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Zuoz 2008

µSR and Superconductors

Masterschool PSI

µSR and Superconductors (SC) Crash course on SC Nanometer scale parameters How to determine them by µSR

o Abrikosov state (Bulk µSR and LEM)o Meissner state (LEM)

Appendices Ginzburg-Landau equation Pairing symmetry Two-gap superconductors

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Zuoz 2008

Crash course on superconductivity

Masterschool PSI

Superconductivity -- Introduction

Temperature dependence of the resistance of a Hg sample

Discovery by Kamerlingh Onnesin 1911 in mercury

Received the Nobel Prize in 1913 for “his investigations on the properties of matter at low temperatures which led, inter alia, to the production of liquid helium”.

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Below the transition temperature a superconductor expels a magnetic field from its inner core (Meissner and Ochsenfeld effect)

Quelle: http://staff.ee.sun.ac.za/wjperold/Research/research.htm

Superconductivity -- Introduction

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Superconductivity -- Introduction

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Superconductivity -- Timeline

1908 Kammerling Onnes: production of liquid helium

1911 Kammerling Onnes: discovery of zero resistance

1933 Meissner and Ochsenfeld: superconductors expell applied magnetic fields (MOE)

1950 Ginzburg and Landau: phenomenological theory of superconductors

1950 Maxwell and Reynolds et al.: isotope effect

1935 F. and H. London: MOE is a consequence of the minimization of the electromagneticfree energy carried by superconducting current

1957 Abrikosov: 2 types of superconductors (magnetic flux)

1957 Bardeen, Cooper, and Schrieffer: BCS theory -- superconducting current as a superfluid of Cooper pairs

1962 Josephson: Joesephson effect

1986 Berdnorz and Müller: High-Tc‘s superconductors

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Lenz-Faraday’s law: currents to keep B constant inside of the sample

Ideal Conductor Superconductor

New thermodynamic state of matter

From: Lecture on Superconductivity, Alexey Ustinov, Uni. Erlangen, 2007

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Main Characteristics:

Is a superconductor “just” an ideal conductor?

New thermodynamic state of matter!

Kamerlingh Onnes Meissner and Ochsenfeld

Zuoz 2008

Nanometer scale parameters

Magnetic penetration depth:

Coherence length:

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Phenomenological London’s equations

F. and H. London, Proc. Roy. Soc. A149, 71 (1935)

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1st Nano-scale Param.: London Penetration Depth

SC

Bz(x)

x

SC

jz(x)

x

Bo

I

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SCOrder

parameter

x

2nd Nano-scale Param.: Coherence Length

The coherence length describes the variation of the fact that thesuperconducting electron density cannot vary abruptly (see Appendix).

In some limiting cases, it can be considered as the typical lengthof the superconducting pair.

The coherence length describes the variation of the fact that thesuperconducting electron density cannot vary abruptly (see Appendix).

In some limiting cases, it can be considered as the typical lengthof the superconducting pair.

Masterschool PSI

Type I and Type II Superconductors

SC

B

x

Free

Ene

rgy D

ensit

y

SC

B

x

Free

Ene

rgy D

ensit

y

Number of superelectrons

Magnetic fluxdensity

Magneticcontribution B2/2o

Condensationenergy -Bc2/2o

Totalenergy

Type I Type II

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Type I ( ) Type II ( )

B

H

H

-M

Hc

Hc

H

TTc

Hc (0)Meissner phase

H

TTc

Hc1 (0)

Hc2 (0)

H

H

-MHc1 Hc2

Hc2Hc1

B

Vortex (Abrikosov) phase

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Vortex (Abrikosov) phase

H

H

TTc

Hc1 (0)

Hc2 (0) B B

Bitter DecorationPb-4at%In rod, 1.1K, 195G U. Essmann and H. Trauble, Phys. Lett 24A, 526 (1967)

Magnetic flux quantum = h/2e = 2.067 · 10-15 Wb

“Flux Line Lattice”

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Zuoz 2008

Abrikosov state (Bulk µSR and LEM)

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p (B

)

Bmin Bsad B Bmax

Muons and Field Distribution in Type II S.C.

Bishop et al., Scientific American 48 (1993)

H

TTc

Hc1 (0)

Hc2 (0) Vortex (Abrikosov) phase

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H

TTc

Hc1 (0)

Hc2 (0)

Vortex (Abrikosov) phase for µSR studies

“Transversal-Field µSR” (TF-µSR)

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Bext

Bext

Since the muon is a local probe, the SR relaxation functionis given by the weighted sum of all oscillations:

Vortex (Abrikosov) phase for µSR studies

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H

Tc

Hc1

Hc2

f (B)

B

f (B)

Bext

T < Tc

T > Tc

Mo3Al2C Bauer et al., Phys. Rev. B 90, 054522 (2014)

T

H

Tc

Hc1

Hc2

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Field Distribution in “Extreme” Type II S.C.

