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Vortex Nernst effect Loss of long-range phase coherence The Upper Critical Field High-temperature Diamagnetism

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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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slide1

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

slide2

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

slide3

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)

slide4

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)

slide5

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

slide7

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

slide8

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

slide9

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

slide10

Vortex signal persists to 70 K above Tc.

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

slide12

Wang, Li, Ong PRB 2006

Vortex-Nernst signal in Bi 2201

slide13

Nernst signal

eN = Ey /| T |

Nernst curves in Bi 2201

Yayu Wang,Lu Li,NPO PRB 2006

underdoped

optimal

overdoped

slide15

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?

slide16

Nernst

region

  • Loss of phase coherence determines Tc
  • Condensate amplitude persists T>Tc
  • Vorticity and diamagnetism in Nernst region
slide17

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?

slide18

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

Diamagnetic currents in vortex liquid

Supercurrents follow contours of condensate

slide19

×

B

m

Cantilever torque magnetometry

Torque on magnetic moment:  = m × B

crystal

Deflection of cantilever:  = k 

slide20

Si single-crystal cantilever

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
slide21

Underdoped

Bi 2212

Wang et al.

Cond-mat/05

Tc

slide23

Magnetization curves in underdoped Bi 2212

Wang et al.

Cond-mat/05

Tc

Separatrix Ts

slide24

y2

y1

y2

y1

Anderson-Higgs mechanism: Phase stiffness

singular phase fluc. (excitation of vortices)

slide27

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

slide29

Wang et al.

Cond-mat/05

slide30

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

slide31

Electron-doped optimal

Hole-doped optimal

Tc

Tc

slide32

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)

slide34

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
slide36

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

slide37

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

slide38

“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.

slide39

+

-

-

+

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

slide40

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

slide41

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)

slide42

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

slide43

Vortex-liquid boundary linear in x as x 0?

dissipative,

vortices mobile

Long-range

phase coherence

Sharp transition in Tc vs x (QCT?)

slide44

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
slide45

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

slide46

Field sensitivity of Gaussian fluctuations

Gollub, Beasley,

Tinkham et al.

PRB (1973)

slide48

x

Abrikosov vortices near Hc2

Upper critical fieldHc2 = f0/2px2

Condensate destroyed when cores touch at Hc2

slide50

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)
slide52

Tc

110K

  • In underdoped Bi-2212, onset of diamagnetic fluctuations at 110 K
  • diamagnetic signal closely tracks the Nernst effect
slide53

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)

slide55

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)

slide56

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

slide57

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

slide58

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

slide59

Wang et al., unpublished

Hc2(0) vs x matches Tonset vs x

slide60

Tco

Overdoped LaSrCuO x = 0.20

H*

Hm

slide61

-M

H

M vs H below Tc

Full Flux Exclusion

Strong Curvature!

Hc1

slide64

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)

slide65

Non-analytic magnetization above Tc

M ~ H1/d

Fractional-exponent

region

slide66

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
slide67

Wang et al. Science (2003)

overdoped

optimum

underdoped

Field scale increases as x decreases

slide69

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

slide70

Temp. dependence of Nernst coef. in Bi 2201 (y = 0.60, 0.50).

Onset temperatures much higher than Tc0 (18 K, 26 K).

slide71

Resistivity is a 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.

slide72

Bardeen Stephen law (not seen)

Resistivity Folly

Ong Wang, M2S-RIO, Physica C (2004)

Hc2

Hc2

Resistivity does not distinguish vortex liquid from normal state

slide73

Isolated off-diagonal Peltier current axy versus T in LSCO

Vortex signal onsets at 50 and 100 K for x = 0.05 and 0.07

slide74

Tco

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
slide75

Nernst effect in optimally doped YBCO

Vortex onset temperature: 107 K

Nernst vs. H in optimally doped YBCO

slide76

Separatrix curve at Ts

Optimum doped

Overdoped

slide77

Vortex Nernst signal

axy = b M

b-1 = 100 K

slide78

n

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?

slide80

Strong correlation in CuO2 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

slide81

Electron-doped optimal

Hole-doped optimal

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