Physica C 426–431 (2005) 74–78

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Fluxon distribution in three-dimensionalsuperconducting networks

Osamu Sato a,*, Masaru Kato b,c

a Department of Liberal Arts, Osaka Prefectural College of Technology, 26-12 Saiwai-cho, Neyagawa, Osaka 572-8572, Japanb Department of Mathematical Sciences, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, Osaka 599-8531, Japan

c CREST-JST, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan

Received 23 November 2004; accepted 15 February 2005

Available online 23 June 2005

Abstract

Fluxon distributions and superconducting transition temperatures in three-dimensional carbon nanotube like super-

conducting networks in a uniform magnetic field were studied using the de Gennes–Alexander network equation. The

field dependence of the transition temperature shows non-periodic behavior. We found enhancement of the transition

temperature at some external fields where fluxons form stable structures.

� 2005 Elsevier B.V. All rights reserved.

PACS: 74.81.Fa; 74.20.De; 74.25.Qt

Keywords: Superconducting network; Fluxon; Superconducting transition temperature

1. Introduction

Fine fabrication has enabled us to control vor-

tex states. Studies of vortex states on superconduc-

tors have spread on artificial meso-structured

superconductors [1,2]. In particular, we expect that

multi-connected superconductors in a magneticfield show various vortex configurations. We have

0921-4534/$ - see front matter � 2005 Elsevier B.V. All rights reserv

doi:10.1016/j.physc.2005.02.022

* Corresponding author. Tel.: +81 72 821 6401; fax: +81 72

821 0134.

E-mail address: gpsato@las.osaka-pct.ac.jp (O. Sato).

reported theoretical results of fluxon distributions

on superconducting networks of two-dimensional

square and honeycomb lattice in magnetic field;

lattice symmetry strongly affects fluxon distribu-

tion [3]. We also have reported fluxon distribution

of C60 like three-dimensional network [4]. Apply-

ing the magnetic field perpendicular to a hexagonor pentagon, symmetries of fluxon distribution

are same as symmetry of the system around an axis

parallel to the magnetic field.

In this paper, we discuss vortex states of

carbon nanotube like structured three-dimensional

ed.

O. Sato, M. Kato / Physica C 426–431 (2005) 74–78 75

superconducting network. The carbon nanotube

like superconducting network is regarded as rolled

honeycomb network. In the two-dimensional hon-

eycomb network, straight-line shape fluxon distri-

butions are obtained by the de Gennes–Alexandertheory [5].

2. Formalism

The development of zigzag-fiber network is

shown in Fig. 1. The dashed line expresses a tube

axis. For preparation of following discussion, wedenote nodal points by three indexes m = 0,1,

2, . . . ,M + 1, n = 1,2,3, . . . ,N + 1, and h = A,B.

Note that there exists only h = B for m = 0, and

h = A for m = M + 1. We regard nodal points

that are denoted by n = N + 1 are identical to

points denoted by n = 1. Each bond has a length

of a, and positions of nodal points rhm;n are

expressed as

(1,1)

(1,2)

(1,N)

(2,N)

A A

A

A

B

(2,1)

(2,2)

(3,1)

(3,2)

development of zigzag

A B

B

A B

BA

A B

B

B

(m,n

A

Fig. 1. A development of carbon nanotube like supercond

‘

rA2l�1;n ¼ 3aðl� 1Þex þffiffiffi3

pa

4 sin p2N

cos2pnN

ey

þffiffiffi3

pa

4 sin p2N

sin2pnN

ez;

rB2l�1;n ¼ 3a l� 2

3

� �ex þ

ffiffiffi3

pa

4 sin p2N

cos2pnN

ey

þffiffiffi3

pa

4 sin p2N

sin2pnN

ez;

rA2l;n ¼ 3a l� 1

2

� �ex þ

ffiffiffi3

pa

4 sin p2N

cos2p nþ 1

2

� �N

ey

þffiffiffi3

pa

4 sin p2N

sin2p nþ 1

2

� �N

ez;

rB2l;n ¼ 3a l� 1

6

� �ex þ

ffiffiffi3

pa

4 sin p2N

cos2p nþ 1

2

� �N

ey

þffiffiffi3

pa

4 sin p2N

sin2p nþ 1

2

� �N

ez.

ð1Þ

x(tube axis)

fiber

)

B

uctive network. We take tube axis along the x-axis.

76 O. Sato, M. Kato / Physica C 426–431 (2005) 74–78

Here, l denotes integer. The tube axis is parallel to

the x-axis. We note that each hexagon is crimped

alongside the longest diagonal. The de Gennes–

Alexander theory leads to coupled equations

1

qi

Xqij¼1

eici;jWj ¼ cosa

nðT Þ �Wi. ð2Þ

Here, qi stands for the number of bonds from nodal

point i to j. The largest eigenvalue (cosða=nðT Þ)

0

0.05

0.1

0.15

0 0.2 0.4 0.6 0.8 1

0

0.05

0.1

0.15

0 0.2 0.4

a b

c

Fig. 2. Magnetic-field dependence of transition temperature:

determines the superconducting transition tempera-

ture, and order parameters at the nodal points Wi

are determined as its eigenvector. The coherence

length n(T) at temperature T behaves nðT Þ ¼n0=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� T =T C

p, where n0 is the coherence length

at zero temperature. The phase factor ci,j is definedby

ci;j ¼2pU0

Z rj

ri

AðrÞ � dr; ð3Þ

0.6 0.8 1

0

0.05

0.1

0.15

0 0.2 0.4 0.6 0.8 1

(a) h = 0, / = 0, (b) h ¼ p2, / = 0, and (c) h ¼ p

4, / = 0.

