Dynamical reentrance and geometry imposed quantization effects in Nb-AlOx-Nb Josephson junction arrays

Abstract

In this paper, we report on different phenomena related to the magnetic properties of artificially prepared highly ordered (periodic) two-dimensional Josephson junction arrays (2D-JJA) of both shunted and unshunted Nb-AlOx-Nb tunnel junctions. By employing mutual-inductance measurements and using a high-sensitive bridge, we have thoroughly investigated (both experimentally and theoretically) the temperature and magnetic field dependence of complex AC susceptibility of 2D-JJA. We also demonstrate the use of the scanning SQUID microscope for imaging the local flux distribution within our unshunted arrays.

Full text

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Dynamical reentrance and geometry imposed quantization
effects in Nb–AlOx–Nb Josephson junction arrays

Fernando M. Araújo-Moreira1 and Sergei Sergeenkov2


1Grupo de Materiais e Dispositivos, Centro Multidisciplinar para o Desenvolvimento de Materiais
Cerâmicos, Departamento de Física, Universidade Federal de São Carlos, 13565-905 São Carlos, SP, Brazil
2Departamento de Física, CCEN, Universidade Federal da Paraíba, Cidade Universitária, 58051-970 João
Pessoa, PB, Brazil


Abstract. In this review, we report on different phenomena related to the magnetic properties of artificially
prepared highly ordered (periodic) two-dimensional Josephson junction arrays (2D-JJA) of both shunted
and unshunted Nb–AlOx–Nb tunnel junctions. By employing mutual-inductance measurements and using a
high-sensitive home-made bridge, we have thoroughly investigated (both experimentally and theoretically)
the temperature and magnetic field dependence of complex AC susceptibility of 2D-JJA.
After brief description of the measurements technique and numerical simulations method, we proceed to
demonstrate that the observed dynamic reentrance (DR) phenomenon is directly linked to the value of the
Stewart-McCumber parameter βC. By simultaneously varying the inductance related parameter βL, we
obtain a phase diagram βC-βL (which demarcates the border between the reentrant and non-reentrant
behavior) and show that only arrays with sufficiently large value of βC will exhibit the DR behavior.
The second topic of this review is related to the step-like structure (with the number of steps n = 4
corresponding to the number of flux quanta that can be screened by the maximum critical current of the
junctions) which has been observed in the temperature dependence of AC susceptibility in our unshunted
2D-JJA with βL(4.2K) = 30 and attributed to the geometric properties of the array. The steps are predicted
to manifest themselves in arrays with βL(T) matching a "quantization" condition βL(0)=2π(n+1).
In conclusion, we demonstrate the use of the scanning SQUID microscope for imaging the local flux
distribution within our unshunted arrays.

1. Introduction

Many unusual and still not completely understood magnetic properties of Josephson
junctions (JJs) and their arrays (JJAs) continue to attract attention of both theoreticians
and experimentalists alike (for recent reviews on the subject see, e.g. Newrock et al 2000,
Araujo-Moreira et al 2002, Li 2003, Kirtley et al 1998, Altshuler and Johansen 2004 and
further references therein). In particular, among the numerous spectacular phenomena
recently discussed and observed in JJAs we would like to mention the dynamic
temperature reentrance of AC susceptibility (Araujo-Moreira et al 2002) closely related
to paramagnetic Meissner effect (Li 2003) and avalanche-like magnetic field behavior of
magnetization (Altshuler and Johansen 2004, Ishikaev et al 2000). More specifically,
using highly sensitive SQUID magnetometer, magnetic field jumps in the magnetization
curves associated with the entry and exit of avalanches of tens and hundreds of fluxons

