Macroscopic quantum phenomena in Josephson structures
The Josephson effect is a probe of unparalleled performances in the study of a variety of macroscopic quantum phenomena. In the present article an overview of important achievements and challenging trends is given referring, in particular, to macroscopic quantum tunneling and energy level quantizati...
Saved in:
| Date: | 2010 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2010
|
| Series: | Физика низких температур |
| Subjects: | |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/117535 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Macroscopic quantum phenomena in Josephson structures / A. Barone, F. Lombardi, G. Rotoli, F. Tafuri // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1098–1106. — Бібліогр.: 63 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-117535 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1175352017-05-25T03:02:33Z Macroscopic quantum phenomena in Josephson structures Barone, A. Lombardi, F. Rotoli, G. Tafuri, F. Quantum coherent effects in superconductors and normal metals The Josephson effect is a probe of unparalleled performances in the study of a variety of macroscopic quantum phenomena. In the present article an overview of important achievements and challenging trends is given referring, in particular, to macroscopic quantum tunneling and energy level quantization. The attention is mainly addressed to high-TC superconducting structures and recent investigations concerning nanostructures. 2010 Article Macroscopic quantum phenomena in Josephson structures / A. Barone, F. Lombardi, G. Rotoli, F. Tafuri // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1098–1106. — Бібліогр.: 63 назв. — англ. 0132-6414 PACS: 74.72.–h, 74.81.Fa, 75.20.–g http://dspace.nbuv.gov.ua/handle/123456789/117535 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Quantum coherent effects in superconductors and normal metals Quantum coherent effects in superconductors and normal metals |
| spellingShingle |
Quantum coherent effects in superconductors and normal metals Quantum coherent effects in superconductors and normal metals Barone, A. Lombardi, F. Rotoli, G. Tafuri, F. Macroscopic quantum phenomena in Josephson structures Физика низких температур |
| description |
The Josephson effect is a probe of unparalleled performances in the study of a variety of macroscopic quantum phenomena. In the present article an overview of important achievements and challenging trends is given referring, in particular, to macroscopic quantum tunneling and energy level quantization. The attention is mainly addressed to high-TC superconducting structures and recent investigations concerning nanostructures. |
| format |
Article |
| author |
Barone, A. Lombardi, F. Rotoli, G. Tafuri, F. |
| author_facet |
Barone, A. Lombardi, F. Rotoli, G. Tafuri, F. |
| author_sort |
Barone, A. |
| title |
Macroscopic quantum phenomena in Josephson structures |
| title_short |
Macroscopic quantum phenomena in Josephson structures |
| title_full |
Macroscopic quantum phenomena in Josephson structures |
| title_fullStr |
Macroscopic quantum phenomena in Josephson structures |
| title_full_unstemmed |
Macroscopic quantum phenomena in Josephson structures |
| title_sort |
macroscopic quantum phenomena in josephson structures |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2010 |
| topic_facet |
Quantum coherent effects in superconductors and normal metals |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117535 |
| citation_txt |
Macroscopic quantum phenomena in Josephson structures / A. Barone, F. Lombardi, G. Rotoli, F. Tafuri // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1098–1106. — Бібліогр.: 63 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT baronea macroscopicquantumphenomenainjosephsonstructures AT lombardif macroscopicquantumphenomenainjosephsonstructures AT rotolig macroscopicquantumphenomenainjosephsonstructures AT tafurif macroscopicquantumphenomenainjosephsonstructures |
| first_indexed |
2025-07-08T12:25:44Z |
| last_indexed |
2025-07-08T12:25:44Z |
| _version_ |
1837081607269253120 |
| fulltext |
© A. Barone, F. Lombardi, G. Rotoli, and F. Tafuri, 2010
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11, p. 1098–1106
Macroscopic quantum phenomena
in Josephson structures
A. Barone
Dip. Scienze Fisiche, Facoltá di Ingegneria, Universitá di Napoli Federico II
Piazzale Tecchio 80, Napoli 80125 and CNR-SPIN Napoli, Italy
E-mail: barone@na.infn.it
F. Lombardi
Quantum Device Physics Laboratory, Dept. of Microtechnology and Nanoscience
Chalmers University of Technology, SE-412 96, Goteborg, Sweden
G. Rotoli and F. Tafuri
Dip. Ingegneria dell'Informazione, Facoltá di Ingegneria, Seconda Universitá di Napoli
via Roma 29, Aversa (CE) 81031 and CNR-SPIN Napoli, Italy
Received April 13, 2010
The Josephson effect is a probe of unparalleled performances in the study of a variety of macroscopic
quantum phenomena. In the present article an overview of important achievements and challenging trends is
given referring, in particular, to macroscopic quantum tunneling and energy level quantization. The attention is
mainly addressed to high- CT superconducting structures and recent investigations concerning nanostructures.
PACS: 74.72.–h Cuprate superconductors;
74.81.Fa Josephson junction arrays and wire networks;
75.20.–g Diamagnetism, paramagnetism, and superparamagnetism.
Keywords: macroscopic quantum phenomena, Josephson structures.
1. Introduction
The importance of the Josephson effect [1–5] is widely
recognized for both the intrinsic relevance in the context of
superconductivity and for the large variety of realized and
potential applications. The Josephson effect still plays a
fundamental role in various challenging topics such as that
of a powerful probe of the symmetry of the order parame-
ter characterizing different classes of superconductive ma-
terials. There are various aspects of the Josephson effect of
paramount importance in the context of macroscopic quan-
tum phenomena. Among these, the quantum decay via ma-
croscopic quantum tunneling (MQT) deserves great atten-
tion as well as the energy level quantization (ELQ) and
other phenomena proper of quantum mechanics at a ma-
croscopic level, as the occurrence of macroscopic quantum
coherence in a Superconducting Quantum Intereference
Device (SQUID). Let us recall that also weakly coupled
Bose-Einstein condensates (BEC) systems are subject of
deep theoretical and experimental investigations on both
interference phenomena and the occurrence of Josephson
effect [6].
