Vortex Pinball Under Crossed ac Drives in Superconductors
with Periodic Pinning Arrays
C. Reichhardt and C.J. Olson
Center for Nonlinear Studies and Applied Physics Division,
Los Alamos National Laboratory, Los Alamos, NM 87545
(Received 8 November 2001; published 7 February 2002)
Vortices
driven with both a transverse and a longitudinal AC drive
which are out of phase are shown to exhibit a novel
commensuration-incommensuration effect when interacting with
periodic substrates.
For different AC driving parameters, the motion of the vortices
forms commensurate orbits with the periodicity of the pinning array.
When the commensurate orbits are present,
there is a finite DC critical depinning threshold, while for the
incommensurate
phases the vortices are delocalized and the
DC depinning threshold is absent.
A wide variety of dynamical systems
can be modeled as a classical
particle moving in a periodic potential.
Examples of such systems
are Josephson-junction (JJ) arrays [1,2],
sliding charge density waves (CDW) [3,4],
atomic friction [5], vortices
in superconductors moving over periodic substrates
[6,7,8,9,10,11,12],
electrons at
low magnetic fields in
antidot arrays [13,14],
as well as ions [15] and colloids
[16] in
optical trap arrays.
When an AC drive is superimposed over the DC drive,
resonances or phase-locking effects can occur which have been
extensively studied in JJ arrays [2] and CDWs [3].
Signatures of the phase locking include oscillations in the
depinning threshold with increasing AC drive amplitude,
and locking of the particle velocity
to a fixed value over a range of the DC drive.
In superconductors
with periodic substrates, phase locking of driven vortices has been
experimentally observed in
samples containing 1D periodic modulations [6].
More recently, experiments, simulations, and theory
have shown that in superconductors with periodic hole arrays,
phase locking or Shapiro steps occur when combined
DC and AC drives are applied.
In all these phase locking systems the AC and DC drives are in
the same direction and the motion can be
considered effectively one-dimensional (1D).
Different effects can arise in a two-dimensional (2D) system.
Recently it was shown that
new kinds of phase-locking effects occur when the AC drive is
applied perpendicular to the DC drive for systems with periodic
substrates [12].
Here we study vortex motion in superconductors with periodic arrays of
pinning sites and three combined driving forces.
A DC drive fdc is applied along the longitudinal direction,
and two AC drives which are 90 degrees out of phase are
added
in the longitudinal
and transverse directions.
In the absence of pinning and at fdc=0, the vortices move in a
circular orbit with radius and eccentricity
determined by the amplitudes and
frequencies of the AC drives.
We focus on a regime just above the first matching field Bϕ,
where each pin is occupied by one vortex and a small
number of additional vortices move in the periodic potential created
by the pinned vortices.
The existence of such interstitial vortices above Bϕ
has been inferred from transport measurements and direct
imaging in samples with periodic holes [7],
and in this same regime
Shapiro steps have previously been observed
[9].
The system we propose has
similarities to the "electron pinball" model studied by
Weiss et al. [13] and others [14], in which
classical cyclotron electron motion is induced by a
magnetic field in samples with periodic arrays of anti-dots.
In these systems, peaks and dips appear in the magnetoresistance
as a function of field. The features are
believed to arise when classically moving electrons
follow pinned or commensurate circular orbits
enclosing integer numbers of dots, in which
the electrons can travel
without scattering off the dot potentials.
At the incommensurate orbits the electrons are
scattered and diffuse throughout the sample.
There are important differences between the vortex
"pinball" system
and the electron systems.
The vortices interact with a long-range smooth
or egg-carton square potential, rather than simple point scatters,
and thus
the moving vortices
must follow square orbits (in the case of square pinning arrays),
rather than circular orbits.
A particularly attractive feature of the vortex system is that the
shape of the orbit can be
carefully controlled experimentally
by changing the ratio of the AC amplitudes, phases or
frequencies. This allows new kinds of
anisotropic orbits and commensuration effects to be produced,
which are not obtainable in the electron pinball models.
We consider a thin superconductor containing 2D vortices,
which is the appropriate model for
the recent experiments in superconductors
with hole and dot arrays.
The vortex-vortex interaction has the form of a
logarithmic potential, Uv = −Avln(r), with the energy normalization
Av = Φ20/8πΛ. Here, Φ0 is the flux
quantum and Λ is the effective 2D penetration depth for
a thin film superconductor.
The overdamped normalized
equation of motion for a single vortex i is
fi = η
dri
dt
= fivv + fivp + fdc + fac = ηvi
(1)
where the damping term η is the Bardeen-Stephen friction.
