Chapter 6...AMS-02 is the Alpha Magnetic Spectrometer, a superconducting particle physics experiment which will be launched on the Space Shuttle and installed on the International

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TT1-Chap6 - 1

Chapter 6

6 Flux Pinning and Critical Currents

6.1 Power Applications of Superconductivity

6.1.1 Examples

6.1.2 Materials Requirements

6.1.3 Superconducting Wires and Tapes

6.2 Critical Current of Superconductors

6.2.1 Depairing Critical Current Density

6.2.2 Depinning Critical Current Density

6.3 Flux Line Pinning

6.4 Magnetization of Hard Superconductors

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TT1-Chap6 - 2

Jc

B T

Jc (T)Jc (B)

Bc2 (T)

Bc2 (0)

Jc (0)

Tc (0)


• power applications require high Tc, Bc2, and Jc

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TT1-Chap6 - 3


• energy transport and storage

SMES (2 MJ)superconducting magnetic energy storage

fault current limiter

6.1.1 Examples

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TT1-Chap6 - 4


Comparison of the amount of space consumed by a superconducting cable (blue) with copper wires carrying the same amount of current

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TT1-Chap6 - 5

• superconducting magnets

magnetic resonance imaging

high energy physics

fusion


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TT1-Chap6 - 6

1m


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TT1-Chap6 - 7

AMS-02 is the Alpha Magnetic Spectrometer, a superconducting particle physics experiment which will be launched on the Space Shuttle and installed on the International Space Station. The project is an international collaboration of 56 research institutes from 16 countries.


http://www.scientificmagnetics.co.uk/special-superconducting-magnets.htm

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TT1-Chap6 - 8


• manufacturability• low cost• availability (sustainability)

• high Tc, Bc2, Jc


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TT1-Chap6 - 9

• important low Tc superconductors

NbTi Nb3Sn

material:

Tc

1:1 alloy

9.6 K

Bc2(T=0) 10.5 - 15 T

intermetallic compound

18 K

23 - 29 T

(application in commercial magnets)

• high Tc superconductors

BSCCO YBCO

material:

Tc

powder in Ag-tube

110 K

Bc2(T=0) 1000 T

thin film on metal tape

91 K

800 T

still under development


material parameters:

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TT1-Chap6 - 10


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TT1-Chap6 - 11


material parameters:

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TT1-Chap6 - 12


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TT1-Chap6 - 15

crit

ical

curr

en

td

en

sity

(n

on

-Cu

) (

A/m

m²)

10

100

1,000

10,000

100,000

1,000,000

0 5 10 15 20 25 30 35

applied field (T)

YBCO: Tape, || Tape-plane, SuperPower

(Used in NHMFL tested Insert Coil 2007)

YBCO: Tape, |_ Tape Plane, SuperPower

(Used in NHMFL tested Insert Coil 2007)

Bi-2212: non-Ag Jc, 427 fil. round wire,

Ag/SC=3 (Hasegawa ASC-2000/MT17-2001)

Nb-Ti: Max @1.9 K for whole LHC NbTi

strand production (CERN, Boutboul '07)

Nb-Ti: Nb-47wt%Ti, 1.8 K, Lee, Naus and

Larbalestier UW-ASC'96

Nb3Sn: Non-Cu Jc Internal Sn OI-ST RRP

1.3 mm, ASC'02/ICMC'03

Nb3Sn: Bronze route int. stab. -VAC-HP,

non-(Cu+Ta) Jc, Thoener et al., Erice '96.

Nb3Sn: 1.8 K Non-Cu Jc Internal Sn OI-ST

RRP 1.3 mm, ASC'02/ICMC'03

Nb3Al: RQHT+2 At.% Cu, 0.4m/s (Iijima et al

2002)

Bi 2223: Rolled 85 Fil. Tape (AmSC) B||,

UW'6/96

Bi 2223: Rolled 85 Fil. Tape (AmSC) B|_,

UW'6/96

MgB2: 4.2 K "high oxygen" film 2, Eom et

al. (UW) Nature 31 May '02

MgB2: Tape - Columbus (Grasso) MEM'06

2212round wire

2223tape B|_

@ 4.2 K unless

otherwise stated

Nb3SnInternal Sn

Nb3Sn1.8 K

2223tape B||

Nb3SnITER

MgB2

film MgB2

tape

Nb3Al

RQHT

1.9 K LHC

Nb-Ti

YBCO B||c

YBCO B||ab


critical current density of superconductors:

