Nernst effect in vortex-liquid state of...

1. Introduction to the Nernst effect2. Vortex signal above Tc3. Loss of long-range phase coherence4. The Upper Critical Field5. The cuprate phase diagram

Talk 2

Nernst effect in vortex-liquid state of cuprates

Boulder, July 2008Supported by NSF-MRSEC, ONR

Boulder School for Condensed Matter and Materials Physics 2008

N. P. Ong

Collaborators: Lu Li, Yayu Wang, Zhuan Xu (Princeton Univ.)S. Uchida (Univ. Tokyo)D. Bonn, W. Hardy, R. Liang (Univ. British Columbia)

holes = 1/2

Phase diagram of Cuprates

T pseudogap

0 0.05 0.25

AF dSC

T*

Tc

Mott insulator

Fermiliquid

doping x

0)( kkkk

k ↑∗

↓−∗+=Ψ ∏ ccveu i

BCSφBCS wave function

Phase φ fixed (phase representation); N fluctuates

[N, φ ] = 1

ku kv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛10

01 φievu kkAnderson

pseudospin

)r(||)r(ˆ φieΨ=Ψ

The phase of macroscopic pair-wave function

φ

S

Vortex in cuprates

CuO2 layers

2D vortex pancake

ξ

Vortex in Niobium

Js

superfluidelectrons(pair condensate)

ξ

Js

b(r)Normal core

H

coherence length ξ

Vortices in type-II superconductors

|Ψ| = Δ

London length λ

b(r)

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

Vortex motion in type II superconductor

Applied supercurrent Js exerts magnus force on vortex core

Velocity gives induced E-field in core (Faraday effect)Current enters core and dissipates (damping viscosity)

Motion of vortices generates observedE-field

Consequence of Josephson equation

Tilt angle of velocity gives negative vortex Hall effect

In clean limit, vortex v is || - Js

E = B x v

(Bardeen Stephen, Nozieres Vinen)

ηρρ /02

Φ== BHH

cNxx

0Φ×=r

sM JF

amplitude fluctuation

F

F

phase fluctuation

ψ2

ψ1

ψ1

ψ2

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

Phase mode θ + EM Fμν

= Massive mode (Meissner effect)

Anderson-Higgs mechanism and phase rigidity

Phase rigidity uniform phase θ |Ψ| eiθ(r)

But phase coherence destroyed by mobile vortices

phase rigidity measured by ρs( )23

21 θρρ ∇= ∫ SrdH

Δθ = 2π

P.W. Anderson Phys. Rev. 1959, RMP 1966

phase-slip and Nernst signal

Passage of a vortex Phase diff. θ jumps by 2π

Integrate VJ to give dc signalprop. to nv

•

= φhJeV2 = 2πh nV

Josephson Eq.

time

Δθ = 2π

ΔVJ

H

Δ−

T

• Baskaran, Zou, Anderson (Sol. St. Comm. 1987)• Doniach, Inui (PRB 1989)• Uemura plot (Nature 1989)

• Emery, Kivelson (Nature 1995)low hole density and high Tccuprates highly suscep. to phase fluctuations

• Corson, Orenstein (Nature 1999)Kinetic inductance meas. at THz freq extends above TcKT physics in ultra-thin film BSCCO

• M. Franz and Z. Tesanovic (1999)Vortex-charge duality, QED3 model

• S. Sachdev (2005)Quantum vortices

Baskaran, Zou, Anderson (Solid State Comm. 1987)Δ

vs. x and the loss of phase coherence in underdoped regime

Emery Kivelson (Nature 1995)Phase fluctuation and loss of coherence at Tc in low (superfluid) density SC’s

M. Renderia et al. (Phys. Rev. Lett. ’02)Cuprates in strong-coupling limit, distinct from BCS limit.

Tesanovic and Franz (Phys. Rev. B ’99, ‘03)Strong phase fluctuations in d-wave superconductor treated by dual mappingto Bosons in Hofstadter lattice --- vorticity and checkerboard pattern

