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1. Vortex Nernst effect2. Loss of long-range phase coherence3. The Upper Critical Field4. High-temperature Diamagnetism
Vorticity and Phase Coherence in Cuprate Superconductors
Yayu Wang, Lu Li, J. Checkelsky, N.P.O. Princeton Univ.M. J. Naughton, Boston College
S. Uchida, Univ. Tokyo S. Ono, S. Komiya, Yoichi Ando, CRI, Elec. Power Inst., Tokyo
Genda Gu, Brookhaven National Lab
Taipeh, June 2006
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holes = 1/2
Phase diagram of Cuprates
T pseudogap
0 0.05 0.25
AF dSC
T*
Tc
Mott insulator
Fermiliquid
doping x
LSCO = La2-xSrxCuO4
Bi 2212 = Bi2Sr2CaCu2O8
Bi 2201 = Bi2-yLaySr2CuO6
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amplitude fluctuation
F
F
phase fluctuation
Anderson-Higgs mechanism: Phase stiffnesssingular phase fluc. (excitation of vortices)
Condensate described by a complex macroscopic wave function
(r) = 1 + i2 = |r| exp[ir]
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Phase rigidity ruined by mobile defects
Long-range phase coherence requires uniform
Phase coherence destroyed by vortex motion
“kilometer of dirty lead wire”
phase rigidity measured by s 23
21 SrdH
Kosterlitz Thouless transition in 2D films (1982)
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Vortex in cuprates
CuO2 layers
2D vortex pancake
Vortex in Niobium
Js
superfluidelectrons
Js
b(r)Normal core
H
coherence length
Vortices, fundamental excitation of type-II SC
London length
b(r)
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upper critical field
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Mean-field phase diagram
H
2H-NbSe2
T
Hc2
Hc1
Tc0
normal
vortex solid
liquid
0
Hm
Meissner state
H
Cuprate phase diagram
4 T
7 Kvortexsolid
vortexliquid
Hc2
Tc
100 T
100 K
Hm
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The Josephson Effect, phase-slippage and Nernst signal
V22 neVJ
JeV2
t
VJ
vortex
2
Ph
ase
diff
ere
nce
Passage of a vortex Phase diff. jumps by 2
Integrate VJ to give dc signalprop. to nv
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Nernst experiment
Vortices move in a temperature gradientPhase slip generates Josephson voltage
2eVJ = 2h nV
EJ = B x v
H
ey
Hm
Nernst signal
ey = Ey /| T |
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Nernst effect in underdoped Bi-2212 (Tc = 50 K)
Vortex signal persists to 70 K above Tc.
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Vortex-Nernst signal in Bi 2201
Wang, Li, Ong PRB 2006
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Nernst curves in Bi 2201
overdoped optimal underdoped
Yayu Wang,Lu Li,NPO PRB 2006
Nernst signal
eN = Ey /| T |
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Spontaneous vortices destroy superfluidity in 2D films
Change in free energy F to create a vortex
F = U – TS = (c – kBT) log (R/a)2
F < 0 if T > TKT = c/kB vortices appear spontaneously
TcMFTKT
0
s
Kosterlitz-Thouless transition
3D version of KT transition in cuprates?
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•Loss of phase coherence determines Tc•Condensate amplitude persists T>Tc• Vorticity and diamagnetism in Nernst region
Nernstregion
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1. Existence of vortex Nernst signal above Tc
2. Confined to superconducting “dome”
3. Upper critical field Hc2 versus T is anomalous
4. Loss of long-range phase coherence at Tc by spontaneous vortex creation (not gap closing)
5. Pseudogap intimately related to vortex liquid state
In hole-doped cuprates
More direct (thermodynamic) evidence?
