µSR and Superconductors · 2018. 12. 10. · 1957 Abrikosov: 2 types of superconductors (magnetic...
Transcript of µSR and Superconductors · 2018. 12. 10. · 1957 Abrikosov: 2 types of superconductors (magnetic...
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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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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
Masterschool PSI
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
Masterschool PSI
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
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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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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)
Masterschool PSI
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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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
Masterschool PSI
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
Masterschool PSI
• 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)