1. Vortices in the Superconducting State of Underdoped High-T c Superconductors A. J. Millis Department of Physics Columbia University Support: NSF-DMR-0338376; and Rutgers Center for Materials Theory L. B. Ioffe Center for Materials Theory Rutgers References: PRB66 094513 2002; JPCS (cond-mat/0112509) CIAR Oct 2003

2. Quantal Vortex Liquid??

3. H c2 (0) T H TcTc Schematic High-T c H-T phase diagram

4. Quantal Vortex Liquid?? H c2 (0) T H TcTc Schematic High-T c H-T phase diagram

5. Quantal Vortex Liquid?? H c2 (0) T H TcTc Schematic High-T c H-T phase diagram T c2 (H)  thermal melting of vortex lattice (esp in underdoped materials)

6. Quantal Vortex Liquid?? H c2 (0) T H TcTc Schematic High-T c H-T phase diagram T c2 (H)  thermal melting of vortex lattice (esp in underdoped materials)

7. Quantal Vortex Liquid?? H c2 (0) T H TcTc Schematic High-T c H-T phase diagram T c2 (H)  thermal melting of vortex lattice (esp in underdoped materials) Why not H c2 (0)  quantal melting of vortex lattice

8. Outline H=0: T c,  S : which states carry the current Vortex statics: length scales T c (H) and  S (H): thermal melting of vortex lattice and ‘Volovik’ depairing H c (T=0): quantal melting of vortex lattice: vortex viscosity and effective magnetic field Summary and confusions.

9. High T c phase diagram: Superconductivity + other phases… Temperature Doping x Assume:  homogeneous superconducting phase with properties which depend smoothly on doping. Ignore other phases. Pseudogap crossover scale Superconducting phase

10. Low energy theory of d-wave SC state Two coupled degrees of freedom: order parameter phase φ Fermions (quasiparticles) c + p => Leading low energy behavior from Lagrangian

11. (Transverse) phase fluctuations: T=0 superfluid stiffness ρ S0 describes H=T=0 ‘bosonic’ superfluid properties (no quasiparticles) Expect ρ S0 ~x E 4 indep of x Bosonic length scale: limit to length on which supercurrent can vary: L  =  S0  d 2 r Q 2 (r) +E 4 Q 4 + … Q=  +ieA Q B 2 ~  S0 /E 4 ~ x

12. Fermions in d-SC: Dirac spectrum v F q.p. fermi velocity (change in energy as move normal to fermi surface) v Δ =dΔ/d(p F θ) fixed by shape of s.c. gap near nodes (change in energy as move along fermi surface) Δ θ θ : position around fermi surface vΔvΔ

13. Mixing Term Quasiparticle (ψ) – phase (φ) coupling involves v F only (not v Δ )

14. Mixing Term Quasiparticle (ψ) – phase (φ) coupling involves v F only (not v Δ ) new parameter Z quasiparticle charge

15. Mixing Term Quasiparticle (ψ) – phase (φ) coupling involves v F only (not v Δ ) new parameter Z quasiparticle charge Z->0 =>quasiparticles turn into ’spinons’ as approach Mott phase. Small Z :‘footprint’ of spin-charge separation

16. Mixing Term Quasiparticle (ψ) – phase (φ) coupling involves v F only (not v Δ ) new parameter Z quasiparticle charge Z->0 =>quasiparticles turn into ’spinons’ as approach Mott phase. Small Z :‘footprint’ of spin-charge separation Corrections: when p,q ~ Δ/v F indep of x

17. Summary: parameters ***Well established values*** ρ S0 : T=0 superfluid stiffness v F : usual fermi velocity roughly ~ doping x 1.8eV-A (indep of x) ***Less well established*** v Δ : opening angle of gap node  indep of x Z: q.p. charge renormalization near gap node  indep of x

18. Statics Integrate out fermions Note: fermions excited by T or supercurrent  H- field (Volovik) T-linear penetration depth Q~B 1/2 so this gives ‘Volovik effect’

19. Implication: ‘Fermionic bounds’ T c = 2  S (T c, H) /  At H=0: T c = 2  S0 / (  + ln(2) Z 2 v F /2v  ~x AT T=0, must have Q < Q F ~ 4  S0 v  / Z 3 v F ~x If Z~1, fermionic bounds more stringent than bosonic bounds (x vs x 1/2 )

20. Determining v Δ Specific heat counts # excited quasiparticles: density of States ~E=> C~T 2 Direct measurments—not yet clearly interpretable Indirect: ‘thermal conductivity’ +’universal limit’ analysis =>v Δ increases as doping goes down (Harris et al PRB 64 06509 ’01; Chiao PRL 82 2943 ’00)