Balents, Sachdev, Fisher et al. (2004)Vorticity and checkerboard in underdoped regime

P. A. Lee, X. G. Wen. (PRL, ’03, PRB ’04)Loss of phase coherence in tJ model, nature of vortex core

Lee, Nagaosa, Wen, Rev. Mod. Phys. (cond-mat/0410***)Good review of phase stiffness, phase fluctuation issues

P. W. Anderson (cond-mat ‘05)Spin-charge locking occurs at Tonset > Tc

Theories on phase fluctuation in cuprates

The Nernst effect of carriers

).(E.J T−∇+= ασtt

).(.E T−∇−= αρtt

( ) )( TE xyxyxy −∂+−= αρρα

Open boundaries, so set J = 0.

)(k

.Bvk

kkxyxy T

fe ll∂∂

×−

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−= ∑ μεε

α0

22

εθπ

∂∂

=∇

≡eTk

TE

e ByN

22

3||

Off-diag. Peltier cond.

Measured Nernst signal

Generally, very small because of cancellation between αxy and σxy

Wang et al. PRB ‘01

The vortex Nernst effect; dominant in vortex liquid state

B

v

E

Gradient drives vortex currentwith velocity v || x

)(v Ts −∇= φη

ηφBs

TEeN =

∇=

||

Force exerted on vortex line by grad T

Line entropy sφ

Balance F by viscous damping

Nernst signal eN

)(F Ts −∇= φ

Moving vortex producesE = B x v

Nernst experiment

Vortices move in a temperature gradientPhase slip generates Josephson voltage

2eVJ = 2πh nVEJ = B x v

H

ey

Hm

Nernst signal

ey = Ey /| T |

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)

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

Vortex signal persists to 70 K above Tc .

Wang, Li, NPOPRB (2006)

OP YBCO UD LSCO

OP Bi2212 UD Bi2212

Nernst contour-map in underdoped, optimal and overdoped LSCO

H*

Hm

Tco

Overdoped LaSrCuO x = 0.20

Optimal, untwinned BZO-grown YBCO

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

Wang et al., PRL ‘02

Nernst contour maps in UD YBCO and OP YBCO

Wang, Li, Ong PRB 2006Contour Map of Nernst Signal in Bi 2201

Spontaneous vortices destroy superfluidity in 2D films

Change in free energy ΔF to create a vortex

ΔF = ΔU – TΔS = (εc – kB T) log (R/a)2

ΔF < 0 if T > TKT = εc /kB vortices appear spontaneously

Δ

TcMFTKT

0

ρs

Kosterlitz-Thouless transition

3D version of KT transition in cuprates?

Kosterlitz Thouless transition in 2D superconductor

Unbinding ofvortex-antivortex

ΔF = U - TSFree energy gain

vortex density

vortex

antivortex

H = ½

ρs d3r

( φ)2

ρs

measures phase rigidityPhase coherence destroyed

at TKTby proliferation of vortices

BCS transition 2D Kosterlitz

Thouless

transition

Δ

Tc

ρs

0

Δ

TMFTKT

nvortex

ρs

0

High temperature superconductors?

Podolsky, Raghu, VishwanathPRL (2007).

Simulation Nernst effect in 2D XY model

•Loss of phase coherence determines Tc•Condensate amplitude persists T>Tc

Wang et al. PRB (2001),NPO et al. Ann. Phys (2004)

• Condensate amplitude persists to Tonset > Tc• Nernst signal confined to SC dome• Vorticity defines Nernst region

upper critical field

The Upper Critical Field -- destruction of pair condensate

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

d-wave symmetry

Cooper pairing in cuprates

ξ +- -

+

Upper critical field

coherence length

Hc2 4 Tesla1040100 Tesla

90572918

NbSe2MgB2Nb3 Sncuprates

ξ (A)o

θθ 2cos)( 0Δ=Δ

20

2 2πξφ

=cH

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

e y (μV

/K)

μ 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

Vortex Nernst signal in overdoped, optim. and underdoped regimes

overdoped optimal underdoped

Wang, Li, NPO PRB 2006

Nernst signal

eN = Ey /| T |

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 (Tc=50K)

μ 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

7570

65

60

55

50 45

40

35

30

25

20

OD-Bi2212 (Tc=65K)

e y (μ

V/K

)

μ 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)

μ 0H (T)

overdoped optimum underdoped

Wang et al. Science (2003)

Resistivity a bad probe

for absence of pair amplitude

Plot of ρ

and ey versus T at fixed H (33 T).