• Ginburg-Landau parameter

• Large range of fields (up to Bc2/4) where London model applies

• Vortex cores well separated and do not interact

• Vortex fields superimpose linearlyd

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d

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By measuring the second moment of the field distribution (for example by µSR), we directly determine the London penetration

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p (B

)

Maisuradze et al.

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Determination of λ by µSR

H

Tc

Hc1

Hc2

f (B)

B

T < Tc

Assume a Gaussian distribution:And since:

One can determine the penetration depthby determining the depolarization rate !!

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Mo3Al2C

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Weak Points of such Analysis

• One assumes that the muon depolarization (and therefore also that the field distribution) can be fitted by a Gaussian

• The field dependence of the second moment of the field distribution is neglected

• The size of the vortex core () is neglected

p (B

)

Bmin Bsad B Bmax

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Gaussian or not Gaussian.. that‘s the question!

• In polycrystals or sintered samples: large density and disorder of pinning sites strong smearing of the field

distribution

• The asymmetry of the field distribution appears in single crystals

Pd-In alloy

Charalambous et al., Phys. Rev. B 66, 054506 (2002)

44 46 48 50

0.00

0.02

0.04

0.06

0.08

Frequency (MHz)

YBa2Cu3O6.97Pümpin et al., Phys. Rev. B 42, 8019 (1990) = 130 (10) nm

811 812 813 814 815 816 817

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Frequency (MHz)

YBa2Cu3O6.95single crystal

811 812 813 814 815 816 817

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Frequency (MHz)

YBa2Cu3O6.95single crystal

Sonier et al., PRL 83, 4156 (1999)

= 150 (4) nm

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0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

aG(t)

time (µs)

Gaussian or not Gaussian.. that‘s the question!

46.5 47.0 47.5 48.0 48.5 49.0-1

0

1

2

3

4

5

6

7

Frequency (MHz)

Example: NbSe2

A Gaussian fit is rather poor but provides a fair approximation of the depolarization

NbSe2B = 3.5 kG

Sonier et al., Rev. Mod. Phys. 72, 769 (2000)

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Constructing the second momentof the field distribution by usingtwo Gaussian components

Example MgB2S. Serventi et al., PRL 93, 217003 (2004)

Gaussian .. or even two Gaussians

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Field Dependence -- Numerical GL Solution

λ = 50 nm, ξ = 20 nm (b = B/Bc2)

Field

Courtesy from A. Maisuradze

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Effect of the coherence length

1

2

B(r) FFT

The fact that the coherence length is finite leads to a “cutoff” of the high terms in the Fourier transform of B(r) when K ~ -1

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More advanced Model

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

5

10

15

20

(M

Hz)

Field (T)

One model (among many others): Modified London Model with Gaussian CutoffE.H. Brandt, J. Low Temp. Phys. 73, 355 (1988)

0.00 0.01 0.02 0.03 0.040.000

0.001

0.002

0.003

0.004

4

2

B/Bc2

= 200 = 25 = 10

Example: MgB2Niedermayer et al., Phys. Rev. B 65, 094512 (2002)

By measuring the field dependence of the second moment of the field distribution by µSR

determination of

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Example YB6

Example: Cubic YB6From the field dependence of µSR depolarization rate (second moment) = 192 nm and = 33 nm

Hillier et al., Phys. Rev. B 42, 8019 (1990)

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Other model: Analytical solution of the Ginzburg-Landau equations considering a Lorentzian function for the order parameter |(r)|2 of an isolated vortex:

J.R. Clem , J. Low Temp. Phys. 18, 427 (1975)Z. Hao et al., Phys. Rev. B 43, 2844 (1991)A. Yaouanc et al., Phys. Rev. B 55, 11107 (1997)

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Zuoz 2008

Meissner state (LEM)

Masterschool PSI

Type I ( ) Type II ( )

B

H

H

-M

Hc

Hc

H

TTc

Hc (0)Meissner phase

H

TTc

Hc1 (0)

Hc2 (0)

H

H

-MHc1 Hc2

Hc2Hc1

B

Vortex (Abrikosov) phase

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H

TTc

Hc1 (0)

Hc2 (0)

B(z)

z

Bext Bext

Superconductorin the Meissner State

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Meissner State Measurements

0 1 2 3 4 5 6 7 8 9 10

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Muo

n Sp

in P

olar

isat

ion

Time (s)

0 1 2 3 4 5 6 7 8 9 10

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Muo

n Sp

in P

olar

isat

ion

Time (s)

0 1 2 3 4 5 6 7 8 9 10

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Muo

n Sp

in P

olar

isat

ion

Time (s)

B(z)

z0

Superconductorin the Meissner State

Bext

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Penetration depth

T Tc

T 0

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Magnetic field profiles in YBa2Cu3O7- and Pb

0 50 100 150

1E-3

0.01YBa2Cu3O7-, T=20K, Tc=87.5K

hext

= 91.5(3) G, 0 = 1.5 nm fixed, 0 = 137(10) nm

hext exp(-z/(T)) 3.4 keV 8.9 keV 15.9 keV 20.9 keV 29.4 keV

B (T

)

z (nm)local: exponential

T.J. Jackson et al., Phys. Rev. Lett. 84, 4958 (2000).A. Suter et al., Phys. Rev. Lett. 92, 087001 (2004).A. Suter et al., Phys. Rev. B72, 024506 (2005).

