I' INTRODUCTION AND OVERVIEW OF THE COURSE

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I. INTRODUCTION AND OVERVIEW OF THE COURSE
A. General phenomenological approach to second
order phase transitions
1. Order parameter field and spontaneous symmetry
breaking
Any second order phase transition is well
described by a). Order parameter field b)
Symmetry group G.
L.D. Landau ( 1937 )
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An example XY- (anti) ferromagnet
Rest of degrees of freedom are irrelevant close
to critical temperature
In plane classical spins of fixed length
Defined on a D dimensional lattice (the type of
lattice and other microscopical details are also
irrelevant).
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1. T0 ordered
large
2. 0ltTltTc ordered
small
3. TgtTc disorderd
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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
is
Symmetry means that the energy of the rotated
state is the same as that of the unrotated one.
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2. Effective free energy near phase transition
Most general functional symmetric under
and space rotations, with smallest
possible powers of
and smallest small number of gradients .
Higher order terms

are expected to be smaller close enough to Tc.
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The remaining coefficients can be expanded around
Tc when close enough to Tc
Now we apply this general considerations to the
superconductor normal metal phase transition.
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B. Ginzburg Landau description of the
superconductor-normal transition
1. Symmetry and order parameter in terms of
microscopic degrees of freedom
The broken symmetry is charge U(1) mathematically
the same symmetry as that of the XY magnet.
Order parameter is
Which is the gap function of BCS or other
microscopic theory.
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Its physical meaning of this complex field can be
better described via modulus and phase
Density of Cooper pairs

• the superconductor phase

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2. Free energy. Influence of magnetic field.
Without external magnetic field the free energy
near transition is
Ginzburg and Landau (1950) generalized this to
the case of arbitrary magnetic field
using gauge invariance of electrodynamics.
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Gauge invariance
Electrodynamics is invariant under local gauge
transformations
This invariance although not a symmetry (only its
global part is) dictates the charge fields
coupling to magnetic field. Minimal substitution
replaces any gradient by a covariant derivative
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The gauge invariance of the gradient term follows
from the linearity of transformation of
Magnetic field
is also gauge invariant.
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Ginzburg Landau equations
Minimizing the free energy with covariant
derivatives one arrives at GL equations. The
nonlinear Schrodinger equation (variation of
)
And the supercurrent equation (variation of A)
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GL equations possess two scales. Coherence length
3. Two characteristic scales
characterizes variations of , while
the penetration depth characterizes
variations of
Both diverge at TTc.
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Ginzburg Landau parameter
The only dimensionless parameter one can
construct from the two lenths is temperature
independent
Properties of solutions crucially depend on GL
parameter. If


(so called type II superconductivity ) there
exist topological solitions the Abrikosov
vortices.
Abrikosov (1957)
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C. Abrikosov Vortices1. Why look for a singular
solution?
Usually one doesnt look for singular solutions
of field equations, however the type II
superconductor case is special. Magnetic field
penetrates the sample as an array of
vortices. One has to look for these solutions due
to combination of four facts. Two crucial and two
technical.
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Mixed state under applied magnetic field
1. Interface energy is negative for type II
superconductors, while positive for the type I.
H
H
Type I Minimal area of domain walls.
Type II Maximal area of domain walls.
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2. Flux quantization.
Division into domains stops due to this.
To minimize potential far from isolated vortex
(where B0 ) one has to fix modulus of order
parameter
The phase however can vary. In order to minimize
the gradient term, one demands
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3. n1 is energetically favoured over ngt2 4. The
region shrinks to a point.
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2. Shape of solution
A Vortex-linear defect.
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Abrikosov vortices in type II superconductors as
seen by electron beam tomography.



KT pair
Tonomuras group PRB43,7631 (1991)
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D. Vortex matter Some properties of vortices
Line energy
To creat a vortex, one has to provide energy per
unit length ( line tension )
Interactions between vortices
They interact with each other via complicated
vectorial force. Parallel straight vortices repel
each other.
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Correlated states of vortex matter
Due to this repulsion they form (usually
hexagonal ) lattice Abrikosov flux line lattice
(FLL)
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Lattice of flux lines as seen by STM and neutron
scattering



Hess et al PRL62,214 (1989)
S.R.Park et.al.,2000 (Brown University)
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Vortex liquid
Due to thermally induced vibrations lattice can
melt into vortex liquid.
The phase diagram becomes more complicated.
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First order melting of the Abrikosov lattice into
a vortex liquid.



Gammel et al PRL80,833 (1998)
Schilling et al Nature 382,791 (1996)
Welp et al PRL76,4809 (1996)
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Two critical fields
first vortex penetrates.
Cores overlap
Vortex dynamics
Vortices move under influence of external
current (due to Lorentz force). Energy is
dissipated in the vortex core. The resistivity is
no longer zero.
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Vortex cutting and entanglement
Vortices may entangle around each other like
polymers, however due to vectorial nature of
their interaction they can also disentangle or
cut each other.
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Disorder
Columnar
Vortices are pinned by disorder. For vortex
systems pinning create a glassy state or viscous
entangled liquid
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Nonlinear IV curve
Current produces expanding vortex loops even in
the Meissner phase leading to nonohmic
broadening of I-V curves
Conclusion in type II superconductors the vortex
degrees of freedom are overwhelmingly important.