University of St. Andrews

Using muons to probe spatial correlations invortex matter systems of superconductors

Steve Lee, David Heron, Stephen Lister,Soumya Ray, Vikash Venkataramana

SUPA, University of St. Andrews, UKG.I. Menon

C.I.T., Taramani, Chennai, India

J. Mesot, R. Gilardi, J. Kohlbrecher

PSI/ETH Zürich, Switzerland

Alan Drew

Université Fribourg, Switzerland

IMSS, Tsukuba, Friday 17th October 2008

Overview• Background to Type II Superconductors• Muon spin rotation (µSR): what we can learn about
superconductivity from the vortex lattice• Vortex correlations in single crystals of anisotropic
superconductors-

La1.9Sr0.1CuO4 (LSCO)-

Bi2.15Sr1.85 CaCu2 O8+δ (BSCCO-2212)-Irradiated Bi2.15Sr1.85 CaCu2 O8+δ (BSCCO-2212)• Summary
IMSS, Tsukuba, Friday 17th October 2008

Background to Type II Superconductors

IMSS, Tsukuba, Friday 17th October 2008

Type II Superconductivity
Hc2(0)

Upper criticalfield Hc2(T)Superconducting
Abrikosov state

H

Flux generated
is one qu antumof f lux
φ = φο= h/2eNormalconductingstate
Vort ex m adeof circulating
currents of
Coope r pairs

B=µoH

Hc1(0)

Superconducting
Meissner state
B=00Vort ex cor ecomp osed ofnormal single
electro nsLower criticalfield Hc1(T)Hc1TcT

Flux line/vortex
IMSS, Tsukuba, Friday 17th October 2008

Abrikosov/Mixed StateH

Vortex Lattice

Local internal fieldH
Field direction

IMSS, Tsukuba, Friday 17th October 2008∼ξCoherence
length ξ:Core size ~ ξ

Penetration depth λ:Magnetic extent~λ∼λSuperconducting
wavefunction

Ψ(r)IMSS, Tsukuba, Friday 17th October 2008

Local internal
field h(r)h(r)∼ξλLayered superconductors e.g. High-Tc cuprates

Superconductingplanes‘Pancake’ vortices

IMSS, Tsukuba, Friday 17th October 2008

Vortex Interactions

Intra-layer repulsion
Inter-layer attraction
Electromagnetic, Josephson

Vortex-pin attraction
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Vortex Interactions and Pinning lead toIrreversibility

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Pinning ‘landscape’

Transition to glass state as scale of disorder increases
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Thermal disruption and melting

IMSS, Tsukuba, Friday 17th October 2008

The Vortex Zoo …..

‘Vortex Matter’

Vortex glass

Links to soft condensed
matter:nematics;

H

entanglement;
discotics;

Vortex liquid
Disorder line
Melting line

glass transitions;colloids;
higher order correlations etc.
IMSS, Tsukuba, Friday 17th October 2008

Vortex solidTMuon Spin Rotation (µSR):
What we can learn about
superconductivity from the vortex lattice

IMSS, Tsukuba, Friday 17th October 2008

Probability

Muon spin rotation µSR

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Internal FieldShape of µSR signal is closely related to spatial distribution of vortices

Probability(‘shape’, ‘size’ and arrangement of vortices themselves)Internal Field
IMSS, Tsukuba, Friday 17th October 2008

If we assume that the vortex lattice is ideal (perfectly ordered), we canextract characteristic length scales such as the magnetic penetration
depth λ(T,B) and the superconducting coherence length ξ(T,B)

Probability

e.g. width (second moment)-2<ΔB2>1/2 (T)∝λ (T)-2

λ (T)∝ns (T)/m*Internal Field

However, vortex lattice may be far from ideal.............
IMSS, Tsukuba, Friday 17th October 2008

Vortex glass

H

Vortex liquid
Disorder line
Melting line

Vortex solid

T

S.L. Lee et al., Phys. Rev. Lett. 71 3862-3865 (1993).

