I. OVERVIEW of the VORTEX PHYSICS
A. General phenomenological approach to
second order phase transitions
1. The order parameter field and spontaneous
symmetry breaking
A second order phase transition is generally well
described phenomenologically if one identifies:
a). The order parameter field
i (x)
b) Symmetry group G and its spontaneous breaking
pattern.
L.D. Landau ( 1937 )
1
Rest of degrees of freedom are “irrelevant” sufficiently
close to the critical temperature Tc. Later, the “relevant”
part, namely the symmetry breaking pattern and
dimensionality was termed “the universality class”.
An example: XY- (anti) ferromagnet
Consider in-plane (planar) classical spins of fixed
length

S 1
defined on the D-dimensional lattice (the type of
lattice and other micriscopic details are also
irrelevant).
2
1. T=0
well ordered
2. 0<T<Tc
ordered


M  S 0
large

M 0
small
3. T>Tc
disordered


M  S 0
3
a. Order parameter : magnetization.

M  (M x , M y )
or, using complex numbers,
  M x  iM y
b . Symmetry : 2D rotations
M x  cos  M x  sin  M y  M x '
M y   sin  M x  cos  M y  M y '
4

Using complex numbers the
symmetry transformation
becomes U(1):
i
 e 
Symmetry means that the energy of the rotated state
is the same as that of the original (not rotated) one.


F (M )  F (M ' )
5
2. Effective free energy near the phase transition
Most general functional symmetric under
  ei
and space rotations, with lowest possible powers of

and lowest number of gradients  is

b(T )

2
F   d x  *   a (T ) *  
( *  ) 
2


D
Higher order terms
 *   , ( * )3 ,....
2
are expected to be smaller close enough to Tc.
6
The remaining coefficients can be expanded around
Tc:
a(T )   (T  Tc )  ...,
b(T )    ...
F
F

T  Tc

T  Tc
Now we apply this general considerations to the
superconductor – normal metal phase transition.
7
B. Ginzburg – Landau description of the
SC-normal transition
1. Symmetry and order parameter in terms of
the microscopic degrees of freedom
a. Order parameter
The complex order parameter is “amplitude of the
Cooper pair center of mass”:
( x )  ( x ),(k )  c  k  c  k 
which is “the gap function” of BCS or any other (no
matter how “unconventional”) microscopic theory .
8
We disregard the quantum fluctuations of the
“bound” state (Cooper pair) and therefore consider
the “Bose condensate” amplitude as a classical field.
The symmetry content of this complex field can be
better specified via modulus and phase:
i ( x )
( x)  ns *( x) e
ns ( x)
ns *( x) 
2
 ( x)
density of the Cooper pairs,
the Bose condensate
the superconductor (the
Josephson, the global U(1) )
phase
9
b. Symmetry
The broken symmetry is charge U(1) mathematically
the same symmetry as that of the XY magnet.
Without external magnetic field the free energy near
transition therefore is:
2


3
2
F [ ]   d x 
 *    (T  Tc ) *   ( *  ) 
2
 2m *

m*  2me
Ginzburg and Landau (1950) postulated a reasonable
way togeneralize this to the case of arbitrary magnetic
field B (x ) :
One is using the “principle” of local gauge invariance
of electrodynamics.
10
2. Influence of magnetic field.
Electrodynamics is invariant under local gauge
transformations:
  ( x )  ei  ( x )  ( x)

e*  2e

c
 ( x)
 A( x)  A( x) 
e*

 x 
11
This invariance although not a symmetry (only the
“global” part of it is) dictates the charge fields
coupling to magnetic field.
To ensure local gauge invariance one makes the
“minimal substitution”, namely replaces any
derivative by a covariant derivative:
e* 

D ( x )     i
A  ( x )
c 

The local gauge invariance of the gradient term
follows from linearity of the transformation of the
covariant derivative:
12

ie * 
c

i ( x )
D  ( x )   
A



(
x
)

