Ratchet Effects for Vortices in Superconductors with
Periodic Pinning Arrays
C. Reichhardt and C.J. Olson Reichhardt
Center for Nonlinear Studies and Theoretical Division,
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Abstract
Using numerical simulations
we show that novel transport phenomena can occur for vortices moving
in periodic pinning arrays when two external perpendicular ac drives
are applied. In particular, we find a ratchet effect where the
vortices can have a net dc drift even in the absence of a dc drive.
This ratchet effect can occur for ac drives which create orbits that break
one or more reflection symmetries.
Keywords: Ratchet effect; Periodic pinning arrays; Vortex motion
Recently there has been considerable interest in using
ratchet effects to control the motion of vortices
in superconductors [1,2,3,4].
In a ratchet, a net dc flow can arise under the application
of a strictly ac drive [5].
Typically, the symmetry breaking which can
allow for this effect is produced by an asymmetric underlying substrate,
such as a saw-tooth potential. However, ratchet effects can occur
in systems with symmetrical substrates when some other
form of symmetry
breaking is introduced.
One possible source of such a symmetry breaking
is the applied ac drive itself. For example, it was recently shown that
a particle moving in a two-dimensional (2D)
periodic substrate can exhibit a ratchet effect
when crossed ac drives are applied, where the ac drives cause the
particle to move in orbits that have broken reflection symmetries [6].
In this paper, we show that for vortices
moving in a periodic pinning array at fields B/Bϕ > 1.0 (where
Bϕ is the field at which each pinning site captures one vortex),
a series of novel dynamical phases which ratchet the vortices can arise
when two external perpendicular ac drives are applied.
Our system can be realized in superconductors with periodic pinning arrays
small enough that only one vortex can be captured per site, so that the
additional vortices sit in the interstitial regions between the pinning
sites [7,8,9,10].
We simulate a thin-film superconductor
containing an N×N square pinning array
with a lattice constant a.
The equation of motion for a vortex i is given by
fi =
dri
dt
= fvv + fivp + fAC.
(1)
The force from the other vortices is
fivv = −∑j ≠ iNv∇iUv(r).
The vortices interact logarithmically via Uv = −ln(r).
The force from the pinning sites is fivp. The pinning
sites are modeled as parabolic traps with a range rp,
where rp/a = 0.1, and maximum pinning force fp.
For the results in this
work, all external drive forces are much smaller than the pinning forces,
so that vortices in the pinning sites remain immobile.
The ac drive is applied in both the x and y directions:
fAC = facx(t)
^
x
+ facy(t)
^
y
.
(2)
For all the ac drives, in the absence of a substrate there is no
dc drift velocity and < fAC > = 0.0.
In the initial configuration,
all the pinning sites are filled with one vortex each,
and the additional vortices are placed
randomly in interstitial locations.
We monitor the long time average vortex velocities.
Figure 1:
The average velocity in the y-direction < Vy > vs B, the
coefficient of the y component of the ac drive, for fixed A = 0.49.
We first consider the case of a system with B/Bϕ = 1.065, so that
the interstitial vortices are far apart and in general do not interact.
We apply an ac drive of the form
fAC = A(sin(ωBt) + sin2(ωAt))∧x + Bcos(ωBt)∧y,
where ωB/ωA = 1.25. In this case a symmetry breaking
arises from the shape of the orbits.
In Fig. 1 we show < Vy >
for a system
where A = 0.49 is fixed and B is varied.
For B < 0.24, the average vortex velocity is zero with the interstitial
vortices moving in closed orbits.
For B > 0.24, there are a series of phases which have
a net dc flux in the y-direction.
This flux can be in either the positive or negative direction.
In general most of the rectified phases give a drift velocity of
0.024. We also find evidence for some rectifying phases that
produce dc drifts that are fractional
multiples of the maximum drift value.
However, these phases appear only for very small
regions of B. Additionally, as B is further increased,
we find regions
where the interstitial vortices become repinned and
< Vy > = 0.0. Each of the rectifying regions corresponds
to different dynamical phases where the vortices move in
distinct periodic orbits.