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were clearly seen in SIS-type arrays (Ishikaev et al 2000). Besides, it was shown that the
probability distribution of these processes is in good agreement with the theory of self-
organized criticality (Jensen 1998). It is also worth mentioning the recently observed
geometric quantization (Sergeenkov and Araujo-Moreira 2004) and flux induced
oscillations of heat capacity (Bourgeois et al 2005) in artificially prepared JJAs as well
as recently predicted flux driven temperature oscillations of thermal expansion coefficient
(Sergeenkov et al 2007) both in JJs and JJAs. At the same time, successful adaptation of
the so-called two-coil mutual-inductance technique to impedance measurements in JJAs
provided a high-precision tool for investigation of the numerous magnetoinductance (MI)
related effects in Josephson networks (Martinoli and Leeman 2000, Meyer et al 2002,
Korshunov 2003, Tesei et al 2006). To give just a few recent examples, suffice it to
mention the MI measurements (Meyer et al 2002) on periodically repeated Sierpinski
gaskets which have clearly demonstrated the appearance of fractal and Euclidean regimes
for non-integer values of the frustration parameter, and theoretical predictions
(Korshunov 2003) regarding a field-dependent correction to the sheet inductance of the
proximity JJA with frozen vortex diffusion. Besides, recently (Tesei et al 2006) AC
magnetoimpedance measurements performed on proximity-effect coupled JJA on a dice
lattice revealed unconventional behaviour resulting from the interplay between the
frustration f created by the applied magnetic field and the particular geometry of the
system. While the inverse MI exhibited prominent peaks at f = 1/3 and at f =1/6 (and
weaker structures at f = 1/9, 1/12, . . ) reflecting vortex states with a high degree of
superconducting phase coherence, the deep minimum at f = 1/2 points to a state in which
the phase coherence is strongly suppressed. More recently, it was realized that JJAs can
be also used as quantum channels to transfer quantum information between distant sites
(Ioffe et al 2002, Born et al 2004, Zorin 2004) through the implementation of the so-
called superconducting qubits which take advantage of both charge and phase degrees of
freedom (see, e.g., Krive et al 2004 and Makhlin et al 2001 for reviews on quantum-state
engineering with Josephson-junction devices).
Artificially prepared two-dimensional Josephson junctions arrays (2D-JJA) consist
of highly ordered superconducting islands arranged on a symmetrical lattice coupled by
Josephson junctions (figure 1), where it is possible to introduce a controlled degree of

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disorder. In this case, a 2D-JJA can be considered as the limiting case of an extreme
inhomogeneous type-II superconductor, allowing its study in samples where the disorder
is nearly exactly known. Since 2D-JJA are artificial, they can be very well characterized.
Their discrete nature, together with the very well-known physics of the Josephson
junctions, allows the numerical simulation of their behavior (see very interesting reviews
by Newrock et al 2000 and by Martinoli et al 2000 on the physical properties of 2D-JJA).

Figure 1. Photograph of unshunted (I) and shunted (II) Josephson junction arrays.

Many authors have used a parallelism between the magnetic properties of 2D-JJA
and granular high-temperature superconductors (HTS) to study some controversial
features of HTS. It has been shown that granular superconductors can be considered as a
collection of superconducting grains embedded in a weakly superconducting - or even
normal - matrix. For this reason, granularity is a term specially related to HTS, where
magnetic and transport properties of these materials are usually manifested by a two-
component response. In this scenario, the first component represents the intragranular
contribution, associated to the grains exhibiting ordinary superconducting properties, and
the second one, which is originated from intergranular material, is associated to the
weak-link structure, thus, to the Josephson junctions network (Clark 1968, Saxena et al
1974, Yu and Saxena 1975, Resnick et al 1981, Sergeenkov 2001, Sergeenkov 2006,
Sergeenkov and Araujo-Moreira 2004, Sergeenkov et al 2007). For single-crystals and
other nearly-perfect structures, granularity is a more subtle feature that can be envisaged
as the result of a symmetry breaking. Thus, one might have granularity on the nanometric
Nb-island
Tunnel
junction
Shunt
resistor
(I)
(II)

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scale, generated by localized defects like impurities, oxygen deficiency, vacancies,
atomic substitutions and the genuinely intrinsic granularity associated with the layered
structure of perovskites. On the micrometric scale, granularity results from the existence
of extended defects, such as grain and twin boundaries. From this picture, granularity
could have many contributions, each one with a different volume fraction (Araujo-
Moreira et al 1994, Araujo-Moreira et al 1996, Araujo-Moreira et al 1999, Passos et al
2000). The small coherence length of HTS implies that any imperfection may contribute
to both the weak-link properties and the flux pinning. This leads to many interesting
peculiarities and anomalies, many of which have been tentatively explained over the
years in terms of the granular character of HTS materials. One of the controversial
features of HTS elucidated by studying the magnetic properties of 2D-JJA is the so-called
Paramagnetic Meissner Effect (PME), also known as Wohlleben Effect. In this case, one
considers first the magnetic response of a granular superconductor submitted to either an
AC or DC field of small magnitude. This field should be weak enough to guarantee that
the critical current of the intergranular material is not exceeded at low temperatures. After
a zero-field cooling (ZFC) process which consists in cooling the sample from above its
critical temperature (TC) with no applied magnetic field, the magnetic response to the
application of a magnetic field is that of a perfect diamagnet. In this case, the
intragranular screening currents prevent the magnetic field from entering the grains,
whereas intergranular currents flow across the sample to ensure a null magnetic flux
throughout the whole specimen. This temperature dependence of the magnetic response
gives rise to the well-known double-plateau behavior of the DC susceptibility and the
corresponding double-drop/double-peak of the complex AC magnetic susceptibility
(Araujo-Moreira et al 1994, Araujo-Moreira et al 1996, Araujo-Moreira et al 1999,
Passos et al 2000, Goldfarb et al 1992). On the other hand, by cooling the sample in the
presence of a magnetic field, by following a field-cooling (FC) process, the screening
currents are restricted to the intragranular contribution (a situation that remains until the
temperature reaches a specific value below which the critical current associated to the
intragrain component is no longer equal to zero). It has been experimentally confirmed
that intergranular currents may contribute to a magnetic behavior that can be either
paramagnetic or diamagnetic. Specifically, where the intergranular magnetic behavior is

