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ELSEVIER Physica C 269 (1996) 330-338

PHYSICA

The interaction between a single vortex and a columnar defect

in the superconducting multilayers

J.L. Chen, T.Y. Tseng *

Department of Electronics Engineering and Institute of Electronics, National Chiao-Tung University, Hsinchu, Taiwan

Received 15 May 1996

Abstract

The interaction energy per unit length Uint(r) between a single vortex and a columnar defect with radius a in the superconducting multilayers is calculated with the help of London theory. We assume that these superconducting layers are coupled by the magnetic field and the interlayer Josephson coupling can be neglected. We obtain a general expression of Uint(r) for any t, where t is half of the spacing between the layers. For t ~ ~, our theory can be reduced to the result of a thin film. But when t becomes smaller, Uint(r) would tend to exhibit the behavior of 3D superconducting bulk.

PACS: 74.60Ge; 74.80.Dm

Keywords: London penetration depth; Multilayers; Thin films; Type-II superconductors

1. I n t r o d u c t i o n

For technical applications of superconductors, the enhancement o f the critical current density (Jc) in high magnetic fields is one of the most important tasks. To obtain a large Jc it is required to introduce strong pinning centers into a superconductor and to enhance the magnetic-flux pinning strength. One of the promising methods of introducing defects inside a superconductor is based on heavy-ion irradiation techniques [1-5]. These nonsuperconducting defects which interact with the vortices would induce screening currents and lead to the so-called electromagnetic pinning [6].

It is necessary to study the structure and interactions o f the vortices with columnar defects in order to understand the effect of columnar defects on enhancing Jc- At first, for understanding the structure o f the vortices, some theory has been developed. Clem et al. have calculated the structure of the vortices in layered superconductors [ 7 - 9 ] and Bulaevskii et al. [10] have calculated the L a w r e n c e - D o n i a c h model [11] in the linear approximation. Recently Benkraouda and Clem [12] have shown the instability of a tilted vortex in the layered superconductors. Next, the interaction between a single vortex with a columnar defect in the 3D type-II superconducting bulk has been calculated by Mkrtchyan and Shmidt [13], and their theory has been generalized to a periodical structure o f columnar defects [14]. On the other hand, for a 2D superconducting thin film, the

* Corresponding author. Fax: + 886 35 724 361.

0921-4534/96//$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved

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J.L. Chen, T.Y. Tseng / Physica C 269 (1996)330-338 331 interaction of a 2D pancake vortex with a circular defect has been studied by Buzdin and Feinberg [15] in the short range and generalized by Chen and Tseng [16] to all ranges. But, unfortunately, as far as we know, the articles about the interation between vortices and columnar defects in the layered superconductors are still deficient.

In this paper, we generalize the result of the interaction of a single 2D pancake vortex with a circular defect in a superconducting thin film [16] to that in superconducting multilayers. We assume that the layered superconductors are coupled by the magnetic field and the interlayer Josephson coupling could be neglected. We improve the mathematical methods developed in our previous work [16] in order to apply them to layered superconductors. For simplicity, we assume the radius of the columnar defect a is a small quantity. This assumption is very reasonable because a ~ 100 ,~ [1-5]. Under these conditions, we obtain a general expression of the interaction energy per unit length Uint(r) for any layer spacing of 2 t and the whole range r > a. The pinning potential and pinning force per unit length are also obtained. In the limit t ~ ~, our theory can be reduced to the result in a thin film [16]. But in the opposite limit, t ~ 0, we obtain the result appropriate for the 3D superconducting bulk [13]. Therefore our theory can successfully explain

how Uint(r)

varies from the 2D superconducting thin film to the 3D superconducting bulk.

2. A single vortex in the superconducting multilayers

We first consider t h a t a single vortex lies along the z-axis in an infinite stack of parallel, thin (Josephson

decoupled) superconducting layers (K >> 1) in the planes z = zm, where zm = (2m + 1)t and m = 0, _ 1, + 2 , - - . . Then the vector potential A 0 satisfies the following equation [17]:

4~r 2 8 ( Z - Z m ) [ q b o ~ O _ A o ] ,

VX V X A o = - j ~ o ' - - (1)

c A 2-rrp

where the screening length A = 2 A2/d plays the role of an effective penetration depth in the superconducting thin film [18-20]. h is the London penetration depth and d is the thickness of the superconducting layers. Because the vector potential A 0 has cylindrical symmetry and we have even and periodical functions of z [i.e.