(
x
)
e


c
e
*



ie *

 i ( x )
     i 
A    i    e
c


 D  ei ( x )
| D( x) |2 | D( x) |2
Magnetic field
  
B    A  Bi   ijk  j Ak
is also gauge invariant.
13
Ginzburg – Landau equations
Minimizing the free energy with covariant derivatives
one arrives at the set of GL equations: the nonlinear
Schrödinger equation (variation with respect to )
2
e* 
2


A     (T  Tc )     0
  i
2m * 
c 
2
and the supercurrent equation (variation of A):
2
c
ie *
e
*
2
*
*
  B  Js  
(    ) 
 A
4
2m *
m*c
14
3. Two characteristic scales
GL equations possess two scales. Coherence length

characterizes variations of  (x) , while the
penetration depth l characterizes variations of B (x )
 (T ) 
2m * (Tc  T )
,
c
m* 
l (T ) 
e * 4 Tc  T 
Both diverge at T=Tc.
15
Ginzburg – Landau parameter
The only dimensionless parameter is the ratio of the two
lengths which is temperature independent:
l (T ) m * c 


 (T ) e * 2
Properties of solutions crucially depend on the GL
parameter. When
1
 
2
(the type II superconductivity ) there exist
“topologically nontrivial” solutions – the Abrikosov
vortices.
Abrikosov (1957)
16
C. Abrikosov Vortices
1. Why look for an asymmetric “singular”
solution?
Normally one doesn’t look for inhomogeneous solutions
in a homogeneous physical situation. Also one prefers to
consider smooth regular one rather than singular
solutions of field equations. Examples: Maxwell,
Schrödinger eqs. etc. However the type II
superconductor case is very special. Homogeneous
external magnetic field does penetrate a sample as an
array.
One has to look for these solutions due to combination of
four facts. Two crucial and two more technical.
17
a. Interface energy is negative for type II
superconductors, while positive for the type I.
Mixed state under applied magnetic field
H
H
Type I :
Type II:
Minimal area of
domain walls.
Maximal area of
domain walls.
18
b. Flux quantization.
Division into domains stops due to this.
To minimize the potential term
far from isolated vortex (where      (Tc  T )
0
B=0 ), one has to optimize the

modulus of the order parameter:
The phase however is free to vary. In order to
minimize the (positive) gradient term, one demands:
D  x    0 De
i ( x )
e *  i ( x )

 0    i
A e
0
c 

e*
 
A
c
Fgrad  0
19
c
   ds A 
e*
C
C


ds

n

0

C
ninteger
hc hc
0 

e * 2e
c. n=1 is energetically favored over n>2
d. The normal core   0 region shrinks to a
point.
20
2. Shape of the vortex solution
B
  0
 l

J
Vortex – a linear
topological defect.
The “singularity”
line   0
r

core
l
21
Abrikosov vortices in type II superconductors as seen
by electron beam tomography.
KT pair
Tonomura et al
PRL66,2519 (1993)
Tonomura et al
PRB43,7631 (1991)
22
D. Overview of properties of vortices and
systems of vortices (vortex matter)
1. Inter-vortex repulsion and the Abrikosov
flux line lattice
Line energy
To create a vortex, one has to provide energy per unit
length ( line tension )
l
 0 
 
