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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,

Brasil

2Grupo de Materiais e Dispositivos, Departamento de Física, UFSCar, Caixa Postal 676,

13565-905, São Carlos, SP, Brasil

Abstract

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: sergei@df.ufscar.br

# Research Leader; E-mail: faraujo@df.ufscar.br

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1. Introduction

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 [12],

flux driven temperature oscillations of thermal expansion coefficient [13], 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 [1]. 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

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example, in rapid single flux quantum logic circuits and programmable Josephson voltage

standards [20,21].

For comparative study of current driven effects in our array, we measured their transport

and magnetic properties. For both purposes, AC current

tItIac

ω

sin)( =

(with amplitude

0<I<5mA and fixed frequency

kHz

40

=

ω

) 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

)(I

χ

with high

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 (

χ

/ )(

χ I

on applied current I as well as

a pronounced resonance-like peak around I=2mA which is clearly seen in Fig.4.

3. Discussion

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

observed behavior.

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

tItIac

ω

sin)( =

parallel to

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the JJA, the total magnetic flux

) t (

Φ

threading the four-junction superconducting loop is

given by )()(

t LI BSt

ac

+=Φ

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]

( )

t

( )

t

[]

( )

L

t

J

2

cos1

2

Φ

+−=Η

ϕ

(1)

which describes the tunneling (first term) and inductive (second term) contributions to the

total energy of the four-junction plaquette. Here, ( )

( )

t

0

2

Φ

Φ

=

t

π

ϕ

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

ϕ

sin

CJ

II =

where

0

/2

Φ=

JIC

π

is the

critical current.

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

ω

/

π

2

τ=

(Cf.[4]):

( )

2

0

2

0

cos

1

τ

)(

=

??

?

??

?

∂

Η

B

∂

−=?

B

t

tdtI

ωχ

τ

(2)

Solid line in Fig.4 shows the best fit of the measured normalized susceptibility

) 0 (

χ

/ )(

χ I

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

r

ωω ≅

, where ω=40kHz is the frequency of the applied observed resonance occurs for

AC current

tItIac

ω

sin)( =

and

( )

I

0

2

π

Φ

??

?

??

?

=

IL

r

ω

ω

ω

is the current induced resonance

frequency with

22

1

τω

ω

+

=

L

L

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

(

ω

−

)

τ

(

ω

)τω

ω

χχ

r

r

I

−

−≈

sin

)(

0

with

0

2

0

2/ Φ=

C IS

πχ

. Recalling

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that

π

2

ωτ

=

and L=64pH, we find that the resonance condition

( ) I

r

ωω ≅

for our array

will be satisfied for the amplitude of the AC current equal to

mAI

2

≅

, in excellent

agreement with the observations (see Fig.4).

Finally, it is important to emphasize that the conventional AC Josephson effect with

frequency

( )

I

?

)(2

I eV

J

=

ω

, 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

( )

ωω >>≈

HzI

J

10

for the estimate of the Josephson frequency in our array.

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4. Summary

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.

Acknowledgments

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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References

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[11] F.M. Araujo-Moreira, S. Sergeenkov, Supercond. Sci. Technol. 21 (2008) 045002.

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[13] S. Sergeenkov, G. Rotoli, G. Filatrella, F.M. Araujo-Moreira, Phys. Rev. B 75 (2007)

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44 (1991) 4601.

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Figure Captions

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 (

χ

/ )(

χ I

(taken at T=4.2K), along with the best fit (solid line) according to Eq.(2).

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Fig.1

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Fig.2

IJ

Iac

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012

I(mA)

345

0

50

100

150

200

250

V(mV)

T(K)

4.2

6.0

Fig.3

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12

012

I(mA)

345

0.8

1.0

1.2

1.4

1.6

1.8

χ(I)/χ(0)

B=0

T=4.2K

Fig.4