Manifestation of geometric resonance in current dependence of AC
susceptibility for unshunted array of Nb-AlOx-Nb Josephson junctions
V. A. G. Rivera1, S. Sergeenkov2,*, E. Marega1 and F.M. Araújo-Moreira2, #
1Instituto de Física de São Carlos, USP, Caixa Postal 369, 13560-970, São Carlos, SP,
2Grupo de Materiais e Dispositivos, Departamento de Física, UFSCar, Caixa Postal 676,
13565-905, São Carlos, SP, Brasil
By improving resolution of home-made mutual-inductance measurements technique, a
pronounced resonance -like structure has been observed in the current dependence of AC
susceptibility in artificially prepared two-dimensional array of unshunted Nb-AlOx-Nb
Josephson junctions (JJA). Using a single-plaquette approximation for JJA model, we were
able to successfully fit our data assuming that resonance structure is related to the
geometric (inductive) properties of the array.
Keywords: Josephson junction arrays; AC susceptibility; inductance related effects
PACS classification codes: 74.25.F-; 74.25.Ha; 74.50.+r; 74.81.Fa
* Corresponding author: phone: (16) 260-8205; fax: (16) 260-4835; E-mail: firstname.lastname@example.org
# Research Leader; E-mail: email@example.com
Probably one of the most promising tendencies in modern development of artificially
prepared Josephson Junction Arrays (JJA) is based on investigation of intricate correlations
between their transport and magnetic properties which include, in particular, simultaneous
measurements of current-voltage characteristics (CVC) and AC susceptibility (for genral
reviews on the subject matter, see, e.g., [1-4] and further references therein). Among the
numerous spectacular phenomena recently discussed and observed in JJA, we could
mention the dynamic reentrance of AC susceptibility [5-11], geometric quantization ,
flux driven temperature oscillations of thermal expansion coefficient , current driven
giant fractional Shapiro steps [14-16], etc. At the same time, successful adaptation of the
so-called two-coil mutual-inductance technique to impedance measurements in JJA
provided a high-precision tool for investigation of the numerous magnetoinductance related
effects in Josephson networks [17-19].
In this Letter we present experimental evidence for manifestation of novel geometric
effects in magnetic response of high-quality ordered array of unshunted Nb-AlOx-Nb
junctions under application of AC current. We observed a pronounced resonance -like
structure in the current dependence of AC susceptibility which was discussed within the
single-plaquette approximation and related to inductance properties of our array.
2. Experimental Results
High quality ordered SIS type unshunted array of overdamped Nb-AlOx-Nb junctions has
been prepared by using a standard photolithography and sputtering technique . It is
formed by loops of niobium islands linked through 100x150 tunnel junctions. The unit cell
of the array has square geometry with lattice spacing a=46µm and a single junction area of
S=5x5 µm2 (see Fig.1). The critical current for the junctions forming the array is IC(T)=150
µΑ at T=4.2K. Given the values of the junction quasi-particle resistance R=10Ω,
inductance L=µ0a=64pH, and capacitance C=1.2fF, the geometrical and dissipation
parameters are estimated to be βL(T)=2πLIC(T)/Φ0=30 and βC(T)=2πCR2IC(T)/Φ0=0.05 at
T=4.2K, respectively. The latter estimate suggests that our array can be effectively used, for
example, in rapid single flux quantum logic circuits and programmable Josephson voltage
For comparative study of current driven effects in our array, we measured their transport
and magnetic properties. For both purposes, AC current
0<I<5mA and fixed frequency
) was applied parallel to the plane of the array
and normally to the Josephson current IJ (see Fig.2 for the sketch of the setup used in the
present experiments). The measurements of CVC V(I) were made using homemade
experimental technique with a high-precision nanovoltmeter [21-23]. Some typical results
for the obtained V(I) dependencies in our array (taken at two distinctive temperatures and
exhibiting a noticeable nonlinear behavior) are shown in Fig. 3.