O. Sato, M. Kato / Physica C 426–431 (2005) 74–78 77

where A(r) is the vector potential, and U0 is the

flux quantum. The external magnetic field is ex-

pressed as H = curlA. According to the Landau

gauge, we put

A ¼ Hz sin/ sin hex þ Hx cos hey þ Hy cos/ sin hez.

ð4Þ

Fig. 3. Fluxon distribution of carbon nanotube like superconducting

top and bottom areas of the tube along the z-axis: (a) H = 0.20a2/U0,

3. Results

Applying a magnetic field along to the z-axis

(perpendicular to the tube axis), we obtain the

superconducting transition temperature of zigzagnetwork (M = 16,N = 6) in the magnetic field as

shown in Fig. 2(a). The largest projective area to

network in a magnetic field of z-axis direction. Shaded areas are

(b) H = 0.48a2/U0, and (c) H = 0.86a2/U0.

78 O. Sato, M. Kato / Physica C 426–431 (2005) 74–78

xy-plane of a hexagon is S1 ¼ 3ffiffiffi3

pa2 cosðp=NÞ=2.

In the N = 6 case, the area is S1 = 2.25a2. The

second and third areas are S2 = 1.77a2 and

S3 = 0.650a2, respectively. When the filling field

of the hexagon S1 is unity, the magnetic field isH = 0.44U0/a

2. In Fig. 1(a), the first principal

matching peak is around H = 0.48U0/a2. The sec-

ond principal matching field is at H = 0.86U0/a2.

If we apply the magnetic filed parallel to the tube

axis, a simple Little–Parks oscillation curve is ob-

tained as shown in Fig. 2(b). A variation range

of the TC curve of (b) is smaller compared with

(a). Since there are multiply stacked superconduc-ting loops around the tube axis, the Meissner cur-

rent per each superconducting wire can be smaller.

Fig. 2(c) shows the transition temperature in the

field of angle h = p/4, u = 0.

We determine a number of fluxon in an arbi-

trary loop from the phase of order parameter.

The phase difference between nodal points ri andrj can be calculated as

ui;j ¼ argWje

ici;j

Wi

� �� ci;j; ð5Þ

where the symbol arg(z) denotes the principal va-

lue of the argument of complex number z. For

an arbitrary loop C in the network, the phasewinding number m(C) that denotes the number

of fluxon passing through the loop C is

mðCÞ ¼ � 1

2p

XChi;ji

ui;j. ð6Þ

The fluxon distribution of zigzag tube network

at H = 0.20U0/a2, 0.48U0/a

2 and 0.86U0/a2 perpen-

dicular to the tube axis are shown in Fig. 3(a)–(c),

respectively. As increasing the field, fluxons enters

from top and bottom points in equilibrium states.

At the first principal matching field H = 0.48U0/a2,

each hexagon except those of faced on zx-plane

contains one fluxon (Fig. 3(b)). At the second prin-

cipal matching field H = 0.86U0/a2, each hexagon

of area S1 and S2 contains two fluxons, and eachS3 contains one fluxon (Fig. 3(c)). On the other

hand, at the field H = 0.62U0/a2 when the transi-

tion temperature is strongly depressed, all hexa-

gons of area S1 contain two fluxons, S2 and S3

contain one fluxon. Fluxon distribution at the field

has a similar symmetry as the distribution at the

second matching field. The ratio of the projection

area of S2/S3 = 2.73 suggests that the Meissner

current of the network at the H = 0.62U0/a2 is lar-

ger than that of the state at the second principalmatching field because of uniformity of the mag-

netic field in space. To keep the uniformity, a num-

ber of the fluxon of a loop should be reciprocally

proportional to the projective area. Up to the first

principal matching field, we can observe low sym-

metrical distributions. This is because, at low field,

we cannot divide fluxons among hexagonal loops

so as to each hexagon has fluxons reciprocal toits projective area.

4. Summary

We have studied transition temperatures and

fluxon distributions of carbon nanotube like super-

conducting networks with finite length in magneticfield. In a magnetic field of perpendicular to the

tube axis, we found principal matching peaks.

The first principal matching peak field is near the

field where the filling field of the largest projective

area of hexagons is unity. Fluxon distributions at

the matching fields and at several fields are ob-

tained. At low field (up to the first matching field),

we found fluxon distribution of low symmetry.

Acknowledgements

We thank T. Ishida and members of his research

group for fruitful discussions. Also we thank Y.

Kayanuma, and other members of quantum phys-

ics research group at Osaka Prefecture University.

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