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paramagnetic, the resulting magnetic susceptibility shows a striking reentrant behavior.
All these possibilities about the signal and magnitude of the magnetic susceptibility have
been extensively reported in the literature, involving both LTS and HTS materials
(Wohlleben et al 1991, Braunich et al 1992, Kostic et al 1996, Geim et al 1998). The
reentrant behavior mentioned before is one of the typical signatures of PME. We have
reported its occurrence as a reentrance in the temperature behavior of the AC magnetic
susceptibility of 2D-JJA (Araujo-Moreira et al 1997, Barbara et al 1999). Thus, by
studying 2D-JJA, we were able to demonstrate that the appearance of PME is simply
related to trapped flux and has nothing to do with manifestation of any sophisticated
mechanisms, like the presence of pi-junctions or unconventional pairing symmetry. To
perform this work, we have used numerical simulations and both the mutual-inductance
and the scanning SQUID microscope experimental techniques.
The paper is organized as follows. In Section 2 we briefly outline the main concepts
related to the mutual-inductance technique. In Section 3 we review the theoretical
background for numerical simulations based on a unit cell containing four Josephson
junctions. In Section 4 we study the origin of dynamic reentrance and discuss the role of
the Stewart-McCumber parameter in the observability of this phenomenon. In Section 5
we present the manifestation of completely novel geometric effects recently observed in
the temperature behavior of AC magnetic response. In Section 6 we demonstrate the use
of scanning SQUID microscope for imaging the local flux distribution within our
unshunted arrays. And finally, in Section 7 we summarize the main results of this work.

2. The mutual-inductance technique

Complex AC magnetic susceptibility is a powerful low-field technique to determine
the magnetic response of many systems, like granular superconductors and Josephson
junction arrays. It has been successfully used to measure several parameters such as
critical temperature, critical current density and penetration depth in superconductors. To
measure samples in the shape of thin films, the so-called screening method has been
developed. It involves the use of primary and secondary coils, with diameters smaller
than the dimension of the sample. When these coils are located near the surface of the

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film, the response, i.e., the complex output voltage V, does not depend on the radius of
the film or its properties near the edges. In the reflection technique (Jeanneret et al 1989)
an excitation coil (primary) coaxially surrounds a pair of counter-wound pick up coils
(secondaries). When there is no sample in the system, the net output from these
secondary coils is close to zero since the pick up coils are close to identical in shape but
are wound in opposite directions. The sample is positioned as close as possible to the set
of coils, to maximize the induced signal on the pick up coils (figure 2).


Figure 2. Screening method in the reflection technique, where an excitation coil (primary) coaxially
surrounds a pair of counter-wound pick up coils (secondaries).


An alternate current sufficient to create a magnetic field of amplitude hAC and
frequency f is applied to the primary coil. The output voltage of the secondary coils, V, is
a function of the complex susceptibility, ´´i´
AC
χ+χ=χ
, and is measured through the usual
lock-in technique. If we take the current on the primary as a reference, V can be
expressed by two orthogonal components. The first one is the inductive component, VL
(in phase with the time-derivative of the reference current) and the second one the
quadrature resistive component, VR (in phase with the reference current). This means that
VL and VR are correlated with the average magnetic moment and the energy losses of the
sample, respectively.
ip
δV

JJA SAMPLE
Primary coil
Secondary coils

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We used the screening method in the reflection configuration to measure χAC(T) of
Josephson junction arrays. Measurements were performed as a function of the
temperature T (1.5K < T < 15K), the amplitude of the excitation field hAC (1 mOe < hAC <
10 Oe), and the external magnetic field HDC (0 < HDC < 100 Oe) parallel with the plane of
the sample (figure 3).


