The Nernst effect in high-temperature superconductors

Transcription

1 The Nernst effect in high-temperature superconductors Iddo Ussishkin (University of Minnesota) with Shivaji Sondhi David Huse Vadim Oganesyan

2 Outline Introduction: - High-temperature superconductors: physics above T c - The Nernst effect: a new puzzle Understanding the Nernst effect in different parts of the phase diagram - Theory of superconducting fluctuations - Critical behavior - Effects of band structure Summary and future directions

3 High-temperature superconductors T T AF Pseudogap Normal Metal x d-wave superconductor What is the nature of the non-superconducting state above T c

4 The pseudogap Pseudogap : a gap is formed along part of the Fermi surface (d-wave) - ARPES: spectral density below Fermi energy, I A(k, ω)f(ω) ω Marshall et al. 96 Other probes: STM, NMR, specific heat, transport

5 The Nernst coefficient: a new puzzle The Nernst effect is almost 0 in conventional metallic state ẑ B ŷ V j = 0 ν = E y ( T ) x B ˆx T v B j 0 E E j = 0 T Metal ν[nv/k T] Aluminum 3.9 Copper 21.6 Lead 0.5 Gold 18.1

6 Nernst effect in superconducting state Below T c, vortices created by magnetic field E T Large Nernst signals generated by vortex flow, 10 3 nv/k T Huebener,... Caroli and Maki, Hu,... Above T c, back to normal metallic behavior

7 E y / T (µ V/K) Surprise in high-temperature superconductors La 2-x Sr x CO 4 x=0.12 T c =28.9K K K T c ν 500 nv K T 35 K Large Nernst signal above T c! K µ 0 H (T) 50 K 60 K 2T c Wang et al. 01

8 Doping dependence of Nernst effect La 2-x Sr x CuO T (K) T onset T c Sr content x Wang et al 01

9 Phase diagram of hole-doped cuprates T T Pseudogap Enhanced Nernst signal AF d-wave superconductor x What is the contribution of superconducting fluctuations?

10 Superconducting fluctuations above T c Ginzburg-Landau free energy: Aslamazov and Larkin, Maki, Thompson,... F = dx [a ψ 2 + b ψ 4 + ( ie A) ψ 2] a T T c controls transition: ψ = 0 for T > T c, ψ 0 for T < T c Thermal fluctuations of the order parameter: Stochastic time-dependent equation ψ t = Γ δf 0 δψ + ζ Γ 0 : relaxation rate of order parameter ζ: Gaussian white noise - introduces fluctuations

11 Gaussian fluctuations above T c Ignoring ψ 4 term linear equation for ψ ψ t = Γ δf 0 δψ + ζ = Γ [ 0 aψ ( ia) 2 ψ ] + ζ Conductivity: Apply E, calculate j e = e ψ ( i e A) ψ + c.c. Result: σ 2D xx ξ 2 1 T T c = Aslamazov-Larkin contribution in microscopic BCS calculation

12 Calculation of Nernst signal We calculate transverse thermoelectric response α xy ( j e tr j Q tr ) = ( σ α α κ ) ( E T ) ν 1 B α xy σ xx Onsager relations α = T α Heat current: j Q = t ψ ( i e A) ψ + c.c.

13 Nernst coefficient in the Gaussian approximation Result with Lawrence-Doniach model (Ginzburg-Landau free energy for layered superconductors) α xy = 1 6π e h ξ 2 ab l 2 B s (2ξc /s) 2 IU, Sondhi, and Huse 02 ξ ab, ξ c correlation lengthes, s - interlayer spacing, l B - magnetic length 2D and 3D Limits: α 2D xy ξ 2 1 T T c α 3D xy ξ 1 T Tc αxy is independent of Γ 0. Great for comparison with experiment! Ratio with magnetization: T α xy M = 1 2

14 Comparison with experiment α xy νσ xx (LSCO, ξ ab (0) = 30 Å, ξ ab /ξ c = 20) Quantitative agreement for overdoped samples! Signal larger for underdoped samples V K T Ω m σxx(ν ν n ) ˆ x = 0.12 x = 0.17 x = (T T c )/T c Very recently: torque magnetization measurements (Wang et al.) T α xy M 0.5 our results 0.6 overdoped 1.0 underdoped

15 Quantitative issues Q: Why is the fluctuation Nernst coefficient large in the cuprates? A: Effect of several factors: layered structure is effectively two-dimensional small interlayer spacing s 10 Å α 2D xy 1/s high normal state resistivity - cuprates are bad conductors ρ n 100 µω cm ν α xy ρ n Should be observable in highly-resistive thin superconducting films

16 Phase diagram of hole-doped cuprates T T Pseudogap Gaussian fluctuations AF d-wave superconductor x What happens at the critical region?