Figure 2: Vortex trajectories (black lines) and pinned vortex positions
(black dots) for the system shown in Fig. 1. (a) B = 0.265,
(b) B = 0.325, (c) B = 0.39, and (d) B = 0.5.
In Fig. 2 we illustrate some of the dynamical phases for
the system in Fig. 1.
The positive rectifying phase at B = 0.265 is shown in Fig. 2(a).
Here a very intricate periodic pattern forms, with the vortices moving in
a long time zig-zag pattern. In Fig. 2(b) the negative rectifying phase
is shown for B = 0.325. Here another distinct
orbit forms with
the vortex moving in lobes that slant alternately.
In Fig. 2(c) we show another positive rectifying mode
at B = 0.39, where the orbit is similar to that in Fig. 1(b) but
the net drift is in the opposite direction. In Fig. 2(d) we illustrate
a non-rectifying orbit. Here the vortices move in complex periodic
closed orbits without a net drift.
Figure 3:
The average velocity in the y direction < Vy > vs A, the ac
amplitude for the x component of the
ac drive, at fixed B = 0.34.
We next consider the case where B is fixed to B=0.34 while A is varied.
In Fig. 3 we plot < Vy > vs A. Here we find similar behavior to
that in Fig. 1, with both positive and negative regions of
net dc drift appearing along with pinned regions. In both Fig. 1 and
Fig. 3, we find no ratchet effect for low values of A or B.
Figure 4: The vortex trajectories (black lines) and
pinned vortex positions (black dots) for the
system in Fig. 3 at: (a) A = 0.256,
(b) A = 0.286, (c) A = 0.289, (d) A = 0.792.
In Fig. 4 we illustrate some of the dynamical orbits for the system in Fig. 3.
In Fig. 4(a), we show the first rectifying phase which occurs at
A = 0.256. Again, very intricate
periodic vortex motions occur. In Fig. 4(b) we plot the positive
rectifying phase that occurs at A = 0.286, where the
net dc motion is in the positive y-direction. The orbit
consists of the vortex moving in loops around a pining site
with a series of much smaller sub-loops. In Fig. 4(c) we
show the pinned phase
for A = 0.289 that occurs just after the phase in
Fig. 4(b). In Fig. 4(d) we illustrate a rectifying phase at
A = 0.792. Here the orbit is much wider in the x direction,
corresponding to the increased amplitude of the x component of the ac drive.
We find similar intricate orbits at the other rectifying phases,
not shown here.
We have also considered other parameters and different ac drive
forms and find similar behaviors,
indicating that the ratchet behaviors we observe are
a very general feature of 2D systems
driven with perpendicular complex ac drives.
In conclusion, we have shown that with perpendicular ac drives
applied to vortices in periodic pinning arrays,
a remarkably rich variety of dynamical phases can be achieved including
ratchet effects.
The ratchet effect arises in this system
even though the periodic
substrate is symmetrical due to a symmetry breaking by the
ac drive itself.
Our results open a new avenue for controlling flux motion in superconductors.
For instance, it may be possible to exercise additional control over the
ratchet effect by changing the periodicity of the pinning array within
a single sample. If, under the same ac drive, one type of pinning
causes the vortices to move in the positive direction,
while another type of pinning causes them to move in
the negative direction, then it should be possible to create a fluxon
focusing effect where vortices can be constricted in small regions.
Conversely, it may be desirable to remove the flux from other
regions, which is useful for certain applications such as
flux sensitive devices.
This work was supported by the US Department of Energy under Contract
No. W-7405-ENG-36.
V. Metlushko, U. Welp, G.W. Crabtree, R. Osgood, S.D. Bader, L.E. DeLong,
Z. Zhang, S.R.J. Brueck, B. Illic, K. Chung, and P.J. Hesketh,
Phys. Rev. B 60, R12 585 (1999).
