Transcript CONSERVATION LAWS AND SOLVABLE SYSTEMS

V. ANISOTROPIC and LAYERED
SUPERCONDUCTORS
A. Some phenomenology
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.
We start with the simplest case of anisotropic GL
theory neglecting layered structure.
1
Various types (old and new) of the “conventional” or
the “BCS” superconductors

Tc ( K )  ( A)


 ( A) Hc1(G) Hc2 (T ) 
Gi
0.4
1 210
 6
Pb 7.2 830 370 800 11 10 5
NbSe2 7.4 77 690 1600 15 9
K3C60 20
YtBC 15
MgB2 25
26
15
2400 130 50
1400 1400 26
90
30
1
2
25
2000 2000 90
150 70
105
106
2
Various types of
1986
High Tc Superconductors “unconventional”
(or “non BCS” SC)
Alex Muller, Georg Bednorz


Tc ( K )  ( A)  ( A) Hc1(G) Hc2 (T )
YBCO6.92 93

d ( A)

Gi
210
 3
15
1440 500
260 100 10
5
YBCO6.7 65 20
LSCCO 34 29
BSCCO 120 25
2550 300
2800 70
2500 150
150 110 12
50 130 13
150 100 18
13 0.1
15 0.1
30 0.5
UPt3 0.5 25
780
2000 2.1
25
-
0.9 10 5
3
B. Anisotropic GL and LawrenceDoniach models
1. Anisotropic GL model
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:
Fgrad 
2
Dz  
2
*
c
2m
2
*
ab
2m
D
x
2
 Dy 
2

The asymmetry factor is defined by
mc *
 * 
mab
5
for YBCO
30
for BSCCO
4
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.
Blatter et al RMP (1994)
Coherence length in the c direction is typically much
2
2
smaller

 c2 


*
2mc Tc  T  
 H
c
c2

0
2 c
  Hc2
while the corresponding
2 *
c
mc 
penetration depth is
2
2






c
larger:
4 e*2//  T  T 
c
5
H
c
c1
 c   H cc1

log 

4c 

  c 
0

Log   
1 

Log   

We don’t have to solve again the GL equations: they do
not change.
It is much easier to create vortices to be oriented in
the ab plane.
Type II:
 ~2,000 A
 ~10 A
 ~ 200 >> c
6
2. The Lawrence - Doniach model
z
y
x
Interlayer
distance d
CuO plane (layer or
bilayer)
Layer
width s
Bi2Sr2Ca1Cu2O8+d
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:
7
 Lawrence-Doniach model

Hamiltonian of LD
model
H LD     dxdy 2m (| Dx |2  | Dy |2 ) 
n
2
ab
a |  n |2  b2 |  n |4  ( B8H )
Order parameter in nth layer
2
(2)
2
2 mc d 2
|  n  n1 |2 
γt :Tunneling factor
d: interlayer spacing
8

