Boulder School for Condensed Matter and Materials Physics 2008 Talk 2 Nernst effect in vortex-liquid state of cuprates 1. Introduction to the Nernst effect 2. Vortex signal above Tc 3. Loss of long-range phase coherence 4. The Upper Critical Field 5. The cuprate phase diagram N. P. Ong Collaborators: Lu Li, Yayu Wang, Zhuan Xu (Princeton Univ.) S. Uchida (Univ. Tokyo) D. Bonn, W. Hardy, R. Liang (Univ. British Columbia) Supported by NSF-MRSEC, ONR Boulder, July 2008
Phase diagram of Cuprates Mott insulator s = 1/2 hole T pseudogap T * AF T c dsc Fermi liquid 0 0.05 0.25 doping x
0 ) ( k k k k k + = Ψ c c v e u i BCS φ BCS wave function Phase φ fixed (phase representation); N fluctuates [N, φ ] = 1 k u k v + + 1 0 0 1 φ i e v u k k Anderson pseudospin (r) (r) ˆ φ i e Ψ = Ψ The phase of macroscopic pair-wave function φ S
Vortices in type-ii superconductors Vortex in Niobium Vortex in cuprates Normal core b(r) CuO 2 layers J s 2D vortex pancake H J s superfluid electrons (pair condensate) ξ ξ coherence length ξ b(r) Ψ = Δ London length λ
Phase diagram of type-ii superconductor 2H-NbSe 2 cuprates 4 T H vortex solid H c1 H m liquid normal H c2 H H c1 vortex solid? vortex liquid H m? 0 T T c0 0 T T c0 Meissner state
Vortex motion in type II superconductor (Bardeen Stephen, Nozieres Vinen) Applied supercurrent J s exerts magnus force on vortex core r F M = J s Φ 0 Velocity gives induced E-field in core (Faraday effect) Current enters core and dissipates (damping viscosity) Motion of vortices generates observed E-field H E = B x v ρ xx = ρ N = BΦ0 / η H Consequence of Josephson equation c2 Tilt angle of velocity gives negative vortex Hall effect In clean limit, vortex v is - J s
Anderson-Higgs mechanism and phase rigidity F F ψ 2 ψ 1 amplitude fluctuation ψ 2 ψ 1 phase fluctuation Anderson-Higgs mechanism: Phase stiffness singular phase fluc. (excitation of vortices) Phase mode θ + EM F μν = Massive mode (Meissner effect)
P.W. Anderson Phys. Rev. 1959, RMP 1966 Phase rigidity uniform phase θ Ψ e iθ(r) H ρ ( ) 2 1 3 d r ρ θ = 2 S phase rigidity measured by ρ s But phase coherence destroyed by mobile vortices Δθ = 2π
phase-slip and Nernst signal H Δ T Δθ = 2π Passage of a vortex Phase diff. θ jumps by 2π Josephson Eq. ev J = hφ 2 = 2πh n V ΔV J time Integrate V J to give dc signal prop. to n v
Baskaran, Zou, Anderson (Sol. St. Comm. 1987) Doniach, Inui (PRB 1989) Uemura plot (Nature 1989) Emery, Kivelson (Nature 1995) low hole density and high Tc cuprates highly suscep. to phase fluctuations Corson, Orenstein (Nature 1999) Kinetic inductance meas. at THz freq extends above Tc KT physics in ultra-thin film BSCCO M. Franz and Z. Tesanovic (1999) Vortex-charge duality, QED3 model S. Sachdev (2005) Quantum vortices
Theories on phase fluctuation in cuprates Baskaran, Zou, Anderson (Solid State Comm. 1987) Δ vs. x and the loss of phase coherence in underdoped regime Emery Kivelson (Nature 1995) Phase fluctuation and loss of coherence at Tc in low (superfluid) density SC s M. 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 mapping to Bosons in Hofstadter lattice --- vorticity and checkerboard pattern Balents, Sachdev, Fisher et al. (2004) Vorticity and checkerboard in underdoped regime P. A. Lee, X. G. Wen. (PRL, 03, PRB 04) Loss of phase coherence in tj model, nature of vortex core Lee, Nagaosa, Wen, Rev. Mod. Phys. (cond-mat/0410***) Good review of phase stiffness, phase fluctuation issues P. W. Anderson (cond-mat 05) Spin-charge locking occurs at T onset > T c
The Nernst effect of carriers J t t = σ.e + α.( T ) Open boundaries, so set J = 0. Wang et al. PRB 01 E E y = t ρ. t α.( T ) ( ρα + ρ )( T ) = α yx yx x Off-diag. Peltier cond. α xy 2 = 2e k 0 f k ε k μ l ε T y v B. k ( l x ) Measured Nernst signal e N E y T = 2 π 3 2 kbt e θ ε Generally, very small because of cancellation between α xy and σ xy
The vortex Nernst effect; dominant in vortex liquid state Moving vortex produces E = B x v Gradient drives vortex current with velocity v x B E v Force exerted on vortex line by grad T F = s φ ( T ) Line entropy s φ Balance F by viscous damping η v = s φ ( T ) Nernst signal e N E e N = = T Bs φ η
Nernst experiment e y H m H Vortices move in a temperature gradient Phase slip generates Josephson voltage Nernst signal e y = E y / T 2eV J = 2πh n V E J = B x v
Nernst effect in LSCO-0.12 Xu et al. Nature (2000) Wang et al. PRB (2001) vortex Nernst signal onset from T = 120 K, ~ 90K above T c`1
Nernst effect in underdoped Bi-2212 (T c = 50 K) Vortex signal persists to 70 K above T c.