Vortex signal is large for T < 26 K, but ρ

is close to normal value ρN above 15 K.

LaSrCuO

0 2 4 6 8 10 12 140.0

0 .2

0 .4

0 .6

0 .8

e y

ρ12K

N dC C O (T c=24 .5K )

e y (μ

V/K

)

μ 0H (T )0 5 1 0 1 5 2 0 2 5 3 0

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

2 2 K

e y

ρ

L S C O (0 .2 0 )

e y(μV

/K)

μ 0H (T )

Resistivity does not distinguish vortex liquid and normal state

Hc2Hc2

Bardeen Stephen law (not seen)

Problems with Flux-flow ResistivityWang, Li, NPO PRB ‘06

NbSe2 NdCeCuO Hole-doped cuprates

Tc0 Tc0Tc0

Hc2 Hc2Hc2

HmHm

Hm

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

Plot of Hm , H*, Hc2 vs. T

• Hm and H* similar to hole-doped

• However, Hc2 has mean-field form

• Vortex-Nernst signal vanishes just above Hc2 line

Vortex Nernst signal in NdCeCuO -- mean-field scenario

Hole-doped optimal Electron-doped optimal

Tc TcStrong phase fluctuations(non Gaussian)

Mean-field like (Gaussianfluctuations above Tc)

Anomalous Nernst signal only in pseudogap state

Wang et al., unpublishedHc2 (0) vs x matches Tonset vs x

Summary

1. Existence of large vortex-Nernst region above Tc dome

2. Transition at Tc not mean-field BCS, but loss oflong-range phase correlation

3. Vortex liquid state extends high above Tc in UD regime

4. Upper critical field Hc2 is very large, 80-150 Tesla

5. Hc2 vs. T dependence not of mean-field BCS form

References (Talk 2)

1. Z. Xu, N.P. Ong, Y. Wang, T. Kakeshita and S. Uchida, Nature 406, 486 (2000).

2. Yayu Wang, Z. A. Xu, T. Kakeshita, S. Uchida, S. Ono, Yoichi Ando, and N. P. Ong, Phys. Rev. B 64, 224519 (2001).

3. Yayu Wang, N. P. Ong, Z.A. Xu, T. Kakeshita, S. Uchida, D. A. Bonn, R. Liang and W. N. Hardy, Phys. Rev. Lett. 88, 257003 (2002)

4. Yayu Wang, S. Ono, Y. Onose, G. Gu, Yoichi Ando, Y. Tokura, S. Uchida, and N. P. Ong, Science, 299, 86 (2003).

5. Yayu Wang, Lu Li and N. P. Ong, Phys. Rev. B 73, 024510 (2006),

6. Daniel Podolsky, Srinivas Raghu and Ashvin Vishwanath, Phys. Rev. Lett. 99, 117004 (2007).

7. V. Oganesyan and Iddo Ussishkin, Phys. Rev. B 70, 054503 (2004).

8. Cigdem Capan and Kamran Behnia et al., Phys. Rev. Lett. 88, 056601 (2002)

9. F. Rullier-Albenque, R. Tourbot, H. Alloul, P. Lejay, D. Colson, and A. Forget,Phys. Rev. Lett. 96, 067002 (2006)

Magnetization in Abrikosov

state

HM

M~ -lnHM = - [Hc2 – H] / β(2κ2 –1)

Hc2Hc1

In cuprates, κ

= 100-150, Hc2 ~ 50-150 T

M < 1000 A/m (10 G)

Area = Condensation energy U

Optimal YBCO3-layer Bi 2223

Wang, Li and NPO, PRB 06