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### 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

• 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

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.

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

criticalpc

at low temperature

p < pc ordered

vortex lattice

p > pc disordered

vortex glass

we will be investigating driven steady states for p > pc

Resistively-Shunted-Junction Dynamics

apply:

current density Ix

response:

voltage/length Vx

vortex line drift vy

Units

voltage/length:

temperature:

current density:

time:

twisted boundary conditions

voltage/length

new variable with pbc

stochastic equations of motion

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

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

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

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

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

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

Correlations between smectic channels

short ranged translational correlations between smectic channels

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

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.

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.

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.

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?

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?

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