1.Vortex Nernst effect 2.Loss of long-range phase coherence 3.The Upper Critical Field 4.High-temperature Diamagnetism 5.KT vs 3DXY: phase-correlation length What lies above : the vortex liquid above T c in cuprate superconductors. Yayu Wang, LuLi, J. Checkelsky, N.P.O. Princeton Univ. M. J. Naughton, Boston College St. Andrews June 2005 S. Uchida, Univ. Tokyo Yoichi Ando, Elec. Power U., Tokyo Genda Gu, Brookhaven S. Onose, Y. Tokura, U. Tokyo B. Keimer, MPI Stuttgart

hole s = 1/2 Phase diagram of Cuprates T pseudogap AF dSC T*T* TcTc Mott insulator Fermi liquid doping x

Vortex in cuprates CuO 2 layers 2D vortex pancake  Gap  (r) vanishes in core  Vortex in Niobium JsJs superfluid electrons  JsJs b(r)b(r) Normal core H

The Josephson Effect, phase-slippage and Nernst signal t  VJVJ vortex 22 Phase difference Passage of a vortex Phase diff.  jumps by 2  Integrate V J to give dc signal prop. to n v

Nernst experiment Vortices move in a temperature gradient Phase slip generates Josephson voltage 2eV J = 2  h n V E J = B x v H eyey HmHm Nernst signal e y = E y /| T |

Nernst effect in underdoped LSCO-0.12 with T c = 29K vortex Nernst signal onset from T = 120 K, ~ 90K above T c`1

Nernst effect in underdoped Bi-2212 (T c = 50 K) Vortex signal persists to 70 K above T c !

Vortex signal above T c0 in under- and over-doped Bi 2212

Phase rigidity Long-range phase coherence requires uniform   e i  r  Phase coherence destroyed by vortex motion    “kilometer of dirty lead wire” phase rigidity measured by  s   Emery, Kivelson, (1995): Spontaneous vortex creation at Tc in cuprates

Spontaneous vortices destroy superfluidity in 2D films Change in free energy  F to create a vortex  F =  U – T  S = (  c – k B T) log (R/a) 2  F T KT =  c /k B vortices appear spontaneously  T c MF T KT 0 ss Kosterlitz-Thouless transition 3D version of KT transition in cuprates?

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

Field scale increases as x decreases overdoped optimum underdoped

eyey PbIn, T c = 7.2 K (Vidal, PRB ’73) Bi 2201 (T c = 28 K, H c2 ~ 48 T) T=8K H c2 T=1.5K HdHd H/H c2 H c2 Upper critical Field H c2 given by e y 0. Hole cuprates --- Need intense fields.

Vortex-Nernst signal in Bi 2201

 LSCO Cooper pairing potential largest in underdoped regime H c2 increases as x decreases (like ARPES gap  0 ) Compare  0 (from H c2 ) with Pippard length  P = hv F /a  0 (a = 3/2) STM vortex core  STM ~ 22 A   (Ding)

NbSe 2 NdCeCuO Hole-doped cuprates T c0 H c2 HmHm HmHm HmHm Expanded vortex liquid Amplitude vanishes at T c0 Vortex liquid dominant. Loss of phase coherence at T c0 (zero-field melting) Conventional SC Amplitude vanishes at T c0 (BCS) vortex liquid vortex liquid

Supercurrents follow contours of condensate H J s = -(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 (Mike Naughton) Capacitive detection of deflection Sensitivity: ~ 5 × emu at 10 tesla ~ 200 times more sensitive than commercial SQUID Micro-fabricated Si single-crystal cantilever Very thin cantilever beam: ~ 5  m H

Tc

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

Magnetization curves in underdoped Bi 2212 Tc Separatrix Ts

At high T, M scales with Nernst signal e N

M(T,H) matches e N in both H and T above T c

Magnetization in Abrikosov state H M M~ -lnH M = - [H c2 – H] /  (2  2 –1) H c2 H c1 In cuprates,  = , H c2 ~ T M < 1000 A/m (10 G) Area = Condensation energy U

H c2 TcTc In conventional type II supercond., H c2 0 T T c - H c2 M M In cuprates, H c2 is unchanged as T T c H c2 TcTc

Resistivity does not distinguish vortex liquid and normal state H c2 Bardeen Stephen law (not seen) Resistivity Folly

T* T onset TcTc spin pairing (NMR relaxation, Bulk suscept.) vortex liquid Onset of charge pairing Vortex-Nernst signal Enhanced diamagnetism Kinetic inductance superfluidity long-range phase coherence Meissner eff. x (holes) Temperature T 0 Phase fluctuation in cuprate phase diagram pseudogap

Doniach Inui (Phys. Rev. B 90) Loss of phase coherence and charge fluctuation in underdoped regime Emery Kivelson (Nature 95) Loss of coherence at Tc in low (superfluid) density SC’s K. Levin (Rev. Mod. Phys. ‘05) 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 mapping to 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 P. W. Anderson (cond-mat ‘05) Spin-charge locking occurs at T onset > T c Relevant Theories

Non-analytic magnetization

Hc1 M vs H below Tc Full Flux Exclusion Strong Curvature! -M H

Strong curvature persists above Tc

Anomalous high-temp. diamagnetic state 1. Vortex-liquid state defined by large Nernst signal and diamagnetism 2.M(T,H) closely matched to e N (T,H) at high T (  is times larger than in ferromagnets). 3.M vs. H curves show H c2 stays v. large as T  T c. 4.Magnetization evidence that transition is by loss of phase coherence instead of vanishing of gap 5.Nonlinear weak-field diamagnetism above T c to T onset. 6.NOT seen in electron doped NdCeCuO (tied to pseudogap physics)

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

Relation between fluctuating M and Nernst current J y =  yx (- T); e N =  xy  xy = -  M Caroli Maki (‘69), Ussishkin, Sondhi (‘02) Fluctuating M generates a transverse charge flow in a gradient For vortices in Bi 2212, 1/  = K For ferromagnet spinel, 1/  = 10 5 K Easy to distinguish between vortex flow and ferromagnetism Recently verified for vortices and ferromagnets

Temp. dependence of Nernst coef. in Bi 2201 (y = 0.60, 0.50). Onset temperatures much higher than T c0 (18 K, 26 K).

H = ½  s d 3 r (  ) 2  s measures phase rigidity Phase coherence destroyed at T KT by proliferation of vortices BCS transition 2D Kosterlitz Thouless transition  T c  s 0  T MF T KT n vortex  s 0 High temperature superconductors?

Plot of H m, H*, H c2 vs. T H m and H* similar to hole-doped However, H c2 is conventional Vortex-Nernst signal vanishes just above H c2 line

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