narrow resonances) 436
B. SINIS junctions, dirty limit 436
1. The general case 436
2. Limit of high temperatures 437
3. Limit of low temperatures 437
4. Limit of small
␥
eff
437
5. Limit of large
␥
eff
437
VII. SFS Josephson Junctions 438
A. Proximity effect in SF bilayer 438
1. The formalism 438
2. Complex coherence length 439
3. Phase variation at the SF interface 440
B. Current-phase relation in SFS junctions: Simple
geometry 440
1. 0-
transitions due to oscillating order
parameter 440
2. 0-
transitions due to phase jumps at the
SF interfaces 441
C. Asymmetric case: SIFS tunnel junction 443
D. Clean SFcFS point contact 443
E. Diffusive SFcFS point contact 444
F. Double-barrier SIFIS junction 445
VIII. Experimental Method of Measuring I
S
(
) and its
Application 446
A. Description of the experimental method 446
B. The range of validity of the method 447
C. Measuring technique and calibration 447
D. Measurements of small critical currents 448
E. Measurement of barrier transparency asymmetry
in SINIS junctions 449
*
Electronic address: A.Golubov@tn.utwente.nl
†
Electronic address: mkupr@pn.sinp.msu.ru
‡
Electronic address: ilichev@ipht-jena.de
REVIEWS OF MODERN PHYSICS, VOLUME 76, APRIL 2004
0034-6861/2004/76(2)/411(59)/$40.00 ©2004 The American Physical Society411
F. Current-phase relation in hybrid S-2DEG-S
junctions 450
G. Current-phase relation anomalies in high-T
c
superconducting junctions 451
1. d-wave effects in high-T
c
superconductors 451
2. Asymmetric 45° grain-boundary junction 452
3. Symmetric 45° grain-boundary junction:
Manifestation of midgap states in the
current-phase relation 453
4. c-axis high-T
c
superconducting junctions 454
5. General remarks 455
IX. Summary and Outlook 455
Acknowledgments 456
Appendix: Microscopic Theory of Superconductivity 457
1. Green’s functions 457
2. The quasiclassical approximation 457
a. Eilenberger equations 457
b. The dirty limit: Usadel equations 458
c. ⌽ parametrization 459
3. Ginzburg-Landau equations 459
4. Critical current in a diffusive point contact:
Derivation of the KO-1 result 460
References 461
I. INTRODUCTION
The Josephson effect was discovered by Brian Joseph-
son in 1962 (see also Josephson, 1964, 1965). The sta-
tionary Josephson effect was first observed experimen-
tally by Anderson and Rowell (1963) and Rowell (1963),
and the nonstationary Josephson effect was observed by
Yanson et al. (1965). Since that time, there has been a
continuously growing interest in the fundamental phys-
ics and applications of this effect. The achievements in
Josephson-junction technology have made it possible to
develop a variety of sensors for detecting ultralow mag-
netic fields and weak electromagnetic radiation; they
have also enabled the fabrication, testing, and applica-
tion of ultrafast digital rapid single flux quantum
(RSFQ) circuits as well as the design of large-scale inte-
grated circuits for signal processing and general purpose
computing (Likharev and Semenov, 1991; Likharev,
1996, 2000; SEMATECH, 2001). In the present litera-
ture, there is no recent overview of this area of physics
available to both experts and people entering the field.
Classical developments are summarized in the review
by Likharev (1979) and several standard textbooks that
deal with the Josephson effect (Barone and Paterno,
1982; Likharev, 1986; Schmidt, 1997), where thorough
treatments are provided of the basic phenomena of the
Josephson effect in tunnel junctions and weak links.
Several recent review articles have been devoted to
applications of the Josephson effect, e.g., in high-T
c
su-
perconductors and Josephson quantum bits.
1
At the
same time, recent progress in the physics of the Joseph-
son effect justifies an overview of the fundamentals of
the Josephson effect on a more general level.
