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Dynamics of vortex matter in Type-II superconductors: Numerical simulations Qing-Hu Chen ( 陈 庆 虎 ) Center for Statistical and Theoretical Condensed Matter Physics, Zhejiang Normal University, Jinhua 321004, P. R. China & Department of Physics, Zhejiang University, Hangzhou 310027, P. R. China CSRC Workshop on Advanced Monte Carlo Methods and Stochastic Dynamics 25 June, 2011

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Outline □ Vortex glass phase transitions in type-II superconductors with strong disorders □ Dynamical melting in high-Tc superconductors with sparse and weak columnar defects □ Theoretical study of Nernst effect in high-Tc cuprate superconductors □ Ratchet effect in two-dimensional Josephson junction arrays

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Part 1: Vortex glass phase transitions in type-II superconductors with strong disorders

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The nature is unclear: Vortex glass ? Or vortex liquid ? B||c

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□ Experiments support this picture ◇ Koch et al., PRL 1989, 1990. ◇ Gammel et al., PRL 1991 ◇ Klein et al., PRB 1998 ◇ Petrean et al., PRL 2000 ◇ Villegas et al., PRB 71, 144522(2005) □ Vortex glass in disordered high-Tc SC in an external field True SC state, vanishing resistivity by diverging energy barriers ◇ Fisher, Phys. Rev. Lett. 62, 1415 (1989) ◇ Fisher, Fisher, and Huse, Phys. Rev. B 43, 130 (1991) □ Questioned ◇ Strachan et al., PRB 2006 ◇ Reichhardt et al., PRL 2000 Tameigai, PRB 2011 Iron-based SC

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Epitaxial films of YBCO in strong fields Koch et al, PRL 90 Fisher, Fisher, and Huse (FFH) Dynamic scaling

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Simulations of vortex glass model □ 3D gauge glass model Katzgraber & Campbell, PRB 69, 094413( 2004) Huse and Seung, PRB 42, R1059(1990) ◇ Strong evidence for a glass transitions ◇ as a model of disordered SC in an applied filed It lacks some of properties and symmetries No net fields ◇ scaling of equilibrium spin-glass susceptibility

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Simulations of overdamped London-Langevin model Reichhardt et al., PRL84, 1994 (2000) Bustingorry et al., PRL96, 027001(2006) ◇ Vortex-glass criticality is arrested at some crossover temperature, instead of transition temperature ◇ Vortex loop excitations is excluded in this model. It is a open question, whether it can adequately describe (non-elastic) vortex glass phase Natterman et al., Advances in Physics 49, 607(2000)

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Equilibrium simulations (unscreened limit) □ Anisotropic 3D XY model with frustration Net field is introduced by frustration Vortex loop is naturally included in XY model ◇ P. Olsson, PRL91, 077002(2003) J ij =J(1+Pε ij ), ε ij is Gaussian random variable with unit variance P=0.4, strong disorder g 2 =40, f=d 2 B/f 0 =1/5

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◇ strong evidence for Vortex glass phase transition Finite size scaling of helicity modulus PRL91, 077002(2003)

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◇ Kawamura, PRB 68, 220502(R) (2003) J ij =[0, 2J] strong disorder f=d 2 B/f 0 =1/4 □ Isotropic 3D XY model with frustration Net field is introduced by frustration Vortex loop is naturally included in XY model

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Finite size scaling of Binder ratio for overlap ◇ Another strong evidence for Vortex glass phase transition

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Failure to scale for helicity modulus for Isotropic system ◇ P. Olsson, PRB 72, 144525(2005) Poor quality of collapse Poor quality of crossing

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Convincing scaling collapse for helicity modulus could not be achieved in Isotropic model possibly due to the small effective randomness in the small system accessed. The dynamical study in the frustrated 3D XY model with and without net fields for strong disorder is so far lacking, which is however more relevant to experiments in the context of Vortex Glass transitions.