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Supercurrents follow contours of condensate
Js = -(eh/m) x ||2 z
Diamagnetic currents in vortex liquid
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Cantilever torque magnetometry
Torque on magnetic moment: = m × B
Deflection of cantilever: = k
crystal
B
m×
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Micro-fabricated single crystal silicon cantilever magnetometer
• Capacitive detection of deflection
• Sensitivity: ~ 5 × 10-9 emu at 10 tesla ~100 times more sensitive than commercial SQUID
• Si single-crystal cantilever
H
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Tc
UnderdopedBi 2212 Wang et al.
Cond-mat/05
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Paramagnetic background in Bi 2212 and LSCO
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Magnetization curves in underdoped Bi 2212
Tc
Separatrix Ts
Wang et al.Cond-mat/05
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amplitude fluctuation
F
F
phase fluctuation
Anderson-Higgs mechanism: Phase stiffnesssingular phase fluc. (excitation of vortices)
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At high T, M scales with Nernst signal eN
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M(T,H) matches eN in both H and T above Tc
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Magnetization in Abrikosov state
HM
M~ -lnH
M = - [Hc2 – H] / (22 –1)
Hc2Hc1
In cuprates, = 100-150, Hc2 ~ 50-150 T
M < 1000 A/m (10 G)
Area = Condensation energy U
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Wang et al. Cond-mat/05
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Mean-field phase diagram
H
2H-NbSe2
T
Hc2
Hc1
Tc0
normal
vortex solid
liquid
0
Hm
Meissner state
H
Cuprate phase diagram
4 T
7 Kvortexsolid
vortexliquid
Hc2
Tc
100 T
100 K
Hm
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Hole-doped optimal Electron-doped optimal
TcTc
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T*
Tonset
Tc
spin pairing(NMR relaxation,Bulk suscept.)
vortex liquid
Onset of charge pairingVortex-Nernst signalEnhanced diamagnetismKinetic inductance
superfluiditylong-range phase coherenceMeissner eff.
x (holes)
Tem
per
atu
re T
0
Phase fluctuation in cuprate phase diagram
pseudogap
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In hole-doped cuprates
1. Large region in phase diagram above Tc domewith enhanced Nernst signal
2. Associated with vortex excitations
3. Confirmed by torque magnetometry
4. Transition at Tc is 3D version of KT transition (loss of phase coherence)
5. Upper critical field behavior confirms conclusion
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End
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d-wave symmetry
Cooper pairing in cuprates
+- -
+
Upper critical field
coherence length
Hc2 4 Tesla1040100 Tesla
90572918
NbSe2MgB2Nb3Sncuprates
(A)o
2cos)( 0
20
2 2
cH
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Contrast with Gaussian (amplitude) fluctuations
In low Tc superconductors,Evanescent droplets of superfluid radius exist above Tc
M’ = 21/2(kBTc / 03/2) B1/2
At Tc, (Schmidt, Prange ‘69)
This is 30-50 times smaller than observed in Bi 2212
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Wang et al. PRL 2005
1. Robustness Survives to H > 45 T. Strongly enhanced by field. (Gaussian fluc. easily suppr. in H).
2. Scaling with Nernst Above Tc, magnetization M scales as eN vs. H and T.
3. Upper critical fieldBehavior of Hc2(T) not mean-field.
“Fluctuation diamagnetism” distinct from Gaussian fluc.
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Signature features of cuprate superconductivity
1. Strong Correlation
2. Quasi-2D anisotropy
3. d-wave pairing, very short
4. Spin gap, spin-pairing at T*
5. Strong fluctuations, vorticity
6. Loss of phase coherence at Tc
+- -
+
Tc
vortex liquid
Hc2
Hm
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Comparison between x = 0.055 and 0.060Sharp change in ground state
Pinning current reduced by a factor of ~100 in ground state
Lu Li et al., unpubl.
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In ground state, have 2 field scales
1) Hm(0) ~ 6 TDictates phase coherence, flux expulsion
2) Hc2(0) ~ 50 TDepairing field. Scale of condensate suppression
Two distinct field scalesM
(A/m
)
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Magnetization in lightly doped La2-xSrxCuO4
5 K5 K
35 K 35 K 30 K
30 K
4.2 K
4.2 K
SC dome
0.03 0.04 0.05 0.06
Lu Li et al., unpubl.