Ao( p, O, z ) = Ao( P, O , - z) and Ao( p, O, z + 2 t ) = Ao( p, O, z)], the general solution of Eq. (1) has the

f o r m [7]

Ao( p, O, z ) = ~of ° d k a o ( k ) J l ( k p ) cosh k( z - 2mt)

Substituting Eq. (2) into Eq. (1), we then obtain

~ ° d k a o ( k ) ( A k sinh k t + cosh kt) Jl( k p ) th°

2 ¢rp "

from which the Hankel transform yields [21]

~b 0 1

a°( k) = 2Ir A k sinh k t + c o s h kt"

From

for ] z - 2rnt l < t. (2)

this result we obtain the magnetic field

qb o fo~ k [ - # p J l ( kp) sinh k( z - 2mt) + ~zJo( kp) cosh k( z - 2mt)]

H° = ~ Jo dk A k sinh kt + cosh kt

for [ z-- 2mt [ < t.

where J , ( x ) is the Bessel function.

(3)

(4)

(3)

332 J.L. Chen, T.Y. Tseng / Physica C 269 (1996) 330-338 The supercurrent density in each superconducting layer is given by

Jso( P, 0, z) = ~ - ~ - ~ A° = "4~ --'5-

Ak tanh kt

+ 1 (6)

The self-energy of a single vortex per unit length Uself is obtained by integrating the supercurrent density Jso [22]

t~ 0 f .

¢b~ fl/~ dk

2

tanh kt

U~lf= ~c JJs° " dtr/2t~-"

167rEtdo

Ak tanh

k t + l '

(7)

where ~(T) is the Ginzberg-Landan coherence length and 2t is the layer spacing. If we substitute tanh x --- x for x _< 1 and tanh x -- 1 for x > 1 into Eq. (7), we have

th 2 1 t

U~,f ~ - - ~ [ln(-~ ) + - ~ l n ( - ~ t t ) ] . (8)

Eq. (8) is an approximate expression of U~elf for any t. In the following paragraphs we discuss this equation in two limits:

(1) t >> A; (2) t << A.

2.1.

t>> A

From EqS. (2-7) we have the vector potential

qb°e° fo~

e-klz-z'lJl(kP)

for

I Z--zml

<_t, (9)

a 0 ( p , 0, z) = ~ dk A k + 1 ' the supercurrent density

Jso(P, 0, z) =

qboC6(Z-Z,n)e~o[H,(A)_Y,(A

)

8

2

~

A

_ 2 ] , (10)

where /4,(x) is the sturve function and Y,(x) is the Neumann function. The self-energy of a single vortex per unit length is

Us, e l f = 16----572, ln[-- 7 . (11)

2.2. t

<< A

We obtain the vector potential

A0( p, 0, z ) = ~b°~° +

2,rrAi I i Ai I for

I z - 2mt l < t,

z --' 2mt) 2 2

(4)

J.L. Chen, T.Y. Tseng /Physica C 269 (1996)330-338 333 the magnetic field

n 0 ( p, 0, z ) = -

z- 2mt)$. [ P)

÷

for

I z - 2mtl <_t,

( z - 2rot) 2 p

2[

6A~ K° ~ - ~ 1 ~ (13)

the supercurrent density

Jso(P,O,z)= qb°ct6(z-zm)

{

(-~11)+ t--~[ P - ~ K o I P I - K

( ) ] } P (14)

4,rr2A~ eo K1 3AII [ 2All ~ All] 1 ~11 " Therefore, the self-energy of a single vortex per unit length is

U~eif= 1 - ~ In , (15)

where All --= ~ is the effective penetration depth [7] in the superconducting multilayers for t << A. K , ( x ) is the modified Bessel function.

As expected, in the limit t >> A, the vector potential A 0 is reduced to the result appropriate for an isolated thin film as indicated in Eq. (9), but in the opposite limit, t << A, we obtain the result appropriate for the 3D superconducting bulk as shown in Eq. (12) in which All is the effective penetration depth. Moreover, we obtain the corrective terms in the order of

t/A = t2/A~l

in Eqs. (12-15) for t << A. Therefore Eqs. (4-7) are valid for a superconductor with any layer spacing.