~ 90

0 50 100 150

1E-3

0.01

0 50 100 150

1E-3

0.01

0 50 100 150

1E-3

0.01Lead, Tc=7.0(2) K, hext = 91.5(3)G,

0 = 90(5)nm, 0 = 58(3)nm

6.66K

6.19K

2.85K

z (nm)

B (T

)

Non-local: non-exponential

~ 0.6

1st experimental determination of x0 in a non-local superconductor; confirmation of some predictions of BCS (~50 years after theory!!)

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AppendixGinzburg-Landau Equations

Coherence Length

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Ginzburg-Landau Equations (1950)

Powerful phenomenological theory, based on the Landau theory of second order transition.

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SCOrder

parameter

x

2nd Nano-scale Param.: Coherence Length

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AppendixPairing symmetry

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T-Dependence of the SC carrier density

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• BCS conventional pairing:isotropic s-wave pairing

• The temperature dependence of the penetration depth provides information on the SC gap function.

Mo3Sb7R. Khasanov et al,

Phys. Rev. B 82, 016501 ( 2009)

B. Mühlschlegel, Z. Phys. 155, 313 (1959)

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High-Tc‘s Cuprates

Example: YBa2Cu3O7-2 CuO2 planesCuO chains: charge reservoirs

From A. Mourachkine Room –Temperature SuperconductivityCambridge Int. Science Publ., 2004

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High-Tc‘s Cuprates

Example: YBa2Cu3O7-2 CuO2 planesCuO chains: charge reservoirs

Charges reservoirUnderdoped Optimally doped Overdoped

Superconductor

Tem

pera

ture

Hole doping in CuO2 planes

Normalmetal?

“Strangemetal”

Mot

tin

sula

tor

Without doping:Band calculation predicts metallic state

BUT:

Strong on-site Coulomb repulsion ( large U ): blocks the motion of the 3d9 Cu-electrons in the CuO2 plane(“Mott insulator”)

AFM via superexchange

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Pairing Symmetry in Cuprates

As the gap disappears along some directions of the Fermi surface (“nodes”), extremely-low-energy quasiparticles excitations (and therefore significant pair-breaking) may occur at very low temperature

BCS High-Tc‘s

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Pairing Symmetry in High-Tc‘s

Single crystal YBa2Cu3O6.95J.E. Sonier et al, PRL 72, 744 (1994)

• BCS conventional pairing:isotropic s-wave pairing

• The temperature dependence of the penetration depth provides information on the SC gap function.

B. Mühlschlegel, Z. Phys. 155, 313 (1959)

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Hirschfeld & Goldenfeld, PRB 48, 4219 (1993)

Pairing Symmetry in High-Tc‘s

µSR penetration depth on single crystal YBa2Cu3O6.95J.E. Sonier et al, PRL 72, 744 (1994)

ab

(µm)

80

0.22

0.20

0.18

0.16

0.146040200

T (K)

Microwave measurementsW.N. Hardy et al., PRL 70, 3999 (1993)

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AppendixTwo-gap superconductivity

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• Metallic SC with highest TcJ. Nagamatsu et al., Nature 410 (2001) 63

• SC mediated by phonon-coupling• two-gap SC ( and )

Two Gaps Superconductivity – MgB2

From A. Mourachkine Room –Temperature SuperconductivityCambridge Int. Science Publ., 2004

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• Metallic SC with highest TcJ. Nagamatsu et al., Nature 410 (2001) 63

• SC mediated by phonon-coupling• two-gap SC ( and )

In-plane E2g Boron mode strongly couples to the boron -band

Two Gaps Superconductivity – MgB2

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Niedermayer et al., Phys. Rev. B 65, 094512 (2002)

Two Gaps Superconductivity – MgB2

Similar results found recently by µSR on the “Sequicarbides” (Ln2C3 with Ln = La, Y) Kuroiwa et al., Phys. Rev. Lett. 100, 097002 (2008)

0 10 20 30 40 500

2

4

6

8

(M

Hz)

T (K)

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Two Gaps Superconductivity – MgB2

Evidence of two coherence lengths from field dependence of the second moment of the FLL

S. Serventi et al., PRL 93, 217003 (2004)