Escape dynamics in moderately damped Josephson junctions (Review Article)
The Josephson effect is a privileged access to the macroscopic quantum nature of superconductors. We review some ideas and experimental techniques on macroscopic quantum decay phenomena occurring in Josephson structures. The attention is mainly addressed to intermediate levels of dissipation which...
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irk-123456789-1171162017-05-21T03:03:17Z Escape dynamics in moderately damped Josephson junctions (Review Article) Massarotti, D. Longobardi, L. Galletti, L. Stornaiuolo, D. Montemurro, D. Pepe, G. Rotoli, G. Barone, A. Tafuri, F. Квантовые когерентные эффекты в сверхпроводниках и новые материалы The Josephson effect is a privileged access to the macroscopic quantum nature of superconductors. We review some ideas and experimental techniques on macroscopic quantum decay phenomena occurring in Josephson structures. The attention is mainly addressed to intermediate levels of dissipation which characterize a large majority of low critical current Josephson devices and are therefore an avoidable consequence of nanotechnology applied more and more to Josephson devices. Phase diffusion phenomena take over thermal activation in some temperature ranges also affecting the transition to macroscopic quantum tunneling, enriching the phase diagram mostly defined by the Josephson energy, the temperature and the level of dissipation. 2012 Article Escape dynamics in moderately damped Josephson junctions (Review Article) / D. Massarotti, L. Longobardi, L. Galletti, D. Stornaiuolo, D. Montemurro,G. Pepe, G. Rotoli, A. Barone, F. Tafuri // Физика низких температур. — 2012. — Т. 38, № 4. — С. 336–347. — Бібліогр.: 85 назв. — англ. 0132-6414 PACS: 74.72.–h, 74.50.+r, 74.40.–n http://dspace.nbuv.gov.ua/handle/123456789/117116 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Квантовые когерентные эффекты в сверхпроводниках и новые материалы Квантовые когерентные эффекты в сверхпроводниках и новые материалы |
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Квантовые когерентные эффекты в сверхпроводниках и новые материалы Квантовые когерентные эффекты в сверхпроводниках и новые материалы Massarotti, D. Longobardi, L. Galletti, L. Stornaiuolo, D. Montemurro, D. Pepe, G. Rotoli, G. Barone, A. Tafuri, F. Escape dynamics in moderately damped Josephson junctions (Review Article) Физика низких температур |
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The Josephson effect is a privileged access to the macroscopic quantum nature of superconductors. We review
some ideas and experimental techniques on macroscopic quantum decay phenomena occurring in Josephson
structures. The attention is mainly addressed to intermediate levels of dissipation which characterize a large
majority of low critical current Josephson devices and are therefore an avoidable consequence of nanotechnology
applied more and more to Josephson devices. Phase diffusion phenomena take over thermal activation in
some temperature ranges also affecting the transition to macroscopic quantum tunneling, enriching the phase diagram
mostly defined by the Josephson energy, the temperature and the level of dissipation. |
| format |
Article |
| author |
Massarotti, D. Longobardi, L. Galletti, L. Stornaiuolo, D. Montemurro, D. Pepe, G. Rotoli, G. Barone, A. Tafuri, F. |
| author_facet |
Massarotti, D. Longobardi, L. Galletti, L. Stornaiuolo, D. Montemurro, D. Pepe, G. Rotoli, G. Barone, A. Tafuri, F. |
| author_sort |
Massarotti, D. |
| title |
Escape dynamics in moderately damped Josephson junctions (Review Article) |
| title_short |
Escape dynamics in moderately damped Josephson junctions (Review Article) |
| title_full |
Escape dynamics in moderately damped Josephson junctions (Review Article) |
| title_fullStr |
Escape dynamics in moderately damped Josephson junctions (Review Article) |
| title_full_unstemmed |
Escape dynamics in moderately damped Josephson junctions (Review Article) |
| title_sort |
escape dynamics in moderately damped josephson junctions (review article) |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2012 |
| topic_facet |
Квантовые когерентные эффекты в сверхпроводниках и новые материалы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117116 |
| citation_txt |
Escape dynamics in moderately damped
Josephson junctions (Review Article) / D. Massarotti, L. Longobardi, L. Galletti, D. Stornaiuolo, D. Montemurro,G. Pepe, G. Rotoli, A. Barone, F. Tafuri // Физика низких температур. — 2012. — Т. 38, № 4. — С. 336–347. — Бібліогр.: 85 назв. — англ. |
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Физика низких температур |
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| fulltext |
© D. Massarotti, L. Longobardi, L. Galletti, D. Stornaiuolo, D. Montemurro, G. Pepe, G. Rotoli, A. Barone, and F. Tafuri, 2012
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4, pp. 336–347
Escape dynamics in moderately damped
Josephson junctions
(Review Article)
D. Massarotti1,2, L. Longobardi3, L. Galletti1,2, D. Stornaiuolo1,2, D. Montemurro4,
G. Pepe1,2, G. Rotoli5, A. Barone1,2, and F. Tafuri5,2,1
1Dip. Scienze Fisiche, Universitá di Napoli Federico II, Piazzale Tecchio 80, Napoli 80125, Italy
2CNR-SPIN, UOS Napoli, Monte S. Angelo via Cinthia, Napoli 80126, Italy
3Department of Physics and Astronomy, Dartmouth College, Hanover NH 03755, USA
4NEST and Scuola Normale Superiore, Piazza San Silvestro 12, I-56127 Pisa, Italy
5Dip. Ingegneria dell'Informazione, Seconda Universitá di Napoli, via Roma 29, Aversa (CE) 81031, Italy
E-mail: tafuri@na.infn.it
Received February 15, 2012
The Josephson effect is a privileged access to the macroscopic quantum nature of superconductors. We re-
view some ideas and experimental techniques on macroscopic quantum decay phenomena occurring in Joseph-
son structures. The attention is mainly addressed to intermediate levels of dissipation which characterize a large
majority of low critical current Josephson devices and are therefore an avoidable consequence of nanotechnolo-
gy applied more and more to Josephson devices. Phase diffusion phenomena take over thermal activation in
some temperature ranges also affecting the transition to macroscopic quantum tunneling, enriching the phase di-
agram mostly defined by the Josephson energy, the temperature and the level of dissipation.
PACS: 74.72.–h Cuprate superconductors;
74.50.+r Tunneling phenomena; Josephson effects;
74.40.–n Fluctuation phenomena.
Keywords: Josephson effect, SQUIDs, macroscopic quantum tunneling.
Сontents
1. Introduction ........................................................................................................................................... 336
2. A ball rolling down a washboard potential: from thermal activation to macroscopic quantum
tunneling ....................................................................................................................................... 337
2.1. Formalism and the first measurements on switching current distributions .................................... 337
2.2. Josephson junctions as a part of a qubit ........................................................................................ 339
3. Phase dynamics in “novel” types of junctions and the moderately damped regime .............................. 339
3.1. Macroscopic quantum phenomena in HTS grain boundary Josephson junctions .......................... 340
3.2. Macroscopic quantum phenomena in HTS intrinsic junctions ...................................................... 341
3.3. The moderately damped regime in low critical currents Josephson junctions ............................... 341
3.4. In the “far” low critical current regime ......................................................................................... 344
3.5. Switching current distribution measurements in “nanostructures” ................................................ 344
Conclusions ............................................................................................................................................... 345
References ................................................................................................................................................. 345
1. Introduction
The Josephson effect [1] and the physics of Josephson
devices [2–4] have been in these last 50 years a continuous
source of inspiration and progress in physics. We have been
learning by investigating a Josephson device, how macro-
scopic coherence propagates in heterostructures [2–5], how
macroscopic and microscopic phases coherently combine
at superconductor/normal metal (S/N) interfaces [6] of a
S/N/S Josephson junction, how a Josephson coupling can
unambiguously identify unconventional symmetry of the
Escape dynamics in moderately damped Josephson junctions
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4 337
order parameter, as occurring in high critical temperature
superconductors (HTS) [7]. The intrinsic and robust quan-
tum nature of the Josephson junctions is also the basis of
well established applications such as the superconducting
quantum interference devices (SQUIDs) [3,8] and of a
prospective hybrid architecture for quantum information
[9]. These are just a few examples chosen in a multitude of
possibilities.
In this contribution we revise some of the main concepts
and experimental procedures on which the studies on escape
dynamics in Josephson devices are funded [10–12]. We look
at escape dynamics aware that material science is not only
offering a variety of novel interfaces and junctions, but
also radically new solutions of synthesizing hybrid Josep-
shon devices taking advantage at the same time of the
progress registered in using nanotechnologies in supercon-
ducting electronics. We are specifically interested in dis-
cussing escape dynamics at intermediate levels of dissipa-
tion (moderately damped regime), which is going to
permeate more and more the nature of futuristic supercon-
ducting hybrid nanostructures.
