Ginzburg–Landau approach for vortex states of superconducting

micro-plates with anti-dots

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

Abstract

We report theoretical results of vortex configurations, and critical fields of regular square micro-plates with 2!2 anti-dots by using the

Ginzburg–Landau theory. To treat inhomogeneousness of the system, we applied the finite element method (FEM). We have calculated the Gibbs

free energies for two types of plates with different anti-dots sizes. We found that 0F0 and 4F0 vortex states are extremely stable states.

q 2006 Elsevier Ltd. All rights reserved.

PACS: 74.20De; 74.76.Kw

Interests of superconducting properties have extended to

micro- or nano-scale dimension. Since superconducting states

have large coherence length compared with that in normal

states, we can easily see quantum coherent effects of

superconducting system. Flux-quantization is one of the

significant superconducting phenomena. In the homogeneous

bulk superconductor of the second kind, magnetic flux that pass

through the superconductor break the superconducting order

around vortex cores and form two-dimensional triangular

lattice. Impurities and defects in a superconductor disturb the

lattice structure. Nowadays, fabrication techniques provide us

various micro-size superconductors with artificially fabricated

shape; in these superconductors, spatially controlled vortex

configurations are realized. In periodic superconducting net-

works and superconducting micro-hole lattice, there are

magnetic fields that we call ‘matching fields’ where the

vortices form stable structure. At the matching fields,

enhancements of the superconducting transition temperature

are observed. In micro-size superconductors, physical proper-

ties are very sensitive to the external field [1].

We have studied vortex states of micro-superconducting

plates with 2!2 anti-dots with use of the finite element method

(FEM) for nonlinear Ginzburg–Landau equations.

0022-3697/$ - see front matter q 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jpcs.2005.10.025

* Corresponding author. Tel.: C81 72 821 6401; fax: C81 72 821 0134.

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

The Gibbs free energy of the system U is

Gðj;AÞZ

ðU

fn Ca jj j2 C

b

2jj j

4

� �dU

C

ðU

1

4mKiZVK

2eA

c

� �j

��������2

Chj j

2

8pK

h,H

4p

� �dU;

(1)where j is the order parameter, A is the vector potential, and fnis the Helmholtz free energy of a normal state. If the

temperature T is near the zero field transition temperature

TC(0), a can be written with the positive coefficient a 0 as aZa 0(TKTC(0)), and b is taken to be positive constant. The

electron mass and charge are denoted by m and e, respectively.

An external magnetic field H is applied perpendicular to the

superconducting plate. The local magnetic field is hZV!A.

From the condition of dGZ0, we get two equationsðU

ðiV ~jK ~A ~jÞðKiVd ~j�K ~Ad ~j

�ÞC ðKiV ~j

�K ~A ~j

�ÞðiVd ~j

�

K ~Ad ~jÞC1

xðTÞ2ð ~j�� ��2K1Þð ~jd ~j

�C ~j

�d ~jÞ

�dUZ 0;

(2)ðU

k2xðTÞ2fðV$ ~AÞðV$d ~AÞC ðV! ~AÞðV!d ~AÞgC jj j2 ~A$d ~A

�

Ki

2f ~j

�V ~jK ~jV ~j

�gd ~A

�dUZ k

2xðTÞ2

ðU

2p

F0

H$ðV! ~AÞdU:

(3)

Journal of Physics and Chemistry of Solids 67 (2006) 476–478

www.elsevier.com/locate/jpcs

Fig. 1. A schematic illustration of a superconducting micro-plate with 2!2

anti-dots. Grayed area shows superconducting area.

Fig. 3. The Gibbs free energy of the plate (B) in the external magnetic field.

