III. VORTICES and THEIR INTERACTIONS in
LONDON APPROXIMATION
A. The isolated vortex solution
1. GL equations in a rotationally invariant
situation
Straight vortex line has
symmetries z-translations + xy
rotations. A clever choice of gauge
should utilize these symmetries.
Since physical quantities depend
on the distance from the center r
only the cylindrical (polar)
coordinates is the natural choice.
1
Using polar coordinates one chooses the following
Ansatz (which includes a choice of the “unitary” gauge):
  0 f (r )ei

ˆ
 A  A(r )
azimuthal
vector field
1 d
B(r ) 
(rA)
r dr
r
1
A(r)   r 'dr ' B(r ')
r0


A

J
tangential vector
field
2
Details: polar coordinates
r
x  r cos 
y  r sin 
The vector potential
x2  y2
y
  arctg
x
Ax   A(r ) sin 
Ay  A( r ) cos 
Partial derivatives

 
r 
y



 2

2
x
x  x r
x  y 

x 2  y 2 r
x
1


  sin 
 cos 
r

r
3
 1


 cos 
 sin 
y r

r
Magnetic field


1


B
Ay 
Ax   sin 
( A cos )  cos ( A cos )
x
y
r

r
1



  cos
(  A sin  )  sin  (  A sin  ) 

r
r

1
1
 cos2   A  cos2  A ' sin 2  A  sin 2   A'
r
r
1
1 d
 A  A ' 
( A  r)
r
r dr
B is indeed a function of r only
4
Supercurrent
2
ie *
e
*
2
*
*
Jx  
(   x    x  ) 
Ax 
2m *
2m * c

ie *
1

 
2 
 i 
i
 
 0  fe   sin 
 cos    fe  c.c.
2m *

r 
 r


e *2
2

A    sin    0 f 2
2m * c
2
e*
1
e
*
2

 0 2 2 f 2   sin   
A    sin    0 f 2
2m *
r
2m * c
e*
2 A 
2 2 1

0 f   
   sin  
m*
 r 0 
Current therefore flows around the vortex.
5
GL equations
Supercurrent equation has the azimuthal component
only
c dB
c d 1 d
 e * 2 2  1 2 A 
J 

(rA)  
f 0  


4 dr
4 dr  r dr
 m*
 r 0 
Similarly the nonlinear Schroedinger equation takes a
form
 1 2 A  2

1
d
df


f  f 3   2  
r
  0
 f 
r



0 
r dr 
dr  

This should be supplemented by a set of four
boundary conditions at the center and far away.
6
2. Boundary condition and asymptotics
near the vortex core.
Near the center one expects a
maximum of magnetic field
B(0) leading to linear
potential:
A
B ( 0)
Anear ( r ) 
r
2
r
Asymptotics of the order parameter at r  0
is assumed to be a power
f near (r )  cr , m  0
m
7
Substituting this single vortex Ansatz into the NLSE
one obtains:
 1  B(0) 2

cr m  c3r 3m   2  
r  cr m  m2cr m2 
0
 r


Leading terms are two:


1  m2 r m2  m  1
The order parameter therefore vanishes at the center of
the vortex core as r for a single fluxon vortex.
Near r=0, we can use an expansion in r.
8
3. Boundary conditions outside the core.
Numerical solution
Far away flux quantization gives
0
hc
B ( r )  0  A far ( r ) 
, 0  * .
2 r
e
The order parameter therefore exponentially approaches
its bulk value in SC
f far (r)  1  conste
r /
Using the four boundary conditions and linearity of
both A and f at origin one can effectively use the
“shooting” method to find the vortex solution
9
Exercise 2: transform the GL equations for a single
vortex into a dimensionless form and solve it
numerically using the shooting method for k1,10.
A good simple fit for order parameter all r is
available:
f (r )  tanh
r

A simple expression for the magnetic field distribution
can be obtained in phenomenologically important case
of strongly type II superconductors using the London
approximation
10
4. The London electrodynamics outside vortex
cores. Magnetic field of a vortex fork  1
Far enough from the vortex
cores one generally makes the
London appr. (even for many
vortices)
  x, y    0e
i  x , y 
Covariant derivative
e*
 

Dx   
i
Ax  
c
 x

e *  i  x , y 

 i 0   ( x, y ) 
Ax  e
c
 x

11
Supercurrent and Londons’ eqs
In this case the supercurrent equation takes a London
form:
ie *
J 
( D *   * D )
2m *
e*
e* 
2 
 i
0   
A   ( x, y )
m*
c 

2
e*
e
*
 i
 0 2 ( x, y ) 
 02 A
m*
m*c
Taking 2D curl of the Maxwell equation


4 
 B 
J
c
12
one obtains for a single vortex phase field:
4
 B
 J 
c
4 c 2
4 e *

    ( x ) 
02 B
e*
m*c
0
4 e *
2
ˆ
 z 2  ( x ) 
0 B

m*c
2
This is transformed into Londons’ equations in the
presence of a straight vortex:
 B
2
1
2
B
0
2
zˆ 2 ( x )
13
Field of a single vortex
The eqs. are mathematically identical to the those for
the Green’s function of the Klein-Gordon eqs and
therefore can be solved by Fourier transform.
 B(k )  eikx B( x )d 2 x



