Talk 1. Quasi-particle charge and heat currents in d -wave superconductor

1. Boulder School for Condensed Matter and Materials Physics 2008 Talk 1. Quasi-particlecharge and heat currents in d-wave superconductor Talk 2. The Nernst effect in vortex liquid state of cuprates Talk 3. Magnetization of vortex liquid state N. P. Ong, Princeton University http://www.princeton.edu/~npo/ Boulder, July 7-11, 2008 Supported by NSF-MRSEC, ONR

2. Boulder School for Condensed Matter and Materials Physics 2008 Talk 1 Quasi-particle charge and heat currents in d-wave superconductor • Introduction : Charge and heat currents • Hall effect, Nernst effect, Thermal Hall effect • 2. The Hall effect in cuprates • 3. Quasiparticles and Thermal Hall conductivity N. P. Ong, Princeton University Collaborators: Lu Li, Joe Checkelsky, Yayu Wang, Kapeel Krishana, Wei Li Lee, Yuexing Zhang, Jeff M. Harris, Y.F. Yan, P.W. Anderson Boulder, July 7-11, 2008 Supported by NSF-MRSEC, ONR

3. The Hall effect J Thermopower bath V heater Seebeck coef.

4. Boltzmann-equation expressions for currents heat charge Charge current conductivity Presence of temp. gradient Thermoelectric current Thermoelectric cond.

5. ek hole particle eF k Quasi-particle excitations in normal state of Fermi liquid particle hole vacancy hk+ = c-k A spin-down vacancy at –k translates to a spin-up hole excitation at k

6. Charge and heat currents in the “excitation” representation Large vacancy population Large vacancy population k k E E(k) E(k) o Large right electron current Large left hole current Both mass (entropy) currents flow to right Charge currents add Mass currents add Mass currents nearly cancel. Difference is the Peltier heat current Charge currents nearly cancel. Difference is the Peltier charge current

7. The Nernst effect (quasiparticles) 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

8. NPO, PRB (1991) The 2D Hall conductivity sxy Boltzmann Eq. Eq. of motion Hall current in 2nd order Gauss mapping to … Area swept out in ell-space!

9. The 2D Hall conductivity sxy swept area sxy is the area swept out by mfp (for arb. anisotropy)

10. Temp. dependences of Hall coef. and Hall angle in YBCO Harris, Yan, NPO, PRB ‘92

11. Similar T dependence of RH seen in LaSrCuO Hwang Batlogg et al., PRL ‘94 Ono, Komiya, Ando, PRB ‘07

12. Kanigel, Campuzano et al. Nature Phys. 2007 ky D(f) f kx node Antinode Fermi arc length T/T*

13. Model: Only states on Fermi arc have long mean free path NPO and Wang unpubl.

14. Fits to T dependence of RH in YBCO 6.63 6.7 6.8 y = 6.95 NPO and Wang unpubl.

15. Fits to cotq and resistivity r

17. b(r) Normal core Js x Vortex in Niobium Vortex in cuprates CuO2 layers superfluid electrons Js H 2D vortex pancake x = 500 A Gap D(r) vanishes in core |Y| = D

18. 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

19. Harris et al. PRB ‘95 Vortex Hall current Vortex Hall sxy is negative. Appearance is abrupt Invert matrix Quasiparticle and vortex Hall conductivities are additive ~H ~ -1/H

20. Thermal Hall conductivity of quasi-particles in cuprates K. Krishana, Yuexing Zhang, J. M. Harris, NPO Problem: Separate the QP current from vortex currents? Monitor thermal currents. kxx vs. T in 90-K YBCO (twinned and untwinned) Is peak from QP or phonons?

21. remove gk = uk c k + vk c-k Excitations of an s-wave superconductor quasiparticles Ek D xk Energy cost Ek = [ xk2 + D2 ]1/2 QP’s cost energy, but increase entropy S(lower free energy F)

22. JQ = ktot (-T) ktot = kel + kph electrons phonons T > Tc, ktot mostly electronic Below Tc, QPs carry heat T << Tc, QP population 0 Phonons are long-lived Heat transport in low-Tc superconductors Heat-current density Condensate does not carry heat (zero entropy) Tc Pb ktot kph 0 T Jericho, Odoni, Ott, PRB ‘85