The force from vortex i on the other vortices is
fivv = −∑j ≠ iNv∇iUv(rij).
The long range
interaction is treated
with a fast converging summation method
[17].
The pinning force fvp arises
from the pinning sites which are modeled as short
range attractive parabolic wells. The pinning sites are placed in a square
array of side L, and each pin has a radius of
rp = 0.15L, which
is within the typical experimental ratios of rp/L = 0.14 to 0.3.
The DC driving term fdc is applied
along the symmetry axis of the
pinning array, in the x-direction.
The
AC driving term is fAC = Asin(ωAt)∧x + Bcos(ωBt)∧y, where
we fix wA/wB = 1.0.
For most of the results presented here we consider the case of a
system containing 64 pins and a
vortex filling fraction
of B/Bϕ=1 + 1/64;
however, we have found the same results for higher filling
fractions (such as B/Bϕ = 2, 7/4, 3/2, and 5/4)
at which the
interstitial vortices form a symmetric pattern so that interstitial vortex
interactions cancel.
The initial vortex position is found by annealing from a high temperature
with no driving and cooling to T=0.
The DC drive is increased in increments of 0.0001 and the
sample is held at each drive for 3 ×105 time steps
to ensure a steady state;
the DC depinning threshold is determined from
the time averaged vortex velocities < Vx > .
Figure 1:
Vortex positions (dots) and trajectories (lines)
for fdc = 0.0 and different isotropic AC amplitudes.
(a) A = 0.16, (b) A = 0.21, (c) A = 0.3, (d) A = 0.355,
(e) A = 0.39, and (f) A = 0.47.
We first consider the case of isotropic AC driving with
A/B = 1.0 and
show that
commensurate-incommensurate
depinning
transitions occur as a function of increasing A
even at fdc=0.
Example vortex trajectories are
illustrated in Fig. 1.
For A = 0.16 [Fig. 1(a)] the vortex is confined to move in a
small circular orbit in the middle of the plaquette. For A = 0.21
[Fig. 1(b)] the AC amplitude is large enough that the
vortex encircles n=1 pin in an orbit that
is slightly tilted due to the
counter-clockwise vortex movement through
the square potential.
In Fig. 1(c), for A = 0.3, the orbit encircles n=4 pins.
In Fig. 1(d),
for A = 0.355, the orbit radius is somewhere between
n=4 and n=9, and
the vortex interacts strongly with the occupied
pins, scattering off them.
The vortex becomes delocalized, and diffuses throughout the sample,
avoiding areas near pins due to the repulsion
from the pinned vortices.
The orbit switches intermittently between n=4 and n=9.
In Fig. 1(e,f) we show localized vortex orbits for
A = 0.39 and 0.47,
with n=9 and n=16, respectively.
We also find a stable orbit with n=25 for A = 0.57.
For 0.42 < A < 0.45, as well as for 0.52 < A < 0.55, the vortex
is delocalized and moves in a manner similar to that shown in Fig. 1(d).
In general
we find localized orbits at values of A for which the vortex
trajectory encloses n=m2 pins, where m is an integer.
By comparison, in
electrons in antidot lattices, Weiss et al. [13]
found commensurate or pinned orbits
when the number of dots encircled was
n = 1, 2, 4, 9, 16, and 21. We do not observe any stable orbits for
n = 2 and 21. In the electron systems [13,14]
these orbits correspond to states where
a portion of the electron orbit closely approaches the
dots, which are point scatters.
In the vortex system, due to the long range
vortex-vortex interactions,
the mobile vortices cannot follow orbits that approach arbitrarily
closely to the pinning sites.
Figure 2:
The DC depinning force Fdp/Fdp0
vs AC amplitude A for the isotropic case A=B.
Fdp0 is the
depinning force for zero AC drive.
Inset: average DC velocity < Vx > vs DC driving force fdc
for an incommensurate orbit A = 0.43 (dotted line) and a commensurate
orbit A = 0.37 (solid line).
In Fig. 2 we show the
DC depinning threshold Fdp/F0dp
vs A
for the isotropic case A=B.
Here, F0dp is the
DC depinning force for A=0.
The commensurate orbits are
pinned at low fdc, producing a series of peaks
in Fdp at the commensurate AC values.
At the incommensurate phases the depinning threshold vanishes.
For 0.0 < A < 0.2 , the vortices move in the n=0
interstitial orbit illustrated in Fig. 1(a).
Fdc decreases as the
AC amplitude increases until reaching a minimum value
at A=0.19, followed by a peak in Fdc
for A = 0.225, which corresponds to an n=1 orbit
[Fig. 1(b)]. The second peak at A = 0.31 corresponds to
the n=4 orbit
[Fig. 1(c)].