The nature of superconductivity in oxide compounds
lies in the background with its still mysterious origin. The
phenomenology of HTS encompasses a wide range of in-
teresting issues at the border of our understanding of solid-
state systems and at the limit of current capabilities of ma-
terial science and nano-technology. The Josephson junc-
tions have been playing an irreplaceable role in defining
crucial properties of HTS. The d-wave order parameter
symmetry (OPS) is probably the most remarkable example
[7,8]. If we imagine to be a few months before the discov-
ery of HTS, who would have imagined that in a few
months a supercurrent would have flown up to about 100
K? Who would have imagined a supercurrent between two
phase coherent electrodes up to about 100 K? What about
the thermal energy, the gap value, the Josephson coupling
energy, the charging energy, the coherence length, the crit-
ical stoichiometry, and so on? These considerations lead to
the first obvious feature, which is independent of the still
Macroscopic quantum phenomena in Josephson structures
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1099
mysterious origin of superconductivity in HTS, of their
very complicate structure, and so on: oxides enlarge the
occurrence of superconductivity to unexpected energy and
length scales. In this short review we try to give the feeling
of novel flavors of HTS on the Josephson effect, with spe-
cial attention to macroscopic quantum phenomena and to
mesoscopic effects.
2. Thermal and macroscopic quantum tunneling
activation
Quantum tunnelling on a macroscopic scale was consi-
dered by Sidney Coleman [9] in the context of ground state
metastability in the cosmological frame. The fate of the false
vacuum, was interpreted as its decay through barrier pene-
tration, toward true vacuum, a more stable state of the Un-
iverse. In the Josephson junction cosmolab analogy, the ma-
croscopic degree of freedom is the relative phase, φ,
between the two weakly coupled superconductors (or the
trapped magnetic flux, Φ, in a rf SQUID superconducting
loop). Let us consider the potential:
= / (2 )( cos( ) ).o COU I I−Φ π φ + φ (1)
(in Fig. 1 a small section of the washboard potential is
shown to focus on a single cell of the periodic structure,
used later to formulate the macroscopic quantum tunneling
problem) given by the sum of the free energy associated to
the Josephson junction barrier and a linear term in φ due
to the bias current I. COI represents the maximum Joseph-
son current. This potential can be also easily derived from
the the resistively and capacitively shunted junction
(RCSJ) model applied to a Josephson junction. The Jo-
sephson inductance LJ and capacitance C act as an anhar-
monic LC resonator (at zero voltage) with resonance fre-
quency 1/2= ( )P JL C −ω (plasma frequency), where =JL
2 2 1/2= / (2 cos( )) = / (2 [ ]) ,O CO O COI I IΦ π φ Φ π − =Pω
2 1/4(2 / ( )) / (1 ( / ) ))CO O COI C I I= π Φ − . Representing the
displacement current by a C capacitor and the sum of the
quasi-particle and insulator leakage current by a resistance
(R), we can devise an equivalent circuit for the junction:
= sin( ) / / .N COI I I V R CdV dt+ φ + + (2)
The second term contains the well-known dc Josephson
equation = sin( )C COI I φ . The noise source IN is asso-
ciated with its shunt resistance.
In the mechanical analogy of this problem we can refer
to a particle of mass 2= ( / 2 )Om CΦ π in such a wash-
board potential (see Eq. (1)) and identify the two states
corresponding to the particle at rest (φ constant) or running
down the slope (φ time-dependent). The motion of the par-
ticle is subject to damping given by 1/ ,Q where
= PQ RCω is the quality factor. Accordingly, from the
constitutive Josephson effect relations, these states will
correspond in the current-voltage (I–V) characteristics to
the zero voltage and the finite voltage state respectively. At
zero temperature such a transition will occur as soon as the
average slope of U(φ) increases up to a value producing the
absence of valleys (i.e. when the current bias reaches the
CI value) so that the particle can run down the slope.
Namely, φ becomes time-dependent and the switching to
the finite voltage state of the I–V curve occurs.
Depending on the entity of dissipation, conditions of
overdamping with single-valued I–V curves to underdamp-
ing generating highly hysteretical I–V curves can be rea-
lized. In the former case, the large dissipation will restore
the V = 0 state as soon as the current bias is reduced down
to the critical value while, in the underdamped regime, the
effect of the junction capacitance is dominant over dissipa-
tion. The mechanical analogy, is obvious referring to the
interplay between the values of the friction and the inertial
mass of the particle.
In the pure thermal regime, the escape rate for weak to
moderate damping ( > 1)Q is determined by the original
Kramer theory as
= exp
2
P
t
B
UA
k T
⎛ ⎞ω Δ
Γ −⎜ ⎟π ⎝ ⎠
where 3/2( ) = (4 2 / 3) (1 )JU a EΔ − γ is the barrier height
and is shown in Fig. 1 for 0= / CI Iγ close to 1, with
0 0= / 2J CE I Φ π . The prefactor A can be specified accor-
dingly to the various damping regimes. The escape rate
will be dominated by MQT at low enough temperature
[11,12]: for > 1Q and γ close to 1 it is approximated by
the expression for a cubic potential:
0.87= exp 1
2
P
q q
P
Ua
Q
⎡ ⎞ω ⎛ ⎞Δ
Γ − + ⎟⎢ ⎜ ⎟π ω ⎝ ⎠⎣ ⎠
U
��
TA
MQT �U
�
�1
��
Fig. 1. Detail of the washboard potential in the RCSJ model for a
finite value of the bias. UΔ represents the energy barrier. TA
stands for thermal activation (dotted line), MQT processes are
indicated for small (solid line) and large (dashed line) dissipation
[15] both from the ground state and the first excited state, respec-
tively. Microwaves can induce a transition from the ground to the
first exited state.
A. Barone, F. Lombardi, G. Rotoli, and F. Tafuri
1100 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11
where 1/2= [864 / ] .q Pa UπΔ ω The occurrence of such a
quantum activation observed in Josephson junctions shows
the validity of quantum mechanics at a macroscopic level
involving indeed a macroscopic variable, namely the rela-
tive phase φ. A complementary quantum phenomenon lies
in the existence of quantized energy levels (ELQ). Evi-
dence of such a feature is provided by the experiments
based on the microwave irradiation with consequent ener-
gy level hopping.
Macroscopic quantum tunnelling in the context of the
Josephson structures were proposed by Anderson [2],
Ivanchenko and Zilberman [11] and by Caldeira and Leg-
gett [12], who gave a quite complete description introduc-
ing the fundamental aspect of the effect of dissipation. The
crossover temperature between the thermal and the quan-
tum regimes is 2 1/2
cr = ( / 2 )[(1 1/ 4 ) 1/ 2 ]P BT k Q Qω π + −
[13]. Below crT quantum effects are dominant over ther-
mal ones [12,14]. A variety of successful experimental
investigations have been carried out to observe MQT and
verify the effect of dissipation in reducing the decay rate
by quantum activation [16,18]. Evidence of the transition
from thermal to quantum activation was clearly shown by
pioneering experiments concerning the measure of the de-
cay rate which decreases with the temperature down to Tcr
while, for cr<T T , a temperature independent activation
prevails. For an excellent description of experiments in this
context the reader is referred to the paper by John Clarke et
al. [19]. A further issue deserving interest lies in the phe-
nomenon of resonant macroscopic quantum tunneling
(RMQT) [20] resulting in the occurrence of sharp voltage
peaks due to a MQT process between levels in neighbour-
ing wells characterized by close energy values. This effect
has been experimentally confirmed by Rouse, Han and
Lukens [21] in a SQUID. Concerning the Macroscopic
Quantum Coherence (MQC), examples of relevant propos-
als and experiments can be found in [22–24].