S.L. Lee et al., Phys. Rev. Lett. 75 922 (1995).
IMSS, Tsukuba, Friday 17th October 2008

Vortex Correlations in Single Crystals of
Anisotropic Superconductors

IMSS, Tsukuba, Friday 17th October 2008

Example: La1.9Sr0.1CuO4 (LSCO)
Increasing anisotropy

YBCO (Tc=93 K)
λ~1400 Å
γ~ 4LSCO (x=0.1) (Tc=29 K)
λ~3000 Å
γ>~ 40

BSCCO-2212 (Tc=85 K)
λ~1800 Åγ>~ 150

λJ = s γ ~ 50Å

λJ = s γ ~ 260 Å

λJ = s γ ~ 2300 Å

λJ << λab

λJ << λabLong λ, modest γ

IMSS, Tsukuba, Friday 17th October 2008

λJ ~ λabInterpretation of second moment (line width)
Disorder of rigid vortices always leads to increase in width of
distribution
Disorder[ΔB]

2= ∑ p(Bi )(Bi − B ) 2Line width€
IMSS, Tsukuba, Friday 17th October 2008

λ>>a

∑ p(Bi )LSCO: line width[ΔB]

2= ∑ p(Bi )(Bi − B ) 212
ΔB 2[ ]∑ p(Bi )IL€€H(G)€Vortex Glass

Vortex lattice
Small-angle neutron scattering pattern

T(K)IMSS, Tsukuba, Friday 17th October 2008

Divakar et al., Phys. Rev. Lett. 92(23) 237004 (2004)

µSR ‘Skewness’

[ΔB]
1

α= [€

α >0

T=6.2 K
B=80 G

0.8

0.4

0.2

0.2

-40-20∑ p(Bi )€B -B (G)020pk

IMSS, Tsukuba, Friday 17th October 2008

4060

α <0

T=6.2 K
B=6000 G

0.6

0.4-603[ ]10.6

012 2ΔB]= ∑ p(Bi )(Bi − B )

p(B)

€p(B)

0.8313 3ΔB0-60-40-20020B -Bpk(G)4060

12ΔB 312ΔB 2

α= [H(G)

[ ]

€€€]

12ΔB 2[ ]Vortex GlassT(K)H(G)

Negative α€€Vortex lattice
Positive αT(K)IMSS, Tsukuba, Friday 17th October 2008

Conventional ‘two body’ structure factor S(q)Second Moment (width)

["B]

222

# $ d qS(q)b (q)S(q)qS(q)=F.T{pair density correlation function g(r)}b(q)=F.T{Flux line local field h(r)} =1⁄(1+q2λ2)
Third Moment (skewness)
33)(ΔB ∝ dq dq S (q ,q )b(q )b(q )b(−q − q )

[ ]

∫∫12

1212

12S ( 3) (q1,q2 ) = N1 δρ (q1 )δρ (q2 )δρ(−q1 − q2 )IMSS, Tsukuba, Friday 17th October 2008

Triplet Structure Factor

Triplet Structure Factor3)(S (q ,q ) = 1 δρ(q )δρ(q )δρ(−q − q )
12

1

N

2

12-information on orientationalcorrelations
12 3
ΔB

α= [€]

12 2ΔB[ ]Vortex Glass
Negative α

H(G)α€

Vortex lattice
Positive αT(K)IMSS, Tsukuba, Friday 17th October 2008
€Magnetic FieldMenon et al., Phys. Rev. Lett. 97 177004 (2006)
12ΔB 3α= [€]

12ΔB 2[ ]High field: glassy order including
non-trivial triangular coordination
extending several lattice spacings

Vortex Glass
Negative αH(G)€Vortex lattice
Positive αT(K)€Low field: ‘trivial’
triangular coordination
Vortex lattice/Bragg glass

IMSS, Tsukuba, Friday 17th October 2008

Typically alpha is positive (α>0) for either highly-ordered orhighly-disordered systems (although disorder reduces value of α)
Theoretically, negative alpha (α<0) occurs only for a very
particular combination of 2-body (S(q)) and 3-body (S(3))
correlations

IMSS, Tsukuba, Friday 17th October 2008

Example: Bi2.15Sr1.85 CaCu2 O8+δ (BSCCO-2212)
Increasing anisotropy

YBCO (Tc=93 K)
λ~1400 Åγ~ 4LSCO (x=0.1) (Tc=29 K)
λ~3000 Å
γ>~ 40

BSCCO-2212 (Tc=85 K)
λ~1800 Å
γ>~ 150λJ = s γ ~ 50Å

λJ = s γ ~ 260 Å

λJ = s γ ~ 2300 ÅλJ << λab

λJ << λabLong λ, modest γ

IMSS, Tsukuba, Friday 17th October 2008

λJ ~ λab12 2ΔBH(G)[ ]Vortex Liquid
Quasi-2d Vortex€GlassILVortex lattice
IMSS, Tsukuba, Friday 17th October 2008

T(K)

Field perpendicular to the planes in only screened
on a length scale ~λab >> s (plane spacing)
Longitudinal fluctuations ‘wash out’
the high fields originating in the coressp(B)!

