1 a. derivation of gl equations macroscopic magnetic field several standard definitions: -field of...
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A. Derivation of GL equations
macroscopic magnetic fieldvolBB
4 4F
H B MB
Several standard definitions:
-Field of “external” currents
M -magnetization
][BF -free energy
II. TYPE I vs TYPE II SUPERCONDUCTIVITY
1.Macroscopic magnetostatics
2
3 1
4G F dx B H
In equilibrium under fixed external magnetic field the relevant thermodynamic quantity is the Gibbs energy:
22 *
3*
2 43
2
( )2
grad
pot c
eF d x i A
m c
F d x T T
Inserting the GL free energy:
one obtains:
2 23 ( )
8 8grad pot
B H HG F F d x
23
8grad pot
BF F F dx
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2. Derivation of the LG equations
By variation with respect to order parameter one obtains the nonlinear Schrodinger equation
22* ** *
( )4 2 * *s
c e eB J i A
m m c
and by variation with respect to vector potential A - the supercurrent equation
222 *
( ) 02 * c
ei A T T
m c
out of five equations only four are independent (local gauge invariance)
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while the magnetization is parallel to it:
*ˆ 0
en i A
c
(**)
(*)
0)(ˆ HBn
The equations should be supplemented by the boundary conditions. The covariant gradient is perpendicular to the surface
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:)(),(* xx and ( ), 1, 2,3.iA x i
)()()( *** xxx
23 * *
23 * *
* *( )( ) ( )
2 *
* *( ) ( )
2 *
grad
grad
e eF d x i A i A
m c c
e eF d x i A i A
m c c
Two components of the complex order parameterfield are varied independently. One of them is:
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23 *
2*
23 *
2*
* *( ) ( )
2 *
*( )
2 *
2 *
2 *
grad
boundary
boundary
e eF d x i A i A
m c c
eds i A
m c
d x D Dm
ds Dm
����������������������������
��������������
Integration by parts of the first term gives
(*)
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* * * * 2 2
* * 2
( )( ) ( )2
( ) 22
pot c
pot c
F T T
F T T
magmag FF
If the full variation G is to vanish, one has to require that both the nonlinear Shroedinger eq. and the boundary condition (*) are satisfied.
Variation with respect to (x) just gives the corresponding complex conjugate equation.
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23 *
23
* *
3
* *( ) ( )
2 *
2 ** * * *
( ) ( )
grad
grad
sgrad
e eF d x i A A i A A
m c c
F d x Ame e e e
i A i Ac c c c
F d x A J
��������������
potpot FF
)(xA
)()()( xAxAxA iii
The variation and supercurrent
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This defines supercurrent
The covariant derivative representation makes its gauge invariance obvious.
** *
*
* 2* * *
* *
( )2( )
*( )
2
gradS
F eJ i D D
mA x
e ei A
m m c
The variation of is identical to that used in derivation of Maxwell equations.
magF
10
3
1 ( ) ( )8
( )4
1 ( )
41
4
1
4
mag ijk j k k jijk k k
iijk j k k
mag i i ijk j k
mag k ijk j i
jk
G A A A A
HA A
G B H A
G d x A B
dS A
( )ijk i iB H
This leads to the supercurrent equation and the boundary condition (**) .
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The boundary condition (*)
*
* **
02S
e hn J i n D n D
m
����������������������������
Supercurrent therefore cannot leave the superconductor through the boundary and therefore circles inside the sample.
ˆ 0n D ��������������
after multiplication by the order parameter field leads to
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The degenerate minima are at
2 40 0 ( )
2pot cH B G F T T
B. Homogeneous and slightly perturbed SC solution
2 20
( )cT T
F
2
08cH
g
0
with free energy density2
20 ( )
2 cg T T
1. Zero magnetic field. Homogeneous solutions.
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222
0 0( )2 8
s cs c n s
F Hf T T g f f g
vol
In addition to the degenerate SC solution (global minima) there exists a nondegeneratetrivial normal solution (a local maximum):
The condensation energy
The free energy density difference between the normal and the superconducting ground states (the condensation energy) is:
0; 0; 0nB H F
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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Assume that variations are along the x direction only and the magnetic field contributions are small :
2 22
2( ) 0
2 * c
dT T
m dx
2. A small inhomogeneity near the SC state
0
( )( )
xx
( )x Is real
Deviations of the order parameter
Defining the normalized order parameter
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22 3
2( ) 0
dT
dx
one is left with a single scale
the coherence length
22 ( )
2 * ( )c
Tm T T
0
one linearizes the (anharmonic oscillator type) equation with
1 11 , 1
For small deviations of from
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1 12
2''
22 1
1 12( ) (1 ) (1 3 ...) 0
dT
dx
2
1( ) ~x
x e
Deviations of from decay exponentially on the scale of correlation length
0
0
x
This corresponds to the “harmonic approximation”.