3. Macroscopic quantum tunneling in HTS
In recent years, the interest in superconducting quantum
devices has been extended to high critical temperature su-
perconductors (HTS) also in view of the possible advantage
of d-wave OPS [7,8] for a quiet qubit [25]. This implies the
possibility of build so-called π-junction devices. The local
magnetization in π-loops, i.e., loops formed with an odd
number of π-junctions, could be used as the states of a qubit
device. The main advantage of such an unconventional qubit
device is that it works in absence of an external field bias.
Such properties of HTS devices could be also related to
the search for a «protected qubit». This last can be traced to
the seminal work of A.Yu. Kitaev [26] and applied to super-
conducting qubits by L.B. Ioffe et al. [25,27]. The basic idea
is that a topological object, say a magnetic «flux» configura-
tion over an array of Josephson junctions could have just at
the classical level before the quantum effects came into play,
has the property of to be insensitive to some perturbations
which are topological invariant of the system.
The use of topology for making a robust qubit, i.e., in-
sensitive to external world, can be found in the experiments
by Wallraff et al. [28]. They have shown that also fluxons in
annular Josephson junctions can behave as quantum objects
at low temperature and could be used in principle as qubits
when subject to a magnetic field induced potential [29].
HTS may be an interesting reference system for novel
ideas on key issues on coherence and dissipation in solid
state systems because of their unusual properties, in partic-
ular the presence of low energy quasi-particles due to
nodes in the d-wave OPS [30,31]. This has represented
since the very beginning a strong argument against the
occurrence of macroscopic quantum effects in these mate-
rials. Quantum tunnelling of the phase leads to fluctuating
voltage across the junctions which excites the low energy
quasi-particles specific for d-wave junctions, causing de-
coherence. Contributions to dissipation due to different
transport processes, such as channels due to nodal quasi-
particles, midgap states, or their combination, have been
identified and distinguished [32–35]. In particular cases,
decoherence times and quality factors were calculated con-
sidering the system coupled to an Ohmic heat bath. It has
also been argued that problems in observing quantum ef-
fects due to the presence of gapless quasi-particle excita-
tions can be overcome by choosing the proper working
phase point [33]. In particular, decoherence mechanisms
can be reduced by selecting appropriate tunnelling direc-
tions because of the strong phase dependence of the quasi-
particle conductance in a d-wave GB junction.
The search of macroscopic quantum effects become
feasible once high quality HTS Josephson junctions
[36,37] with significant hysteresis in the current-voltage
characteristics were available. We can distinguish two
classes of experiments, which are based on two different
complementary types of junctions: 1) MQT and ELQ
[30,31] on off-axis YBCO grain boundary biepitaxial JJs,
where the experiment has been designed to study d-wave
effects with a lobe of the former electrode facing the node
of the latter; 2) MQT and ELQ on intrinsic junctions on
single crystals of different materials [38,39], where d-wave
are expected to play a minor role [33,34]. The experiments
using GBs are more complicated because of the complexi-
ty of these junctions, but are very complete and allow to
address relevant issues on the effects of a d-wave OPS on
dissipation and coherence. Only GBs junctions can be
more easily integrated into circuits.
The GB biepitaxial junctions [40,41] used in [30,31]
had reproducible hysteretic behavior up to 90%. A specific
feature of these structures is the use of a (110)-oriented
CeO2 buffer layer, deposited on (110) SrTiO3 substrates.
YBCO grows along the [001] direction on the CeO2 seed
layer, while it grows along the [103]/[013] direction on
SrTiO3 substrates [41,42]. The presence of the CeO2 pro-
Macroscopic quantum phenomena in Josephson structures
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1101
duces an additional 45° in-plane rotation of the YBCO
axes with respect to the in-plane directions of the substrate
(Fig. 2,a). Atomically flat interfaces can be achieved in
appropriate conditions [40]. As a consequence, the GBs are
the product of two 45° rotations, a first one around the c-
axis, and a second one around the b-axis. This configura-
tion produces a 45° misorientation between the two elec-
trodes to enhance d-wave order parameter effects, by vary-
ing the interface orientation.
In addition the possibility to tune the critical current CI
through the interface orientation θ in complete agreement
with the predictions of a d-wave OPS (see Fig. 2,b) [41]
allows to select the junction for the MQT experiment
knowing the OPS configuration exactly. Specific angle
orientations can favor both junctions with a Fraunhofer-
like pattern (Fig. 2,c) and the spontaneous generation of
fractional vortices (Fig. 2,d) [43,37]. The suitable junction
can be therefore selected for the experiment. Since the in-
terest was mostly focused in those features that are distinct
from the case of low cT superconductor (LTS) junctions,
namely effects due to OPS, and dissipation due to low
energy quasi-particles, the junction in the tilt configuration
(angle θ = 0°) turns out to be the most interesting case for
the MQT and ELQ experiments. This configuration (lobe
to node) maximizes d-wave induced effects and allows
explore the effects of low energy quasi-particles.
Some new features of Josephson dynamics could be ac-
cessible in HTS junction configurations, as, for instance,
the role of Andreev bound states [44] and the intrinsic
doubly degenerate fundamental state [36,37]. The last is
due to unconventional Josephson Current-Phase Relation
(CPR) which shows the presence of higher harmonics
(sin 2 )ϕ caused by the d-wave symmetry of the order pa-
rameter [45]. The dynamics of a current biased JJ also
strongly depends also on the CPR. Up to now, the junction
features, which induce the sin 2φ component, are not un-
ambiguously identified in a system characterized by a facet-
ing of the grain boundary line [45]. A detailed description of
the features of a JJ assuming the presence of both first and
second harmonic components in the CPR (we neglect higher
harmonics due to our low junction barrier transparency) is
outside the scope of this review [30,46].
3.1. Experiments on YBCO biepitaxial Josephson
junctions
We follow [30] and [31] in reporting on the first expe-
rimental measurements on MQT in HTS JJs, where all
details can be found. The escape rate of the superconduct-
ing phase φ from a local minimum in the washboard poten-
tial into the running state as a function of temperature has
been investigated in analogy with experiments on low-Tc
junctions. Figure 3 shows a set of switching current proba-
bility distributions as a function of temperature for the bi-
epitaxial JJ. In the inset of Fig. 3 the switching current
probability distribution measured at T = 0.019 K is re-
ported along with the original I–V.