For the delocalized incommensurate orbits [Fig. 1(d)] of
0.34 < A < 0.36, Fdp vanishes.
A finite Fdp
is regained when n=9 orbits
appear for 0.36 < A < 0.42.
As A continues to increase, Fdp is nonzero for A values
at which the stable n=16 and n=25 orbits are observed,
with Fdp=0 in portions of the regions between these values
of A.
We have checked the oscillatory behavior for different values of
L and find the same general behavior.
Since the radius of the vortex orbit
R ∼ A/ωA,
a similar series of peaks in Fdc
occurs if A is fixed and 1/ωA is varied.
In the inset of Fig. 2 we show
typical curves of the average DC
velocity < Vx > vs fdc for an unpinnned incommensurate orbit
at A = 0.43 and a pinned commensurate orbit at A = 0.37,
illustrating
the well defined sharp depinning threshold for the pinned orbits.
The maximum Fdp values
in Fig. 2 do not decrease with
increasing A, indicating that all of the commensurate orbits are
pinned equally well.
In addition the boundaries between the pinned and
unpinned regions are sharp.
Thus the behavior of Fdp vs A
clearly differs from that associated with Shapiro steps
[1,2,10],
where the depinning threshold oscillates
with A according to
a Bessel function J0(A), the peaks in the depinning
force are smooth, and the peak Fdp value
gradually decreases with A.
Figure 3:
The DC depinning force Fdp/Fdp*
vs AC amplitude B/A for the anisotropic case
where A = 0.225 and B is varied. Fdp* is the
depinning force for B = 0.0.
We next consider anisotropic or elliptical orbits for A ≠ B.
In Fig. 3 we show
Fdp/Fdp* vs B/A for
fixed A = 0.225 and varying B, where Fdp* is the
depinning force for B = 0.
Again we find strongly pinned orbits, indicated by
peaks in Fdp;
however, in this case the pinned orbits occur when
the vortex orbit encircles n=m pins.
If a higher value of A is chosen such that
the orbit encircles
two pinning sites in the transverse direction in one period,
commensurate orbits that encircle n=2m pins appear.
The peaks in the depinning curve shown here are much
more symmetric than those in
Fig. 2. This may be due
to the fact that the elliptical orbits are
less perturbed by the square pinning potential
than the isotropic circular orbits.
Figure 4:
Vortex positions (dots) and trajectories (lines) for fixed
A = 0.225 and (a) B = 0.225, (b) B = 0.285 and (d) B = 0.35.
In (c) a sliding orbit is shown for B = 0.3 and fdc > Fdp.
In Fig. 4 we present several anisotropic commensurate orbits
at the depinning peaks shown in Fig. 3.
Here the orbits for Fig. 4(a,b,d)
encircle n=1, 2, and 3 pins, respectively.
This trend continues for the higher peaks.
In Fig. 4(c) we also show a sliding orbit at
B = 0.3 with fdc above depinning.
We now discuss experimental systems in which these phases
can be observed.
For superconductors with periodic antidot arrays, the pin geometry
should be chosen such that the pins have vortex saturation numbers of one,
so that above Bϕ the additional vortices will sit in the
interstitial regions.
For pins with higher saturation numbers,
commensuration effects should still be observable at higher
matching fields when interstitial vortices start to appear.
Samples with very low intrinsic pinning should be used to
enhance the effect.
Samples in which Shapiro steps for moving interstitial vortices
have been observed would be ideal.
As in the case of Shapiro steps [10],
the variation of the depinning force versus AC drive amplitude
should be most pronounced at
filling fractions where the interstitial vortices form a symmetrical
pattern and the interstitial vortex interactions effectively cancel,
such as at B/Bϕ = 2.0, 1.75, 1.5, and 1.25.
Using samples such those in Ref.[9]
where Shapiro steps are observed,
for a pinning lattice spacing of a = 2 μm,
ω = 2πν, and ν = 40 MHz,
commensuration effects
should be observable with applied crossed AC currents from
0 to 10 Ic where Ic is the critical current.
Superconductors with rectangular and triangular
(rather than square) pinning arrays should
also exhibit these phenomena; however,
the stable orbits would encircle different numbers of
pins than the orbits described here.
In superconductors with
magnetic dot arrays, the dots can act as
long range repulsive sites, so that
pinned orbits of the type described here
could be observed at vortex filling fractions such
as B/Bϕ=1 and 1/2.
In addition, superconductors with 2D smoothly modulated surfaces
should also produce commensurate and incommensurate orbits.
