Vortices in layered high-Tc superconductors

PHYSICA

PhysicaC 178 (1991) 161-170 North-Holland

Vortices in layered high-Tc superconductors K.H. Fischer

Institutj~r FestkOrperforschungder KFA Jiilich, W-5170Jiilich, Germany Received 20 March 1991

We investigate the properties of a point-like vortex, vortex-antivortex pairs and vortex lines in layered superconductors. The formation energy of a single point-like vortex turns out to diverge logarithmically with the size of the sample. The spontaneous creation of vortex-antivortex pairs leads to a Kosterlitz-Thouless transition (at a temperature TKT) that is similar to the transition in 2D superconductors. The interaction between pairs in different planes is small. A single straight vortex line and a vortex line pair have the same energy as in continuous superconductors. However, the stiffness of a single vortex line in a layered superconductor is considerably smaller than in conventional systems, and the line "melts" in the absence of pinning at TKr.

I. Introduction The discovery o f high-Tc superconductors [ 1 ] has led to much activity in the investigation o f flux line motion. F o r three-dimensional superconductors, the basic concepts which emerged from these investigations include melting o f the flux line lattice either very near to the lower critical field He, [2,3] (with a possible transition to a hexatic flux liquid phase [ 4 , 5 ] ) or for higher fields [6,10] and the appearance o f a glass-like vortex state [ 11-15 ] that is separated from a vortex liquid by a critical (or crossover) line He(T) in the m a g n e t i c - f i e l d - t e m p e r a t u r e plane [ 11,15,16 ]. This has also been described as a depinning transition at the line H¢(T) [ 1 7 - 1 9 ] which separates the flux creep region from a region o f thermally activated flux flow ( T A F F ) [ 20 ]. This "critical" line coincides with the "irreversibility line" [ 1,17 ], below which one has r e m a n e n t magnetization with a In t decay law. At present it is not clear whether this line is described better as " d e p i n n i n g " or " m e l t i n g " o f flux lines, although there seems to be some evidence in favour o f d e p i n n i n g [21 ]. One expects that the free-energy barriers due to pinning are not all equal but have a certain distribution [22,23 ]. Recently, the concept o f scaling has been applied to flux line pinning [ 1 1 - 1 6 ] , and there is experimental evidence for a distribution o f free energy barriers that is based on scaling and leads below the line H~(t) to

a vanishing linear resistivity. Above H e ( T ) , the highest barriers vanish, and one has the T A F F region with a linear resistivity. The concepts of melting o f the flux line lattice due to thermal fluctuations and o f a vortex glass due to pinning with a transition into a vortex fluid have been reviewed in ref. [15 ] and the more traditional depinning theory in ref. [20]. High-Tc superconductors are strongly anisotropic and actually have a layered structure. If the interlayer distance d is larger than the coherence length perpendicular to the layers ~ a model for a homogeneous superconductor becomes inadequate. (In Y B a C u O this ratio is about 2 at T = 0, and for the Biand Tl-based c o m p o u n d s considerably larger.) Instead one has superconducting layers o f thickness do with weak Josephson coupling between them. Such a m o d e l was discussed first by Lawrence and Doniach ( L D ) [24] and later by K l e m m [25] and Efetov [26]. Based on this model, Feigelman et al. [ 27], Volkov [28] and A r t e m e k o and Kruglov [29] calculated the properties o f various vortex configurations in layered superconductors. However, these authors a p p l i e d a " c o n t i n u u m " a p p r o x i m a t i o n , in which the supercurrents are allowed to exist also between the layers. Their basic equation is a generalized L o n d o n equation for the vector potential AI= (Ax, Ay) and vortex points as sources in the planes z~=nd (eq. ( 3 ) o f [27] and eq. ( 1 ) o f [29] ),

0921-4534/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved

162

K. tf. Fischer / I brtwes tn layered tf7S( '~

All--22V2AII = v d

~

d(_--z,,)Vrl~0,,.