Fig. 2. (a) Sketch of the grain boundary structures in biepitaxial
CeO2 (b)-based out-of-plane biepitaxial junctions. The presence of
the CeO2 produces an additional 45° in-plane rotation of the YBCO
axes with respect to the in-plane directions of the substrate.
(b) Normalized critical current density JC vs angle θ for two sets
of c-axis tilt biepitaxial YBCO junctions, with width 10 μm (trian-
gles) and 4μm (stars). The solid lines connecting the symbols are
guides to the eye. The dotted line is the Sigrist–Rice-like formula
assuming pure 2 2x yd
−
pairing symmetry in this geometry [41]. In
the bottom inset the scheme of the junction is reported for three
different angles, along with the d-wave profiles of the two elec-
trodes Adapted from [41]. (c) I–V curves as a function of the mag-
netic field. A Fraunhofer profile of the critical current is visible.
The misorientation angle is in this case 60°. (d) Scanning SQUID
microscope image of a 200× 200 μm2 area, enclosing tilt-tilt and
twist-tilt in CeO2-based biepitaxial GBs.The GBs are marked by
the presence of spontaneous currents. The sample was cooled and
imaged at T = 4.2 K in nominally zero field. Adapted from [43].
M
�20 40 60 80
0
0.2
0.4
0.6
0.8
1.0
(001) YBaCuO
a
b
c
c
(110) CeO
SrTiO3 (110) substrate
a
b
a
b
modified d-wave
(103) YBaCuO
50
0
H (G)
–100
100V, mV
–1
1
c
d
–50
I,
A
�
�, deg
I
/I
C
CM
A
X
BP 10 m�
BP 4 m�
(001)
YBCO
(103)
YBCO
A. Barone, F. Lombardi, G. Rotoli, and F. Tafuri
1102 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11
The measured σ saturates below 50 mK, indicating a
crossover from the thermal to the MQT regime. An estima-
tion of R 100 Ω results for the electrode impedance us-
ing a microstrip transmission line model and SC 1.6 pF
(which is not far from the rough estimate of C obtained
from the hysteresis in the dc-I–V curve [47]). An estimation
of JC can be obtained by using Eq. (2) in MQT regime
[48]. The extracted JC ∼ 0.22 pF value gives a plasma
frequency / 2Pω π 2.6 GHz and a quality factor larger
than 1 in the quantum regime. The observed crossover tem-
perature ( 50T mK) between the thermal and the quan-
tum regimes is consistent with the predicted values from
1 ( / 2 )D
c P BT kω π [13,48].
To rule out that the saturation of σ is due to any spu-
rious noise or heating in the measurement setup the switch-
ing current probability distributions were measured for a
reduced critical current ( 0 =CI 0.78 A) by applying an
external magnetic field B = 2 mT. The width for = 2B mT
are shown in the inset of Fig. 4. The data in the presence of
a magnetic field clearly show a smaller width, which does
not saturate down to the base temperature.
In the temperature regime between 50 and 100 mK the
data start to follow the well known 2/3σ dependence due
to thermal escape [49] (see the dashed line in Fig. 4).
However, for 110T ∼ mK there is an hump, i.e., a transi-
tion to a 2/3σ dependence with lower values of (solid
line). These larger values in the low temperature region
correspond to an enhanced thermal escape rate. A possible
explanation of this effect is the onset of a second harmonic
component in the CPR at low temperatures, due to the low
junction barrier transparency [30].
Apart from being one of the keys to have low barrier
transparency, another important consequence of c-axis tilt
is the presence of a significant kinetic inductance in the
modeling of YBCO JJ. Indeed, in these junctions the pres-
ence of a kinetic inductance and a stray capacitance deter-
mine the main difference in the washboard potential mak-
ing the system behavior depending on two degrees of
freedom [31]. The YBCO JJ is coupled to this LC-circuit
(Fig. 5,a) and the potential become two-dimensional (2D).
Similar behavior has been observed in a low-TC dc super-
conducting quantum interference device [50].
Fig. 3. Switching current probability distribution for 0 =CI 1.40
μA at B = 0 T for different bath temperatures bathT . In the inset
the switching current probability distribution measured at T =
= 0.019 K is reported along with the original I–V. Adapted from
Bauch et al. [30].
100
80
60
40
20
0
0.8 0.9 1.0 1.1 1.2 1.3
P,
A
�
–
1
V, mV
0.9
0.7
0.5
0.3
0.21
0.1
0.019
1.0
2.00 1.0
2.0
0
0.019 K
I,
A
�
Fig. 4. The measured σ saturates below 50 mK, indicating a
crossover from the thermal to the quantum regime. The width σ
for B = 2 mT and the data for B = 0 mT are shown in the inset.
Adapted from Bauch et al. [30].
60
40
30
20
10
T , mKbath
0 200 400 600 800
14
13
12
11
10
9
20 40 60
T*
50
T, mk
,
n
A
Fig. 5: Measured switching current probability distribution P(I)
in presence of microwaves at a frequency of 850 MHz and tem-
perature T = 15 mK. The applied power at the room temperature
termination varies from –20 to –14 dBm. On the right the MQT
processes correspond to the switching current probability distri-
bution P(I). Adapted from Bauch et al. [31]
a
L1 C1
Y I
1
0 �0
1
0
�1
�0
1
0
�1
0 20 40 60 80
1.28
1.26
1.24
–20 dBm
–17 dBm
–16 dBm
–14 dBm
Distribution ( ), 1/ AP I �
S
w
it
ch
in
g
cu
rr
en
t,
I,
A
�
b
Macroscopic quantum phenomena in Josephson structures
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1103
The coupling with LC can be described by the follow-
ing 2D potential [48,51]:
21( , ) / = ( ) cos
2S J j S S jU Eφ φ φ −φ − γφ − φ
β
where 0= (2 / )S j S SI Lφ φ + π Φ is the phase difference
across the shunt capacitor SC and SI the current in the
inductance .SL It has been shown that the barrier height
UΔ is the same found in the 1D case [51,48]. The 2D
potential modifies the value of the crossover temperature
and, in general, of both thermal and MQT escape rates. In
particular in the experiment reported in [31] the LC values
was 0 07.2( / 2 = 7.2 0)S C JL I LΦ π and 7.3 .S JC C
In analogy to the LTS case the normalized bias current
γ is ramped from zero to a value near to 1, at finite temper-
ature, the junction may switch into a finite voltage state for
a bias current < 1. This corresponds to the particle escaping
from the well either by a thermally activated process or by
tunneling through the barrier potential (MQT). In the pure
thermal regime, the escape rate for weak to moderate
damping ( > 1Q ) is determined by
2= exp
2
D R
t t
B
Ua
k T
⎛ ⎞ω Δ
Γ −⎜ ⎟π ⎝ ⎠
where 3/2= (4 2 / 3) (1 )JU EΔ − γ is the barrier height for
γ close to 1, and Rω is the attempt frequency in the well.