The purpose of the present review is to provide a the-
oretical basis for the dependence of the supercurrent I
S
on the phase difference
and to discuss the forms this
dependence takes in Josephson junctions of different
types: superconductor-normal-superconductor (SNS),
superconductor-insulator-superconductor (SIS), double-
barrier (SINIS), superconductor-ferromagnet-super-
conductor (SFS), and superconductor two-dimensional
electron gas superconductor (S-2DEG-S) junctions,
and superconductor-constriction-superconductor (ScS)
point contacts. Unconventional symmetry in the order
parameter of a high-T
c
superconductor, as manifested in
the current-phase relation (C⌽R) will also be discussed.
Recently, a new experimental tool has been developed
(Rifkin and Deaver, 1976; Il’ichev, Zakosarenko, IJssel-
stein, et al., 1998a; Il’ichev, Zakosarenko, Schultze, et al.,
2000) and applied to the study of the C⌽R for a variety
of Josephson junctions. We shall describe this method
and its practical applications. It is important that the
method has a resolution permitting the study of weak
links with Josephson energies smaller than the thermal
energy. We shall illustrate by a number of examples the
importance of these experimental studies for fundamen-
tal physics and applications. The examples include mea-
surements of small critical currents in Josephson junc-
tions, a calibration of second oxidation in SINIS
junctions, and studies of hybrid S-2DEG-S Josephson
devices, C⌽R anomalies in high-T
c
superconducting
junctions, physics of
states in SFS junctions and qubit
structures.
The main emphasis in this review is on the general
nature of the Josephson effect and on fundamental
physical mechanisms that control the C⌽R. At the same
time, some details are provided regarding types of weak
links and their fabrication. References to original ex-
perimental papers are given in the appropriate places.
II. dc JOSEPHSON EFFECT
Josephson (1962) predicted that a supercurrent I
S
could exist between two superconductors separated by a
thin insulating layer and that its value would be propor-
tional to the sine of the difference
⫽
1
⫺
2
of the
phases of the superconductor order parameters
⌬
1
exp
兵
i
1
其
and ⌬
2
exp
兵
i
2
其
,
I
S
共
兲
⫽ I
C
sin
, (1)
the so-called dc Josephson effect. The maximum current
I
C
in that C⌽R is the critical current. Further studies
have shown (de Gennes, 1966; for a review see Likharev,
1979) that the effect extends beyond Josephson’s predic-
tions and can exist if superconductors are connected by
a ‘‘weak link’’ of any physical nature (normal metal,
semiconductor, superconductor with smaller critical tem-
perature, geometrical constriction, etc.)
The C⌽R is an important characteristic of a Joseph-
son junction. In only a few cases does the C⌽R reduce
1
See, for example, Kupriyanov and Likharev, 1990; Braginski,
1992; Sigrist and Rice, 1995; van Harlingen, 1995; Delin and
Kleinsasser, 1996; Kashiwaya and Tanaka, 2000; Tsuei and
Kirtley, 2000; Lo
¨
fwander et al., 2001; Makhlin et al., 2001;
Hilgenkamp and Mannhart, 2002.
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Golubov, Kupriyanov, and Il’ichev: The current-phase relation in Josephson junctions
Rev. Mod. Phys., Vol. 76, No. 2, April 2004
to the familiar sinusoidal form of Eq. (1), which is ordi-
narily used to study the dynamics and ultimate perfor-
mance of analogous and digital devices based on Joseph-
son junctions (see van Duzer and Turner, 1981; Barone
and Paterno, 1982; Likharev, 1986; Gallop, 1991; Or-
lando and Delin, 1991; Tinkham, 1996; Schmidt, 1997;
Kadin, 1999).
The physics of the dc Josephson effect can be under-
stood if we take into account that a quasiparticle located
in the weak link cannot penetrate directly into a super-
conductor if its energy is smaller than the superconduct-
ing energy gap. However, another form of charge trans-
port, the so-called Andreev reflection (Andreev, 1964),
can occur. An electron with momentum k impinging on
one of the interfaces is converted into a hole moving in
the opposite direction, thus generating a Cooper pair in
a superconductor. This hole is consequently Andreev re-
flected at the second interface and is converted back to
an electron, leading to the destruction of a Cooper pair
(see Fig. 1). As a result of this cycle, a pair of correlated
electrons is transferred from one superconductor to an-
other, creating a supercurrent flow across a junction.
Since Andreev reflection amplitudes depend on the cor-
responding phases
1,2
, the resulting current depends on
the phase difference
, thus leading to the dc Josephson
effect.