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TT1-Chap6 - 17

10

100

1000

10000

0 5 10 15 20 25 30 35 40 45

applied field (T)

J E(A

/mm

²)

YBCO Insert Tape (B|| Tape Plane)

YBCO Insert Tape (B Tape Plane)

MgB2 19Fil 24% Fill (HyperTech)

2212 OI-ST 28% Ceramic Filaments

NbTi LHC Production 38%SC (4.2 K)

Nb3Sn RRP Internal Sn (OI-ST)

Nb3Sn High Sn Bronze Cu:Non-Cu 0.3

YBCO B|| Tape Plane

YBCO B Tape Plane

2212

RRP Nb3Sn

Bronze

Nb3SnMgB2

Nb-Ti

SuperPower

tape used in

record breaking

NHMFL insert

coil 2007

18+1 MgB2/Nb/Cu/Monel

Courtesy M. Tomsic,

2007

427 filament strand

with Ag alloy outer

sheath tested at

NHMFL

Maximal JE for

entire LHC NbTi

strand production

(–) CERN-

T. Boutboul '07,

and (- -) <5 T data

from Boutboul et

al. MT-19, IEEE-

TASC’06)

Complied from

ASC'02 and

ICMC'03 papers

(J. Parrell OI-

ST)

4543 filament High Sn

Bronze-16wt.%Sn-

0.3wt%Ti (Miyazaki-

MT18-IEEE’04)


“engineering” critical current density of superconductors:

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TT1-Chap6 - 19

December 2002 - Compiled by Peter J. Lee

University of Wisconsin-Madison

Applied Superconductivity

Center

crit

ical

cu

rre

nt

de

nsi

ty (

A/m

m²)

10

100

1 000

10 000

0 5 10 15 20 25 30applied field (T)

Nb3Al

Nb-Ti

2212 Round Wire

2223

At 4.2 K Unless

Otherwise Stated

1.8 K

Nb-Ti-Ta

Nb3Sn Internal Sn

1.8 K

Nb3SnBronze Process

Nb3AlITER

2 K

Nb-Ti-Ta

Nb-Ti: Example of Best Industrial Scale Heat Treated

Composites ~1990 (compilation)

Nb-Ti(Fe): 1.9 K, Full-scale multifilamentary billet for

FNAL/LHC (OS-STG) ASC'98

Nb-44wt.%Ti-15wt.%Ta: at 1.8 K, monofil. high field

optimized, unpubl. Lee et al. (UW-ASC) ‘96

Nb-37Ti-22Ta: at 2.05 K, 210 fil. strand, 400 h total HT,

Chernyi et al. (Kharkov), ASC2000

Nb3Sn: Bronze route VAC 62000 filament, non-Cu

0.1µW·m 1.8 K Jc, VAC/NHMFL data courtesy M. Thoener.

Nb3Sn: Non-Cu Jc Internal Sn OI-ST RRP #6555-A, 0.8mm,

LTSW 2002

Nb3Al: Nb stabilized 2-stage JR process (Hitachi,TML-

NRIM,IMR-TU), Fukuda et al. ICMC/ICEC '96

Nb3Al: JAERI strand for ITER TF coil

Bi-2212: non-Ag Jc, 427 fil. round wire, Ag/SC=3

(Hasegawa ASC2000+MT17-2001)

Bi 2223: Rolled 85 Fil. Tape (AmSC) B||, UW'6/96

Bi 2223: Rolled 85 Fil. Tape (AmSC) B|_, UW'6/96


“engineering” critical current density of 100 m cable

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TT1-Chap6 - 20


fabrication of superconducting wires:

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TT1-Chap6 - 21

superconducting wires:NbTi, Nb3Sn in Cu-matrix

cable from high-Tc

superconductor


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TT1-Chap6 - 22


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TT1-Chap6 - 23

pipe coating

stainless pipe

electrical insulation

high temperaturesuperconductor

LN2

LN2

Sumitomo


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TT1-Chap6 - 24


high-Tc wires:

can be made

• Ag is too expensive and too soft• HTS have higher critical fields

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TT1-Chap6 - 25

Preparation of multi-filamentary BiSrCaCu-oxid (2223) tapes in Ag/AgMg-sheath by the powder in tube method Investigation of the structural and superconducting properties

cross-section of 61 filamentary tape

600m tape on a coil for Jc-measurement


http://www.ifw-dresden.de/

http://www.ifw-dresden.de/

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TT1-Chap6 - 26


HTS cable

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TT1-Chap6 - 27


The "AmpaCity" project has been kicked off: The RWE Group and its partners are just about to replace a 1-kilometre-long high-voltage cable connecting two transformer stations in the Ruhr city of Essen with a state-of-the-art superconductor solution. This will mark the longest superconductor cable installation in the world. As part of this project, the Karlsruhe Institute of Technology will analyse suitable superconducting and insulating materials.

The three-phase, concentric 10 kV cable will be produced by Nexans and is designed for a transmission capacity of 40 megawatts.

19. January 2012

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TT1-Chap6 - 28

evaporator230 mm

• low cost process for large area deposition of HTS films

• high quality and reproducibility

heater700°C

O2 oxygen

Y

rotating

vacuum pumpto

vacuum

substrate

Ba

Cu

"Garching-technology": http://www.theva.com/


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TT1-Chap6 - 29


coated conductor:

superconducting film onflexible steel tape

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TT1-Chap6 - 30


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TT1-Chap6 - 31


pulsed laser

deposition

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TT1-Chap6 - 32

fiber for IR laser heating

substrate manipulators

excimer laser optics

target manipulators

pyrometer

casing of RHEED screen and camera

operator tool

AFM/STM system

atomic oxygen source

Laser Molecular Beam Epitaxy, Garching 2006

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TT1-Chap6 - 33

L-MBE: Fully automated target holder

K.-W. Nielsen,S. Geprägs,Th. Brenninger

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TT1-Chap6 - 34

Fully automated control of beam profile

S. Geprägs,M. Opel,Th. Brenninger

computer controlled variation of beam profile and power density

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TT1-Chap6 - 36

6.1.4 Superconducting Bulk Material

bulk superconductors as “permanent magnets”

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TT1-Chap6 - 37


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TT1-Chap6 - 38


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TT1-Chap6 - 39

maximum velocity:

581 km/h (02. 12. 2003)

Jap. Yamanashi MAGLEV-System

MLX01

(42.8 km long test track between Sakaigawaand Akiyama)


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TT1-Chap6 - 41

6.2 Critical Current of Superconductors

• increase of supercurrent density results in increase of velocity of superconducting electron

critical current density: kinetic energy = binding energy of Cooper pairs

• increase of supercurrent results in Lorentz force on flux lines in mixed stateof type-II superconductors

critical current density: Lorentz force = pinning force

depairing critical current density

depinning critical current density

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TT1-Chap6 - 42


• minimization of free enthalpy of superconductor: integration of enthalpy density over whole volume of superconductorminimization by variation of Y and A

revision: Ginzburg-Landau theory:

1. GL-equation

2. GL-equation

2nd characteristic length scale

GL coherence length

Ginzburg-Landau equations:

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TT1-Chap6 - 43


• consider a thin wire with diameter d << xGL

no amplitude variation of order parameter Y across wire

• superconducting material is assumed homogeneous• same current density along the wire

no amplitude variation of order parameter Y along the wire

• we use 1. and 2. GL equation:

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TT1-Chap6 - 44


• we obtain:

condensation energy per Cooper pair

reduction of is just proportional to ratio of kinetic and condensation energy

order parameter decreases due to additional kinetic energy of pairs

• expression for current density:

• determine maximum of Js by setting :

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TT1-Chap6 - 45


• T-dependence of Jc determined by T-dependence of lL and xGL:

close to Tc:

• we can use

• note: according London theory we would expect Jc = Hcth/lL

(London theory does not take into account reduction of OP with increasing Js)

Pb:Bcth ≈ 80 mT, lL ≈ 40 nm Jc ≈ 8 x 1011 A/m²

Nb:Bcth ≈ 200 mT, lL ≈ 40 nm Jc ≈ 2 x 1012 A/m²

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TT1-Chap6 - 46


• Gedanken experiment: what is the critical current of a Pb rod with large diameter d ?

d = 1cm

critical current Ic = Jc ∙ A = Jc ∙ p (d/2)² ≈ 6 x 107 A ????

