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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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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

  • Loss of long-range phase coherence

  • The Upper Critical Field

  • High-temperature Diamagnetism

Taipeh, June 2006


Mott insulator

T*

T

pseudogap

Tc

Fermi

liquid

AF

dSC

0

0.25

0.05

doping x

Phase diagram of Cuprates

s = 1/2

hole

LSCO = La2-xSrxCuO4

Bi 2212 = Bi2Sr2CaCu2O8

Bi 2201 = Bi2-yLaySr2CuO6


Condensate described by a complex macroscopic wave function

Y(r) = Y1 + iY2 = |Y(r)| exp[iq(r)]

y2

y1

y2

y1

Anderson-Higgs mechanism: Phase stiffness

singular phase fluc. (excitation of vortices)


q

q

q

q

q

q

Phase rigidity ruined by mobile defects

Long-range 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 type-II SC

Vortex in cuprates

Vortex in Niobium

CuO2 layers

superfluid

electrons

Js

2D vortex pancake

H

coherence length x



normal

liquid

Hm

Hc2

vortex solid

Hc1

0

Tc0

T

Mean-field phase diagram

Cuprate phase diagram

2H-NbSe2

4 T

100 T

Hc2

H

H

vortex

liquid

Hm

Tc

vortex

solid

100 K

7 K

Meissner state


Phase difference

vortex

2p

f

Integrate VJ to give dc signal

prop. to nv

VJ

t

The Josephson Effect, phase-slippage and Nernst signal

Passage of a vortex

Phase diff. f jumps by 2p


Nernst signal

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 Bi-2212 (Tc = 50 K)


Wang, Li, Ong PRB 2006

Vortex-Nernst signal in Bi 2201


Nernst signal

eN = Ey /| T |

Nernst curves in Bi 2201

Yayu Wang,Lu Li,NPO PRB 2006

underdoped

optimal

overdoped


rs

D

0

TKT

TcMF

Kosterlitz-Thouless 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?


Nernst

region

  • Loss of phase coherence determines Tc

  • Condensate amplitude persists T>Tc

  • Vorticity and diamagnetism in Nernst region


In hole-doped cuprates

  • Existence of vortex Nernst signal above Tc

  • Confined to superconducting “dome”

  • Upper critical field Hc2 versus T is anomalous

  • Loss of long-range phase coherence at Tc

  • by spontaneous vortex creation (not gap closing)

  • 5. Pseudogap intimately related to vortex liquid state

More direct (thermodynamic) evidence?


Js = -(eh/m) x |Y|2 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 


Micro-fabricated single crystal silicon cantilever magnetometer

H

  • Capacitive detection of deflection

  • Sensitivity: ~ 5 × 10-9 emu at 10 tesla

  • ~100 times more sensitive than commercial SQUID


Underdoped

Bi 2212

Wang et al.

Cond-mat/05

Tc



Magnetization curves in underdoped Bi 2212

Wang et al.

Cond-mat/05

Tc

Separatrix Ts


y2

y1

y2

y1

Anderson-Higgs mechanism: Phase stiffness

singular phase fluc. (excitation of vortices)



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 = 100-150, Hc2 ~ 50-150 T

M < 1000 A/m (10 G)

Area = Condensation energy U


Wang et al.

Cond-mat/05


normal

liquid

Hm

Hc2

vortex solid

Hc1

0

Tc0

T

Mean-field phase diagram

Cuprate phase diagram

2H-NbSe2

4 T

100 T

Hc2

H

H

vortex

liquid

Hm

Tc

vortex

solid

100 K

7 K

Meissner state


Electron-doped optimal

Hole-doped optimal

Tc

Tc


Phase fluctuation in cuprate phase diagram

spin pairing

(NMR relaxation,

Bulk suscept.)

T*

pseudogap

Tonset

Onset of charge pairing

Vortex-Nernst signal

Enhanced diamagnetism

Kinetic inductance

TemperatureT

vortex liquid

Tc

superfluidity

long-range phase coherence

Meissner eff.

0

x (holes)


  • In hole-doped cuprates

  • 1. Large region in phase diagram above Tc dome

  • with enhanced Nernst signal

  • Associated with vortex excitations

  • Confirmed by torque magnetometry

  • Transition at Tc is 3D version of KT transition

  • (loss of phase coherence)

  • 5. Upper critical field behavior confirms conclusion



x

+

o

-

-

x (A)

+

Hc2

4 Tesla

40

10

100 Tesla

Cooper pairing in cuprates

d-wave 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 30-50 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 mean-field.