This topic has been of high interest since the early eigh-
ties, with the first experimental evidence of quantum beha-
vior in a Josephson tunnel junction displaying energy le-
vels quantization and macroscopic quantum tunneling,
which showed unequivocally that the difference in the
phase of the Ginzburg–Landau [13] wavefunction in the
two electrodes of a Josephson junction behaves as a quan-
tum variable. This concept has more recently been ex-
tended to the implementation of superconductive quantum
bits and in a whole series of beautifully designed experi-
ments displaying a high degree of control of the energy
level landscape.
2. A ball rolling down a washboard potential: from
thermal activation to macroscopic quantum tunneling
2.1. Formalism and the first measurements on switching
current distributions
Considerable insights into the nonlinear dynamics of a
Josephson junction can be gained by realizing that the equ-
ation resulting from the resistively shunted junctions (RSJ)
model [2–4]
2 22
0 0
2
1 = 0,
2 2
C U
R tt
Φ Φ∂ φ ∂φ ∂⎛ ⎞ ⎛ ⎞+ +⎜ ⎟ ⎜ ⎟π π ∂ ∂φ⎝ ⎠ ⎝ ⎠∂
(1)
where
0= / (2 )( cos( ) )cU I I−Φ π φ + φ (2)
with 0Φ the flux quantum and cI the critical current,
describes the motion of a ball moving on the “tilted
washboard” potential U (see Figs. 1,a,b). The term in-
volving the capacitance C represents the mass of the
particle, the 1/ R term represents the damping of the mo-
tion, and the average “tilt” of the washboard is propor-
tional to .I− The strength of the friction can be also ex-
pressed through the junction quality factor = ,pQ RCω
where 1/2 2 1/4= (2 / ) (1 )p ceI Cω −ι is the plasma fre-
quency being = / .cI Iι
When the bias current is ramped from = 0ι to <1,ι the
junction is in the zero-voltage state in absence of thermal
and quantum fluctuations and the particle is confined to one
of the potential wells, where it oscillates back and forth at
the frequency .pω At finite temperature, the junction may
switch into a finite voltage state for a bias current < 1.ι This
corresponds to the particle escaping from the well either by
a thermally activated (TA) process [14] or by tunneling
through the barrier potential, known as macroscopic quan-
tum tunneling (MQT) [5,9,11,12] (see Fig. 1,c). This only
occurs if the kinetic energy gained from running “down
hill” in the tilted-washboard potential is not all dissipated,
but enough remains to carry the representative point over
the next “hill”. The Q quality factor measures the damping
of the plasma oscillation by the effective shunt resistance R
of the junction and estimates the number of periods during
which the oscillation energy will be dissipated. Once the
phase point gets over a hill by fluctuations, it keeps run-
ning, provided that the damping is below some critical
value. Following an event of escape the particle may travel
down the potential for a few wells and then be retrapped in
one of the following minima of the potential [15]. At low
bias the process of escape and retrapping may occur mul-
tiple times generating extensive diffusion of the phase until
an increase of the tilt of the potential, due to a change in
the bias current, raises the velocity of the particle and the
junction can switch to the running state [15–19].
Fig. 1. (a) 3-dim view of wash-board potential in the RSJ model
as a function of the phase and of the bias current. The current
spans from 0 to .cI In (b) JE is half of the value used in (a),
and this favors phase diffusion regime (see text). (c) A 2-dim cut
from (b) for a fixed value of the current. TA stands for thermal
activation (black dotted line), while MQT for macroscopic quan-
tum tunneling (red dotted line). Once the particle/phase over-
comes the barrier, it rolls in the running state. Retrapping
processes may happen for intermediate levels of dissipation de-
termining a phase diffusion (PD) regime.
MQT
U
U
I
a
b
cPD
Running
state
TA
�
�
D. Massarotti, L. Longobardi, L. Galletti, D. Stornaiuolo, D. Montemurro, G. Pepe, G. Rotoli, A. Barone, and F. Tafuri
338 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4
In the pure thermal regime, the escape rate for weak to
moderate damping ( > 1Q ) is determined by [14]
( )( ) = exp ,
2
p
t t
B
U II a
k T
ω ⎛ ⎞Δ
Γ −⎜ ⎟π ⎝ ⎠
(3)
where 3/2( ) = ( 4 2 / 3)(1 )JU I EΔ −ι is the barrier height
for ι close to one and = / 2J cE I e is the Josephson
energy. The escape rate will be dominated by MQT at low
enough temperature [5,20]: for > 1Q and ι close to one it
is approximated by the expression for a cubic potential
( ) 0.87( ) = exp 7.2 1 ,
2
p
q q
p
U II a
Q
⎡ ⎤ω ⎛ ⎞Δ
Γ − +⎢ ⎥⎜ ⎟π ω ⎝ ⎠⎢ ⎥⎣ ⎦
(4)
where 1/2= (864 / ) .q pa UπΔ ω
The MQT rate is affected by dissipation, the irreversible
energy transfer between the system and the environment,
because of the damping dependent factor [5,20,21].
It is convenient to express the thermal and the quantum
escape through an escape temperature escT defined as
,
esc
( )( ) = exp .
2
p
t q
B
U II
k T
ω ⎡ ⎤Δ
Γ −⎢ ⎥π ⎣ ⎦
(5)
This is made possible because in both the classical and
quantum regimes escT is very nearly independent of the
bias current. The crossover temperature crT between the
thermal and quantum regimes is given by
2 1/2
cr = ( / 2 ){(1 1/ 4 ) 1/ 2 }.p BT k Q Qω π + −
An analytical expression for the retrapping rate from
the resistive to the superconducting state is known only for
strongly underdamped JJs:
2 2
0
0 2
( )
= exp ,
2 2
J JR R
R p
c B B c
E E QI I I I
I k T k T I
⎡ ⎤⎛ ⎞− −
⎢ ⎥Γ ω − ⎜ ⎟⎜ ⎟π ⎢ ⎥⎝ ⎠⎣ ⎦
(6)
where RI is the fluctuation-free retrapping current, 0pω
and 0Q are the plasma frequency and quality factor for
I = 0, respectively.
The behavior of the phase difference ϕ is deduced
from measurements of the escape rate Γ of the junctions
from its zero-voltage state. To determine the escape rate
104–105 events are typically collected for each set of pa-
rameters. The resulting distribution of the switching prob-
ability ( )P I is used to compute the escape rate out of the
zero-voltage state as a function of the bias current I fol-
lowing Fulton and Dunkleberger [10]:
( )
1( ) = ln ,
( )
i I
i I I
P I
dII
I dt P I
≥
≥ +Δ
⎛ ⎞
⎜ ⎟
Γ ⎜ ⎟Δ ⎜ ⎟
⎝ ⎠
∑
∑
(7)
where /dI dt is the current ramp rate and IΔ is the chan-
nel width of the analog-to-digital converter.
The very first experiments on MQT in a Josephson
junction were carried out by Voss and Webb [22] and by
Jackel et al. [23], while related experiments on a junction
inserted in a superconducting loop were realized by de
Bruyn Ouboter et al. [24], Prance et al. [25] and Dmitrenko
et al. [26]. Switching current distribution histograms
present a characteristic behavior reported schematically in
Fig. 2. By lowering the temperature T the histograms move
to higher currents and their width σ scales with the tem-
perature down to cr .T For cr< ,T T σ is independent of T.
The temperature dependence of the effect of damping on
the tunneling has been addressed by later experiments
[27,28]. Devoret, Martinis and Clarke [11,12] have estab-
lished a detailed conceptual and experimental protocol to
follow to prove the macroscopic quantum nature of tunne-
ling and its crossover to the thermal regime, used in most
of later experiments. It has been clearly addressed the
problem of the complex impedance presented to the junc-
tion at microwave frequencies by the wires directly con-
nected to it or by any circuit in its vicinity, and classical
phenomena have been used to measure all relevant para-
meters of the junction in situ [11,12]. The relevant parame-
ters of the junction (critical current and shunting admit-
tance) were determined in situ in the thermal regime from
the dependence of Γ on bias current and from resonant
activation in the presence of microwaves. The shunting
capacitance dominated the self-capacitance of the junction,
while the bias circuitry determined the shunting conduc-
tance, respectively. The magnetic field has been used as a
knob to tune the crossover temperature by changing the
critical current and therefore the plasma frequency.
