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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 1. Vortex Nernst effect 2. Loss of long-range phase coherence 3. The Upper Critical Field 4. High-temperature Diamagnetism Taipeh, June 2006 Phase diagram of Cuprates Mott insulator s = 1/2 hole T* T pseudogap Tc Fermi liquid LSCO = La2-xSrxCuO4 AF dSC Bi 2212 = Bi2Sr2CaCu2O8 0 0.05 0.25 Bi 2201 = Bi2-yLaySr2CuO6 doping x Condensate described by a complex macroscopic wave function Y(r) = Y1 + iY2 = |Y(r)| exp[iq(r)] F F y2 y1 amplitude fluctuation y2 y1 phase fluctuation Anderson-Higgs mechanism: Phase stiffness singular phase fluc. (excitation of vortices) Phase rigidity ruined by mobile defects Long-range phase coherence requires uniform q “kilometer of dirty lead wire” q q q q Hr 1 d 3r r S (q ) 2 phase rigidity measured by rs 2 Phase coherence destroyed by vortex motion q q Kosterlitz Thouless transition in 2D films (1982) Vortices, fundamental excitation of type-II SC Vortex in Niobium Vortex in cuprates CuO2 layers b(r) Normal core superfluid electrons Js 2D vortex pancake H Js x x coherence length x b(r) |Y| D London length l upper critical field Mean-field phase diagram Cuprate phase diagram 2H-NbSe2 4T 100 T normal Hc2 liquid H Hm H Hc2 vortex vortex solid liquid Hm Hc1 0 T Tc0 Tc vortex 100 K 7K solid Meissner state The Josephson Effect, phase-slippage and Nernst signal Passage of a vortex Phase diff. f jumps by 2 vortex 2eVJ f 2 Phase difference f 2eVJ 2 nV VJ Integrate VJ to give dc signal prop. to nv t Nernst experiment ey Hm H Vortices move in a temperature gradient Nernst signal Phase slip generates Josephson voltage ey = Ey /| T| 2eVJ = 2h nV EJ = B x v Nernst effect in underdoped Bi-2212 (Tc = 50 K) Vortex signal persists to 70 K above Tc. Wang, Li, Ong PRB 2006 Vortex-Nernst signal in Bi 2201 Nernst curves in Bi 2201 Yayu Wang,Lu Li,NPO PRB 2006 optimal underdoped overdoped Nernst signal eN = Ey /| T| 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 rs D 0 TKT TcMF 3D version of KT transition in cuprates? Nernst region •Loss of phase coherence determines Tc •Condensate amplitude persists T>Tc • Vorticity and diamagnetism in Nernst region In hole-doped cuprates 1. Existence of vortex Nernst signal above Tc 2. Confined to superconducting “dome” 3. Upper critical field Hc2 versus T is anomalous 4. 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? Diamagnetic currents in vortex liquid Supercurrents follow contours of condensate Js = -(eh/m) x |Y|2 z Cantilever torque magnetometry Torque on magnetic moment: t = m × B crystal B t f × m Deflection of cantilever: t = k f Micro-fabricated single crystal silicon cantilever magnetometer H • Si single-crystal cantilever • Capacitive detection of deflection • Sensitivity: ~ 5 × 10-9 emu at 10 tesla ~100 times more sensitive than commercial SQUID Underdoped Wang et al. Bi 2212 Cond-mat/05 Tc Paramagnetic background in Bi 2212 and LSCO Magnetization curves in underdoped Bi 2212 Wang et al. Cond-mat/05 Tc Separatrix Ts F F y2 y1 amplitude fluctuation y2 y1 phase fluctuation Anderson-Higgs mechanism: Phase stiffness singular phase fluc. (excitation of vortices) At high T, M scales with Nernst signal eN M(T,H) matches eN in both H and T above Tc Magnetization in Abrikosov state M Hc1 Hc2 H 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 