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

2. Physics Department, University of California, Davis, CA 95616

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

**We study stochastic transport of fluxons in superconductors by alternating
current (AC) rectification. Our system provides a fluxon pump, "lens", or fluxon
"rectifier" because the applied electrical AC is transformed into a net DC motion
of fluxons. Thermal fluctuations and the asymmetry of the ratchet channel walls
of the simulated samples induce this "diode" effect, which can have important
applications in devices, like SQUID magnetometers. While many of the behaviors
observed are consistent with previously studied one-dimensional (1D) ratchets,
certain features are unique to this novel 2D geometric rectifier.**

Unless otherwise specified, each figure refers to simulations conducted
in the following way: Initially fluxons were randomly placed inside the
channel and subjected to an alternating current along *y *--- producing
a square-wave Lorentz driving force along *x* with F º
f_{L} / f_{0} = 15, and period P = 2 t
(here,
t
= 100 MD steps). This means that in the absence of thermal noise (T = 0)
and channel walls, a single fluxon would alternate
traveling 15 l (where l
is the Landau penetration depth) in one direction
(e.g., +*x*), and then 15 l in the other
(-*x*). The period 9 l of the horizontal
ratchet teeth was such that the driving force F = 15 should be
sufficient to allow for a rectification, or diode effect, because it moves the
fluxon back and forth through the bottlenecks indicated in the inset of
Fig. 1. Ten runs, of 250 000 MD steps each, were used to find the average current and standard deviation for each plotted point. When a
standard deviation is not shown, it was significantly smaller than the
plotted point. All samples had a 2D geometric ratchet made of
asymmetric walls. The simulated sample was an 18 l
by 18 l square, with periodic boundary conditions.
The channel is 7 l wide, with saw-teeth of period
9 l (four teeth per unit cell) and slope 1/3
(except in Fig. 5 where geometry was varied). This leads to a bottleneck
that is one l wide. This construction
is a novel 2D ratchet that uses geometry, instead of potential energy,
to cause rectification.

** Temperature
dependence of the fluxon rectification.** --- Figure 1 shows the
rectified average fluxon velocity

These simulations clearly indicate an optimal or "resonant" temperature
regime in which the DC fluxon velocity is maximized by the fluxon pump
or diode. This optimal temperature regime can be explained as a trade-off
between allowing the fluxons to fully explore the ratchet geometry (i.e.,
T must not be too low) and washing out the driving force (i.e., T must
not be too large). At low temperatures, the alternating driving force
will cause a fluxon to migrate to the center of the channel and no longer
be impeded or assisted by interactions with the geometry; while at high
temperatures the driving force becomes irrelevant and thus the fluxon is
no longer pushed through bottlenecks regularly. The *<v>(T)*
curve for many randomly placed fluxons is similar to the single fluxon
case, but the magnitude of *<v>(T)* decreases when the fluxon density
B increases. This is because the repulsive force produced by a large
number of fluxons act to restrict each other's motion. Note that
at the optimal temperature, the fluxons travel nearly 15 l
every 100 MD steps. The general shape of the curve was established
with two fine scans with one and fifty fluxons. Conformity of other
field strengths to this curve was judged by performing a coarser scan,
then connecting the points with a spline. The small dip in the *<v>*
for low T will be discussed separately (see Fig. 3 below).

** Driving
period dependence of the flux pump.** ---
Average fluxon velocity

Fig. 2(a) shows *<v>(P)* for six different combinations of driving
force amplitudes F and temperatures T for a single fluxon. Interestingly,
as the period P of the driving force was varied from low to high (from
P = 20 MD steps to 2000 MD steps)
*no optimal peak *in *<v>*
was discovered. This is in contrast with previous work on ratchets,
which provide a peak in *<v>* versus either P or frequency.
Instead, we find that while a high frequency driving force yields a very
small ratchet velocity, the velocity quickly converges to a nearly stable
value with increasing P. This result corresponds to the idea that
a fluxon must be forced through the bottleneck of the geometric 2D ratchet
in order for it to be a rectifier. At high frequencies, a fluxon
does not travel far enough to interact with the bottleneck. Once
the frequency is low enough to force it through a bottleneck, however,
the net velocity changes little by forcing it through further bottlenecks.

All of the simulations in Fig. 2(a) were conducted at relatively low
temperatures to allow the effects of the driving force to dominate.
In addition to trying F = f_{L} / f_{0} = 15, the value
typically used in this paper, two combinations of driving force and temperature
magnitudes were tried that both also had an F/T ratio of 3/1. While
varying P at the low-T regime had little effect upon the period dependence
curve, varying F very clearly did have an effect: The higher F, the greater
*<v>*.

Fig. 2(b) shows the dependence of *<v>* on P, for T = F / 3
= 5 with many fluxons instead of one. Initially, fifty randomly placed
fluxons were simulated. As with the single fluxon case, a high frequency
driving force resulted in a low *<v>* because the fluxons were
not being forced through the ratchet bottleneck. Increasing P produced
a rapidly plateauing *<v>*. Once the shape of the curve was
determined, similar sets of observations were made of different fluxon
densities B at six different periods. These simulations of many fluxons
demonstrate that, as with one fluxon, there was *no* optimal peak.

The inset demonstrates a few constructs made possible by ratchets.
By coupling two ratchets that rectify in opposite directions, fluxon lenses
that could either (c) disperse or (d) concentrate fluxons in chosen regions
of a sample can be created. Corners (e) can also be constructed.
Such remarkable devices, and modifications of them, would allow the transport
of fluxons along complicated nanofabricated channels: a microscopic network
of fluxon channels in superconducting devices. This type of configuration
could be very important to get rid of unwanted, trapped flux in SQUID magnetometers,
and also move fluxons along channels in devices.

* Fluxon
pump effect versus field. *--- By varying the density of fluxons
B, we found a maximum in

If a fluxon happens to remain for some time in the center of a bottleneck,
unlikely to happen at high-enough temperatures, then it will not be rectified;
while the potential barrier rectifies at every period. Increased fluxon
density again increases resistance, reducing *<v>*.

** Geometry
dependence. **--- To determine the generality of the simulation results
presented here with respect to varied ratchet geometries, in Fig. 5 we
compare

* Corresponding author.