Explicit expression for 2D thermal prefactor 2D
ta can be
found in [48]. In the limit of large SL and SC (as in expe-
riment [31]) this correction is small and it is mainly due to
the shift of the attempt frequency which is lower than stan-
dard JJ plasma frequency .Pω
The escape rate will be dominated by MQT at low
enough temperature for > 1Q the expression for a 2D po-
tential is:
2 536= exp 1
2 5 2
D JP
q q
P S
LUa
L
⎡ ⎤⎛ ⎞ω Δ
Γ − +⎢ ⎥⎜ ⎟π ω⎢ ⎥⎝ ⎠⎣ ⎦
where
1/2
5
= 864 ( / ) 1 .
2
J
q P
S
L
a U
L
⎡ ⎤⎛ ⎞
π Δ ω +⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦
So the MQT
rate is reduced by the factor 5 / 2J SL L [48]. From above
Eqs. (1) and (2) also an expression for ( )Tσ the HMFW of
( )P I switch distribution can be numerically calculated
and compared with experiments.
The theory reported in Ref. 48 is in excellent agreement
with experimental escape rates, though, at the moment,
cannot explain the hump structure of σ near 0.1 K. The
problem can be related to a correct description of dynami-
cal/thermal population of excited states in the metastable
well, at present neglected in the LC-circuit model [48].
The LC-circuit model, the so-called «shell» circuit, is
also of great importance to explain the Energy Level Quan-
tization (ELQ) experiment reported in Ref. 31. Micro-
waves at frequency rfω were transmitted to the junction
via a simple dipole antenna at a temperature below .cT
When rfω of the incident radiation (or multiples of it)
coincides with the bias current-dependent level separation
of the junction, 10 ( ) = rfmν γ ω , the first excited state is
populated. Here, m is an integer number corresponding to
an m-photon transition from the ground state to the first
excited state. Fig. 5,b shows the evolution of the switch-
ing-current histogram as a function of the applied micro-
wave power for the = 3m three photon process.
At low power values (–20 dBm), the escape is basically
from the ground state, since the occupation probability of
the first excited state is negligible. When the applied power
is increased (–17 dBm and –16 dBm), the first excited state
starts to be populated. Then in the histogram two peaks ap-
pear corresponding to tunneling from both the first excited
1Γ and ground 0Γ states. The escape from the first excited
state is exponentially faster and dominates, and the switch-
ing current distribution is again single peaked at –14 dBm.
From the Lorentzian-shape of the escape rate, a Q value of
the order of 40 is extracted [31], comparable with the first
best results obtained in LTS junctions.
Specific effects related to stray capacitance and large
kinetic inductance have been discussed both in the original
paper [31] and in subsequent papers [48].
The observation of quantum tunneling, narrow width of
excited states, and a large Q value support the notion of
«quiet» qubits based on d-wave symmetry superconductor,
but the meaning of the experiments goes beyond. There
may be some mechanism preventing the low-lying quasi-
particles in the d-wave state from causing excessive dissipa-
tion. It could be also supposed the presence of some kind of
condensation mechanism of quasiparticles, in general
agreement with the HTS SU(2) slave-boson model, where
the physical properties of the low lying quasiparticles are
found to resemble those in BCS theory [52]. The existence
of a subdominant imaginary s-wave component of the order
parameter inducing a gapped excitation spectrum could be
another possible explanation, probably more related to the
presence of the junction interface. This last possibility has
been discussed in various experiments available in literature,
but there is neither a convincing reproducible proof nor a
neat definition of the controllable experimental conditions
which lead to this effect [36,37].
4. Mesoscopic effects and coherence in HTS
nanostructures
Nanotechnology can provide another path to study co-
herence and quasi-particle relaxation processes in HTS.
The ultimate limit of GB performances also in terms of
yield and reproducibility, will be possibly achieved when
A. Barone, F. Lombardi, G. Rotoli, and F. Tafuri
1104 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11
junction dimensions get closer to the characteristic scaling
lengths of HTS (i.e., coherence length, charge domains,
and so on) and to the typical size of GB facets, which are
one of the main sources of the lack of uniformity of the
transport properties of GBS junctions. Nanoscale junctions
have therefore full potentials to isolate intrinsic features
of HTS systems, and represent the ideal tool to better ad-
dress the interesting topic of coherence in strongly corre-
lated d-wave superconductors. Benefits will be obviously
extended to applications based on HTS junctions.
The first studies on bicrystal submicron JJs have given
encouraging results as the reduction of decoherence [53],
the presence of the 2ϕ component [54], and of Andreev
bound states [55].
Recently submicron biepitaxial junctions have been rea-
lized down to about 500 nm by using both e-beam lithogra-
phy and C and Ti masking [56]. This step is even more sig-
nificant because applied to the off-axis biepitaxial junctions,
which have shown the macroscopic quantum effects, and to
be sensitive to directional transport along the lobes or the
nodes of the d-wave OPS. Yield and reproducibility have
been improved on this width scale. These improvements
reflect the advances registered in patterning simple nano-
bridges, which have been reproducibly scaled on c-axis
YBCO down to about 100 nm [57]. Studies on flux dynam-
ics have been also realized in nanorings of inner and exter-
nal radius of about 150 nm and 300 nm, respectively [58].
This classical controllable «top-down» approach is
going to be accompanied by some sort of «bottom-up»
techniques, which fund on the intrinsic nature of GB. The
complex growth process may determine self-assembled
nanochannels of variable dimensions, ranging typically
from 20 nm to 200 nm, often «enclosed» in macroscopic
impurities. Even if this very last technique is not ideal on
the long range for applications, since it needs an additional
critical step to locate the nanobridges and etch the HTS
thin film, it can be really helpful to understand the ultimate
limit of the junction performances and to understand the
transport mechanisms.
An example of how to use the natural self-assembling to
extract information on the physics of HTS Josephson junc-
tions, has been recently given on a study on universal con-
ductance fluctuations (UCF) in magnetic field in YBCO
biepitaxial Josephson junctions. This is of relevance to
investigate coherent quantum behavior in HTS [59,60].
