1. Electrons on a triangular lattice in Na-doped Cobalt Oxide Yayu Wang, Maw Lin Foo, Lu Li, Nyrissa Rogado, S. Watauchi, R. J. Cava, N.P.O. Princeton University 1.Frustration on triangular lattice 2.Large thermopower in Na x CoO 2 3.ARPES 4.Hall effect 5.Phase diagram Supported by NSF, ONR

2. Geometrical Frustration on triangular lattice H = -J S i S j Antiferromagnetic Ising model ? Impossible to have AF alignment on all 3 bonds Ground state is disordered and highly degenerate Resonating valence bond model(s) 1971, 1987 Spin Ice in pyrochlores 1998 Frustrated magnetic states in spinels 1999 (i,j)

3. Na x CoO 2 Na ions (dopants) sandwiched btw layers of tilted CoO 2 octahedra Co ions define a triangular lattice tilt Octahedra tilted to form a layer building block Na Co Terasaki, Uchinokura 1997

4. as grown Na Co

5. Resistivity of Na x CoO 2 (x ~ 0.71) Terasaki et al., PRB 1997 Wang et al. Nature ‘03

6. Susceptibility of insulators vs metals 0T  ~ 1/ T Curie  Pauli susceptibility Susceptibility  = dM/dH In metals,  small and indept of T 0T 1/  free spins AF spins In Antiferromagnets  = C/(T +  )  = T (Neel temp) N energy DOS kT

7. AF Neel temperature T N ~ 60-100 K Magnitude of  implies Co 4+ ions spin S = ½ Co 3+ is diamagnetic (S = 0), Co 3+ Co 4+ Susceptibility  has Curie-Weiss form

8. Metallic resistivity but antiferromagnetic in spin response (Curie-Weiss Metal)

9. Thermopower and Peltier coef. Ratio of currents J Q /J =  (Peltier coeff) S =  / T = J Q / JT J JQJQ Heat current density J Q accompanies charge current density J holes E

10. Large thermopower ~10 times Sommerfeld value at 300 K Sommerfeld Terasaki et al Phys. Rev. B (1997) S Large thermopower S of Na x CoO 2 x = 0.71

11. Classical gas J = nev J Q = n k B T v Peltier coef.  = J Q /J = k B T/e Seebeck coef. S =  / T = k B /e e kBTkBT Natural unit of S k B /e = 86  V/K  Semiconductor J Q = n v  S = (k B /e)(  /k B T)  Thermopower

12. Thermopower of conventional metals Fermi Gas in E field Charge currents add mass currents cancel Heat currents cancel S = J Q /JT strongly suppressed S ~ (k B /e) (T/T F ) ~ 86 x 10 -2  V/K “Excitation picture” (k)(k)  E Fermi level particle hole particle excitations hole excitations vacancies E d k = c -k T F ~ 50,000 K S virtually indepndnt of H

13. In-plane field H || - T Strong field suppression of Thermopower T - B V S Field dependence of S in Na x CoO 2 Wang et al. Nature ‘03

14. Spin contribution to thermopower (Chaikin Beni, 1976) J JQJQ J = nev Spin entropy per carrier = k B log 2 J Q = nv k B T log 2 S = J Q /JT = (k B /e) log 2 ~ 60  V/K Not signif. in conv. metals

15. S(H,T) curve is a function of H/T only S(H)/S(0) - 1

16. Conclusion : 1. Spin entropy is the source for enhanced thermopower 2. Key for new thermo-electric materials -- Spin S(H)/S(0) - 1 Wang et al. Nature ‘03

17. Co 3d states In Na x CoO 2, hole density n h = 1-x

18. Na x CoO 2 Multiple electronic phases vs. Na content Superconductivity

19. Water intercalated superconductor Na x CoO 2 ·y H 2 O, x ~ 0.35, y ~ 1.30 Superconductor with T c ~ 4.5K Takada et al., Nature (2003). pairing symmetry: s, p or d-wave? Why is water essential? What is pairing mechanism: e-ph or e-e or magnetic?

