1 / 34

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

In equilibrium under fixed external magnetic

field the relevant thermodynamic quantity is the

Gibbs energy

Inserting the GL free energy

one obtains

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)

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.

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

Integration by parts of the first term gives

()

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.

The variation and supercurrent

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.

This leads to the supercurrent equation and

the boundary condition () .

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.

B. Homogeneous and slightly perturbed SC solution

1. Zero magnetic field. Homogeneous solutions.

The degenerate minima are at

with free energy density

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.

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

one is left with a single scale

the coherence length

For small deviations of from

one linearizes the (anharmonic oscillator

type) equation with

This corresponds to the harmonic approximation.

Deviations of from decay

exponentially on the scale of correlation length

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

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.

Anderson Higgs mechanism

In unitary gauge (and absence of topological

chargeflux) order parameter Y can be made real

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.)

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.

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.

C. The SC-normal domain wall surface energy.

- Extreme type II case the energy gain due to

magnetic field penetration into SC

Assume first

In the SC region but

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!

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

Details of solution

Multiplying the eq. by and

integrate over x with boundary conditions

one obtains

The energy per unit area

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.

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

Using l(T) as a unit of length this becomes

The boundary conditions still are

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.

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

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.

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