Ajay Kumar GhoshJadavpur University

Kolkata, India

Vortex Line Ordering in the Driven 3-D Vortex Glass

Vortex Wrocław 2006

Stephen TeitelUniversity of Rochester

Rochester, NY USA

Peter OlssonUmeå University

Umeå, Sweden

Outline

• The problem: driven vortex lines with random point pins

• The model to simulate: frustrated XY model with RSJ dynamics

• Previous results

• Our resultseffects of thermal vortex rings on large drive meltingimportance of correlations parallel to the applied field

• Conclusions

• 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.

uniform drivingforce

Driven vortex lines with random point pinning

For strong pinning, such that thevortex lattice is disordered in equilibrium, how do the vortex linesorder when in a driven steady statemoving at large velocity?

3D Frustrated XY Model kinetic energy of flowing supercurrentson a discretized cubic grid

coupling on bond i

phase of superconducting wavefunction

magnetic vector potential

density of magnetic flux quanta = vortex line densitypiercing plaquette of the cubic grid

uniform magnetic field along z directionmagnetic field is quenched

weakly coupled xy planes

constant couplings between xy planes || magnetic field

random uncorrelated couplings within xy planes disorder strength p

T Temperature

p disorder strengthvortex lattice

liquid

1st order melting

vortex glass

2nd order glass

pc

Equilibrium Behavior

critical pc

at low temperature

p < pc orderedvortex lattice

p > pc disorderedvortex glass

we will be investigating driven steady states for p > pc

Resistively-Shunted-Junction Dynamics••••••••••••••••RInoise••xθiθ +i Itotal

Units

current density:

time:

voltage/length:

temperature:

apply: current density Ix

response:voltage/length Vx

vortex line drift vy

twisted boundary conditions

voltage/length

new variable with pbc

stochastic equations of motion

RSJ details

Previous Simulations

Domínguez, Grønbech-Jensen and Bishop - PRL 78, 2644 (1997)

vortex density f = 1/6, 12 ≤ L ≤ 24, Jz = J weak disorder ??

moving Bragg glass - algebraic correlations both transverse and parallel to motionvortex lines very dense, system sizes small, lines stiff

Chen and Hu - PRL 90, 117005 (2003)

vortex density f = 1/20, L = 40, Jz = J weak disorder p ~ 1/2 pc

at large drives moving Bragg glass with 1st order transition to smecticsingle 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 density f = 1/20, L = 40, Jz = J strong disorder p ~ 3/2 pc

at large drives moving Bragg glass with 1st order transition to smecticsingle 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

Parameters

p = 0.15 > pc ~ 0.14

Jz = J

L up to 96

vortex density f = 1/12

ground state p = 0Ix Vx

vortex linemotion vy

Quantities to Measurestructural

dynamic use measured voltage drops to infer vortex linedisplacements

ln S(k, kz=0)

Ix Vx

vortex linemotion vy

Phase Diagram

Previous results of Chen and Hu p ~ 1/2 pc 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

melting of ordered state upon increasing current I is due to proliferation of thermally excited vortex rings

I

F(R) ~ RlnR - IR2

ring expands when

R > Rc ~ 1/Ifrom superfluid 4He

rings proliferate

when F(Rc) ~ T ~ 1/I

Tm

+−

Disordered state above 1st order melting Tm

Tm

ln S(k)

vortex linemotion vy

When we increase the system size, the height of the peaksin S(k) along the kx axis do NOT increase only short⇒ranged translational order.

⇒ disordered state is anisotropic liquid and not a smectic

I = 0.48, T = 0.13

Ordered state below 1st order melting Tm

ln S(k)

vortex line motion vy

C(x, y, z=0)

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

⇒ ordered state is a smectic not a moving Bragg glass

I = 0.48, T = 0.09

Correlations between smectic channels

short ranged translational correlations between smectic channels

Correlations within a smectic channel

C(x, y, z=0)

exp decay to const > 1/4algebraic decay to 1/4

correlations within channel are either long ranged, or decay algebraically with a slow power law ~ 1/5

averaged over 40random realizations

Correlations along the magnetic fieldsnapshot 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.

Dynamics

y0(t) y3(t) y6(t) y9(t)

center of mass displacementof vortex lines in each channel

Channels diffuse with respect to one another.

Channels may have slightly different average velocities.

Such effects lead to the short range correlations along x.

36 x 96 x 96

Dynamics and correlations along the field direction z

See group of strongly correlated channels moving together.

Smectic channels that move together are channels in which vortex linesdo not wander much as they travel along the field direction z.

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 along x.

smaller system: 48 x 48 x48

Behavior elsewhere in ordered driven state

I=0.48, T=0.07, L=6020 random realizations

transverse correlations

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 ~ LThese are realizations where all channels have “straight” lines along z.

More ordered state at low T? Or finite size effect?

So far analysis was for I=0.48, T=0.09 just below peak in Tm(I)

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 (shortrange transverse order)

• Importance of vortex line wandering along field direction for decoupling ofsmectic channels

• Moving Bragg glass at lower temperature? or finite size effect?