1.Vortex Nernst effect 2.Loss of long-range phase coherence 3.The Upper Critical Field 4.High-temperature Diamagnetism 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 Taipeh, June 2006

hole s = 1/2 Phase diagram of Cuprates T pseudogap AF dSC T*T* TcTc Mott insulator Fermi liquid doping x LSCO = La 2-x Sr x CuO 4 Bi 2212 = Bi 2 Sr 2 CaCu 2 O 8 Bi 2201 = Bi 2-y La y Sr 2 CuO 6

    Anderson-Higgs mechanism: Phase stiffness singular phase fluc. (excitation of vortices) Condensate described by a complex macroscopic wave function  (r) =  1 + i  2 = |  r  | exp[i  r  ]

Phase rigidity ruined by mobile defects Long-range phase coherence requires uniform  Phase coherence destroyed by vortex motion    “kilometer of dirty lead wire” phase rigidity measured by  s   Kosterlitz Thouless transition in 2D films (1982)

Vortex in cuprates CuO 2 layers 2D vortex pancake  Vortex in Niobium JsJs superfluid electrons  JsJs b(r)b(r) Normal core H coherence length  Vortices, fundamental excitation of type-II SC  London length b(r)

upper critical field

Mean-field phase diagram H 2H-NbSe 2 T H c2 H c1 T c0 normal vortex solid liquid 0 HmHm Meissner state H Cuprate phase diagram 4 T 7 K vortex solid vortex liquid H c2 TcTc 100 T 100 K HmHm

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 Bi-2212 (T c = 50 K) Vortex signal persists to 70 K above T c.

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

Nernst curves in Bi 2201 overdoped optimal underdoped Yayu Wang,Lu Li,NPO PRB 2006 Nernst signal e N = E y /| T |

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 Vorticity and diamagnetism in Nernst region Nernst region

1.Existence of vortex Nernst signal above T c 2.Confined to superconducting “dome” 3.Upper critical field H c2 versus T is anomalous 4.Loss of long-range phase coherence at T c by spontaneous vortex creation (not gap closing) 5.Pseudogap intimately related to vortex liquid state In hole-doped cuprates More direct (thermodynamic) evidence?

Supercurrents follow contours of condensate 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 Capacitive detection of deflection Sensitivity: ~ 5 × emu at 10 tesla ~ 100 times more sensitive than commercial SQUID Si single-crystal cantilever H

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

Paramagnetic background in Bi 2212 and LSCO

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

    Anderson-Higgs mechanism: Phase stiffness singular phase fluc. (excitation of vortices)

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

Wang et al. Cond-mat/05

Mean-field phase diagram H 2H-NbSe 2 T H c2 H c1 T c0 normal vortex solid liquid 0 HmHm Meissner state H Cuprate phase diagram 4 T 7 K vortex solid vortex liquid H c2 TcTc 100 T 100 K HmHm

Hole-doped optimal Electron-doped optimal TcTc TcTc

T*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

In hole-doped cuprates 1. Large region in phase diagram above Tc dome with enhanced Nernst signal 2.Associated with vortex excitations 3.Confirmed by torque magnetometry 4.Transition at Tc is 3D version of KT transition (loss of phase coherence) 5. Upper critical field behavior confirms conclusion

End

d-wave symmetry Cooper pairing in cuprates  Upper critical field coherence length H c2 4 Tesla Tesla NbSe 2 MgB 2 Nb 3 Sn cuprates  (A) o

Contrast with Gaussian (amplitude) fluctuations In low Tc superconductors, Evanescent droplets of superfluid radius  exist above Tc M’ = 2  1/2 (k B T c /  0 3/2 ) B 1/2 At T c, (Schmidt, Prange ‘69)  This is times smaller than observed in Bi 2212

Wang et al. PRL Robustness Survives to H > 45 T. Strongly enhanced by field. (Gaussian fluc. easily suppr. in H). 2. Scaling with Nernst Above T c, magnetization M scales as e N vs. H and T. 3. Upper critical field Behavior of H c2 (T) not mean-field. “Fluctuation diamagnetism” distinct from Gaussian fluc.

