I' OVERVIEW of the VORTEX PHYSICS

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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 b) Symmetry group
G and its spontaneous breaking pattern.
L.D. Landau ( 1937 )
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An example XY- (anti) ferromagnet
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.
Consider in-plane (planar) classical spins of
fixed length
defined on the D-dimensional lattice (the type of
lattice and other micriscopic details are also
irrelevant).
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1. T0 well ordered
large
2. 0ltTltTc ordered
small
3. TgtTc disordered
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a. Order parameter magnetization.
or, using complex numbers,
b . Symmetry 2D rotations
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Using complex numbers the symmetry transformation
becomes U(1)
Symmetry means that the energy of the rotated
state is the same as that of the original (not
rotated) one.
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2. Effective free energy near the phase transition
Most general functional symmetric under and space
rotations, with lowest possible powers of
and lowest number of gradients is
Higher order terms

are expected to be smaller close enough to Tc.
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The remaining coefficients can be expanded around
Tc
Now we apply this general considerations to the
superconductor normal metal phase transition.
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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
which is the gap function of BCS or any other
(no matter how unconventional) microscopic
theory .
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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
density of the Cooper pairs, the Bose condensate
the superconductor (the Josephson, the global
U(1) ) phase
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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
Ginzburg and Landau (1950) postulated a
reasonable way to generalize this to the case of
arbitrary magnetic field
One is using the principle of local gauge
invariance of electrodynamics.
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2. Influence of magnetic field.
Electrodynamics is invariant under local gauge
transformations
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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
The local gauge invariance of the gradient term
follows from linearity of the transformation of
the covariant derivative
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Magnetic field
is also gauge invariant.
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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 Y)
and the supercurrent equation (variation of A)
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3. Two characteristic scales
GL equations possess two scales. Coherence length
characterizes variations of , while
the penetration depth l characterizes variations
of
Both diverge at TTc.
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Ginzburg Landau parameter
The only dimensionless parameter is the ratio of
the two lengths which is temperature independent
Properties of solutions crucially depend on the
GL parameter. When


(the type II superconductivity ) there exist
topologically nontrivial solutions the
Abrikosov vortices.
Abrikosov (1957)
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1. Why look for an asymmetric singular
solution?
C. Abrikosov Vortices
Normally one doesnt 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.
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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 Minimal area of domain walls.
Type II Maximal area of domain walls.
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b. Flux quantization.
Division into domains stops due to this.
To minimize the potential term far from isolated
vortex (where B0 ), 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
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C
c. n1 is energetically favored over ngt2 d. The
normal core region shrinks to a
point.
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2. Shape of the vortex solution
Vortex a linear topological defect.
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Abrikosov vortices in type II superconductors as
seen by electron beam tomography.
KT pair
Tonomura et al PRB43,7631 (1991)
Tonomura et al PRL66,2519 (1993)
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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 )
Therefore vortices enter an infinite sample
only when field exceeds certain value
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Interactions between vortices
They interact with each other via a complicated
vector-vector 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)
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Two critical fields

As a result the phase diagram of type II SC is
richer than that of the two-phase type I
first vortex penetrates.
cores overlap
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Two theoretical approaches to the mixed state

Just below vortex cores almost overlap.
Instead of lines one just sees array of
superconducting islands
Lowest Landau level appr. for constant B
Just above vortices are well separated and
have very thin cores
London appr. for infinitely thin lines
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2. Thermal fluctuations and the vortex liquid
In high Tc SC due to higher Tc, smaller x 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.
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First order melting of the Abrikosov lattice
Magnetization
Specific heat


Schilling et al Nature (1996,2001)
Zeldov et al Nature (1995)
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Metastable states zero field cooled and field
cooled protocols result in different states.
Neutron scatering in Nb Ling et al (2000)
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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
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3. Disorder and the vortex glass
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.
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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)
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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.
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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
In 2D thermal fluctuations generate a curious
Kosterlitz Thouless vortex plasma exhibiting
many unique features well understood theoretically
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Unstable normal domain under homogeneous quench
splits into vortex-antivortex (KT) plasma
Kirtley,Tsuei and Tafuri (2003)
Polturak, Maniv (2004)
Scanning SQUID magnetometer
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Spontaneous flux in rings
Kirtley,Tsuei and Tafuri (2003)
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Vortex front propagation is normally shock wave
like, but occasionally creates avalanches
after 50 ns
after 10 s
Magneto-optics in YBCO films, 10K, B30mT, size
2.3x1.5 mm
Boltz et al (2003)
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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
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Dependence on magnetic history the field cooled
and the field cooled with return protocols result
in different states.
Transport in Nb
Reversible region
Irreversible region
Andrei et al (2004)
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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.