OFFPRINTSuperconducting networks with the proximity
effectS. Tsuchiya and S. Tanda
EPL, 94 (2011) 47008

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May 2011
EPL, 94 (2011) 47008
doi: 10.1209/0295-5075/94/47008

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Superconducting networks with the proximity effect
S. Tsuchiya1(a) and S. Tanda1,2
1

Department of Applied Physics, Hokkaido University - Sapporo 060-8628, Japan
Center of Education and Research for Topological Science and Technology, Hokkaido University
Sapporo 060-8628, Japan2

received 30 August 2010; accepted in final form 14 April 2011
published online 18 May 2011
PACS
PACS

74.45.+c – Proximity eﬀects; Andreev reflection; SN and SNS junctions
74.78.-w – Superconducting films and low-dimensional structures

Abstract – We report on the first observation of a novel type of superconducting proximity
network using a superconductor-normal metal bilayer. Little-Parks oscillation measurements showthat the superconducting current flows through a path enclosed by the edge rather than by thecenter of the Pb/Au wire in the network. Furthermore, several peaks were observed in a power
spectrum analysis. We observed that the sequence of these peaks and that of the monolayer network
were connected by the power function, which is a factor of the line width, SBn = αn−2 SAn . Thissuggests that even in a proximity network vortices are arranged in a way identical to a monolayer
network.
c EPLA, 2011
Copyright Introduction. – A network is a topological conceptthat represents the connectivity between nodes andbranches. This concept has been applied in various
fields to, for example, an electrical circuit, a molecular
structure, and computer networks. Their properties are
beneficially aﬀected by the connectivity of the network.
However, networks provide no information about theirlength, width, and the curves of branches.
Superconducting networks, which consist of multiply
connected thin superconducting wires, are typical examples of such networks. In superconducting networks, thenetwork structure has a beneficial eﬀect on the physicalproperties, because this system is sensitive to the phasecoherence of the order parameter over the network. In
fact, a phase interference phenomenon, which is known as
Little-Parks oscillation, is driven by a magnetic field [1].
Characteristic vortex configurations occur in various
network geometries [2–13]. So we can observe the eﬀect
on the network as dips or cusps of variations in themagnetic field responses.
Monolayer wire networks have generally been used inprevious research. In such systems, the amplitude of theorder parameter |Ψ| has the same value along the width
direction because the line width d is suﬃciently smaller
than the coherence length ξ(T ) and the penetration depthλ(T ) (d ≪ ξ, λ). However, for a bilayer network using a(a) Present adress: National Institute for Materials Science Tsukuba, Ibaraki 305-0003, Japan; E-mail: TSUCHIYA.Satoshi@nims.
go.jp

superconductor-normal metal wire, it is still not clear
whether or not the above condition is appropriate [14].
When a superconductor and a normal metal are joinedwith a good electrical contact, the superconductivity is
weakened and induced in the normal metal. This is known
as the proximity eﬀect [15]. In this case, since |Ψ| can be
modified along the width direction, novel network eﬀects
and size eﬀects are expected to occur.
In this paper, we report on a novel kind of superconducting proximity network realized by using a superconductornormal metal bilayer. Little-Parks oscillation results showthat a superconducting current ﬂows through a pathenclosed by the edge rather than by the center of the wire,
due to the proximity eﬀect. Additionally, even in a bilayer
network, vortices are arranged in a way that is analogous
with a superconducting monolayer network. The contribution of the proximity eﬀect is that it provides a vortex
configuration with topological stability.
Experimental. – For comparison, we fabricated two
types of honeycomb network using standard electron
beam lithography methods. One consisted of a lead (Pb)monolayer wire, and the other of a Pb-gold (Au) bilayer
wire. A 0.01 µm thick Au layer and a 0.1 µm thick Pb layer
were thermally evaporated on a SiO2 substrate and thenthe resist was lifted oﬀ. Figure 1 shows a scanning electron
microscope (SEM) image of the sample. The samples haveabout 2500 cells with a lattice constant of 2 µm and a line
width of 0.2 µm.