17 Critical behavior of superconducting transition Landau-Ginzburg-Wilson functional, with complex order parameter ψ F {ψ} = dx ( r 0 ψ 2 + u 0 ψ 4 + ψ 2) Strongly type-ii limit (κ = λ/ξ 1) 3D-XY universality class From theory of critical phenomena: - Critical exponents: e.g., correlation length: ξ T T c ν Mean field theory: ν = 1/2 3D-XY: ν Scaling: e.g., free energy density f = ξ d F(Bξ 2 ) Magnetic susceptibility (T > T c ): χ = 2 f/ B 2 B=0 ξ 4 d

18 Critical dynamics Model A dynamics (Hohenberg and Halperin): ψ t = Γ δf 0 δψ + ζ = Γ [ 0 r0 ψ + 2u 0 ψ 2 ψ 2 ψ ] + ζ Dynamical scaling: dynamical critical exponent τ ξ z z = 2 + O(ɛ 2 ) (Halperin, Hohenberg, & Ma 72) Conductivity: j e = ξ d+1 J e ( Eξ 1+z, Bξ 2) 3D σ xx 1 (T T c ) 0.64 σ xx ξ 2+z d 1 (T T c ) (2+z d)ν σ xx 1 T Tc Fisher, Fisher, & Huse 91

19 Scaling relations for thermal transport Heat current operator: j Q = t ψ ψ + c.c. Scaling relation: j Q = ξ d Q J e ( Eξ 1+z, Bξ 2) What is the dimension d Q appearing in the scaling relation? - Mean field theory: d Q = d + 1 In the critical regime: no anomalous dimension d Q = d + 1 IU, Huse, and Sondhi 04 - No correction in ɛ-expansion (to second order) - Hidden conservation law

20 Critical behavior for the Nernst experiment Transverse thermoelectric response: α xy B ξ2+z d 1 (T T c ) (2+z d)ν α xy 1 (T T c ) 0.67 Gaussian fluctuations Nernst coefficient: ν α xy σ xx B const. Ratio with magnetization: α xy M ξz 2 1 (T T c ) (z 2)ν T α xy M 1 (T T c ) 0.01 T α xy M = 1 2

21 Phase diagram of hole-doped cuprates T T Pseudogap Gaussian fluctuations Critical behavior AF d-wave superconductor x What is the origin of the enhanced Nernst signal in underdoped samples?

22 d-density wave scenario Explain pseudogap with d-wave density wave ordering Chakravarty et al 01 T c k+q c k cos k x cos k y Q = (π, π) DDW Modified dispersion contains nodal points AF SC x DDW+SC ɛ k y k x

23 Nernst effect of quasiparticles Quasiparticle Nernst effect (another way to see why it is small) ν = π2 3 k 2 B T eb Θ H µ Drude model: Θ H = ω c τ Passing through the topological transition: Hall angle changes sign at the node: Θ H 1/µ m = k/v µ Θ H ɛ k y µτ k x

24 Nernst coefficient at the topological transition ν Peak at the node of Nernst coefficient: Oganesyan and IU, 04 µτ Possible applications: - d-density wave scenario for pseudogap: NO! chemical potential away from node - Nodal metal at very underdoped samples? Taillefer et al 03 - Other materials: single layers of graphite Geim 04

25 Scenario of strong superconducting fluctuations T T In underdoped regime: - Tc MF well separated from T c - T c suppressed by phase fluctuations Emery and Kivelson 95 AF T MF c T MF c and T are distinct temperature scales d-wave superconductor x Other fluctuation phenomena: - Magnetization: very recently observed - Conductivity: not oberved Fast relaxation of order parameter

26 Summary T T T MF c Gaussian fluctuations AF SC Critical behavior Not d-density wave quasiparticles Strong superconducting fluctuations? x Future directions: Kosterlitz-Thouless transition; description in terms of vortices Recently: Large Nernst effect in heavy fermion, organics,...