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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
Complementarity Muon Training Course Feb 2005
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)
Complementarity Muon Training Course Feb 2005
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
Complementarity Muon Training Course Feb 2005
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
Complementarity Muon Training Course Feb 2005
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)
Complementarity Muon Training Course Feb 2005
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
Complementarity Muon Training Course Feb 2005
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
Complementarity Muon Training Course Feb 2005
The flux latticeThe flux lattice
22µµmm
10µ10µmm
Complementarity Muon Training Course Feb 2005
The flux line latticeThe flux line lattice
Lowerdensity offlux lines
Lowerdensity offlux lines
Defectsand
disorder
Defectsand
disorder
Hexagonallattice
Hexagonallattice
flux linecurvatureflux line
curvature
Complementarity Muon Training Course Feb 2005
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=
Complementarity Muon Training Course Feb 2005
Small angle neutron scatteringSmall angle neutron scattering
R
scatteringangle 2θ
sample
multi-detector
64x64cm2
incidentneutron beam
B
L
Complementarity Muon Training Course Feb 2005
Flux Flux distrbutionsdistrbutions
Complementarity Muon Training Course Feb 2005
Probing the flux lattice with muonsProbing the flux lattice with muons
100nm100nm
Complementarity Muon Training Course Feb 2005
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=∆
Complementarity Muon Training Course Feb 2005
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
Complementarity Muon Training Course Feb 2005
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
Complementarity Muon Training Course Feb 2005
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
Complementarity Muon Training Course Feb 2005
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)
Complementarity Muon Training Course Feb 2005
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
Complementarity Muon Training Course Feb 2005
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
Complementarity Muon Training Course Feb 2005
Uemura’s Uemura’s universal universal correlationscorrelations
(PRL 66 (1991) 2665)
Complementarity Muon Training Course Feb 2005
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
Complementarity Muon Training Course Feb 2005
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)
Complementarity Muon Training Course Feb 2005
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 ≤
Complementarity Muon Training Course Feb 2005
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
Complementarity Muon Training Course Feb 2005
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
Complementarity Muon Training Course Feb 2005
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