J.F. Wambaugh^{1},
F.
Marchesoni^{1,2}, and
Franco
Nori^{1*}

1. Department
of Physics, University of Michigan, Ann Arbor, Michigan 48109-1120

2. Istituto Nazionale di Fisica della
Materia, Universit'a di Camerino, Camerino, I-62032, Italy

**We study via computer simulations the motion of magnetic flux-line
vortices (fluxons) confined to a straight pin-free channel in a strong-pinning
superconducting sample. We find that, when a constant current is
applied across this system, a very unusual oscillatory shearing manifests,
in which the fluxons at the edges of the channel repeatedly trail behind
and then suddenly leapfrog past the inner rows. For small enough
driving forces, the oscillatory shearing dynamic phase is replaced by a
continuous shearing phase in which the distance between initially-nearby
fluxons grows in time, quickly destroying the order of the lattice.**

* Simulation
---
*Unless
otherwise specified, each figure refers to simulations conducted in the
following way: Initially fluxons were placed uniformly across the
sample in a minimum-energy, field-cooled triangular lattice and the fluxons
in the channel were subjected to a current along

We want to monitor the time evolution of *v*(*y*), since shear
implies a non-zero gradient along *y* of *v*(*y*).
We have found that binned velocity snapshots, like the ones shown in Fig.
2, are useful because they illustrate the shear-induced presence of gradients
in *v*(*y*). Thus, we binned the fluxon velocity along
*y*
and integrated the velocities along *x*.

The two sequences [(a--c) and (d--f) in Fig.~2] of velocity profiles across a channel are successive velocity snapshots. These snapshots are made every 500 MD steps during the course of a simulation: (a--c) shows three successive snapshots, while in (e--f) every third snapshot is shown. These two particular chronological sequences show two different types of behaviors that we have found when a perfect triangular lattice of fluxons is placed and then forced to move on a superconducting sample with a straight channel.

*Continously Sheared Dynamic Phase for Low Driving*

Sequence (a--c) in Fig. 2 corresponds to a relatively weak driving force of magnitude F = 0.02, and displays the expected "beer-belly" curved velocity profile. Here the shear between the edge fluxons of the channel and the pinned fluxons outside the channel induces a curvature in the velocity profile. In (a), the edge rows, interacting with the infinitely pinned fluxons experience drag, resulting in the inner row moving slightly faster. The interaction between the inner (bin 9) and outer (bins 7 and 11) fluxon rows pulls the outer rows along, eventually pushing outwards a few fluxons in the edge rows---as they catch on the pinned fluxon potential, as shown in (b). This outward displacement of a few edge fluxons creates defects in the moving fluxon lattice and also the appearance, in the velocity profile in (b), of moving fluxons (in bins 6 and 12) where there were previously none. For this low density of fluxons, and for our chosen fine-grained bins, there are no fluxons in bins 8 and 10. A higher density example will be discussed below. In (c), the velocity profile has remained "fanned out" as the fluxons continue to be driven. The original triangular lattice arrangement has not been preserved because the fluxon lattice has been continuously sheared, producing many defects. Indeed, the distance between initially nearby fluxons grows with time, for low driving forces.

*Oscillatory Shearing Dynamic Phase for Higher Driving*

At a higher driving force, however, a novel dynamic fluxon phase is
observed. Sequence (d--f) in Fig. 2 has the same number of fluxons
as in (a--c), but driven by a higher driving force; F = 1 instead of F
= 0.02. Now, instead of allowing fluxons to slip from the outer rows
as they catch on the interactions with pinned fluxons, the outer rows of
fluxons actually *leapfrog* forward past the inner row, by temporarily
moving more rapidly than the inner row. Here, (e) clearly shows the
outer rows actually move faster than the inner rows. Eventually,
the faster edge rows slow down and, as shown in (f), this results in a
brief effective uniform velocity across the channel.

This unexpected cycle repeats periodically and it is even more pronounced when there are higher fluxon densities, corresponding to higher magnetic fields. The three-row arrangement shown here is just the simplest example that illustrates this novel fluxon dynamic phase.

* Further
Characterization *--- Figure 3 shows the temporal evolution of the
difference in velocities

** Leapfrogging
Edge Fluxons at Higher Fluxon Densities** --- The very unusual cycle
of "trail-behind" and leapfrogging edge fluxons seen in Fig. 2 very clearly
persists at much higher field strengths, as indicated in Fig. 4.
There, the density of fluxons is about four times higher than in the case
shown in Fig. 2.