Due to the electron-hole interference in the quantum
well, formed by the pairing potentials of the supercon-
ducting electrodes, standing waves with quantized en-
ergy E
AB
appear in the weak-link region The corre-
sponding quantum states are referred to as Andreev
bound states. The physics of Andreev bound states in
Josephson junctions has been studied extensively, start-
ing with the pioneering work of Kulik (1969).
It follows from the microscopic theory of supercon-
ductivity (see review by Lambert and Raimondi, 1998;
Belzig et al., 1999; Kopnin, 2001) that in stationary situ-
ations the supercurrent across a Josephson junction,
I
S
共
兲
⬀
冕
⫺ ⬁
⬁
dE
关
1⫺ 2 f
共
E
兲
兴
Im
兵
I
E
共
兲
其
, (2)
depends on the electron energy distribution function
f(E) and the spectral current Im
兵
I
E
其
. The spectral cur-
rent incorporates information on the energy distribution
of the Andreev bound state in a junction. Im
兵
I
E
其
de-
pends on the distance d, between the superconductors
and the transport parameters of the junction’s materials
(resistivities
1,2
, Fermi velocities
v
F1,2
, and interface
parameters).
In structures where the momentum of an electron in
the weak-link region is a good quantum number (so-
called ‘‘clean’’ Josephson junctions), Andreev bound
states form a regular structure in energy and Im
兵
I
E
(
)
其
is
peaked at the corresponding energies (Kulik, 1969; Ishii,
1970; Bardeen and Johnson, 1972; Bagwell, 1992; Schu
¨
s-
sler and Kummel, 1993; Tang, Wang, and Zhang, 1996;
Tang, Wang, and Zhu, 1996).
An increase of the degree of disorder in the weak link
leads to a broadening and decrease of the amplitudes of
the peaks. The disorder generates a distribution of the
lengths of the electron trajectories in the weak link.
Therefore Im
兵
I
E
(
)
其
for a disordered junction is a
weighted average of the ballistic result over the corre-
sponding distribution. This makes the spectral current
Im
兵
I
E
(
)
其
a continuous function of energy. Other
sources of the broadening of Andreev bound states are
the Doppler shifts of Andreev bound-state energies due
to current flow (see, for example, Fogelstro
¨
m et al.,
1997) and many-body lifetime effects (Freericks et al.,
2002).
The electron energy distribution function in Eq. (2)
defines the population of Andreev bound states at a
given temperature. Thus Eq. (2) shows that the whole
supercurrent I
S
(
) is the sum of the partial currents
transported via Andreev bound states. Therefore one
can modify the shape of I
S
(
) in two ways: (a) by modi-
fying the spectral current Im
兵
I
E
(
)
其
, changing the mate-
rial parameters or the geometry of a junction; (b) by
manipulating the occupation numbers of Andreev
bound states, i.e., creating a nonequilibrium distribution
function f(E) in a weak link.
Nonequilibrium effects in weak links have been inten-
sively studied by many authors.
2
These effects occur ei-
ther in the nonstationary regime, when I⬎ I
C
and a volt-
age is generated across the junction (stimulation by
current), under microwave irradiation, or in multitermi-
nal structures, when the junction is in a stationary re-
gime but voltage is applied via additional terminals.
In this review we shall restrict ourselves to the station-
ary Josephson effect in two-terminal weak links when
nonequilibrium effects do not play a role. This corre-
sponds to the above-mentioned case (a), when the C⌽R
is determined by the spectral current Im
兵
I
E
(
)
其
, depend-
2
See, for example, Zaitsev, 1976, 1993, 1995; Artemenko
et al., 1979; Zaikin and Zharkov, 1981; Aslamazov and Lem-
pitskii, 1982; Lempitskii, 1983; Zaikin 1983, 1988; Aslamazov
and Volkov, 1986; van Wees et al., 1991; Shumeiko et al., 1993;
Gorelik et al., 1995; Volkov, 1995; Chang and Bagwell, 1997;
Samuelsson et al., 1997, 2000, 2001; Volkov and Takayanagi,
1997; Morpurgo et al., 1998; Tolga Ilhan et al., 1998; Wilhelm
et al., 1998, 2000; Yip, 1998; Baselmans et al., 1999, 2001;
Kutchinsky et al., 1999, 2000; Neurohr et al., 1999; Heikkila
¨
et al., 2000, 2002; Seviour and Volkov, 2000; Sun et al., 2000;
Baselmans, Heikkila, et al. 2002; Baselmans, van Wees, et al.,
2002; Huang et al., 2002; Zhu et al., 2002; Brinkman et al.,
2003.