Signature features of cuprate superconductivity 1. Strong Correlation 2. Quasi-2D anisotropy 3. d-wave pairing, very short  4. Spin gap, spin-pairing at T* 5. Strong fluctuations, vorticity 6. Loss of phase coherence at Tc TcTc vortex liquid H c2 HmHm

Comparison between x = and Sharp change in ground state Pinning current reduced by a factor of ~100 in ground state Lu Li et al., unpubl.

In ground state, have 2 field scales 1) H m (0) ~ 6 T Dictates phase coherence, flux expulsion 2) H c2 (0) ~ 50 T Depairing field. Scale of condensate suppression Two distinct field scales M (A/m)

Magnetization in lightly doped La 2-x Sr x CuO 4 5 K 35 K 30 K 4.2 K SC dome Lu Li et al., unpubl.

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

The case against inhomogeneous superconductivity (granular Al) 1.LaSrCuO transition at T = 0 much too sharp 2.Direct evidence for competition between d-wave SC and emergent spin order 3. In LSCO, H c2 (0) varies with x

Competing ground states Abrupt transition between different ground states at x c = 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 (H c2 ~ T) iv) Cooling to 0.5 K tips balance against phase coherence.

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

Vortex signal above T c0 in under- and over-doped Bi 2212 Wang et al. PRB (2001)

 Abrikosov vortices near H c2 Upper critical field H c2 =  0 /2  2 Condensate destroyed when cores touch at H c2

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)

Nernst contour-map in underdoped, optimal and overdoped LSCO

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

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. Wang et al. Science (2003)

Vortex-Nernst signal in Bi 2201

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

Phase diagram of type-II superconductor H 2H-NbSe 2 cuprates vortex solid vortex liquid ? ? HmHm 0 T T c0 H H c1 T H c2 H c1 T c0 normal vortex solid liquid 0 HmHm 4 T Meissner state

 Superconductivity in low-Tc superconductors (MF) Energy gap  Pairs obey macroscopic wave function Phase  important in Josephson effect Phase Cooper pairs with coherence length  Quasi-particles TcTc Gap  Temp. T amplitude

 = m p x B + MV x B Van Vleck (orbital) moment m p 2D supercurrent Torque magnetometry   V =  c H x B z –  a H z B x + M B x M eff =  / VB x =  p H z + M(H z ) H M  mpmp c, z  H mpmp M Exquisite sensitivity to 2D supercurrents

Wang et al., unpublished H c2 (0) vs x matches T onset vs x

H*H* HmHm T co Overdoped LaSrCuO x = 0.20

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

Strong curvature persists above Tc

M ~ H 1/  M non-analytic in weak field

Fit to Kosterlitz Thouless theory  = -(k B T/2d  0 2 )   2   = a exp(b/t 1/2 ) Strongly H-dependent Susceptibility  = M/H Susceptibility and Correlation Length

Non-analytic magnetization above Tc M ~ H 1/  Fractional-exponent region

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

Field scale increases as x decreases overdoped optimum underdoped Wang et al. Science (2003)

Optimal, untwinned BZO-grown YBCO

Nernst effect in LSCO-0.12 vortex Nernst signal onset from T = 120 K, ~ 90K above T c`1 Xu et al. Nature (2000) Wang et al. PRB (2001 )

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

Resistivity is a bad diagnostic for field suppression of pairing amplitude Plot of  and e y versus T at fixed H (33 T). Vortex signal is large for T < 26 K, but  is close to normal value  N above 15 K.

Resistivity does not distinguish vortex liquid from normal state H c2 Bardeen Stephen law (not seen) Resistivity Folly Ong Wang, M 2 S-RIO, Physica C (2004)

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

Contour plots in underdoped YBaCuO 6.50 (main panel) and optimal YBCO 6.99 (inset). T co Vortex signal extends above 70 K in underdoped YBCO, to 100 K in optimal YBCO High-temp phase merges continuously with vortex liquid state

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

Separatrix curve at T s Optimum doped Overdoped

Vortex Nernst signal  xy =  M  -1 = 100 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?

Strong correlation in CuO 2 plane Cu 2+ Large U charge-transfer gap  pd ~ 2 eV Mott insulator metal? doping t = 0.3 eV, U = 2 eV, J = 4t 2 /U = 0.12 eV J~1400 K best evidence for large U antiferromagnet Hubbard

Hole-doped optimal Electron-doped optimal

Overall scale of Nernst signal amplitude