21. Measuring Z “Ferrel-Glover-Tinkham” (f) sum rule: ρ total = ρ S + ρ N Total (low energy) charge response Superfluid part of charge response Normal fluid (quasiparticle) part of charge response ρ total is conserved as T changes =>decrease in ρ S implies increase in ρ N (has been checked in optimal YBCO: Hosseini et al PRB 01)

22. Z II Quasiparticle charge response depends on BOTH number (v F /v Δ ) and on ‘charge’ Z e If v Δ known then ρ S (T)=>Z e

23. In pictures Δ θ Small T>0: # q.p. ~ T 2 /(v F v Δ ) current per q.p. ~Z v F /T Coupling to field: Z ρ n ~ Z 2 T(v F /v Δ ) qp here excited from condensate

24. ‘Conventional’ d-superconductor ρ S0 ~v F p F : all carriers near fermi surface condense into s.c. state and contribute to supercurrent Z~1: No significant charge renormalization Δ θ qp near nodes excited out of condensate ρSρS T

25. ‘Conventional’ d-superconductor ρ S0 ~v F p F : all carriers near fermi surface condense into s.c. state and contribute to supercurrent Z~1: No significant charge renormalization Δ θ more qp excited out of cond. ρSρS T

26. ‘Conventional’ d-superconductor ρ S0 ~v F p F : all carriers near fermi surface condense into s.c. state and contribute to supercurrent Z~1: No significant charge renormalization Δ θ almost all qp out of condensate ρSρS T Gap begins to decrease

27. ‘Conventional’ d-superconductor ρ S0 ~v F p F : all carriers near fermi surface condense into s.c. state and contribute to supercurrent Z~1: No significant charge renormalization => Superfluidity dies when particles all over FS are excited above the gap Δ θ ρSρS T T=T c no particles left in condensate TcTc Δ Initial slope => ρ S (T)= ρ S0 [1-T/Δ]

28. It’s different in underdoped high-T c ! ρ S0 strongly renormalized From freq. dep conductivity (Bonn, Czech Jnl Phys 46 3195 ‘96) Optimally doped YBCO: ρ S0 = ρ S,band /7= v F p F /3 Underdoped YBCO: ρ S0 = ρ S,band /11= v F p F /5 Note T c ~  S => transition driven by loss of phase stiffness (Uemura; Kivelson)

29. Proximity to Mott phase reduces low frequency conductivity and therefore ρ S0 (S. Uchida et al PRB43 7942 ’91)(J Orenstein et al PRB42 6342 ’90) More or less: low frequency peak condenses into ρ S0

30. Two classes of theory ‘Brinkman-Rice’ Quasiparticles get slow: v F ->0 as x->0 ‘RVB/Gauge theory’ Quasiparticle lose charge (turn into ‘spinons’) Z->0 as doping->0 In both cases, ρ S ->0 as doping->0 AND dρ S /dT ->0 as doping->0

31. ‘Conventional’ Mott insulating superconductor ρ S0 ~v F * p F : all carriers near fermi surface condense into s.c. state and contribute to supercurrent Z(x): Doping dependent charge renormalization v F * (x): Doping dependent fermi velocity Δ θ almost all qp out of condensate ρSρS T/T c ρ S (T) ->0 when excite all fermi surface over the gap Initial slope => ρ S (T)= ρ S0 [1-T/Δ]

32. It’s different in underdoped high-T c ! ρ S0 strongly renormalized Initial slope dρ S /dT does not change much (P. Lee) From freq. dep conductivity (Bonn, Czech Jnl Phys 46 3195 ‘96) Optimally doped YBCO: ρ S0 = ρ S,band /7= v F p F /3 Underdoped YBCO: ρ S0 = ρ S,band /11= v F p F /5

33. Underdoped high T c superconductor ρ S0 small but slope not small v F does not change much Nor does Z?? ρSρS TcTc

34. Underdoped high T c superconductor ρ S0 small but slope not small v F does not change much Nor does Z?? Δ θ ρSρS TcTc ρ S reduced almost to 0 after exciting only q.p. near nodes!?

35. Underdoped high T c superconductor ρ S0 small but slope not small v F does not change much Nor does Z?? Δ θ ρSρS TcTc ρ S reduced almost to 0 after exciting only q.p. near nodes!? =>shaded regions do not carry (much) supercurrent

36. Underdoped high T c superconductor ρ S0 small but slope not small v F does not change much Nor does Z?? Δ θ ρSρS TcTc ρ S reduced almost to 0 after exciting only q.p. near nodes!? =>shaded regions do not carry (much) supercurrent ?Supercurrent carried only by near-node states?

