1. 高温超導的量子磁通状態和相変 胡 暁 計算材料科学研究中心 物質・材料研究機構、日本筑波    hc/2e |||| B JsJs  Vortex states and phase transitions in high-Tc superconductivity Xiao Hu National Institute for Materials Science, Tsukuba, Japan

2. Summary Outline Introduction Melting of flux line lattice in HTSC ♣ B || ab plane ♣ impacts of point defects ♣ B || c axis ab plane c axis

3. GL theory for superconductivity Superconductivity order parameter:  e i  GL free energy functional:  =-  ’(1-T/T c ) Two length scales : (i)Correlation length of SC order parameter:  ~1/√(1-T/T c ) (ii) Penetration depth of magnetic field: ~1/√(1-T/T c ) GL number:  (i)  <<1(ii)  >1(iii)  >>1

4. MF phase diagram: two 2 nd order trs. |||| B JsJs  Meissner phase Mixed phase Normal phase H T H c2 H c1 ♣ H c1 =ln  0 /4  2 ♣ H c2 =  0 /2  2 Flux quantization:    hc/2e ♣  =(  0 /4  ) 2 lnk ♣ V(r)=2(  0 /4  ) 2 ln( /r) Broken symmetry: (i) U(1) gauge symmetry (ii) translational symmetry Self energy and repulsion of vortices: penetration of flux Type-II SC :  >1/√2 Vortex & flux quantum in type-II SC

5. Extremely type-II SC: >>  Layer structure & high anisotropy: Importance of thermal fluctuations Experimental observations pancake vortex ab plane c axis H.Safar et al. 1993 A.Schilling et al. 1997E.Zeldov et al. 1995 Vortex states in HTSC

6. (i)Elastic theory for flux lines: C 11 & C 44 & C 66 (ii) Renormalization group:  -expansions Lack of a good theory for 1 st order transitions! + Lindemann criterion for melting physical but phenomenological  =6-D=3>>1 difficult to control the RG flows Theoretical approaches

7. where defined on 3-dim grids of simple cubic lattice: unit length d Derivable from Ginzburg-Landau Lawrence-Doniach model degrees of freedom: A ◆◆◆  Superconductivity order parameter:  e i  3D anisotropic, frustrated XY model

8. Extremely type-II superconductors:  d v  d v ~d  √ f Vortex as topological singularity of phases:  B |||| dvdv d Flux line Magnetic induction: B || c axis f=Bd 2 /  0 =1/25 A=-r×B/2=(-yB/2,xB/2,0) Vortex and flux line

9. System size: L xy =50, L z =40 Boundary condition: periodic in 3 directions Typical process of MC simulations : ◆ generate a random configuration of phase variables at a high T ◆ cool system according to the Metropolis scheme ♣ search lattice structure ♣ measure T dependence of quantities MC simulation steps: ◆ equilibriution: 50,000 sweeps ◆ measurement: 100,000 sweeps ◆ around the transition point: ~ 10 7 sweeps Anisotropy:  2 =10 Temperature skips: ◆ ◆ around T m :  T=0.1J/k B  T=0.001J/k B Monte Carlo simulations

10. To achieve equilibrium in reasonable time: To simulate the fluctuations sufficiently: A big system A small system Slow annealing Quick annealing We know the spatial and temporal scales only after the phenomenon is understood. A serious trade off! We don ’ t even know if the Hamiltonian is sufficient! Can call them approximations? We deal with Hamiltonian. Good computer! Good luck! Good physics! Simulation: theory or experiment?

11. Finite size scaling: C + ≈ 18.5k B C - ≈ 17.5k B C max ≈ 23k B Q ≈ 0.07k B T m  T ≈ 0.008T m First order thermodynamic phase transition

12. Note |  | is finite even for T>T m U(1) gauge symmetry is broken at T=T m Phase stiffness & conductance Normal to superconductivity transition

13. Eliminate possibilities of : disentangled flux-line liquid; supersolid Translational symmetry is also broken at T m. T<T m T>T m Melting of flux line lattice

14. T~0 Abrikosov flux line lattice Flux line liquid ~ spaghetti T≥T m Real space snapshot

15. Mechanism of melting: FL entanglement

16. Clausuis-Clapeyron relation: Lindemann number: c L =0.18 Melting line: B liquid >B solid Same as water! Length scale:  d Competition: Elastic energy Thermal fluc. B-T phase diagram: melting line