FIG. 1. Formation of Andreev levels in a Josephson junction.
An electron e and the Andreev-reflected hole h are shown. A
pair of correlated electrons is transferred from the left super-
conductor to the right one, creating a supercurrent flow across
a junction.
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Golubov, Kupriyanov, and Il’ichev: The current-phase relation in Josephson junctions
Rev. Mod. Phys., Vol. 76, No. 2, April 2004
ing on material parameters, junction geometry, and cur-
rent flow.
A. General properties of the Josephson current-phase
relation
There are several properties of the C⌽R that are
rather general and depend neither on the junction’s ma-
terials and geometry nor on the theoretical model used
to describe the processes in the junction.
(1) A change of phase of the order parameter of 2
in
any of the electrodes is not accompanied by a
change in their physical state. Consequently this
change must not influence the supercurrent across a
junction, and I
S
(
) should be a 2
periodic func-
tion,
I
S
共
兲
⫽I
S
共
⫹2
兲
. (3)
(2) Changing the direction of a supercurrent flow across
the junction must cause a change of the sign of the
phase difference; therefore
I
S
共
兲
⫽⫺I
S
共
⫺
兲
. (4)
Note that Eq. (4) is violated in superconductors with
broken time-reversal symmetry, leading to sponta-
neous currents. These effects have been discussed
for Josephson junctions between superconductors
with unconventional pairing symmetry (Geshken-
bein et al., 1986, 1987; Yip, 1995; Fogelstrom et al.,
1997; Tanaka and Kashiwaya, 1997b; Sigrist, 1998)
and for superconductor-ferromagnet-supercon-
ductor (SFS) junctions (Krawiec et al., 2002). Dis-
cussion of spontaneous currents and their mecha-
nisms is beyond the scope of this review and Eq. (4)
is fulfilled in all considered cases.
(3) A dc supercurrent can flow only if there is a gradient
of the order-parameter phase. Hence, in the absence
of phase difference,
⫽ 0, there should be zero su-
percurrent,
I
S
共
2
n
兲
⫽0, n⫽ 0,⫾ 1,⫾ 2,.... (5)
(4) It follows from (1) and (2) that the supercurrent
should also be zero at
⫽
n,
I
S
共
n
兲
⫽0, n⫽ 0,⫾ 1,⫾ 2,...; (6)
therefore it is sufficient to consider the I
S
(
) only in
the interval 0⬍
⬍
.
As follows from Eqs. (1)–(4), I
S
(
) can in general be
decomposed into a Fourier series (see, for example,
Tanaka and Kashiwaya, 1997b)
I
S
共
兲
⫽
兺
n⭓1
兵
I
n
sin
共
n
兲
⫹ J
n
cos
共
n
兲
其
, (7)
where I
n
and J
n
are coefficients to be determined. The
J
n
vanish if time-reversal symmetry is not broken.
The free energy E
J
of a Josephson junction is gener-
ally given by an integral (ប/2e)
兰
0
I
S
(
)d
. If the C⌽R
is sinusoidal, I
S
(
)⫽ I
C
sin
(Fig. 2, curve a), the de-
pendence of E
J
on
has the standard form
E
J
共
兲
⫽
បI
C
2e
共
1⫺ cos
兲
, (8)
also well known as the 2
-periodic ‘‘washboard poten-
tial.’’ The standard junction has an energy minimum at
⫽ 0 when there is no current flowing across the junc-
tion. For junctions with a small capacitance, the charging
energy becomes important and should be added to
E
J
(
) (Likharev, 1986; Tinkham, 1996).
Possible types of C⌽R in Josephson junctions are
shown in Fig. 2. Curve a is the standard sinusoidal C⌽R.
