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1

Ginzburg-Landau Theory

- flux quantisation and critical fields

Prof. Damian Hampshire Durham University, UK

Rigi Kaltbad Switzerland June 2007

2

Superconductivity

• ITER is a fabulous opportunity in superconductivity• Thanks to Pierluigi for arranging the school

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i) Microscopic theory – describes why materials are superconducting

There are two main theories in superconductivity:

ii) Ginzburg-Landau Theory – describes the properties of superconductors in magnetic fields

4

Ginzburg-Landau theory

• This is a phenomenological theory, unlike the microscopic BCS theory. Based on a so-called phenomenological order parameter.

• Not strictly an ab initio theory, but essential for problems concerning superconductors in magnetic fields.

• G-L was the first theory to explain the difference between Type I and Type II superconductivity, and enable the calculation of two critical fields Hc1 and Hc2.

5

Outline of the Lecture• Ginzburg-Landau theory • The two G-L equations• The basic phenomenology – Type I and Type II

superconductors• Time dependent G-L theory• Ginzburg-Landau and Pinning• Conclusions

6

Ginzburg-Landau Theory

2 4 2 - 2e ) + d 1 1f = + + (-i2 2m

α ψ β ψ ψ∇ ∫A H B

Ginzburg and Landau (G-L) postulated a Helmholtz energy density for superconductors of the form:

where α and β are constants and ψ is the wavefunction. α is of the form α’(T-TC) which changes sign at TC

7

The two Ginzburg-Landau Equations• Functional differentiation w.r.t. ψ and A gives the two G-L

equations (coupled partial differential equations):

G-L I

G-L II

( ) 02i-21 22 =ΨΨ+Ψ+Ψ−∇ βαAem

( )2

2* *i 4e eJ Am m

= − Ψ ∇Ψ −Ψ∇Ψ − Ψ

8

London Theory – Type I superconductors

• The London brothers -

sL Jt

E∂∂

μ= 20λ

sL JB ∧∇μ−= 02λ

*s

*

L enm

0

2

μ=λ

BBL

22 1

λ=∇

Substituting the second London equation into one of Maxwell’s equations we get the Meissner state:

where

9

Magnetisation of superconductors

-Hc

Hc1 Hc Hc2

Type IType II

M

H0

Tc

H0T

Hc1

Hc

Hc2

Meissner

Mixed Normal

The magnetisation (M) of a type I and type II superconductor, with the same Hc as a function of applied magnetic field (Ho). Inset: The phase diagrams of type I and type II superconductors.

10

The structure of the fluxon

• The order parameter, magnetic field (B) and supercurrent density (JC) as a function of distance (r) from the centre of an isolated vortex (κ ≈ 8). The order parameter squared is proportional to the density of superelectrons.

0

ψ∞

2μ0Hc1

0 ξ λ

Jθ

Bz

Jθ(r)

Bz(r)

r

|ψ(r)|

11

The Mixed state

λ: depth of the current flowd: fluxon – fluxon spacingIf B increases, d decreases.

12

The Mixed State in NbSe2

• (a) (b)• The hexagonal Abrikosov lattice: (a) contour diagram of the order parameter

from Kleiner et al.; the lines also represent contours of B and streamlines of J. (b) Scanning-tunnelling-microscope image of the flux lattice in NbSe2(1 T, 1.8 K) from Hess et al. the vortex spacing is ~479Å.

13

The Mixed State in Nb

Vortex lattice in niobium – the triangular layout can clearly be seen. (The normal regions are preferentially dark because of the ferromagnetic powder). – Consider flux box.

14

Flux Quantisation

• By considering the persistent current as a wave, phase coherence demands (n: integer) - leads to flux quantisation. k: momentum of the superelectrons.

15

G-L Theory – Type I and Type II superconductors

2 emξ

α=

204em

eβλ

μ α=

Ginzburg-Landau theory predicts that a superconductor should have two characteristic lengths:

Penetration depth

Coherence length

The Ginzburg-Landau parameter

This ratio, κ, distinguishes Type-I superconductors, for which κ ≤ 1/√2, from Type-II superconductors which have higher κ values.

λκξ

=

16

Thermodynamic Critical Field, HC

• In Type I superconductors HC is the critical field at which superconductivity is destroyed − ψ drops abruptly to zero in a first-order phase transition.

