高温超導的量子磁通状態和相変

1. f0=hc/2e x |y| l B Js 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 Vortex states and phase transitions in high-Tc superconductivity Xiao Hu National Institute for Materials Science, Tsukuba, Japan

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

3. a=-a’(1-T/Tc) GL free energy functional: Two length scales: • Correlation length of SC order parameter:x~1/√(1-T/Tc) (ii)Penetration depth of magnetic field:l~1/√(1-T/Tc) GL theory for superconductivity Superconductivity order parameter:Y=|Y|eij GL number:k=l/x (i) k<<1 (ii) k >1 (iii) k>>1

4. Vortex& flux quantum in type-II SC Type-II SC : k=l/x>1/√2 penetration of flux Flux quantization: f0=hc/2e H Normal phase Mixed phase x Hc2 |y| l B Meissnerphase Hc1 Js T Self energy and repulsion of vortices: ♣e=(f0/4pl)2lnk ♣V(r)=2(f0/4pl)2ln(l/r) MF phase diagram: two 2nd order trs. ♣Hc1=lnkf0/4pl2 ♣Hc2=f0/2px2 Broken symmetry: (i) U(1) gauge symmetry (ii) translational symmetry

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

6. Theoretical approaches • Elastic theory for flux lines: • C11&C44 & C66 + Lindemann criterion for melting physical but phenomenological (ii) Renormalization group: e-expansions e=6-D=3>>1 difficult to control the RG flows Lack of a good theory for 1st order transitions!

7. Superconductivity order parameter: Y=|Y|eij where Derivable from Ginzburg-Landau Lawrence-Doniach model j |y| A degrees of freedom: ◆ ◆ ◆ 3D anisotropic, frustrated XY model defined on 3-dim grids of simple cubic lattice: unit length d

8. Magnetic induction: B || caxis f=Bd2/f0=1/25 A=-r×B/2=(-yB/2,xB/2,0) Vortex as topological singularity of phases: Flux line d |y| l B x dv Vortex and flux line Extremely type-II superconductors: l>>dv>>x , dv~d/√f

9. Temperature skips:◆ ◆aroundTm: DT=0.1J/kB DT=0.001J/kB Monte Carlo simulations Boundary condition: periodic in 3 directions System size:Lxy=50, Lz=40 Anisotropy:g2=10 Typical process of MC simulations: ◆generate a random configuration of phase variables at a highT ◆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: ~ 107 sweeps

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

11. C+≈18.5kB C-≈17.5kB Cmax≈23kB Q≈0.07kBTm DT≈0.008Tm First order thermodynamic phase transition Finite size scaling:

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

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

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

15. Mechanism of melting: FL entanglement

16. Melting line: B-T phase diagram: melting line Lindemann number:cL=0.18 Clausuis-Clapeyron relation: Bliquid>Bsolid Same as water! Competition: Elastic energy Thermal fluc. Length scale: gd

17. Comparison with experiments YBCO:d=12Å, lab(0)=1000Å, g=8, Tc=92K, k=100 by Schilling et al. BSCCO:d=15Å, lab(0)=2000Å, g=150, Tc=90K, k=100 by Zeldov et al. Ref. XH, S. Miyashita & M. Tachiki, PRL 79, p.3498 (1997); PRB 58, p.3438 (1998)

18. Q Phase transition under B || ab plane ? For H=0(1)g<∞: 2nd order transition in 3D XY universalityclass (2)g=∞: KT transitions in decoupled layers Phase transition for B || ab plane Intrinsic pinning of CuO2 layer to Josephson vortices translational symmetry along c axis is broken a priori cf. B ||c axis: 2-dim symmetry in ab plane 1st order melting Difficulty in experiments: high anisotropy requiring very accurate alignment of magnetic field with CuO2 layers

19. Decoupling transition: Td SC transition: Tc Mikheev & Kolomeisky, 1991 Korshunov & Larkin, 1992 KT transition Blatter et al. 1991 r Liquid → Smectic → Solid two-step freezing Balents & Nelson, 1994 Theories Tc>Td There should be no decoupling, provided that Josephson vortices are confined by CuO2 layers for T≤Td. Hopping of Josephson flux line via a pancake pair Binding & unbinding of pancake pairs Two 2nd order transitions

