1

Most of the type II superconductors are anisotropic. In extreme cases of layered high Tc materials like BSCCO the anisotropy is so large that the material can be considered two dimensional. It is important to distinguish the anisotropy in directions parallel and perpendicular to the magnetic field direction.

V. ANISOTROPIC and LAYERED SUPERCONDUCTORS

We start with the simplest case of anisotropic GL theory neglecting layered structure.

A. Some phenomenology

2

Various types (old and new) of the “conventional” or the “BCS” superconductors

7.2 830 370 800 - 0.4 1

7.4 77 690 1600 15 9 11

20 26 2400 130 50 90 1

15 15 1400 1400 26 30 2

25 25 2000 2000 90 150 70

( )cT K )(

A )(

A 1( )cH G 2( )cH T

Pb

2NbSe603CK

YtBC

2MgB

Gi6210

510610

510

3

93 15 1440 500 260 100 10 5

65 20 2550 300 150 110 12 13 0.1

34 29 2800 70 50 130 13 15 0.1

120 25 2500 150 150 100 18 30 0.5

0.5 25 780 2000 2.1 25 - 0.9

( )cT K )(

A )(

A 1( )cH G 2( )cH T

6.92YBCO

LSCCO

BSCCO

3UPt

Gi( )d A

6.7YBCO

3210

510

Various types of “unconventional”

(or “non BCS” SC)

1986High Tc Superconductors

Alex Muller, Georg Bednorz

4

for YBCO

First we assume that the material is rotationally symmetric in the plane perpendicular to magnetic field. While the potential and magnetic terms are always symmetric, the gradient term generally is not:

The asymmetry factor is defined by

for BSCCO

2 2

222

* *2 2grad z x yc ab

F D D Dm m

*

*c

ab

m

m

5

30

1. Anisotropic GL model

B. Anisotropic GL and Lawrence-Doniach models

5

One repeats the calculations in the anisotropic (and even in the “tilted” geometry when magnetic field is not oriented parallel to one of the symmetry axes) using scaling transformations.

2 2

2*

02 2

2

2

cc c

cc c

c

m T T

H H

Coherence length in the c direction is typically much

smaller

while the corresponding penetration depth is larger:

Blatter et al RMP (1994)

//

2 *2 2

*24c

cc

c m

e T T

6

0 11 log 1

4

cc c cc

c c

LogHH

Log

It is much easier to create vortices to be oriented in the ab plane.

We don’t have to solve again the GL equations: they do not change.

AA

~2,000 ~10

Type II: ~ 200 >> c

7

However when the material consists of well separated superconducting layers, the continuum field theory might not approximate the situation well enough: one should use the LD tunneling model:

x

yz

Interlayer

distance d

CuO plane (layer or bilayer)

Layer width s

2. The Lawrence - Doniach model

BiBi22SrSr22CaCa11CuCu22OO8+8+

8

Lawrence-Doniach model

Hamiltonian of LD model

8)(4

22

2

12

222

2

2

22

||||

||)|||(|

HB

nb

n

nndmyxmn

LD

a

DDdxdyHcab

Order parameter in nth layer

γt :Tunneling factor

d: interlayer spacing

(2)

9

2 2

22 2

1* 2 *, 2 2i i x i y i

i c abx y

F d D Dm d m

Criterion of applicability of GL for layered material is when coherence length in the c direction is not smaller than the interlayer spacing:

2 4

2c i iT T

( ) / ( )cT T d

finite differences can be replaced by derivatives and sums by an integral

10

5, 10

15 , ( 0) 6c

d A

A T A

YBCO

GL still OK

30, 18

25 , 4.6c

d A

A A

BSCCO

Anisotropic GL invalid

The condition is obeyed in most low Tc materials and barely in optimal doping YBCO at temperatures not very far from Tc, but generally not obeyed in BSCCO and other high Tc superconductors

11

Until now we have assumed that the system is in plane O(2) rotationally symmetric:

jiji xRx Real materials are usually not symmetric. However if the material is “just” fourfold ( ) symmetric

2/Ryx

y x R

xx

yy In YBCO there is sizable explicit O(2) ( in plane )

breaking due to the d-wave character of pairing. However asymmetry is not always related to the non s – wave nature of pairing.