( 1)

Here, 2 is the bulk London penetration depth in the ab-plane, qbo = h c / 2 e the flux quantum, ~0,,the phase of the order parameter in the nth layer and V H the gradient components in the ab-plane. In this paper, we show that the exact equations reads

(2) with the effective penetration depth A = 2 2 / d . The difference between eqs. ( 1 ) and (2) leads to rather different properties. The solution (2) of the Ginzburg-Landau ( G L ) free energy was investigated first by Efetov [26]. Some of his results have recently been rederived by Buzdin and Feinberg [30] in part of their work. These authors define instead ofA =Z2/ d the effective penetration depth 2~n-=2~/do with the layer thickness do and the London penetration depth 2~ in a film (see also [31,32] ). Equation (2) states with 4~.

rot h = - - j ,

(3)

h=rotA ,

(4)

C

and C

j=

47t V2A,

divA=0,

(5a) (5b)

that the supercurrents are restricted to the planes z = z , , . A finite but small thickness do of the layers is irrelevant, as has been shown by Pearl [ 33 ] for a single layer with do << ~, i.e. for a two-dimensional ( 2 D ) superconductor. Such a 2D model has a KosterlitzThouless ( K T ) transition [ 3 4 - 3 9 ] : There is a spontaneous creation and annihilation of point-like vortex-antivortex pairs due to thermal fluctuations. Above the Kosterlitz-Thouless temperature TKj these pairs unbind and form a "neutral" gas. Here we show that such a KT transition also exists in 3D layered superconductors under certain conditions, in agreement with experimental data [ 4 0 48] and results based on the approximation (1) [27,29]. Our calculation is based on expressions for

the vector potential A u and tbr the energy for an arbitrary configuration of vortices derived in refs. [26,30], from which we obtain properties of a point vortex, vortex-antivortcx pairs, a vortex line and the interaction between such lines. It turns out that a point vortex has a formation energy proportional to In R where R is a sample radius in the ab-plane. This makes the formation of a 2D flux lattice in a single plane in an external field energetically unfavourable. However, the formation of fairly decoupled 2D lattices in all planes [49] might still be possible. The formation energy of vortex-antivortex pairs in a single plane turns out to be finite. The interaction energy between pairs in different planes is small and the system has a KT transition, similar to that in a 2D superconductor. In principle, such a KT transition can be destroyed by Josephson currents, but a crude estimate shows that this should not be the case at least for the T1- and Bi-based compounds, which have extremely high anisotropies. Since the interaction between pairs in different planes is small, the formation of vortex rings extending over several planes [27,49] is rather unlikely. A small local displacement Pn in the plane n of a straight flux line in the .c-direction leads to a deformation energy that is approximately proportional to p~, as in the continuous superconductor. However, the prefactor or stiffness turns out to be a factor ( d / Z) 2~ 10 -4 to 10 -6 smaller. As a consequence, such a flux line will be extremely unstable against thermal fluctuations. Large displacements lead to a deformation energy proportional to lnpn and the flux line will "melt" or "evaporate" exactly at the temperature '~KT for the unbinding of vortex-antivortex pairs. This result is essentially unchanged if one considers more extended displacements o f the flux line (which is indeed a line o f point vortices). An additional vortex-antivortex pair will be attracted by the vortex line, which lowers its formation energy. The magnetic field of the line enhances the creation and unbinding o f such pairs, and the line will serve as a " s o u r c e " o f vortex pairs. A similar effect is known for vortex pairs in an external field [431. This paper is organized as follows: In section 2 we rederive the basic equations from the GL free-energy functional. In particular, the G L equation for the order parameter differs from that derived by others

K.H. Fischer/ Vorticesin layeredHTSCs

163 Zn+ i

[27,31,49] and is identified with the condition d i v j = 0. In section 3 we consider various point vortex configurations and in section 4 vortex lines. The results are summarized in section 5.

+ ~2m 5 ( 1 - c o s ( ~ o . + , - ~ o . - c2e

f dzA~))] Zn

if d3r(h-Hex) 2,

+~

(7)

where we have introduced the London penetration depth in the ab-plane

2. Basic equations In the LD model a superconductor is described by superconducting planes n with interlayer distance d and with the order parameter q / , ( p ) = I~%(P)I ×exp[i~0,(p)] with p=(x, y). From the conventional GL free energy [50] one obtains by discretization of the kinetic energy in the z-direction [24] (h=l)

( m£2 ~ '/2 2 = k47tnse2 ]

where ns is the 3D density of superconducting electrons. Variation of eq. ( 7 ) with respect to A II(P, z) yields with divA = 0 eq. (2), and variation with respect to Az(p, z) the Josephson current density between the layers n + 1 and n,

j.(p, z) =

G(T,H~x, {q/,})=d~ ~ d2p[al~u,12+½fll~u,[4

(8)

¢

- ~ V 2 A=(p, z)

(9)

and

l (-iv,,-IA)~,. ~

j.(p, 2 , ) = f i

+ 2m*

Zn+ 1

2

sin t2,.

