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Introduction to the Nernst effect Vortex signal above Tc Loss of long-range phase coherence

Boulder School for Condensed Matter and Materials Physics 2008. Talk 2 Nernst effect in vortex-liquid state of cuprates. Introduction to the Nernst effect Vortex signal above Tc Loss of long-range phase coherence The Upper Critical Field The cuprate phase diagram. N. P. Ong

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Introduction to the Nernst effect Vortex signal above Tc Loss of long-range phase coherence

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  1. Boulder School for Condensed Matter and Materials Physics 2008 Talk 2 Nernst effect in vortex-liquid state of cuprates • Introduction to the Nernst effect • Vortex signal above Tc • Loss of long-range phase coherence • The Upper Critical Field • 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

  2. Mott insulator T* T pseudogap Tc Fermi liquid AF dSC 0 0.25 0.05 doping x Phase diagram of Cuprates s = 1/2 hole

  3. The phase of macroscopic pair-wave function + S f BCS wave function Anderson pseudospin Phase f fixed (phase representation); N fluctuates [N, f] = 1

  4. b(r) Normal core Js x x b(r) |Y| = D London length l Vortices in type-II superconductors Vortex in cuprates Vortex in Niobium CuO2 layers Js 2D vortex pancake superfluid electrons (pair condensate) H coherence length x

  5. 4 T normal ? ? liquid vortex liquid H Hm Hc2 vortex solid vortex solid Hm Hc1 Hc1 0 Tc0 0 Tc0 T T Phase diagram of type-II superconductor cuprates 2H-NbSe2 H Meissner state

  6. Vortex motion in type II superconductor (Bardeen Stephen, Nozieres Vinen) Applied supercurrent Js exerts magnus force on vortex core Velocity gives induced E-field in core (Faraday effect) Current enters core and dissipates (damping viscosity) Motion of vortices generates observed E-field E = B x v Consequence of Josephson equation Tilt angle of velocity gives negative vortex Hall effect In clean limit, vortex v is || - Js

  7. Anderson-Higgs mechanism and phase rigidity y2 y1 y2 y1 Anderson-Higgs mechanism: Phase stiffness singular phase fluc. (excitation of vortices) Phase mode q + EM Fmn = Massive mode (Meissner effect)

  8. P.W. Anderson Phys. Rev. 1959, RMP 1966 |Y| eiq(r) Phase rigidity  uniform phase q phase rigidity measured by rs But phase coherence destroyed by mobile vortices Dq = 2p

  9. T D- = 2ph nV Dq = 2p DVJ Integrate VJ to give dc signal prop. to nv time phase-slip and Nernst signal Passage of a vortex  Phase diff. q jumps by 2p H Josephson Eq.

  10. 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

  11. Theories on phase fluctuation in cuprates Baskaran, Zou, Anderson (Solid State Comm. 1987) D 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 Tonset > Tc

  12. The Nernst effect of carriers Wang et al. PRB ‘01 Open boundaries, so set J = 0. Off-diag. Peltier cond. Measured Nernst signal Generally, very small because of cancellation between axy and sxy

  13. The vortex Nernst effect; dominant in vortex liquid state E B v Moving vortex produces E = B x v Gradient drives vortex current with velocity v || x Force exerted on vortex line by grad T Line entropy sf Balance F by viscous damping Nernst signal eN

  14. Nernst signal ey = Ey /| T | Vortices move in a temperature gradient Phase slip generates Josephson voltage 2eVJ = 2ph nV EJ = B x v Nernst experiment ey Hm H

  15. Xu et al. Nature (2000) Wang et al. PRB (2001) Nernst effect in LSCO-0.12 vortex Nernst signal onset from T = 120 K, ~ 90K above Tc`1

  16. Vortex signal persists to 70 K above Tc. Nernst effect in underdoped Bi-2212 (Tc = 50 K)

  17. OP YBCO UD LSCO Wang, Li, NPO PRB (2006) OP Bi2212 UD Bi2212

  18. Nernst contour-map in underdoped, optimal and overdoped LSCO

  19. Tco Overdoped LaSrCuO x = 0.20 H* Hm

  20. Optimal, untwinned BZO-grown YBCO

  21. Nernst contour maps in UD YBCO and OP YBCO Tco Contour plots in underdoped YBaCuO6.50 (main panel) and optimal YBCO6.99 (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 Wang et al., PRL ‘02

  22. Contour Map of Nernst Signal in Bi 2201 Wang, Li, Ong PRB 2006

  23. Kosterlitz-Thouless transition rs D 0 TKT TcMF Spontaneous vortices destroy superfluidity in 2D films Change in free energy DF to create a vortex DF = DU– TDS = (ec – kBT) log (R/a)2 DF < 0 if T > TKT = ec/kB vortices appear spontaneously 3D version of KT transition in cuprates?

