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A. Derivation of GL equations

macroscopic magnetic fieldvolBB

4 4FH B MB

Several standard definitions:

-Field of “external” currents

M -magnetization

][BF -free energy

II. TYPE I vs TYPE II SUPERCONDUCTIVITY1.Macroscopic magnetostatics

2

3 14

G 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 Am c

F d x T T

Inserting the GL free energy:

one obtains:

2 23 ( )

8 8grad potB H HG F F d x

23

8grad potBF 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 * *sc 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 Tm c

out of five equations only four are independent (local gauge invariance)

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while the magnetization is parallel to it:

*ˆ 0en i Ac

(**)

(*)

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 Am c c

e eF d x i A i Am 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 Am c c

eds i Am 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 Am c c

F d x Ame e e ei 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 DmA x

e ei Am m c

The variation of is identical to that used in derivation of Maxwell equations.

magF

10

3

1 ( ) ( )8

( )4

1 ( )41

41

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 (*)

*

* ** 0

2Se hn J i n D n Dm

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 gvol

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 ( ) 02 * c

d T Tm 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( ) 0dTdx

one is left with a single scale

the coherence length2

2 ( )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 ...) 0dTdx

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 *ec B Am c

0

22 2

220

* ( ) ( )4 *m cA B B T Be

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 mTe 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 Am c

e Am 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

Be Am 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

0 cB 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 gradH 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 00 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 )ddx

2 ddx

( 0) 0( ) 1xx

one obtains2

2 2 41 12 2

ddx

2

2 21 (1 )2

ddx

Details of solution

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2

( ) tanh 12

x

xxx e

21 (1 )2

ddx

The energy per unit area

22 2 2

2 4 ( )2 * 2 8 8s c

d H B HG dx T Tm dx

2

2 2 20 0

0

2 (1 )sdG g g dxdx

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12 2 2 2

00 0

12

0 0 00

2 (1 ) 2 (1 )

42 2 (1 ) 2 1.89 03

dxg dx dd

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)()(

TT

crit 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

*'' 02 * 2 *

*''4 *

ceT T A

m m cc eA A

m c

3. General case

ˆ( ) ( ) ,B x B x z

,ˆ)()( yxAxA

)()()(' xBxAdxdxA

Set of GL equations (the solution (x) is real) is

A convenient choice of gauge for the 1D problem:

or using dimensionless functions

*ea Ac

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

10 ' 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 gm

Nonlinear Schrodinger equation 2

22 4 02 *

a Dm

simplifies the expression:2

4 4 20

( ) (1 2 ')2 8

cH 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 ' (*)2

a 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 ) '2

a a

2 12 ' '

2a a

2 21 1'' ' ' (1 )22

a 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

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