20. Details of Monte Carlo simulation c x Magnetic field: B|| y axis A=(0,0,-xB) f=Bd2/f0=1/32 Anisotropy:g=8 System size: Lx*Ly*Lz=384d*200d*20d # of flux lines = 240 Periodic boundary conditions

21. 1st order phase transition intrinsic pinning Fp I Response to applied current FL I Ohmic resistivity FL f=1/32 g=8 ◆ ◆ Ux~UyUc=0 non-Ohmic resistivity cf. universal jump of helicity modulus at KT transition: U/kBTKT=2/p

22. kc=p/10d kx=p/192d Structure of Josephson vortex lattice

23. kc=p/10d kx=p/192d Melting of Josephson vortex lattice 2 １

24. Tricritical point:gtc=9~10@f=1/32 2nd order melting for large anisotropy f=1/32 Other parameters: ♣ g=7,6,…1forf=1/32 1st order melting gtcincreases asfdecreases ♣f=1/25, 1/36

25. MF theory for flux line lattice melting: 3 1 2 1st order melting, as B|| c axis 2nd order melting Simulations give9<gtc<10 Mechanism of the tricritical point Invariant unit cell forg>8atf=1/32 Balance of inter-vortex repulsions (2d)2=d2+(d/2fg)2 Tricritical Point! Forg>gtc, fluctuations along c axis are essentially suppressed by layers 3rd order terms exist forg<gtc suppressed forg>gtc Numerically,gtc=16/√3≈ 9.24 @ f=1/32

26. Ratio of collisions and hoppings J J/g2 J/g2 J Hopping of Josephson flux lines Thermal excitations ◆Tmis high ~J/kB ◆excitation energy of Josephson vortices is small ~ J/γ2 cf. B || c axis Observation of hopping of Josephson flux lines atT<Tm

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

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

29. B 2nd order melting Tricritical point 1st order melting T TKT Tc TKT~0.89J/kB B-T phase diagram for B||ab plane Competition: elastic vs. thermal Pinning effect of CuO2 layers commensuration effect suppress c-axis fluctuations 2nd order melting tricritical point Small B: Bm~ (kBTm/J)-2× f0/gd2 length scales: gd & d Bm,ab/Bm,c~g Large B:Tm→TKT decoupled limit

30. Impacts of point defects: Larkin theory Hel=c∑ijuiuj Hp=-∫dDrV(r)r(r) ui=xi-Ri0 r(r)=∑id(r-Ri0-ui) u(Ra)-u(0)~a: lattice spacing a ep=-VRaD/2r0 eel=c(a/Ra)2RaD=cRaD-2a2 Ra~a[c2aD/(Vr0)2]1/(4-D) Larkin length Arbitrarily weak disorders destroy lattice order for D<4! Similar arguments in other systems such as CDW etc. “Linearized” Larkin model B(r)≡‹[u(r)-u(0)]2›~(r/Ra)4-D C(r)~exp(-B)~exp[-(r/Ra)4-D]

31. B Random manifold Dislocation Ra Larkin Alnr r2n r Rc Ra H liquid Vortex glass T Impacts of point defects Asymptotic C(r)~1/rh Quasi LRO! No positional order in D=3 Gauge glass: H=∑ijcos(fi-fj-Aij) 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

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

33. interaction J Details of our approach filling factor:f=1/25 anisotropy parameter:g=20 density of point defects:p=0.003 interaction (1-e)J with probability p MC sweeps:equilibriution: 4~8*107 measurement: 2~4*107 system size: Lxy=50, Lz=40 s.c. lattice & p.b.c. Model with point defects

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

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

36. Bragg glass Vortex liquid Structure factors Bragg glass: as perfect as a lattice Global minimum Energy landscape Dynamics

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

38. ●SC achieved only atT=Tg (<Tsl) ◆ 2nd order like glass transition Liquid to slush transition ●1st order phase transition atTsl ●trace of thermal melting of Bragg glass

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

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

41. B || c axis: 1st 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! Summary Melting of flux line lattice in HTSC Computer simulations New concept New paradigm