Wang et al. Cond-mat/05 Mean-field phase diagram Cuprate phase diagram 2H-NbSe2 4T 100 T normal Hc2 liquid H Hm H Hc2 vortex vortex solid liquid Hm Hc1 0 T Tc0 Tc vortex 100 K 7K solid Meissner state Hole-doped optimal Electron-doped optimal Tc Tc Phase fluctuation in cuprate phase diagram spin pairing (NMR relaxation, T* Bulk suscept.) pseudogap Temperature T Tonset Onset of charge pairing Vortex-Nernst signal Enhanced diamagnetism Kinetic inductance vortex liquid Tc superfluidity long-range phase coherence Meissner eff. 0 x (holes) 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 Cooper pairing in cuprates d-wave symmetry + - - x coherence length + D(q ) D 0 cos 2q Upper critical field f0 H c2 2x 2 cuprates Nb3Sn MgB2 NbSe2 18 29 57 90 o x (A) Hc2 100 Tesla 40 10 4 Tesla 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‟ = 21/2(kBTc / f03/2) B1/2 This is 30-50 times smaller than observed in Bi 2212 “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. 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 Hc2 vortex liquid Hm Tc 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 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) Magnetization in lightly doped La2-xSrxCuO4 Lu Li et al., unpubl. SC dome 0.03 0.04 0.05 0.06 4.2 K 5K 5K 30 K 35 K 35 K 30 K 4.2 K dissipative, vortices mobile Long-range phase coherence Sharp transition in Tc vs x (QCT?) Vortex-liquid boundary linear in x as x 0? 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, Hc2(0) varies with x Competing ground states 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. Field sensitivity of Gaussian fluctuations Gollub, Beasley, Tinkham et al. PRB (1973) Vortex signal above Tc0 in under- and over-doped Bi 2212 Wang et al. PRB (2001) Abrikosov vortices near Hc2 x Upper critical field Hc2 = f0/2x2 Condensate destroyed when cores touch at Hc2 Anomalous high-temp. diamagnetic state 1. Vortex-liquid state defined by large Nernst signal and diamagnetism 2. M(T,H) closely matched to eN(T,H) at high T (b is 103 - 104 times larger than in ferromagnets). 3. M vs. H curves show Hc2 stays v. large as T Tc. 4. Magnetization evidence that transition is by loss of phase coherence instead of vanishing of gap 5. Nonlinear weak-field diamagnetism above Tc to Tonset. 6. NOT seen in electron doped NdCeCuO (tied to pseudogap physics) Nernst contour-map in underdoped, optimal and overdoped LSCO Tc 110K • In underdoped Bi-2212, onset of diamagnetic fluctuations at 110 K • diamagnetic signal closely tracks the Nernst effect PbIn, Tc = 7.2 K (Vidal, PRB ‟73) Bi 2201 (Tc = 28 K, Hc2 ~ 48 T) 2.0 T=1.5K T=8K 1.5 e y ( m V/K ) 1.0 ey Hd 0.5 Hc2 Hc2 0.0 0 10 20 30 40 50 60 0.3 1.0 m 0 H (T) H/Hc2 • Upper critical Field Hc2 given by ey 0. Wang et al. Science (2003) • Hole cuprates --- Need intense fields. Vortex-Nernst signal in Bi 2201 NbSe2 NdCeCuO Hole-doped cuprates Hc2 Hc2 Hc2 vortex vortex liquid liquid Hm Hm Hm Tc0 Tc0 Tc0 Expanded vortex liquid Vortex liquid dominant. Conventional SC Amplitude vanishes at Loss of phase coherence Amplitude vanishes Tc0 at Tc0 (zero-field melting) at Tc0 (BCS) Phase diagram of type-II superconductor 2H-NbSe2 cuprates 4T normal ? ? liquid H Hm H vortex liquid Hc2 vortex solid vortex Hc1 solid Hm Hc1 0 T Tc0 0 T Tc0 Meissner state Superconductivity in low-Tc superconductors (MF) Cooper pairs with