Last time: BCS in addition to heavy-fermion superconductorsBardeen-Cooper Schrieffer (conve

Last time: BCS in addition to heavy-fermion superconductorsBardeen-Cooper Schrieffer (conve www.phwiki.com

Last time: BCS in addition to heavy-fermion superconductorsBardeen-Cooper Schrieffer (conve

Cohen, Gary, News Director has reference to this Academic Journal, PHwiki organized this Journal Last time: BCS in addition to heavy-fermion superconductorsBardeen-Cooper Schrieffer (conventional) superconductorsDiscovered in 1911 by Kamerlingh-OnnesFully gapped Bogoliubov quasiparticle spectrumImportant effectsVanishing resistivityMeissner effect (London penetration depth)Coherence effects (coherence length)Heavy-fermion superconductorsDiscovered by Steglich et al. in 1979Key ingredientsLattice of f-electronsConduction electronsMultiple superconducting phasesCuprates overviewIntroduction in addition to PhenomenologyExperimentsPseudogapStripesNodal quasiparticlesIntroduction to Resonating Valence Bond (RVB)Phase fluctuations vs. competing orderNumerical techniquesSingle hole problemSlave particles in addition to gauge fieldsMean field theoryU(1) gauge theoryConfinement physicsI. Introduction (cuprates)DiscoveryBednorz in addition to Müller reported Tc 30 K in Ba-doped La2CuO4 in 1986Highest BCS superconductor was Nb3Ge with Tc = 23.2 KN2 barrier Tc > 77 K in YBCO“Universal” phase diagram

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II. Basic electronic structure of the cupratesLattice, bonding, in addition to dopingRelevant energy scales:t hopping energyUd double-occupancy penaltyLa2CuO4: La3+, Cu2+, O4–; 1 hole doped by La3+ Sr2+La1.85Sr0.15CuO4 (LSCO) Tc 40 KYBa2Cu3O7: Y3+, Ba2+, Cu2+, O4–; already hole doped!YBa2Cu3O7– (YBCO) Tc 93 K II. Basic electronic structure of the cupratesTheoretical modelingThe “t-J model” HamiltonianProjection operator P restricts the Hilbert space to one which excludes double occupation of any siteNext-nearest (t) in addition to next-next-nearest (t) hopping gives better fits to dataA non-zero t accounts as long as asymmetry in electron in addition to hole doped systemsWeak coupling between CuO2 layers gives non-zero TcCuprates are “quasi-2D” 2D layer describes the entire phase diagramCuprates overviewIntroduction in addition to PhenomenologyExperimentsPseudogapStripesNodal quasiparticlesIntroduction to Resonating Valence Bond (RVB)Phase fluctuations vs. competing orderNumerical techniquesSingle hole problemSlave particles in addition to gauge fieldsMean field theoryU(1) gauge theoryConfinement physics

III. Phenomenology of the underdoped cupratesMagnetic propertiesNMR/Knight shift on YBCO (Tc = 79 K)s is T-independent from 300 K to 700 Ks drops below Heisenberg model expectation be as long as e TcStrongly points to singlet as long as mation as origin of pseudogapA. The pseudogap phenomenon in the normal stateIII. Phenomenology of the underdoped cupratesSpecific heatLinear T-dependence of specific heat coefficient above Tc as long as YBa2Cu3O6+y as long as different y; optimally doped curves in the inset as long as La2-xSrxCuO4 as long as different x; overdoped curves in the inset at Tc reduces with decreasing dopingA. The pseudogap phenomenon in the normal stateIII. Phenomenology of the underdoped cupratesDC ConductivityAnomalous linear-T “normal” state resistivityAC ConductivityIn-plane (CuO2 plane) conductivity (a) only gapped below TcPerpendicular conductivity (c) gapped in the pseudogap phaseA. The pseudogap phenomenon in the normal state

III. Phenomenology of the underdoped cupratesARPESSuperconducting gap exhibits nodesPseudogap opens at (/a, 0)Luttinger’s theorem Fermi surface volume = 1 – xSpectral weight in coherence peak vanishes with decreasing hole dopingA. The pseudogap phenomenon in the normal stateIII. Phenomenology of the underdoped cupratesSTMSurface inhomogeneity in the gap functionSTM sees two dips first dip is indication of pseudogap stateA. The pseudogap phenomenon in the normal stateCuprates overviewIntroduction in addition to PhenomenologyExperimentsPseudogapStripesNodal quasiparticlesIntroduction to Resonating Valence Bond (RVB)Phase fluctuations vs. competing orderNumerical techniquesSingle hole problemSlave particles in addition to gauge fieldsMean field theoryU(1) gauge theoryConfinement physics

III. Phenomenology of the underdoped cupratesStripe orderObserved in LSCO at doping of x = 1/8Charge density wave (CDW) periodicity = 4Spin density wave (SDW) periodicity = 8Neutron scatteringScattering peak at q = (/2, /2)Incommensurability () scales with doping (x)“Fluctuating stripes” apparently invisible to experimental probesFluctuating stripes “may” explain pseudogap in addition to superconductivityB. Neutron scattering, resonance in addition to stripesCuprates overviewIntroduction in addition to PhenomenologyExperimentsPseudogapStripesNodal quasiparticlesIntroduction to Resonating Valence Bond (RVB)Phase fluctuations vs. competing orderNumerical techniquesSingle hole problemSlave particles in addition to gauge fieldsMean field theoryU(1) gauge theoryConfinement physicsIII. Phenomenology of the underdoped cupratesVolovik effectShift in quasiparticle energiesC. Quasiparticles in the superconducting stateOriginal quasiparticle spectrumNodal quasiparticle disperses like “normal” currentPhase winding around a vortex