47008-p1

S. Tsuchiya and S. Tanda
(b)
0.25

B=0G

10 (c)

Magnetic flux (Gauss)(a)

T = 4.3 K

R/RN

R/RN

1

0.2

0.15

0

Slope = 2.67 Gauss / Number

04

Temperature (K)

5

0

10

Magnetic flux (Gauss)0

Index number of dip position

5

Fig. 3: (a) Temperature dependence of the sample resistancenormalized by RN for the bilayer network. Tc was observedat around 4.3 K. (b) The magnetic flux dependence of thesample resistance normalized by RN for the bilayer network.
The arrows indicate periodic dips. (c) The index number of thedip positions as a function of magnetic flux. The slope of theline is 2.67 G.
Fig. 1: SEM image of a honeycomb network with about 2500
cells, lattice constant of 2 µm and line width of 0.2 µm.
(b)

0.9

10 (c)

Magnetic flux (Gauss)

(a)

R/RN

0.9

0.8

0.8

0.7

0.7

B=0G

0.6

0.6

R/RN

1

T = 7.2 K

07

Temperature (K)0Magnetic flux (Gauss)

010Magnetic flux (Gauss)

80

Slope = 2.13 Gauss / Number
05

Index number of dip position

Fig. 2: (a), (b) The magnetic flux dependence of the sample
resistance normalized by RN for the monolayer network. The
arrows indicate periodic dips. The inset shows the temperaturedependence of the normalized resistance. Tc was observed ataround 7.2 K. (c) The index number of the dip positions as afunction of magnetic flux. The slope of the line is 2.13 G.

The Little-Parks oscillation is a powerful tool for investigating the configuration of vortices in a network. A
Little-Parks oscillation is a periodic variation in Tc with amagnetic field induced by superconducting ﬂuxoid quantization [1]. Specifically, when the temperature is near Tc ,the phase coherence of the order parameter is stretchedover the entire system. Hence the Tc variation is aﬀectedby the vortex configuration. Experimentally a Little-Parks
oscillation of the Tc can be observed as a periodic variation in resistance with a magnetic field at a fixed temperature, near the midpoint of the normal-to-superconducting
transition. We measured the Little-Parks oscillation of
our samples using a 12.5 Hz four-terminal ac resistancebridge with an excitation voltage of 10 µV. In addition, we
performed a spectral analysis using the maximum-entropy
method (MEM) to explore certain periods.
Results and discussion. – First, we investigatedthe monolayer network. The inset in fig. 2(b) shows thetemperature dependence of the sample resistance at zeromagnetic field, normalized by RN . The measured Tc isapproximately 7.2 K. This value is in good agreement with
the Tc of bulk Pb. Figures 2(a) and (b) show the magnetic
ﬂux dependence of the normalized resistance in the −10
to 0 and 0 to 10 G ranges at 7.2 K. We found periodic dips

as indicated by the arrows. Figure 2(c) shows the index
number of the dip positions as a function of the magnetic
ﬂux. The slope shows an oscillation period of 2.13 G. The
area estimated from the periodicity is 9.72 µm2 , and thiscorresponds to a hexagonal unit cell enclosed by the centerof the wire of the network. This value compares well withthe value of 9.85 µm2 obtained from a SEM observation
with 1.3% accuracy. Thus, the period corresponds to one
ﬂux quantum Φ0 = /2e per unit cell.
In contrast, a diﬀerent period of oscillation was obtained
for the bilayer network. Figure 3(a) shows the temperaturedependence of the normalized resistance at zero magnetic
field resulting in a Tc of approximately 4.3 K. This reduction in Tc suggests that the proximity eﬀect is well inducedin the bilayer network [16]. Figure 3(b) shows the magnetic
ﬂux dependence of the normalized resistance at 4.3 K. The
linear background is the result of the temperature ﬂuctuation with a long period and is related to the used temperature regulation system. Since the ﬂuctuation period ismuch larger than the measurement time, the ﬂuctuation
background changes gradually and linearly in these experiments. Therefore, this background is not intrinsic to themagnetoresistance experiments and does not inﬂuence ourconclusions. Periodic dips were found as indicated by thearrows. The slope of the line is 2.67 G as shown in fig. 3(c).
The area estimated from the period is 7.75 µm2 and corresponds to a hexagonal unit cell enclosed by the edge
rather than by the center of the wire in the network. Thisvalue compares well with the value of 7.56 µm2 obtained
from a SEM observation with 2.4% accuracy. Hence thearea of the vortex is diﬀerent from that of the monolayer network, although both networks have the same
honeycomb design. In this case, surface superconductivity can occur, which is nucleated near the sample boundary or around the holes in mesoscopic superconducting
systems [17–19]. Generally, surface superconductivity is
not expected to occur with a network since the wires of the
network are much shorter than the coherence length. In
fact, the results for the monolayer network do not indicate
surface superconductivity because the current ﬂows in the
middle of the wire. However, with the bilayer network, the47008-p2

