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. Vortex Nernst effect
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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
T*
T
pseudogap
Tc
Fermi
liquid
AF
dSC
0
0.25
0.05
doping x
Phase diagram of Cuprates
s = 1/2
hole
LSCO = La2xSrxCuO4
Bi 2212 = Bi2Sr2CaCu2O8
Bi 2201 = Bi2yLaySr2CuO6
Condensate described by a complex macroscopic wave function
Y(r) = Y1 + iY2 = Y(r) exp[iq(r)]
y2
y1
y2
y1
AndersonHiggs mechanism: Phase stiffness
singular phase fluc. (excitation of vortices)
q
q
q
q
q
Phase rigidity ruined by mobile defects
Longrange phase coherence requires uniform q
“kilometer of dirty lead wire”
phase rigidity measured by rs
Phase coherence destroyed by vortex motion
Kosterlitz Thouless transition in 2D films (1982)
b(r)
Normal core
Js
x
x
b(r)
Y = D
London length l
Vortices, fundamental excitation of typeII SC
Vortex in cuprates
Vortex in Niobium
CuO2 layers
superfluid
electrons
Js
2D vortex pancake
H
coherence length x
liquid
Hm
Hc2
vortex solid
Hc1
0
Tc0
T
Meanfield phase diagram
Cuprate phase diagram
2HNbSe2
4 T
100 T
Hc2
H
H
vortex
liquid
Hm
Tc
vortex
solid
100 K
7 K
Meissner state
vortex
2p
f
Integrate VJ to give dc signal
prop. to nv
VJ
t
The Josephson Effect, phaseslippage and Nernst signal
Passage of a vortex
Phase diff. f jumps by 2p
ey = Ey / T 
Vortices move in a temperature gradient
Phase slip generates Josephson voltage
2eVJ = 2ph nV
EJ = B x v
Nernst experiment
ey
Hm
H
Vortex signal persists to 70 K above Tc.
Nernst effect in underdoped Bi2212 (Tc = 50 K)
VortexNernst signal in Bi 2201
eN = Ey / T 
Nernst curves in Bi 2201
Yayu Wang,Lu Li,NPO PRB 2006
underdoped
optimal
overdoped
rs
D
0
TKT
TcMF
KosterlitzThouless transition
Spontaneous vortices destroy superfluidity in 2D films
Change in free energy DF to create a vortex
DF = DU– TDS = (ec – kBT) log (R/a)2
DF < 0 if T > TKT = ec/kB vortices appear spontaneously
3D version of KT transition in cuprates?
region
In holedoped cuprates
More direct (thermodynamic) evidence?
Js = (eh/m) x Y2 z
Diamagnetic currents in vortex liquid
Supercurrents follow contours of condensate
×
B
m
Cantilever torque magnetometry
Torque on magnetic moment: = m × B
crystal
Deflection of cantilever: = k
Microfabricated single crystal silicon cantilever magnetometer
H
M(T,H) matches eN in both H and T above Tc
Magnetization in Abrikosov state
M
H
Hc1
Hc2
M =  [Hc2 – H] / b(2k2 –1)
M~ lnH
In cuprates, k = 100150, Hc2 ~ 50150 T
M < 1000 A/m (10 G)
Area = Condensation energy U
Condmat/05
liquid
Hm
Hc2
vortex solid
Hc1
0
Tc0
T
Meanfield phase diagram
Cuprate phase diagram
2HNbSe2
4 T
100 T
Hc2
H
H
vortex
liquid
Hm
Tc
vortex
solid
100 K
7 K
Meissner state
Phase fluctuation in cuprate phase diagram
spin pairing
(NMR relaxation,
Bulk suscept.)
T*
pseudogap
Tonset
Onset of charge pairing
VortexNernst signal
Enhanced diamagnetism
Kinetic inductance
TemperatureT
vortex liquid
Tc
superfluidity
longrange phase coherence
Meissner eff.
0
x (holes)
+
o


x (A)
+
Hc2
4 Tesla
40
10
100 Tesla
Cooper pairing in cuprates
dwave symmetry
coherence length
Upper critical field
cuprates
NbSe2
MgB2
Nb3Sn
57
18
29
90
Contrast with Gaussian (amplitude) fluctuations
In low Tc superconductors,
Evanescent droplets of
superfluid radius x
exist above Tc
x
At Tc, (Schmidt, Prange ‘69)
M’ = 2p1/2(kBTc / f03/2) B1/2
This is 3050 times smaller than observed in Bi 2212
“Fluctuation diamagnetism” distinct from Gaussian fluc.
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 field
Behavior of Hc2(T) not meanfield.


+
Hc2
vortex
liquid
Hm
Tc
Signature features of cuprate superconductivity
1. Strong Correlation
2. Quasi2D anisotropy
3. dwave pairing, very short x
4. Spin gap, spinpairing at T*
5. Strong fluctuations, vorticity
6. Loss of phase coherence at Tc
Comparison between x = 0.055 and 0.060
Sharp change in ground state
Lu Li et al., unpubl.
