Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 1/30 Mutual influence of vortices and quasiparticles in high-temperature superconductors Predrag Nikolić and Subir Sachdev pnikolic@fas.harvard.edu Harvard University
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. /30 Overview Quantum vortices in high-temperature superconductors introduction experimental signatures Vortex dynamics introduction; microscopic model mass renormalization and friction quasiparticle mediated interactions Quasiparticles near a vortex electronic LDOS comparison with experiments
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 3/30 Phenomenology of Superconductivity BCS theory in metals attractive interactions between electrons Cooper pairs = charged superfluid BCS: BEC: Excitations: quasiparticles plasmons vortices
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 4/30 Phenomena and Motivation Applicability of BCS theory: conventional superconductors... microscopic description high-temperature superconductors... phenomenology High-Tc puzzles transport in the normal phase competing orders vortex core structure unified picture: quantum vortices Quasiparticles are the key to vortex dynamics
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 5/30 Crucial Experiment: Nernst Effect Vortices move in thermal gradient Lorentz force gives rise to perpendicular voltage Vortices exist in the normal phase, even at T= 0 Nernst effect 180 160 La -x Sr x CuO 4 140 10 T T* onset T (K) 100 80 10 0 Yayu Wang, Lu Li, 40 500 T c 0 N. P. Ong ; Phys. Rev. B 0 73, 04510 (006) 0.00 0.05 0.10 0.15 0.0 0.5 0.30 60 50 100 Sr content x
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 6/30 Quantum Vortices on a Lattice Theory of competing orders near a D superfluid-mott transition bosons on a lattice, fractional hole doping duality: vortices on dual lattice, external flux = doping Hofstadter: degenerate vortex flavors = density waves T. Hanaguri, et al. Nature 430, 1001 (004) L. Balents, L. Bartosch, A. Burkov, S. Sachdev, K. Sengupta A. Melikyan, Z. Tešanović
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 7/30 Quantum Vortices and Fermions Quantum d-wave vortices? small vortex cores light vortices? nearly frictionless vortex dynamics? BCS: vortices are classical Vortex quantum fluctuations: resonant scattering of quasiparticles sub-gap peaks in LDOS B.W.Hoogenboom, et al. Phys.Rev.Lett. 87, 67001 (001)
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 8/30 Quasiparticles and Vortex Dynamics Conventional BCS superconductors (s-wave) bound core states (Caroli, de Gennes, Matricon) large core traps many quasiparticles = semi-classical vortex dynamics High-Tc superconductors (d-wave) no bound core states (Wang, McDonald; Franz, Tešanović) small cores but, gapless nodal quasiparticles... = quantum vortex dynamics?
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 9/30 Vortex dynamics in clean d-wave superconductors (contribution of nodal quasiparticles)
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 10/30 Vortex Dynamics. m v u v = hρ s ẑ (u v v s ) D(u } {{ } v v n ) D } {{ } ẑ (u v v n ) du v +F }{{} ext Magnus force quasiparticle friction impurity friction F ext... all external forces, vortex-vortex interactions, etc. Magnus Force (Galilean invariance): F M = hρ s u v
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 11/30 Vortex Mass Bare hydrodynamic mass (any superfluid) neutral: m v = E v log ( ) R s ξ charged: m v is finite due to screening 01 111 000 1111 0000 Fermionic superfluids quasiparticles play very important role in s-wave superconductors vortex mass total mass of core states vortex friction total friction of core states d-wave: scattering of extended states
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 1/30 Microscopic Model Main focus: role of nodal quasiparticles in vortex dynamics quasiparticle dynamics in presence of a vortex Ingredients: a single vortex nodal quasiparticles: gapless Dirac fermions
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 13/30 Model: Bogoliubov-de Gennes H=H v (r v )+ nodes d rψ (r)h BdG (r,r v )Ψ(r) Linearized Bogoliubov-de Gennes Hamiltonian + Franz-Tešanović transformation: v f (p x + a x ) v (p y + a y ) H BdG = v (p y + a y ) v f (p x + a x ) + mv 1 0 fv x 0 1 Berry phase effects Doppler shift a(r)= ẑ ˆr (ξ 0) r v(r)= π d ( k ik ẑ m (π) k 1 ) 1 1+λ k e ikr λ ξ φ ^ z
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 14/30 Quasiparticle Contribution Approximations: ignore quasiparticle interactions, disorder... nd order expansion in vortex displacement from the origin linear response to external oscillating force Main idea: dr v (τ)dψ DΨe dτd rψ ( τ +H BdG (r v ) )Ψ = dr v (τ)e S v[r v (τ)] integrate out massive Dirac quasiparticles, then set their mass to zero dω [ S v = F (ω) r v (ω) + F (ω)iẑ ( r π v(ω) r v (ω) ) ]
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 15/30 Summary of Results Vortex dynamics due to nodal quasiparticles: F (ω)= η ω +A 1 ω ln( ω )+ m vω + A ω 3 ; F (ω)=0 Analytical results with Doppler shift ignored: ( ) η = π 1 6 + 1 T Ohmic dissipation at T 0 v v f ( ) A 1 = ln() 1 4 + 1 T ln(t) anomalous term at T 0 v v f ( ) m v 0.05 Λ vortex mass at T = 0 1 v f + 1 v A... a universal function of v f v sub-ohmic dissipation Doppler shift introduces no qualitative changes.
