Vortex Line Ordering in the Driven 3-D Vortex Glass

1. Vortex Line Ordering in the Driven 3-D Vortex Glass Peter Olsson Umeå University Umeå, Sweden Ajay Kumar Ghosh Jadavpur University Kolkata, India Stephen Teitel University of Rochester Rochester, NY USA Vortex Wrocław 2006

2. Outline • The problem: driven vortex lines with random point pins • The model to simulate: frustrated XY model with RSJ dynamics • Previous results • Our results effects of thermal vortex rings on large drive melting importance of correlations parallel to the applied field • Conclusions

3. Driven vortex lines with random point pinning For strong pinning, such that the vortex lattice is disordered in equilibrium, how do the vortex lines order when in a driven steady state moving at large velocity? • Koshelev and Vinokur, PRL 73, 3580 (1994) motion averages disorder⇒ shaking temperature ⇒ ordered driven state • Giamarchi and Le Doussal, PRL 76, 3408 (1996) transverse periodicity ⇒ elastically coupled channels ⇒ moving Bragg glass • Balents, Marchetti and Radzihovsky, PRL 78, 751 (1997); PRB 57, 7705 (1998) longitudinal random force remains⇒ liquid channels ⇒ moving smectic • Scheidl and Vinokur, PRE 57, 2574 (1998) • Le Doussal and Giamarchi, PRB 57, 11356 (1998) We simulate 3D vortex lines at finite T > 0.

4. coupling on bond im phase of superconducting wavefunction magnetic vector potential density of magnetic flux quanta = vortex line density piercing plaquette a of the cubic grid uniform magnetic field along z direction magnetic field is quenched constant couplings between xy planes || magnetic field random uncorrelated couplings within xy planes disorder strength p weakly coupled xy planes 3D Frustrated XY Model kinetic energy of flowing supercurrents on a discretized cubic grid

5. Equilibrium Behavior criticalpc at low temperature p < pc ordered vortex lattice p > pc disordered vortex glass we will be investigating driven steady states for p > pc

6. Resistively-Shunted-Junction Dynamics apply: current density Ix response: voltage/length Vx vortex line drift vy Units voltage/length: temperature: current density: time:

7. RSJ details twisted boundary conditions voltage/length new variable with pbc stochastic equations of motion

8. Previous Simulations Domínguez, Grønbech-Jensen and Bishop - PRL 78, 2644 (1997) vortex densityf = 1/6, 12 ≤ L ≤ 24, Jz = J weak disorder ?? moving Bragg glass - algebraic correlations both transverse and parallel to motion vortex lines very dense, system sizes small, lines stiff Chen and Hu - PRL 90, 117005 (2003) vortex densityf = 1/20, L = 40, Jz = J weak disorder p ~ 1/2 pc at large drives moving Bragg glass with 1st order transition to smectic single system size, single disorder realization, based on qualitative analysis of S(k) Nie, Luo, Chen and Hu - Intl. J. Mod. Phys. B 18, 2476 (2004) vortex densityf = 1/20, L = 40, Jz = J strong disorder p ~ 3/2 pc at large drives moving Bragg glass with 1st order transition to smectic single system size, single disorder realization, based on qualitative analysis of S(k) We re-examine the nature of the moving state for strong disorder, p > pc, using finite size analysis and averaging over many disorders

9. Parameters ground state p = 0 p = 0.15 > pc ~ 0.14 Jz = J Lup to 96 vortex densityf = 1/12 Ix Vx vortex line motion vy Quantities to Measure structural dynamicuse measured voltage drops to infer vortex line displacements

10. Ix Vx vortex line motion vy ln S(k, kz=0) Phase Diagram

11. Previous results of Chen and Hup ~ 1/2pc weak disorder a, b - “moving Bragg glass” algebraic correlations both transverse and parallel to motion a´, b´ - “moving smectic” We will more carefully examine the phases on either side of the 1st order transition

12. Digression: thermally excited vortex rings Tm melting of ordered state upon increasing current I is due to proliferation of thermally excited vortex rings ring expands when R > Rc ~ 1/I I from superfluid 4He rings proliferate whenF(Rc) ~ T ~ 1/I F(R) ~ RlnR- IR2

13. When we increase the system size, the height of the peaks in S(k) along the kx axis do NOT increase ⇒ only short ranged translational order. ⇒ disordered state is anisotropic liquid and not a smectic Disordered state above 1st order melting Tm I = 0.48, T = 0.13 ln S(k) vortex line motion vy Tm

14. vortex line motion vy C(x, y, z=0) ⇒ordered state is a smecticnot a moving Bragg glass Ordered state below 1st order melting Tm I = 0.48, T = 0.09 ln S(k) Bragg peak at K10⇒ vortex motion is in periodically spaced channels peak at K11 sharp in ky direction ⇒ vortex lines periodic within each channel peak at K11 broad in kx direction ⇒ short range correlations between channels

15. Correlations between smectic channels short ranged translational correlations between smectic channels

16. C(x, y, z=0) averaged over 40 random realizations exp decay to const > 1/4 algebraic decay to 1/4 Correlations within a smectic channel correlations within channel are either long ranged, or decay algebraically with a slow power law ~ 1/5

17. Correlations along the magnetic field snapshot of single channel z ~ 9 As vortex lines thread the system along z, they wander in the direction of motion y a distance of order the inter-vortex spacing. Vortex lines in a given channel may have a net tilt.

18. 36 x 96 x 96 y0(t) y3(t) y6(t) y9(t) center of mass displacement of vortex lines in each channel Dynamics Channels diffuse with respect to one another. Channels may have slightly different average velocities. Such effects lead to the short range correlations along x.

19. Dynamics and correlations along the field direction z smaller system:48 x 48 x48 See group of strongly correlated channels moving together. Smectic channels that move together are channels in which vortex lines do not wander much as they travel along the field directionz. Need lots of line diffusion along z to decouple smectic channels. As Lz increases, all channels decouple. Only a few decoupled channels are needed to destroy correlations alongx.

20. Behavior elsewhere in ordered driven state transverse correlations I=0.48, T=0.07, L=60 20 random realizations So far analysis was forI=0.48,T=0.09 just below peak inTm(I) Many random realizations have short ranged correlations along x. These are realizations where some channels have strong wandering along z. Many random realizations have longer correlations along x; x ~ L These are realizations where all channels have “straight” lines along z. More ordered state at low T? Or finite size effect?

21. Conclusions For strong disorder p > pc (equilibrium is vortex glass) • Driven system orders above a lower critical driving force • Driven system melts above an upper critical force due to thermal vortex rings • 1st order-like melting of driven smectic to driven anisotropic liquid • Smectic channels have periodic (algebraic?) ordering in direction parallel to motion, short range order parallel to applied field; channels decouple (short range transverse order) • Importance of vortex line wandering along field direction for decoupling of smectic channels • Moving Bragg glass at lower temperature? or finite size effect?