Superconducting networks with the proximity eﬀect
10

0.02

0.01

A2

A3

0
1

2

3

Power spectrum (a.u.)

Power spectrum (a.u.)

Monolayer network

0

B1

(a)

A1

0.03

(b)
Bilayer network

5

B2

0

0

1/B (1/Gauss)

1

B3

2

3

1/B (1/Gauss)

Fig. 4: (a) Power spectrum of the monolayer network. The
fundamental peak labeled A1 corresponds to a period of 2.11 G.
The second strongest peak labeled A2 and the third labeled
A3 correspond to periods of 0.67 and 0.45 G, respectively.
(b) Power spectrum of the bilayer network. The fundamental
peak labeled B1 corresponds to a period of 2.71 G. The second
strongest peak labeled B2 and the third labeled B3 correspondto periods of 0.68 and 0.36 G, respectively.
Table 1: The period and the corresponding area.

Period (G)Area (µm2 )

A1
2.11
9.81

A2
0.67
30.90

A3
0.45
46.00

B1
2.71
7.64

B2
0.68
30.44

B3
0.36
57.50

proximity eﬀect modifies the superconducting parameterof the Pb-Au bilayer in terms of the coherence length,
penetration depth, and extrapolation length [14]. Therefore, the surface superconductivity provides a possible
explanation for these results.
The spectrum analysis was performed using the MEM
to explore the network eﬀects. Figure 4(a) shows the power
spectrum of the monolayer network, and the complete
results are summarized in tables 1 and 2. The fundamental
peak labeled A1 corresponds to a period of 2.11 G, andis consistent with the previous result of 2.13 G. The
corresponding area SA1 is 9.81 µm2 . In addition, we
observed the second and third strongest peaks, which are
0.67 G and 0.45 G and labeled A2 and A3 , respectively.
The corresponding areas SA2 and SA3 are 30.90 and
46.00 µm2 , respectively. The area ratio is related to thefilling ratio of the vortex Φ/Φ0 , which is the magnetic
ﬂux Φ in units of the ﬂux quantum Φ0 per unit cell.
SA2 /SA1 and SA3 /SA2 are 3.15 and 1.49, respectively.
In particular, since SA2 /SA1 is close to 3, the period of
A2 corresponds in a recent report [20] to the downward
cusp in the magnetic field response at Φ/Φ0 = 1/3. Thus,
these peaks are attributed to the eﬀect of the honeycomb
network.
Several peaks were also observed with the bilayer
network, as shown in fig. 4(b). The fundamental peak B1
corresponds to a period of 2.71 G and is consistent with theprevious result of 2.67 G. The corresponding area SB1 is
7.64 µm2 . The second and third strongest peaks labeled
B2 and B3 correspond to periods of 0.68 and 0.36 G,
respectively. The corresponding areas SB2 and SB3 are
30.44 and 57.50 µm2 . The ratios SB2 /SB1 and SB3 /SB2