(10)

Here,

• e*

Zn+ I

2e

Zn

+~

1

J dzA.(p,z)

(11)

Zn

f d3r ( h - H ~ x ) 2 .

(6)

Here, h and Hex are the internal and external fields, respectively. There is experimental evidence that one has in high-Tc superconductors similar Cooper pairs with e*=2e, m * = 2 m and M * = 2 M as in conventional ones. High-To superconductors are extreme cases of type-II superconductors with a GL parameter 2/~>> 1 where ~ is the coherence length in the ab-plane. Apart from the small vortex cores with radius ~, one has I q/,I 2=ns=const and has to consider in (6) the phase (0, only. The strong anisotropy is hidden in eq. (6) in the different masses m and M where M/m= (2e/2) z is about 25-30 for YBaCuO and about 3500-5000 for Bi and TI compounds. We have from eq. (6)

G(T, Hex{~O,})= q~2

[/V

_ 2~

2

is the gauge-invariant phase difference, eq)o J c - 8~2Ajff,

J.J=d(M/m) 1/2

(12)

the maximum Josephson current density, and the corresponding penetration depth, respectively. As expected, a large anisotropy leads to a small Josephson current density Jc. Variation of eq. (7) with respect to the order parameter ~o,(p) yields 2x divlt (Vii ~0, - ~oo A, (p, z, ) ) = - 2 j - 2 (sin 12, - sin g2,_ l ) .

(13)

Equation (13) differs from the usual sine-Gordon equation [27,31,49,51] by the second term on the r.h.s. [ 52 ], and includes - with eq. ( 11 ) - coupling between the phases in three planes. In fact, eq. ( 13 ) is identical to the current conservation d i v j = d i v j , +djz/dz--O, where from eq. (lO)

K.H. Ftscher/ l'ortwesin layeredttTSC~

164 d/:=d d:

Ail(q,k)= f d2pdze-i(qP+~:>Arl(p,:),

I(j:(p,z,,)-j:(p,:,,_l )

i - " ~ (sin~2,,-sin£2,, d

(19)

and ,)

(14)

A(q, k)= f d2pdze -i(qp+/'=> E (~(2 n

and where divjl I is obtained by integrating eq. (2) over the interval z,, + d/2. Equation (2) then leads witheqs. (5) and (12) to

- : )Al,(p.:)= ~ e

'~:"A,~ll(q).

(20)

with A,,ll(q)~Ail(q, :,,). One has from cq. (2) with eqs. (17) to (20)

divjll (p, :,, ) - 2~ div~l(Vll(P"(P)-d , ~sAil(p,z,,))

(15)

and with eq. ( 13 ) to d i v j = 0. In the isotropic continuum limit one has 2j~s= d and div(V(p(r) - (2e/ c)A(r)) =0.

All(q, k ) =

6(q, k)-`i, (q, k) A(q2+k2)

(21)

and with `i(q, k)=,ill(q, k - 2xn/d) [53] ,ill (q, k ) = d - 1 }~ AiI(q, k - 2xn/d) tt

(~(q,k)-,iu(q,k)) sinh qd 2Aq (cosh qd- cos kd) "

3. Point vortices

(22) A vortex in a single plane of thickness do--~0 has a core of the area x~ 2 and is point-like for a small coherence length { (actually, it looks like a "'pancake"). In this section, we consider various configurations of point vortices. The solution of eq. (2) for an arbitrary configuration of point vortices has been presented in refs. [26] and [30] and, in order to make the paper selfcontained, we summarize the main steps of their calculation. One defines the "London vector"

Equations (21) and (22) lead with

F(q, k) =2Aq cosh q d - cos kd sinh qd

( 23 )

to

~(q, k) F(q, k) AIj(q,k)=A(q2+k2) ( l + F ( q , k ) )

(24)

From eq. (22) we have [54] 2~/d

q)o

(P"(P) = ~-x ,~. e" Vll~°n(P-P('?) '