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Vortex-liquid boundary linear in x as x 0?
Sharp transition in Tc vs x (QCT?)
dissipative,vortices mobile
Long-rangephase coherence
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The case against inhomogeneous superconductivity(granular Al)
1. LaSrCuO transition at T = 0 much too sharp
2. Direct evidence for competition between d-wave SCand emergent spin order
3. In LSCO, Hc2(0) varies with x
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Competing ground states
Abrupt transition between different ground states at xc = 0.055
1. Phase-coherent ground state (x > 0.055)Cooling establishes vortex-solid phase; sharp melting field
2. Unusual spin-ordered state (x < 0.055)
i) Strong competition between diamagnetic state and paramagnetic spin ordering
ii) Diamagnetic fluctuations extend to x = 0.03
iii) Pair condensate robust to high fields (Hc2~ 20-40 T)
iv) Cooling to 0.5 K tips balance against phase coherence.
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Gollub, Beasley,Tinkham et al.PRB (1973)
Field sensitivity of Gaussian fluctuations
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Vortex signal above Tc0 in under- and over-doped Bi 2212Wang et al. PRB (2001)
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Abrikosov vortices near Hc2
Upper critical field Hc2 = 0/22
Condensate destroyed when cores touch at Hc2
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Anomalous high-temp. diamagnetic state
1. Vortex-liquid state defined by large Nernst signal and diamagnetism
2. M(T,H) closely matched to eN(T,H) at high T ( is 103 - 104 times larger than in ferromagnets).
3. M vs. H curves show Hc2 stays v. large as T Tc.
4. Magnetization evidence that transition is by loss of phase coherence instead of vanishing of gap
5. Nonlinear weak-field diamagnetism above Tc to Tonset.
6. NOT seen in electron doped NdCeCuO (tied to pseudogap physics)
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Nernst contour-map in underdoped, optimal and overdoped LSCO
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• In underdoped Bi-2212, onset of diamagnetic fluctuations at 110 K
• diamagnetic signal closely tracks the Nernst effect
110K
Tc
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ey
PbIn, Tc = 7.2 K (Vidal, PRB ’73) Bi 2201 (Tc = 28 K, Hc2 ~ 48 T)
0 10 20 30 40 50 600.0
0.5
1.0
1.5
2.0
ey (V
/K)
0H (T)
T=8K
Hc2
T=1.5K
Hd
0.3 1.0H/Hc2
Hc2
• Upper critical Field Hc2 given by ey 0.
• Hole cuprates --- Need intense fields.Wang et al. Science (2003)
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Vortex-Nernst signal in Bi 2201
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NbSe2 NdCeCuO Hole-doped cuprates
Tc0 Tc0Tc0
Hc2 Hc2Hc2
Hm
HmHm
Expanded vortex liquid Amplitude vanishes at Tc0
Vortex liquid dominant.Loss of phase coherenceat Tc0 (zero-field melting)
Conventional SCAmplitude vanishesat Tc0 (BCS)
vortex liquid
vortex liquid
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Phase diagram of type-II superconductor
H
2H-NbSe2cuprates
vortex solid
vortexliquid
??
Hm
0 T Tc0
H
Hc1
T
Hc2
Hc1
Tc0
normal
vortex solid
liquid
0
Hm
4 T
Meissner state
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Superconductivity in low-Tc superconductors (MF)
Energy gap
Pairs obey macroscopic wave function
Phase important in Josephson effect
Phase
Cooper pairs with coherence length
)(||)(ˆ rr ie
Quasi-particles
||
Tc
Gap
Temp. T
amplitude
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= mp x B + MV x B
Van Vleck (orbital) moment mp
2D supercurrent
Torque magnetometry
V = cHx Bz – aHz Bx + M Bx
Meff = / VBx = p Hz + M(Hz)
H
M
mp
c, z
H
mp
MExquisite sensitivity to 2D supercurrents
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Wang et al., unpublished
Hc2(0) vs x matches Tonset vs x
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H*
Hm
Tco
Overdoped LaSrCuO x = 0.20
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Hc1
M vs H below TcFull Flux Exclusion
Strong Curvature!