Structural investigations allow first to locate macroscopic
impurities which enclose the conducting channel, whose
size is roughly confirmed by the period of the magnetic
pattern of the critical current. At low temperatures, quan-
tum coherence can be monitored in the conductance G of
a normal metallic sample of length xL attached to two
reservoirs [61,62]. The electron wave packets that carry
current in a diffusive wire have minimum size of the order
of > >> .T xL L l Here l is the electron mean free path in
the wire and TL is the thermal diffusion length (D is the
diffusion constant). The first inequality is satisfied at rela-
tively low temperatures as far as 2<< /B C xk T D Lε ≈
being Cε the Thouless energy. Conductance fluctuations
become appreciable at low temperatures, in the whole
magnetic field range. At low voltages ( << CeV ε ), the
system is in the regime of universal conductance fluctua-
tions: the variance 2< >g of the dimensionless conduc-
tance 2= / (2 / )g G e is of order of unity. The fluctua-
tions are nonperiodic, and have all the typical charac-
teristics of mesoscopic fluctuations [61,62]. Studies have
been carried out at different voltages and non-equilibrium
conditions. An energy scale of the order of 1 meV arises
naturally from the analysis of the autocorrelation function
of the conductance as a function of the voltage [59,60].
This has been identified as the Thouless energy Cε , and its
value is consistent with a size of the channel of the order of
100 nm. This is proportional to the inverse time an electron
spends in moving coherently across the mesoscopic sam-
ple. Quasi-particles seem to travel coherently across the
junction even if >> CV ε . Hence, microscopic features
of the weak link appear as less relevant, in favor of meso-
scopic, non local properties. In this case, the quasi-particle
phase coherence time ϕτ does not seem to be limited by
energy relaxation due to voltage induced nonequilibrium.
The remarkably long lifetime of the carriers, found in these
experiments, appears to be a generic property in high-TC
YBCO junctions as proved by optical measurements [63]
and by macroscopic quantum tunneling [30,31].
5. Conclusions
The great interest of the results reviewed in this chapter
lies in the combination of the stimulating subject of macros-
copic quantum phenomena (MQT, ELQ) with the new im-
portant clues that such phenomena provide on underlying
aspects of the physics of HTS. We have focused on macros-
copic quantum decay phenomena, as one of the most excit-
ing expressions of the Josephson effect. A system which
displays macroscopic quantum effects despite the presence
of nodes in the order parameter symmetry and therefore of
low energy quasi-particles, raises several challenging issues
on dissipation mechanisms and on the peculiar coherence
phenomena occurring in Josephson systems and in HTS. We
believe that the progress in quantum engineering and in na-
notechnologies will represent an invaluable additional drive
force to further address advanced topics on the Josephsin
effect and on macroscopic quantum phenomena. Once the
true «intrinsic» transport channels across junctions are con-
trolled and isolated from «extrinsic» contributions, which
result from the complex morphology of the junctions (facet-
ing, and so on), not only we will get closer to the basic fea-
tures of HTS (possibly stripes, spin-charge separation, ...)
but we will be also entitled to have a more complete scena-
rio on the Josephson effect, analogies and differences be-
tween HTS and LTS JJs [1–5].
Macroscopic quantum phenomena in Josephson structures
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1105
We acknowledge support from EC Strep Project MI-
DAS–Macroscopic Interference Devices for Atomic and
Solid State Physics: Quantum Control of Supercurrents.
We would like to thank Thilo Bauch, John R. Kirtley,
Daniela Stornaiuolo and Arturo Tagliacozzo for valuable
discussions.
1. B.D. Josephson, Phys. Lett. 1, 251 (1962).
2. P.W. Anderson, Special Effects in Superconductivity, in:
Lectures on the Manybody Problem, E.R. Caianiello (ed.),
Vol. II p. 113, Academic Press, Ravello (1963).
3. I.O. Kulik and K. Yanson, The Josephson Effect in
Superconductive Tunneling Structures (Israel Program of
Scientific Translations), Jerusalem (1972).
4. A. Barone and G. Paternó, Physics and Applications of the
Josephson Effect, John Wiley, New York (1982).
5. K.K. Likharev, Dynamics of Josephson Junctions and
Circuits, Gordon and Breach, New York (1986); K.K.
Likharev, Rev. Mod. Phys. 51, 101 (1979).
6. A. Barone, Weakly Coupled Macroscopic Quantum Systems:
Likeness with Difference, I.O. Kulik and R. Ellialtioglu
(eds.), Kluwer Academic Publishers (2000), p. 301.
7. D.J. Van Harlingen, Rev. Mod. Phys. 67, 515 (1995).
8. C.C. Tsuei and J.R. Kirtley, Rev. Mod. Phys. 72, 969 (2000)
and Refs. reported therein.
9. S. Coleman, Phys. Rev. D15, 2929 (1977).
10. A.J. Leggett, Macroscopic Quantum Systems and the Quantum
Theory of Measurements, Supplement of Prog. Theor. Phys. 69,
80 (1980); A.J. Leggett, Quantum Tunneling of a Macroscopic
Variable in Quantum Tunneling in Condensed Media, Yu.
Kagan and A.J. Leggett (eds.), Elsevier Science Publ. (1992)
and Refs. reported therein.
11. Yu.M. Ivanchenko and L.A. Zilberman, Sov. Phys. JETP 28,
1272 (1969).
12. A.O. Caldeira and A.J. Leggett, Phys. Rev. Lett. 46, 211 (1981).
13. H. Grabert and U. Weiss, Phys. Rev. Lett. 53, 1787 (1984).
14. A.I. Larkin and Yu.N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 85,
1510 (1983) [Sov. Phys. JETP 58, 876 (1983)]; Phys. Rev.
B28, 828 (1983).
15. A. Barone and Yu.N. Ovchinnikov, J. Low Temp. Phys. 55,
297 (1984); Yu.N. Ovchinnikov and A. Barone, J. Low
Temp. Phys. 67, 323 (1987); A. Barone, J. Supercond:
Incorporating Novel Magnetism 17, 585 (2004).
16. R.F. Voss and R.A. Webb, Phys. Rev. Lett. 47, 265 (1981);
L.D. Jackel, J.P. Gordon, E.L. Hu, R.E. Howard, L.A. Fetter,
D.M. Tennant, R.W. Epworth, and J. Kurkijarvi, Phys. Rev.
Lett. 47, 697 (1981); S. Washburn, R.A. Webb, R.F. Voss,
and S.M. Farris, Phys. Rev. Lett. 54, 2712 (1985); D.B.