(16)

A,,ll(q) = d i

dkeikna`i(q' k) o

which describes point vortices with sign er= +1 al the positions pO in the plane n. For a single vortex at p0 = 0 one has Vll~0n(p) =eo/P where eo and ep are unit vectors in the coordinates p--(p,O). For a sufficiently fast decaying current densfty, the magnitude of the London vector is just q)o: One has from eqs. (2) and (5) for a sufficiently large loop in the p plane fdsAil=cI)=CI)o. We define the Fourier transforms

O(q,k)= Ze-i~="q~,(q),

z,=nd

(17)

n

0~(q) = f d 2 p e d

~qPo,,(P),

(18)

= ~. O,,,(q)W,,,~(q),

(25)

m

with

~.~(q)=

sinhqd (G a - ( G q - I )l/2)l ...... i

2Aq

(G~,- 1 )1/2

(26)

and Gu = cosh qd+ (sinh qd)/2Aq.

(27)

(Note that our characteristic length A =).2/d differs from that in ref. [30] ). Following Pearl [33], we define the 2D currenl density 1,, in layer n by

K.H. Fischer/ Vorticesin layered HTSCs

j(p,z)=

~J.(p)6(z-z.),

2

(28)

n

which leads with eqs. (2) and (5) to

~-l

2

~ln

C

J n ( q ) = ~ - ~ [(J~(q)-A~ll(q) ] •

,f

G(T. n e x ) + ~

d3rh2(p,z)

+ 2-~A c 2 ~ I d2PJ]

1

(30)

d2

(31)

d2q 2 [

-O,(-q).A,l~(q)] .

(32)

For a single point vortex in the plane n = 0 at p ° = 0 one has from eqs. (16) and (18)

¢~,,(p)=eo 2~p S,,,o ,

(33)

@~(q) =ieo @° 8~,o - # ( q ) 8 ~ , o , q

(34)

where eo is a unit vector perpendicular to q and e~. For dq << 1 or distances p >> d one has from eqs. (25), (26) and (34) with A=22/d

¢,od

apart from corrections of the order (d/2)2<< 1. Hence, the free energy of a single point vortex diverges for R-~ov and cannot be created by an external field. This is in contrast to the formation energy of a single vortex in a thin layer of thickness do, for which it is of the order [33] (37)

According to eq. (36), a 2D vortex lattice in a single plane o f a 3D layered superconductor also cannot exist, in contrast to a vortex lattice in a 2D superconductor. However, it is still possible that a 3D vortex lattice decays into fairly independent 2D lattices in all planes [49], since a single flux line is fairly unstable (see section 4). The current density in the plane n = 0 is easily calculated from eqs. (29) and (35). Apart from corrections of order (d/2)2 we have for p >> d, J o ( p ) - -e o

cq~o 1

87~2A P ,

(n=O).

(38)

For the magnetic field in the z-direction hz(q, k)=iqXAil(q , k) one has from (24) with dll(p , z) =eoAo(p, z)

3. I. Single point vortex

A~=o(q) =e~ 2q2 "

(36)

Iv ~ \~-n / ~5 l n ~ ) .

where we have omitted the core energies. Partial integration of the first term leads with eqs. (4), (28) and (29) to [30,55]:

G(T,"~):(-~c)~ f ~¢~(q)'J.(-q)

,

(29)

For a given distribution of vortices ( 16 ), the current density (29) is completely determined by eqs. (25) and (26). The change of free energy of the superconductor due to the vortices consists of the magnetic and kinetic energies. One has with (28) [ 33,50 ]

1

165

(35)

This leads with eq. (32) to the free energy (R is the sample radius)

h=(p, z) = qb° 1 r= (p2+z2) ~/2 4~A r ' (p >> d, d<< z << 2ab) •

( 39 )

In contrast to eq. (39), the result of refs. [27,29] based on eq. (1) contains an additional screening factor e x p ( - r / 2 ) due to the fact that the current is not restricted to the planes 8 ( z - z n ) . The currents in the planes n ~ 0 can be calculated from the vector potential Ajj. One has from eqs. (19), (23) and (24)

q~o ( [ z l - r )

Au(p, z) = - c o 4hA

P

,

(40)

which leads to eq. (39), to the additional field component in direction eo

K.It. Fischer/ I brnce.sin layered 1t7S( ~

166

~o 1 ( h,,(p,z)=~Ap\~7-_l -;),

(z4-nd).