In a further series of experiments, the existence of
quantized energy levels in the potential well of the junction
was demonstrated spectroscopically [11,12]. The escape
rate from the zero-voltage state was increased when the
microwave frequency Ω corresponded to the energy dif-
ference between two adjacent energy levels. A crucial
Fig. 2. Schematic representation of SCD measurements from the
thermal regime to macroscopic quantum tunneling.
P
ro
b
ab
il
it
y,
a
rb
.
u
n
it
s
Current, arb. units
< Tcr
Increasing
temperatures
Escape dynamics in moderately damped Josephson junctions
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4 339
point is that the anharmonic nature of the well, which re-
sults from the nonlinear inductance of Josephson junctions,
causes the energy spacing to decrease as the quantum
number progressively increases, so each transition has a
distinct frequency. The transition may involve more than
one photon at once, thus called multiphoton transition,
which has been observed experimentally [29].
This set of experiments clearly proved that ϕ is a quan-
tum variable. Although this system contains a large num-
ber of atomic constituents, it is atomlike in the sense that it
has a single degree of freedom behaving quantum mechan-
ically. Thermal energy must be sufficiently low to avoid
incoherent mixing of eigenstates, and the macroscopic de-
gree of freedom must be sufficiently decoupled from other
degrees of freedom for the lifetime of the quantum states to
be long on the characteristic time scale of the system [5].
2.2. Josephson junctions as a part of a qubit
Superconducting qubits are solid state electrical circuits
that exploits the numerous advantages of using supercon-
ductors, as well as the macroscopic quantum behavior of
devices based on Josephson tunnel junctions, to engineer
quantum states suitable for quantum computing. Supercon-
ductors are particularly fit to be used for quantum inte-
grated circuits since they allow for design and fabrication
using techniques that can be borrowed from conventional
integrated circuits, and they have intrinsically low dissipa-
tion and low noise due to the fact that the experiments are
often conducted at millikelvin temperatures. Furthermore,
as explained in the previous section, the Josephson poten-
tial is highly anharmonic, which allows to selectively ex-
cite the states used as basis of the qubit, and to avoid con-
tributions from other energy levels.
Macroscopic quantum coherence, i.e., the superposi-
tion of two macroscopically distinctive quantum states
1|Ψ 〉 and 2| Ψ 〉 in the form 1 2| = | |Ψ〉 α Ψ 〉 +β Ψ 〉 [30]
is a fundamental component of quantum computation,
and was first demonstrated by Nakamura et al. in 1997
with the first experiment on a charge qubit [31], showing
spectroscopically the superposition of the Cooper-pair
states | n〉 and | 1n + 〉 , where the integer n is the quantum
number specifying the number of Cooper pairs. Demon-
strations of the superposition of states in a flux qubit by
the Stony Brook [32] and Delft [33] groups followed. A
flux qubit consists of a superconducting loop interrupted
by one [32] or three [33] Josephson junctions. The two
quantum states consist of supercurrent flowing in an anti-
clockwise or clockwise direction or, equivalently, flux
pointing up and flux pointing down, respectively. The
Saclay group [34] realized in 2002 a qubit, named “quan-
tronium”, in which two small junctions are connected by
a superconducting island, involving the superposition of
the Cooper-pair states | n〉 and | 1n + 〉 . In the phase qubit
realized by Martinis et al. in 2002 [35], the relevant quan-
tum states are the ground state and the first excited state,
and the final device is basically the same used earlier to
observe quantized energy levels [11,12].
These first experiments also classify the main three dif-
ferent types of qubit, i.e., charge, flux and phase, and have
opened the way to a vast series of studies [9]. Measurements
in the time domain for instance followed in order to deter-
mine the dynamical behavior of a qubit [36–38]. These ex-
periments aimed to measure Rabi oscillations, spin echoes
and Ramsey fringes require manipulating the state of the
qubit by using appropriate microwave pulses, which are also
the operative tools to implement single-qubit gates for quan-
tum computing. Macroscopic resonant tunneling (MRT)
when energy levels in each well are aligned, turns out to be
an “accessible” reliable tool to perform on a routine basis the
task to extract all devices quantum parameters, flux noise,
and for a diagnostics of fabrication processes and materials
[39,40]. This represents another example of how “quantum
measurements” become more and more accessible, and a
tool to drive material science choices on the nature and the
performances of the junctions.
Another big force in driving research in the supercon-
ducting quantum measurements/qubit field has been to
engineer systems in order to make them as less as possible
sensitive to decoherence. This work has led to remarkable
increases in decoherence times compared with those of
early devices [9]. The development of more advanced
charge qubits such as the transmon [41] and the quantro-
nium [34] are examples of improved charge qubits, where
attention is paid to protect the devices from low-frequency
noise and from electrons moving among defects. Future
implementations of quantum information processing will
try to promote hybrid systems. The idea is to combine mi-
croscopic systems such as atoms or spins which are natu-
rally well decoupled from their environment and can reach
extremely long coherence times, with more macroscopic
objects such as superconducting circuits which are strongly
coupled to electromagnetic fields, making them easy to
entangle although with shorter coherence times [42].
3. Phase dynamics in “novel” types of junctions and the
moderately damped regime
The idea that performances of qubits could be improved
and optimized by a suitable design of the ensemble qu-
bit/cavity (circuit) was accompanied by the awareness that
some limits on coherence are imposed by intrinsic dissipa-
tion due to the “chemistry” of the junctions, of the barrier
interfaces and of the materials composing them. Progress in
engineering new materials into junctions and in understand-
ing and more and more controlling the physics of interfaces
may offer novel solutions for junctions of superior quality
and complementary functionalities, and therefore may lead
in the long run to improve specific qubit performances. In
other words, material science contributes to develop solu-
tions for hybrid systems for quantum computation.
D. Massarotti, L. Longobardi, L. Galletti, D. Stornaiuolo, D. Montemurro, G. Pepe, G. Rotoli, A. Barone, and F. Tafuri
340 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4
In this section we focus on how experiments on the
junction macroscopic quantum behavior have been ex-
tended to novel types of structures and materials. Switch-
ing current distribution (SCD) measurements in these last
years have turned to be standard tools to investigate phase
dynamics in unconventional and hybrid systems and nano-
structures. High critical temperature superconductors
(HTS) are an example of unconventional systems, because
of the d-wave order parameter symmetry (OPS) and of the
presence of low-energy quasiparticles [7], which are ex-
pected to induce high level of dissipation and as a conse-
quence to spoil macroscopic quantum coherence.
For low critical temperature superconductors, once the
barrier thickness and the critical current density ( cJ ) have
been fixed, a reduction in its size unavoidably leads to a
lowering of the critical current and determines a quite dif-
ferent phase dynamics “re-normalized” to the new scaling
energy. Lower critical currents cI result in lower Joseph-
son energies ,JE and higher levels of dissipation are ex-
pected. The range of the energy dynamical parameters is
significantly enlarged, and it is technologically easier to
reproducibly realize nontrivial configurations. The pio-
neering studies of Kautz and Martinis [15] and Iansiti et al.
[43] on small junctions where EJ could be significantly
lowered, can be now supported and developed by different
types of junctions of quite different sizes. These devices
are characterized by intermediate levels of dissipation
(moderately damped regime) and by phase diffusion phe-
nomena. The low cJ limit seems to be characteristic also
of all futuristic nanohybrids devices incorporating nano-
wires, and the moderately damped regime is intrinsically
more common than it could be expected.
In summary we will refer as novel types of junctions,
those composed of novel materials, or devices scaled to the
nanosize or based on novel design concepts, as intrinsic
junctions in HTS or junctions using nanowires as barriers.
3.1. Macroscopic quantum phenomena in HTS grain
boundary Josephson junctions
The first examples of unconventional systems are given
by HTS devices [44], biepitaxial grain boundary (GB)
YBCO Josephson junctions [45,46] and a variety of intrinsic
junctions, built on high-quality single crystals [47,48].
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 [7,49].
Low-energy quasiparticles have represented since the very
beginning a strong argument against the occurrence of ma-
croscopic quantum effects in these materials. Quantum tun-
nelling of the phase leads to fluctuating voltage across the
junctions which excites the low-energy quasiparticles specif-
ic for d-wave junctions, causing decoherence. Contributions
to dissipation due to different transport processes, such as
channels due to nodal quasiparticles, midgap states, or their
combination, have been identified and distinguished [50].
