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Dynamics of vortex matter in Type-II superconductors: Numerical simulations

CSRC Workshop on Advanced Monte Carlo Methods and Stochastic Dynamics 25 June, 2011. Dynamics of vortex matter in Type-II superconductors: Numerical simulations. Qing-Hu Chen ( 陈 庆 虎 )

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Dynamics of vortex matter in Type-II superconductors: Numerical simulations

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  1. CSRC Workshop on Advanced Monte Carlo Methods and Stochastic Dynamics 25 June, 2011 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

  2. 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

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

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

  5. □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) □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) Tameigai, PRB 2011 Iron-based SC □Questioned ◇Strachan et al., PRB 2006 ◇Reichhardt et al., PRL 2000

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

  7. Simulations of vortex glass model □3D gauge glass model ◇ scaling of equilibrium spin-glass susceptibility 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

  8. 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)

  9. 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) Jij=J(1+Pεij), εij is Gaussian random variable with unit variance P=0.4 , strong disorder g2=40, f=d2B/f0=1/5

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

  11. □Isotropic 3D XY model with frustration Net field is introduced by frustration Vortex loop is naturally included in XY model ◇ Kawamura, PRB 68, 220502(R) (2003) Jij=[0, 2J] strong disorder f=d2B/f0=1/4

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

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

  14. 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.

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

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

  17. Depinning transition: T=0 □continuous depinning transition

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

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

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

  21. Depinning transition: T=0 □continuous depinning transition

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

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

  24. Discussion and comparison □Both models with different parameters and disorder realization: gives non-Arrhenius creep Equilibrium state is vortex glass □Luo and Hu: PRL, 2007 weak pinning non-Arrhenius creep Equilibrium state is Bragg 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

  25. 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）

  26. Three-dimensional XY spin glass: Vortex glass in high-Tc superconductors with d-wave symmetry Qing-Hu Chen, Phys. Rev. B 80, 144420(2009) □The Hamiltonian [Young’s group, PRL 90, 227203(2003).] Jij: 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.

  27. 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).) ◇ Chiral glass correlation function ◇ Spin glass correlation Function

  28. □Most theory done for the Ising (Si =± 1) spin glass. Clear evidence for finite TSG. Best evidence: finite size scaling (FSS) of correlation length (Ballesteros et al. PRB 62, 14237 2000.) □Many experiments closer to a vector spin glass Si. Theoretical situation is less clear: ◇Old Monte Carlo: TSG, if any, seems very low, probably zero. ◇Kawamura and Li, PRL 87, 187204(2001): TSG = 0 but transition in the “chiralities”, TCG > 0. This implies spin–chirality decoupling. ◇But: possibility of finite TSG 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.

  29. 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]

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

  31. □Dynamical scaling 64x64x64 ◇ Perfect collapse ◇ Strachan et al. (PRB 06): perfect collapse is not sufficient ◇convexity-concavity criterion to identify the Tg ◇ 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.

  32. 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! • 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!

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

  34. 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.

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

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

  37. Motion direction Y Transverse direction X Profiles of Bragg peaks Ix0=5, iy0=8 □QLRO in transverse direction □“LRO” in motion direction

  38. 1st order dynamical phase transition □structure factor at T=0.18 Moving smectic or liquid □Tm≈ 0.173J/kB p=1/250

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

  40. Moving Bose glass I=0.5, T=0.12 □ structure factor absence of 6 Bragg peaks □ real-space distribution moving □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.

  41. Motion direction Y Transverse direction X Profiles of Bragg peaks Ix0=11, iy0=0

  42. Motion direction Y Transverse direction X Profiles of Bragg peaks Ix0=5, iy0=8

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

  44. 1st order dynamical phase transition □I=0.5 □ Tm ≈ 0.162J/kB p=1/25 □T<Tm, although 6 peaks absent superconducting coherence along c axis remains

  45. 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.

  46. 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.

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

  48. Nernst effect in high-Tc superconductors in experiments N. P. Ong, Yayu Wang, Z. A. Xu, Princeton University HOT COLD Nernst coefficient □Anomalous large Nernst signal eN extending from below Tc0 to above Tc0, in LSCO, Bi-2201, YBCO

  49. 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

  50. Modeling □2D Josephson junction arrays (QHC, Tang, Tong , PRL 2001) f = a2B/Φ0, filling factor; B=∇XA □The dynamical equation for the phase: TDGL dynamics □Open BC are taken along x direction: temperature gradient in x direction vortex moving along x direction. Fluctuating twist BC along y axis

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