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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

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

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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

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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

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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

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(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

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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

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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

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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

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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?

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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

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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

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

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

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Mechanism of melting: FL entanglement

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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

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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

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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

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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

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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

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◆ ◆ 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

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

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

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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

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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

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Thermal excitations Ratio of collisions and hoppings Observation of hopping of Josephson flux lines at T

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No 2 nd order phase transition, provided no hopping T c

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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

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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

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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

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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

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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

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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

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

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

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

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

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● 1 st order phase transition at T sl ● SC achieved only at T=T g (

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

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● 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

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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

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