Shwartz, B. Sen, C.N. Archie, and J.E. Lukens, Phys. Rev.
Lett. 54, 1547 (1985); M.H. Devoret, J.M. Martinis, and J.
Clarke, Phys. Rev. Lett. 55, 1908 (1985); W. der Boer and R.
de Bruyn Ouboter, Physica 98B, 185 (1980); D.W. Bol, R.
van Weelderen and R. de Bruyn Ouboter, Physica 122B, 2
(1983); D.W. Bol, J.J.F. Scheffer, W. Giele, and R. de Bruyn
Ouboter, Physica 113B, 196 (1985); R.J. Prance, A.P. Long,
T.D. Clarke, A. Widom, J.E. Mutton, J. Sacco, M.W. Potts,
G. Negaloudis, and F. Goodall, Nature 289, 543 (1981); I.M.
Dimitrenko, V.A. Khlus, G.M.Tsoi, and V.I. Shnyrkov, Fiz.
Nizk. Temp. 11, 146 (1985) [Sov.J. Low Temp. Phys. 11, 77
(1985)]; J.M. Martinis, M.H. Devoret, and J. Clarke, Phys.
Rev. B35, 4682 (1987); U. Weiss, H. Grabert, and S.
Linkwitz, J. Low Temp. Phys. 68, 213 (1987).
17. Josepshon Effect-Achievements and Trends ISI-85 Torino, A.
Barone (ed.) World Scientific (1985) and Refs. reported
therein.
18. A.J. Leggett, S. Chakravarty, A.T. Dorsey, M.P.A. Fischer,
A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1 (1987).
19. J. Clarke, A.N. Cleland, M.H. Devoret, D. Esteve, and J.M.
Martinis, Science 29, 992 (1988).
20. A.V. Zhuravlev and A.B. Zorin, Fiz. Nizk. Temp. 16, 184
(1990) [Sov. J. Low Temp. Phys. 16, 102 (1990)]; J.M.
Schmidt, A.N. Cleland, and J. Clarke, Phys. Rev. B43, 229
(1991).
21. R. Rouse, S. Han, and J.E. Lukens, Phys. Rev. B75, 1614
(1995).
22. C. Cosmelli, P. Carelli, M.G. Castellano, F. Chiariello, G.D.
Palazzi, R. Leoni, and G. Torrioli, Phys. Rev. Lett. 82, 5357
(1999). For aspects of the sample preparation see also M.G.
Castellano et al. J. Appl. Phys. 80, 2928 (1996).
23. Y. Nakamura, Y. Pashkin, and J.S. Tsai, Nature 398, 786
(1999).
24. J.R. Friedman, V. Patel, W. Chen, S.K. Tolpygo, and J.E.
Lukens, Nature 406, 43 (2000).
25. L.B. Ioffe, V.B. Geshkenbein, M.V. Feigel’man, A.L.
Fauchere, and G. Blatter, Nature 398, 679 (1999); C.C.
Tsuei and J.R. Kirtley, Physica (Amsterdam) 367C, 1
(2002); B. Doucot, M.V. Feigel’man, and L.B. Ioffe, Phys.
Rev. Lett. 90, 107003 (2003).
26. A.Yu. Kitaev, preprint quant-ph/9707021 at xxx.lanl.gov,
(1997).
27. G. Blatter, V.B. Geshkenbein, and L.B. Ioffe, Phys. Rev.
B63, 174511 (2001).
28. A. Wallraff, A. Lukashenko, J. Lisenfeld, A. Kemp, M.V.
Fistul, Y. Koval, and A.V. Ustinov, Nature, 425, 155, (2003).
29. A. Kemp, A. Wallraff, and A.V. Ustinov, Phys. Status Solidi
233, 472 (2002).
30. T. Bauch, F. Lombardi, F. Tafuri, G. Rotoli, A. Barone, P.
Delsing, and T. Cleason, Phys. Rev. Lett. 94, 087003 (2005).
31. T. Bauch, T. Lindström, F. Tafuri, G. Rotoli, P. Delsing,
T. Claeson, and F. Lombardi, Science 311, 5757 (2006).
32. Y.V. Fominov, A.A. Golubov, and M.Y. Kupriyanov, JETP
Lett. 77, 587 (2003).
33. M.H.S. Amin and A.Y. Smirnov, Phys. Rev. Lett. 92, 017001
(2004).
34. S. Kawabata, S. Kashiwaya, Y. Asano, and Y. Tanaka, Phys.
Rev. B70, 132505 (2004).
35. S. Kawabata, S. Kashiwaya, Y. Asano, and Y. Tanaka, Phys.
Rev. B72, 052506 (2005); T. Yokoyama, S. Kawabata,
T. Kato, and Y. Tanaka, Phys. Rev. B76, 134501 (2007).
36. H. Hilgenkamp and J. Mannhart, Rev. Mod. Phys. 74, 485
(2002).
A. Barone, F. Lombardi, G. Rotoli, and F. Tafuri
1106 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11
37. F. Tafuri and J.R. Kirtley, Rep. Prog. Phys. 68, 2573 (2005).
38. K. Inomata, S. Sato, K. Nakajima, A. Tanaka, Y. Takano,
H. B. Wang, M. Nagao, H. Hatano, and S. Kawabata, Phys.
Rev. Lett. 95, 107005 (2005).
39. X.Y. Jin, J. Lisenfeld, Y. Koval, A. Lukashenko, A.V.
Ustinov, and P. Mueller, Phys. Rev. Lett. 96, 177003 (2006).
40. F. Tafuri, F. Miletto Granozio, F. Carillo, A. Di Chiara, K.
Verbist, and G. Van Tendeloo, Phys. Rev. B59, 11523 (1999).
41. F. Lombardi, F. Tafuri, F. Ricci, F. Miletto Granozio, A.
Barone, G. Testa, E. Sarnelli, J.R. Kirtley, and C.C. Tsuei,
Phys. Rev. Lett. 89, 207001 (2002).
42. F. Miletto Granozio, U. Scotti di Uccio, F. Lombardi, F.
Ricci, F. Bevilacqua, G. Ausanio, F. Carillo, and F. Tafuri,
Phys. Rev. B67, 184506 (2003).
43. F. Tafuri, J.R. Kirtley, F. Lombardi, and F. Miletto
Granozio, Phys. Rev. B67, 174516 (2003).
44. C.-R. Hu, Phys. Rev. Lett. 72, 1526 (1994); T. Löfwander,
V.S. Shumeiko, and G. Wendin, Supercond. Sci. Technol.
14, R53 (2001); S. Kashiwaya and Y. Tanaka, Rep. Prog.
