Vortex structures in small superconducting disks

26Dcccmber1994 PHYSICS LETTERS A

Phys~sLeRe~A196 (1994) 267-271

ELSEVIER

Vortex structures in small superconducting disks A . I . B u z d i n a,b, j . p . B r i s o n c • SPSMS/DRFMC, Centre d'Etudes Nucldaires de Grenoble, 38054 Grenoble Cedex 9, France b Institutefor High Pressure Physics, Troitsk 142092, Russian Federation ¢ Centre de Recherches sur les Trks Basses Tempdratures, CNRS, B.P. 166, 38042 Grenoble Cedex 9, France

Received 7 October 1994; acceptedfor publication 21 October 1994 Communicatedby J. Flouquet

Abstract

Vortex structures in a small superconducting disk in the low field regime are considered, using an image method. The magnetization and the optimum vortex configuration are calculated as well as the entrance field.

1. Introduction

Modern microfabrication techniques enable one to create very small superconducting structures with unique properties [ 1 ]. For example, the behaviour of such structures in an external magnetic field is strongly influenced by the boundary conditions and it has been shown that the upper critical field H¢2 of a small superconducting disk has an oscillatory dependence on its radius [2]. A similar behaviour of the critical field for the appearance of superconductivity near a small hole has been predicted [ 3 ] and recently observed experimentally [ 4 ]. We consider here the properties of such small disks in the low field regime, in contrast to previous experimental and theoretical work in the high field regime [ 2 ]. More precisely, we are interested in the behaviour of the Abrikosov vortices near the lower critical field H¢l of the disk. We restrict ourselves to the case of a disk of radius R < 2eff, where ;tefr (see, for example, Ref. [ 5 ] ) is the effective London penetration depth for a thin film: ;tCfr=A2/d (2 is the usual London penetration depth and d is the film thickness). In such a situation the interaction between vortices

is logarithmic and this enables us to use an analogy with the electrostatic interaction between charged threads. The image method [ 6 ] allows us to solve exactly the problem of intervortex interactions for circular boundary conditions for which a normal component of the current at the disk's edge is absent. The application of this method to the interaction of a vortex with a cylindrical defect has already been studied in Ref. [7]. Note that the first critical field H¢1 for a superconducting disk of arbitrary dimension has been calculated [ 8 ] but the method used was too complicated to permit an analysis of the vortex structure near H¢1. In the case of a superconducting cylinder the vortex configurations for up to four vortices have been calculated numerically for several ratios R / 2 [9 ]. In Section 2 we present the formulation of the problem and the image method. In Section 3 the energies of different vortex configurations are compared and the critical fields of interconfignration transitions are calculated, as well as the magnetization curves. In the conclusion, we briefly discuss the generalization of our approach to higher densities of vortices and experimental techniques for the observation of the vortex configurations.

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A.L Buzdin, £P. Brison / PhysicsLettersA 196 (1994) 267-271

268

2. Free energy of arbitrary vortex configuration

leads to the following simple expression for the energy of the disk,

The main influence of the restriction R <2,ff is to suppress the screening effect of the superconducting currentsjs: the magnetic field will be almost identical inside and outside the disk. In such a case also, because of the absence of demagnetizing fields, the boundary conditions are the same for a disk and a cylinder (with the condition R <2 for the cylinder), namely no normal components ofj~ at the surface. The following is then valid for both geometries. We start with the Gibbs free energy in a constant external magnetic field H (perpendicular to the disk) in the London approximation (see, for example, Ref. [5]),

,J

G= ~

[ ( B - H ) 2 + 2 2 ( c u r l B ) 2 1 d3r,

(1)

where B=curlA is the local magnetic field. In the presence of vortices the London equation for the superconducting current ]~ may be written as

C

L = 4--~ ( ~ - A ) ,

(2)

where

• = ~+,(p-p,),

i

~±,(p)=(0,

+0o/2rtp). (3)

G= ~ d ~ (~l~--4ext)2 d2p.

(6)

Note that this is equivalent to neglecting the screening field magnetic energy in ( 1 ) and keeping only the kinetic energy contribution of is to the Gibbs free energy. On the disk boundary, the current Js must have a tangential component only. To solve this boundary condition problem we use an image method, Let us consider first one vortex at some point P0 (Fig- 1 ). As the vortex currentjsoc ~ + t ( p - # o ) decreases as 1/ ]P-Po] with the distance we may use an analogy with the electrostatics: the field of a charged thread has the same spatial dependence. The image method [6] gives the exact solution of the problem of the interaction between a charged thread at Po and a metallic cylinder - to obtain the field distribution inside the cylinder it is necessary to add the field of an imaginary thread with opposite charge placed at a distance p~ =po(R/po) 2. In our case (for more details see Ref. [ 7 ] ) it means that we must add an antivortex at point pt, leading to ~ = ~ + 1(J~-Po) -{-~ - i ( P - P l ) •

(7)