2
 2
2
2
F   d   * 2  i   i 1  * Dx  i  D y  i
2mc d
2mab
i
x, y 
2


4
 T  Tc  i  i 
2

2
Criterion of applicability of GL for layered material is
when coherence length in the c direction is not smaller
than the interlayer spacing:
 (T ) /   c (T )  d
finite differences can be replaced by derivatives and
sums by an integral
9
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
YBCO
  5,d  10 A
  15 A, c (T  0)  6 A
GL still OK
BSCCO
  30,d  18 A
  25 A, c  4.6 A
Anisotropic GL
invalid
10
3. Fourfold anisotropy
Until now we have assumed that the system is in plane
O(2) rotationally symmetric:
xi  Rij x j
Real materials are usually not symmetric. However
if the material is “just” fourfold ( D4 ) symmetric
R / 2 
x y
y  x
R
x  x
y y
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.
11
There is no quadratic in covariant derivative
terms that break O(2) but preserve D 
4
to include effects of O(2) breaking, one has to use
“small” or “irrelevant” four derivative terms.
Di
There are three such terms
2
( D  D ) , D  ( D  D ) , ( D  D )
2
x
2
y
2
z
2
x
2
y
*
2
y
2
x
2
but only the last breaks the O(2) and is thereby a
“dangerous irrelevant”. One therefore adds the
following gradient term:
2
4
2
2
Fgrad   ( D y  Dx )
 '  m * e * H
With dimensionless constant
characterizing the strength of the rotational asymmetry
12
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 D2 symmetric rhombic lattices.
Structural phase transitions in vortex lattices
Most remarkable phenomenon structural phase
transition. Body centered rectangular lattice
becomes square ( D4  D2 )
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C. Vortices in thin films and layered SC
1. Pearl’s vortices in a thin film
Magnetic monopole field
z
s
Anti-monopole field
y
x
Pearl’s solution for
thin film
14
2D London’s equation inside the film, z=0
0
1
2
 B  2 B   2 zˆd 2 ( x )


0
 A   A 
  
2
2
2
Where  is the polar angle (see the derivation of the
vortex Londons’ eq. in part
 I). Now I drop curl
using Londons’ gauge A  0
A    A   (r )
2
2
Where the vector field  ( r ) is defined by
15




 0 zˆ  r
 (r ) 

2
y
2

r

 
r
0
x
 (r ) 
2r

For s   ,, J and A almost do not vary inside
the film as function of z. The 2D supercurrent density
consequently is:
j ( r )  J ( r , z  0)s 
c
2eff
( ( r )  A( r , 0))
Where the effective 2D penetration depth is defined
2
by
2
eff 
s
 
16
Since the current flows only inside the film, the
Maxwell equation in the whole space is:
4
2
 B 
J 
d ( z )[ (r )  A(r )]
c
eff
2
2
 A
d ( z )[ ( r )  A( r )]
eff
Two different Fourier transforms
3D : A(q, k ) 

ei ( qr  kz ) A( r , z )
x, y, z
1
iqr
2 D :a (q ) 
A
(
q
,
k
)
dk

A
(
r
,
z

0)
e
r
2 
17
The 3D equation takes a form:
(q  k ) A(q, k ) 
2
2
A(q , k ) 
2
eff
2
eff
0 zˆ  q
 (q )  a (q )  , (q ) 


2 q2
1
a (q )   (q ) 
2
2 

q k
Integrating over k, one gets:


1
a (q )   (q ) 
dk


2
2
k



eff  q  k

1 1
 a (q )   (q ) 


eff q 
1
2
A( q , k )  a ( q ) 
1
1
a (q )   (q )
1  qeff
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
q
Substituting back into eq.(*) and performing the k and
the angle integrations one obtains the vector potential:
A( q , k ) 
2
eff
1
q2  k 2
 1

 1  (q )

1  qeff

2q ( q )
 0 zˆ  q
1
  2

2
2
2


q
q

k
1

q

q

k



 1  qeff 
eff
eff 
The magnetic field z component in the film is:
i
0
Bz (q, z  0)   q  A(q, k ) 
2 k
1  eff q
The effective penetration depth indeed describes
magnetic field scale in thin fielm
19
For example, the flux crossing the film within radius
r is:
r
for r  eff
0
(r ) 
eff
eff  for r  

eff
 0 1 

r 

Performing the k and angle integration in the inverse
3D Fourier transform one obtains:

A (r , z )   0 
For r 
0
q z
dq
e
J1 (qr)
2
1  eff q
effthis gives monopole field:

0
B
2

z r
3
z r
20
Supercurrent

j (q ) 
c
2eff
 
 
 
c (q ) eff q
[ (q )  a (q )] 
2eff 1  eff q
c0 
2
j ( r) 
H1 ( r / eff )  Y1 ( r / eff )  
2 
8eff 

c 0
4 eff
2

c 0 1
4 2 r 2
1
r
for
for
  r  eff
r  eff
21


Force that a vortex at r  0 exerts on a vortex at r
is:


