Theory of superconducting transition of micro-hole lattices

Osamu Sato a,b,*, Masaru Kato a

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

Accepted 13 November 2001

Abstract

The phase transition temperature and configurations of fluxoids in the superconducting honeycomb network are

studied theoretically using the de Gennes–Alexander nodal equation. In the network with the periodic boundary

condition, the transition temperature curve (U–Tc curve) shows a dip-structure at the points U=U0 ¼ 1=3, 2=5, 3=7, 1=2,4=7, 3=5, and 2=3 in the broad structure of Little–Parks oscillation. In the network with edge, the dip-structure in the

U–Tc curve is vanished. Fluxoid-patterns of the network are also discussed.

� 2002 Elsevier Science B.V. All rights reserved.

PACS: 74.20.De; 74.76.)wKeywords: Transition temperature; de Gennes–Alexander theory; Micro-hole lattice; Superconducting film

It has been realized matching problem betweenthe lattice periodicity and the external magneticflux U by the development of technology. Re-cently, the superconductive transition temperature(Tc) and magnetic properties of the triangularmicro-hole lattice on Pb film (TriMHoLP) in themagnetic field have been extensively studied [1,2].In the TriMHoLP, experimental results show thatthe behavior of transition temperature in the mag-netic field (U–Tc curve) has many dips, and alsoit has the spontaneous magnetization under thezero magnetic field. Since the TriMHoLP structuremakes a frustration in vortex patterns, peculiar phys-ical properties are expected. If the width of the

superconducting film between holes is narrow,order of coherence length n, the system might beregarded as a superconductive network. Therefore,we treat TriMHoLP as a honeycomb supercon-ductive network (HSN). In the superconductivenetwork model, correlations between fluxoids inadjacent hexagons are very strong, because super-currents that accompany adjacent fluxons mustpass through a branch that is shared by both ofthem.

Let us consider an HSN with length of a brancha in the external magnetic field H ¼ Hez. Themagnetic flux pass through per unit hexagon isU ¼ ð3

ffiffiffi3

p=2Þa2H . We introduce lattice vectors a1,

a2 and divide nodal points to two sets, A, B.

a1 ¼ffiffiffi3

paey ; a2 ¼

3

2aex �

ffiffiffi3

p

2aey ; ð1Þ

rAðm; nÞ ¼ ma1 þ na2 � 12ex; ð2Þ

Physica C 378–381 (2002) 341–343

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*Corresponding author. Address: Department of Mathe-

matical Sciences, Osaka Prefecture University, 1-1 Gakuen-cho,

Sakai 599-8531, Japan. Tel.: +81-722-54-9368; fax: +81-722-54-

9916.

E-mail address: ctsato@ms.osakafu-u.ac.jp (O. Sato).

0921-4534/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

PII: S0921-4534 (02 )01440-5

rBðm; nÞ ¼ ma1 þ na2 þ 12ex: ð3Þ

Superconductive order parameters at nodes rAðm;nÞ and rBðm; nÞ are expressed by DAðm; nÞ andDBðm; nÞ, respectively. We also use simple descrip-tions i; j; . . . to specify the node instead of pairs ofindex ðm; nÞ; ðm0; n0Þ; . . . and symbols A, B. Accord-ing to the dGA theory [3,4], dGA equation forHSN at node i isXj

Dj expðici;jÞ ¼ Di

Xj

cos h; ð4Þ

whereP

j means the summation over the neigh-boring nodes of i, and h is defined by h ¼ a=nðT Þ.The Landau Ginzburg coherence length is nðT Þ ¼n0=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� T=T0

pnear the transition temperature of

zero-field T0 The phase caused by external mag-netic field is expressed by

ci;j ¼2pU0

Zi!j

Adr;

where U0 ¼ hc=2e is the flux quanta.We consider two types of HSN, i.e. HSN with

periodic boundary condition and with fixed bound-ary condition (or network with edge).

