department of physics and astronomy university of leeds · department of physics and astronomy...

µµSR SR andandSuperconductivitySuperconductivity

Bob Cywinski

Department of Physics and AstronomyUniversity of Leeds

Leeds LS2 9JT

Muon Training Course, February 2005

Complementarity Muon Training Course Feb 2005

“Superconductivity isperhaps the mostremarkable physicalproperty in theUniverse”

David Pines


The superconducting elementsThe superconducting elements

Li Be0.026

B C N O F Ne

Na Mg Al1.1410

Si P S Cl Ar

K Ca Sc Ti0.3910

V5.38142

Cr Mn Fe Co Ni Cu Zn0.8755.3

Ga1.0915.1

Ge As Se Br Kr

Rb Sr Y Zr0.5464.7

Nb9.5198

Mo0.929.5

Tc7.77141

Ru0.51

7

Rh0.03

5

Pd Ag Cd0.56

3

In3.429.3

Sn3.7230

Sb Te I Xe

Cs Ba La6.0110

Hf0.12

Ta4.483

83

W0.0120.1

Re1.420

Os0.65516.5

Ir0.141.9

Pt Au Hg4.153

41

Tl2.3917

Pb7.1980

Bi Po At Rn

Transition temperatures (K)Critical magnetic fields at absolute zero (mT)

Nb(Niobium)

Tc=9KHc=0.2T

Fe(iron)Tc=1K

(at 20GPa)

Fe(iron)Tc=1K

(at 20GPa)

Superconductivity in alloys and oxidesSuperconductivity in alloys and oxides

1910 1930 1950 1970 1990

20

40

60

80

100

120

140

160

Su

per

con

du

ctin

g t

ran

siti

on

tem

per

atu

re (

K)

Hg Pb NbNbCNbC NbNNbN

V3SiV3Si

Nb3SnNb3Sn Nb3GeNb3Ge(LaBa)CuO(LaBa)CuO

YBa2Cu3O7YBa2Cu3O7

BiCaSrCuOBiCaSrCuO

TlBaCaCuOTlBaCaCuO

HgBa2Ca2Cu3O9HgBa2Ca2Cu3O9

HgBa2Ca2Cu3O9(under pressure)HgBa2Ca2Cu3O9(under pressure)

Liquid Nitrogen temperature (77K)


H3C

H3CCH3

CH3

HS

Se

TMTSF

BEDT-TTF

EE

Families of SuperconductorsFamilies of Superconductors

A

DD

A. High temperature cupratesB. BuckminsterfullerenesC. Chevrel phases (MxMo6X8)D. RNi2B2CE. Organics (eg (BEDT-TTF)2X

B

C


The two characteristic length scalesThe two characteristic length scales

ξ - the coherence length

The length scale overwhich the superconductingwave function Ψ varies

λ - the penetration depth

The length scale overwhich the flux densityvaries


Type I superconductivity; Type I superconductivity; ξξ>>λλ

This is the intermediate state

Creating a surfacebetween normal andsuperconducting regionscosts energy…..

…. hence relatively few thicknormal “domains” form(under conditions of largedemagnetising factors)


Type II superconductivity; Type II superconductivity; λλ>>ξξ

This is the mixed state

Here the surface energy isnegative, ie flux penetrationoccurs spontaneously toreduce energy

Therefore as many smallregions of normal domains(flux lines) form as possible


The mixed state in Type II superconductorsThe mixed state in Type II superconductors

BHc1< H <Hc2

The bulk is diamagnetic but it isthreaded with normal cores

The flux within each core is generatedby a vortex of supercurrent

Hc2Hc1H

0

-M


The flux latticeThe flux lattice

22µµmm

10µ10µmm


The flux line latticeThe flux line lattice

Lowerdensity offlux lines

Lowerdensity offlux lines

Defectsand

disorder

Defectsand

disorder

Hexagonallattice

Hexagonallattice

flux linecurvatureflux line

curvature


Field distributionsField distributionsIn the London limit (ξ is very small) the variation of fieldaround a vortex is