London theory:

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TT1-Chap6 - 47

d = 1cm

lL ≈ 40 nm

critical current:

Ic = Jc ∙ A = Jc ∙ p d lL ≈ 1 x 103 A

• supercurrent flow only within surface layer of thickness lL


technical critical current density:

Jc = Ic / ∙ p (d/2)² ≈ 10 A/mm²

comparable to Cu-wire

• solution: - use multifilament wire with d < lL

difficult to fabricate

- use type-II superconductor

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TT1-Chap6 - 48


• type-II superconductors:partial field penetration above Bc1

Bi > 0 for Bext > Bc1

Shubnikov phase between Bc1 ≤ Bext ≤ Bc2

upper and lower critical fields Bc1 and Bc2

200 nm

• in mixed state field penetratesthe superconductor

current flow is not restrictedto thin surface layer

high values of Bc2

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TT1-Chap6 - 49


• extreme type-II superconductors (k >> 1) have very high Bc2 high field operation

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TT1-Chap6 - 50


• how does the transport current interact with the vortex lattice ??

µ0Hext

H(r)y0

y(r)

0 x l r

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TT1-Chap6 - 51

- force on single pair:

fL results in force on charge carriers, which cannot leave conductor

flux motion perpendicular to applied transport current


• interaction of a transport current with the vortex lattice:

Fnetto

total current density

for charge density r:

force density charge density

Jtr

Jcirc

Bx

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TT1-Chap6 - 52


Jtr

Jcirc

B small total superfluid velocity small total FL

large total superfluid velocity large total FL

net force

core offlux line

• analogy to Magnus force

• origin of force acting on a flux line (plausibility check)

• Benoulli‘s principle:an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy

• without transport current: zero net force• contributions of circulating current cancel

+ =

+ =

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TT1-Chap6 - 53

0

• FL causes motion of the flux line- what is the velocity vL of the flux line (depends on damping)- what is the work done by the Lorentz force

homogeneous


• calculation of the total Lorentz force on single flux line:

• force per unit length of flux line:

Jtr

FL

B

L

B(r)y0

y(r)

0r

x l

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TT1-Chap6 - 54


• dissipation by moving flux line:- motion with velocity vL induces electric field:- E is parallel to Jtr acts like resistive voltage

• Faraday's law of induction:The induced electromotive force (EMF) in any closed circuit is equal to the time rate of change of the magnetic flux through the circuit.

• electromotive force (EMF):

flux density inside loop doesnot change

(no flux change)

vL

A

B

JtrE

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TT1-Chap6 - 55

sample widthsample length cross-sectional area

superconductor shows resistance: flux-flow resistance


• generated power for - single flux line:- N flux lines:

voltage drop along sample

• power balance:

L

bB

other point of view: power balance

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TT1-Chap6 - 56

• assumption: single flux line in sample

• phase difference between sample ends:

• integration path 2:

• note: only the phase factor eiq must be unambiguous


• phase change due to flux motion:

2p

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TT1-Chap6 - 57

• crossing of single flux line changes phase differenceby 𝜑 = 𝜃2 − 𝜃1 = 2𝜋

• required time for crossing: 𝛿𝑡 = 𝑏/𝑣𝐿

q1

q2

temporal change of phase difference


• moving flux line:

vL

N

b

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TT1-Chap6 - 58

- we make use of

"2. Josephson Equation“


• relation between change of phase difference and electric field:

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TT1-Chap6 - 59

- behind the flux-line: |Ψ| increases

recombination of pairs by phonon emission:

(a) pair breaking and recombination:

Ψ 𝑥, 𝑡

x

pair breakingrecombination

vL

- in front of flux-line: |Ψ| decreases

pairs have to break up

- pair breaking due to absorption of phonons:

- finite phonon lifetime delays thermal equilibrium:

irreversible process friction, viscous flow

- electric energy is transferred to heat


• power dissipation during vortex motion:

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TT1-Chap6 - 60

(b) eddy current losses:

eddy current with ohmic losses

core of the flux line is considered as normal conductor

• both mechanisms: E vL friction force

• balance between Lorentz and friction force: vL becomes stationary

vL Jtr vL E

• we expect: Jtr E Ohm‘s law zero critical current density

B(x,t)

x

dB/dt

vF


• power dissipation during vortex motion:

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• Ic depends on defect density

• inhomogeneities pin flux-lines: flux pinning

• vL = 0 is caused by flux pinning no work, no dissipation

no voltage drop

• experimental result:

„technically usable“ critical current

ideal superconductor

superconductor with low defect density

superconductor with high defect density

V

IIc


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precipitatewith small

or normalconducting

• at precipitate: vortex core causes no additional loss in condensation energy

x

condensation energyis lost

„pinning force"

• most effective, if defect size x

• motion of vortex core costs energy (condensation energy)

• effective binding forced at precipitate


• pinning mechanism:

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r

Vpin

• pinning force: reduce rp to increase Fp

optimum: rp ≈ x (vortex size ≈ pin size)

reduced condensation energy at defect

rpcondensation energy:

homgeneous for ideal superconductor

• for high-temperature superconductors: x ≈ 1 nm, Econ large

very small pinning sites required large pinning force

• problem for HTS: thermally activated escape of flux line (thermally activated flux flow: TAFF)

(x small and T large)

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• so far: pinning of a single flux line

• now: pinning of a flux line lattice

complicated problem:- flux line lattice is elastic object

(stiffness of the lattice, flux lines can bend)- pinning potential is disordered

• example: net pinning force of completely stiff flux line lattice by statistical pinningpotential vanishes flux line lattice has to deform to adopt to pinning potential even if a single flux line does not sit in a potential well, it is pinned by

the interaction with the other flux lines (rigidity of the lattice) collective pinning theory

see e.g.Vortices in high-temperature superconductorsG. Blatter, M. V. Feigel'man, V. B. Geshkenbein, A. I. Larkin, and V. M. VinokurRev. Mod. Phys. 66, 1125 (1994)

https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.66.1125

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important for superconducting thin films

relative length difference of flux line at different positions may be large

x

surface

surface roughness


• surface pinning:

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• decrease of Jc with increasing B: - several flux lines per pinning site

• decrease of Jc with increasing T: - thermally activated flux motion- reduced condensation energy

E

JJc


• Jc as a function of temperature and applied magnetic field:

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• pinning prevents flux motion, i.e. penetration and exit of flux lines

• Bi is inhomogeneous within the sample

hysteresis loop !

• Bc1 and Bc2 stay the same

remanentflux

Bi

BaBc1 Bc2-Bc1-Bc2

virgin curve

ideal SC


• flux line pinning in an external field:

magnetization curve

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• sample surface: jump ideal magnetization curve

B

SLx

Ba < Bc1 (Meißner)

Ba Bc2

Ba increases

Bi

BaBc1 Bc2

• within superconductor: field gradient gradient of flux density

• gradient decreases with increasing magnetic field

• flux lines repel each other: motion if repulsion > pinning force


• field distribution in sample:

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Bi

Bc1 Bc2-Bc1

Ba

B

x0

• gradient changes sign

• Ba < - Bc1: flux lines with opposite direction penetrate

recombination with frozen-in flux lines inside the superconductor


• field distribution in sample (demagnetization):

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• flux gradient shielding current

Ampère‘s law

• macroscopic average:

• here:

• for small Bi,z/x: J < Jc flux lines are pinned x

y

• for large Bi,z/x: J > Jc flux lines move

"critical state"

• note:

x

B

• motion until J = Jc

measurement of Bi,z(x)/x: Jc

Jc decreases with increasing Bi smaller slope


• Bean model:

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• magneto-optical imaging of flux distribution

Meißner state

critical state

remanent state

B

SL x

B

x0

SL

Documents

Chapter 6...AMS-02 is the Alpha Magnetic Spectrometer, a superconducting particle physics experiment which will be launched on the Space Shuttle and installed on the International