Pinning current reduced by a factor of ~100 in ground state
In ground state, have 2 field scales
1) Hm(0) ~ 6 T
Dictates phase coherence, flux expulsion
2) Hc2(0) ~ 50 T
Depairing field. Scale of condensate suppression
M (A/m)
Magnetization in lightly doped La2xSrxCuO4
Lu Li et al., unpubl.
SC dome
0.03
0.05
0.04
0.06
4.2 K
5 K
5 K
30 K
35 K
30 K
35 K
4.2 K
Vortexliquid boundary linear in x as x 0?
dissipative,
vortices mobile
Longrange
phase coherence
Sharp transition in Tc vs x (QCT?)
The case against inhomogeneous superconductivity
(granular Al)
Competing ground states
Vortex signal above Tc0 in under and overdoped Bi 2212
Wang et al. PRB (2001)
Abrikosov vortices near Hc2
Upper critical fieldHc2 = f0/2px2
Condensate destroyed when cores touch at Hc2
Tc
110K
T=8K
Hd
Hc2
0.3
1.0
H/Hc2
PbIn, Tc = 7.2 K (Vidal, PRB ’73)
Bi 2201 (Tc= 28 K, Hc2 ~ 48 T)
ey
Hc2
Wang et al. Science (2003)
NbSe2
NdCeCuO
Hc2
Hc2
Hc2
vortex
liquid
vortex
liquid
Hm
Hm
Hm
Tc0
Tc0
Tc0
Vortex liquid dominant.
Loss of phase coherence
at Tc0 (zerofield melting)
Expanded vortex liquid
Amplitude vanishes at Tc0
Conventional SC
Amplitude vanishes
at Tc0 (BCS)
normal
?
?
liquid
vortex
liquid
H
Hm
Hc2
vortex solid
vortex
solid
Hm
Hc1
Hc1
0
Tc0
0
Tc0
T
T
Phase diagram of typeII superconductor
cuprates
2HNbSe2
H
Meissner state
Gap D
Tc
Temp. T
Superconductivity in lowTc superconductors (MF)
Cooper pairs with coherence length x
Quasiparticles
Energy gap D
Pairs obey macroscopic wave function
Phase
amplitude
Phase q important in Josephson effect
c, z
H
q
mp
M
t
mp
H
M
Torque magnetometry
Van Vleck (orbital) moment mp
t= mpx B + MV x B
2D supercurrent
t/V = ccHx Bz – caHz Bx + M Bx
Meff = t / VBx = DcpHz + M(Hz)
Exquisite sensitivity to 2D supercurrents
Hc2(0) vs x matches Tonset vs x
M ~ H1/d
Susceptibility and Correlation Length
Strongly Hdependent
Susceptibility c = M/H
Fit to
Kosterlitz Thouless theory
c = (kBT/2df02) xKT2
xKT = a exp(b/t1/2)
Plot of Hm, H*, Hc2 vs. T
Wang et al. PRB (2001)
Nernst effect in LSCO0.12
vortex Nernst signal onset from T = 120 K, ~ 90K above Tc`1
Temp. dependence of Nernst coef. in Bi 2201 (y = 0.60, 0.50).
Onset temperatures much higher than Tc0 (18 K, 26 K).
Resistivity is a 0.50).bad diagnostic for field suppression of pairing amplitude
Plot of r and ey versus T at fixed H (33 T).
Vortex signal is large for T < 26 K, but r is close to normal value rN above 15 K.
Bardeen Stephen law (not seen) 0.50).
Resistivity Folly
Ong Wang, M2SRIO, Physica C (2004)
Hc2
Hc2
Resistivity does not distinguish vortex liquid from normal state
Isolated offdiagonal Peltier current 0.50).axy versus T in LSCO
Vortex signal onsets at 50 and 100 K for x = 0.05 and 0.07
T 0.50).co
Contour plots in underdoped YBaCuO6.50 (main panel) and optimal
YBCO6.99 (inset).
Nernst effect in optimally doped YBCO 0.50).
Vortex onset temperature: 107 K
Nernst vs. H in optimally doped YBCO
n 0.50).
vortex
D
0
T
T
T
c
KT
MF
H = ½rsd3r ( f)2
r
r
s
s
2D Kosterlitz Thouless transition
BCS transition
D
0
Phase coherence destroyed at TKT
by proliferation of vortices
rs measures phase rigidity
High temperature superconductors?
Strong correlation in CuO 0.50).2 plane
Cu2+
Large U
chargetransfer
gap Dpd ~ 2 eV
best evidence
for large U
metal?
Mott insulator
antiferromagnet
J~1400 K
doping
Hubbard
t = 0.3 eV, U = 2 eV, J = 4t2/U = 0.12 eV
Electrondoped optimal 0.50).
Holedoped optimal