21. T-linear Hall coefficient Yayu Wang, 03 R H conv. metal

22. High-frequency R H * in tJ model (B.S. Shastry ‘93, ‘03)  H ~ i(  t) 3 exp(i  )  ~ (  t) 2 R* H ~  H /H  2 ~ (  t) -1 Hopping Hall current in triangular lattice (Holstein, ‘61)  H ~ t 12 t 23 t 31 ~ i t 3 exp(i  ) Why is R H T-linear?  t << 1 (  = 1/T) T-linear Peierls phase     t 12 1 3 2 t 13 M2S-RIO conf. Rio de Janeiro, May 28 th 2003 (N.P. Ong)

23. ARPES: Weak quasiparticle dispersion Single-particle hopping : t < 0 and |t| ~ 10 meV (bandwidth < 100 meV) Momentum Kinetic energy (eV) Small bandwidth  Low degeneracy T Z. Hasan et al. (PRL ‘04)

24. Fermi Surface of Na 0.71 CoO 2 measured by ARPES Large hole-like FS Hopping integral t ~ 10 meV Fermi velocity < 0.4 eV.A Hasan et al.

25. Behavior of quasi-particles versus temperature Resistivity is T-linear below 100K ARPES Quasiparticles are coherent only below 150K

26. Na x CoO 2 Multiple electronic phases vs. Na content as grown Foo et al. PRL ‘04 Insulating state

27. Reduce the Na content by a series of chemical de-intercalation x = 0.75, as grown crystals of Floating zone or flux method x = 0.68: NaClO 3 in water x = 0.50: I 2 in Acetonitrile x = 0.31: Br 2 in Acetonitrile Fine-tuning of Na content in Na x CoO 2 single crystals Foo et al., condmat/0312174 (2003), PRL ‘04 Stronger oxidation agent High-quality crystals with Na content 0.31 < x < 0.75

28. Calibration of the Na content vs. c-axis lattice parameter Calibration procedure treat powder and crystals under same conditions powder x-ray diffraction to get c-axis lattice constant ICP-AES to determine the Na contents of powders x vs. c-axis calibration curve from the c-axis of crystal, extract the Na content

29. x = 0.50 (1/2): Two kinks at T c1 =88K and T c2 =53K in  Resistivity shows insulating behavior below T=53K

30. An unexpected insulator at x = ½

31. Electron diffraction at 300K shows the superlattice formed by the Na ions, consistent with a zig-zag order Zendbergen et al., condmat/0403206 (2004)

32. Thermal Conductivity Hall coefficient Foo et al., PRL ‘04

33. 0

35. Na x CoO 2 Multiple electronic phases vs. Na content as grown Foo et al. PRL ‘04 Spin ordered

36. S x = 0.71 x = 0.88 Sommerfeld Further enhancement of thermopower

37. P = S 2  x ~ 0.85 x = 0.71

38. Unusual electronic behavior in Na x CoO 2 Strongly correlated s = ½ holes hopping on triangular lattice Paramagnetic Metal (x ~ 1/3) High conductivity, superconducting with H 2 O intercalatn. Charge-ordered Insulator (x = ½) Na ion ordering, hole ordering (stripes?), giant thermal conductivity Curie-Weiss metal (x ~ 2/3) Curie-Weiss susceptibility, metallic cond., large thermopower from spin entropy, T-linear Hall coef. Spin Ordered Phase (x > ¾) Even larger thermopower, field-induced metamagnetism

40. x = 0.31 (~ 1/3), parent compound of the SC  is T-independent, not Curie-Weiss M-H curves are linear at low T, no ferromagnetic order Magnetic properties rather normal

41. x = 0.31 (~ 1/3): Smaller high temperature thermopower Smaller Hall coefficient, weaker T-dependent larger hole concentration (~3x10 22 /cm 3 ) and reduced correlation Consistent with ARPES (MZ Hasan et al., and Hong Ding et al.)