The search of macroscopic quantum effects become
feasible once high-quality HTS Josephson junctions [44]
with significant hysteresis in the current–voltage (I–V)
characteristics were available. We can distinguish two
classes of experiments, which are based on two different
complementary types of junctions: 1) MQT and energy
level quantization (ELQ) [45,46] 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
[47,48], where d-wave effects are expected to play a minor
role [50]. The experiments using GBs are more complicated
because of the complexity 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.
In HTS Josephson junctions, when a lobe of the OPS on
one side of the junction meets a lobe on the other side, a
larger critical current density ( )cJ is measured, differently
from the case of a lobe facing a node [7,44,51]. If we pro-
duce d-wave junctions with different interface orientations,
as for instance made possible by the biepitaxial technique,
a wide range of different OPS configurations are realized,
representing an additional knob to tune phase dynamics.
The interest for this “intrinsically encoded phase dynam-
ics” cannot be disjointed from the general problem of un-
derstanding dissipation in systems with very low energy
and potentially highly dissipative quasiparticles, as occur-
ring in d-wave systems. Since the frequency dependent
quality factor ( ) = ( )pQ R Cω ω ω [15] is a measure of dis-
sipation in the JJ, while the critical current cI is an easily
accessible parameter, the dissipative essence strongly de-
pends on the value of the effective frequency dependent
resistance ( )R ω and of shunting capacitance C, which in
turn depend on several interplaying effects, such as circuit
impedance, subgap resistance and stray capacitance. Since
these parameters are not easily accessible, a reliable way
able to estimate the frequency dependent quality factor
( )Q ω is of great interest.
The GB biepitaxial junctions [52,53] used in [45,46]
had reproducible hysteretic behavior up to 90%. A specif-
ic 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 [53,54]. The presence of the CeO2 pro-
duces an additional 45° in-plane rotation of the YBCO
axes with respect to the in-plane directions of the sub-
strate. Atomically flat interfaces can be achieved in ap-
propriate conditions [52]. This configuration produces a
45° misorientation between the two electrodes to enhance
d-wave order parameter effects, by varying the interface
orientation.
Escape dynamics in moderately damped Josephson junctions
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4 341
The junction in the tilt configuration (angle θ = 0°)
turns out to be the most interesting case for the MQT and
ELQ experiments. This lobe to node configuration max-
imizes d-wave induced effects and allows to explore the
effects of low-energy quasiparticles. The switching current
probability distributions as a function of temperature for
the biepitaxial JJ substantially follows what commonly
measured on LTS JJs, with a saturation of the measured σ
below 50 mK [45], which corresponds to the crossover
temperature from the thermal to the MQT regime and is
consistent with the predicted values. The change of crT
through an external magnetic field is an important confir-
mation of the occurrence of MQT. Values of R 100 Ω
and JC ∼ 0.22 pF can be obtained from the measurements
with a plasma frequency / 2pω π 2.6 GHz and a quality
factor of the order of 10 [45,46,55]. C-axis tilt is mostly
responsible for low-barrier transparency and leads to the
presence of a significant kinetic inductance in the model-
ing of YBCO JJ. In these junctions the presence of a kinet-
ic inductance and a stray capacitance determine the main
difference in the washboard potential making the system
behavior depending on two degrees of freedom [46,55].
The YBCO JJ is coupled to this LC-circuit and the poten-
tial become two-dimensional (2D).
In summary macroscopic quantum phenomena have ob-
served also in HTS in extremely unfavorable conditions
of lobe-node configuration and therefore nominally in
presence of low-energy quasiparticles [45,46].
3.2. Macroscopic quantum phenomena in HTS intrinsic
junctions
Experiments on Bi2Sr22CaCu2O8 intrinsic Josephson
junctions (IJJ) [47,48] have been aimed to increase the cros-
sover temperature cr( )T and to clarify the nature of IJJs, ra-
ther than raising novel themes of coherence in d-wave sys-
tems. In these junctions the nodes of the d-wave order
parameter are not expected to affect significantly MQT.
Josephson coupling between CuO2 double layers has
been proved, and most of the materials behaved like
stacks of S–I–S JJs with effective barriers of the order of the
separation of the CuO2 double layers (1.5 nm) ( cJ typically
103 A/cm2) [56,57]. IJJs have a much higher Josephson
coupling energy than GB junctions, the I–V curves exhi-
bited large hysteresis and multiple branches, indicative of a
series connection of highly capacitive junctions. Practical
realizations of IJJ have been designed in order to nominal-
ly avoid heating effects [18,57]. However, at high-voltages
caution is required when extracting information because of
possible unavoidable heating problems. crT has been re-
ported to be about 800 mK, remarkably higher than those
usually found in LTS systems. By using microwave spec-
troscopy, the unique uniform array structure of intrinsic
Josephson junction stacks have been considered responsi-
ble for a remarkable enhancement of the tunneling rate
[48]. This enhancement adds a factor of approximately N
2
to the quantum escape rate of a single Josephson junction,
also resulting in a significant increase of cr ,T where N is
the number of the junctions in the stack. This effect can be
caused by large quantum fluctuations due to interactions
among the N junctions [48]. Intrinsic junctions have been
also studied in the moderately damped regime as discussed
in the next section.
3.3. The moderately damped regime in low critical
currents Josephson junctions
The possibility to reproducibly achieve low critical cur-
rents in submicron junctions promoted studies on mod-
erately damped junctions (1 < Q < 10). The same regime
can be controllably induced in larger junctions in case of
low critical current densities, or with lower reproducibility
in junctions with larger intrinsic dissipation levels, as oc-
curring in HTS systems.
As it can be seen in Fig. 1,c, the diffusive regime is cha-
racterized by the onset of multiple retrapping processes in
subsequent potential wells. As roughly sketched in Fig. 1,b
and by the Eq. (6), a decrease of the Josephson energy JE
and of the quality factor Q enhances the retrapping proba-
bility ,RΓ causing multiple retrapping phenomena in the
switching dynamics.
As opposed to the strongly underdamped Josephson
junctions case, well described by the analysis of Fulton and
Dunkleberger [10] in the thermal region, overdamped junc-
tions show nonhysteretic behavior, with a finite voltage on
the supercurrent branch of the I–V characteristic, asso-
ciated with thermally activated phase diffusion, and ther-
mal fluctuations leading to very much smaller variations in
the switching behavior. If the damping around the junction
plasma frequency is sufficiently high, at low bias the phase
particle will always retrap in a local minima after escape
and a finite resistance phase-diffusion branch appears on
the I–V curve of the junction (the appearance of a small
voltage, prior to switching to a voltage on the order of
twice the superconducting gap ).Δ Phase diffusion (PD) in
junctions with hysteretic I–V characteristics has been dis-
cussed in literature [15,58] and has been associated with
frequency-dependent damping, so that junctions are under-
damped at high frequencies but are in the overdamped lim-
it at low frequencies, respectively.
Phase diffusion process has reappeared in a different
regime in magnetometers with much larger and unshunted
junctions used for qubits readout [17]. The retrapping
process is affected by the frequency dependent impedance
of the environment of the dc SQUID. For > 1Q with rela-
tively small ,cI such that >> ,J B cE k T E∼ it has been
shown that a regime exists where escape does not lead to a
finite-voltage state, but rather to underdamped phase diffu-
sion [16]. In addition to the usual crossover between MQT
and TA behavior, the transition from TA behavior to un-
derdamped phase diffusion has been also observed, result-
ing in a more diversified ( , )B Jk T E phase diagram [16].
D. Massarotti, L. Longobardi, L. Galletti, D. Stornaiuolo, D. Montemurro, G. Pepe, G. Rotoli, A. Barone, and F. Tafuri
342 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4
The transition from TA to underdamped phase diffusion
regime is marked by the collapse of the width of the
switching distributions and the turn over temperature *T is
defined as the temperature at which the width σ of the
switching current distribution reaches the maximum value
(see below). The change in the sign of the derivative of the
second moment of the distribution and a modification of
the shape of the distributions at temperatures around *T
turn to be distinctive signatures of the PD regime. The
phase diagram in [16] defines the relation between JE and
the turn over temperature *T and indicates how the transi-
tion from TA to underdamped phase diffusion can be tuned
by the critical current of the JJ.