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This is the Londons’ equation, valid beyond GL theory, everywhere close to deep inside the superconductor. Taking a curl, one obtains:
The magnetic field penetration profile
In the supercurrent equation the magnetic field cannot be neglected. However in this case one can neglect setting
2 20*
4 *
ecB A
m c
0
22 2
220
*( ) ( )
4 *
m cA B B T B
e
0
1
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The solution of the linear Londons’ eq. is also exponential:
/0( ) xB x B e
The magnetic field decays exponentially inside superconductor on the scale of magnetic penetration depth.
2
22
*( )
4 * c
c mT
e T T
The relevant scale here is the penetration depth
B0B
x
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22 *
*
22 *2 2*
2 4
2
2
( )2
grad
pot c
eF i A
m c
eA
m c
F T T
In unitary gauge (and absence of topological charge=flux) order parameter can be made real
2 2( )
8 8grad pot
B H HG F F
Anderson – Higgs mechanism
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In harmonic approximation one expands to second order around the SC state
0 1 1; 0A A
and obtains (up to a constant) following quadratic terms (linear terms generally vanish due to eq. of motion or GL eqs.):
2 2 2 2 21 1 0 1*
2*22 2 10 1* 2
2 2 2 21 1*
2 2 21 1
( ) 32
2 8
2 ( )2
1( )
8
harmonic c
c
c
F T Tm
BeA
m c
T Tm
B T T A
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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
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.
0A ��������������
and the phase of order parameter
The situation is sometimes termed “spontaneous gauge symmetry breaking”.
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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.
0
Type Ismall
interface > 0
NSC
cB H
Type IIlarge
interface < 0
NSC
cB H0
Beyond 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
In the SC region but
/ 1, 0, 0
0cB H
Assume first
cH B H normal superconductor
0 Gn
Gs
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0 0sG L g g
The energy gain is therefore:
2 2
0 0
( )0 0 0
8 8n pot grad
H B Hg f f g g
On the SC side assume that one still can use the Londons asymptotics with :
2 2
2 22 / / 2 /
0 0 0
0 0
( )
8 8
28 8
s pot grad
L Lx x xc c
x x
H B HG f L f L dx
H Hg L e dx e e dx g L g
0 cB H
cB H H
The Gibbs free energy density in the N part (assuming
) is the same as is homogeneous SC
Highly unusual!
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In the junction region but
1, 0; 0;
0 0B
0g The energy loss of the condensation energy
naively is:
In the opposite case
2. Extreme type I limit: the energy loss due to order parameter depression near N
Less naively one solves the anharmonic oscillator type equation exactly:
0 cH B H
GnGs supernormal
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Multiplying the eq. by and integrate over x with boundary conditions
22 2
2(1 )
d
dx
2d
dx
( 0) 0
( ) 1
x
x
one obtains2
2 2 41 1
2 2
d
dx
22 21
(1 )2
d
dx
Details of solution
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2
( ) tanh 12
x
x
xx e
21(1 )
2
d
dx
The energy per unit area
22 2 2
2 4 ( )
2 * 2 8 8s c
d H B HG dx T T
m dx
2
2 2 20 0
0
2 (1 )s
dG g g dx
dx
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12 2 2 2
0
0 0
12
0 0 0
0
2 (1 ) 2 (1 )
42 2 (1 ) 2 1.89 0
3
dxg dx d
d
g d g g
Therefore in type I SC the behavior is as expected: one
has to pay energy in order to create interfaces.
0( ) g
1)(
)(
T
Tcrit
For the domain wall energy
changes sign. Type II SC unlike any other material,
likes to create domain walls.
Summing up naively the two contribution we obtain the interface energy
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2 2
3 22
22
*'' 0
2 * 2 *
*''
4 *
c
eT T A
m m c
c eA A
m c
3. General case
ˆ( ) ( ) ,B x B x z
,ˆ)()( yxAxA
)()()(' xBxAdx
dxA
Set of GL equations (the solution (x) is real) is
A convenient choice of gauge for the 1D problem:
or using dimensionless functions
*ea A
c
and
0
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2 3 2
2 2
'' 0
''
a
a a
0
' 1/ 2cB H a
The boundary conditions still are:
/ ,x
1
0 ' 0B a
/ ,x
Using as a unit of length this becomes2 3 2
2
'' 0
''
a
a a
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A simplified expression for the domain wall energy
2 2
22 402 2 * 8 4
c
x
B B Ha D g
m
Nonlinear Schrodinger equation 2
22 4 02 *
a Dm
simplifies the expression:2
4 4 20
( )(1 2 ')
2 8cH B
g dx a
Exercise 1: solve the GL equations for S-N numerically using the shooting method for
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4. For what the interface energy vanishes?
1/ 2
Obviously in (*) vanishes if the integrand
vanishes 22
4 11 2 ' 0 ' (*)
2a a
It turns out that for the exact solution (which is not known analytically) obeys it!
For this particular value of the GL equations
(with x in units of takes a form:
3 2
2
2 '' 0 (1)
'' (2)
a
a a
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2 21'' (1 ) '
2a a
2 1
2 ' '2
a a
2 21 1'' ' ' (1 )
22a a a
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.2
1
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Summary
1. Order parameter changes on the scale of coherence length , while magnetic field on the scale of the penetration depth . The only dimensionless quantity is
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 at which a SC becomes type II is
2
1