The last expression is in polar coordinates. The resulting London vector ~ ( p ) is the sum of vectors ~ + corresponding to vortices ( + ) and/or antivortices ( - ) located at points Pi in the plane of disk, and Oo=hc/2e is the flux quantum. Taking into account the Maxwell equation for the field ( B - H ) created by the supercurrent, 47t. c u r l ( B - H ) = -c-A,

(4)

we may present the energy ( 1 ) in the form

G=-~-~

(~-A+a)'(~-A)dZp,

(5)

where d is the disk thickness and a=A-.4ext is the vector-potential of the field ( B - H ) created by the current j, and the integration is performed over the disk surface. For a small disk (R<<2eff) the field screening is weak and it is possible to neglect a ( a _~ (R/J.eff) 2 A ), i.e. to put A " A e x t = ½H×P which

Fig. 1. The vortex is situated inside the disk of radius R, at a distance go from the center. The imagevortex used to satisfythe circular boundary conditions on the superconducting current is located at a distance Pl outside the disk.

A.L Buzdin,J.P.Brison/ PhysicsLettersA 196(1994)267-271 It is easy to verify that this choice of • really gives a vanishing normal component of the current at the disk edge. For arbitrary vortex configurations in the disk, the London vector in the representation (3) is obtained by adding for each vortex ~ + ~ inside the disk a corresponding antivortex ~ _ 1 outside. To calculate the Gibbs free energy (6) it is convenient to introduce the "auxiliary field" f directed along the z axis (perpendicular to the disk),

f= Z z ~ l n ( A / l p - p , [ ) . Note that

~=curlf

(8)

and

269

where h=HltR2/(bo is the dimensionless external magnetic field. It is obvious that when the field increases the first vortex must appear at the center of the disk and with the help of ( 11 ) we easily obtain the energy g~ of such a configuration,

gl =ln(R/~)-h+h2/4 ,

(13)

Note that d(Oo/47dO21n(R/~) is just the energy of the vortex at the center of the disk. From the condition g~ =go we immediately obtain the field for the first vortex to appear as

cud q ~ = c u r l ( c u d f ) =

Y~ z~,J(p-p~). With the help of (8) and using the

h~=ln(R/~), i.e. Hl=n-~Eln(R/~),

(14)

equality q~-cuff f = f . cuff ~ - div ( • × f ) ,

(9 )

we may write ¢262d2p= ~~pj
~j)f(R),

(10)

w h e r e f ( R ) = Ypj
G=8-~L~
--

-

2 pj
-

.

(11)

The summation here is performed only over the real vortices inside the disk whereas all vortices contribute t o f Note that the fieldfcreated by a vortex at its center diverges which merely means that as usual [ 5 ], we must use as a cutoff the superconducting coherence length ~. Expression ( 11 ) is the central result of this section - it provides us with a rather simple algebraic formula for the energy of any vortex structure.

which coincides with the result of Ref. [ 8 ] for the limit R<
gN--go =Nln(g/~) - N ( N - 1 ) ln(x) N N-I ln(1 + x 4 - 2 x 2 cos(2rcn/N)_~ + 2 ~=1 4 sin2(nn/N) ] -Nh ( 1 - x 2) + N l n ( 1 - x 2) ,

where x=po/R is the dimensionless distance of the vortices from the center. Minimization of ( 15 ) for x gives the relation between the magnetic field and distance x,

hN= N - 1 2X 2 3. Different

vortex

(15')

N-I

x2_cos(2rtn/N)

n=lE1 +X4--2X 2 cos(27rn/N)

configurations l

In the absence of vortices the dimensionless energy go of the screening currents is 167t2~.2

7~2H2R4

go= doE Go=

402

hE

- 4 '

(12)

+ _x------1 ~ .

•

(15")

The corresponding formula for an additional vortex at the center may be written as

gNa-go=gN+ln(R/~)-2Nln(x) - h ,

(16')

A.L Buzdin, J.P. Brison / Physics Letters A 196 (1994) 267-271

270

065oB .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

o.s5 0.5

14

h -15

'"

i . . . .

I . . . .

i

. . . .

I . . . .

I . . . .

I

. . . .

Fig. 3. Minimum distance I between vortices as a function of the reduced field. For the configuration (5, 1 ), this minimum distance is the distance between the central vortex and the peripheral ones. For (7, 1 it is the distance between two adjacent peripheral vortices.

I ' ' '

-20

~. -2s

-0.5 ~ -1

...'Sill,.,: ........

11 11.5 12 12.5 13 13.5 14 14.5 15

~ ~

- 1.5

h Fig. 2. Free energy as a function of the reduced field h=HTt,R2/ 0o for different vortex configurations shown on the right handside oftbe figure. (a) From 1 to 4 vortices. (b) From 5 to 8 vortices. In (a) gl is least steep, for g2, g3, g4 the steepness increases. In (b) g5 is least steep, for gs,], g6,b gT,~ the steepness increases. hN,1 =hN-[- I / X 2 •

hi= In(R/~)= 4

-2

0

~

~

'

~

, , : , ~ , : . , . . . . .i .i .i 5 10 I1

i

i

15

Fig. 4. Reduced magnetization (m) of the disk as a function of the reduced field h. The magnetization M i s equal to Mom where Mo=F(~odR2; 167t22). Note that each vortex entry produces a positive jump of the magnetization.

