ginzburg-landau theory - durham universitycommunity.dur.ac.uk/superconductivity.durham... ·...
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Ginzburg-Landau Theory
- flux quantisation and critical fields
Prof. Damian Hampshire Durham University, UK
Rigi Kaltbad Switzerland June 2007
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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
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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.
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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
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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
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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
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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
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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.
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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)|
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The Mixed state
λ: depth of the current flowd: fluxon – fluxon spacingIf B increases, d decreases.
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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.
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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.
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Flux Quantisation
• By considering the persistent current as a wave, phase coherence demands (n: integer) - leads to flux quantisation. k: momentum of the superelectrons.
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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
=
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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
=
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Lower and upper critical fields
1 ln2C
cH
H κκ
≈
22 1c C CH H Hκ κ≈ ≈
00 2 22cH
φμπξ
=
0cH
αμ β
=
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Time evolution of order parameterFlux entering a superconductor
This simulation is simply applying an instantaneous fields of 0.5Hc2.
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Time evolution of order parameter
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Time evolution of order parameter
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Time evolution of order parameter
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Time evolution of order parameter
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Time evolution of order parameter
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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
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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
= =
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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.
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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’)
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Model for a polycrystalline superconductor
• A collection of truncated octahedra – Avoids the
problematic straight channels of a simple cubic system.
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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
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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)
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Layout of granular system
Superconductor Weak superconductor Normal metal Edges matched by periodic boundary conditions
a
b
x
y
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Time dependant Ginzburg-Landau theory provides the framework for understanding flux pinning
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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
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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
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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
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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
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Order Parameter at 0.942 Hc2
TDGL 0.942Hc2 Psi2 Enlarged.avi
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Normal Current at 0.942 Hc2
TDGL 0.942Hc2 J Enlarged.avi
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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/
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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)