First, in the HSN with periodic boundary con-dition, we can expand the order parameter alongthe y-direction as Dðm; nÞ ¼ ei gmgðnÞ. We get ei-genvalue equations,

gðBÞðnÞ þ cðnÞgðBÞðn� 1Þ ¼ 3 cos hgðAÞðnÞ; ð5Þ

gðAÞðnÞ þ cðnþ 1ÞgðAÞðnþ 1Þ ¼ 3 cos hgðBÞðnÞ:

ð6Þ

Here, we take Landau gauge A ¼ Hzey , and coef-ficients of above equations are given,

cðnÞ ¼ exp½�ð2n� 1Þp iU=2U0 � iq�þ exp½ð2n� 1Þp iU=2U0�:

Generally, many eigenvalues exist for each valueof external magnetic field, known as the Hofstad-ter’s diagram [5]. In order to obtain the criticaltemperature, it is sufficient to know the maximumeigenvalue for each value of external magneticfield.

In the infinite HSN with periodic boundarycondition, we get U–Tc curve from Eqs. (5) and (6)

(Fig. 1, curve A). There are dips at U=U0 ¼ 1=2,1=3, 2=3, 2=5, 3=5, 3=7, and 4=7. The suppressionsof critical temperature are explained by the in-crease of the kinetic energy. And the fluxoids ar-rangements at these values of U have relativelylower kinetic energy.

Distribution of fluxoids can be determined bythe phase of the order-parameter. As shown inFig. 2, fluxoids through the HSN with periodicboundary condition form parallel lines. In thehoneycomb network, a hexagon shares all of sixsides with six adjacent hexagons. A fluxoid througha hexagon strongly interacts with fluxoids throughadjacent hexagons by the supercurrent in sharedsides. Because of this frustration, fluxoids formparallel lines. On the other hand, in the squarelattice, one square shares four sides with nearestfour squares, and shares only points with nextnearest four squares, thus the frustration is can-celled easily.

Next, we consider honeycomb networks withedge. We show U–Tc curve in the case of fourhexagons (two hexagons each in a1 and a2 direc-tions) and 2500 hexagons (50 hexagons in each a1and a2 directions), which are shown in Fig. 1. Inthe case of four hexagons, U–Tc curve consists offive curves, which are expressed five stable fluxoids

Fig. 1. U–Tc curve for A: HSN with periodic boundary condi-

tion, B: 2500 hexagons with edge, C: four hexagons with edge.

342 O. Sato, M. Kato / Physica C 378–381 (2002) 341–343

patterns in each 0, 1, 2, 3, and four fluxons in thenetwork. In the case of 2500 hexagons, the dipstructure vanishes in U–Tc curve. It suggests thatthe effect of the edge is very important in the su-perconductive network; the super-current at theedge of the network controls the total magneticfluxoids pass through the network in order tominimize the free energy. Configurations of flux-oids of 400 hexagons case are shown in Fig. 3.When it reaches certain strength of the externalmagnetic field, fluxoids tend to line up along theedge.

In conclusion, we studied the superconductingtransition temperature and the configuration ofthe fluxoids of HSN using the dGA theory. In theHSN with periodic boundary condition, U–Tccurve has the dip-structure at some simple rationalnumbers of U=U0 and fluxoids tend to form par-allel lines. In HSN with edge, the dip structure

disappears due to the effect of the edge and flux-oids tend to line up along the edge.

References

[1] M. Yoshida, T. Ishida, K. Okuda, Physica C 357–360 (2001)

608.

[2] T. Ishida et al., Physica C 357–360 (2001) 604.

[3] P.G. de Gennes, C.R. Acad. Sci. B 292 (1981) 9.

[4] S. Alexander, Phys. Rev. B 27 (1983) 1541.

[5] D.R. Hofstadter, Phys. Rev. B 14 (1976) 2239.

Fig. 3. Fluxoid-patterns of HSN (400 hexagons) with edge:

U=U0 ¼ 0:05 (a), 0.10 (b), 0.30 (c), 0.46 (d). A circle denotes a

fluxoid.

Fig. 2. Fluxoid-patterns of the HSN with periodic boundary

condition: U=U0 ¼ 1=2 (a), 2=5 (b). A black circle denotes a

fluxoid.

O. Sato, M. Kato / Physica C 378–381 (2002) 341–343 343