λπλΦ= r

K2

)r(B o2o

where Ko is a Hankel function of zero order, and

λ<<<<ξ

λπλΦ⇒ rfor

rln

2)r(B

2o

λ>>λ−λπλ

Φ⇒ rfor)rexp(r

2)r(B

2o

For fields somewhat above Hc1 typical flux line separationis a<λ and flux lines overlap

Since each flux line carries one flux quantum Φo (=h/2e)the average internal flux density for triangular and squareflux lattices are respectively

2o

T a23

BΦ= 2

oS a

BΦ=


Small angle neutron scatteringSmall angle neutron scattering

R

scatteringangle 2θ

sample

multi-detector

64x64cm2

incidentneutron beam

B

L


Flux Flux distrbutionsdistrbutions


Probing the flux lattice with muonsProbing the flux lattice with muons

100nm100nm


Calculating the field distributionCalculating the field distributionFor an ideal flux lattice the internal flux density is periodic and

)rGiexp(bB)r(BG

G ⋅= ∑where G is a reciprocal lattice vector of the flux lattice

In the London limit the fourier components 22GG1

1b

λ+=

The second moment of the field distribution is given by

( ) 2/1

cellunit

222 drB)r(BB 21

−=∆ ∫

Therefore2/1

0G

2G

2 bBB 21

=∆ ∑

≠

2/1

0G222 )G1(

1B

λ+= ∑

≠

For H>>Hc1 =

ξλ

πλΦ

ln4 2

o (so that λ .G >> 1) we find

2/1

4

2o2 00371.0

B 21

λΦ=∆


Relating Relating σσ to the penetration depth to the penetration depthThe effects of distortions and defects in the flux latticeoften smear the flux distribution, p(B), and lead to aGauusian form for p(B)

In this case the transverse field relaxation, Gx(t) is alsoGaussian: 22t

x e)t(G σ−=

2

22 2

Bµγσ=∆and

with σ in µs-1 and λ in nm.2

75780λ

=σ

The gaussian approximation thus provides a simplerelationship between σ and λ, -and is often a reasonable approximation

But ideally Max Ent should be used to extract the field profile


LuNiLuNi22BB22C C TTcc=16K=16K

Time (µs)

0.0 0.5 1.0 1.5 2.0

Asy

mm

etry

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

T=17K

0.0 0.5 1.0 1.5 2.0

-0.10

-0.05

0.00

0.05

0.10

Time (µs)

Asy

mm

etry

T=12.2K

0.0 0.5 1.0 1.5 2.0

-0.05

0.00

0.05

Time (µs)A

sym

met

ry

T=1.4K

λ=226nm


Temperature dependence of Temperature dependence of λλ

Temperature (K)

0 5 10 15

σ (µ

s-1)

0.0

0.5

1.0

1.5

LuNi2B2C

40mT

N=4.7

(Y0.5La0.5)Ni2B2C40mT

Temperature (K)

0 1 2 3 4 5 6 7

σ(µs

- 1)

0.0

0.1

0.2

0.3

N=2.0

Temperature (K)

0 2 4 6 8 10

Dep

olar

isat

ion

rate

, σ (

µ s- 1

)

0

1

2

3

4

Y(Ni1-xCox)2B2C

x=0.05

x=0.10

N=3.1

N=2.0

40mT

−σ=σ

N

CTT

1)0()T(

Hillier and Cywinski


YBYB66, , TTcc=6K=6K

Field (T)

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Dep

olar

isat

ion

rate

, σ (µ

s-1 )

0.0

0.5

1.0

1.5

YB6T=1.4K

22

22

GG1

)4Gexp(b

λ+ξ−=

As the applied field increases theeffects of the core size, ξ, mustbe taken into account

is a reasonable approximation,yielding both λ and ξ from σ

λ=192nm, ξ = 33nm

Asy

mm

etry

-0.2

-0.1

0.0

0.1

0.2

Time (µs)