To measure the current induced response of complex AC susceptibility
precision, we used a home-made susceptometer based on the so-called screening method in
the reflection configuration [5,6,9-12]. We observed a characteristic oscillating dependence
of the zero-field (B=0) normalized susceptibility
) 0 (
on applied current I as well as
a pronounced resonance-like peak around I=2mA which is clearly seen in Fig.4.
Turning to the interpretation of the obtained experimental results, it is important to mention
that magnetic field dependence of the critical current of the array (taken at T=4.2K) on DC
magnetic field B (parallel to the plane of the sample) exhibits [5-11] a sharp Fraunhofer-
like pattern characteristic of a single-junction response, thus proving a rather strong
coherence within array (with negligible distribution of critical currents and sizes of the
individual junctions) and hence the high quality of our sample.
Due to the well-defined periodic structure of our array, it is quite reasonable to assume that
the experimental results could be understood by analyzing the dynamics of just a single unit
cell (plaquette) of the array. As we shall see, theoretical interpretation of the presented here
experimental results based on single-loop approximation is in excellent agreement with the
In our analytical calculations, the unit cell is the loop containing four identical Josephson
junctions. If we apply a DC magnetic field B and an AC current
the JJA, the total magnetic flux
) t (
threading the four-junction superconducting loop is
given by )()(
t LI BSt
where L is the loop (plaquette) inductance, and S=5x5 µm2 is
a single junction area (see Fig.1).
To properly treat the current mediated magnetic properties of the system, let us introduce
the following model Hamiltonian [4,12]
which describes the tunneling (first term) and inductive (second term) contributions to the
total energy of the four-junction plaquette. Here, ( )
is the gauge-invariant
superconducting phase difference across the ith junction [4-12], and the corresponding
Josephson current (shown in Fig.2) is given by
The seeking dependence of zero-field (B=0) susceptibility on the amplitude of the
applied current I can be defined as a time average over the period
Solid line in Fig.4 shows the best fit of the measured normalized susceptibility
) 0 (
using Eq.(2) with χ(0)=−0.72SI, L=64pH, and ω=40kHz. A remarkable agreement between
the experimental points and theoretical curve justifies a posteriori the adopted here single
plaquette approximation scenario. Furthermore, a careful analysis of Eq.(2) reveals that the
( ) I
, where ω=40kHz is the frequency of the applied observed resonance occurs for
is the current induced resonance
being the dynamical inductance of the plaquette [2,17,18].
More precisely, near the resonance, Eq.(2) can be approximated (with good accuracy) by
the Fraunhofer-type dependence
and L=64pH, we find that the resonance condition
( ) I
for our array
will be satisfied for the amplitude of the AC current equal to
, in excellent
agreement with the observations (see Fig.4).
Finally, it is important to emphasize that the conventional AC Josephson effect with
, related to the above-discussed nonlinear CVC law V(I) in our
array, is too high to account for the observed resonant structure of AC susceptibility.
Indeed, according to Fig.3, the resonant current of I=2mA corresponds to the voltage of
V=175mV (see the intersection of dotted lines in Fig.3 for T=4.2K) which gives
for the estimate of the Josephson frequency in our array.
In summary, we reported on observation of pronounced resonance-like structure in current
dependence of AC susceptibility for SIS-type ordered array of unshunted Nb-AlOx-Nb
Josephson junctions. The origin of the observed phenomenon was discussed within a
single-plaquette approximation and attributed to manifestation of geometric (inductance
related) properties of the array.
We are very grateful to R.S. Newrock, P. Barbara and C.J. Lobb for helpful discussions and
for providing high quality SIS samples. This work has been financially supported by the
Brazilian agencies CNPq, CAPES and FAPESP.
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Fig.1: SEM photography of a single plaquette (consisting of four junctions) for the studied
array of unshunted Nb-AlOx-Nb Josephson junctions.
Fig.2: A simplified sketch of the experimental setup showing the directions of applied Iac
and Josephson IJ currents in the array.
Fig.3: Current-voltage characteristics of the array taken at two distinctive temperatures.
Fig.4: The current dependence of the normalized zero-field susceptibility
) 0 (
(taken at T=4.2K), along with the best fit (solid line) according to Eq.(2).
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