4D

3. Fourfold anisotropy

12

to include effects of O(2) breaking, one has to use “small” or “irrelevant” four derivative terms.

iD

There is no quadratic in covariant derivative terms that break O(2) but preserve

4D

There are three such terms222*222222 )(,)(,)( xyyxzyx DDDDDDD

24 2 2( )grad y xF D D

but only the last breaks the O(2) and is thereby a “dangerous irrelevant”. One therefore adds the following gradient term:

Hem **' With dimensionless constant characterizing the strength of the rotational asymmetry

13

Most remarkable phenomenon structural phase transition. Body centered rectangular lattice becomes square )( 24 DD

This term leads to anisotropic shape of the vortex and an angle dependent vortex – vortex interaction leading to emergence of lattices other than hexagonal: the symmetric rhombic lattices.

Structural phase transitions in vortex lattices2D

14

Pearl’s solution for thin film

C. Vortices in thin films and layered SC

1. Pearl’s vortices in a thin film

Anti-monopole field

Magnetic monopole field

s x

yz

15

2D London’s equation inside the film, z=0

Where is the polar angle (see the derivation of the vortex Londons’ eq. in part I). Now I drop curl using Londons’ gauge

0A

2 2 ( )A A r

Where the vector field is defined by ( )r

2 202 2

1ˆ ( )B B z x

2 2 0

2A A

16

( ) ( , 0) ( ( ) ( ,0))2 eff

cj r J r z s r A r

For , and almost do not vary inside the film as function of z. The 2D supercurrent density consequently is:

Where the effective 2D penetration depth is defined by

,s J

20 ˆ

2)(

r

rzr

rr

2)( 0

A

22eff s

r

y

x

17

Since the current flows only inside the film, the Maxwell equation in the whole space is:

4 2( )[ ( ) ( )]

eff

B J z r A rc

2 2( )[ ( ) ( )]

eff

A z r A r

Two different Fourier transforms( )

, ,

3 : ( , ) ( , )i qr kz

x y z

D A q k e A r z

12 : ( ) ( , ) ( , 0)

2iqr

r

D a q A q k dk A r z e

18

Integrating over k, one gets:

2 2

2 1( , ) ( ) ( )

eff

A q k a q qq k

2 2 02

ˆ2( ) ( , ) ( ) ( ) , ( )

2eff

z qq k A q k q a q q

q

2 2

1 1 1( , ) ( ) ( ) ( )

2

1 1( ) ( )

effk

eff

A q k a q dk a q qq k

a q qq

1( ) ( )

1 eff

a q qq

The 3D equation takes a form:

19

Substituting back into eq.(*) and performing the k and the angle integrations one obtains the vector potential:

0( , 0) ( , )2 1z k

eff

iB q z q A q k

q

q

The magnetic field z component in the film is:

2 2

02 2 2 2

2 1 1( , ) 1 ( )

1

ˆ2 ( ) 1

1 1

eff eff

effeff eff

A q k qq k q

q q z q

qq k q q k q

The effective penetration depth indeed describes magnetic field scale in thin fielm

20

For example, the flux crossing the film within radius r is:

eff

r

0

0 1 eff

r

)(reffr

effr

for

for

30

2 r

r

z

zB

q

eqrJ

dqzrA

eff

zq

1)(

2),( 1

0

0

For this gives monopole field:effr

Performing the k and angle integration in the inverse 3D Fourier transform one obtains:

21

Supercurrent

q

qqcqaq

cqj

eff

eff

effeff

12

)()]()([

2)(

01 12

2( ) ( / ) ( / )