The critical current I
C
⫽ max
关
I
S
(
)
兴
can be achieved at
both
max
⭐
/2 and
max
⭓
/2 (the curves b and c, re-
spectively).
An interesting special case is the junction with I
C
⬍ 0
(Fig. 2, curve d), the so-called
junction (Bulaevskii
et al., 1977). According to Eq. (8), such a junction has an
energy minimum at
⫽
, i.e., it provides a phase shift
of
in the ground state.
A
junction may be used as the phase inverter in
superconducting digital circuits. The so-called comple-
mentary Josephson digital devices (e.g., the
SQUID),
were discussed by Terzioglu and Beasley (1998). Ioffe
et al. (1999), Blais and Zagoskin (2000), and Zagoskin
(2002) proposed
junctions as candidates for engineer-
ing a quantum two-level system, or qubit, which is the
basic element of a quantum computer. Blatter et al.
(2001) suggested structures including arrays of conven-
tional (0) and
junctions for the realization of a ‘‘quiet’’
phase qubit. A topologically stable qubit, based on a
triangular or more complicated
junction array, was dis-
cussed by Blatter et al. (2001) as well.
In some special cases I
S
(
) may cross the horizontal
axis at a position in between
⫽ 0 and
⫽
, as shown
in Fig. 2, curve e. The energy-phase relation in this case
has two minima at
⫽ 0 and
⫽
.
FIG. 2. Various types of current-phase relation (C⌽R): curve
a, the standard sinusoidal C⌽R; curves b and c, deviations
from the standard C⌽R, when the critical current is achieved
at
max
⭐
/2 and
max
⭓
/2, respectively; curve d,a
junction:
curve e has the C⌽R whose energy-phase relation has two
minima at
⫽ 0 and
⫽
; curve f, multivalued C⌽R that
does not correspond to a true Josephson effect and may have
various causes (see the text).
414
Golubov, Kupriyanov, and Il’ichev: The current-phase relation in Josephson junctions
Rev. Mod. Phys., Vol. 76, No. 2, April 2004
One might consider the possibility of a multivalued
C⌽R with I
S
(
n)⫽0, as shown in Fig. 2, curve f. How-
ever, structures with a multivalued C⌽R cannot be re-
garded as real Josephson junctions. Such a current-
phase relation may be realized either due to Abrikosov
vortices (see Kupriyanov et al., 1975; Likharev, 1979) or
phase-slip centers inside the junction (Ivlev and Kopnin,
1984; Martin-Rodero et al., 1994; Sols and Ferrer, 1994).
The C⌽R in this case is controlled by completely differ-
ent physical processes than in the standard Josephson
junctions.
I
S
(
) dependences of the type shown in Fig. 2 by
curves b and c may be roughly described by the expres-
sion (see Likharev, 1976, 1979; Zubkov et al., 1981;
Schu
¨
ssler and Ku
¨
mmel, 1993)
I
£
共
兲
⫽ I
C
sin
关
⫺ £I
£
共
兲
兴
, (9)
by which a weak link is represented as an ideal Joseph-
son junction with I
S
(
)⬀sin
connected in series with a
nonlinear inductance £. This inductance can be either
positive (Fig. 2, curve b) or negative (Fig. 2, curve c)
and is a consequence of the specific properties of a su-
perconducting condensate. The presentation of the C⌽R
in the form of Eq. (9) may be useful for the analysis of a
mode of operation of a system containing several Jo-
sephson junctions with a nonsinusoidal C⌽R.
B. Basic Josephson structures
The Josephson effect may be observed in a variety of
structures. To realize such structures it is enough to fab-
ricate a ‘‘weak’’ place interrupting the supercurrent flow
in a superconductor or suppress the ability of a super-
conductor to carry a current, e.g., by deposition of a
normal metal on its top, by implantation of impurities
within a restricted volume, or by changing the geometry
of a sample. All of these possibilities have been exten-
sively discussed in reviews and textbooks (see Likharev,
1979, 1986; van Duzer and Turner, 1981; Barone and
Paterno, 1982; Gallop, 1991; Orlando and Delin, 1991;
Tinkham, 1996; Schmidt, 1997; Kadin, 1999). Among
them only a few configurations have importance for
practical applications. These are point contacts, tunnel
junctions, sandwiches, and variable-thickness bridges
(having a normal metal, a semiconductor, or a weak fer-
romagnet as a weak-link material), and double-barrier
structures.