 log  
 4l 
 
Therefore vortices enter an infinite sample only when
field exceeds certain value
2
23
Interactions between vortices
They interact with each other via a complicated vectorvector force. Parallel straight vortices repel each other
forming highly ordered structures like flux line lattice
(as seen by STM and neutron scattering).
S.R.Park et al
(2000)
Pan et al
(2002)
24
Two critical fields
As a result the phase diagram of type II SC is richer
than that of the two-phase type I
H c1 
H
Hc2
Normal
0
4l 2
first vortex
penetrates.
 0 cores overlap
Hc2 
2 2
Mixed
Hc1
T
Meissner
H c 2 2l 2
 2  2 2  1
H c1 
Tc
25
Two theoretical approaches to the mixed state
Just below H c 2 vortex cores almost overlap.
Instead of lines one just sees array of
superconducting “islands”
Hc2
Mixed
Lowest Landau level
appr. for constant B
Hc1
Meissner
Just above H c1vortices are well
separated and have very thin cores
Tc
London appr. for
infinitely thin lines
26
2. Thermal fluctuations and the vortex liquid
In high Tc SC due to higher Tc, smaller  and high
anisotropy thermal fluctuations are not negligible.
Thermally induced vibrations of the flux lattice can
melt it into a “vortex liquid”.
The phase diagram becomes
more complicated.
H
Hc2
FLL
Normal
Vortex
liquid
H c1
Meissner
Tc
T
27
First order melting of the Abrikosov lattice
Magnetization
Zeldov et al
Nature (1995)
Specific heat
Schilling et al
Nature (1996,2001)
28
Metastable states: zero field cooled and field
cooled protocols result in different states.
Neutron scatering in Nb
Ling et al (2000)
29
Vortex “cutting” and entanglement
Vortices can entangle around each other like polymers,
however due to vectorial nature of their interaction
they can also “disentangle” or “cut each other”.
There are therefore profound differences compared
to the physics of polymers
30
3. Disorder and the vortex glass
point
Columnar
Vortices are typically pinned by disorder. For vortex
systems pinning create a glassy state or viscous
entangled liquid. In the glass phase material becomes
superconducting (zero resistance) below certain
critical current Jc.
31
Imperfections act as pinning centers of vortices
STM of both
the pinning
centers (top)
and vortices
(bottom)
Pan e al
PRL (2000)
Disappearance of Bragg
peaks as the disorder
increases
Gammel et al PRL (1998)
32
4. Vortex dynamics
Vortices move under influence of external current (due
to the Lorentz force).
Field driven flux motion
probed by STM
on NbSe2
A.M.Troianovski (2004)
The motion is generally friction dominated. Energy is
dissipated in the vortex core which is just a normal
metal. The resistivity of the flux flow is no longer zero.
33
Vortex loops, KT pairs and avalanches
Current produces expanding vortex loops even in the
Meissner phase leading to non-ohmic “broadening” of
I-V curves
J ext
V
 const / T I 
  e
I
In 2D thermal fluctuations generate a curious
Kosterlitz – Thouless vortex plasma exhibiting many
unique features well understood theoretically
34
Unstable normal domain under homogeneous
quench splits into vortex-antivortex (KT) plasma
Kirtley,Tsuei and
Tafuri (2003)
Polturak, Maniv (2004)
Scanning SQUID magnetometer
35
Spontaneous flux in rings
Kirtley,Tsuei
and Tafuri
(2003)
36
Vortex front propagation is normally shock wave like,
but occasionally creates avalanches
after
50 ns
after
10 s
Boltz et al (2003)
Magneto-optics in YBCO
films, 10K, B=30mT, size
2.3x1.5 mm
37
5. Vortex dynamics in the presence of disorder
Disorder profoundly affects dynamics leading to the
truly superconducting vortex glass state in which
exhibits irreversible and memory dependent
phenomena (like aging).
Magneto-optics in Nb
Johansson et al (2004)
It became perhaps the most convenient playground to
study the glass dynamics
38
Dependence on magnetic history: the field cooled and the
field cooled with return protocols result in different
states.
Transport in Nb
Reversible
region
Voltage (V)
30
B
30
FCW
FCW
FCW
4.2K
80%
20
20
10
10
4.90 K
FC
0
0.01
FC
Irreversible
region
0.1
Time(ms)
1
FC
0
0.01
0.1
6.30 K
T
Andrei et al (2004)
1
Time(ms)
39
Summary
1. In extreme type II superconductors the “topological”
vortex degrees of freedom dominate most of the
macroscopic magnetic and transport properties.
2. One can try to use the GL theory to describe these
degrees of freedom.
3. Experiments suggest that in new high Tc SC thermal
fluctuations are important as well as disorder.
4. The vortex matter physics is quite unique, well
controlled experimentally and may serve as a
“laboratory” to test a great variety of theoretical
ideas.
40