3. A single vortex in the superconducting multilayers with a columnar defect

We next consider a columnar defect with radius a in an infinite stack of parallel, thin (Josephson decoupled) superconducting layers in the planes z = zm, and a single vortex is located at a distance r from the center of the defect. Both columnar defects and vortices are aligned along the z-axis (as shown in Fig. 1). For simplicity we assume a << min(A, All) in this paper. Then the vector potential A satisfies the equations [16]

4"rr

2 6 ( z - z , ) [ p o $ ~ A l

for p > a,

V x V × A = I j s =

c A 2 ~ R

VX V X A = 0 for p < a, (16)

where the center of the columnar defect is located at (0, 0, z) which corresponds to the z-axis of the cylindrical coordinates ( p, 0, z), and the single vortex is located at (r, 0, z) which corresponds to the z-axis of the other cylindrical coordinates (R, ~b, z). In order to solve Eq. (16), we would like to generalize the mathematical methods applicable in a thin film [16] to the multilayers. The solution of Eq. (16) includes two parts and can be written as A( p, 0, z ) = A 0 + A H , where A 0 is the vector potential of the single vortex and has been solved and given in Section 2. A H is the homogeneous solution of Eq. (16), and is caused by the screening current of the columnar defect. On the basis of Eqs. (2) and (4) in Section 2, we have

~b0e ~ f ~ d k J l ( k R ) c o s h

k( z - 2rot)

(5)

334

J J,. Chen, T.Y. Tseng /Physica C 269 (1996)330-338

Z

1 /

z = ' 3 t /

II

Fig. 1. A columnar defect with radius a lies in the Josephson decoupled superconducting multilayers in the planes z = zm = (2m + 1)t, and a single vortex is located at a distance r from the center of the defect.

N o w we transform the coordinates from ( R , th, z) to ( p , 0, z) in Eq. (17). B y the summation theorem, the Bessel function

Jo(kR)

can be expanded in the following form [23]

Jo(kR) =Jo(kp)Jo(kr)

+ 2 E

J.(kp)J.(kr)

cos

nO.

(18)

n = l

Taking the partial derivatives

O/Op

and 0/O0 on both sides o f Eq. (18), we have

( p - - r cos O)

] - J l ( k R )

=Jl(kP)Jo(kr) + ~_, [Jn+l(kp) -Jn_l(kP)]J.(kr)

cos

nO.

(19)

n = l

r sin 0

R

Jl(kR) = ~-" [J~+l(kP) +J"-l(kP)]J"(kr)

sin

nO.

(20)

n = l

The angular unit vector in ( R , ~b, z), ~6 can be expressed as

- r sin O ~ p + ( p - r sin O)eo

= R ' ( 2 1 )

From Eqs. ( 1 9 - 2 1 ) , we have

oo 1

Jl( kR)e6=2Jo( kr)Lo( p,O; k) + ~_. J.( kr)L,( p, O; k),

(22)

(6)

J.L. Chen, T.Y. Tseng / Physica C 269 (1996) 330-338 335

where

Ln( p, O; k ) = -R6[ Jn+ ,( kp) + Jn_,( kp)]sin nO + eo[ Jn+,( kp) - Jn_,( kp)]cos nO,

n = 0 , 1 , 2 . . . . (23)

Substituting Eq. (22) into Eq. (17), we have

q~o ( ~

cosh k( z - 2mt) [1

~

]

A0( p. 0, z) = ~'~ "o dkAksinh kt+cosh

kt -~Jo(kr)Lo( p, O; k) + E Jn(kr)L.( p, O; k)

.=1

for

I z - 2mt l < t.

(24) From Eq. (24) we find that

A '= dPo~c,/2"rrR-A o

can be expanded in a series of the orthogonal functions

Un( p, O, Z; r),

where

Un( p, O, z; r)

is the solution of the equation

V× VU n = - 2 6 ( z ) U n / A

and can be written as

= ~b 0 / - ~ [ cosh

k ( z - 2mt) ]

v.(

p, 0, z ; r )

~-~-w Jo [ 1 -

Aksinh

k t + c o s h

kt

.i

Jn( kr)Ln( p, O; k).

(25)

In Eq. (25) r is regarded as a parameter. Then we can construct the homogeneous solution of Eq. (16), A H, in the region of interest p > a with the expansions of Un( p, 0, z; a) in a series as follows:

~o(r)

A n ( p , O, z) =

2

V°( p' O, z; a) + E an(r)Vn( p, O, z; a)

n = l

for

p > a and [ z - 2rnt [ <_ t.