(16")

jumps of the magnetization of the disk, The results o f the calculation o f the energy for vortex configurations for up to 7 vortices are presented in Figs. 2a and 2b. We see that up to N = 5 the vortices form only polygons, while for N = 6 it is more favorable to have a pentagon and one vortex at the center though the energy difference between this configuration and the hexagon is very small ( o f the order of 0.1% of go). The numerical calculations [9] for a superconducting cylinder for N = 4 show that a square has a lower energy than a triangle with an additional vortex at the center. This is in agreement with our result. Increasing the external field, the intervortex distance decreases (Fig. 3) but when a new vortex enters the disk, this distance may decrease as well. The entrance o f the vortices is associated with

OG

M - - - T9 =M0m, OodR 2 M o = 16~z22, m = -

Og O-h"

(17)

The field dependence of the magnetization is presented in Fig. 4. It is clearly seen that the entrance of each additional vortex is marked by a positive j u m p o f the magnetization. In the high field regime (not treated here), the magnetization will approach that of the Abrikosov vortex lattice. Let us address briefly the question of a surface barrier for the disk. The energy o f one vortex at an arbitrary point X = p o / R inside the disk is given by expression (15') f o r N = 1 (Fig. 5),

A.L Buzdin, J.P. Brison / PhysicsLettersA 196 (1994) 267-271 3

'

,

i

,

r

'

i

'

i

'

i

'

i

'

i

,

i

,

i

'

.

0

,I=

~

-3

• -- h=4

h--6

- -

-4

I

0

0,2

0,4

0,6

,

I

h=8 ,

,

0,8

X

Fig. 5. Free energy of one vortex as a function of its distance x=po/ R from the center of the disk for different magnetic fields, above and below h~ = I n ( R / l ) . The height of the surface bander decreases when increasing the field.

gl-go=ln(R/~)-h(1-x2)+ln(l-x2)

•

(18)

Naturally its m i n i m u m corresponds to x = 0 (a vortex at the center o f the disk) and near the edge, there is a surface barrier similar to the Bean-Levingston barrier [10] for a flat surface. Note that this minim u m exists until the field h = 1. At even lower fields, it becomes a maximum. This means that due to the geometrical barrier, the vortex can be trapped at the center o f the disk in a field below the first critical one which is h~ = l n ( R / ~ ) . As in the case o f a bulk superconductor, the entrance field he is determined by the vortex-image interaction and is little affected by the presence of other vortices. In our case, the surface barrier disappears only in a field h e ~ R / ~ , i.e. He ,,, ~ o / Tt(.~R

(19)

and He is much larger than H~, except in a narrow region close to To. But near To, in the case of a thin superconducting disk and depending on the parameters of the system, the appearance o f a KosterlitzThouless transition is possible. This circumstance may then strongly influence the physical picture described here.

4. C o n c l u s i o n

Our approach shows a simple way to calculate the energies of any vortex configuration in a small super-

271

conducting disk and to trace the transition from low field structures to the Abrikosov vortex lattice which must appear in high fields. The representation ( 11 ) for the energy of an arbitrary vortex configuration in our opinion provides a good basis for computer calculations of different complicated vortex structures in the intermediate field range. Tunneling microscopy may be the most convenient technique to observe the vortex configurations in mesoscopic systems. Magnetization measurements will not be sensitive to the different vortex configurations (absence o f screening and demagnetizing effects). An additional difficulty for "real samples" is the influence o f pinning effects which may strongly influence the real vortex distribution, and of surface barrier effects for the determination of H~. Finally, for the case o f vortices in neutral systems like vortices in liquid 4He (see also Ref. [8] ) our image method is perfectly applicable without any restriction on the ratio R/2eff (no screening effects whatever the distance) and gives a powerful tool to treat vortex interactions in a circular geometry.

Acknowledgement

We thank A. Huxley for reading the manuscript and useful remarks.

References

[ 1] B. Kramer, ed., Quantum coherence in mesoscopicsystems (Plenum, New York, 1991). [2] O. Buisson, P. Gandit, R. Rammal, Y.Y. Wang and B.

Pannetier, Phys. Lett. A 150 (1990) 36. [3] A. Buzdin, Phys. Rev. B 47 (1993) 11416. [ 4 ] A. Bezryadin and B. Pannetier, submitted to J. Low Temp. Phys. [5] A.A. Ahrikosov, Fundamentals of the theory of metals (North-Holland, Amsterdam, 1988). [6]L.D. Landau and E.M. Lifshitz, Electrodynamics of continuous media (Pergamon, Oxford, 1983). [ 7 ] A. Buzdin and D. Feinberg, to bc published in Physica C, Proceedings of the M2SHTSC 1994 Conference, Grenoble. [8] A.L. Fetter, Phys. Rcv. B 22 (1980) 1200. [ 9 ] G. Bobel, Nuovo Cimento 38 (1966) 1741. [ 10] C.P. Bean and J.B. Levingston, Phys. Rcv. Lett. 12 (1964) 14.