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supercurrent normal current voltage at site i □ The dynamical equation for the phase Resistivity-shunted-junction dynamics Current I Plaquette Modeling □ Hamiltonian: B || c axis

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Model I: Ani Model I: Ani sotropic □ IV characteristics 100x100x60, f=1/5 Same model and parameters in PRL91, 077002(2003) ◇ Perfect collapse □ Dynamical scaling

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Depinning transition: T=0 □ continuous depinning transition

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Creep and Scaling Analysis □ Ccritical force and exponent at □ Scaling function: From second-order phase transitions Fisher, 1985 Luo & Hu, 2007

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Scaling plot □ Scaling function: □ Creep law □ exponent non-Arrhenius

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64x64x64, f=1/4 Same model and parameters in PRB 68, 220502(R) (2003) Model II: Model II: Isotropic □ IV characteristics □ Dynamical scaling ◇ Perfect collapse

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Depinning transition: T=0 □ continuous depinning transition

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Creep and Scaling analysis □ Critical force and exponent at □ Scaling function: From second-order phase transitions

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Scaling plot □ Scaling function: □ Creep law □ exponent non-Arrhenius

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□ Luo and Hu: PRL, 2007 weak pinning non-Arrhenius creep Equilibrium state is Bragg glass Discussion and comparison □ Both models with different parameters and disorder realization: gives non-Arrhenius creep Equilibrium state is vortex glass strong pinning Arrhenius creep □ Remark: Luo & Hu, molecular dynamical simulation without vortex loops : Creep in Low Tc SC Anderson-Kim theory Present, 3D XY model with strong disorder: Vortex glass vortex loop excitations are included

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Summary For both model parameters □ Vortex glass transitions ◇ Strong evidence for finite temperature vortex glass transition ◇ Nearly perfect collapse of IV data in dynamic scaling. ◇ The transition temperatures, static exponents are in excellent agreement with previous equilibrium studies. The dynamic exponent is new and compatible with experiments □ Depinning and creep of vortex matter in the vortex glass state. ◇ A genuine continuous depinning transition at T=0. ◇ a non-Arrhenius creep motion ◇ contrary to recent molecular dynamical simulations for strong disorder Qing-Hu Chen, Phys. Rev. B 78 ， 104501 （ 2008 ）

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□ The Hamiltonian [Young’s group, PRL 90, 227203(2003).] J ij: zero mean and standard deviation unit The nature of d-wave symmetry will changes the sign of the coupling between XY spins, while the spin angle denotes the phase of the superconducting order parameters. Three-dimensional XY spin glass: Vortex glass in high-Tc superconductors with d-wave symmetry Qing-Hu Chen, Phys. Rev. B 80, 144420(2009)

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◇ Chiral glass correlation function Chirality □ Unfrustrated: Thermally activated chiralities (vortices) drive the Kosterlitz-Thouless-Berezinskii transition in the 2d XY ferromagnet. □ Frustrated: Chiralities are quenched in by the disorder at low-T because the ground state is non-collinear. Define chirality by: (Kawamura ， Phys. Rev. B 36, 7177(1987).) ◇ Spin glass correlation Function

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□ Most theory done for the Ising (S i =± 1) spin glass. Clear evidence for finite T SG. Best evidence: finite size scaling (FSS) of correlation length (Ballesteros et al. PRB 62, 14237 2000.) □ Many experiments closer to a vector spin glass S i. Theoretical situation is less clear: ◇ Old Monte Carlo: T SG, if any, seems very low, probably zero. ◇ Kawamura and Li, PRL 87, 187204(2001): T SG = 0 but transition in the “chiralities”, T CG > 0. This implies spin–chirality decoupling. ◇ But: possibility of finite T SG raised by various authors, e.g. Maucourt &Grempel, Akino & Kosterlitz, Granato, Matsubara et al. Nakamura and Endoh (non-equilibrium MC) proposed a single transition for spins and chiralities. ◇ Hence, situation confusing.