Phys. 63, 1641 (2000).
45. E. Ilichev, V. Zakosarenko, R.P.J. Ijsselsteijn, H.E. Hoenig,
V. Schultze, H.G. Meyer, M. Grajcar, and R. Hlubina, Phys.
Rev. B60, 3096 (1999); E.E. Il’ichev, M. Grajcar, R.
Hlubina, R.P.J. IJsselsteijn, H.E. Hoenig, H.-G. Meyer, A.
Golubov, M.H.S. Amin, A.M. Zagoskin, A.N. Omelyan-
chouk, and M.Yu. Kupriyanov, Phys. Rev. Lett. 86, 5369
(2001); T. Lindstrom, S.A. Charlebois, A.Ya. Tzalenchuk, Z.
Ivanov, M.H.S. Amin, and A.M. Zagoskin, Phys. Rev. Lett.
90 117002 (2003).
46. The dynamics of a JJ can be described by assuming the
presence of both first and second harmonic components
in the CPR (higher harmonics can be neglected because
of low junction barrier transparency). defining =I
1(sin sin 2 )I a= φ − φ where 2 1= /a I I , and 2I and 1I are
the second and first harmonic components. At a given value
of a, the dynamics is equivalent [47] to that of a particle of
mass mφ moving in a washboard potential ( ) =U a
1 0 0 1( / 2 )(( / ) cos ( / 2)cos2 )CI I I a= − Φ π γφ + − φ , where
1 1 0= / 2E I Φ π is the first harmonic of the Josephson energy
and 0Φ is the flux quantum. The normalized bias current
= / 0CI Iγ determines the tilt of the potential, where
0 1= max(sin sin 2 )CI I aφ − φ is the maximum critical
current of the JJ. For zero bias and > 0.5a the potential has
the shape of a double well. In the case 0.5 < < 2a the phase
will always escape into the running state from the lower
lying well of the tilted double-welled washboard potential,
as could be confirmed by performing numerical simulations
of the Langevin equation of motion. More details can be
found in: A.Ya. Tzalenchuk, T. Lindström, S.A. Charlebois,
E.A. Stepantsov, Z. Ivanov, and A.M. Zagoskin, Phys. Rev.
B68, 100501 (2003).
47. W.C. Stewart, Appl. Phys. Lett. 12, 277 (1968); D.E.
McCumber, J. Appl. Phys. 39, 3113 (1968).
48. S. Kawabata, T. Bauch, and T. Kato, Phys. Rev. B80,
174513 (2009).
49. L.D. Jackel, J.P. Gordon, E.L. Hu, R.E. Howard, L.A. Fetter,
D.M. Tennant, R.W. Epworth, and J. Kurkijärvi, Phys. Rev.
Lett. 47, 697 (1981).
50. F. Balestro, J. Claudon, J.P. Pekola, and O. Buisson, Phys.
Rev. Lett. 91, 158301 (2003).
51. G. Rotoli, T. Bauch, T. Lindström, D. Stornaiuolo, F. Tafuri,
and F. Lombardi, Phys. Rev. B75, 144501 (2007).
52. X.G. Wen and P.A. Lee, Phys. Rev. Lett. 80, 2193 (1998).
53. A.Ya. Tzalenchuk, T. Lindström, S.A. Charlebois, E.A.
Stepantsov, Z. Ivanov, and A.M. Zagoskin, Phys. Rev. B68,
100501 (2003).
54. E. Il’ichev, M. Grajcar, R. Hlubina, R.P.J. IJsselsteijn, H.E.
Hoenig, H.-G. Meyer, A. Golubov, M.H.S. Amin, A.M.
Zagoskin, A.N. Omelyanchouk, and M.Yu. Kupriyanov, Phys.
Rev. Lett. 86, 5369 (2001).
55. G. Testa, A. Monaco, E. Esposito, E. Sarnelli, D.J. Kang,
S.H. Mennema, E.J. Tarte, and M.G. Blamire, Appl. Phys.
Lett. 85, 1202 (2004); G. Testa, E. Sarnelli, A. Monaco, E.
Esposito, M. Ejrnaes, D.J. Kang, S.H. Mennema, E.J. Tarte,
and M.G. Blamire, Phys. Rev. B71, 134520 (2005).
56. D. Stornaiuolo, G. Rotoli, K. Cedergen, T. Bauch, F. Lombardi,
and F. Tafuri, to be published in J. Appl. Phys. (2010);
D. Stornaiuolo, E. Gambale, T. Bauch, D. Born, K. Cedergren,
D. Dalena, A. Barone, A. Tagliacozzo, F. Lombardi, and F.
Tafuri, Physica C468, 310 (2008).
57. G. Papari, F. Carillo, D. Born L. Bartoloni, E. Gambale, D.
Stornaiuolo, P. Pingue, F. Beltram, and F. Tafuri, IEEE
Trans. on Appl. Supercond.19, 183 (2009).
58. F. Carillo, G. Papari, D. Stornaiuolo, D. Born, D.
Montemurro, P. Pingue, F. Beltram, and F. Tafuri, Phys.
Rev. B81, 054505 (2010).
59. A. Tagliacozzo, D. Born, D. Stornaiuolo, E. Gambale, D.
Dalena, F. Lombardi, A. Barone, B.L. Altshuler, and F.
Tafuri, Phys. Rev. B75, 012507 (2007).
60. A. Tagliacozzo, F. Tafuri, E. Gambale, B. Jouault, D. Born,
P. Lucignano, D. Stornaiuolo, F. Lombardi, A. Barone, and
B.L. Altshuler, Phys. Rev. B79, 024501 (2009).
61. B.L. Altshuler and A.G. Aronov, in: Electron-Electron
Interaction in Disordered Systems, A.L. Efros and M. Pollak
Elsevier (eds.), Amsterdam (1985); P.A. Lee and T.V.
Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985); P.A. Lee
and A.D. Stone, Phys. Rev. Lett. 55, 1622 (1985); P.A. Lee,
A.D. Stone, and H. Fukuyama, Phys. Rev. B35, 1039 (1987);
B. Al’tshuler and P. Lee, Phys. Today 41, 36 (1988); R.A.
Webb and S. Washburn, Phys. Today 41, 46 (1988); For a
review see Mesoscopic Phenomena in Solids, B.L. Altshuler,
P.A. Lee, and R.A. Webb (eds.), North-Holland, New York,
(1991).
62. Y. Imry, Introduction to Mesoscopic Physics, Oxford
University Press (1997).
63. N. Gedik, J. Orenstein, R. Liang, D.A. Bonn, and W.N.
Hardy, Science 300, 1410 (2003).
|