Figure 3. Sketch of the experimental setup, where the excitation field
ac
h
and the external magnetic field
dc
H
are respectively perpendicular and parallel to the plane of the sample.




3. Numerical simulations: theoretical background

We have found that all the experimental results obtained from the magnetic
properties of 2D-JJA can be qualitatively explained by analyzing the dynamics of a single
unit cell in the array (Araujo-Moreira et al 1997, Barbara et al 1999).


Figure 4. Unit cell of the array, containing a loop with four identical junctions.
h ac
H DC

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In our numerical simulations, we model a single unit cell as having four identical
junctions (see figure 4), each with capacitance CJ, quasi-particle resistance RJ and critical
current IC. If we apply an external field of the form:
) tcos(hH
AC ext
ω=
(3.1)
then the total magnetic flux,
TOT
Φ
, threading the four-junction superconducting loop is
given by:
LI
EXTTOT
+Φ=Φ
(3.2)
where
EXT
2
0EXT
Ha
µ=Φ
is the flux related to the applied magnetic field with
0
µ being
the vacuum permeability, I is the circulating current in the loop, and L is the inductance
of the loop. Therefore the total current is given by:
2
2
0
0
22
)( sin)()(
dt
d
C
dt
d
R
tTItI
i
j
i
j
iC
φ
π
φ
π
φ
Φ
+
Φ
+=
(3.3)
Here,
) t (
iφ
is the superconducting phase difference across the ith junction,
0
Φ is the
magnetic flux quantum, and IC is the critical current of each junction. In the case of our
model with four junctions, the fluxoid quantization condition, which relates each ) t (
iφ
to
the external flux, reads:
0
22
Φ
Φ
+=
TOT
i
n
ππ
φ
(3.4)
where n is an integer and, by symmetry, we assume that (Araujo-Moreira et al 1997,
Barbara et al 1999 ) :
i4321
φ≡φ=φ=φ=φ
(3.5)
In the case of an oscillatory external magnetic field of the form of Eq. (3.1), the
magnetization is given by:
2
0a
LI
M
µ
=
(3.6)
It may be expanded as a Fourier series in the form:
∑
=
n
∞
ωχ+ωχ=
0
"
n
'
nAC
)]tn ( sin) t n cos([h) t ( M
(3.7)

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We calculated ' χ and "
χ through this equation. Both Euler and fourth-order Runge-
Kutta integration methods provided the same numerical results. In our model we do not
include other effects (such as thermal activation) beyond the above equations. In this
case, the temperature-dependent parameter is the critical current of the junctions, given to
good approximation by (Meservey 1969, Sergeenkov et al 2007):



 ∆
2




∆
∆
=
Tk
)T(
tanh
) 0 (
)T(
) 0 (
C
I)T(I
B
C
(3.8)
where






−
∆=∆
T
TT
2 . 2tanh) 0 ()T(
C

(3.9)
is the analytical approximation of the BCS gap parameter with
CBTk 76. 1) 0 (
∆=
.
We simulated
1 χ as a function of temperature and applied magnetic fields keeping
in mind that
1 χ depends on the geometrical parameter
L
β (which is proportional to the
number of flux quanta that can be screened by the maximum critical current in the
junctions), and the dissipation parameter
C
β (which is proportional to the capacitance of
the junction)
0
C
L
)T( LI
Φ
2
)T(
π
=β
(3.10)
0
2
JC
Φ
C
J
RCI2
)T(
π
=β

(3.11)

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4. On the origin of dynamic reentrance