8 eff effeff

cj r H r Y r

220

20

1

4

1

4

r

c

r

c

eff

for

for

effr

effr

22

The potential energy therefore is:r

rrj

crjn

crf z

)()(ˆ)( 00

)]()([8

)( 00

20int

effeffeff

rY

rHrV

0rForce that a vortex at exerts on a vortex at

is:r

02s

effeff

effeff

Log for rr

for rr

2

00 4

where the standard unit of the line energy is used

23

Energy to create a Pearl vortex is

The film therefore behaves as a superconductor with

The two features, logarithmic interaction and finite creation energy make statistical mechanics of Pearl’s vortices subject to thermal fluctuations a very nontrivial 2D system.

02 log effE s

eff

eff

How to make a good type II superconductor from a type I material?

24

s d

“Pancake” vortex

2

//

2,

2eff

eff

d

d

Pearl’s region

2. “Pancake” vortices in layered superconductors

Two magnetic field scales

25

Fourier transform for one pancake vortex in the layer n=0

:),( ii nr

)()(ˆ

2),(

])[(

20

2

i

i

i

i

n

nnrr

rrzrn

AnszdA

London’s eqs. for a pancake vortex centered at

20 ˆˆ

2),(),(

q

qzrneekq

n

rqiiknd

rdzdrzAeekqA rqiikz ),(),(

26

)],(),([)(),( 222 kqkqAdekqkqAn

insk

),(2

1)( kqAedkqa

n

insk

2 2

2 1( , ) ( ) ( )insk

neff

A q k a q e qq k

( )2 2

,

1( ) ( , )

2

1 1[ ] ( ) [ ] ( )

insk

m

insk i n m sk

km n meff

a q dk e A q k

e a q e qq k

27

Magnetic field extends beyond the Pearl’s region:

Total flux through cillinder of height z and radius r is:

0

2 20 //// // //

1( , ) exp / exp / /

2 eff

A r z z r zr

2 20// //2 2

1( , ) exp / /

2zeff

B r z r zr z

2 20// // //2 2

1( , ) exp / exp / /

2reff

zB r z z r z

r r z

2 20 //// // //( , ) exp / exp / /

eff

r z z r z

Flux through the central layler where core is located

28

Current in the central layer

Due to squeezing of magnetic field the cutoff disappeared for all distances !

///0 0 //2

1( ) ( ,0) 1 (1 )

2 2 4r

eff eff eff

c cj r A r e

r r

In higher layers:

eff

Interaction in the same plane

2 2// /// ( ) /0 //

2 2

1( , )

4nd nd r

eff

cj r ns e e

r

int0( ) 2 / ( / )V r s Log r r O s

r

29

Energy of a single pancake vortex is logarithmically infinite in infrared: cannot be isolated.

R

dE log2~ 0

Pancake vortices in different layers also interact:

int0

//

( , 0)2

s sV r z

//

r

//

log//

r

ez

for

for

// rz

//r

30

The Ginzburg-Landau string tension is recovered in the case of straight vortices with and replacing and .

// //

0

nr

H Pancake vortices in neighbour planes attract each other.

Abrikosov flux line in layered superconductors

31

iqr

effeff

eqqsq

zqzdqzrA

20

0 )()coth(21

),(

2),(

)sinh(

)sinh()(sinh[),(

qz

qzezsqzqz

Qs

zQe2//

2 1)sinh()cosh()cosh(

qQ

q

qsqsQs

eff

…for sz 0

32

Summary

1. In thin films the field “leaks” out and the vortex envelop (effective penetration depth becomes large. The material becomes therefore more type II and interaction acquire longer range. 22

eff d

2. Layered SC (a superlattice) causes interaction between vortices (which become “pancakes”) to be truly long range logarithmic.

3. While moderately anisotropic layered SC still can be described by the anisotropic GL theory for strongly anisotropic ones Lawrence-Doniach tunneling theory should be used.