Both width σ and skewness
α are reduced

B!
IMSS, Tsukuba, Friday 17th October 2008λab

12 3ΔB

α= [12 2ΔB]

Reduced 3bodycorrelations[ ]Vortex Liquid€ α>0

α<0Vortex Glass
α<0Residual medium range 3body in-plane correlations

α>0Vortex lattice
IMSS, Tsukuba, Friday 17th October 2008

T(K)

7K
12K
30K
40K

1.00.5

<∆B >

3 1/3

/<∆B >

2 1/2

α

0.0

-0.5

1000

2000

3000
Field / GH(G)IMSS, Tsukuba, Friday 17th October 2008

4000

5000

6000

BSCCO with Columnar Defects Bφ = 2000 G
Irradiation with heavy ions (17.7 GeV U ions)
Columnar Defects

IMSS, Tsukuba, Friday 17th October 2008

The Matching Field Bφ : Applied for for which the areal
density of vortex lines = density of ‘columnar defects’
B=Bφ

IMSS, Tsukuba, Friday 17th October 2008

If positions of vortices were truly random at B= Bφ, μSR
line width would extremely large, and S(q) would betrivial.

In reality there are always some 2-body correlations even at Bφsee e.g. S.L. Lee et al., a Phys. Rev. Lett. 81 5209 (1998).

IMSS, Tsukuba, Friday 17th October 2008

B=2Bφ=4000 G

B=Bφ=2000 G
S(q)
S(q)

q(Å-1)q(Å-1)

Structure Factors-Measured by Small-angle neutron scattering (SANS)
IMSS, Tsukuba, Friday 17th October 2008

["B]

222

# $ d qS(q)b (q)Related to 2-body correlations
12 2ΔBH(G)[ ]€BφBose GlassT(K)At low temperature optimal disruption to in-plane order at B=Bφ

Width is always large in irradiated sample:Defects keep the vortices ‘rigid’

H(G)

€[ ]

Pristine (unirradiated)

H(G)Irradiated

1
2 2
ΔB

BφT(K)
IMSS, Tsukuba, Friday 17th October 2008

T(K)

12ΔB 3

α= []

[ ]IL

1. α>0 everywhere
2. At low T, α has
minimum at Bφ: implies 3body correlations alsostrongly suppressed
3. At high T and B, 3body correlations seemwell-developed above
irreversibility line: newphase??

Bφ

€H(G)

€12
ΔB 2T(K)
IMSS, Tsukuba, Friday 17th October 2008

4. At high T and low B,
minimum in α and widthat B~ Bφ⁄3 - Bφ⁄2:
entropically driven
disorder?

12ΔB 3α= []12ΔB 2

12ΔB 2[ ][ ]1.8x1072K
78K
79K
81K

1.0

1.665K
68K72K
75K
81K

€

1.4

0.9

IL/T

1/30.7

<∆B >

3

1.2

1.0

<∆B >

0.82 1/22/ <∆B >

1/2

€

-3

0.8

0.6

0.6
0.5

0.4
1000

2000

Field / G

3000

4000

1000

2000

3000

Field / G

4000

5000

Mimima develop at temperatures above the irreversbility line (IL)where the vortices become much more mobile
IMSS, Tsukuba, Friday 17th October 2008

6000

Competition in free energy between
interlayer coupling (Josephson, dipolar) andgain in entropy as pinning ‘sites’ start tobecome filled.

αH(G)BφT(K)

SummaryBulk µSR is useful for probing the vortex state in single crystals ofsuperconducting materials.
µSR may even yield information on classical three body correlations indisordered vortex systems, which are hard to measure by other methods.
Characteristic length scales (ξ, λ) can be extract if one fully undertands thevortex state and how to model it. The state of order of the vortex system
has a profound influence on the moments of the µSR line shapes andquantitative interpretation of these moments must be carefully considered.

IMSS, Tsukuba, Friday 17th October 2008