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L. W. Lee and A. P. Young, Phys. Rev. Lett. 90, 227203 2003. There seems a intensely competition for the lattice sizes accessible, the record until now is L=48 [ Phys. Rev. B 80, 024422 2009 ]

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□ IV characteristics 64x64x64 □ Finite Tg □ nonlinear resistivity YBCO Yamao et al.,, J. Phys. Soc. Jpn. 68, 871( 1999).

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64x64x64 ◇ Perfect collapse □ Dynamical scaling ◇ Strachan et al. (PRB 06): perfect collapse is not sufficient ◇ convexity-concavity criterion to identify the T g ◇ z, ν agree with exp: Koch et al.,, PRL. 63, 1511 1989; Klein et al, PRB 58,12411 1998; Petrean et al., PRL. 84, 5852 2000.

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Summary □ Vortex glass transitions ◇ Strong evidence for finite temperature vortex glass transition ◇ Nearly perfect collapse of IV data in dynamic scaling. ◇ The exponents are compatible with experiments □ the XY spin glass model may capture the essential transport feature in high-Tc cuprate superconductors with d-wave symmetry. The spin-chirality decoupling scenario in the XY spin glass H. Kawamura: PRL 102, 027202(2009) Yes! A.P. Young, PRB 78, 014419 (2008). No! □ Our results can be interpreted in terms of both the phase-coherence (spin-glass) transition and the chiral- glass transition. Qing-Hu Chen, Phys. Rev. B 80, 144420(2009) : No!

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Part 2. Dynamical Melting in High-Tc Superconductors with Sparse and Weak Columnar Defects

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Introduction □ In experiments, columnar defects are introduced to high-Tc superconductors by heavy-ion irradiation to increase the critical current. Civale et al., Phys. Rev. Lett. 67, 648(1991). □ A Bose glass (BG) phase: Tilt modulus diverges D. R. Nelson and V. M. Vinokur, Phys. Rev. B 48, 13 060 (1993). ◇ moving BG phase E. Olive et al., Phys. Rev. Lett. 91, 037005 (2003) □ Interstial liquid: some flux lines traped L Radzihovsky, Phys. Rev. Lett. 74, 4923 (1995) Banerjee et al., Phys. Rev. Lett. 90, 087004(2003); 93, 097002(2004) ◇ Moving phase has not been studied □ Bragg-Bose glass (BBG) phase with sparse and weak columnar defects Y. Nonomura and X. Hu, Europhys. Lett. 65, 533(2004) ◇ However, the nature of this phase driven by external current is not clear.

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Columnar Defects (CD) Parameters Density of CD (1) p=1/250 p/f=0.08 (2) p=1/25 p/f=0.8 System size 40 x 40 x 40 Phase diagram PRL 91, 037005

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I=0.5, T=0.16 □ real-space distribution □ structure factor Moving Bragg-Bose glass current direction Vortex moving direction The red circle is vortex; the blue square is CD (1) Results for p=1/250

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Profiles of Bragg peaks Transverse direction X Motion direction Y □ QLRO in transverse direction □ “LRO” in motion direction Ix 0 =5, iy 0 =8

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1 st order dynamical phase transition □ T m ≈ 0.173J/k B p=1/250 □ structure factor at T=0.18 Moving smectic or liquid

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I=0.5, T=0.16 (2) Results for p=1/25 □ structure factor Moving Bose glass □ real-space distribution

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I=0.5, T=0.12 □ real-space distribution □ structure factor absence of 6 Bragg peaks Moving Bose glass □ In the Moving Bose glass at T=0.12 and T=0.16, CD Pinning in transverse direction is not effective. But along the moving direction, it plays important role, resulting in more topological defects and suppress the correlation along the moving direction. moving

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Profiles of Bragg peaks Transverse direction X Motion direction Y Ix 0 =11, iy 0 =0

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Profiles of Bragg peaks Transverse direction X Motion direction Y Ix 0 =5, iy 0 =8

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I=0.5, T=0.18 □ real-space distribution □ structure factor Moving smectic or liquid

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□ I=0.5 1 st order dynamical phase transition □ T m ≈ 0.162J/k B p=1/25 □ T

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Mechanism of dynamic melting □ The density of dislocations in Moving Bose glass phase is higher than that in Moving Bragg glass phase The CD pinning along the moving direction is more effective with the increase of CD density, resulting in more dislocations.