OP YBCO UD LSCO Wang, Li, NPO PRB (2006) OP Bi2212 UD Bi2212
Nernst contour-map in underdoped, optimal and overdoped LSCO
Overdoped LaSrCuO x = 0.20 H* H m T co
Optimal, untwinned BZO-grown YBCO
Nernst contour maps in UD YBCO and OP YBCO Contour plots in underdoped YBaCuO 6.50 (main panel) and optimal YBCO 6.99 (inset). Vortex signal extends above 70 K in underdoped YBCO, to 100 K in optimal YBCO High-temp phase merges continuously with vortex liquid state T co Wang et al., PRL 02
Contour Map of Nernst Signal in Wang, Bi 2201 Li, Ong PRB 2006
Kosterlitz-Thouless transition Spontaneous vortices destroy superfluidity in 2D films Change in free energy ΔF to create a vortex ΔF = ΔU TΔS = (ε c k B T) log (R/a) 2 ΔF < 0 if T > T KT = ε c /k B vortices appear spontaneously ρ s Δ 0 T KT T c MF 3D version of KT transition in cuprates?
Kosterlitz Thouless transition in 2D superconductor vortex density antivortex vortex Unbinding of vortex-antivortex Free energy gain ΔF = U - TS
BCS transition 2D Kosterlitz Thouless transition n vortex Δ Δ ρ s ρ s 0 T c 0 T KT T MF H = ½ ρ s d 3 r ( φ) 2 ρ s measures phase rigidity Phase coherence destroyed at T KT by proliferation of vortices High temperature superconductors?
Simulation Nernst effect in 2D XY model Podolsky, Raghu, Vishwanath PRL (2007).
Loss of phase coherence determines Tc Condensate amplitude persists T>Tc Wang et al. PRB (2001), NPO et al. Ann. Phys (2004)
Condensate amplitude persists to T onset > T c Nernst signal confined to SC dome Vorticity defines Nernst region
The Upper Critical Field -- destruction of pair condensate upper critical field
Phase diagram of type-ii superconductor 2H-NbSe 2 cuprates 4 T H vortex solid H c1 H m liquid normal H c2 H H c1 vortex solid? vortex liquid H m? 0 T T c0 0 T T c0 Meissner state
Cooper pairing in cuprates d-wave symmetry + - - ξ coherence length + Δ θ ) = Δ cos 2θ ( 0 Upper critical field H c 2 = φ0 2 2πξ cuprates Nb 3 Sn MgB 2 NbSe 2 18 29 57 90 o ξ (A) H c2 100 Tesla 40 10 4 Tesla
PbIn, T c = 7.2 K (Vidal, PRB 73) Bi 2201 (T c = 28 K, H c2 ~ 48 T) T=1.5K 2.0 1.5 T=8K ey e y (μv/k) 1.0 H d H c2 0.5 H c2 0.3 1.0 H/H c2 0.0 0 10 20 30 40 50 60 μ 0 H (T) Upper critical Field H c2 given by e y 0. Hole cuprates --- Need intense fields. Wang et al. Science (2003)
Vortex-Nernst signal in Bi 2201
Vortex Nernst signal in overdoped, optim. and underdoped regimes Wang, Li, NPO PRB 2006 overdoped optimal underdoped Nernst signal e N = E y / T
Wang et al. Science (2003) overdoped optimum underdoped e y (μv/k) 3.5 3.0 2.5 2.0 1.5 OD-Bi2212 (T c =65K) 45 50 40 55 35 60 30 3.5 3.0 2.5 2.0 1.5 OPT-Bi2212 (T c =90K) 90 75 70K 80 85 3.0 2.5 2.0 1.5 UD-Bi2212 (T c =50K) 40K 45 50 55 60 1.0 65 25 1.0 95 1.0 65 0.5 70 75 20 80 0.0 85 90 0 5 10 15 20 25 30 μ 0 H (T) 0.5 100 105 0.0 110 0 5 10 15 20 25 30 μ 0 H (T) 70 0.5 75 80 90 100 0.0 0 5 10 15 20 25 30 μ 0 H (T) Field scale increases as x decreases