(41)

and to hp=0 in the planes z=nd. The current density (28) in the planes n ¢ 0 is

cqbo pd J,,(p)=eo(4n)eAr~,,

(n~-O).

(42)

According to eqs. ( 3 8 ) - ( 4 2 ) the currents and fields decay extremely slowly. In particular, the p ~-decrease of Jo(p) and h:(p,O) eqs. (38) and (39) leads to the infinite formation energy (36). A single vortex point might be an unphysical quantity, but its properties are important for understanding a vortex-antivortex pair in which one has a superposition of the currents and fields (38) to (42) for two vortices of opposite sign at different sites (see section 3.2). In order to estimate the importance of the Josephson current density, we replace j- eq. (10) by its upper limit ( 12 ). The current density ( 38 ) becomes of the order of the 2D current density j~d if p > p o = ( M / m ) d = (2,,/2)~d. The anisotropy factor M / m is about 25 in YBaCuO and 2500 to 3000 in the Bi- and Tl-based compounds [15]. In the latter systems the influence of the Josephson coupling should then be small. In the case of YBaCuO, one has to calculate the Josephson current density j: explicitly in order to find its impact on the formation of point vortices.

3.2. I brtex-antivorlex pair For a pair consisting of a vortex at site p = 0 and a vortex with opposite flux direction (antivortex) at p=p~ in the plane n = 0 , one has from eqs. (16), (18) and (34)

O,,(q)

=O(q)

( 1 - e -'q'pt ) (~,,,o •

(43)

This leads with eqs. (25) and (26) to the total free energy 4 q~o ~2 o

-Jo(qp,))(1-Woo(q)),

(44a)

where Jo(x) is a Bessel function of first kind. In contrast to the energy G~, eq. (36), G2(p~ ) diverges only

for Pl .,~c. One approximately

has

(;~(P~)+k4nJ A

x~/"

for

d<<2

and

p~ >>d

which agrees with the result of the approximation ( 1 ) [27] and, apart from the different characteristic length ).cfr=22/do, with the result for a single layer of thickness d0 [33] (see also ref. [30] ). However, for a single layer one has G2 (p,) - I n p~ only for p~ << 2,,~ and G2(p~) ~ p ~ for p~ >>;%ff, whereas eq. (44b) holds for all distancespt >>d. This is consistent with the finite formation energy (37) of a single vortex in a thin layer, in contrast to the infinite formation energy (36) in a layered superconductor. Vortex-antivortex pairs can be created by thermal fluctuations. If the Josephson current can be neglected (see section 3.1 ), these pairs dissociate at the KT temperature

/"uT~'=k4n} 2,t('J~ji ~ ~Q'-(r~,),

!45)

where Q corresponds to the electric charge in a 2D Coulomb gas [ 39 ]. Equation (45) is derived from the mean squared distance between the members of a pair [ 34] ~

3

2[]Q2tnlsl

( P T ) = l :l~d- Pl P~e . d e p l e 2/~Q21npi

=g

:2 f l Q 2 - 1 flQ2_ 2 .

(46)

which diverges at the temperature Twr. In an improved theory [ 34,39,48 ] one introduces a dielectric function c(p~ ) which takes into account the screening by other pairs. Such a KT transition in high-7] superconductors has been observed by many groups [ 40-47 ]. The theory [ 38,39,48 ] predicts for 2D systems a nonlinear voltage-current relation l ' - I ''<~ with a universal j u m p a ( T k l ) = 3 , a ( T > 7"KI ) = 1. Indications for such a j u m p have been found in YBaCuO and in the T1, Bi and Er compounds [4047]. In all cases, the sample thickness do was large compared to ~,. (typically 2X 102-104 /k) and the systems were indeed three-dimensional. A KT transition in a 3D layered superconductor is certainly not trivial, since a vortex-antivortex pair induces currents in all planes, in contrast to a single layer. Since an external current I breaks up pairs of finite size (which leads to I'~F'), a corresponding

K.H. Fischer/ Vorticesin layeredHTSCs anomaly should even be observable if the Josephson current destroys the sharp KT transition due to the infinite separation of the pair members. One might ask if a KT transition is destroyed by the interaction between pairs in different planes. (Pairs with members in different planes have an energy proportional to In R and can be excluded [ 30 ] ). Two pairs in the layers n = 0 and n = m with the same size Pl at the sites (0, p~ ) and (P2, Pl "l-p2) (Fig. 1 ) have the interaction energy from eqs. (25), (26) and (32),

layer [34-39,48] for a 3D layered superconductor.