23. Dirac-like spectrum of QP at nodes D(f) = D0 cos 2f Quasiparticle dispersion E vs. k is linear.

24. Ek node D0 xk Ek node xk Quasiparticles in a d-wave superconductor gap max. (antinode) antinode D(f) = D0 cos 2f

25. Heat current in normal and superconducting states Ek - T ek- m x-k xk 0 -kF kF 0 nodal QP’s fk Q-ptcl mass current Q-hole mass current ptcl mass current hole mass current hole excitations Fermi Surface electrons

26. Separating electronic and phonon k in 93-K YBa2Cu3O7 ka = ke + kph

27. Quasiparticles in the CuO2 plane vortex Dirac quasiparticle H

28. Hall thermal current from asymmetric scattering of QP by vortex bath heater warm cool heater scattering of phonons: no asymmetry asymmetric scattering of QP by vortex

29. counter-moving qp counter-moving qp co-moving co-moving vs vs Doppler shift QP excitations J = Js + Jqp (Js || -Jqp) In a supercurrent vs, energy of counter-moving QP’s lowered.

30. Origin of QP Hall current vs Doppler shift lowers energy of counter-moving QPs counter-moving vortex core Scattered QP’s flow against vs circulation Incident QP’s co-moving Skew scattg cross-section (Durst,Vishwanath, Lee, PRL ‘03) sH ~ T /H1/2 sH T/H1/2

31. Thermal Hall Conductivity kxy in high-purity YBCO7 (normal state)

32. Thermal Hall Conductivity kxy In high-purity YBCO7 • 1) Hall signal much larger below Tc • 2) Giant increase in initial slope 85 to 40 K • 3) Strongly non-linear in H

33. Thermal Hall Conductivity kxy in high-purity YBCO7 (12.5 to 35 K)

35. Plot initial slope limB-0 kxy/B vs. T. Initial slope increases by 1000 between 85 and 30 K Steep increase in QP mean-free-path ~ 120

36. QP mean-free-path l derived from Hall angle Increases by 120 from 85 to 30 K Abrupt increase at Tc (coherence effect?)

37. kxy = C0 (TH)1/2 F(x) ~ x Simon-Lee scaling kxy = T2 F( H1/2/T )

38. Calculated fits to Kxx and Kxy (Adam Durst, Ashvin Vishwanath, P.A. Lee, 2003)

39. Durst, Vishwanath, Lee kxx = cevFl ~ T2T-1 kxy = kxx tan q = kxx nVsHl ~ T2T-1 . H .TH-1/2. T-1 ~ (TH)1/2 Explains observation kxy = C0 (TH)1/2

40. Quasi-particles are hole like in UD YBCO y=6.60

41. Summary • Below Tc, we observe • . 1000-fold increase in kxy (weak field) • . 200-fold increase in QP mfp l. • (80 Angstrom to 2 microns) • . Giant anomaly in ktot is entirely from QP. • .Steep increase in mfp starts just below Tc • (conflicts with ARPES) • . Intriguing scaling behavior in kxy (Simon-Lee) • . No evidence (yet) for Landau quantization

42. References for Lect. 1 (website http://www.princeton.edu/~npo/) 1. N. P. Ong, Phys. Rev. B 43, 193 (1991). 2. J.M. Harris, Y.F. Yan, and N.P. Ong, Phys. Rev. B 46, 14293 (1992). 3. J.M. Harris, Y.F. Yan, O.K.C. Tsui, Y. Matsuda and N.P. Ong, Phys. Rev. Lett. 73, 1711 (1994). 4. J. M. Harris, N. P. Ong, P. Matl, R. Gagnon, L. Taillefer, T. Kimura and K. Kitazawa, Phys. Rev. B 51, 12053 (1995). 5. K. Krishana, J. M. Harris, and N. P. Ong, Phys. Rev. Lett. 75, 3529 (1995). 6. Y. Zhang, N.P. Ong, Z.A. Xu, K. Krishana, R. Gagnon, and L. Taillefer, Phys. Rev. Lett. 84, 2219 (2000). 7. Y. Zhang, N.P. Ong, P. W. Anderson, D. A. Bonn, R. X. Liang, and W. N. Hardy, Phys. Rev. Lett. 86, 890 (2001). 8. Adam C. Durst, Ashvin Vishwanath, and Patrick A. Lee, Phys. Rev. Lett. 90, 187002 (2003).

43. x Vortices in cuprates CuO2 planes 2D vortex pancake Gap amplitude vanishes in core |Y| = D

44. x0 Js(r) Js(r) x* f0 H* = 2px*2 Cheap, fast vortices A new length scale x* x0 D(r) D(r) 0 0 Is H* determined by close-packing of fat vortices?