11

Last lecture (#3):Last lecture (#3):Flux quantisation in multiples of φ0 – a very intriguing property.Vortices in type-II superconductors and the vortex lattice leadto a very intricate magnetisation curve:

In the vortex state we still have a finite sc order parameter ψacross most of the material – but no perfect Meissnerdiamagnetism. One obvious question to ask:

is the resistance still zero?

-µ0M

BEBc1 Bc2Bc

“vortex rush” M vertical

2nd order M linear

Meissner state

)(8

2200

B

BM c

ln

!"

#µ $%

EBM =! 0µVortex state

22

Lecture 4:Lecture 4:

• Nucleation of superconductivity at Bc2

• Critical fields – a summary• The B-T phase diagram for type-I and type-II sc• Vortex-current and vortex-vortex interactions – interpretation

in terms of the “Magnus effect” and topological charge• Vortex pinning, flux flow, and finite resistance• Critical currents• Experimental observations – hysteresis and vortex liquid state• A few words on applications: superconducting wires and

superconducting high field magnets

• Literature: Waldram chs. 4, 5 & 18, plus Schmidt ch.3

33

At Bc2 we have a 2nd order phase transition into the normalstate. Since |ψ| goes to zero continuously at Bc2, we considerthe linearised GL equation, i.e., the Schrödinger equation

This leads to Landau eigenvalues satisfying

where n is an integer and ωc = 2eB/m.

Bc2 is the maximum field where thiscondition is satisfied (n = kz = 0)

as given in lecture 3 (with η = 1).

LandauTubes

in k-space

An Improved Model for An Improved Model for BBcc22

0)2(2

1 2=++!" #$$eAi

mh

022

122

=++!"

#$%

&+ '(

m

kn z

c

hh

222

||or 0

2

1 0

!"

#$$% ===+

hh

e

mBcc

kz

44

In summary, when λ >> ξ the critical fields are given by:

A quantum of flux φ0 is distributed over a circle of radius of orderλ at Bc1 and of order ξ at Bc2 (at Bc2 the vortex cores essentiallytouch).

The temperature dependences of the critical fields depend onα = a(T – Tc) near Tc. Thus, the fields go to zero linearly as Tgoes to Tc. The GL model involves a low order expansion in ψand in (T – Tc) and thus cannot describe temperaturedependences in the low temperature limit. A more completedescription of the low T behaviour requires a microscopic theory.

)a(T-Tnmne

mcs

s

==== !"

!

!#

µ$

|| ||2

4

22

22

0

h

and 4

, 2

2212122

00ccccc BBBBB ===

!"

#

!$

#

55

BB--TT Phase Diagrams Phase Diagrams

1st order

Meissner state

2nd order

B

T T

B

Meissner state

Vortex state

2nd order

Type-I Type-II

Tc

Tc

66

Critical Current in Type-I SuperconductorsCritical Current in Type-I Superconductors

Consider first the simplest case of a conventional type-I super-conductor. There is an intrinsic limit to how much supercurrentcan be sustained: the field produced by the current must beless than or of the order of Bc. From the figure below the criticalcurrent Ic is given by

Since the current is within the penetration depth λ ofthe surface, the critical current density Jc is thus

Silsbee’s rule for the depairing current density

if λ << r.

A more detailed analysis gives the same qualitativeresult.

0

0 2or

2 µ

!

!

µ ccc

c BrIB

r

I""

!µ!" 02

ccc

B

r

IJ =#

Js

B

λr

77

Critical Current in Type-II SuperconductorsCritical Current in Type-II SuperconductorsVortex-Current InteractionsVortex-Current Interactions

In a type-II superconductor we need to consider the effects of asteady current on vortex lines.

