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

1. 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

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 LSCO = La2-xSrxCuO4 Bi 2212 = Bi2Sr2CaCu2O8 Bi 2201 = Bi2-yLaySr2CuO6

3. 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)

4. 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)

5. 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

6. upper critical field

7. 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

8. 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

9. 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

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

12. Wang, Li, Ong PRB 2006 Vortex-Nernst signal in Bi 2201

13. Nernst signal eN = Ey /| T | Nernst curves in Bi 2201 Yayu Wang,Lu Li,NPO PRB 2006 underdoped optimal overdoped

15. 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?

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

17. 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?

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

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

20. 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

21. Underdoped Bi 2212 Wang et al. Cond-mat/05 Tc

22. Paramagnetic background in Bi 2212 and LSCO

23. Magnetization curves in underdoped Bi 2212 Wang et al. Cond-mat/05 Tc Separatrix Ts

24. y2 y1 y2 y1 Anderson-Higgs mechanism: Phase stiffness singular phase fluc. (excitation of vortices)

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

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

27. 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

29. Wang et al. Cond-mat/05

30. 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

31. Electron-doped optimal Hole-doped optimal Tc Tc

32. 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)

34. 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

35. End

36. 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

37. 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

38. “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.

39. + - - + 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

40. 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

41. 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)

42. 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

43. Vortex-liquid boundary linear in x as x 0? dissipative, vortices mobile Long-range phase coherence Sharp transition in Tc vs x (QCT?)

44. 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

45. 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

46. Field sensitivity of Gaussian fluctuations Gollub, Beasley, Tinkham et al. PRB (1973)

47. Vortex signal above Tc0 in under- and over-doped Bi 2212 Wang et al. PRB (2001)

48. x Abrikosov vortices near Hc2 Upper critical fieldHc2 = f0/2px2 Condensate destroyed when cores touch at Hc2

50. 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)