1
 ikx
2
B
(
x
)

e
B
(
k
)
d
k

2 
(2 )

  k  
2
2
0
 B (k )   2

14
B( k ) 
0
 2  k 2   2 
which has a pole. Inverse Fourier transform therefore
will fall off exponentially:
0
B( x) 
(2 ) 2
0

(2 ) 2

e  ikx
k 2   2
2


0
0
k e  ikr cos
r 
d  dk 2

K0  
2
2
k 
2

0
where K 0 is the Hankel function
15
The core cutoff

B (r ) 
0
log k
2
2

 0 2 Log ( r /  )
2
0    


2 2  2 r 
Exponential tail

r 
  r  
1/ 2
e r / 
r  
Most of the flux for k1 passes
log k
2 0
 0
through the r region. The core 
2
2
2

2
k
region fraction is insignificant:
16
The supercurrent distribution
Taking a derivative the supercurrent is calculated
r 
~ 0

c dB  e * 1
J (r )  e * vs (r ) 

  r  
4 dr  m * r
r / 


e
r  

One observes a rather long range decrease of the
supercurrent between the coherence length and the
penetration depth distances.
17
5. Vortex carrying multiple flux quanta
   0 f ( r )ein
Then in the Laplacian we
will have to replace
1
n2

r
r
and asymptotics at r=0 changes to:
f～r n
Core is much larger. As a result these vortices have
larger energy and are difficult to find.
18
E. The line energy and interaction
between vortices
1. Line Energy for k
1
The vortex line energy density  is defined as the
Gibbs energy of vortex solution minus g s  gn   g0 .
Neglecting the core and the condensation energy, we
have:
 

out of
coreD
C1
C2
[ f grad  g magn ]d x
2
D
19


*
In the London
1  2 m
2
2 2


J

B
d x
limit (  0 )
s
2

*
8



D  e 0 
cov. gradient is


proportional to
1
2
2
2
2
supercurrent:

[

(


B
)

B
]
d
x

8 D
This replaces the Maxwell energy. Integration by parts
gives
1
2
2

B
[
B




(


B
)]
d
x

8 D
2

8

C1 C2


 z  B  (  B )  dS


20
Using the Londons equation
 B
2
1
2
B
0
2
zˆ 2 ( x )
One sees that the bulk integral vanishes and the inner
boundary gives
2
1

B


2
2

B 0 ( x )d x 
B( r )
2 r 


8 D
8 
r
 r 
To calculate the
derivative one uses
magnetic field in the
intermediate region
dB
0 1
0


|
2
dr 2 r r  2 2
21
0
2
1

B ( )2

 0 B(0) 
2
8
2  8
2
    0  2

 0 
 
Log    
  Log  

 4 
    4 
 
Hc2

4 2 Log (k )  4 2 g 0 Log (k )   0 Log (k )
8
Consistency check: contribution of the core to energy
is indeed small for k1,but just logarithmically
2

g0 thecore area    g0
2
22
2. Interaction between two straight vortices
Consider two parallel straight vortices
x
x1
r  x2  x1
x2
The London equation is linear in magnetic field.

Therefore within range of its validity r  
B ( x )  Bv  x  x1   Bv  x  x2

23
The interaction line energy (potential) between two
straight vortices is defined by
F  F ( x1 , x2 )  F ( x1 )  F ( x2 )
Neglecting cores and using the trick of integration by
part as before one obtains from the London equation
with two sources
1
2
F ( x1 , x2 ) 

B
(
x
)[

(
x

x
)


(
x

x
)]
d
x
0
1
2

2 D
2
 [   z  B  (  B)   dS

8 C C 

1

2
24
To estimate the multiple
internal boundary
contribution, we first
approximate the
derivatives
C1
C2
D
  B |C1     B ( x1 )  B ( x2 )  |C1
0  1 1 

  C1
2 
2   r 
Since we will always (while using Londons appr.)
assume r>> the last term which is Powerwise
small in 1/k will be dropped
25
The two solitons energy is therefore proportional to
magnetic field
0 2
F12 
[ B( x1 )  B( x2 )]
2
8 
0
 [ Bv ( x1  x1 )  Bv ( x1  x2 )  Bv ( x2  x1 )  Bv ( x2  x2 )]
8
The interaction energy is
0
 02
F 
Bv (r ) 
K0  r /   =  0 K0  r /  
2 2
4
8 
26
Force per unit length:
1
 ,
r
d
Force  
F 
dr
x1 ( )
x1 ( )
  r  
  