4. Vortex lines

For a single flux line in the z-direction with @,(q) =@(q) the sum (25) with (26) can be performed and leads with (29) and (31) to the vector potential [ 30 ] A
u(p,,p2)= (q}o) \ ~ j 2 ~2 ! qdqwmo(q ) [2Jo(qp2) -Jo(qlPl -P2

[)-Jo(qlP~

't-P2 I)]

In the special case P2----0 Pl >> ~ and attract with the energy

(~o~22d( U(p~,0)=-

\~-1

~

•

167

sinh qd 2Aq( Gq - 1) '

(49)

to the 2D current density

(47)

d<
c

4o

and to the free energy per layer

I

.(~0. 2 ~

G,L=IL=2(.~X)

d)"

(50)

J~(q) = e 0 ~ 2Aq+ coth qd/2 '

f o

1-

dq 2Aq+ coth qd/2 "

(51a)

Equation (51a) leads in the limit k>>d to ×ln(~),

(me0).

(48a)

In the opposite limit ~<
f(J~0"~2 1 d/g

U(pl,p2)= ~-)

~ 1 -

d)m ~1_2)2 --

.

(48b)

With d/2(0) ~ 10 -2 to 10 -3, this interaction is much smaller than the vortex-antivortex energy. As a consequence, clusters of pairs in different planes can exist only at very low temperatures and break up far below TKT- Pairs in different planes will also not influence the screening significantly and one can take over the calculation of the KT transition for a single

F~

,~

P2

I

n=0

dI

i

Fig. 1. Two vortex-antivortex pairs in different planes.

(,lb,

The result (51 b) agrees exactly with the free energy of a flux line of the length d in a homogeneous superconductor [50] and for d=do with the formation energy (37) of a vortex in a single layer. As a consequence, there is no difference between the lower critical field He, in a homogeneous and a layered superconductor. A pair of parallel or antiparallel flux lines at distance p~ has the free energy G2L =2IL +--U(pl

)

(52)

and the interaction energy

2 2 ( p ) ( 2, U(p,)= (q)o) ~ -AKo-~ n=m

,I

((po~2 1

>> d)

(53,

In deriving eq. (53), we used the representation of the modified Bessel function Ko(x) [56]

Jo(aX) Ko(ak) = ii aXXxz +k~ . o

(54)

Equation (53) also agrees with the interaction energy of two flux lines in a homogeneous supercon-

K.tt. /"ts<'he#"/ 17.'ricesin layeredttTS"(;s

168

ductor, and one has the same type o f Abrikosov-fiux line lattice in both systems. In order to investigate the stability of a single vortex line, we consider a local displacement p~ in the plane #7= 0. Such a displacement can be represented by an additional v o r t e x - a n t i v o r t e x pair with the m e m b e r s at the sites p = p ' - , O and p = p ' +p~ (fig. 2 ). This flux line configuration has the free energy

G(pl)=ll +Gie+G2(Pi)=ll +6G(pl),

(55)

with the line energy It, cq. ( 5 1 b ) , the line-pair energy G~p and the pair energy G2 (Pl), eq. ( 4 4 b ) . One has ~i A]) which reduces p~ (d<
for

-In a

small

,

(56a)

displacement

(,nQ~:)+ l).