coherence length x Quasi-particles x Energy gap D Pairs obey macroscopic wave function ˆ Y (r ) | Y | e i q ( r ) Phase amplitude |Y| D Gap D Phase q important in Josephson effect Temp. T Tc Torque magnetometry c, z H Van Vleck (orbital) moment mp q mp t = mp x B + MV x B 2D supercurrent M t/V = ccHx Bz – caHz Bx + M Bx t Meff = t / VBx = Dcp Hz + M(Hz) mp H Exquisite sensitivity to 2D supercurrents M Wang et al., unpublished Hc2(0) vs x matches Tonset vs x Overdoped LaSrCuO x = 0.20 H* Hm Tco M vs H below Tc Full Flux Exclusion Strong Curvature! Hc1 -M H Strong curvature persists above Tc M non-analytic in weak field M ~ H1/d Susceptibility and Correlation Length Strongly H-dependent Susceptibility c = M/H Fit to Kosterlitz Thouless theory c = -(kBT/2df02) xKT2 xKT = a exp(b/t1/2) Non-analytic magnetization above Tc M ~ H1/d Fractional-exponent region 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 Wang et al. Science (2003) overdoped optimum underdoped 3.5 OD-Bi2212 ( T c =65K) 3.5 OPT-Bi2212 ( T c =90K) 3.0 UD-Bi2212 ( T c =50K) 40K 3.0 75 45 50 3.0 70K 2.5 45 2.5 40 55 2.5 50 35 80 2.0 2.0 e y ( m V/K) 55 60 2.0 30 90 1.5 85 1.5 60 1.5 1.0 65 25 1.0 1.0 65 95 0.5 70 70 0.5 100 0.5 75 75 20 80 80 105 0.0 85 0.0 90 110 100 90 0.0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 m 0 H (T) m 0 H (T) m 0 H (T) Field scale increases as x decreases Optimal, untwinned BZO-grown YBCO Nernst effect in LSCO-0.12 Xu et al. Nature (2000) Wang et al. PRB (2001) vortex Nernst signal onset from T = 120 K, ~ 90K above Tc`1 Temp. dependence of Nernst coef. in Bi 2201 (y = 0.60, 0.50). Onset temperatures much higher than Tc0 (18 K, 26 K). Resistivity is a bad diagnostic for field suppression of pairing amplitude Plot of r and ey versus T at fixed H (33 T). Vortex signal is large for T < 26 K, but r is close to normal value rN above 15 K. Resistivity Folly Bardeen Stephen law (not seen) Ong Wang, M2S-RIO, Physica C (2004) 1.0 N d CCO ( T c =24.5K) 0.8 LSCO (0.20) 0.8 12K 0.6 r 22K r e y( m V/K) e y (m V/K) 0.6 0.4 0.4 ey ey 0.2 0.2 Hc2 Hc2 0.0 0.0 0 2 4 6 8 10 12 14 0 5 10 15 20 25 30 m 0 H (T) m 0 H (T) Resistivity does not distinguish vortex liquid from normal state 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 Contour plots in underdoped YBaCuO6.50 (main panel) and optimal YBCO6.99 (inset). • Vortex signal extends above 70 K in underdoped YBCO, to 100 K in optimal YBCO • High-temp phase merges continuously with vortex liquid state Tco Nernst effect in optimally doped YBCO Vortex onset temperature: 107 K Nernst vs. H in optimally doped YBCO Separatrix curve at Ts Optimum doped Overdoped Vortex Nernst signal axy = b M b-1 = 100 K BCS transition 2D Kosterlitz Thouless transition n vortex D D r r s s 0 Tc 0 TKT TMF H = ½ rs d3r ( f)2 Phase coherence destroyed at TKT rs measures phase rigidity by proliferation of vortices High temperature superconductors? Strong correlation in CuO2 plane Cu2+ Large U charge-transfer gap Dpd ~ 2 eV metal? best Mott insulator evidence antiferromagnet J~1400 K for large U doping H t c is c js U ni ni Hubbard i , j ,s i t = 0.3 eV, U = 2 eV, J = 4t2/U = 0.12 eV Hole-doped optimal Electron-doped optimal Overall scale of Nernst signal amplitude

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posted: | 11/29/2011 |

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