(26) The supercurrent density Js is given by

is( o. 0, z) = c ~ ( z - zm) 4,03,

A

27rqrA 2-rrR

qboC~( Z - Zm) fo~

ktanh kt

( 1 j

=

4--~ ~

dkAktanh k t + l

"2[ °(kr) - a ° ( r ) J ° ( k a ) ] L ° ( p' O; k)

+ E [Jn(kr) - ° G ( r ) J n ( k a ) ] L n ( P , O; k)

f o r p > a , (27)

n = l

where an(r), can be determined by boundary conditions.

Because the radial component of Js vanishes at p = a ÷ =

a + ~(T), an(r)

(n v~ 0) can be determined as

Jn( ka)J,( kr)tanh kt

f

Jo dk

~

~--t-t-~i

an(r) =

~o

Jn(ka)Jn(ka+)tanh kt

' n = 1, 2, 3 , - - - for r > a; (28)

fo

dk

Aktanh kt + 1

ct0(r) can be determined by the condition that the total magnetic flux trapped in the superconducting multilayers (including the defect) is equal to ~b o. The z component of the magnetic field

Hz( p < a, z = z,,)

is distributed uniformly and the radial and angular components of H ( p < a, z = z,,) vanish because V × H --

4~rjJc

= 0 for p < a and z = z,,. By integrating Eq. (16) along the circular contour of the defect in z =

Zm

just as was done in Ref. [13], we have

~p A . dl='rra2Hz(p<a, z = z , , ) .

(29) =a

(7)

336

J.L. Chen, T.Y. Tseng /Physica C 269 (1996)330-338 Substituting Eqs. (24) and (26) into Eq. (29), we obtain

f o dk AktanhaJ°( kr)kt + 1 [lkaJ°( ka) - Jl( ka)]

ao(r ) =

~

aJo(ka+)

for r > a. (30)

fo

dk Aktanh kt + 1 [½kaJ°( ka) - Jl( ka)] + 1

The free energy

U(r)

per unit length is given by

6o fJs"

d o ' = Use,f+ Uint(r),

U(r) =

(31)

where the self-energy of a single vortex per unit length U~elf is given in Eq. (7), and the interaction energy per unit length

Uint(r) between

single vortex and columnar defect in the superconducting multilayers is given as

Uint( r ) =

t k/ [

1

62 fo dk

Oto(r)Jo(ka)Jo(kr )

+ 2 •

ot,(r)S,(ka)Jn(kr ) .

(32)

16~2t

Aktanh kt + 1

Because we assumed a << min(A, All) previously, the first term in the right-hand side of Eq. (32) is much smaller than the second term, and therefore it can be neglected. We consequently get the general expression of Uint(r) for any t:

fo ~ dkJ,( ka) J,( kr )tanh kt

(33)

Ui,t(r)

=

8'rr2t n= 1

62 ~, a , ( r )

Aktanh kt + 1

'

where

a,(r)

is given in Eq. (28). We can examine Eq. (33) in the two limits t--+ ~ and t--+ 0. In the limit t ~ ~, Eq. (33) reduces to the results of the superconducting thin film which we had been obtained in Ref. [16]. In the opposite limit t ~ 0, Eq. (33) then approaches the results of Mkrtchyan and Shmidt [13] appropriate for the 3D superconducting bulk with effective penetration depth All = q-~'. Since the high-T c superconductor possesses a multilayer structure, and the spacing between the CuO 2 layers is very short (N 10 .~), we are interested in the limit of this small layer spacing. Taking the limit t << A, we can calculate Eq. (33) up to the order of

t / A

(t=/A~) as in the following:

6 g ( a / 2 a , , ) K" ~ 1 - + - -

Uint(r)

4~2A~ n--

~1 n.---~(n: 1").~

for a << All and t << A,

t2r K._,(r/All

) ] 3A~

K.(r/Atl )

]

(34) where the first term of the right-hand side in Eq. (34) is equal to the result of Mkrtchyan and Shmidt [13], and the other terms in Eq. (34) are the correction up to the order of

t2/A~l.

Taking the two limits r

<<

All and r >> All, Eq. (34) can be rewritten as

1-

ln(1-

for a<r<<A,,,

Uint(r)

= ~ ~ 4"rrAII ] ~ 3All ] ~ -~" '

!