56b

Apart from a logarithmic correction, this energy is quadratic in Pl and can be c o m p a r e d with the elastic energy of a flux line in a homogeneous superconductor. For a d i s p l a c e m e n t of the form u ( z ) = ( d - Izl )pl/d for - d < z < d and zero otherwise, the latter is a factor ( 2 / d ) 2 larger than the energy ( 5 6 b ) . Hence, a flux line in a layered superc o n d u c t o r is much less stable to thermal fluctuations than in a homogeneous system. This is true, in particular, if one considers large displacements. For distances p~ >>2 the Bessel function Ko in ( 5 6 a ) can be ignored and 5G describes essentially the energy o f an i n d e p e n d e n t v o r t e x - a n t i v o r t e x pair which disintegrates at TkT, eq. (4.5). Hence the line " m e l t s " or "'evaporates" at this temperature. A similar conclu-

n=0

Fig. 2. Vortex line and vortex-antivortex pair.

sion recently was reached in refs. [29] and [31 ]. We still have to consider the possibility that a flux line decays at a lower t e m p e r a t u r e due to more extended fluctuations. Since pairs in different planes are very loosely bound (see section 3.2 ), one expects that this will not be the case. Bulaevskii et al. [31 ] find indeed that a long-wave distortion o f a flux line in harmonic a p p r o x i m a t i o n is unimportant. Here we investigate the case of two adjacent lattice points that are displaced by the same distance p~. The displacement energy turns out to be 28G(p~) with 6G(p~). eq. ( 5 6 b ) , apart from a prefactor ( 1 - d / 2 ) in front of the logarithm in eq. ( 5 6 b ) . This means that vortex points of a flux line fluctuate fairly independently and that the flux line '~mclts" locally. So far we have considered thermal fluctuations of a flux line. In the case o f a straight flux line and an additional pair with p' ¢ 0 in fig. 2 one has instead o f e q . (55) the total energy G(1),I / ) = I n + ( i ~ ( p l ) + ~(~n (Pl, P' )

(57)

with the line-pair interaction energy for p > > d ,

(58) The energy (58) is negative and decreases exponentially for large distances. Hence. the presence of a vortex line enhances the f o r m a t i o n and depairing o f v o r t e x - a n t i v o r t e x pairs due to their interaction with the magnetic field o f the line.

5. Conclusions

In this paper we have investigated vortices in layered superconductors, and we now s u m m a r i z e the results. (a) The formation energy o f a single point vortex is p r o p o r t i o n a l to In R, where R is the sample radius in the ab-plane, making the formation o f a 2D vortex lattice in a single plane impossible. This is in contrast to a single layer o f thickness do << ~c. For the latter, the corresponding energies are finite and an Abrikosov lattice exists in a certain f i e l d - t e m p e r a ture range. Since a single flux line in a layered superconductor can easily be deformed, a quasi 2D flux

K.H. Fischer / Vortices in layered HTSCs

line lattice with correlations within the planes but strong disorder in z-direction [27,49,57] is possible. (b) The formation free energy of a single pointlike vortex-antivortex pair is finite. The formation of such pairs increase the entropy of the system. Such pairs are created by thermal fluctuations and disintegrate at the Kosterlitz-Thouless temperature TKx. Evidence for such a transition in 3D high-To superconductors has indeed been observed [40-47]. In 2D systems TKx is determined by do, in 3D systems by the interlayer distance d. Since a pair induces currents in all neighbouring planes, the existence of such a KT transition in a 3D system is not trivial. It is possible since the coupling with pairs in other planes is extremely small and since (at least in the Bi- and Tl-based compounds) the coupling due to Josephson currents is negligible. (c) Flux lines in layered compounds with d<<2 have the same formation free energy and the same interaction energy as in continuous superconductors. Hence they penetrate at the same critical field H,:, and form the same Abrikosov lattice. However, their elastic energy with respect to distortion is a factor (d/2) 2 smaller than in conventional superconductors. This leads for larger local displacements to the melting of a single flux line at TKx. Since we have ignored in the calculation of this melting temperature the interaction between flux lines, this effect can possibly be observed only very near to He,. Flux line configurations are strongly influenced by pinning, which we have ignored completely in this paper. It is well possible that pinning as well as flux line interactions prevent the melting discussed above. In addition, one has to consider the Josephson current j_-, which increases with decreasing anisotropy and which finally leads to the vanishing of all 2D effects in an anisotropic superconductor.

Acknowledgements I would like to thank D. Feinberg, L.N. Bulaevskii, N. Schopohl and in particular V.L. Pokrovsky for many stimulating discussions and valuable comments. I am grateful to the ILL Grenoble for its warm hospitality during my stay at Grenoble, where this work was begun.

169

Note added in proof After submission of this paper I received a preprint from J.R. Clem [ 59], who considers a similar model, however, with layers of finite thickness do. For do--,0 part of his results agrees with ours.

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