0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.2

-0.1

0.0

0.1

0.2

0.3

-0.2

-0.1

0.0

0.1

0.2

Field (mT)0 35 40 45

P(B)

0

5

10

15

20

(a)

(b)

(c)

4.4K

1.4K

A D Hillier and R CywinskiApplied Mag Res 13 (1997)


Amorphous Amorphous ZrZr-Fe - using max -Fe - using max entent

Field (mT)

38 39 40 41 42

P(B

)

0

20

40

60

80

100

Field (mT)

39 40 41 42

P(B

)

0

25

50

75

100

a-Zra-Zr7676FeFe2424

Tc = 1.7K , λ = 500nm, ξ = 29nm

Manuel and Kilcoyne, 2002


Flux lattice meltingFlux lattice melting

The effects of vortex latticemelting in the hightemperature can be measuredvia the asymmetry of P(B), asdefined by

α = <∆B3>1/3/ <∆B2>1/2

The field dependence of themelting temperature showsgood agreement with theory

Lee et al PRB 55 (1997) 5666


Uemura’s Uemura’s universal universal correlationscorrelations

(PRL 66 (1991) 2665)


The The Uemura Uemura plotplot

2/1

ees

e

l1

rn4m/*m

)0(

ξ+π

=λ

The general London formula gives

so in the clean limit (ξ/le <<1), we have

*m)0(n

)0( s∝σ

Note that for a quasi-2d non-interacting electron gas the Fermitemperature is given by

*mn

)(Tk d2s2FB π= h

Uemura’s result therefore implies a direct correlationbetween TC and TF


The The Uemura Uemura plotplot

*mn)3)(2/(Tk 3/2

s3/222

FB π= h

For a 3d system, the Fermi temperature is given by

2

3/1e

2B

32

m*nk3 h

π=γ

As the Sommerfeld constant, γ, is given by

λ(0) , and hence σ(0), can then be combined with γ toprovide

4/14/3FB )0(Tk −γσ∝

This is the basis of the generalised Uemura plot of Tc v TF

Uemura and CywinskiMuon Science p165-172 (Editors: S L Lee, S HKilcoyne and R Cywinski, IOP Publishing 1999)


The The Uemura Uemura plot: evidence forplot: evidence forunconventional superconductivity?unconventional superconductivity?

*m/nTk 32

s2

BB h≈

Uemura et al PRL 66 (1991) 2665

101

TT

1001

F

C <<

31

TT

B

C ≤


Spin fluctuations as the pairing mechanismSpin fluctuations as the pairing mechanism

Within the framework ofSelf ConsistentRenormalsation (SCR)theory this suggests thatthe same antiferromagneticspin fluctuations may beresponsible in all systems Nakamura, Moriya and Ueda

J Phys Soc Japn 65 (1996) 4026

Zr2Rh

Experiment shows that Tc

also correlates with theeffective “spin fluctuation”temperature, as measuredwith neutrons, in many“exotic” superconductors:

Tc/To ~1/30


Cold muons and superconductivityCold muons and superconductivity

PSI

Muons are cryogenicallymoderated and energyselected to tune localisationdepth within the sample:

E(keV) R(nm) ∆R(nm)

0.010 0.5 0.3

0.100 2.1 1.3

1.0 13.1 5.4

10.0 75.0 18.0

30.0 244.0 36.0

See Morenzoni in “Muon Science”eds Lee, Kilcoyne and Cywinski, 1998


Cold muons at PSICold muons at PSI

….but the efficiency is very low (~10-5)

Complementarity Muon Training Course Jan 2003

Flux penetration with cold muonsFlux penetration with cold muons

Niedermayer, Forgan et alPRL83, 3935, 1999

YBa2Cu2O7

Ag

simulated measured

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