In [18] a systematic analysis has been carried out on
different types junctions, all with low values of the criti-
cal current: low ohmic Nb–Pt–Nb (S–N–S) junctions,
Nb–CuNi–Nb (S–F–S) junctions with a diluted ferromag-
netic alloys, Nb–InAs–Nb (S–2D electron gas–S) struc-
tures and BiSrCaCuO (2212) IJJ. The damping parameter
of the junctions has been tuned in different manners by
changing temperature, magnetic field, and gate voltage and
introducing a ferromagnetic layer or in situ capacitive
shunting. The phenomenon of an unexpected collapse of
switching current fluctuations with increasing T is ex-
plained by the interplay of two counteracting consequences
of thermal fluctuations [18]. On one hand, thermal fluctua-
tions assist in premature switching into the resistive state
and, on the other hand, help in retrapping back to the su-
perconducting state [18]. In other words, temperature does
not only provide energy for excitation of a system from
equilibrium state but also enhances the rate of relaxation
back to the equilibrium. Multiple-retrapping processes in a
hysteretic BiSrCaCuO (2212) IJJ with a high tunneling
resistance have been reported to govern the switching from
a resistive state in the phase-diffusion regime into the qua-
siparticle tunneling state [59]. The frequency-dependent
junction quality factor, representing the energy dissipation
in a phase-diffusion regime, determines the observed tem-
perature dependence of the switching current distribution
and the switching rate. Phase dynamics has been investi-
gated also in underdamped ferromagnetic JJ by measuring
the switching probability in both the stationary and nonsta-
tionary regimes [60,61]. The junction is Nb–Al2O3–PdNi
(10% Ni)–Nb. Incomplete relaxation leads to dynamical
phase bifurcation. Bifurcation manifests itself as a prema-
ture switching, resulting in a bimodal switching distribu-
tion [60]. Escape rate measurements at temperatures T
down to 20 mK show that the width of the switching cur-
rent histogram decreases with temperature and saturates
below T = 150 mK in Nb/Al2O3/Cu40%Ni60%/Nb (super-
conductor–insulator–ferromagnet–superconductor) hetero-
structures [61].
Distinctive fingerprints of phase diffusion can be found
in the temperature dependence of switching current distri-
bution (SCD) histograms and of their width .σ An exam-
ple is shown in Fig. 3 for NbN/MgO/NbN JJs [62,63].
SCD curves are reported for zero (H = 0) and for finite
(H = 6.09 G) magnetic field in Figs. 3,a and 3,b, respec-
tively. The temperature dependence of σ is given in the
upper frame of Fig. 4. We can clearly distinguish at lower
temperatures the transition from MQT to TA. When in-
creasing the temperature, histograms tend to become more
symmetric and shrink rather than broaden with a conse-
quent increase of their maximum amplitude [62,63]. This
translates in a characteristic dependence of ,σ i.e, appear-
ance of an anticorrelation between the temperature and the
width of the switching distributions [16–19,62,63]. After
the MQT saturation at the lowest temperatures (in this case
about 100 mK), σ follows the expected 2/3T dependence,
deviations are evident in proximity and above the crossov-
er temperature *T where the temperature derivative of
( )Tσ becomes negative. The MgO barrier is 1 nm thick in
this case, providing cJ of about 3 A/cm2. The total current
cI is about 2 μA and falls under the criteria of the mod-
erately damped regime. The experimental data are well
Fig. 3. Switching current probability distribution at = 0H (a)
and = 6.09GH (b) of a NbN/MgO/NbN JJ [62,63].
H=6 .09G
30
30
20
20
10
10
0
0
0.8 1.0 1.2 1.4 1.6 1.8
P
ro
b
ab
il
it
y
d
en
si
ty
,
A
–
1
�
P
ro
b
ab
il
it
y
d
en
si
ty
,
A
–
1
�
Switching current, A�
Switching current, A�
0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
a
b
Escape dynamics in moderately damped Josephson junctions
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4 343
reproduced by the expected values, as shown in the lower
frame of Fig. 4, calculated on the basis of the physical ar-
guments of phase diffusion [63].
The simulations are based on the recent work on phase
diffusion by Fenton and Warburton (FW) [19]. The phase
difference ( )tϕ is a solution of the following Langevin
differential equation:
/ = 0.tt t NQϕ + ϕ + +ι ι (8)
Times t are normalized to 1/ ;pω ι is the bias current
normalized to critical current 0cI and Nι is a Gaussian
correlated thermal noise current, i.e.:
0 0( ), ( ) = 2 / ( ).N N B ct t k T QI t t′ ′π Φ δ −ι ι
Stochastic dynamics is simulated by integrating the above
Langevin equation by a Bulirsh–Stoer integrator using
as noise generator the cernlib routine RANLUX [64].
The magnetic field works as a knob to tune *T and pro-
vides an additional validity test for the estimate of
= 2.7 0.1Q ± [62].
Phase diffusion also appears in the escape rates Γ ,
shown in the inset of Fig. 4 as a function of the ratio be-
tween the barrier height UΔ and the thermal energy. The
escape rates are calculated from the switching distributions
using Eq. (7). In the thermal activation regime the distribu-
tions are asymmetric and skewed to the left, and Γ values
all fall onto the same line, as it is the case for the reported
data from T = 0.3 K to 1.56 K. Retrapping processes cause
a progressive symmetrization of the switching distribution,
as it can be seen in Figs. 3,a and 3,b, and a bending in the
Γ vs / ,BU k TΔ as shown in the inset of Fig. 4 [62,63].
The symmetrization of the switching distribution due to
the interplay between escape and retrapping events can be
clearly observed by plotting, as a function of temperature,
the skewness of the distributions ,γ i.e., the ratio 3
3 /m σ
where 3m is the third central moment of the distribution.
For the lowest temperatures we obtain = 1,γ − which is
consistent with the case of switching current distributions
in the quantum or thermal regime. As the temperature in-
creases the distributions become more and more symmetric
as γ tends to zero [62,63]. It should be noted that for these
data the temperature *T at which the derivative of ( )Tσ
changes sign is equal to (1.62±0.3) K and that the skew-
ness starts increasing already at about 1.2 K, which is a
clear indication that the onset of retrapping phenomena
occurs well below *T [19].
The very good fitting of experimental curves [62,63]
using the FW approach [19] confirms the occurrence of a
multiple-retrapping regime with a large number of escapes
of duration of 1
R
−Γ , and in particular the relevance of the
dependence of retrapping probability on time. The experi-
mental results are well described by a numerical model
involving a frequency independent damping which demon-
strates an efficient way to estimate the dissipation in mod-
erately damped JJs.
A nonexhaustive list of Josephson devices that have dis-
played a similar PD regime is reported in Table 1 along with
the most relevant device parameters. Independently of the
physical size of the device, all the junctions exhibiting phase
diffusion over a large range of materials and geometry have a
low critical current, 1 < Q < 6 and *12 < ( ) / < 18,BU I k TΔ
which are therefore relevant parameters signaling the in-
surgence of multiple escape and retrapping in a washboard
potential.
Fig. 4. Top frame: temperature dependence of the standard devia-
tion σ of the switching distributions for H = 0 G (squares) and
H = 6.09 G (circles) shown in Fig. 3. Bottom frame: a numerical
simulation of the data. In the inset escape rates (symbols) as a
function of the barrier height at zero magnetic field for tempera-
tures near *T are shown, together with numerical simulations
(dashed lines).
4040
4040
3535
3535
3030
3030
2525
2525
2020
2020
1515
1515
σ,σ,
n
A
n
A
σ,σ,
n
A
n
A
TT, K, K
HH = 0= 0
HH = 6.09 G= 6.09 G
0.10.1 11
II
II
PPssww
PPssww
ΓΓ,
s,
s––
11
101055
101044
101033
101022
101011
101000
1414 1616 1818 2020 2222
ΔΔUU kk TT//
BB
TT = 2.35 K= 2.35 K
2.12 K2.12 K
1.86 K1.86 K and 1.56 Kand 1.56 K
0.3 K0.3 K
D. Massarotti, L. Longobardi, L. Galletti, D. Stornaiuolo, D. Montemurro, G. Pepe, G. Rotoli, A. Barone, and F. Tafuri
344 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4
Table 1. Comparison of device parameters
* In this paper the authors estimated the fit parameters to be tem-
perature dependent. Here we report the values at the lowest ex-
perimental temperature T = 1.5 K.
** This is the value of the critical current at the turn over tempera-
ture * = 75KT .
3.4. In the “far” low critical current regime
The possibility to have extremely low critical current
density can be functional to investigate phase dynamics at
extreme conditions. An example is given by a recent expe-
riment on submicron Nb/AlOx/Nb junctions [64]. Data
show an anomalous ( )Tσ dependence with a negative
/d dTσ over the entire temperature range. This regime
can be achieved by engineering junctions with lower criti-
cal current and junction capacitance, such that the ratio
0 / ,cI C which regulates cr ,T is constant and the turn over
temperature *T is lower or comparable to the quantum
crossover temperature crT [11]. Another example is given
by Yoon et al. [65]. They have engineered Al/AlOx/Al JJs
in order to obtain low critical currents, of about 400 nA,
and low capacitance, of about 40 fF, at the same time. In
this way they have observed that TA is completely sup-
pressed since *T is lower than cr .T On the other hand, by
adding a shunting capacitance in the device circuit, TA
regime is recovered, showing that the shunting capacitance
can be used in order to tune the phase dynamics.