37. Crucial issue: v  Δ θ Big gap: density wave (charge, spin, stripe, …) Extra gap—superconducting origin v Δ : unrelated to gap max; doping dependent vΔvΔ Example: density wave => doping-dependent ‘conducting arc’

38. If v  doping independent, then must assume angle-dependent ‘Z’ Δ θ Possibility: Z->0 as move away from diagonal: range where Z~1 decreases as reduce doping?? Angle-dependent spin- charge separation?? Z(θ) Half width of Z~1 region: ~doping??

39. Summary: Part I High T c superconductors doped Mott insulators Three classes of incipient Mott physics: ‘Brinkman-Rice’ v F ->0: disagrees with photoemission ‘Slave boson’ Z->0 uniformly: disagrees with  S (T) Conducting Arc: no theory of angle-dependent Z—but apparently supported by data Rest of talk: mainly assume fermionic bounds dominate.

40. Implication for Vortices Two definitions of core size  Δ(x)/Δ(  ) r/   From quasiparticle gap r/  current From supercurrent j(x) In conventional superconductors—dfns are equivalent Generically in doped Mott insulator, expect  current diverges as doping x->0 while   remains finite.

41. Vortex: circulating current Q~1/r =>current-induced pairbreaking cuts off J as r->0 Current maximal at r=  current Ex: at T=0 if fermionic bound relevant (neglect E 4 )  S ~x =>  current ~1/x. For opt YBCO:  current ~100Z 3 A Sensitivity to Z unfortunate

42. Experimental Evidence muon spin rotation Sonier et al PRL 79 2875 ‘97 YBCO-6.6  current ~50Å YBCO-6.95  current ~20Å Sonier et al PRL 83 4165 ‘99 Note! Strong field dependence not yet understood

43. Importance of  current Many ‘transport’ phenomena (H cII (T=0); boundary of ‘paraconductivity’ region of strong superconducting fluctuations are determined by condition “When vortex cores overlap”

44. Importance of  current Many ‘transport’ phenomena (H cII (T=0); boundary of ‘paraconductivity’ region of strong superconducting fluctuations are determined by condition “When vortex cores overlap” =>Important question ??What do you mean by core??

45. Importance of  current

46. Many ‘transport’ phenomena (H cII (T=0); boundary of ‘paraconductivity’ region of strong superconducting fluctuations are determined by condition “When vortex cores overlap” =>Important question ??What do you mean by core?? I will argue: should use  current

47. Dissipation caused by moving vortex (after Tinkham, Superconductivity) Vortex—center position X vort, (slow) velocity V Supercurrent j~1/(R- X vort ) => dj/dt ~ V/(R- X vort ) 2 But j=  -A is a vector potential so dj/dt=dA/dt is an electric field => In presence of ‘normal fluid conductivity’  n, power P dissipated by moving vortex is P=  d 2 r  n E 2 =  d 2 r  n V 2 /(R- X vort ) 4 ~ (  n /  2 current ) V 2 Bardeen-Stephen Result!! Note!! Use current-defined core size

48. Z->0: U(1) RVB Gauge theory L B Ioffe and A I Larkin PRB39 8988 ’89 L B. Ioffe and AJM PRB66 094513 ‘02 Separate electron into charge e ‘holon’ b charge 0 ‘spinon’ f Assume ‘d-RVB’ pairing of spinons Introduce ‘internal gauge field ‘a’ to project out unphysical degrees of freedom ‘spinon’ velocity v 1 set by superexchange J constant as x->0

49. H in superconducting state 3 terms + 1 constraint Boson stiffness  B (r) length scale x -1/2 size x Spinon stiffness  F (r) length scale v 1 /  size p F v 1 Mixing term: spinons feel a which couples to boson Constraint: boson, spinon currents equal, opposite

50. Long distance theory eliminate a Physical superfluid stiffness ~x Z ~x (q.p. -> spinons) Main unphysical feature Structure of H phase => conventional 2  0 vortices Ioffe/Larkin PRB ‘98; Han & Lee PRL 85 1100 (‘00) Physical s.c. phase 

51. Structure of vortex: Length scale associated with current: ~x -1/2 (see also Franz/ Tesanovic PRB63 064515 ’01) Short length scale boson physics solve for boson amplitude  (r) and gauge field a(r)