are 3.98 and 1.89, respectively. These ratios are obviously
diﬀerent from those of the monolayer network. Therefore
these results indicate the realization of a novel network
including the proximity eﬀect.
We should comment on the extra peaks in the bilayer
results of the MEM analysis. In the MEM analysis, weneed to appropriately determine the correlation range(distance), which is related to spectral resolution, since thepeaks that appeared in the MEM spectrum can depend on
the value of the correlation range. Therefore, the MEM
analysis should be tested for various correlation ranges,
and peaks that are independent of the correlation range
should be detected as intrinsic peaks. As regards our MEM
analysis results, the B1 , B2 , and B3 peaks are kept for
any correlation range and the other peaks are not. So weconclude that the B1 , B2 , and B3 peaks are intrinsic tothe spectrum of the bilayer network.
In what follows we discuss the vortex configuration ofthese networks at each observed peak. With the monolayer
network, vortices are commensurately arranged with theirbase structures at rational Φ/Φ0 . The geometric ratios
SA2 /SA1 and SA3 /SA2 are expected to be 3 and 3/2
because of the honeycomb network. Therefore, the vortex
configurations at Φ/Φ0 = 1 corresponding to peak A1 , atΦ/Φ0 = 1/3 corresponding to peak A2 , and at Φ/Φ0 =2/9 corresponding to peak A3 are constructed as shownin fig. 5(a), (b), and (c), respectively. The unit cells
occupied by vortices are shaded in the figure. The dashedline denotes the path through which the superconducting
current ﬂows. At peak A1 , vortices are placed in all
the unit cells enclosed by the center of the wire. At
peak A2 , one vortex is allocated to every three unit
cells. This pattern is commensurately matched with theunderlying honeycomb lattice and energetically stable
over the whole system. In the same way, a stable vortex
configuration is constructed at peak A3 . In comparison
with a previous report [20], the 2/9 peak of A3 appears
and the 2/5, 1/2 and 3/5 peaks are absent as filling
factors in our experimental data. This disagreement can be
attributed to the sample boundary. The boundary of oursamples is approximately hexagonal. Since the total sizeof our samples is smaller than that reported in previous
work [13,20], the boundary conditions of the sample canhave a considerable eﬀect on the vortex configuration in
the network. As regards filling factors 2/5, 1/2 and 3/5,
the vortex configuration with the hexagonal symmetry is
not constructed. On the other hand, at the 2/9 peak,the vortex configuration is constructed while maintaining
the hexagonal symmetry in the same way as the 1 and
1/3 peaks. In our case, the vortex configuration with the
hexagonal symmetry can be energetically stable due tothe boundary eﬀect. Thus the hexagonal symmetry of thesample edge can induce the 2/9 peak instead of the 2/5,
1/2 and 3/5 peaks. When the sample size becomes large,the 2/5, 1/2 and 3/5 peaks can appear since the boundary
condition cannot inﬂuence the vortex configuration in the
network.

47008-p3

S. Tsuchiya and S. Tanda

Table 2: The area ratio of the experimental value, geometrically expected value, and our findings.

Experimental value
Geometrically expected value
Our findings

Monolayer network

SA2 /SA1
3.15
3–

SA3 /SA2
1.49
3
2

–

(a)

2
3

(b)

(c)

1

: A1

0

: B1

1
3

1
1.28

0

( 31 ) : B 2

2
3

1
1.28

0

( 92 ) : B 3

(d)

1
3

: A2

0

(e)

2
9

: A3

0

(f)

1
0

SB3 /SB2
1.89
3
2

1.48 × 1.28

SA1 /SB1
1.28––

The value of 1.48 is nearly equal to the SA3 /SA2 valueof 1.49. The area ratio of the bilayer network consistsof two parts. One is the area ratio of the monolayer
network, and the other is the factor of the line width.
These relations strongly suggest that the features ofthe monolayer network could appear even in the bilayer
network. That is, the A1 , A2 and A3 peaks are expected
to correspond to the B1 , B2 and B3 peaks, respectively.
Therefore, the vortex configurations at the magnetic ﬂux
corresponding to peaks B1 , B2 and B3 are considered to
be as shown in fig. 5(d), (e) and (f), respectively. Even in
the bilayer network, vortices are arranged in a way thatis analogous with the monolayer network, and the path of
the superconducting current is enclosed by the edge of thewire. However, this relation is not derived from a simple
discussion of the vortex area. We speculate that the surface
superconductivity may be induced by the proximity eﬀect
to avoid total energy loss in the entire system.
In addition, a relational expression between SBn and
SAn is derived from these findings as follows:

Bilayer network

1
1.28

1
3

SB2 /SB1
3.98
3
3.11 × 1.28

SBn = αn−2 SAn .
Fig. 5: (Colour on-line) (a)–(c) Vortex configuration of themonolayer network at a magnetic flux corresponding to peaks
A1 , A2 , and A3 , respectively. The plaquettes occupied by thevortices are shaded in the figure. The dashed lines denote theloops of the vortices. A vortex consists of a loop enclosed by
the center of the wire. (d)–(f) The vortex configuration of thebilayer network at a magnetic flux corresponding to peaks B1 ,B2 , and B3 , respectively. A vortex consists of a loop enclosedby the edge of the wire. The arrows and the accompanying
values indicate the area ratios.

On the other hand, with the bilayer network, thevortex configuration cannot be constructed from a simple
discussion of the vortex area because the area ratio is
diﬀerent from that with the monolayer. In other words,
the peaks in the spectrum cannot be explained by thecalculation of the geometrical area of the vortex. The
geometric area ratios SB2 /SB1 and SB3 /SB2 are expected
to be 3 and 3/2, respectively, which are far from theexperimental values.
Surprisingly, we found a factor that is associated withthe line width in the SB2 /SB1 and SB3 /SB2 ratios (seetable 2). The SA1 /SB1 ratio, namely the ratio of the area
enclosed by the edge of the wire and the center of thewire, is around 1.28. Then SB2 /SB1 is equal to 3.11 × 1.28.
The value of 3.11 is very close to the SA2 /SA1 value of
3.15. Furthermore SB3 /SB2 is calculated as 1.48 × 1.28.

(1)Here SAn and SBn represent the sequences {SA1 , SA2 ,SA3 , · · ·} and {SB1 , SB2 , SB3 , · · ·}, respectively. α denotes
SA1 /SB1 , in other words, the factor of the line width andn is an integer. It is noteworthy that n appears in a powerindex of α. This fact indicates the contribution of theproximity eﬀect. If only a simple eﬀect of the line widthis provided, n cannot appear in the power index. In thiscase, the factor should be a constant α−1 for any n.
The power function of n also suggests the topologicalstability of the vortex configuration. Because n is considered to be an integer, each state on each n is topologicallystable. In fact, according to our vortex configuration
model, vortices are correlated with each other andnetworked on the basis of a triangular lattice that isdual to the honeycomb lattice. The network of arranged
vortices is topologically stable because it is maintained
regardless of the base network line width. Therefore, anovel kind of superconducting network with the proximityeﬀect is realized in a superconductor-normal metal bilayer
network.
Summary. – We demonstrated a new kind of superconducting network with the proximity eﬀect by using PbAu bilayer wires. Little-Parks oscillation results showedthat the vortex consists of a loop enclosed by the edgeof the wire rather than by the center of the wire. Thisis because the surface superconductivity may be induced

47008-p4

Superconducting networks with the proximity eﬀect
by the proximity eﬀect to avoid the total energy loss ofthe entire system. Furthermore, we observed not only thefundamental peak but also the second and third strongest
peaks in the power spectrum analysis. Even in the bilayer
network, stable vortex configurations were found to appear
at each peak in the same way as in the monolayer networkfrom a comparison with a Pb monolayer network. The
contribution of the proximity eﬀect was found to be thepower function αn−2 in the relation between the SBn andSAn sequences. This indicates the topological stability ofthe vortex configuration. In fact, the network of arranged
vortices, which is based on a dual network, is maintained
regardless of the line width of the base network. This
topological stability of the vortex configuration could beapplied to an antidot array system. Furthermore, theeﬀects of the line width could be studied by controlling
the proximity eﬀect.
∗∗∗
We are grateful to K. Inagaki, Y. Asano, and S.
Uji for useful discussions. We also thank M. Tsubota,
S. Takayanagi, K. Yamaya, and N. Matsunaga forexperimental support. This work was supported by aGrant-in-Aid for the 21st Century COE program “Topological Science and Technology” from the Ministry of
Education, Culture, Sport, Science and Technology ofJapan.
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