III. Phenomenology of the underdoped cupratesNodal quasiparticlesUniversal conductivity per layerC. Quasiparticles in the superconducting stateAntinodal gap obtained from extrapolationPhenomenological expression as long as linear-T superfluid densityLondon penetration depth shows = constantSlave boson theory predicts xCuprates overviewIntroduction in addition to PhenomenologyExperimentsPseudogapStripesNodal quasiparticlesIntroduction to Resonating Valence Bond (RVB)Phase fluctuations vs. competing orderNumerical techniquesSingle hole problemSlave particles in addition to gauge fieldsMean field theoryU(1) gauge theoryConfinement physicsIV. Introduction to RVB in addition to a simple explanation of the pseudogapResonating Valence Bond (RVB)Anderson revived RVB as long as the high-Tc problemRVB state “soup” of fluctuating spin singlets

IV. Introduction to RVB in addition to a simple explanation of the pseudogapDeconfinement of “slave particle”We can “split” an electron into charge in addition to spin degrees of freedomPurely spin degrees of freedom “spinons”IV. Introduction to RVB in addition to a simple explanation of the pseudogapResonating Valence BondAnderson revived RVB as long as the high-Tc problemPotential explanation of the pseudogap phaseHoles confined to 2D layersVertical motion of electrons needs breaking a singlet a gapped excitationCuprates overviewIntroduction in addition to PhenomenologyExperimentsPseudogapStripesNodal quasiparticlesIntroduction to Resonating Valence Bond (RVB)Phase fluctuations vs. competing orderNumerical techniquesSingle hole problemSlave particles in addition to gauge fieldsMean field theoryU(1) gauge theoryConfinement physics

London penetration depth inferred from SR rateV. Phase fluctuation vs. competing orderFactors influencing TcLondon penetration depth as long as field penetration perpendicular to the ab planeIndication of intralayer bose condensation of holes from SRV. Phase fluctuation vs. competing orderTc as a function of phase stiffnessPhase stiffness of the order parameterA. Theory of TcThe BKT transition; energy of a single vortexRelation between phase stiffness in addition to TcCheap vorticesSuppose TMF is described by the st in addition to ard BCS theoryEF Ec kBTc Pseudogap mostly superconducting Ec is clearly not of order EFEc Tc Ks notion of strong phase fluctuations is applicable only on a temperature scale of 2TcV. Phase fluctuation vs. competing orderNernst effectTransverse voltage due to longitudinal thermal gradient in the presence of a magnetic fieldNernst region as second type of pseudogap explained by phase fluctuationsThe first type of pseudogap explained by singlet as long as mationB. Cheap vortices in addition to the Nernst effect

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Cuprates overviewIntroduction in addition to PhenomenologyExperimentsPseudogapStripesNodal quasiparticlesIntroduction to Resonating Valence Bond (RVB)Phase fluctuations vs. competing orderNumerical techniquesSingle hole problemSlave particles in addition to gauge fieldsMean field theoryU(1) gauge theoryConfinement physicsVI. Projected trial wavefunctions in addition to other numerical resultsAnderson’s original RVB proposalThe Gutzwiller projection operatorProjection operator too complicated to treat analyticallyProperties of the trial wave function evaluated using Monte Carlo samplingWave function ansatz SC: superconducting without antiferromagnetism SC+AF: superconducting with antiferromagnetism SF: staggered-flux without antiferromagnetism SF+AF: staggered-flux with antiferromagnetism ZF: zero-fluxVI. Projected trial wavefunctions in addition to other numerical resultsd-wave BCS trial wavefunctionA. The half-filled caseStaggered flux stateSU(2) symmetry

Cuprates overviewIntroduction in addition to PhenomenologyExperimentsPseudogapStripesNodal quasiparticlesIntroduction to Resonating Valence Bond (RVB)Phase fluctuations vs. competing orderNumerical techniquesSingle hole problemSlave particles in addition to gauge fieldsMean field theoryU(1) gauge theoryConfinement physicsVII. The single hole problemVacancy in an “antiferromagnetic sea”Dynamics of a single holeUsing self-consistent Born approximation, in addition to ignoring crossing magnon propagators, self-consistent equation as long as the hole propagator isARPES sees two peaks in A(k, ) in addition to hole quasiparticle peaks centered atThese can be understood as the “string” excitation of the hole moving in the linear confining potential due to the AF backgroundVII. The single hole problemVacancy in an “antiferromagnetic sea”ARPES sees two peaks in A(k, ) in addition to hole quasiparticle peaks centered atThese can be understood as the “string” excitation of the hole moving in the linear confining potential due to the AF backgroundThe hole must retrace its path to “kill” the string holes are localizedDo holes really conductYes! A hole does not necessarily need to retrace its path without raising energy

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