The interaction between the supercurrent and the magnetic fieldleads to a Lorentz force on the supercurrent. For an isolatedvortex and homogeneous background transport current Jtr, sothat Js = Jvortex + Jtr, the total force per unit lengthalong the vortex reduces to

This force is analogous to the Magnus forceon a rotating football:

!

f = Js " B d2r = Jtr " B d2r = Jtr " (#0

)

B )$$

B

Jtr

fJvortex

Js

Js

88

It can be shown that the form of the interaction remains thesame in a more general situation. Jtr is then the current densityat the vortex core due to all current sources other than thevortex itself.1

One interesting consequence of the Lorentz force is that twovortices in a type-II superconductor repel each other, as can beseen from their current patterns:2

1 O. Narayan, J. Phys. A 36, L373 (2003)2 Jtr(r) = φ0 K1(r/λ) / (2π µ0λ3) (appendix, lecture 3), so that theforce is f = φ 02 K1(r/λ) / (2π µ0λ3) per unit length.

B

Jtr

f

Jvortex

99

Pict

ure

cre

dits:

M. Sriniv

asar

ao

Comment on Topological ChargeComment on Topological ChargeWe may think of a vortex as a topological defect in thesuperconducting phase. In general topological defects interactand their interaction can be thought of as arising from theireffective charge, the topological charge. In the case of asuperconductor, the charge is the winding number, Δθ/2π, of thephase around the vortex core. Defects with charges of the samesign usually repel each other (e.g., two superconductingvortices), while defects with charges of opposite sign attract.

Equal charges:energy penaltyrepulsion

Two interacting topologicaldefects in a liquid crystal

Oppositecharges:annihilationattraction

1010

Dissipation Due to Vortex MotionDissipation Due to Vortex MotionReconsider the single vortex in a homogeneous transport currentJtr. If the Lorentz force acting on the vortex leads to vortexmotion at velocity v, this implies that the vortex is subject to apower input per unit volume of vortex of characteristic radius rB

The power is supplied by an electric field ε defined by

P = ε Jtr and from above, ε = B v

which implies that the wire is resistive, withresistivity

The key point is that vortex motionleads to dissipation, and the zeroresistance state breaks down!

vBJv

r

J

r

vfP trtr

BB

22

0 ===!

"

! Lorentz forceper unit volume

trtr J

Bv

J==

!" v

B

Jtr

fJvortex

Js

Js

1111

Vortex Flow in the Presence ofVortex Flow in the Presence ofa Transport Current a Transport Current JJtrtr

BE

Jtr

BE

Defect (possible pinning centre)

vortexv

Jtr

The critical current for dissipationless flow in the vortex state is dependent on vortex pinning by defects! (Appendix 1)

1212

Recall the ideal magnetisation curve:

-µ0M

Vortex pinning has a profound influence on the kind of M(BE)curve we actually observe in experiment – strong hysteresis(Appendix 2):

Hysteresis Hysteresis Due to Vortex PinningDue to Vortex Pinning

zero fieldcooled

BE

BE

-µ0M

ΔM

1313

Pict

ure

cre

dit:

Cav

endis

h L

abora

tory

, A

.P.

Mac

kenzi

e, S

.R.

Julia

n e

t al

.

Irreversibility Line fora Cuprate Superconductor

(Tl2Ba2CuO6, 10mK to 16K)

Mag

net

ic F

ield

(Tes

la)

Temperature (K)

The critical lines The critical lines BBcc11 and and BBcc22 vs. vs. TT do not provide the do not provide thewhole story for the whole story for the BB--TT phase diagram: phase diagram:

1414

Irreversibility LineIrreversibility Line

the irreversibility line. It isparticularly noticeable in thecuprates probably becauseof the high transitiontemperature, low supercon-ducting coherence lengthand low dimensionality. Theregion below Bc2 and aboveBirr may be thought of as avortex liquid. A descriptionof this state requires aconsideration of the effectsof fluctuations in the GLmodel, which is beyond thescope of this course.