Junctions with intrinsically low critical current densities
could represent an interesting term of comparison to study
these kinds of unconventional regimes using standard
micrometer junctions. Recently YBCO biepitaxial JJs have
been engineered on (La0.3Sr0.7)(Al0.65Ta0.35)O3 (LSAT)
substrates, responding to the needs of improving biepitaxi-
al performances at microwaves [66]. These junctions are
characterized by higher c NI R values and by lower values
of capacitance. Since LSAT substrates have a lower dielec-
tric constant with respect to SrTiO3 substrates, this struc-
ture allows to isolate GB contribution from stray capacit-
ance, and tends to favor moderately damped regime [66].
In a recent study [67] a direct transition from PD regime to
MQT has been demonstrated in these junctions. This has
been made possible by selecting junctions where the cros-
sover temperature crT is higher than *.T In these condi-
tions thermal escape is dominated by the other contribu-
tions in their respective ranges. This behavior is easily
visualized in Fig. 5 through a schematic representation of
the switching current distribution as a function of tempera-
ture. Differently from the low dissipation case (see Fig. 2),
the SCD broadens when lowering the T and corresponding-
ly the peak intensity decreases. Monte Carlo simulations of
the thermal behavior of σ in the PD regime could be used
in order to estimate the junction quality Q also in the mod-
erately damped regime.
The whole set of data collected in these subsections is a
solid framework where most of the phase dynamics can be
easily classified through a study of switching current dis-
tributions. A phase diagram valid in a large range of dissi-
pation conditions emerges as a functional guide to classify
all types of behaviors and as reference for phase dynamics
of novel types of junctions.
3.5. Switching current distribution measurements in
“nanostructures”
SCD measurements have recently performed on a series
of different nanostructures. Some of them are junctions and
then can be easily classified in the schemes described above,
and more specifically in the moderately damped regime.
Some of them are simple nanowires. We believe a subtle
path exists between these different systems with analogies
Author Device structure R,
Ω
0cI ,
μA
C,
fF
Q
(I=0)
Longobardi
et al. [63]
NbN/MgO/NbN JJ 65 1.91 300 2.7
Kivioja
et al. [16]
Al/AlOx/Al dc SQUID 500 0.2 100 3.9
Kivioja
et al. [16]
Al/AlOx/Al JJ 230 0.63 130 3.6
Männik
et al. [17]
Nb/AlOx/Nb dc SQUID 70 4.25 90 2.4
Männik
et al. [17]
Nb/AlOx/Nb dc SQUID 70 2.9 260 3.3
Bae et al.
[59]*
Bi-2212 intrinsic JJ 62 1.26 330 2.2
Yu et al.
[64]
Nb/AlOx/Nb JJ 1800 0.122 20 4.8
Yu et al.
[64]
Nb/AlOx/Nb JJ 315 0.48 77 3.3
Krasnov
et al. [18]
S–2DEG–S 10 37 200 1.6
Krasnov
et al. [18]
Bi-2212 intrinsic JJ 40 80** 685 5.6
Fig. 5. Schematic representation of SCD measurements in the
case of direct transition from MQT to phase diffusion regime.
<Tcr
Pr
ob
ab
ili
ty
, a
rb
. u
ni
ts
Current, arb. units
Escape dynamics in moderately damped Josephson junctions
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4 345
and quite distinctive features. When considering that a
microbridge of width of the order of the coherence length
behaves as a Josephson junction [3,4], SCD measurements
will turn to be more and more a direct way of discriminating
the phase dynamics and the transport in nontrivial cases,
which are going to be more and more common with ad-
vances in nanopatterning superconductors at extreme scales.
Supercurrent passes in graphene sheets comprised in be-
tween superconducting electrodes [68]. This is one of the
nanostructured proximity-coupled Josephson systems based
on conducting spacers, able in principle to be electrically
tuned. Other possibilities other than graphene are offered by
nanowires [70,71], carbon nanotubes [72,73] and nanocrys-
tals [74]. In Lee et al. [68] graphene is attached to PbIn elec-
trodes separated by a trench of 300 nm. PbIn superconduct-
ing electrodes significantly enhances the critical current cI
compared with commonly used Al (as high as 6 μA in high-
ly doped regions). The crossover from the classical to quan-
tum regime is controlled by the gate voltage and has been
found surprisingly high of the order of a few hundreds mK.
Q factor is about 5–6 for all voltages. Capacitance is for
instance about 35 fF at V = –60 V and seems to be not re-
lated to self-heating [75] but consistent with an effective
capacitance eff Th/ NC R E= due to diffusive motion
of quasiparticles in graphene [76] Th(E is the Thouless
energy). Phase diffusion regime has been found for all gate
voltages with *T ranging from about 1 K (V = 0) to 2 K
(V = –60 V). Fingerprints of the Thouless energy [77] and of
the minigap, commonly observed in the transport properties
of nanostructures [78,79] should also somehow manifest in
the switching dynamics in nanostructures.
Stochastic dynamics of superconductive-resistive swit-
ching in hysteretic current-biased superconducting nano-
wires undergoing phase-slip fluctuations is a topic of
growing interest. Recent studies have reported phase-slip
induced switching in superconducting nanowires [80–83].
In Mo79Ge21 nanowires of lenght ranging from 100 nm up
to 200 nm [80], SDM have been used to investigate the
behavior of individual quantum phase-slip events at high
bias currents, observing a monotonic increase of σ with
decreasing temperature. In Al nanowires [81] of width less
than 10 nm and length ranging from 1.5 to 10 μm (with
critical currents of the order of a few μA), fluctuations in
the average critical current exhibit three distinct regions of
behaviors and are nonmonotonic in temperature. Saturation
is present well below the critical temperature ,cT σ in-
creases as 2/3T at intermediate temperatures, and a col-
lapse is present close to .cT The relationship between in-
dividual phase slips and switching has been also
theoretically investigated [84] in order to provide a tool to
study phase slips, to help establish whether they are caused
by thermal fluctuations or by macroscopic quantum tunne-
ling [85]. It has been found that although several phase-slip
events are generally necessary to induce switching, there is
an experimentally accessible regime of temperatures and
currents for which just one single phase-slip event is suffi-
cient to induce switching, via the local heating it causes.
Conclusions
We have focused on macroscopic quantum decay phe-
nomena, as one of the most exciting expressions of the
Josephson effect. We have in particular reviewed experi-
ments on unconventional systems where Josephson junc-
tions are characterized by intermediate levels of dissipa-
tion. For unconventional types of junctions, we intend both
d-wave HTS JJs and low-Jc NbN JJs, or devices scaled to
the nanosize or incorporating superconductors and nano-
wires or flakes as barriers. Switching current distribution
measurements are a direct way of discriminating the phase
dynamics and the transport also in nontrivial cases of mod-
erate damping, which are going to be more and more
common with advances in nanopatterning superconductors
and in materials science with novel possibilities of synthe-
sizing also hybrid coplanar systems. A wide vision on ma-
croscopic quantum phenomena in a variety of complemen-
tary systems including d-wave junctions can promote novel
arguments on the interplay of coherence and dissipation in
solid state systems.
Acknowledgments
This contribution initially intended to celebrate 50 years
from the discovery of the Josephson effect, also stands
now as the last paper of our friend and maestro Antonio
Barone. Antonio passed away before He could see the final
line of the work, that however responds to the common
initial planning. This manuscript is now also in his honor,
touching one of his favorite topics, macroscopic quantum
phenomena in Josephson junctions. This work is supported
by MIUR PRIN 2009 under the project SuFET based on
nanowires and HTS. We also acknowledge partial support
by STREP MIDAS, Macroscopic Interference Devices for
atomic and Solid State Physics: Quantum Control of Su-
percurrents and by a Marie Curie International Reintegra-
tion Grant No. 248933 hybMQC within the 7th European
Community Framework Programme.
1. B.D. Josephson, Phys. Lett. 1, 251 (1962).
2. I.O. Kulik and K. Yanson, The Josephson Effect in
Superconductive Tunneling Structures, Israel Program of
Scientific Translations (Jerusalem) (1972).
3. A. Barone and G. Paternó, Physics and Applications of the
Josephson Effect, John Wiley, New York (1982).