+

-

-

+

Hc2

vortex

liquid

Hm

Tc

Signature features of cuprate superconductivity

1. Strong Correlation

2. Quasi-2D anisotropy

3. d-wave pairing, very short x

4. Spin gap, spin-pairing 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


Two distinct field scales

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 La2-xSrxCuO4

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


Vortex-liquid boundary linear in x as x 0?

dissipative,

vortices mobile

Long-range

phase coherence

Sharp transition in Tc vs x (QCT?)


The case against inhomogeneous superconductivity

(granular Al)

  • LaSrCuO transition at T = 0 much too sharp

  • Direct evidence for competition between d-wave SC

  • and emergent spin order

  • 3. In LSCO, Hc2(0) varies with x


  • 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.

Competing ground states


Field sensitivity of Gaussian fluctuations

Gollub, Beasley,

Tinkham et al.

PRB (1973)


Vortex signal above Tc0 in under- and over-doped Bi 2212

Wang et al. PRB (2001)


x

Abrikosov vortices near Hc2

Upper critical fieldHc2 = f0/2px2

Condensate destroyed when cores touch at Hc2


  • Anomalous high-temp. diamagnetic state

  • Vortex-liquid state defined by large Nernst signal and diamagnetism

  • M(T,H) closely matched to eN(T,H) at high T (b is 103 - 104 times larger than in ferromagnets).

  • M vs. H curves show Hc2 stays v. large as T Tc.

  • Magnetization evidence that transition is by loss of phase coherence instead of vanishing of gap

  • Nonlinear weak-field diamagnetism above Tc to Tonset.

  • NOT seen in electron doped NdCeCuO (tied to pseudogap physics)



Tc

110K

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

  • diamagnetic signal closely tracks the Nernst effect


T=1.5K

T=8K

Hd

Hc2

0.3

1.0

H/Hc2

  • Upper critical Field Hc2 given by ey 0.

  • Hole cuprates --- Need intense fields.

PbIn, Tc = 7.2 K (Vidal, PRB ’73)

Bi 2201 (Tc= 28 K, Hc2 ~ 48 T)

ey

Hc2

Wang et al. Science (2003)



Hole-doped cuprates

NbSe2

NdCeCuO

Hc2

Hc2

Hc2

vortex

liquid

vortex

liquid

Hm

Hm

Hm

Tc0

Tc0

Tc0

Vortex liquid dominant.

Loss of phase coherence

at Tc0 (zero-field melting)

Expanded vortex liquid

Amplitude vanishes at Tc0

Conventional SC

Amplitude vanishes

at Tc0 (BCS)


4 T

normal

?

?

liquid

vortex

liquid

H

Hm

Hc2

vortex solid

vortex

solid

Hm

Hc1

Hc1

0

Tc0

0

Tc0

T

T

Phase diagram of type-II superconductor

cuprates

2H-NbSe2

H

Meissner state


x

Gap D

Tc

Temp. T

Superconductivity in low-Tc superconductors (MF)

Cooper pairs with coherence length x

Quasi-particles

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


Wang et al., unpublished

Hc2(0) vs x matches Tonset vs x


Tco

Overdoped LaSrCuO x = 0.20

H*

Hm


-M

H

M vs H below Tc

Full Flux Exclusion

Strong Curvature!

Hc1




Susceptibility and Correlation Length

Strongly H-dependent

Susceptibility c = M/H

Fit to

Kosterlitz Thouless theory

c = -(kBT/2df02) xKT2

xKT = a exp(b/t1/2)


Non-analytic magnetization above Tc

M ~ H1/d

Fractional-exponent

region


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


Wang et al. Science (2003)

overdoped

optimum

underdoped

Field scale increases as x decreases



Xu et al. Nature (2000)

Wang et al. PRB (2001)

Nernst effect in LSCO-0.12

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, M2S-RIO, Physica C (2004)

Hc2

Hc2

Resistivity does not distinguish vortex liquid from normal state


Isolated off-diagonal 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).

  • Vortex signal extends above

  • 70 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 0.50).

Vortex onset temperature: 107 K

Nernst vs. H in optimally doped YBCO


Separatrix curve at T 0.50).s

Optimum doped

Overdoped


Vortex Nernst signal 0.50).

axy = b M

b-1 = 100 K


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

charge-transfer

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


Electron-doped optimal 0.50).

Hole-doped optimal



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