• At the thermodynamic critical field HC, the Gibbs free energies for superconducting and normal phases are equal. For the superconducting phase B = 0, and for the normal phase ψ = 0

0cH

αμ β

=

17

Lower and upper critical fields

1 ln2C

cH

H κκ

≈

22 1c C CH H Hκ κ≈ ≈

00 2 22cH

φμπξ

=

0cH

αμ β

=

18

Time evolution of order parameterFlux entering a superconductor

This simulation is simply applying an instantaneous fields of 0.5Hc2.

19

Time evolution of order parameter

20

Time evolution of order parameter

21

Time evolution of order parameter

22

Time evolution of order parameter

23

Time evolution of order parameter

24

Flux nucleation and entry into a superconductor

• Contour plot of a superconductor bounded by an insulating outer surface. The applied magnetic field was increased to above the initial vortex penetration field (i.e. to Hp + 0.01Hc2)

-60 -40 -20 0 20 40 60-50

-30

-10

10

30

50

25

Other Ginzburg-Landau predictions

( )( )

20 22 1

c

A

B BMμ

κ β−

= −− ( )

4

22A

ψβ

ψ=

The magnetization of in the mixed state near Bc2 is given by:

Surface barriers 3 21.69C CB B=

Depairing current density: 2 222

20.385

3 3c c

DH H

Jκ ξκ ξ

= =

26

Josephson diffraction

0.00 0.01 0.02 0.03 0.04 0.050.00

0.02

0.04

0.06

0.08

0.10Junction Thickness 0.5ξ

Crit

ical

Cur

rent

Den

sity

(Hc2

/κ2 ξ

)

Applied Field (Hc2)0.00 0.01 0.02 0.03 0.04 0.05

0.000

0.005

0.010

0.015

0.020

0.025

0.030Junction Thickness 1.5ξ

Applied Field (Hc2)

Crit

ical

Cur

rent

Den

sity

(Hc2

/κ2 ξ

)

0.00 0.01 0.02 0.03 0.04 0.050

1x10-3

2x10-3

3x10-3

Junction Thickness 4.5ξ

Applied Field (Hc2)

Crit

ical

Cur

rent

Den

sity

(Hc2

/κ2 ξ

)

0.00 0.01 0.02 0.03 0.04 0.050

1x10-4

2x10-4

3x10-4

4x10-4

Junction Thickness 9.5ξ

Applied Field (Hc2)

Crit

ical

Cur

rent

Den

sity

(Hc2

/κ2 ξ

)

Jc computed for an ρN = 10ρS junction in a 30ξ-wide κ = 5 superconductor.

27

Ginzburg-Landau predictions -restricted dimensionality behaviour

• Behaviour of thin films - A thin film has a much higher critical field (if the field lines are parallel to the film), than a bulk superconductor. This is predicted by Ginzburg-Landau theory.

• Anisotropic Ginzburg-Landau theory - It is possible to extend Ginzburg-Landau theory to anisotropic systems (eg high-Tc superconductors). This shows that Hc is isotropic, but Hc2 is not.

• Lawrence-Doniach theory - 2D version of GL theory. Predicts that Hc2 is higher for layered superconductors, and that at low T’s, fluxons are locked parallel to planes (a kind of ‘transverse Meissner effect’)

28

Flux Pinning using G-L theory

( )2

1 22 22

0

14

cp

BF b b

Dμ κ= −

( )222

0

31

16c

pB

F b bD

πμ κ

= −

( )( )

52 1 24 2 2

2200 0

2 12.8 10 11

cp

BF b b

ad

πμ κ φ

−= × −−

( )( )

52 3 24 2 2

22 200 0

2 1 0.29 13.9 10 exp 131

cp

B b bF b ba bd

πμ κ φ κ

− ⎛ ⎞− − ⎟⎜= × −⎟⎜ ⎟⎜⎝ ⎠−

( )( )( )

222

2 20 0

1exp 1 0.29 18 3

cp

B bF b b bD a bπμ κ κ

⎛ ⎞− ⎟⎜= − −⎟⎜ ⎟⎜⎝ ⎠−

( )52

23 1.2922

0 0

21.5 10 1cp

BF b bπ

μ κ φ−≈ × −

2 4

3 20 44 6616p p

p

n fF

a C C=

Force Fp =JC.B per unit volumeModel

Pinning at surfaces25 (Dew-Hughes 1974)

Pinning at surfaces27 (Dew-Hughes 1987)

Flux shear past point pinning sites30 (Kramer, Labusch31 C66)

Flux shear past point pinning sites30 (Kramer, Brandt33 C66)

Maximum shear strength of lattice27 (Dew-Hughes 1987)

Flux shear along grain boundaries in 2D105 (Pruymboom 1988)

Collective pinning model34 (Larkin-Ovchinnikov)

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Reversible Magnetic Properties of ‘Perfect’ Superconductors

• Below Hc, Type I superconductors are in the Meissner state: current flows in a thin layer around the edge of the superconductor, and there is no magnetic flux in the bulk of the superconductor. (Hc : Thermodynamic Critical Field.)