4. K.K. Likharev, Dynamics of Josephson Junctions and Cir-
cuits, New York: Gordon and Breach (1986); K.K. Likharev,
Rev. Mod. Phys. 51, 101 (1979).
5. A.O. Caldeira and A.J. Leggett, Phys. Rev. Lett. 46, 211
(1981); A.J. Leggett, J. Phys. (Paris) Colloq. 39, C6–1264
(1980); A.O. Caldeira and A.J. Leggett, Ann. Phys. (N.Y) 149,
374 (1983).
D. Massarotti, L. Longobardi, L. Galletti, D. Stornaiuolo, D. Montemurro, G. Pepe, G. Rotoli, A. Barone, and F. Tafuri
346 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4
6. A.F. Andreev, Zh. Eksp. Teor. Fiz. 46, 1823 (1964) [Sov.
Phys. JETP 19, 1228 (1964)]; I.O. Kulik, Sov. Phys. JETP 30,
944 (1969).
7. C.C. Tsuei and J.R. Kirtley, Rev. Mod. Phys. 72, 969 (2000).
8. T. Van Duzer and C.W. Turner, Principles of Supercon-
ductive Devices and Circuits, New York, Elsevier (1991).
9. J. Clarke and F. Wilhelm, Nature 453, 1031 (2008).
10. T.A. Fulton and L.N. Dunkleberger, Phys. Rev. B9, 4760
(1974).
11. M.H. Devoret, J.M. Martinis, and J. Clarke, Phys. Rev. Lett.
55, 1908 (1985).
12. J.M. Martinis, M.H. Devoret, and J. Clarke, Phys. Rev. B35,
4682 (1987); J.M. Martinis, M.H. Devoret, and J. Clarke,
Phys. Rev. Lett. 55, 1543 (1985).
13. V.L. Ginzburg and L.D. Landau, Zh. Eksperim. i Teor. Fiz.
20, 1064 (1950).
14. H.A. Kramers, Physica (Utrecht) 7, 284 (1940).
15. R.L. Kautz and J.M. Martinis, Phys. Rev. B42, 9903 (1990);
J.M. Martinis and R.L. Kautz, Phys. Rev. Lett. 63, 1507
(1989).
16. J.M. Kivioja, T.E. Nieminen, J. Claudon, O. Buisson, F.W.J.
Hekking, and J.P. Pekola, Phys. Rev. Lett. 94, 247002 (2005).
17. J. Männik, S. Li, W. Qiu, W. Chen, V. Patel, S. Han, and
J.E. Lukens, Phys. Rev. B71, 220509 (2005).
18. V.M. Krasnov, T. Golod, T. Bauch, and P. Delsing, Phys.
Rev. B76, 224517 (2007); V.M. Krasnov, T. Bauch, S.
Intiso, E. Hurfeld, T. Akazaki, H. Takayanagi, and P.
Delsing, Phys. Rev. Lett. 95, 157002 (2005).
19. J.C. Fenton, and P.A. Warburton, Phys. Rev. B78, 054526
(2008).
20. Yu.M. Ivanchenko and L.A. Zilberman, Zh. Eksp. Teor. Fiz.
55, 2396 (1968) [Sov. Phys. JETP 28, 1272 (1969)].
21. 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).
22. R.F. Voss and R.A. Webb, Phys. Rev. Lett. 47, 265 (1981).
23. 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).
24. W. der Boer and R. de Bruyn Ouboter, Physica B98, 185
(1980); D.W. Bol, R. van Weelderen, and R. de Bruyn Ouboter,
Physica B122, 2 (1983); D.W. Bol, J.J.F. Scheffer, W. Giele,
and R. de Bruyn Ouboter, Physica B113, 196 (1985).
25. R.J. Prance, A.P. Long, T.D. Clarke, A. Widom, J.E. Mut-
ton, J. Sacco, M.W. Potts, G. Negaloudis, and F. Goodall,
Nature 289, 543 (1981).
26. I.M. Dmitrenko, V.A. Khlus, G.M. Tsoi, and V.I. Shnyrkov,
Fiz. Nizk. Temp. 11, 146 (1985) [Sov. J. Low Temp. Phys. 11,
77 (1985)].
27. S. Washburn, R.A. Webb, R.F. Voss, and S.M. Farris, Phys.
Rev. Lett. 54, 2712 (1985).
28. D.B. Schartz, B. Sen, C.N. Archie, and J.E. Lukens, Phys.
Rev. Lett. 55, 1547 (1985).
29. A. Wallraff, T. Duty, A. Lukashenko, and A.V. Ustinov,
Phys. Rev. Lett. 90, 370031 (2003).
30. A.J. Leggett, in Chance and Matter: Les Houches Session
XLVI, J. Souletie, J. Vannimenus, and R. Stora (eds.), Else-
vier, Amsterdam (1987), p. 395.
31. Y. Nakamura, C.D. Chen, and J.S. Tsai, Phys. Rev. Lett. 79,
2328 (1997); Y. Nakamura, Y.A. Pashkin, and J.S. Tsai,
Nature 398, 786 (1999).
32. J.R. Friedman, V. Patel, W. Chen, S.K. Tolpygo, and J.E.
Lukens, Nature 406, 43 (2000).
33. C.H. van der Wal, A.C.J. ter Haar, F.K. Wilhelm, R.N.
Schouten, C.J.P.M. Harmans, T.P. Orlando, S. Lloyd, and
J.E. Mooij, Science 290, 773 (2000).
34. D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C.
Urbina, D. Esteve, and M.H. Devoret, Science 296, 886
(2002); G. Ithier, E. Collin, P. Joyez, P.J. Meeson, D. Vion,
D. Esteve, F. Chiarello, A. Shnirman, Y. Makhlin, J.
Schriefl, and G. Schon, Phys. Rev. B72, 134519 (2005).
35. J.M. Martinis, S. Nam, J. Aumentado, and C. Urbina, Phys.
Rev. Lett. 89, 117901 (2002).
36. P. Bertet, I. Chiorescu, G. Burkard, K. Semba, C.J. Harmans,
D.P. DiVincenzo, and J.E. Mooij, Phys. Rev. Lett. 95,
257002 (2005).
37. M. Steffen, M. Ansmann, R. McDermott, N. Katz, R.C.
Bialczak, E. Lucero, M. Neeley, E.M. Weig, A.N. Cleland,
and J.M. Martinis, Phys. Rev. Lett. 97, 050502 (2006).
38. Y. Nakamura, Y.A. Pashkin, T. Yamamoto, and J.S. Tsai,
Phys. Rev. Lett. 88, 047901 (2002).
39. D.A. Bennett, L. Longobardi, Vijay Patel, Wei Chen, D.V.
Averin, and J.E. Lukens, Quantum Inf. Process. 8, 217 (2009).
40. L. Longobardi, Studies of Quantum Transitions of Magnetic
Flux in a rf SQUID Qubit, LAP Lambert Academic Publish-
ing (2010).
41. J. Koch, T.M. Yu, J. Gambetta, A.A. Houck, D.I. Schuster, J.
Majer, A. Blais, M.H. Devoret, S.M. Girvin, and R.J. Scho-
elkopf, Phys. Rev. A76, 042319 (2007).
42. Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya,
N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques,
A. Dreau, J.-F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Es-
teve, and P. Bertet, Phys. Rev. Lett. 107, 220501 (2011).
43. M. Iansiti, A. Johnson, W.F. Smith, H. Rogalla, C.J. Lobb,
and M. Tinkham, Phys. Rev. Lett. 59, 489 (1987); M. Iansiti,
M. Tinkham, A.T. Johnson, W.F. Smith, and C.J. Lobb,
Phys. Rev. B39, 6465 (1989).
44. F. Tafuri and J.R. Kirtley, Rep. Prog. Phys. 68, 2573 (2005).
45. T. Bauch, F. Lombardi, F. Tafuri, G. Rotoli, A. Barone,
P. Delsing, and T. Cleason, Phys. Rev. Lett. 94, 087003 (2005).
46. T. Bauch, T. Lindström, F. Tafuri, G. Rotoli, P. Delsing, T.
Claeson, and F. Lombardi, Science 311, 5757 (2006).
47. 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).
48. X.Y. Jin, J. Lisenfeld, Y. Koval, A. Lukashenko, A.V. Usti-
nov, and P. Mueller, Phys. Rev. Lett. 96, 177003 (2006).
49. P.W. Anderson, Science 288, 480 (2000); Handbook of High-
Temperature Superconductivity: Theory and Experiment, J.S.
Brooks and J. Robert Schrieffer (eds.), Springer (2006).