We shall begin our analysis by considering the dc Jo-
sephson effect in point contacts in Sec. III. The point
contact is a structure with strong supercurrent concen-
tration and provides the simplest system in which the
Josephson effect may be observed. A point contact may
be fabricated by placing a superconducting tip on top of
a bulk superconductor or by depositing a supercon-
ductor on top of a superconductor-dielectric bilayer with
a submicrometer hole in the insulator which defines a
small contact area. Contacts based on a two-dimensional
electron gas are also close to this type of structure. Point
contacts are well-defined systems in which the funda-
mental physics of the Josephson effect can be most eas-
ily studied theoretically and verified experimentally, be-
cause all nonlinear, nonstationary, and nonequilibrium
processes are localized within the weak link, while the
electrodes may be considered as undisturbed (i.e., in
equilibrium).
Fabrication and measurements of ultrasmall super-
conducting point contacts, in which the constriction size
is reduced towards atomic dimensions, was reported by
several groups in atomic break junctions and nanotubes
(Muller et al., 1994; van der Post et al., 1994; Vleeming
et al., 1994; Scheer et al., 1997, 2001; Ludoph et al., 2000;
Buitelaar et al., 2002, 2003; Agrait et al., 2003). Such
atomic contacts have proven a rich test bed for concepts
from mesoscopic physics like multiple Andreev reflec-
tion, shot noise, conductance quantization and fluctua-
tions, and dynamical Coulomb blockade.
In tunnel junctions, discussed in Sec. IV, the weak
place is formed by a dielectric layer separating two su-
perconducting electrodes. An ideal tunnel junction is
characterized by a sinusoidal C⌽R. In many real tunnel
junctions, such as those based on NbAl/AlO
x
/Nb tech-
nology (Gurvitch et al., 1983), the base electrode has the
form of a NbAl bilayer. The proximity effect may
strongly influence the supercurrent in such junctions,
while the C⌽R remains sinusoidal (see Golubov, Ku-
priyanov, and Lukichev, 1984; Golubov and Kupriyanov
1988, 1989; Golubov, Gurvitch, et al., 1993; Golubov,
Houwman, et al., 1995). AlO
x
forms a thin dielectric
layer with a high potential barrier having an approxi-
mately trapezoidal barrier profile (Tolpygo, Cimpoiasu,
et al., 2003). In order to increase the current density up
to the values of 100–200 kA/cm
2
, required for fabrica-
tion of high-J
C
, intrinsically shunted tunnel junctions
(where J
C
is the critical current density), this barrier can
be made extremely thin, of the order of a few atomic
layers, by decreasing the degree of Al oxidation to a
level at which the formation of a disordered region is
expected rather than a high-quality AlO
x
barrier. The
properties of these junctions should be close to those of
superconductor-correlated metal-superconductor struc-
tures studied theoretically by Nikolic
´
et al. (2001) and
Freericks et al. (2002). The transport properties of high-
J
C
tunnel junctions were analyzed by Naveh et al. (2000)
and Rippard et al. (2002), who have shown that elec-
tronic transport across these structures may be domi-
nated by resonant tunneling via localized states in the
disordered region.
High-quality Nb-based tunnel junctions have also
been fabricated with AlN
x
barriers (see Iosad et al.,
2003; Lapitskaya et al., 2003). In tunnel structures hav-
ing NbN electrodes, either TiN
x
(see, for example, Ish-
izaki et al., 2003; Takeda et al., 2003; Uzawa et al., 2003;
Wang, Saito, et al., 2003) or MgO (see, for example,
Johnson et al., 2003) barriers have been successfully
used for fabrication of SIS mixers and RSFQ circuits.
There exists another type of tunnel junction consisting
of homogeneous electrodes divided by a thick and rela-
tively low potential barrier (as in high-T
c
junctions with
PrBaCuO barriers; see, e.g., Gao et al., 1990). The situ-
ation in structures with extended barriers may be more
415
Golubov, Kupriyanov, and Il’ichev: The current-phase relation in Josephson junctions
Rev. Mod. Phys., Vol. 76, No. 2, April 2004