• In Type II superconductors, between the lower critical field (Hc1), and the upper critical field (Hc2), magnetic flux penetrates into the sample, giving a “mixed” state.

30 -0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0 0.2 0.4 0.8 1

Applied field H (H c 2)

Mag

netiz

aion

M(H

c2)

Magnetization Characteristic100ξ × 80ξ, κ = 5

0.6

Reversible Magnetization Loop• The reversible response of a superconductor

,)12(

)(2

Aβκ −−

−=HΗ

M C2SC 2C0 Hμ

1C0 Hμ

31

M vs. H for a Superconductor Coated With a Normal Metal, κ = 5

• H = 0.05 Hc2

• The material is in the Meissner state

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0 0.2 0.4 0.6 0.8 1

Applied field H (H c 2)

Mag

netiz

aion

M (H

c2)

Magnetization Characteristic100ξ × 80ξ, κ = 5

32

Magnetization Loop

• H = 0.15 Hc2

• The material is in the mixed state

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0 0.2 0.4 0.6 0.8 1

Applied field H (H c 2)

Mag

netiz

aion

M (H

c2)

Magnetization Characteristic100ξ × 80ξ, κ = 5

33

Magnetization Loop

• H = 0.40 Hc2

• Note the nucleation of fluxons at the superconductor-normal boundary

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0 0.2 0.4 0.6 0.8 1

Applied field H (H c 2)

Mag

netiz

aion

M (H

c2)

Magnetization Characteristic100ξ × 80ξ, κ = 5

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Magnetization Loop

• H = 0.70 Hc2

• In the reversible region, one can determine κ

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0 0.2 0.4 0.6 0.8 1

Applied field H (H c 2)

Mag

netiz

aion

M (H

c2)

Magnetization Characteristic100ξ × 80ξ, κ = 5

35

Magnetization Loop

• H = 0.90 Hc2

• The core of the fluxons overlap and the average value of the order parameter drops

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0 0.2 0.4 0.6 0.8 1

Applied field H (H c 2)

Mag

netiz

aion

M (H

c2)

Magnetization Characteristic100ξ × 80ξ, κ = 5

36

Magnetization Loop

• H = 1.00 Hc2

• Eventually the superconductivity is destroyed

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0 0.2 0.4 0.6 0.8 1

Applied field H (H c 2)

Mag

netiz

aion

M (H

c2)

Magnetization Characteristic100ξ × 80ξ, κ = 5

37

Magnetization Loop

• H = 0.50 Hc2

• Note the Abrikosov flux-line-lattice with hexagonal symmetry

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0 0.2 0.4 0.6 0.8 1

Applied field H (H c 2)

Mag

netiz

aion

M (H

c2)

Magnetization Characteristic100ξ × 80ξ, κ = 5

38

Magnetization Loop

• H = 0.00 Hc2

• A few fluxons remain as the inter-fluxon repulsion is lower than the surface pinning

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0 0.2 0.4 0.6 0.8 1

Applied field H (H c 2)

Mag

netiz

aion

M (H

c2)

Magnetization Characteristic100ξ × 80ξ, κ = 5

39

Time-dependent Ginzburg-Landau equations

• These equations were postulated by Schmid (1966), and then derived using microscopic theory in the gapless case by Gor’kov and Eliashberg (1968)

22

20

*2

0 0

1 21 0

1 Re 22

c

e

eTT i

ee i

ϕξ

ϕμ λ

∇ ∂⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞Δ − − Δ + − Δ + + =⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ∂⎛ ⎞ ⎛ ⎞= Δ ∇ − Δ − ∇ +⎜ ⎟ ⎜ ⎟⎜ ⎟ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠

A

J A

1 2 ∆

σ

eiD t

tA

40

Uncertainty/Complexity in G-L theory

• Thermodynamics is seductive but very complex. • There are number of different approaches to deriving the

G-L equations (DeGennes, Campbell, Clem, Pippard, Jackson and Landau and Lifshitz)

• There is confusion in the literature about the length scales on which G-L applies

• TDGL for real systems has only been possible in the last 3 - 5 years

41

Model for a polycrystalline superconductor

• A collection of truncated octahedra – Avoids the

problematic straight channels of a simple cubic system.

