Superconductivity and its applications - UNIGESuperconductivity: Physics and Applications Chapter 4...

Carmine SENATORE

Superconductivity and its applications

Lecture 2

Département de Physique de la Matière QuantiqueUniversité de Genève

Previously, in lecture 1

If it is a superconductor, then…

temperature

resi

stiv

ity

Tc

0

1

T < Tc

H < Hc

2 3

Type-I and Type-II superconductorsPreviously, in lecture 1

H

-4πM

Hc0 H < Hc

Type-IHc 10-3 – 10-2 T

H

-4πM

Hc20 Hc1H < Hc1 Hc1 <H < Hc2

Type-IIHc2 10 – 102 T

Previously, in lecture 1

2

8c

S N

HG G

cH Hif1)

S NS S2) higher order in the superconducting state

3) From the 1st London equation * 2

*24 L

S

m c

n e

with

penetration depth

h

z0

insulator superconductor

H L

z

h(z) He

Landau theory of the order-disorder transition

The order parameter is a quantity which is zero in one phase

(usually above the critical point) and non-zero in the other

Example: para- / ferro- magnetic transition

order parameter( )

( )(0)

M Tm T

M

1i

i

MV

where

T > TCuriem = 0

T < TCuriem 0


The Landau expansion for the free energy density close to TCurie

2 40

1( , ) ( , )

2F r T F r T m m

At the equilibrium 0F

m

32 2 0m m

2 0m m

Two solutions: 0m 2m

and


and are function of the temperature

20 1 2

1( ) ( ) ( ) ...

2Curie CurieT T T T T

20 1 2

1( ) ( ) ( ) ...

2Curie CurieT T T T T

From the minimization conditions

1 CurieT T

0

Two solutions: 0m 2m

and


2 m

Temperature dependence of the order parameter close to TCurie

112

1 2

0Curiem T T

1950: Ginzburg-Landau Theory of Superconductivity

1962 2003

Ginzburg-Landau Theory of Superconductivity

In the G-L theory ( )sn r where ( ) ( ) ir r e

2 4( , ) ( , )

2S NF r T F r T

In the London model ns does not depend on the position

The superfluid density ns is the order parameter

2( )r

Free energy expansion in the zero-field case

2 2

*2m

rigidity of the order parameter


2 4( , ) ( , )

2S NF r T F r T

Free energy expansion in the zero-field case

2 2

*2m

Free energy expansion for B ≠ 0

qp p A

c B A In electrodynamics where

*ei i a

c For the momentum operator

2* 2

2 4

*

1( , ) ( , )

2 82S N

e hF r T F r T i a

cm

2* 2

2 4

*

1( , ) ( , )

2 82S N

e hF r T F r T i a

cm


Ginzburg-Landau Theory of Superconductivity2

* 22 4

*

1( , ) ( , )

2 82S N

e hF r T F r T i a

cm

2* 2

2 4

*

1( ) ( , )

2 82S N

V

e hF T dr F r T i a

cm

* * * , , , ,S S SF F a a F a

*

V

SF dr a

Energy density

( ) ( , )S S

V

F T dr F r T Energy

Minimization procedure variations of , *, and a

0SF

*0SF

0SF

a

And set

Fosshein & Sudbø, Superconductivity: Physics and Applications, pp. 117-120

The two Ginzburg-Landau equations

0SF

*0SF

0SF

a

2*

2

*

1 0

2

ei a

cm

1st G-L equation

* *2

2* ** *2

e eJ a

m i m c

2nd G-L equation

Zero-field case deep inside superconductor2

* 22 4

*

1( , ) ( , )

2 82S N

e hF r T F r T i a

cm

2 4

2S NF F

becomes

2*

2

*

1 0

2

ei a

cm

And the 1st G-L equation

2 0 becomes

The energy density

2

0 Two solutions: and

Zero-field case deep inside superconductor

On the sign of and

2 1 1 2

4

3

2

1

α > 0 andβ < 0

2 1 1 2

12

10

8

6

4

2

α ≤ 0 and β < 0

FS FS

2 1 1 2

2

4

6

8

10

12

2 1 1 2

1

2

3

4

α 0 and β > 0 α < 0 andβ > 0

FS FS


2

112

1 2

0cT T

2 4

2S NF F

21

2

The solution for the superconducting state

1 cT T 0 Choosing and

and


2 4

2S NF F

21

2

Relation between , and Hc

2

8c

S N

HF F

and

22

2 8cH

Thus

Zero-field case near superconductor boundary2

*2

*

1 0

2

ei a

cm

The 1st G-L equation

becomes 22

*

1 0

2i

m

We define f

2

where

In one dimension2 2

3* 2

02

d ff f

m dx

22 2

2(1 ) 0

d ff f

dx with

22

*2m

0 0f x with the boundary condition

Zero-field case near superconductor boundary

22 2

2(1 ) 0

d ff f

dx with

22

*

1( )

1 /2 ( ) c

TT Tm T

tanh2

xf

tanh2

x

The solution is

x0


The coherence length is*2m

H 0 case near superconductor boundary

The 2nd G-L equation * *2

2* ** *2

e eJ a

m i m c

*22

*

eJ a

m c

*22

*

eh

m c

4 h J

c

2 2 0h h with* 2

2 *24

m c

e

If is small *2

2

*

eJ a

m c see C.P. Poole, Superconductivity (2007), pp. 150-154

H 0 case near superconductor boundary

2 2 0h h with* 2

2 *24

m c

e

h

x0


H ( )x

h x He

* 2

*24L

S

m c

n e

The London penetration depth is

Flux exclusion and retention

Flux exclusion & retention : the resistenceless circuitThe total magnetic flux threading a closed resistanceless circuit cannot change so long as the circuit remains resistanceless.

1. The circuit is cooled below Tc in an applied field Ba . The magnetic flux in the circuit is = A Ba

2. The value of Ba is changed

adB dIA RI L

dt dt ( ) (t) constantaLI t AB

Flux quantization

The 2nd G-L equation * *2

2* ** *2

e eJ a

m i m c

2* * 2 ( )i r ( )( ) i rr e

* *2

2 2

* *2

2

e eJ i a

m i m c

*

2 **2

m c ca J

ee

*

*2 2 *

m Jdl cadl dl

e e

Flux quantization

*

*2 2 *

m Jdl cadl dl

e e

S

adl a ndS

( )S

S

h ndS h is a single-value function

2dl n

*

*2 2 *( )S

m Jdl hch n

e e

*( )S

hch n

e 0n

Calculation of the domain-wall energy

<<

At a domain wall

magnetic energy is gained

condensation energy is lost

2

18

cHE A

2

28

cHE A

2

8cH

E A

0 2

08

cHE A

<<

Calculation of the domain-wall energy

Ginzburg-Landau parameter

1

2 0E

Type-I superconductor

1

2 0E

Type-II superconductor

Bibliography

Fosshein & SudbøSuperconductivity: Physics and ApplicationsChapter 4

TinkhamIntroduction to SuperconductivityChapter 4

De GennesSuperconductivity of Metals and AlloysChapter 6

Rose-Innes & RhoderickIntroduction to SuperconductivityChapter 1

Superconductivity and its applications - UNIGESuperconductivity: Physics and Applications Chapter 4...

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