Resistivity a bad probe for absence of pair amplitude Plot of ρ and e y versus T at fixed H (33 T). Vortex signal is large for T < 26 K, but ρ is close to normal value ρ N above 15 K. LaSrCuO
0.8 N d CCO (T c =24.5K) e y (μv/k) 0.6 0.4 12K Problems with Flux-flow Resistivity Bardeen Stephen law (not seen) Wang, Li, NPO PRB 06 1.0 LSC O (0.20) ρ e y (μv/k) 0.8 0.6 0.4 22K ρ e y e y 0.2 H c2 0.2 H c2 0.0 0 2 4 6 8 10 12 14 μ 0 H (T) 0.0 0 5 10 15 20 25 30 μ 0 H (T) Resistivity does not distinguish vortex liquid and normal state
NbSe 2 NdCeCuO Hole-doped cuprates H c2 Hc2 H c2 vortex liquid vortex liquid H m H m H m T c0 Conventional SC Amplitude vanishes at T c0 (BCS) T c0 Expanded vortex liquid Amplitude vanishes at T c0 T c0 Vortex liquid dominant. Loss of phase coherence at T c0 (zero-field melting)
Vortex Nernst signal in NdCeCuO -- mean-field scenario Plot of H m, H*, H c2 vs. T H m and H* similar to hole-doped However, H c2 has mean-field form Vortex-Nernst signal vanishes just above H c2 line
Anomalous Nernst signal only in pseudogap state Hole-doped optimal Electron-doped optimal T c Strong phase fluctuations (non Gaussian) T c Mean-field like (Gaussian fluctuations above Tc)
H c2 (0) vs x matches T onset vs x Wang et al., unpublished
Summary 1. Existence of large vortex-nernst region above T c dome 2. Transition at Tc not mean-field BCS, but loss of long-range phase correlation 3. Vortex liquid state extends high above Tc in UD regime 4. Upper critical field H c2 is very large, 80-150 Tesla 5. H c2 vs. T dependence not of mean-field BCS form
References (Talk 2) 1. Z. Xu, N.P. Ong, Y. Wang, T. Kakeshita and S. Uchida, Nature 406, 486 (2000). 2. Yayu Wang, Z. A. Xu, T. Kakeshita, S. Uchida, S. Ono, Yoichi Ando, and N. P. Ong, Phys. Rev. B 64, 224519 (2001). 3. Yayu Wang, N. P. Ong, Z.A. Xu, T. Kakeshita, S. Uchida, D. A. Bonn, R. Liang and W. N. Hardy, Phys. Rev. Lett. 88, 257003 (2002) 4. Yayu Wang, S. Ono, Y. Onose, G. Gu, Yoichi Ando, Y. Tokura, S. Uchida, and N. P. Ong, Science, 299, 86 (2003). 5. Yayu Wang, Lu Li and N. P. Ong, Phys. Rev. B 73, 024510 (2006), 6. Daniel Podolsky, Srinivas Raghu and Ashvin Vishwanath, Phys. Rev. Lett. 99, 117004 (2007). 7. V. Oganesyan and Iddo Ussishkin, Phys. Rev. B 70, 054503 (2004). 8. Cigdem Capan and Kamran Behnia et al., Phys. Rev. Lett. 88, 056601 (2002) 9. F. Rullier-Albenque, R. Tourbot, H. Alloul, P. Lejay, D. Colson, and A. Forget, Phys. Rev. Lett. 96, 067002 (2006)
Magnetization in Abrikosov state M H c1 H c2 H M~ -lnh M = - [H c2 H] / β(2κ 2 1) In cuprates, κ = 100-150, H c2 ~ 50-150 T M < 1000 A/m (10 G) Area = Condensation energy U
Optimal YBCO Wang, Li and NPO, PRB 06 3-layer Bi 2223