42

Obtaining bulk Jc

• The Bean critical state model is used to obtain Jc.

• A symmetrical straight-line Bean profile is fitted to the calculated B-field data

• The E-field is calculated from the ramp rate by Maxwell’s equations. -60 -40 -20 0 20 40 60

0.28

0.29

0.30

0.31

region over which J is calculated

DIff

used

regi

on

DIff

used

regi

on

Loca

l Fie

ld (B

c2)

y-coordinate (ξ)

Down ramp at 0.302Hc2

Up ramp at 0.298Hc2

43

Branching test runs – up, fast

3000 3500 4000 4500 5000 5500 60002

4

6

8

10

12

Emax = +5.33 × 10−4Hc2ρS/κ2ξ

Diffusive resistivity ≈ 31ρS

C

urre

nt D

ensi

ty (1

0−3 H

c2/κ

2 ξ)

Time (t0)

Applied Field Main-line Data Branch Data

0.20

0.25

0.30

0.35

0.40

0.45

App

lied

Fiel

d (H

c2)

44

Layout of granular system

Superconductor Weak superconductor Normal metal Edges matched by periodic boundary conditions

a

b

x

y

45

Time dependant Ginzburg-Landau theory provides the framework for understanding flux pinning

46

Flux motion in low fields (0.43 HC2) along grain boundaries

(a) Order parameter (b) Normal current

-50 0 50

-100

-50

0

50

100

150

20 30 40

-40

-30

-20

a)

-50 0 50

-100

-50

0

50

100

150

b)

0

0.01

0.1

1

10^-7

10^-6

10^-5

10^-4

10^-3

47

Order Parameter at 0.43 Bc2• The motion of flux through the system takes

place predominantly along the grain boundaries.

• TDGL 0.430Hc2 Psi2 Enlarged.avi

48

Normal Current at 0.430 Hc2• The normal current movie shows

that dissipation above Jc occurs principally in the grain boundaries.

• TDGL 0.430Hc2 J Enlarged.avi

49

Flux motion in high fields into the interior of grain boundaries

(a) Order parameter (b) Normal current-50 0 50

-100

-50

0

50

100

150

20 30 40

-40

-30

-20

0

0.01

0.1

1

a)

-50 0 50

-100

-50

0

50

100

150

b)

10^-7

10^-6

10^-5

10^-4

10^-3

50

Order Parameter at 0.942 Hc2

TDGL 0.942Hc2 Psi2 Enlarged.avi

51

Normal Current at 0.942 Hc2

TDGL 0.942Hc2 J Enlarged.avi

52

Conclusion

• Ginzburg-Landau theory is a triumph of physical intuition. It is the central theory for understanding the properties of superconductors in magnetic fields

• Bibliography/electronic version of the talk is available at: http://www.dur.ac.uk/superconductivity.durham/

53

Excellent books for students new to SuperconductivityC P Poole, H A Farach and R J Creswick, Superconductivity (Academic Press Inc, San

Diego, California, 1995)

Excellent books for more established post-graduates in superconductivityM Tinkham, Introduction to Superconductivity (McGraw-Hill Book Co., Singapore, 1996)C P Poole, Handbook of superconductivity (Academic press, 2000)M N Wilson, Superconducting Magnets (Oxford University Press, 1986)J B Ketterson and S N Song, Superconductivity (Cambridge University Press, 1999)J R Waldram, Superconductivity of Metals and Cuprates (IOP Publishing Ltd., London, 1996)

Excellent Books for G-L theoryP G De Gennes, Superconductivity of Metals and Alloys (Addison Wesley Publishing

Company, Redwood City, California, 1989)D R Tilley and J Tilley, Superfluidity and Superconductivity (IOP publishing Ltd., Bristol, 1990)

Excellent Reference Books:D Cardwell and D Ginley, Handbook of Superconducting Materials (IOP, Bristol, 2003)B Seeber, Handbook of Applied Superconductivity (Institute of Physics, Bristol, 1998)E M Landau, I M Lifshitz and L P Pitaevskii, Electrodynamics of Continuous Media

(Butterworth Heinneman, 1960)E Landau, E Lifshitz and L Pitaevskii, Electrodynamics of Continuous Media (Butterworth

Heinneman, 2002)J D Jackson, Classical Electrodynamics (John Wiley and Sons, New York, 1999)G Woan, (CUP, Cambridge, 2003)