According to the current paradigm, paramagnetic Meissner effect (PME) can be
related to the presence of π-junctions (Braunisch et al 1992, Li 2003, Kostic et al 1996,
Ortiz et al 2001, Passos et al 2000, Lucht et al 1995, Li and D. Dominguez 2000) , either
resulting from the presence of magnetic impurities in the junction (Bulaevskii et al 1977,
Kusmartsev 1992) or from unconventional pairing symmetry (Kawamura and Li 1996).
Other possible explanations of this phenomenon are based on flux trapping (Chen et al
1995) and flux compression effects (Terentiev et al 1999). Besides, in the experiments
with unshunted 2D-JJA, we have previously reported (Araujo-Moreira et al 1997,
Barbara et al 1999) that PME manifests itself through a dynamic reentrance (DR) of the
AC magnetic susceptibility as a function of temperature. These results have been further
corroborated by Nielsen et al (2000) and De Leo et al (2001) who argued that PME can
be simply related to magnetic screening in multiply connected superconductors. So, the
main question is: which parameters are directly responsible for the presence (or absence)
of DR in artificially prepared arrays?
Previously (also within the single plaquette approximation), Barbara et al (1999)
have briefly discussed the effects of varying
L
β on the observed dynamic reentrance with
the main emphasis on the behavior of 2D-JJA samples with high (and fixed) values of
C
β . However, to our knowledge, up to date no systematic study (either experimental or
theoretical) has been done on how the
C
β value itself affects the reentrance behavior. In
this section, by a comparative study of the magnetic properties of shunted and unshunted
2D-JJA, we propose an answer to this open question. Namely, by using experimental and
theoretical results, we will demonstrate that only arrays with sufficiently large value of
the Stewart-McCumber parameter
C
β will exhibit the dynamic reentrance behavior.
To measure the complex AC susceptibility in our arrays we used a high-sensitive
home-made susceptometer based on the so-called screening method in the reflection
configuration (Jeanneret et al 1989, Araujo-Moreira et al 2002), as shown in previous
sections. The experimental system was calibrated by using a high-quality niobium thin
film. To experimentally investigate the origin of the reentrance, we have measured
)T( ' χ

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for three sets of shunted and unshunted samples obtained from different makers
(Westinghouse and Hypress) under the same conditions of the amplitude of the excitation
field
ac
h (1 mOe <
ac
h <10 Oe), external magnetic field
dc
H (0 <
dc
H < 500 Oe) parallel
to the plane of the sample, and frequency of AC field
f2π=ω
(fixed at f = 20 kHz).
Unshunted 2D-JJAs are formed by loops of niobium islands linked through Nb-AlOx-Nb
Josephson junctions while shunted 2D-JJAs have a molybdenum shunt resistor (with
Ω≈
2 . 2Rsh
) short-circuiting each junction (see figure 1). Both shunted and unshunted
samples have rectangular geometry and consist of
150 100×
tunnel junctions. The unit
cell for both types of arrays has square geometry with lattice spacing
m46a
µ≈
and a
single junction area of
2
m55
µ×
. The critical current density for the junctions forming the
arrays is about 600A/cm2 at 4.2 K. Besides, for the unshunted samples 30)K 2 . 4 (
C
≈β

and 30)K2 . 4 (
L
≈β
, while for shunted samples 1)K2 . 4 (
C
≈β
and 30)K 2 . 4 (
L
≈β
where
L
β and
C
β are given by expressions (3.10) and (3.11), respectively. There,
pF58 . 0Cj≈

is the capacitance,
Ω≈
4 . 10Rj
the quasi-particle resistance (of unshunted array), and
A150)K 2 . 4 (IC
µ≈
the critical current of the Josephson junction.
0
Φ is the quantum of
magnetic flux.

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0.20.40.60.81.0
-0.8
-0.6
-0.4
-0.2
0.0
unshunted(a)
T/TC
hac=10 mOe
χ'(SI)
hac=100 mOe

0.4 0.60.8 1.0
-0.15
-0.13
-0.10
-0.08
-0.05
-0.03
0.00
shunted
(b)
hac=10 mOe
hac=200 mOe
hac=25 mOe
χ'(SI)
T/TC

Figure 5. Experimental results for
)H,h , T( '
χ
dcac
: (a) unshunted 2D-JJA for
=
ac
h
10 and 100 mOe;
(b) shunted 2D-JJA for
=
ac
h
10, 25, and 200 mOe. In all these experiments
0Hdc=
. Solid
lines are the best fits (see text).