52. Vortex Equation of Motion j Viscosity:  = (  /  2 current ) vortex density n vort   t X vort =(z x j) E vort =n vort j/  Current j E tot =(1/  n +n vort /  )j = n vort  2 current (1+ n vort  2 current ) j nn

53. Big, Fast Vortices Large  2 current (big vortex) => small  (fast vortex) Fast vortex => no s.c. contribution to conductivity Current-defined cores overlap:  --normal Quasiparticle-defined cores overlap: DOS-normal

54. Nernst Effect Wang PRL 88 257003 ‘02 T > H cII (T) (melt vortex lattice): resistivity quickly reverts to normal state value; Nernst coeff e y (  V  /  T) more slowly TT Idea: vortex has entropy =>thermal gradient  T drives vortices =>  V 

55. Nernst Effect TT Idea: vortex has entropy =>thermal gradient  T drives vortices =>  V  Speculation: vortex entropy associated with quasiparticles, hence with quasiparticle-defined core Vortex conductivity associated with current-defined core => Naturally e y (T),  (T) differ Subtleties: cf I Ussiskin et. al.

56. Summary II: Vortex statics Generically expect: current-defined vortex core size diverges as Mott insulator phase is approached. (Lee & Wen PRL 78 4111 97; PRB64 224517 ’00) ???x -1/2 or x -1 ??? ??Fermionic or bosonic?? Diverging size: implications for dissipation; paraconductivity; Nernst Experimental evidence??

57. T c2 (H) (resistive) is thermal melting of vortex lattice. H Raise T at fixed H

58. Simple argument Abrikosov lattice, spacing b Intervortex force K~  S Position fluctuations ~T/K ‘Lindemann criterion’ =c L 2 b 2 c L 2 = 0.01—0.1 b

59. Blatter: T c2 (H)   S (H, T c2 )/14 H Here T<H so use  S (H, 0) Low T resistive critical field line maps out H- dependence of  S (A~1.5-1.7)

60. Quantal Melting of Vortex Lattice ‘Semiquantum’ argument: Abrikosov lattice, T=0 spacing b Position fluctuations ‘Lindemann criterion’ =c L 2 b 2 c L 2 = 0.01—0.1 b

61. Need quantum fluctuations of vortices vortex dynamics Add to previous action the term i  n  t  Treat dissipation properly

62. Vortex Equation of Motion: 2 terms Dissipation: coupling of vortex to spinon continuum Vortex hall effect—from term i  t  in superfluid action (cf Geshkenbein, Ioffe Larkin, PR55 3173 ’97)

63. Vortex Hall Effect Superfluid action, condensate density n S superfluid = i  n  t  + ….  Move vortex around circle of area A phase of point inside winds by  Action increases by   An =>vortex feels field B eff =   n =>Measure  by vortex Hall effect

64. Value of  i  n c  t   acceleration of vortex Galilean invariance, T=0—accelerating vortex drags all particles with it =>n c =n total  =1 Doped Mott insulator. T=0: effective Galilean invariance at low energy => n c =x  =1 Conventional sc, near Tc: ‘2 fluid’:  ~(T c /E F ) RVB: E F  J and T c set by phase fluct=> use  not T c. D-wave nodes=>quasiparticles even at low T (high B) suggests  /J except perhaps as T->0, B->0 Crossover not well understood Data (Ong):  0)

65. Estimate of fluctuations Equation of motion Force—from other vortices in lattice. Proportional to superfluid stiffness  S Constant K 0 <<1 F V =K X V K=K 0 n V  S 2 limits: “fermion”—dissipative (  ) term dominates “boson”—nondissipative (  ) term dominates

66. Quantal Melting of Flux Lattice: Boson Limit Boson limit (x->0;  - >0) Vortices: dissipationless particles in high B eff.  Wigner crystal to ‘vortex FQHE’ transition c L 2 = 0.01 => n V  0.1 (B eff /  0 ) <<x =>melting when n v  2 current  0.1

67. Quantal Melting of Flux Lattice: Fermion Limit =(1/   ) ln(1+  /K) Note crucial role of dissipation Lindemann estimate ~ QQ  H2H2 00 F(H  2 /  0 ) F: function describing decrease in dissipation for r<  Expect F(y) ~ y a with a>1 => lattice melts when current-defined cores touch

68. Nature of melted phase?? ‘Boson’ limit: QHE of vortices=>insulator (duality) Fermion limit: ‘soup’ of overdamped vortices. Likely to be almost classical (quantum diffusion for overdamped vortices: R~ln[t] so it takes ridiculously long for them to entangle). Note also: quantal transition is first order!!