-M
H
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Strong curvature persists above Tc
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M ~ H1/
M non-analytic in weak field
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Fit to Kosterlitz Thouless theory
= -(kBT/2d02) 2
= a exp(b/t1/2)
Strongly H-dependentSusceptibility = M/H
Susceptibility and Correlation Length
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Non-analytic magnetization above Tc
M ~ H1/
Fractional-exponentregion
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Plot of Hm, H*, Hc2 vs. T
• Hm and H* similar to hole-doped
• However, Hc2 is conventional
• Vortex-Nernst signal vanishes just above Hc2 line
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0 5 10 15 20 25 300.0
0.5
1.0
1.5
2.0
2.5
3.0
100908075
65
60
55
50
45
70
40KUD-Bi2212 (T
c=50K)
0H (T)
0 5 10 15 20 25 30
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
908580
75
70
65
60
55
5045
40
35
30
25
20
OD-Bi2212 (Tc=65K)
ey (V
/K)
0H (T)
Field scale increases as x decreases
0 5 10 15 20 25 30
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
105110
9085
80
75
95
100
70K
OPT-Bi2212 (Tc=90K)
0H (T)
overdoped optimum underdoped
Wang et al. Science (2003)
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Optimal, untwinned BZO-grown YBCO
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Nernst effect in LSCO-0.12
vortex Nernst signal onset from T = 120 K, ~ 90K above Tc`1
Xu et al. Nature (2000)
Wang et al. PRB (2001)
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Temp. dependence of Nernst coef. in Bi 2201 (y = 0.60, 0.50).
Onset temperatures much higher than Tc0 (18 K, 26 K).
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Resistivity is a bad diagnostic for field suppression of pairing amplitude
Plot of and ey versus T at fixed H (33 T).
Vortex signal is large for T < 26 K, but is close to normal value N
above 15 K.
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0 2 4 6 8 10 12 140.0
0.2
0.4
0.6
0.8
ey
12K
NdCCO (T
c=24.5K)
ey (V
/K)
0H (T)
0 5 10 15 20 25 300.0
0.2
0.4
0.6
0.8
1.0
22K
ey
LSCO (0.20)
ey(
V/K
)
0H (T)
Resistivity does not distinguish vortex liquid from normal state
Hc2Hc2
Bardeen Stephen law (not seen)Resistivity Folly
Ong Wang, M2S-RIO, Physica C (2004)
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Isolated off-diagonal Peltier current xy versus T in LSCO
Vortex signal onsets at 50 and 100 K for x = 0.05 and 0.07
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Contour plots in underdoped YBaCuO6.50 (main panel) and optimalYBCO6.99 (inset).
Tco
• Vortex signal extends above70 K in underdoped YBCO,to 100 K in optimal YBCO
• High-temp phase merges continuously with vortex liquid state
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Nernst effect in optimally doped YBCO
Nernst vs. H in optimally doped YBCO Vortex onset temperature: 107 K
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Separatrix curve at Ts
Optimum doped Overdoped
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Vortex Nernst signal
xy = M
-1 = 100 K
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H = ½ s d3r ( )2
s measures phase rigidityPhase coherence destroyed at TKT
by proliferation of vortices
BCS transition 2D Kosterlitz Thouless transition
Tc
s
0
TMFTKT
nvortex
s
0
High temperature superconductors?
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Strong correlation in CuO2 plane
Cu2+Large U
charge-transfer gap pd ~ 2 eV
Mott insulatormetal?
doping
t = 0.3 eV, U = 2 eV, J = 4t2/U = 0.12 eV
J~1400 K
best evidence for large Uantiferromagnet
,, ji iiiji nnUcctH Hubbard
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Hole-doped optimal Electron-doped optimal
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Overall scale of Nernst signal amplitude