Carmine SENATORE

Superconductivity and its applications

Lecture 4

http://supra.unige.ch

Group of Applied SuperconductivityDepartment of Quantum Matter Physics

University of Geneva, Switzerland

Type-II superconductors: Mixed state and quantized vortices

• Magnetic flux penetrates beyond Hc1

1

2E 0

wall energy

• Being the wall energy negative, the system prefers to maximize the walls

• The entering flux is fractionated in

vortices, each one carrying a flux

quantum hc

e 0

2

Previously, in lecture 3

Previously, in lecture 3From the Ginzburg-Landau equations

In a type-II superconductor, the critical fields are related to the characteristic lengths

cc

HH ln ln

0

1 24 2

c cH H

0

2 22

2

The structure of the vortex lattice

r(r ) tanh

2h(r ) ln .

r

02

0 122

From the solution of the linearized 1st G-L equation at Hc2

n

nqx

H0

2

n nC C and

Interaction between vortices

vortex h

0

1 08

and in the case of 2 vortices

vortices h r h r h r h r

02 1 1 1 2 2 1 2 2

8

h r h r

0 01 1 1 22

8 4

vortex interaction 12

In lecture 3, we found

Interaction between vortices

The force of vortex 1 on vortex 2 is

The obvious generalization to an arbitrary array is

s ˆf J zc

0

Js is the total supercurrent due to all other vortices Jarray + any

transport current Jext at the vortex core position.

Obviously, at equilibrium

array ˆJ zc

0 0

interaction ˆf J r zc

02 1 2

Vortex motion and dissipation: Flux Flow

Let’s focus on the effects of a transport current Jext

On a single vortex

extJc

ˆf z

0

On the vortex lattice

v ext vF f zn f J nc

0 ˆ

ext

BJ

c

Therefore, vortices tend to move transverse to Jext . If v is their velocity

vE B

c DISSIPATION !!

Vortex motion and dissipation: Flux Flow

Bardeen and Stephen [PR 140 (1965) A1197] showed that the vortex

velocity v is dumped by a viscous drag term

ext LJ vc

0

And ff

EB

J c

0

2

vE B

c

The following form is predicted for

c

n

B

c

0 2

2

And thus ff n

c

B

B

2

Normal fraction in the

SC, occupied by the

vortex cores

condensation condensation magG G defect G vortex G

condensation magG G defect G

Vortex-defect interaction

vortex defect

Force to extract the vortex from the defect pf U r

Defects are impurities, grain boundaries and any spatial inhomogeneity, whose

size is comparable with and

extf Jc

0

Vortex-defect interaction

Force exerted from Jext

pf

Pinning Force exerted

from defects

cJc

0

Jc is the critical current density

pf f v 0 0If then and

pf f v 0 0then andIf

Only superconductors with defects are truly superconducting ( = 0 ) !!

Critical state: the Bean model

BF J

c H J

c

4+

z

yx

BFor an infinite slab in parallel field

JBF

c

dBB

dx

1

4c

P

J BF

c

In the critical state pF Fc

dBJ

dx c

4

N.B. The critical current density Jc is different from the depairing current Jd

Depairing current * cs d d

Hn m v J

c

22 2 2

2

1 2

2 8c

d

HJ c

4

vc

dnJ

dx c

0

4

C.P. Bean, Phys. Rev. Lett. 8 (1962) 250

Critical state: the Bean model

*cB J a

c

4

Critical state: Pinning strength

The critical current density Jc depends on the type, size and distribution of the

pinning centers

Critical state: Reversing field

The shaded area

corresponds to the sample

magnetization

Critical state: Hysteresis loop

*B B0

1

2

*B B0

5

4

*B B0 3

cM J a

Critical state: Hysteresis loop

Critical state: Hysteresis and losses

*

B

B 0

Q MdB B 20

Superconducting wires are multifilamentary. WHY ?

