Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42Slide Number 43Slide Number 44Slide Number 45Slide Number 46Slide Number 47Slide Number 481. Introduction to the Nernst effect2. Vortex signal above Tc3. Loss of long-range phase coherence4. The Upper Critical Field5. The cuprate phase diagramTalk 2Nernst effect in vortex-liquid state of cupratesBoulder, July 2008Supported by NSF-MRSEC, ONRBoulder School for Condensed Matter and Materials Physics 2008N. P. OngCollaborators: Lu Li, Yayu Wang, Zhuan Xu (Princeton Univ.)S. Uchida (Univ. Tokyo)D. Bonn, W. Hardy, R. Liang (Univ. British Columbia)holes = 1/2Phase diagram of CupratesTpseudogap00.050.25AFdSCT*TcMott insulatorFermiliquiddoping x0)(kkkkk↑∗↓−∗+=Ψ∏ccveuiBCSφBCS wave functionPhase φfixed (phase representation); N fluctuates[N, φ] = 1kukv+⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛1001φievukkAndersonpseudospin)r(||)r(ˆφieΨ=ΨThe phase of macroscopic pair-wave functionφSVortex in cupratesCuO2 layers2D vortex pancakeξVortex in NiobiumJssuperfluidelectrons(pair condensate)ξJsb(r)Normal coreHcoherence length ξ Vortices in type-II superconductors|Ψ| = ΔLondon length λb(r)Phase diagram of type-II superconductorH2H-NbSe2cupratesvortex solidvortexliquid??Hm0TTc0HHc1THc2Hc1Tc0normalvortex solidliquid0Hm4 TMeissner stateVortex motion in type II superconductorApplied supercurrent Js exerts magnus force on vortex coreVelocity gives induced E-field in core (Faraday effect)Current enters core and dissipates (damping viscosity) Motion of vortices generates observedE-fieldConsequence of Josephson equationTilt angle of velocity gives negative vortex Hall effectIn clean limit, vortex v is || - JsE = B x v(Bardeen Stephen, Nozieres Vinen)ηρρ/02Φ== BHHcNxx0Φ×=rsMJFamplitude fluctuationFFphase fluctuationψ2ψ1ψ1ψ2Anderson-Higgs mechanism: Phase stiffnesssingular phase fluc. (excitation of vortices)Phase mode θ+ EM Fμν = Massive mode (Meissner effect)Anderson-Higgs mechanism and phase rigidityPhase rigidity Æ uniform phase θ|Ψ| eiθ(r)But phase coherence destroyed by mobile vorticesphase rigidity measured by ρs()2321θρρ∇=∫SrdHΔθ= 2πP.W. Anderson Phys. Rev. 1959, RMP 1966phase-slip and Nernst signalPassage of a vortex ÆPhase diff. θ jumps by 2πIntegrate VJ to give dc signalprop. to nv•=φhJeV2= 2πh nVJosephson Eq.timeΔθ= 2πΔVJHΔ−T• Baskaran, Zou, Anderson (Sol. St. Comm. 1987)• Doniach, Inui (PRB 1989)• Uemura plot (Nature 1989)• Emery, Kivelson (Nature 1995)low hole density and high Tccuprates highly suscep. to phase fluctuations• Corson, Orenstein (Nature 1999)Kinetic inductance meas. at THz freq extends above TcKT physics in ultra-thin film BSCCO• M. Franz and Z. Tesanovic (1999)Vortex-charge duality, QED3 model• S. Sachdev (2005)Quantum vorticesBaskaran, Zou, Anderson (Solid State Comm. 1987)Δ vs. x and the loss of phase coherence in underdoped regimeEmery Kivelson (Nature 1995)Phase fluctuation and loss of coherence at Tc in low (superfluid) density SC’sM. 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 mappingto Bosons in Hofstadter lattice --- vorticity and checkerboard patternBalents, Sachdev, Fisher et al. (2004)Vorticity and checkerboard in underdoped regimeP. A. Lee, X. G. Wen. (PRL, ’03, PRB ’04)Loss of phase coherence in tJ model, nature of vortex coreLee, Nagaosa, Wen, Rev. Mod. Phys. (cond-mat/0410***)Good review of phase stiffness, phase fluctuation issuesP. W. Anderson (cond-mat ‘05)Spin-charge locking occurs at Tonset > TcTheories on phase fluctuation in cupratesThe Nernst effect of carriers).(E.J T−∇+=ασtt).(.E T−∇−=αρtt())( TExyxyxy−∂+−=αρραOpen boundaries, so set J = 0.)(k.BvkkkxyxyTfe ll∂∂×−⎟⎟⎠⎞⎜⎜⎝⎛∂∂−=∑μεεα022εθπ∂∂=∇≡eTkTEeByN223||Off-diag. Peltier cond.Measured Nernst signalGenerally, very small because of cancellation between αxy and σxyWang et al. PRB ‘01The vortex Nernst effect; dominant in vortex liquid stateBvEGradient drives vortex currentwith velocity v || x)(v Ts−∇=φηηφBsTEeN=∇=||Force exerted on vortex line by grad TLine entropy sφBalance F by viscous dampingNernst signal eN)(F Ts−∇=φMoving vortex producesE = B x vNernst experimentVortices move in a temperature gradientPhase slip generates Josephson voltage2eVJ = 2πh nVEJ = B x vHeyHmNernst signaley = Ey /| T |Nernst effect in LSCO-0.12vortex Nernst signal onset from T = 120 K, ~ 90K above Tc`1Xu et al. Nature (2000)Wang et al. PRB (2001)Nernst effect in underdoped Bi-2212 (Tc = 50 K)Vortex signal persists to 70 K above Tc .Wang, Li, NPOPRB (2006)OP YBCOUD LSCOOP Bi2212 UD Bi2212Nernst contour-map in underdoped, optimal and overdoped LSCOH*HmTcoOverdoped LaSrCuO x = 0.20Optimal, untwinned BZO-grown YBCOContour plots in underdoped YBaCuO6.50 (main panel) and optimalYBCO6.99 (inset).Tco• Vortex signal extends above70 K in underdoped YBCO,to 100 K in optimal YBCO • High-temp phase merges continuously with vortex liquid stateWang et al., PRL ‘02Nernst contour maps in UD YBCO and OP YBCOWang, Li, Ong PRB 2006Contour Map of Nernst Signal in Bi 2201Spontaneous vortices destroy superfluidity in 2D filmsChange in free energy ΔF to create a vortexΔF = ΔU – TΔS = (εc – kB T) log (R/a)2ΔF < 0 if T > TKT = εc /kB vortices appear spontaneouslyΔTcMFTKT0ρsKosterlitz-Thouless transition3D version of KT transition in cuprates?Kosterlitz Thouless transition in 2D superconductorUnbinding ofvortex-antivortexΔF = U - TSFree energy gainvortex densityvortexantivortexH= ½ ρs d3r ( φ)2ρs measures phase rigidityPhase coherence destroyed at TKTby proliferation of vorticesBCS transition2D Kosterlitz Thouless