Escape dynamics in moderately damped Josephson junctions
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4 347
50. Y.V. Fominov, A.A. Golubov, and M.Y. Kupriyanov, JETP
Lett. 77, 587 (2003); M.H.S. Amin and A.Y. Smirnov, Phys.
Rev. Lett. 92, 017001 (2004); S. Kawabata, S. Kashiwaya, Y.
Asano, and Y. Tanaka, Phys. Rev. B70, 132505 (2004); ibid.
B72, 052506 (2005); T. Yokoyama, S. Kawabata, T. Kato,
and Y. Tanaka, Phys. Rev. B76, 134501 (2007).
51. H. Hilgenkamp and J. Mannhart, Rev. Mod. Phys. 74, 485
(2002).
52. F. Tafuri, F. Miletto Granozio, F. Carillo, A. Di Chiara, K.
Verbist, and G. Van Tendeloo, Phys. Rev. B59, 11523 (1999).
53. 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).
54. 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).
55. G. Rotoli, T. Bauch, T. Lindstrom, D. Stornaiuolo, F. Tafuri,
and F. Lombardi, Phys. Rev. B75, 144501 (2007).
56. R. Kleiner, F. Stenmeyer, G. Kunkel, and P. Muller, Phys.
Rev. Lett. 68, 2394 (1992).
57. R. Kleiner and P. Muller, Phys. Rev. B49, 1327 (1994); H.B.
Wang, P.H. Wu, and T. Yamashita, Appl. Phys. Lett. 78,
4010 (2001).
58. D. Vion, M. Gotz, P. Joyez, D. Esteve, and M.H. Devoret,
Phys. Rev. Lett. 77, 3435 (1996).
59. Myung-Ho Bae, M. Sahu, Hu-Jong Lee, and A. Bezryadin,
Phys. Rev. B79, 104509 (2009).
60. I. Petkovic and M. Aprili, Phys. Rev. Lett. 102, 157003 (2009).
61. J. Pfeiffer, T. Gaber, D. Koelle, R. Kleiner, E. Goldobin, M.
Weides, H. Kohlstedt, J. Lisenfeld, A.K. Feofanov, and A.V.
Ustinov, Preprint Cond Mat. 0903.1046; J. Pfeiffer, M.
Kemmler, D. Koelle, R. Kleiner, E. Goldobin, M. Weides,
A.K. Feofanov, J. Lisenfeld, and A.V. Ustinov; Phys. Rev.
B77, 214506 (2008).
62. L. Longobardi, D. Massarotti, G. Rotoli, D. Stornaiuolo, G.
Papari, A. Kawakami, G.P. Pepe, A. Barone, and F. Tafuri,
Appl. Phys. Lett. 99, 062510 (2011).
63. L. Longobardi, D. Massarotti, G. Rotoli, D. Stornaiuolo, G.
Papari, A. Kawakami, G.P. Pepe, A. Barone, and F. Tafuri,
Phys. Rev. B84, 184504 (2011).
64. H.F. Yu, X.B. Zhu, Z.H. Peng, Ye Tian, D.J. Cui, G.H.
Chen, D.N. Zheng, X.N. Jing, Li Lu, S.P. Zhao, and Siyuan
Han, arXiv:1101.2250v1 [cond-mat.supr-con] (2011).
65. Y. Yoon, S. Gasparinetti, M. Mttnen, and J.P. Pekola, J.
Low. Temp. Phys. 163, 164 (2011).
66. D. Stornaiuolo, G. Papari, N. Cennamo, F. Carillo, L.
Longobardi, D. Massarotti, A. Barone, and F. Tafuri,
Supercond. Science and Technology 24, 045008 (2011),
D. Stornaiuolo et al., unpublished (2012).
67. L. Longobardi, D. Massarotti, D. Stornaiuolo, L. Galletti, G.
Rotoli, and F. Tafuri, unpublished (2012).
68. G.-H. Lee, D. Jeong, J.-H. Choi, Y.-J. Doh, and H.-J. Lee, Phys.
Rev. Lett. 107, 146605 (2011); D. Jeong, J.-H. Choi, G.-H. Lee1,
S. Jo, Y.-J. Doh, and H.-J. Lee, Phys. Rev. B83, 094503 (2011).
69. C. Ojeda-Aristizabal, M. Ferrier, S. Gueron, and H. Bou-
chiat, Phys. Rev. B79, 165436 (2009); I.V. Borzenets, U.C.
Coskun, S.J. Jones, and G. Finkelstein, Phys. Rev. Lett. 107,
137005 (2011).
70. Y.J. Doh, J.A. van Dam, A.L. Roest, E.P.A.M. Bakkers, L.P.
Kouwenhoven, and S. De Franceschi, Science 309, 272 (2005).
71. J. Xiang, A. Vidan, M. Tinkham, R.M. Westervelt, and C.M.
Lieber, Nature Nanotech. 1, 208 (2006).
72. P. Jarillo-Herrero, J.A. van Dam, and L.P. Kouwenhoven,
Nature 439, 953 (2006).
73. J.P. Cleuziou, W. Wernsdorfer, V. Bouchiat, T. Ondarcuhu,
and M. Monthioux, Nature Nanotech. 1, 53 (2006).
74. G. Katsaros, P. Spathis, M. Stoffel, F. Fournel, M. Mongillo,
V. Bouchiat, F. Lefloch, A. Rastelli, O.G. Schmidt, and S.
De Franceschi, Nature Nanotech. 5, 458 (2010).
75. H. Courtois, M. Meschke, J.T. Peltonen, and J.P. Pekola,
Phys. Rev. Lett. 101, 067002 (2008).
76. L. Angers, F. Chiodi, G. Montambaux, M. Ferrier, S. Gureron,
H. Bouchiat, and J.C. Cuevas, Phys. Rev. B77, 165408 (2008).
77. J.T. Edwards and D. Thouless, J. Phys. C: Solid State Phys.
5, 807 (1972); D.J. Thouless, Phys. Rev. Lett. 39, 1167
(1977); Y. Imry, Introduction to Mesoscopic Physics, Oxford
University Press (1997); C.W.J. Beenakker, Rev. Mod. Phys.
69, 731 (1997); B.L. Altshuler and P.A. Lee, Phys. Today
41, 36 (1988); R.A. Webb and S. Washburn, Phys. Today 41,
46 (1988).
78. M. Vinet, C. Chapelier, and F. Lefloch, Phys. Rev. B63,
165420 (2001); A.K. Gupta, L. Cretinon, N. Moussy, B.
Pannetier, and H. Courtois, Phys. Rev. B69, 104514 (2004);
H. le Sueur, P. Joyez, H. Pothier, C. Urbina, and D. Esteve,
Phys. Rev. Lett. 100, 197002 (2008),
79. P. Lucignano, D. Stornaiuolo, F. Tafuri, B.L. Altshuler, and
A. Tagliacozzo, Phys. Rev. Lett. 105, 147001 (2010).
80. M. Sahu, M.H. Bae, A. Rogachev, D. Pekker, T.C. Wei, N.
Shah, P.M. Goldbart, and A. Bezryadin, Nature Phys. 5, 503
(2009).
81. P. Li, P.M. Wu, Y. Bomze, I.V. Borzenets, G. Finkelstein,
and A.M. Chang, Phys. Rev. Lett. 107, 137004 (2011).
82. Peng Li, Ph.M. Wu, Yu. Bomze, I.V. Borzenets, G. Finkelstein,
and A.M. Chang, Phys. Rev. B84, 184508 (2011).
83. M. Tinkham, J.U. Free, C.N. Lau, and N. Markovic, Phys.
Rev. B68, 134515 (2003).
84. N. Shah, D. Pekker, and P.M. Goldbart, Phys. Rev. Lett. 101,
207001 (2008).
85. W.A. Little, Phys. Rev. 156, 396 (1967); J.S. Langer and V.
Ambegaokar, Phys. Rev. 164, 498 (1967); D.E. McCumber
and B.I. Halperin, Phys. Rev. B1, 1054 (1970); N. Giordano,
Phys. Rev. Lett. 61, 2137 (1988); A. Bezryadin, C.N. Lau,
and M. Tinkham, Nature 404, 971 (2000); C.N. Lau, N.
Markovic, M. Bockrath, A. Bezryadin, and M. Tinkham,
Phys. Rev. Lett. 87, 217003 (2001); A. Rogachev, A.T.
Bollinger, and A. Bezryadin, Phys. Rev. Lett. 94, 017004
(2005); F. Altomare, A.M. Chang, M.R. Melloch, Y. Hong,
and C.W. Tu, Phys. Rev. Lett. 97, 017001 (2006).
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