A. Derivation of GL equations

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A. Derivation of GL equations
II. TYPE I vs TYPE II SUPERCONDUCTIVITY
1.Macroscopic magnetostatics
Several standard definitions
macroscopic magnetic field
-Field of external currents
-magnetization
-free energy
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In equilibrium under fixed external magnetic
field the relevant thermodynamic quantity is the
Gibbs energy
Inserting the GL free energy
one obtains
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2. Derivation of the LG equations
By variation with respect to order parameter
Y one obtains the nonlinear Schrodinger equation
and by variation with respect to vector
potential A - the supercurrent equation
out of five equations only four are
independent (local gauge invariance)
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The equations should be supplemented by the
boundary conditions. The covariant gradient is
perpendicular to the surface
()
while the magnetization is parallel to it
()
Note that the external magnetic field enters
boundary conditions only magnetic field is a
topological charge.
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Details of the derivation of the set of GL
equations and boundary conditions
We have to vary with respect to five independent
fields
and
Two components of the complex order
parameter field are varied independently. One of
them is
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Integration by parts of the first term gives
()
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If the full variation dG is to vanish, one
has to require that both the nonlinear
Shroedinger eq. and the boundary condition ()
are satisfied.
Variation with respect to Y(x) just gives the
corresponding complex conjugate equation.
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The variation and supercurrent
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This defines supercurrent
The covariant derivative representation makes
its gauge invariance obvious.
The variation of is identical to
that used in derivation of Maxwell equations.
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This leads to the supercurrent equation and
the boundary condition () .
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The boundary condition ()
after multiplication by the order parameter field
leads to
Supercurrent therefore cannot leave the
superconductor through the boundary and therefore
circles inside the sample.
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B. Homogeneous and slightly perturbed SC solution
1. Zero magnetic field. Homogeneous solutions.
The degenerate minima are at
with free energy density
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The condensation energy
In addition to the degenerate SC solution
(global minima) there exists a
nondegeneratetrivial normal solution (a local
maximum)
The free energy density difference between the
normal and the superconducting ground states (the
condensation energy) is
where Hc is defined as the thermodynamic
critical field. As will be clear later at this
field nothing special happens in type II
superconductor.
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2. A small inhomogeneity near the SC state
Deviations of the order parameter
Assume that variations are along the x
direction only and the magnetic field
contributions are small
Is real
Defining the normalized order parameter
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one is left with a single scale
the coherence length
For small deviations of from
one linearizes the (anharmonic oscillator
type) equation with
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This corresponds to the harmonic approximation.
Deviations of from decay
exponentially on the scale of correlation length
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The magnetic field penetration profile
In the supercurrent equation the magnetic
field cannot be neglected. However in this case
one can neglect setting
This is the Londons equation, valid beyond
GL theory, everywhere close to
deep inside the superconductor. Taking a curl,
one obtains
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The relevant scale here is the penetration depth
The solution of the linear Londons eq. is
also exponential
The magnetic field decays exponentially
inside superconductor on the scale of magnetic
penetration depth.
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Anderson Higgs mechanism
In unitary gauge (and absence of topological
chargeflux) order parameter Y can be made real
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In harmonic approximation one expands to
second order around the SC state
and obtains (up to a constant) following
quadratic terms (linear terms generally vanish
due to eq. of motion or GL eqs.)
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In the normal phase one has three massless
excitation fields two transverse polarizations
of photon (use, for example the Colomb gauge
and the phase of order parameter
In the SC phase the situation changes
dramatically due to mixing all the excitations
become massive. In the unitary gauge this is seen
as a three component massive vector field A.
The situation is sometimes termed spontaneous
gauge symmetry breaking.
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Beyond perturbation theory
Of course when deviations are not small like
in the SC-N junction one has to consider both the
order parameter and the magnetic field
simultaneously and go beyond the perturbation
theory.
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C. The SC-normal domain wall surface energy.

1. Extreme type II case the energy gain due to
   magnetic field penetration into SC

Assume first
In the SC region but
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The Gibbs free energy density in the N part
(assuming ) is the same as
is homogeneous SC

On the SC side assume that one still can use
the Londons asymptotics with
The energy gain is therefore
Highly unusual!
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2. Extreme type I limit the energy loss due to
order parameter depression near N
In the opposite case
In the junction region but
The energy loss of the condensation energy
naively is
Less naively one solves the anharmonic
oscillator type equation exactly
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Details of solution
Multiplying the eq. by and
integrate over x with boundary conditions
one obtains
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The energy per unit area
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Therefore in type I SC the behavior is as
expected one has to pay energy in order to
create interfaces.
Summing up naively the two contribution we obtain
the interface energy
For the domain wall
energy changes sign. Type II SC unlike any other
material, likes to create domain walls.
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3. General case
A convenient choice of gauge for the 1D problem
Set of GL equations (the solution Y(x) is real) is
or using dimensionless functions
and
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Using l(T) as a unit of length this becomes
The boundary conditions still are
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A simplified expression for the domain wall energy
Nonlinear Schrodinger equation
simplifies the expression
Exercise 1 solve the GL equations for S-N
numerically using the shooting method for
k.5,1,10.
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4. For what k the interface energy vanishes?
Obviously s in () vanishes if the integrand
vanishes
It turns out that for the
exact solution (which is not known analytically)
obeys it!
For this particular value of k the GL equations
(with x in units of l) takes a form
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Substituting the zero interface requirement
() into the second eq.(2) one gets
Differentiating it and using () again one gets
eq.(1)
The value therefore separates
between type I and type II.
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Summary
1. Order parameter changes on the scale of
coherence length x, while magnetic field on the
scale of the penetration depth l. The only
dimensionless quantity is kl/x. 2. The interface
energy between the normal and the superconducting
phases in type II SC is negative. This leads to
energetic stability of an inhomogeneous
configuration. 3. The critical value of the
Ginzburg parameter k at which a SC becomes type
II is