42. x = 0.31 (~ 1/3)  is T-independent, non Curie-Weiss  smaller, T 2 at low T S small, ~34  V/K at 300K R H weaker T-dependence Paramagnetic (T 2 ) metal More like conventional metal x = 0.71 (~ 2/3)  Curie-Weiss, AF interaction  is T-linear at low T S large, ~90  V/K at 300K R H strong T-linear Curie-Weiss metal Strong magnetic interaction and electron correlation

43. Sodium ion ordering versus x Lynn, Cava et al.

44. S have giant negative values below T c1 The number of holes are strongly reduced, the residual charge carriers seem to be electron like

45. R H becomes negative and the amplitude is 100 times larger charge density reduces by ~ 100 times particle-hole symmetry at low T

46. Possible charge ordering in Na x CoO 2 x = 1/2 electron hole electron hole    x = 1/3 electron hole  3 a   3 a x = 2/3 electron hole  3 a   3 a

47. NaCoO 2 : 1 pair of electron per Co site, band insulator. CoO 2 : 1 electron per Co site, Mott insulator? 0 < x < 1/4 No results, Doped Mott Insulator? 3/4 < x < 1 Magnetic ordering? Motohashi et al., PRB (2003) Sugiyama et al., condmat (2003) Bayrakci et al., condmat (2003) x ~ 1/3 More like conventional metal x ~ 2/3 Strong magnetic interaction and electron correlation 1/4 < x < 1/3 dome shape SC Schaak et al., Nature (2003) X = 1/2 Charge ordered insulator Maw-lin Foo et al., condmat (2003) To appear in PRL (2004)

48. Good conductor  is T-linear below 100 K Hall coefficient n 2D ~ 4× 10 22 /cm 2

49. x = 0.31 (~ 1/3), parent compound of the SC Better metal,  is smaller that x = 0.71 R ~ T 2 below 30K,  ~ 10  cm at 4K More like a conventional Fermi liquid 2 2

50. Thermoelectric and Peltier effects Ratio of currents J Q /J =  (Peltier coeff) S =  / T = J Q / JT specimen JQJQ n p J Thermoelectric cooler J JQJQ Heat current density J Q accompanies charge current density J holes

51. Foo et al., PRL ‘04 Systematic change vs x except at x = ½ Susceptibility Resistivity

52. Na CoO 2 Transition metal oxide tunable carrier density Quasi-2D: ρ c /ρ ab ~200 at 4K Triangular Co lattice with AF interaction—Frustrated magnetic system Enhanced thermopower I. Terasaki et al (1997) Superconductivity K. Takada et al (2003) Co

53. Strong-Correlation System: Kubo formula : Free Spin model: P.M.Chaikin et al (1976),,

54. Close fit using free-spin model From fit: Landé factor g ~ 2.2

55. S total = S 0 - S wire Phosphor bronze wire is H-independent All obs. field dependence from Na x Co 2 O 4 T- ThTh TcTc T0T0 V phosphor -bronze

56. S ~ T/T F S ~  /T   T T metal semiconductor Difficult to have ZT larger than 0.01 below 200 K. 1000  V/K 5  V/K Materials Constraint

57. Thermal conductivity: mostly from phonons Much larger thermal conductivity: longer phonon mean free path Na ion ordering: reduced scattering by disordered Na ions in the Na layer charge ordering: steep increase below 88K, reduced electron-phonon scattering in the CoO 2 plane,

58. Na x CoO 2 Several electronic phases vs. Na content Foo et al. PRL (’04)

59. Low-energy electronic structure of Na 0.7 CoO 2 ARPES work by Hasan Group : cond-mat/0308438 Results on Na 0.7 CoO 2 : OWeak quasiparticle dispersion : narrow bandwidth OSignatures of Strong Correlation (Large Hubbard U) OFermisurface : Large rounded Hole-like, small v f (anisotropic) OThermal behavior of QPs: coherent QP only below 150K

60. Strong Correlation (Large Hubbard U) U ~ 5 eV Resonance profile of valence satellite  a measure of Hubbard U ~ 5 eV Narrow band

64.  Pauli T (Curie Weiss) In Na x CoO 2 (x = 0.70), susceptibility implies spin-1/2 local moments, instead of degenerate electron gas T resistivity Na x CoO 2 : Curie-Weiss Metal

69. Magneto-resistance also from Spin effect, similar anisotropy between in-plane and c-axis.

70. Two figures of merit for thermoelectrics 2. The ZT number ZT = (S 2 /  )(T/  ) S = thermopower  = resistivity  = thermal conductivity ZT ~ 1 in Bi 2 Te 3 (at 300 K) 1. Power factor S 2 /  ONR workshop 2004, Tampa Maximize ZT and minimize resistivity (physically conflicting demands)  T max = ½ ZT 2 Max temp. difference