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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