2  2 r 
1/ 2
x2x(() )
2
e r /  ,
r  
Parallel vortices repel, antiparallel attract, however the
picture is more complicated
than that: the force between
curved vortices is of the
vector-vector type
27
3. Vortices as line - like objects
Curved Abrikosov vortices in London approximation
are infinitely thin elastic lines x ( ) with interaction
energy
Eint


d
x

d
x
1
2
0
d x1  d x 2  x1  x2 / 
 x1  x2 /  


e

e
 x1  x2

2 
x1  x2
1 , 2


Brandt, JLTP (1991)
Interaction is therefore mainly magnetic, hence pair wise (superposition principle).
28
4. Lorentz force of a current on the fluxon.
Magnetic field affects current (moving charges) via
the Lorentz force




J
ˆ
  B  0


f



f L  J ( x) 
c
Current consequently applies a force in the opposite
direction on fluxon due to Newton’s 3rd law.
29


In particular, force of vortex at x1 on vortex at x2
can be written as:


 

f 12  J 1 ( x 2 ) 
c
0

x1
FL
The same logic leads
to repultion between
a vortex and an
antivortex pointing
to a vector – vector
type of interaction
J
FL
JV
Ao, Thoules, PRL 70, 2159 (93)
30
5. Flux flow and dissipation.
The Lorentz force on vortices
which causes their motion is
balanced in the stationary flow
state by the friction force due
to gapless excitations in the
vortex cores. The vortex mass
is negligible.
Phenomenologically the
friction force is described (in
2D) by:
d
f dissipation   x  v
dt
31
The overdamped dynamics results in motion of vortices
with a constant velocity
0 JB

f dissipation  v  f L  J  v 
c
c
across the boundary of length L. It produces the flux
change
B
   vtL 
 0  vLBt
0
Leading, using Maxwell eqs., to the voltage
0
1  1
v
V
 vLB E  B  2 J B
c t
c
c
c 
32
which in turn
implies a finite flux
flow Ohmic
resistivity
0
r 2 B
c 
Unless some other force like pinning obstructs the
motion, the SC loses its second “defining” property:
zero conductivity
33
The phenomenological Bardeen – Stephen model
Let us assume that the dissipation which happens
mainly in the normal cores is the same as in normal
metal with resistivity r n . The fraction of area covered
by the cores is proportional to B:
B
 2   B /  0   H
c2
2
The resistivity therefore is the same fraction of the
normal state resistivity
B
r
rn
H c2
34
When the magnetic field reaches H c 2the cores cover the
whole area and one is supposed to recover the whole
normal state conductivity. This fixes the coefficient.
Now the friction constant can be estimated:
0B
B
0H c2
r 2 
rn    2
c  H c 2
c rn 
We will return to this later using the time dependent GL
eqs.
How fast vortices can move?
Within the Bardeen – Stephen
model the vortex velocity is
cr n
v
J
H c 2
35
For the Nb films
r n  105 A / cm2
H c 2  5T
J d  3106 A / cm 2
J c  105 A / cm 2
One gets velocities of 20m/sec and 600m/sec
for the critical and the depaitring current values of J
respectively
36
For YBCO film
r n  2  104 A / cm 2 ; H c 2  100T
J d  108 A / cm 2 ; J c  104 A / cm 2
One gets velocities of 20m/sec and 200km/sec.
Boltz et al (2003)
37
6. Simulation of vortex arrays
Given all the forces one can simulate the vortex system
using Runge – Kutta … When random disorder or
thermal fluctuations are important they are introduces
via random potential or force respectively (the
Langeven method). The problem becomes that of
mechanics of points or lines.


 x a  J ( xa )    0 K0  xa  xb   U pin  xa    a
c
Here the gaussian (usually white noice) Langeven
random force represents thermal fluctuations
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 ai  bj  T  ij ab
Pinning force is assumed to be well represented by a
gaussian random pinning potential with certain
correlator:
U pin  x  U pin  y   k  x  y 
Fangohr et al (2001)
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Some sample results in 2D
The I-V curves at different
temperatures. Critical current
Dynamical phase diagram
in 2D
Koshelev (1994)
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Structure functions and hexatic
order
Fangohr et al (2001)
Hellerquist et al (1996)
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Summary
1. An isolated Abrikosov vortex carries in most cases
one unit of magnetic flux. It has a normal core of
radius x and the SC magnetic “envelope” of the
size  carrying a vortex of supercurrent.
2. It has a small inertial mass and the creation energy
(chemical potential) 
3. Parallel vortices repel each others, while curved ones
interact via direction dependent vector force.
4. Interact with electric current in the mixed state. The
current might induce the flux flow with finite resistance.
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Details: Singular functions
x
The polar angle function  ( x, y )  arctg
y
at the origin x=y=0 and has a “mild” singularity- a cut at
y=0.
For singular functions generally ( x  y   y  x )  0
In particular
 iji j  2 ( x )
(2)
To prove this let us
take integral d 2 x
over arbitrary circle.

F
R C
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
area
or

d 2 x ij  i F j 
Fi ( x )dxi
circumference
2
d
 x  F 
F
A
C
for function
dx
Stokes Theorem
F   
Using derivatives formula in polar coordinates one
finds that the line integral is:
2  0
This is true for any
R0
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