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Since our shunted and unshunted samples have the same value of
L
β and different
values of
C
β , it is possible to verify the dependence of the reentrance effect on the value
of the Stewart-McCumber parameter. For the unshunted 2D-JJA (figure 5a) we have
found that for an AC field lower than 50 mOe the behavior of
)T ( ' χ
is quite similar to
homogeneous superconducting samples, while for
ac
h > 50 mOe (when the array is in the
mixed-like state with practically homogeneous flux distribution) these samples exhibit a
clear reentrant behavior of susceptibility (Araujo-Moreira et al 1999). At the same time,
the identical experiments performed on the shunted samples produced no evidence of any
reentrance for all values of
ac
h (see figure 5b). It is important to point out that the
analysis of the experimentally obtained imaginary component of susceptibility
)T ( "
χ

shows that for the highest AC magnetic field amplitudes (of about 200 mOe) dissipation
remains small. Namely, for typical values of the AC amplitude,
ac
h = 100 mOe (which
corresponds to about 10 vortices per unit cell) the imaginary component is about 15 times
smaller than its real counterpart. Hence contribution from the dissipation of vortices to
the observed phenomena can be safely neglected.
0.40.60.81.0
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
Hdc=30.5 Oe
Hdc=26 Oe
Hdc=19.5 Oe
Hdc=13 Oe
Hdc=0 Oe
χ'(SI)
T/TC

Figure 6. Experimental results for
)H,h , T( '
χ
dcac
for unshunted 2D-JJA for
=
dc
H
0, 13, 19.5, 26, and
30.5 Oe. In all these experiments
=
ac
h
100 mOe. Solid lines are the best fits (see text).

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To further study this unexpected behavior we have also performed experiments
where we measure
)T ( ' χ
for different values of
dc
H keeping the value of
ac
h constant.
The influence of DC fields on reentrance in unshunted samples is shown in figure 6. On
the other hand, the shunted samples still show no signs of reentrance, following a familiar
pattern of field-induced gradual diminishing of superconducting phase (very similar to a
zero DC field flat-like behavior seen in figure 5b).
To understand the influence of DC field on reentrance observed in unshunted
arrays, it is important to emphasize that for our sample geometry this parallel field
suppresses the critical current
C I of each junction without introducing any detectable flux
into the plaquettes of the array. Thus, a parallel DC magnetic field allows us to vary
C I
independently from temperature and/or applied perpendicular AC field. The
measurements show (see figure 6) that the position of the reentrance is tuned by
dc
H .
We also observe that the value of temperature
min
T (at which
)T( ' χ
has a
minimum) first shifts towards lower temperatures as we raise
dc
H (for small DC fields)
and then bounces back (for higher values of
dc
H ). This non-monotonic behavior is
consistent with the weakening of
C I and corresponds to Fraunhofer-like dependence of
the Josephson junction critical current on DC magnetic field applied in the plane of the
junction. We measured
C I from transport current-voltage characteristics, at different
values of
dc
H at T = 4.2 K and found that
)K2 . 4T( '
χ=
, obtained from the isotherm T =
4.2 K (similar to that given in figure 6), shows the same Fraunhofer-like dependence on
dc
H as the critical current )H(I
dcC
of the junctions forming the array (see figure 7). This
gives further proof that only the junction critical current is varied in this experiment. This
also indicates that the screening currents at low temperature (i.e., in the reentrant region)
are proportional to the critical currents of the junctions. In addition, this shows an
alternative way to obtain )H(I
dcC
dependence in big arrays. And finally, a sharp
Fraunhofer-like pattern observed in both arrays clearly reflects a rather strong coherence
(with negligible distribution of critical currents and sizes of the individual junctions)
which is based on highly correlated response of all single junctions forming the arrays,

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thus proving their high quality. Such a unique behavior of Josephson junctions in our
samples provides a necessary justification for suggested theoretical interpretation of the
obtained experimental results. Namely, based on the above-mentioned properties of our
arrays, we have found that practically all the experimental results can be explained by
analyzing the dynamics of just a single unit cell in the array.


Figure 7. The critical current
C I (open squares) and the real part of AC susceptibility ' χ (solid triangles)
H
for T=4.2K (Araujo-Moreira et al 1999). as a function of DC field
dc

To understand the different behavior of the AC susceptibility observed in shunted
and unshunted 2D-JJAs, in principle one would need to analyze in detail the flux
dynamics in these arrays. However, as we have previously reported (Araujo-Moreira et al
1999), because of the well-defined periodic structure of our arrays (with no visible
distribution of junction sizes and critical currents), it is reasonable to expect that the
experimental results obtained from the magnetic properties of our 2D-JJAs can be quite
satisfactory explained by analyzing the dynamics of a single unit cell (plaquette) of the
array. An excellent agreement between a single-loop approximation and the observed
behavior (seen through the data fits) justifies a posteriori our assumption. It is important
to mention that the idea to use a single unit cell to qualitatively understand PME was first
suggested by Auletta et al (1994, 1995). They simulated the field-cooled DC magnetic
susceptibility of a single-junction loop and found a paramagnetic signal at low values of
external magnetic field.