Bi2223

Bi2212

MgB2

Nb3Sn

NbTi

Superconducting wires are multifilamentary. WHY ?

cM J d d is the filament size

cM nJ d

d is the filament size

n is the number of filaments

M

B

With the subdivision of the superconducting layer in filaments, hysteretic

losses are reduced but the critical current density Jc is unchanged

Extracting Jc from magnetization

Knowing that it is possible to calculate the

expression of M for a given, constant JV

M r J dVV

1

2

Sphere c

MJ

R

3

Slab in perpendicular field

Slab in parallel field

Cylinder in parallel field

Cylinder in perpendicular field

c

MJ

d

2

c

MJ

b b a

3

3

c

MJ

R

3

2

c

MJ

R

3

8

ab

Field dependence of Jc

-10 -8 -6 -4 -2 0 2 4 6 8 10-3

-2

-1

0

1

2

3

m [

emu

]

B [Tesla]

T = 7.5 K

Nb3Sn

M M B c c J J B

Field dependence of Jc

0 1 2 3 4 5 65

10

100

@4.2K

M

[e

mu/c

m3]

B [Tesla]

LHC dipole NbTi strand

0 1 2 3 4 5 6 7 80.5

1

10

50

Jc [

kA

/mm

2]

B [Tesla]

Reference, transport measurement

LHC dipole NbTi strand

@4.2K

c

MJ

R

3

8

-6 -4 -2 0 2 4 6-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

m [

emu

]

B [Tesla]

LHC dipole NbTi strand

@4.2K

Temperature dependence of Jc

-10 -8 -6 -4 -2 0 2 4 6 8 10-3

-2

-1

0

1

2

3

m [

emu

]

B [Tesla]

T = 7.5 KNb3Sn

-10 -8 -6 -4 -2 0 2 4 6 8 10-3

-2

-1

0

1

2

3

10 K

m [

emu

]

B [Tesla]

T = 7.5 KNb3Sn

-10 -8 -6 -4 -2 0 2 4 6 8 10-3

-2

-1

0

1

2

3

12.5 K10 K

m [

emu

]

B [Tesla]

T = 7.5 KNb3Sn

-10 -8 -6 -4 -2 0 2 4 6 8 10-3

-2

-1

0

1

2

3

15 K12.5 K

10 K

m [

emu

]

B [Tesla]

T = 7.5 KNb3Sn

M M B,T c c J J B,T

E-J relation for a type-II superconductor

E

J0

Jc

pinning: Bean Model

E

J0

Jc

ff n

c

B

B

2

Bean Model + flux flow

E-J relation for a superconductor in the mixed state

E

J

0

ff n

c

B

B

2

w/o pinning: flux flow

The effect of thermal excitation was not yet included

Beyond Bean: Thermally activated flux creep

BU/ k TR e 0

In the presence of an external current

B B BU / k T U / k T U / k TR R R e e e

0

Thermally activated flux creep

B B BU / k T U / k T U / k TR R R e e e

0

U U J BU U / k TR R e

0

In a first approximation

where

The B-profile relaxes with t

Thermally activated flux creep

BU U J / k TR R e

0

E B

x t

+

In the infinite slab – parallel field geometry

E vB

2) J is constant in space

Thermally activated flux creep: Anderson-Kim model

The solution for J is Bk T tJ t J ln

U t

0

0 0

1

In the Anderson-Kim model where U U B,T0 0

JU U

J

0

0

1

P.W. Anderson and Y.B. Kim, RMP 36 (1964) 39

BH

0

1) B << Hd 2

Thermally activated flux creep: Anderson-Kim model

Bk T tJ t J ln

U t

0

0 0

1

The relaxation rate S is directly proportional to T and inversely proportional to U0

BdM k TS

M d lnt U

0 0

1

M J

The current density J and the magnetization M both relax with the logarithm of time

Measuring S as a function of B and T, we have an experimental access to the pinning energy

U U B,T0 0

Magnetic relaxation experiments

1 10 100 1000 10000650

700

750

800

850

900

@ 10K, 3T

MgB2

MgB2 + 10wt% SiC

MgB1.9

C0.1

J c [A

/mm

2]

time [s]

M

B

Typical values of the activation energy U0 span in the range 10-100 meV

(~100-1000 K)

Bibliography

Fosshein & SudbøSuperconductivity: Physics and ApplicationsChapter 8

TinkhamIntroduction to SuperconductivityChapter 5