Mutual influence of vortices and quasiparticles in high-temperature superconductors

Transcription

1 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 Harvard University

2 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

3 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

4 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

5 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 La -x Sr x CuO T T* onset T (K) Yayu Wang, Lu Li, T c 0 N. P. Ong ; Phys. Rev. B 0 73, (006) Sr content x

6 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

7 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, (001)

8 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?

9 Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 9/30 Vortex dynamics in clean d-wave superconductors (contribution of nodal quasiparticles)

10 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

11 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 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

12 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

13 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

14 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) ) ]

15 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 T Ohmic dissipation at T 0 v v f ( ) A 1 = ln() 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.

16 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 ) 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

17 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 La -x Sr x CuO T T* onset T (K) Yayu Wang, Lu Li, T c 0 N. P. Ong ; Phys. Rev. B 0 73, (006) Sr content x

18 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?

19 Influence of vortex quantum fluctuations on nodal quasiparticles Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 19/30

20 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.

21 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

22 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 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 ε/ ω v

23 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

24 Mutual influence of vortices and quasiparticles in high-temperature superconductors p. 4/30 Full Quasiparticle LDOS Energy scans at different radii: 0.5 α = α = ρ ω v 0.3 ρ ω v ε/ω v ε/ω v sub-gap peak due to resonant scattering? no bound states in d-wave vortex cores...

25 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 α E core [mev] T T 4 T 6 T YBCO 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

26 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

27 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

28 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

29 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

30 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