  24. Kosterlitz Thouless transition in 2D superconductor vortex density antivortex vortex Unbinding of vortex-antivortex DF = U - TS Free energy gain

  25. n vortex D 0 T T T c KT MF H = ½rsd3r ( f)2 r r s s 2D Kosterlitz Thouless transition BCS transition D 0 Phase coherence destroyed at TKT by proliferation of vortices rs measures phase rigidity High temperature superconductors?

  26. Simulation Nernst effect in 2D XY model Podolsky, Raghu, Vishwanath PRL (2007).

  27. Wang et al. PRB (2001), NPO et al. Ann. Phys (2004) • Loss of phase coherence determines Tc • Condensate amplitude persists T>Tc

  28. Condensate amplitude persists to Tonset > Tc • Nernst signal confined to SC dome • Vorticity defines Nernst region

  29. The Upper Critical Field -- destruction of pair condensate upper critical field

  30. 4 T normal ? ? liquid vortex liquid H Hm Hc2 vortex solid vortex solid Hm Hc1 Hc1 0 Tc0 0 Tc0 T T Phase diagram of type-II superconductor cuprates 2H-NbSe2 H Meissner state

  31. x + o - - x (A) + Hc2 4 Tesla 40 10 100 Tesla Cooper pairing in cuprates d-wave symmetry coherence length Upper critical field cuprates NbSe2 MgB2 Nb3Sn 57 18 29 90

  32. T=1.5K T=8K Hd Hc2 0.3 1.0 H/Hc2 • Upper critical Field Hc2 given by ey 0. • Hole cuprates --- Need intense fields. PbIn, Tc = 7.2 K (Vidal, PRB ’73) Bi 2201 (Tc= 28 K, Hc2 ~ 48 T) ey Hc2 Wang et al. Science (2003)

  33. Vortex-Nernst signal in Bi 2201

  34. Vortex Nernst signal in overdoped, optim. and underdoped regimes Nernst signal eN = Ey /| T | Wang, Li, NPO PRB 2006 underdoped optimal overdoped

  35. Wang et al. Science (2003) overdoped optimum underdoped Field scale increases as x decreases

  36. Resistivity a bad probefor absence of pair amplitude LaSrCuO Plot of r and ey versus T at fixed H (33 T). Vortex signal is large for T < 26 K, but r is close to normal value rN above 15 K.

  37. Problems with Flux-flow Resistivity Bardeen Stephen law (not seen) Wang, Li, NPO PRB ‘06 Hc2 Hc2 Resistivity does not distinguish vortex liquid and normal state

  38. Hole-doped cuprates NbSe2 NdCeCuO Hc2 Hc2 Hc2 vortex liquid vortex liquid Hm Hm Hm Tc0 Tc0 Tc0 Vortex liquid dominant. Loss of phase coherence at Tc0 (zero-field melting) Expanded vortex liquid Amplitude vanishes at Tc0 Conventional SC Amplitude vanishes at Tc0 (BCS)

  39. Vortex Nernst signal in NdCeCuO -- mean-field scenario Plot of Hm, H*, Hc2 vs. T • Hm and H* similar to hole-doped • However, Hc2 has mean-field form • Vortex-Nernst signal vanishes just above Hc2 line

  40. Anomalous Nernst signal only in pseudogap state Tc Hole-doped optimal Electron-doped optimal Tc Mean-field like (Gaussian fluctuations above Tc) Strong phase fluctuations (non Gaussian)

  41. Wang et al., unpublished Hc2(0) vs x matches Tonset vs x

  42. Summary • 1. Existence of large vortex-Nernst region above Tc dome • Transition at Tc not mean-field BCS, but loss of • long-range phase correlation • Vortex liquid state extends high above Tcin UD regime • Upper critical field Hc2 is very large, 80-150 Tesla • Hc2 vs. T dependence not of mean-field BCS form

  43. 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)

  44. Magnetization in Abrikosov state M H Hc1 Hc2 M = - [Hc2 – H] / b(2k2 –1) M~ -lnH In cuprates, k = 100-150, Hc2 ~ 50-150 T M < 1000 A/m (10 G) Area = Condensation energy U

  45. Wang, Li and NPO, PRB 06 Optimal YBCO 3-layer Bi 2223

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