69. Summary: Melting Key length—current- defined core size Key parameter: non- dissipative coefficient  Key question: what is a ‘quantum-melted vortex lattice’ Gauge theory—unphysical features—but ‘existence proof’ that a theory exists with these features First order Second order H T

70. Limits of Long Wavelength Action We wrote: S=  S0 Q 2 +fermion terms Theory breaks down when corrections to  S0 become of same order as  S0. Two kinds of corrections Q 4 (higher order gradients  short range ‘bosonic’ physics Fermionic excitations

71. Thus S =>  S0 Q 2 +fermion terms + E Q Q 4 +… Bosonic corrections important when Q 2 ~  S0 / E Q or T~  S0 Fermionic corrections important when ??Which happens first?? Data suggest fermionic physics more important

72. Conclusions I Simple parametrization of low-T d-superconductor : v F, v ,  S, Z Experiment: Z~1 (problem for gauge th.) current is only carried by near-zone-diagonal states (angle-dependent spin-charge separation??) Data not (yet) fully consistent Δ θ Current only from these

73. Conclusions II Vortex size as defined from supercurrent crucial for transport, H cII (T=0) Explanation for Nernst vs resistivity Quantal melting of vortex lattice Vortex hall coefficient  ; charge renormalization Z--boson vs fermion physics j(x) Δ(x)/Δ(  )

74. Dissipation

75. Current-defined core radius

76. Dissipation (Effective charge) 2 Current-defined core radius

77. Dissipation (Effective charge) 2 Current-defined core radius Factors from detailed calculation

78. Dissipation (Effective charge) 2 Current-defined core radius Factors from detailed calculation Key point—dissipation ~  -2

79. Value of  : II => interpolation formula: higher doping, smaller  : fermion physics Lower doping, larger  : boson physics Experiment (vortex hall angle):   small in all high-Tc=> no ‘boson physics’ visible?? ??at very small doping, low T, clean samples??

80. Phase diagram—II: ‘Pseudogap’ M. Suzuki, PRL 85 4787 (2000) (tunnelling— similar observations in photoemission) ‘ Temperature Doping ‘Pseudogap’—gap Δ PG in electronic DOS at T>T c Note T-scale not too high

81. ‘Pseudogap’ vs Superconducting Gap T Doping V. M.Krasnov et al PRL 86 2657 (2001) ‘ Superconducting gap—feature Δ SC appearing inside pseudogap Δ PG Δ SC K Lang et al Nature p. 412 v. 425 (2001) Inhomogeneity in SC gap

82.  indep of x

83. Vortex properties (assume 2d system for rest of talk) Length scale in ρ(r) must diverge as approach Mott phase (Lee & Wen PRL 78 4111 97; PRB64 224517 ’00)  Scale over which supercurrent can vary must diverge as doping x->0  Implication for vortex core size

84. Current carried by states far from nodes? Possible test: ρ S (T?) T-dependence not useful: transition is driven by thermal phase fluctuations when ρ S (T c )=2 T c /π (Uemura; Emery/Kivelson) ‘KT line’ ρ S (T)=2T/π for bilayer system But: quantal flucts weaker than thermal flucts….

85. Possible better test : ρ S (B,T=0) For a 2d vortex lattice, if Z is constant (A=1.5… for square or triangle lattice) ??? does this formula predict H cII (T=0) in underdoped If H cII (T=0) is larger than this formula—angle-dependent Z? ρSρS B 1/2 ???At what field does vortex lattice quantum melt???

86. The Paradox: Corson et. al. cond-mat/9810280 Superfluid stiffness v. temp Vortices proliferate: whole sample is ‘vortex core’ for Tc<T<<T* d-wave like gap s-c transition ‘K-T’-like ‘Pseudogap’: =>’Pseudogap==(phase) fluctuating s.c. IF SAMPLE IS ‘ALL CORE’: WHY DOES GAP PERSIST TO HIGH T?

87. Quantal Melting of Flux Lattice: Fermion Limit Fermion limit (  important) Ratio: gap to (small # times) superfluid stiffness. Very small in conventional SC; order 1 in high-T c

88. Quantal Melting of Flux Lattice: Fermion Limit Fermion limit (  important) Ratio: gap to (small # times) superfluid stiffness. Very small in conventional SC; order 1 in high-T c Ratio: vortex spacing to current-defined vortex size

89. Quantal Melting of Flux Lattice: Fermion Limit Fermion limit (  important) Ratio: gap to (small # times) superfluid stiffness. Very small in conventional SC; order 1 in high-T c Ratio: vortex spacing to current-defined vortex size Conventional SC: cores touch when lattice melts High-T c : melting when current-defined cores touch