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Summary □ At low temperature and low density of CD, a moving ordered phase with hexagonal Bragg peaks has been found. With increases of temperature, a moving smectic appears via a first-order phase transition. □ In Moving Bose glass phase, although 6 hexagonal Bragg peaks has been not found, the superconducting coherence along c axis remains. It can also decay to a moving smectic or liquid with increase of temperature.

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Part 3. Theoretical study of Nernst effect in high-Tc superconductors Nernst effect in two-dimensional Josephson junction arrays: Modeling the vortex Nernst effect in high-Tc superconductors ?

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Nernst effect in high-Tc superconductors in experiments N. P. Ong, Yayu Wang, Z. A. Xu, Princeton University Nernst coefficient □ Anomalous large Nernst signal e N extending from below T c0 to above T c0, in LSCO, Bi-2201, YBCO HOTCOLD

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Z. A. Xu et al., Nature(London) 406, 486(2000) Y. Y. Wang, L. Li, and N. P. Ong, Phys. Rev. B 73, 024510(2006). Optimally doped & Underdoped Bi-2201

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□ The dynamical equation for the phase: TDGL dynamics Modeling □ 2D Josephson junction arrays (QHC, Tang, Tong, PRL 2001) □ Open BC are taken along x direction: temperature gradient in x direction vortex moving along x direction. Fluctuating twist BC along y axis f = a 2 B/Φ 0, filling factor; B= ∇ XA

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□ qualitatively consistent with the experiment by Ong’s group even in 2D model (f<1/4) □ The excitations of vortex and anti-vortex in 2D is the possible origin of the anomalous Nernst effect. N. P. Ong’s group, PRB 73, 024510(2006). Our results Qing-Hu Chen, arXiv:0811.4145 f=0, T KT =0.8904

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Nernst effect Our numerical data L=32 L=64 Andersson et al, PRB 81, 060508 (2010) L=16 Qualitatively different! Qing-Hu Chen, arXiv:0811.4145 Only f<0.4

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Anomalous negative Nernst effect at f close to but smaller than 1/2 vacancy The vacancy moving to the low T regime = Vortex moving to the high T regime a sign reversal of the Nernst signal! Only exist in square lattice JJ arrays A1-A3: general events A4-A6: rare events

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□ Consistent with the experimental observations in some high-Tc cuprate superconductors (e.g. Bi2201) qualitatively. □ The excitations of vortex and anti-vortex in 2D is the possible origin of the anomalous Nernst effect. □ The inter-layer Josephson coupling in BSCCO only renormalizes the inplane coupling strength and may not play essential role in this phase-disordering scenario. □ Detailed mechanism for anomalous negative Nernst effect near f=1/2 and f=1/3 in JJ arrays is proposedSummary Qing-Hu Chen, Wei Zhou, Fei Qi, new version for arXiv:0811.4145 is in preparation

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Part 4. Ratchet effect in two-dimensional Josephson junction arrays In collaboration with Qing-Miao Nie (ZUT) Wei Zhou, Fei Qi

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Marconi, PRL 98 047006(07) Rectification Effect Current Reverse D. E. Shalom et al, PRL 94, 177001 (2005). JJAs

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Ratchet effect for DC currents Two typical values : f=8/128 21/128 Reverse System size: 128 X 128

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The vortex distribution at f=8/128 v + > v - +x -x Average vortex number

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The vortex distribution at f=21/128 v + < v -

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AC vs DC Ratchet Effect AC DC DC Without Ratchet DC With Ratchet Current Reverse Still ongoing

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Ratchet VxVx VyVy input signal V in =V x output signal V out =V y0 -V y V y0 is the voltage in the y direction at I x = 0. Amplification Rate A v =10~100 Diode, triode ?

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Thank you very much for your attention!

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