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

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

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]

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)

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)

upper critical field

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

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

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 |

Nernst effect in underdoped Bi-2212 (Tc = 50 K)

Vortex signal persists to 70 K above Tc.

Vortex-Nernst signal in Bi 2201

Wang, Li, Ong PRB 2006

Nernst curves in Bi 2201

overdoped optimal underdoped

Yayu Wang,Lu Li,NPO PRB 2006

Nernst signal

eN = Ey /| T |

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?

•Loss of phase coherence determines Tc•Condensate amplitude persists T>Tc• Vorticity and diamagnetism in Nernst region

Nernstregion

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?

Supercurrents follow contours of condensate

Js = -(eh/m) x ||2 z

Diamagnetic currents in vortex liquid

Cantilever torque magnetometry

Torque on magnetic moment: = m × B

Deflection of cantilever: = k

crystal

B

m×

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

Tc

UnderdopedBi 2212 Wang et al.

Cond-mat/05

Paramagnetic background in Bi 2212 and LSCO

Magnetization curves in underdoped Bi 2212

Tc

Separatrix Ts

Wang et al.Cond-mat/05

amplitude fluctuation

F

F

phase fluctuation

Anderson-Higgs mechanism: Phase stiffnesssingular phase fluc. (excitation of vortices)

At high T, M scales with Nernst signal eN

M(T,H) matches eN in both H and T above Tc

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

Wang et al. Cond-mat/05

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

Hole-doped optimal Electron-doped optimal

TcTc

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

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

End

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

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

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.

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

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.

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

)

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.

Vortex-liquid boundary linear in x as x 0?

Sharp transition in Tc vs x (QCT?)

dissipative,vortices mobile

Long-rangephase coherence

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

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.

Gollub, Beasley,Tinkham et al.PRB (1973)

Field sensitivity of Gaussian fluctuations

Vortex signal above Tc0 in under- and over-doped Bi 2212Wang et al. PRB (2001)

Abrikosov vortices near Hc2

Upper critical field Hc2 = 0/22

Condensate destroyed when cores touch at Hc2

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)

Nernst contour-map in underdoped, optimal and overdoped LSCO

• In underdoped Bi-2212, onset of diamagnetic fluctuations at 110 K

• diamagnetic signal closely tracks the Nernst effect

110K

Tc

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)

Vortex-Nernst signal in Bi 2201

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

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

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

= 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

Wang et al., unpublished

Hc2(0) vs x matches Tonset vs x

H*

Hm

Tco

Overdoped LaSrCuO x = 0.20

Hc1

M vs H below TcFull Flux Exclusion

Strong Curvature!

-M

H

Strong curvature persists above Tc

M ~ H1/

M non-analytic in weak field

Fit to Kosterlitz Thouless theory

= -(kBT/2d02) 2

= a exp(b/t1/2)

Strongly H-dependentSusceptibility = M/H

Susceptibility and Correlation Length

Non-analytic magnetization above Tc

M ~ H1/

Fractional-exponentregion

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

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)

Optimal, untwinned BZO-grown YBCO

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)

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 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.

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)

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

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

Nernst effect in optimally doped YBCO

Nernst vs. H in optimally doped YBCO Vortex onset temperature: 107 K

Separatrix curve at Ts

Optimum doped Overdoped

Vortex Nernst signal

xy = M

-1 = 100 K

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?

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

Hole-doped optimal Electron-doped optimal

Overall scale of Nernst signal amplitude