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 16/30 Comparison With Semiclassics Doppler: T = 0 vortex mass due to quasiparticles: s( α D ) 5 4 3 1 4 6 log( α D ) Vortex mass forα D = v F m v = s(α D) 8 ( α D + 1 α D v : ) m e Semi-classical: infra-red divergent vortex mass infra-red cut-off: inter-vortex separation, m v 1 B characteristic length-scale is absent (gapless Dirac qp.) = spatial variations of potential are large Kopnin, Volovik
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 17/30 Implications for the Normal State small vortex motion damping small vortex mass (of the order of electron mass) Consequences: large vortex quantum fluctuations = DW order mostly responding to Magnus force = flux-flow resistivity Nernst effect 180 160 La -x Sr x CuO 4 140 10 T T* onset T (K) 100 80 10 0 Yayu Wang, Lu Li, 40 500 T c 0 N. P. Ong ; Phys. Rev. B 0 73, 04510 (006) 0.00 0.05 0.10 0.15 0.0 0.5 0.30 60 50 100 Sr content x
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 18/30 Interactions Between Vortices y Doppler shift due to one vortex presents a α nodal x chemical potential for Dirac quasiparticles near the other vortex Effects: dissipation to quasiparticles in vortex-vortex scattering vortex lattice orientation pinned to the substrate magnetic field dependent dynamics relevant for flux-flow regime?
Influence of vortex quantum fluctuations on nodal quasiparticles Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 19/30
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 0/30 Model Vortex localization: by neighboring vortices in a vortex lattice by a pinning impurity Model: vortex in a harmonic trap: H v = p v m v + 1 m vω vr v Vortex is coupled to Dirac quasiparticles: H=H v (r v )+ d rψ (r)h BdG (r,r v )Ψ(r) nodes Vortex position r v and momentum p v are operators. Vortex mass m v, and trap frequencyω v are known parameters.
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 1/30 Perturbation Theory Vortex zero-point motion: H 0 =ω v b µb µ + d rψ V 0 Ψ Resonant scattering : H 1 = d rψ ( V µ b µ+ h.c. ) Ψ + quasiparticle propagator vortex propagator ν simple scattering ν Small parameter: nd order scattering µ ν µ ν α= m vv f ω v 1
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. /30 Quasiparticle LDOS ρ(ǫ, r) spectral weight in the quasiparticle Green s function ρ(ǫ, r)= ω v v f n=0 ( ǫ α n F n, ǫr ) ;α ω v v f Effect of the vortex zero-point quantum motion: ρ 0 / ω v 0.4 ρ 0 m v ω v 0.3 0. 0.1 ε/ m v ω v α α α α α α α =5 =1 =0.5 =0.1 =0.05 =0.01 =0.005 finite LDOS at the origin no zero-energy peak 0.5 1 1.5.5 ε/ ω v
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 3/30 Effects of Resonant Scattering One-loop correction to LDOS:ρ 1 (ǫ, r) main peak secondary features discontinuity atǫ=ω v
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 4/30 Full Quasiparticle LDOS Energy scans at different radii: 0.5 α =0.3 0.6 α =1 0.4 0.5 ρ ω v 0.3 ρ ω v 0.4 0. 0.3 0. 0.1 0.1 0.5 1 1.5 0.5 1 1.5 ε/ω v ε/ω v sub-gap peak due to resonant scattering? no bound states in d-wave vortex cores...