17. B=8T T m [K] T m [K]  S[k B /vortex]  B[G] simulation 81 81 0.55 0.55 0.19 0.19 experiment 79 79 0.4 0.4 0.25 0.25 YBCO: d=12 Å, ab (0)=1000 Å,  =8, T c =92K,  =100 by Schilling et al. BSCCO: d=15 Å, ab (0)=2000 Å,  =150, T c =90K,  =100 by Zeldov et al.B=160G T m [K] T m [K]  S[k B /vortex]  B[G] simulation 65 65 0.22 0.22 0.12 0.12 experiment 65 65 0.4 0.4 0.35 0.35 Ref. XH, S. Miyashita & M. Tachiki, PRL 79, p.3498 (1997); PRB 58, p.3438 (1998) Comparison with experiments

18. Q Phase transition under B || ab plane ? cf. B ||c axis: 2-dim symmetry in ab plane 1 st order melting For H=0 (1)  <∞: 2 nd order transition in 3D XY universality class (2)  =∞: KT transitions in decoupled layers translational symmetry along c axis is broken a priori Difficulty in experiments: high anisotropy requiring very accurate alignment of magnetic field with CuO 2 layers Intrinsic pinning of CuO 2 layer to Josephson vortices Phase transition for B || ab plane

19. Korshunov & Larkin, 1992Mikheev & Kolomeisky, 1991 There should be no decoupling, provided that Josephson vortices are confined by CuO 2 layers for T ≤ T d. Decoupling transition: T d SC transition: T c T c >T d r Hopping of Josephson flux line via a pancake pair Binding & unbinding of pancake pairs KT transition Blatter et al. 1991 Liquid → Smectic → Solid two-step freezing Balents & Nelson, 1994 Two 2 nd order transitions Theories

20. Magnetic field: B || y axis x c System size: # of flux lines = 240 Periodic boundary conditions Anisotropy:  =8 A=(0,0,-xB) f=Bd 2 /  0 =1/32 L x *L y *L z =384d*200d*20d Details of Monte Carlo simulation

21. ◆ ◆ Response to applied current I FpFp FLFL non-Ohmic resistivity I FLFL Ohmic resistivity f=1/32  =8  x ~  y  c =0 intrinsic pinning cf. universal jump of helicity modulus at KT transition:  k B T KT =2/  1 st order phase transition

22. k c =  /10dk x =  /192d Structure of Josephson vortex lattice

23. k c =  /10dk x =  /192d １ 2 Melting of Josephson vortex lattice

24. 2 nd order melting for large anisotropy Tricritical point:  tc =9~10 @ f=1/32 ♣ f=1/25, 1/36  tc increases as f decreases f=1/32 Other parameters: ♣  =7,6,…1 for f=1/32 1 st order melting

25. Mechanism of the tricritical point Tricritical Point! Invariant unit cell for  >8 at f=1/32 Balance of inter-vortex repulsions (2d) 2 =d 2 +(d/2f  ) 2 For  >  tc, fluctuations along c axis are essentially suppressed by layers 1 2 3 MF theory for flux line lattice melting: 3 rd order terms exist for  <  tc Numerically,  tc =16/√3≈ 9.24 @ f=1/32 Simulations give 9<  tc <10 1 st order melting, as B|| c axis 2 nd order melting suppressed for  >  tc

26. Thermal excitations Ratio of collisions and hoppings Observation of hopping of Josephson flux lines at T<T m ◆ excitation energy of Josephson vortices is small ~ J/γ 2 ◆ T m is high ~J/k B J J J/  2 cf. B || c axis Hopping of Josephson flux lines

27. No 2 nd order phase transition, provided no hopping T c <T KT ~0.89J/k B KT transition: Liquid → Smectic → Solid: two 2 nd order transitions 2 nd order melting: modified MF theory Balents & Razihovsky Simulation: Hopping is observed at T<T m Simulation: T m >T KT Simulation: 2 nd order tr. is observed only when every block layer is occupied, and is single Theories revisited

