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Ginzburg-Landau equations

• Order parameter• Free energy• Ginzburg-Landau equations• Critical field of a film• Type-II superconductors• Upper critical field• Lower critical field• Vortex structure• Vortex motion. Pinning

Ginzburg-Landau theory

BCS: microscopic, very powerful – but too complicated

London equations: describe a very special situation ξ δ

Way out: to develop a theory based on Landau theoryof second order phase transitions.

Applicability range: close to the transition temperature.No restrictions concerning the nature of (conventional) superconductors.

Idea: order parameter is a complex number – wave function of Cooper pairs.

( )rΨ

Free energy

Free energy – functional of the order parameter.

2 22 4 1 2

2 4 8s nb e HF F dV a i A

m cτ

π

⎧ ⎫⎪ ⎪⎛ ⎞= + Ψ + Ψ + − ∇− Ψ +⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

∫

Energy of the normal state

Expansion at the critical

point

Energy of magnetic

field

Fluctuation contribution

(“kinetic energy of Cooper pairs”)

Without fluctuations:

20

/0

c

c

a b T TT T

τ− <⎧Ψ = ⎨ >⎩

( ) /c cT T Tτ = −

Ginzburg-Landau equations

Minimizing the free energy:

2 22 4 1 2

2 4 8s nb e HF F dV a i A

m cτ

π

⎧ ⎫⎪ ⎪⎛ ⎞= + Ψ + Ψ + − ∇− Ψ +⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

∫

With respect to *Ψ2

2 1 2 04

ea b i Am c

τ ⎛ ⎞Ψ + Ψ Ψ+ − ∇− Ψ =⎜ ⎟⎝ ⎠

Boundary condition: 2 0en i Ac

⎛ ⎞− ∇− Ψ =⎜ ⎟⎝ ⎠

Ginzburg-Landau equations

Minimizing the free energy:

2 22 4 1 2

2 4 8s nb e HF F dV a i A

m cτ

π

⎧ ⎫⎪ ⎪⎛ ⎞= + Ψ + Ψ + − ∇− Ψ +⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

∫

With respect to A4H jcπ

∇× =

“Maxwell equation”

( )2

2* * 22ie ej Am mc

= − Ψ ∇Ψ−Ψ∇Ψ − Ψ

Rescaling GL equations

3/ 20 0 0/ , / , / , 2 2r r H H H H a bδ π τΨ→Ψ Ψ → → =

202 /eH cκ δ=The rescaled equations contain only one constant!

( )( )

21 2

1

2* *

( ) 0

0 at the surface

2

i A

n i A

iA A

κ

κ

κ

−

−

− ∇− Ψ−Ψ+Ψ Ψ =

− ∇− Ψ =

∇×∇× = − Ψ ∇Ψ−Ψ∇Ψ − Ψ

2

Critical field of a film

Steps of the calculation:- Write the energy difference between S and N phases;- Find at which field the energies are the same (depends on- Solve GL equations and find the order parameter;- Find the critical field

Ψ)

Results: Thick films, d δ (1 / )c cbH H dδ= + 0 / 2cbH H=

Thin films d δ 2 6 /c cbH H dδ=cH

cbH

/ dδ

Critical field of a film

Let us now look at the details.

constΨ ≈Thick films d δ 2 1Ψ ≈

Thin films d δ 0Ψ =

s nF F−

cbH 2Ψ

GL equations for 1κ give

First-order phase transition

Second-order phase transition

At certain thickness the transition order changes 5cd δ=d δ

H

s nF F−

cbH 2Ψ

d δ

H

Critical field of a film

Type-I superconductors: 1st order phase transition in the bulkis realized by splitting into N and S layers

Interface energy: positiveCritical field of one layer is higher

Optimal thickness of a layer

Intermediate state!

Type II superconductors: does not work!

Interface energy: negative

Formation of infinitely thin layers is preferable

First-order phase transition is shifted to infinitely strong fields!

Type-II superconductors

Phase transition must be of the second order and occurs in the interval of fields

2cH H> (Upper critical field)Even a small piece of superconductor is unstableNormal metal

1cH H<(Lower critical field)Fully developed Meissner effectSuperconductor

1 2c cH H H< < Mixed stateNo Meissner effect

Upper critical field

Two possible mechanisms:- Spins of two electrons in a Cooper pair become parallel – too strong fields- Larmor radius of a Cooper pair becomes shorter than ξ

Larmor radius: Lcp creH eHξ

⊥= <

p⊥ -component of momentum of the pair transverse to the magnetic field

2 2L ccr H H

eξ

ξ< ⇒ < ∼ 2c cb cbH H Hκ= >

Lower critical field

What happens below 2 ?cHMagnetic field penetrates the superconductor

Persistent currents (no decay): vortices!

Quantum vortices:H

ρ

v Bohr-Sommerfeld quantization:

2

2 2 2

n pdq

m vdl mv

π

πρ

=

= = ⋅

∫∫

/ 2v n mρ= - quantized velocity

n=1: energetically the most favorable ξ ρ δ

3

Lower critical field

Vortex energy (per unit length):

H

ρ

v

/ 2v n mρ= ξ ρ δ

2 222 2 ln ln

2 4 4s s s

sn n nE mv d

m m

δ

ξ

π πδπρ ρ κξ

= ⋅ = =∫

Vortex magnerization (per unit length):

22 22 2s s

sn n eeM v d

c mc

δ

ξ

πρ πρ ρ δ= ⋅ =∫Magnetic energy: -MH

Lower critical field: the first vortex is formed

2 21 / ( / ) ln ln /c cb cbH E M c e H Hδ κ κ κ= = <∼

H

κ1/ 2

cbH

Isolated vortex

Ψ

ρξ δ

H

What is the magnetic flux in a vortex?

( )2

2* * 22ie ej Am mc

= − Ψ ∇Ψ−Ψ∇Ψ − Ψ

Current:

ie θΨ = Ψ2

2 2e ej Am mc

θ⎛ ⎞

= Ψ ∇ −⎜ ⎟⎝ ⎠

Far from the vortex core: no current

Let us integrate over a contour around the vortex center.

2 2e edl Adlc c

θ Φ∇ = =∫ ∫ , /s sn c eπΦ = Φ Φ ≡ -superconducting

flux quantum

Vortex core

Vortex lattice

232 4

s a HΦ=

Close to lower critical field – sparse vortices

Form a triangular vortex lattice

Lattice period: determined by H

Per plaquette: half a vortex a

Total flux per plaquette:

2cH : vortices overlap 22 /c sa Hξ ξ⇒ Φ∼ ∼

21 / ; /c s cb sH Hδ δξΦ Φ∼ ∼

Vortex motion

F j H∝ ×

If current is passed through asuperconductor in a mixed state:Lorentz force acts at the core electrons

a

Resistance!! 2/N cH Hρ ρ∼

I

Vortex motion

transverse to the current

Motion of magnetic flux with a constant velocity

Electric field1E H vc

= ×

Like if produced by vortex cores.

Pinning

Does it mean Type-II superconductors have resistance down to very low fields?

No, vortices do not move (at weak currents)

Pinning:

Is more favorablethan

It costs energy to depin a vortex.