6202

F

t 2

,2. l

(35)

/ ~ e - 2 " / a , , | l - ---v + ' ' /

(8)

J.L. Chen, T.Y. Tseng / Physica C 269 (1996)330-338 337

The pinning potential per unit length for the single vortex, Upin(a), is defined by Upin(a ) = Uint(r-- a +)

= -- 1 -- In for a << All and t << A. The pinning force per unit length is

f p = -- V Uint( r) l r=a+

= - - 1 - - ~ f o r a < < A l l and t < < A .

(36)

(37)

4. Conclusions

In this paper we generalize the mathematical methods which we had developed to apply in the superconduct- ing thin film [16] to the Josephson decoupled superconducting multilayers. The interaction energy per unit length, U i n t ( r ) , between a single vortex and a columnar defect is consequently derived for the whole range r > a, where a is the radius of the columnar defect and we assume a is small for simplicity.

It is interesting that our theory cannot only reduce to the results of an isolated thin film which we had derived in Ref. [16] in the limit t ---> o% but also our theory corresponds to the result of Mkrtchyan and Shmidt [13] in the limit t ~ 0, where t is half of the spacing between the layers. For r >> A the interaction energy per unit length Uint(r) exhibits 2D behavior. B u t Uint(r) more likely exhibits the behavior of the 3D superconducting bulk for smaller t. Therefore our theory can be applied to superconducting multilayers with any layer spacing.

Finally we calculate Uint(r) up to the order of t / A in the limit t < < A (the condition of high-T~ superconductors). Under this condition we also calculate the pinning potential and pinning force per unit length in our system and we believe that this work is useful to realize the pinning mechanism of high-T~ superconductors.

Acknowledgement

The authors gratefully acknowledge the financial support from the National Science Council of Taiwan under contract no. NSC85-2112-M009-037.

References

[1] M. Konezykowski et al., Phys. Rev. B 44 (1991) 7167. [2] L. Civale et al., Phys. Rev. Lett. 67 (1991) 648. [3] W. Gerhauser et al., Phys. Rev. Lett. 68 (1992) 879.

[4] A. Legris, F. Rullier-Albenque and P. Lejay, Phys. Rev. B 48 (1993) 10634. [5] H. Dai et al., Science 265 (1994) 1552.

[6] A. Buzdin and D. Feirberg, Physica C 235-240 (1994) 2755. [7] J.R. Clem, Phys. Rev. B 43 (1991) 7837.

[8] J.R. Clem, M.W. Coffey and Z. Hao, Phys. Rev. B 44 (1991) 2732. [9] L. Bulaevskii and J.R. Clem, Phys. Rev. B 44 (1991) 10234.

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338 J.L. Chen, T.Y. Tseng / Physica C 269 (1996)330-338

[11] W.E. Lawrence and S. Doniach, Low Temp. Phys. LT-12, ed. E. Kanda (Keigaku, Tokyo, 1970) p. 361. [12] M. Benkraouda and J.R. Clem, Phys. Rev. B 53 (1996) 438.

[13] G.S. Mkrtehyan and V.V. Schraidt, Zh. Eksp. Teor. Fiz. 61 (1971) 367 [Sov. Phys. JETP 34 (1972) 195]. [14] I.B. Khalfin and B.Ya. Shapiro, Physica C 207 (1993) 359.

[15] A. Buzdin and D. Feinberg, Physica C 256 (1996) 303. [16] J.L. Chen and T.Y. Tseng, Phys. Rev. B 54 (1996) 502. [17] A.L. Fetter and P.C. Hohenberg, Phys. Rev. 159 (1967) 330.

[18] J. Pearl, Low Temp. Phys. LT-9, part A, eds. J.G. Daunt, D.O. Edwards, F.J. Milford and M. Yaqub (Plenum, New York, 1965) p. 566.

[ 19] J. Pearl, Appl. Phys. Lett. 5 (1964) 65.

[20] J. Pearl, PhD. thesis, Polytechnic Institute of Brooklyn, USA, 1965. [21] J.D. Jackson, Classical Electrodynaraics (Wiley, New York, 1962) p. 77. [22] F. London, Superfluids I, (Wiley, New York, 1950) p. 76.

數據

Fig.  1. A columnar defect with radius  a  lies in the Josephson decoupled superconducting multilayers in the planes  z =  zm = (2m +  1)t, and  a single vortex is located at a distance r  from the center of the defect

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