28. Tricritical magnetic induction Ref. XH & M.Tachiki, PRL 85, p.2577 (2000) 1 st order melting 2 nd order melting Tricritical point T KT T B TcTc T KT ~0.89J/k B  =8 d=12 Å B tc ≒ 50 Tesla YBCO BSCCO B tc ≒ 1.7 Tesla  =150 d=15 Å B-T phase diagram for B||ab plane

29. 1 st order melting 2 nd order melting Tricritical point T KT T B TcTc T KT ~0.89J/k B 2 nd order melting Competition: elastic vs. thermal commensuration effect Pinning effect of CuO 2 layers suppress c-axis fluctuations length scales:  d & d Small B: B m ~ (k B T m /J) -2 ×  0 /  d 2 B m,ab /B m,c ~  tricritical point Large B: T m →T KT decoupled limit B-T phase diagram for B||ab plane

30. H p =- ∫ d D rV(r)  (r) H el =c ∑ ij u i u j u i =x i -R i 0  (r)= ∑ i  (r-R i 0 -u i ) u(R a )-u(0)~a: lattice spacing a  el =c(a/R a ) 2 R a D =cR a D-2 a 2  p =-VR a D/2  0 R a ~a[c 2 a D /(V  0 ) 2 ] 1/(4-D) Arbitrarily weak disorders destroy lattice order for D<4! “ Linearized ” Larkin model B(r) ≡ ‹ [u(r)-u(0)] 2 › ~(r/R a ) 4-D C(r)~exp(-B)~exp[-(r/R a ) 4-D ] Larkin length Similar arguments in other systems such as CDW etc. Impacts of point defects: Larkin theory

31. RaRa Dislocation Gauge glass: H= ∑ ij cos(  i -  j -A ij ) No positional order in D=3 liquid Vortex glass T H Physics: for a lattice of flux lines, one line does not have to make displacement much larger than a to pass through a particularly favorable region of disorders, because of periodicity. cf. a single flux line Asymptotic Alnr B r RaRa RcRc Larkin Random manifold r 2 C(r)~1/r  Quasi LRO! Impacts of point defects

32. Q How many phases? Characters of phase transitions? How to understand them in a unified scheme? Bragg glass B T vortex liquid 1 st order Q: phase? Q: phase transition? Q: 1 st order? Bragg glass quasi long-range correlation free of dislocations Bragg glass melting thermal fluctuations intensity of pins Competitions: elasticthermalpinning Effects of point pins: B||c axis

33. interaction J interaction (1-  )J with probability p Details of our approach filling factor: f=1/25 anisotropy parameter:  =20 system size: L xy =50, L z =40 s.c. lattice & p.b.c. density of point defects: p=0.003 MC sweeps: equilibriution: 4~8*10 7 measurement: 2~4*10 7 Model with point defects

34. Ref. Y. Nonomura & XH: PRL 86, p.5140 (2001)  -T phase diagram

35. Same as the melting of Abrikosov lattice Thermal melting of Bragg glass

36. Structure factors Bragg glass Vortex liquid Bragg glass: as perfect as a lattice Global minimumEnergy landscapeDynamics

37. ● 1 st order phase transition cancellation ◆◆ pinning energyelastic energy ● phase boundary almost parallel to T axis e[J] Defect-induced melting of Bragg glass

38. ● 1 st order phase transition at T sl ● SC achieved only at T=T g (<T sl ) ◆ 2 nd order like glass transition ● trace of thermal melting of Bragg glass Liquid to slush transition

39. ● sharp jump in the density of dislocations at T sl Ref. Kierfeld & Vinokur, 2000 Liquid to slush transition

40. ● Bouquet et al. Nature 411, p.448 (2001) ● a step-like anomaly in C left ● no vortex loop blowout ●  -function peak in C ● same in jump of disl. density suppressed above  ≈ 0.15 critical endpoint! Ref. Kierfeld & Vinokur, 2000 high fields Trace of BrG melting Crossover?! Like liquid-gas line of water! Point pins create attractive force! Critical endpoint and beyond

41. Summary Melting of flux line lattice in HTSC Computer simulations B || c axis: 1 st order, FLL to entangled liquid B || ab plane: tricritical point Impacts of point defects under B || c axis Belong to the category of theory Try to break the frontier! Try to go beyond theory! New concept New paradigm