0
0
r
ˆ z  j (r ) 
f (r ) 
n
j ( r )
c
c
r
The potential energy therefore is:
2
0
r
r
int
V (r ) 
[H0 (
)  Y0 (
)]
8eff
eff
eff
eff

Log
 forr  eff


r
 2s 0
 eff  forr  
eff

 r
where the standard unit of
the line energy is used
 0 
0  

 4 
2
22
How to make a good type II superconductor
from a type I material?
Energy to create a Pearl vortex is
eff
E  2s 0 log

The film therefore behaves as a superconductor with
 eff
eff

 

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.
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2. “Pancake” vortices in layered superconductors
“Pancake” vortex
s  d  
Pearl’s region
Two magnetic field scales
// 
d eff
2
,eff
2 2

d
24
London’s eqs. for a pancake
 vortex centered at
(ri , ni ) :





     A   dd ( z  n s)[ A   ]
2
n


 ( n, r ) 

i


ˆ  ( r  ri )
0 z
d ( n  ni )


2
2
r  ri
Fourier transform for one pancake vortex in the layer
n=0
 
 

 0 zˆ  qˆ
iknd iqr
 (q , k )   e
e  (n, r ) 
2
2

q
n
 
 


 ikz  iqr
A( q , k )   e
e
A( z , r ) dzd r
25
 
 
 
2
2
2
insk
A(q, k ) (q  k )   de [ A(q, k )   (q, k )]

 
n
1
insk
a (q ) 
dk e A(q, k )

2
n
 1
2 

insk 
A(q, k )  
 a( q )    e   ( q )  2
2
eff 
q

k
 n


1
a(q )   dk  einsk A(q , k ) 
2
m

1
1 
insk
i ( n  m ) sk

[ e ]a (q )  [ e
] (q ) 

2
2

eff k q  k  m
n,m

26


0// 1
2
2

A (r, z ) 
exp   z / //   exp   r / //    z / // 


2eff r
Magnetic field extends beyond the Pearl’s region:
0
1
2
2

Bz ( r, z ) 
exp   r / //    z / // 
2
2


2eff r  z
0 1 
z
2
2 

Br (r, z ) 
exp   z / //   2 2 exp   r / //    z / //   


2eff r 
r z
Total flux through cillinder of height z and radius r is:


0//
2
2

(r, z) 
exp   z / //   exp   r / //    z / // 


eff
0
Flux through the central layler where core is located 
27
Current in the central layer

c  0
c0 1  //

 r / //
j ( r ) 
 A ( r,0)   2
(1  e
)
1 

2eff  2 r
 4 eff r  eff

Interaction in the same plane
V (r)  2s 0  Log  r / r   O(s / )
int
Due to squeezing of magnetic field the cutoff
disappeared for all distances   r !
eff
In higher layers:
c0// 1   nd / //

j (r, ns)   2 2 e
e
4 eff r 
( nd )2  r 2 / //


28
Pancake vortices in different layers also interact:
r
 //
s s
V (r, z  0)   0
2 //
int

e
z  r  //
for
z
//
log
r
//
for
r  //
Energy of a single pancake vortex is logarithmically
infinite in infrared: cannot be isolated.
E ~ 2d 0 log
R

29
Abrikosov flux line in layered
superconductors
 
rn  0
H
Pancake vortices in
neighbour planes
attract each other.
The Ginzburg-Landau string tension is recovered in the
case of straight vortices with // and //
replacing  and 
.
30

0
z ( q, z )
iqr
e
2
2

1

2
q

cot
h(
qs
)

(
q

)
eff
eff
0
sinh[q( s  z )  e Qs sinh(qz)
z (q, z ) 
for 0  z  s
sinh(qz)
…
A (r , z )   dq
e
Q z
cosh(Qs)  cosh(qs) 
sinh(qs)
Q
qeff
q2 
1
2//
31
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.
2 2
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.
32