Vortex Nernst effect Loss of long-range phase coherence The Upper Critical Field High-temperature Diamagnetism KT vs 3DX

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

2. 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

3. b(r) Normal core Js x x Vortex in cuprates Vortex in Niobium CuO2 layers superfluid electrons Js H 2D vortex pancake Gap D(r) vanishes in core |Y| = D

4. 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

5. 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

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

7. Vortex signal persists to 70 K above Tc ! Nernst effect in underdoped Bi-2212 (Tc = 50 K)

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

10. q q q q q q Phase rigidity |Y| eiq(r) Long-range phase coherence requires uniform q “kilometer of dirty lead wire” phase rigidity measured by rs Phase coherence destroyed by vortex motion Emery, Kivelson, (1995): Spontaneous vortex creation at Tc in cuprates

11. 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?

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

13. overdoped optimum underdoped Field scale increases as x decreases

14. 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

15. Vortex-Nernst signal in Bi 2201

16. Hc2increases as x decreases • (like ARPES gap D0) • Compare x0 (from Hc2) with • Pippard length • xP = hvF/aD0 (a = 3/2) • STM vortex core • xSTM ~ 22 A LSCO D D0 (Ding) Cooper pairing potential largest in underdoped regime

17. 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)

18. Js = -(eh/m) x |Y|2 z Diamagnetic currents in vortex liquid H Supercurrents follow contours of condensate

19.  × B  m Cantilever torque magnetometry Torque on magnetic moment:  = m × B crystal Deflection of cantilever:  = k 

20. Micro-fabricated Si single-crystal cantilever • Very thin cantilever beam: ~ 5 m Micro-fabricated single crystal silicon cantilever magnetometer(Mike Naughton) H • Capacitive detection of deflection • Sensitivity: ~ 5 × 10-9 emu at 10 tesla • ~200 times more sensitive than commercial SQUID

21. Tc

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

24. Magnetization curves in underdoped Bi 2212 Tc Separatrix Ts

26. At high T, M scales with Nernst signal eN

27. M(T,H) matches eN in both H and T above Tc

28. 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

31. Hc2 Hc2 M T Tc- Tc In conventional type II supercond., Hc2 0 Hc2 Hc2 M Tc In cuprates, Hc2 is unchanged as T Tc

33. Bardeen Stephen law (not seen) Resistivity Folly Hc2 Hc2 Resistivity does not distinguish vortex liquid and normal state

34. 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 Temperature T vortex liquid Tc superfluidity long-range phase coherence Meissner eff. 0 x (holes)

35. Relevant Theories 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 Tonset > Tc

36. Non-analytic magnetization

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

38. Strong curvature persists above Tc

39. 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)

40. End

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

42. Jy = ayx (- T); eN = raxy Relation between fluctuating M and Nernst current Caroli Maki (‘69), Ussishkin, Sondhi (‘02) axy = -b M Fluctuating M generates a transverse charge flow in a gradient Recently verified for vortices and ferromagnets For vortices in Bi 2212, 1/b = 50-100 K For ferromagnet spinel, 1/b = 105 K Easy to distinguish between vortex flow and ferromagnetism

43. Temp. dependence of Nernst coef. in Bi 2201 (y = 0.60, 0.50). Onset temperatures much higher than Tc0 (18 K, 26 K).

44. 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?

45. 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

46. 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