The chronological sequence of "snapshots" in velocity space in Fig. 4 depicts the time evolution of the velocity of fluxons across the channel. This situation is similar to Fig. 2 (d--f), but now there is a much higher field strength: 340 fluxons have now been placed in a triangular lattice on the sample with a straight central channel. This combination of field strength and channel width results in the channel being filled with seven horizontal rows of seventeen fluxons each.

The behavior exhibited in Fig. 4 is the result of inner rows of fluxons
being locked together with the outer rows despite the interactions of the
outer rows with the pinned fluxons outside the zero-pinning channel.
Intuitively one would expect that profiles like (a), (b) and (f) are the
norm---the outer rows snag on the pinned fluxons and lag behind the less
restricted inner rows. The surprising behavior is that profiles like
(c) and (d) indicate that the slowed fluxons in the outer rows suddenly
"catch up" or *leapfrog* by momentarily exceeding the velocity of
fluxons in the inner rows. Panel (e) in Fig. 4 shows that this can
also result in a much flatter velocity profile than otherwise expected,
albeit just briefly.

Simulations at both field strengths discussed here (80 and 340 fluxons
on the sample) demonstrate continuous shearing under low driving forces,
but unusual oscillatory shearing under higher driving forces. In
addition to higher field strengths, simulations with smaller Dt's
were used, producing the same results (e.g., 0.001 instead of 0.01).
The animation shows the change in velocity profile during a simulation
with smaller Dt (the velocities are larger than
those in the snapshots because the particular simulation animated was conducted
with F = 1 instead of F = .2 as in the snapshots).

** Velocity
versus Time for Several Driving Forces **--- Figure 5 shows the fluxon
velocity, now averaged over the

The lowest curve shown in Fig. 5 corresponds to a weak-driving case;
here with force F = 0.02. First, there is a transient. Afterwards,
the average velocity of the entire lattice, not just the edge fluxons anymore,
shows marked oscillations with time. This corresponds to a "stick-slip"-type
motion of the fluxon lattice *impinging upon a succession of potential
energy bottlenecks as it is pushed through the channel*. Notice
that strictly speaking, the "stick" phase is really moving, and never stuck.
Thus, it is more appropriate to describe it as a phase with an oscillating
sine-like average velocity. Notice also that the velocity oscillations
in this low-driving "continuous shearing" dynamic phase span about 1/3
of an order of magnitude (the vertical axis has a logarithmic scale).

The highest curve in Fig. 5 shows a typical large-driving, F = 1, case. There, the system is in the oscillating shearing dynamic phase, and the average velocity of the entire system oscillates, as shown in the magnification of this plot located in the lower right inset. The average velocity appears to be constant in the main panel of Fig. 5 because its vertical axis spans five decades!

To monitor the crossover between these two dynamic phases, Fig. 5 also shows an intermediate curve, where the driving force was slowly increased from an initial value of F = 0.02 to a final value of F = 1. While the average velocity oscillations of the continuous shearing phase, the low-driving-force-regime, initially persist, it is clear that the amplitude of the velocity oscillations decrease until there is no discernable oscillation (without magnification) in the logarithmic plot.

In other words, when the average velocity of the moving fluxons is plotted
over time, while the driving force is slowly increased, the average velocity
eventually becomes relatively stable, in the sense that there is not much
overall relative shifting in the lattice. Thus, the lattice, while
sheared with oscillating velocities,

is not completely torn apart at higher driving forces---because of
the periodic "leapfrog" jumps of the trailing edge fluxons.

For the measurements shown in Fig. 5, the average displacement of all moving fluxons was recorded every 500 MD steps, over the course of 250,000 MD step simulations. We also studied many more values for the driving force, including very high values. For all cases where F was approximately larger than 0.05, oscillating shearing was found.

Since the oscillations in the average velocity are caused by the interaction
between the moving and pinned fluxons, we have found that the frequency
of the oscillations depends on the density of the fluxons (they are inversely
proportional). Thus, careful measurements of the average velocity
(e.g., as a voltage), might provide an indirect way of determining the
fluxon density and magnetic field strength. Indeed, in Fig. 3, the
system that produces the signal (c) is the same one as in (b) but with
about four times the number of vortices. Notice how the period decreases.