Other promising systems
in which these phases could be observed include 2D JJ
arrays, 2D atomic friction models, and colloids on
optical pin-scapes [16].
A possible application of our results would be
particle segregation in multi-species
systems, such as colloids on periodic substrates,
where the different species have
different mobilities.
Here it should be possible
to tune the AC drive such that one species would
be pinned and another depinned.
The species could then be segregated
with a DC drive.
To summarize,
we have numerically studied the motion of vortices
interacting with a 2D
periodic potential created by immobile vortices located at
pinning sites placed in a square array. We apply
two AC drives,
perpendicular to one another and out of phase by 90°,
such that the vortices move in a circle in the absence
of a substrate potential.
For AC drives of equal amplitude,
as a function of increasing AC amplitude
we find a series of
pinned orbits enclosing n=m2 pinning sites. Each of these orbits
has a finite depinning threshold to an additional applied DC force.
At AC amplitude values
between these pinned orbits, the vortices are delocalized
and diffuse through the sample, and the DC depinning threshold is zero.
Experimentally these states can be observed as a series of pinned and
non-pinned regions as a function of AC amplitude or frequency.
For anisotropic AC drives,
we find a series of asymmetric pinned orbits
which enclose n=m pinning sites.
We call our model the vortex pinball model in analogy to the
electron pinball system for electron cyclotron motion in anti-dot arrays.
We also suggest other systems in which these
phases can be observed experimentally.
Acknowledgements: This work was supported by the US Dept. of Energy
under contract W-7405-ENG-36.
S.P. Benz, M.S. Rzchowski, M. Tinkham, and C.J. Lobb,
Phys. Rev. Lett. 64, 693 (1990);
K.H. Lee, D. Stroud and J.S. Chung, Phys. Rev. Lett. 64, 962 (1990);
M. Octavio, J.U. Free, S.P. Benz, R.S. Newrock, D.B. Mast, and C.J. Lobb,
Phys. Rev. B 44, 4601 (1991);
D. Dominguez and J.V. Jose, Phys. Rev. Lett. 69, 514 (1992).
R.E. Thorne, W.G. Lyons, J.W. Lyding, J.R. Tucker, and J. Bardeen,
Phys. Rev. B 35, 6360 (1987);
M.J. Higgins, A.A. Middleton, and S. Bhattacharya, Phys. Rev. Lett. 70,
3784 (1993).
A.T. Fiory, A.F. Hebard, and S. Somekh, Appl. Phys. Lett. 32, 73
(1978);
M. Baert, V.V. Metlushko, R. Jonckheere, V.V. Moshchalkov, and
Y. Bruynseraede, Phys. Rev. Lett. 74, 3269 (1995);
K. Harada, O. Kamimura, H. Kasai, T. Matsuda, A. Tonomura,
and V.V. Moshchalkov, Science 274, 1167 (1996);
V.V. Metlushko, U. Welp, G.W. Crabtree, Z. Zhang, S.R.J. Brueck,
B. Watkins, L.E. DeLong, B. Ilic, K. Chung, and P.J. Hesketh,
Phys. Rev. B 59, 603 (1999).
C. Reichhardt, C.J. Olson and F. Nori, Phys. Rev. Lett. 78, 2648 (1997);
C. Reichhardt and F. Nori, Phys. Rev. Lett. 82, 414 (1999);
V.I. Marconi and D. Dominguez, Phys. Rev. Lett. 82, 4922 (1999);
G. Carneiro, Phys. Rev. B 62, R14661 (2000);
V. Gotcheva and S. Teitel, Phys. Rev. Lett. 86, 2126 (2001).
J.I. Martín, M. Vélez, J. Nogués, and I.K. Schuller,
Phys. Rev. Lett. 79, 1929 (1997);
D.J. Morgan and J.B. Ketterson, Phys. Rev. Lett. 80, 3614 (1998);
J.I. Martín, M. Vélez, A. Hoffmann, I.K. Schuller,
and J.L. Vicent, Phys. Rev. Lett. 83, 1022 (1999);
M.J. Van Bael, J. Bekaert, K. Temst, L. Van Look, V.V. Moshchalkov,
Y. Bruynseraede, G.D. Howells, A.N. Grigorenko, S.J. Bending,
and G. Borghs, Phys. Rev. Lett. 86, 155 (2001).
R. Fleischmann, T. Geisel and R. Ketzmerick,
Phys. Rev. Lett. 68, 1367 (1992);
J.H. Smet, K. von Klitzing, D. Weiss, and W. Wegsheider,
Phys. Rev. Lett. 80, 4538 (1998);
J. Wiersig and K.H. Ahn, Phys. Rev. Lett. 87, 026803 (2001).