O. Sato, M. Kato / Journal of Physics and Chemistry of Solids 67 (2006) 476–478 477

Here, j and A are rewritten by ~jZj=ffiffiffiffiffiffiffiffiffiffiaj j=b

pand

~AZ2pA=F0, respectively. The flux quantum is denoted by

F0. We divided the region U into small right isosceles

triangular elements and studied numerically by the two-

dimensional FEM. In the method, we imposed a boundary

condition of A$nZ0, where n is the normal vector of the

super-vacuum boundary. In the calculation, we used the

Galerkin’s method of the first-order in solving a coupled

Eqs. (2) and (3).

We studied two types of superconducting square plate with

2!2 anti-dots (A) and (B). A schematic illustration of the plate

is shown in Fig. 1. In both of them, a length of sides of plates is

a, the GL parameter is set to kZ0.5, and the coherent length is

xZ0.1a. The plate (A) has a hole width of bZ6a/24Z0.25a, a

width of superconducting area is dZ4a/24y0.17a. Similarly,

the geometric parameters of plate (B) are bZ7a/23y0.30a and

dZ3a/23y0.13a. In the plate (A), numbers of elements and

nodal points of division are 1152 and 625, in the plate (B), 1058

and 529, respectively.

The Gibbs free energies of the vortex states at the

temperature TZ0.9TC(0) are shown in Figs. 2 and 3. The

first critical fields HC1 of plate (A) and plate (B) are 2.4F0/a2

and 2.0F0/a2, respectively. We found that large hole-size shifts

Fig. 2. The Gibbs free energy of the plate (A) in the external magnetic field.

HC1 smaller. In both cases, odd numbered fluxon states are very

unstable, on the other hand, 0 and 4 fluxon states are stable for

the wide range of magnetic field, because the plate has 4-fold

rotational symmetry. In Fig. 4, we show relative amplitude of

order parameter jjj/jj0j of plate (A), 0F0, F0, 3F0, and 4F0

states at TZ0.9TC(0). Here, j0 is the order parameter at zero-

field. In the 0F0 state, jjj has a slightly small value at the outer

boundary of the plate, where the Meissner current flows to

expel the field. In F0 and 3F0 states, because jjj has no

symmetry, the states have a disadvantage for the free energy. In

the 4F0 state, though jjj has a very small spatial variation as

seen in the 0F0 state, jjj is uniformly suppressed by the

external field. Berdiyorov et al. studied the superconducting

plate with several anti-dots patterns [2]. They solved non-linear

Fig. 4. Spatial variations of order parameters of the plate (A) in the external

magnetic field H. (a) Ha2/F0Z1.8, (b) Ha2/F0Z3.0, (c) Ha2/F0Z4.2 and (d)

Ha2/F0Z5.0. A black arrow expresses a fluxon.

O. Sato, M. Kato / Journal of Physics and Chemistry of Solids 67 (2006) 476–478478

GL equation numerically with use of the finite difference

method (FDM). Our results of the behavior of the free energy

have qualitative agreement with that of their results at the

phase boundary.

In our previous paper, we imposed additional condition in the

FEM calculation that j is continuous at boundaries of

superconducting region and hole region [3]. The condition

corresponds to that of super-normal metal boundary. Only four-

fold symmetrical fluxon patterns were found in this condition.

For example, near and above the first critical field, one fluxon

pass through the central point of the plate (not pass through hole

region). The first critical field is about HC1Z5F0/a2 which is

almost same value as superconducting simple plate.

In summary, we calculated vortex states of superconducting

micro-plates with 2!2 anti-dots. Stabilities of fluxon

arrangements of the system depend on its symmetry; odd

numbered fluxons states are unstable.

Acknowledgements

We thank T. Ishida and members of his research group for

fruitful discussions.

References

[1] T. Puig, E. Rossel, L. Van Bael, V.V. Moshchalkov, Y. Bruynseraede,

Phys. Rev. B 58 (1998) 5744.

[2] G.R. Berdiyorov, B.J. Baelus, M.V. Milosevic, M. Peeters, Phys. Rev. B 68

(2003) 174521.

[3] Osamu Sato, Masaru Kato, Physica C 412–414 (2004) 262.