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 5/30 Further Tests Scaling of the sub-gap peak: ε c /ω v 0.8 0.6 0.4 0. 0.1 0. 0.3 0.4 α E core [mev] 16 14 1 10 8 6 4 1 T T 4 T 6 T YBCO 0 0 10 0 30 40 50 60 p [mev] ǫ c αω v ω v 0 ǫ c 0 measure LDOS discontinuities = trapping potential measure vortex size (STM) = zero-point amplitude together: determine bothω v spectrum, m v
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 6/30 Conclusions Vortices are quantum particles in clean d-wave superconductors Nernst effect, CDW, LDOS near vortex cores microscopic theory of quasiparticle contribution to vortex dynamics finite and small vortex mass dissipation: super-ohmic at T= 0, Ohmic at T> 0 influence of vortex quantum fluctuations on electronic spectra no zero-energy peak sub-gap resonant peaks in LDOS
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 7/30 The Big Picture S v = [ m 0 dτ v ( drv (τ) dτ ) +i dr ] v(τ) dτ A 0 (r v (τ)) +i d rdτa µ (r,τ)j vµ (r,τ) S A = d ( kdω 1 [ k 8π 3 8π A τ (k,ω) +ω A i (k,ω) ] ρ s [ + e k δ i j k ] ) ik j 4π k A i ( k, ω)a j (k,ω) S Ψ = i { d rdτ α µ (r,τ)j vµ (r,τ)+ i π ǫ µνλa µ ν α λ iψγ µ ( µ ia µ )Ψ+ iv F Ψγ 0 Ψ ( ) } y A τ τ A y 4πρ s
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 8/30 Without Doppler Shift... Rescale coordinates and use gauge invariance to make the Hamiltonian isotropic: p x + a x p y + a y H= p y + a y p x + a x The spectrum is gapless:ǫ q,l,k = qk q=±1... charge (particle or hole state) l Z... angular momentum k>0... radial wavevector Wavefunctions are: ψ q,l,k (r,φ)= 1 4π ǫ J l+ 1 (kr)e i(l 1)φ iq ǫ J l 1 (kr)e ilφ ǫ Jl 1 (kr)e i(l 1)φ iq ǫ J l+ 1 (kr)e ilφ, l<0, l>0
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 9/30 Effects of the Core Special attention is needed for zero angular momentum there are two square-integrable solutions for ψ both cannot be included = over-complete set of states both cannot be excluded = incomplete set of states must introduce a parameter: θ ψ q,0,k (r,φ)= 1 4π sinθ ǫ J 1 (kr)e iφ iq ǫ J 1 (kr) + cosθ ǫ J 1 (kr)e iφ iq ǫ J 1 (kr) θ captures all details of the vortex core! if all of flux is inside a finite disc,θ=0 work withθ=0: no qualitative changes forθ 0
Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 30/30 Transition Matrix Elements No Doppler shift,θ=0, node p k f ˆx: U 1, = d rψ 1 (r) ψ (r) = 1 8 e iπ 4 (q q 1 ) Non-zero only forσ=l l 1 =±1. ( σ ˆx + i ŷ ) U 1, v F v U 1, = 4 ( k 1 k δ(ǫ ǫ 1 ) C k1 ) σl 1 σ ǫ1 +ǫ k k1 Θ (σ(k k k 1 )), l> σ+1 4 k 1 k δ(ǫ 1 ǫ ) C σ ( k1 k ) σl 1 ǫ1 +ǫ k1 k Θ (σ(k 1 k )),l< σ+1 σq 1 q π ǫ 1 +ǫ ǫ 1 ǫ + 1 π C σ ( k1 ) σl 1 ǫ1 +ǫ k k1 log ( ) k 1 k k k 1 +k, l= σ+1 C σ = q,σ=1 q 1,σ= 1