In the cuprates in particular the critical current Jc is found tovanish at fields far below Bc2. The zero resistance state withfinite Jc is found only below a new critical field Birr. In the figurethe line Birr(T), below which the resistivity vanishes, is known as

Jc = 0|ψ| > 0

|ψ| = 0

Jc > 0|ψ| > 0 P

ictu

re c

redits:

Cave

ndis

h L

abora

tory

, J.

R.

Cooper

Vortex Liquid

Vortex Lattice

1515

Bc2(T)Magnetic Field – Temperature PhaseDiagram of Cuprate Superconductors

Birr(T)

Bc1(T)

IrreversibilityLine

Pict

ure

cre

dit:

E.M

. Fo

rgan

et

al.

1616

The use of superconducting wires in DC magnets cooled by liquidhelium is now at an advanced stage. Applications include:

• Magnetic resonance Imaging (MRI) systems– a few Tesla• Magnetic cooling to nK – less than 10T• Research magnets – up to 25T (2012)

Use NbTi up to 9T – ductile metal, severecold working to introduce pinning centres.

Use Nb3Sn up to 22T (45T for hybrids) –brittle material, needs to be wound inprecursor form and then fired as a solenoid.

For higher fields, to 25T and beyond, high-Tcmaterials are used, but this is still an activearea of R&D. Some main challenges: toreduce brittleness, improve effectiveness ofpinning and reduce risk of thermal runaway.

ATLA

S s

ole

noid

(CERN

)M

RI

Some Applications of Type-II SuperconductorsSome Applications of Type-II Superconductors

1717

The key to the zero resistance state in high fields is theseemingly unexciting issue of vortex pinning.

Vortices can be pinned by various types of defects in the crystal,such as dislocations or grain boundaries. At a defect |ψ| may besuppressed anyway, so having the vortex core at the defectgives you “two normal regions for the price of one” !energetically favourable. This is a complex problem involving anumber of other factors. It is enough to stress that theeffectiveness of vortex pinning is crucial to attaining high criticalcurrents and low dissipation.

The Lorentz force per volume goes as Jtr B, i.e., it rises as Bincreases. Once it exceeds the pinning force density of thevortex lattice, the material will lose its zero resistance.Generally, we expect the critical value of Jtr when this happensto scale as

Appendix 1: Vortex PinningAppendix 1: Vortex Pinning

BJc

1!

1818

This is roughly consistent with experiment except at low fieldswhere not all pinning centres carry a vortex core and at highfields near Bc2 when Jc clearly vanishes. Also, we get anenhancement of Jc at intermediate fields when the averagepinning centre separation matches the average separation of thevortices. This enhancement in Jc is known as the peak effect(see figure below).

Jc can be made quite high bymetallurgical processes.Paradoxically, defects are good.*Typical values that can beachieved are 5x105 A/cm2 (NbTiat 4 T and 4.2 K) and 106 A/cm2

(YBCO films at 10 T and 77 K).

*Also note that empiricallyimpurities tend to reduce ξ andhence increase Bc2.

Pict

ure

cre

dits:

J. Pet

erm

ann /

A.

M.

Cam

pbel

l &

J.

E. Eve

tts

enhancement

1919

Appendix 2: The Bean ModelAppendix 2: The Bean Model

The Bean Model offers a crude description of the hysteresis loopof a thin slab of a type-II superconductor. The model assumes:

• |∂B/∂x| = µ0|Js| is equal to the critical value µ0Jc (taken to be a constant) and B = BE at the surface;

• when ramping up BE (blue lines in figure on p20) the flux enters at the surface and when ramping down (red lines) the flux leaves near the surface.

There is a critical value of BE where the field B penetrates to thecentre of the slab (near the third blue line from bottom). Whenthe field BE is reduced to zero the flux remains trapped inside thesuperconductor (bottom red line).

2020

The Bean ModelThe Bean Model

x

ramp up

ramp down

B Type-II Superconductor

Start ramp up End

ramp down

Endramp up

Ramp down