C. Reichhardt, C. J. Olson, and Franco Nori
Department of Physics, The University of Michigan, Ann Arbor,
Michigan 48109-1120
(Received 23 December 1996)
We present results from extensive simulations of driven vortex lattices interacting with periodic arrays of pinning sites. Changing an applied driving force produces a rich variety of novel dynamical plastic flow phases which are very distinct from those observed in systems with random pinning arrays. Signatures of the transition between these different dynamical phases include sudden jumps in the current-voltage curves as well as marked changes in the vortex trajectories and vortex lattice order. Several dynamical phase diagrams are obtained as a function of commensurability, pinning strength, and spatial order of the pinning sites. [S0031-9007(97)02855-X]
PACS numbers: 74.60.Ge, 74.60.Jg
The dynamics of driven vortex lattices interacting with quenched disorder have recently attracted considerable attention. Both theoretical [1-5] and experimental [6] studies have suggested the exciting possibility that these systems exhibit novel dynamic phase transitions between different flow regimes as a function of driving force. In samples with random pinning, evidence for a transition from a plastic flow phase to a strongly driven ordered homogeneous phase has produced current debate over whether the strongly driven phase forms a moving crystal [2] or moving ordered glass [3,4]. The dynamic phases may also be relevant to many other systems, such as Josephson junction arrays, charge-density waves, and electron crystals. Although much work has been done on dynamic phases in systems with random disorder, the case of periodic pinning has not been addressed. We show that driven vortex lattices interacting with periodic pinning exhibit a number of novel plastic flow phases which are not observed in random pinning arrays. Further, the onset of these different phases produces microscopic features in the vortex structure and flow patterns and gives rise to very pronounced features in macroscopically measurable current-voltage curves.
General interest in periodic arrays of pinning sites has increased now that it is possible to construct samples with well defined periodic pinning structures in which the microscopic pinning parameters, such as size, depth, periodicity, and density, can be carefully controlled [7-9]. Interesting commensurability effects are observed both in magnetization measurements [7] and with direct imaging [8]. Periodic pinning arrays are also of technological importance since they can produce higher critical currents than an equal number of randomly placed pins [9].
When the pinning radius is much smaller than the lattice spacing,
the "disorder" in the system can be fine-tuned by
changing the commensurability. At B/=1, where
is the field at which the number of vortices Nv
equals the number of pinning sites Np, the vortex lattice
locks into a periodic pinning array [7-9]
and the pinning force is maximized. For
B/
> 1, the vortex
lattice contains two species of vortices: the pinned vortices
that are commensurate with the pinning array, and the generally more
weakly pinned interstitial vortices that are caged by vortices at
the pinning sites. When B/
< 1,
the vortex lattice contains
a well defined number of vacancies which have their own dynamical
behavior.
In order to study the two-dimensional (2D) dynamics of rigid flux lines driven over periodic arrays of pinning sites, we have performed a large number of T = 0, current-driven molecular dynamics (MD) simulations. Unlike previous current-driven simulations [1,2,4,5], we examine the effects of periodic pinning arrays rather than random arrays, and cover a much larger range of the microscopic pinning and system parameters, allowing us to construct a series of detailed dynamical phase diagrams.
We numerically integrate the overdamped equations of motion
[10]:
fi=fivv+fivp+fd=vi.
Here fi is the total force acting on vortex i
and we take
=1.
The force from the other vortices is
where K1(r/
)
is a modified Bessel function,
is the penetration depth, and
=(ri-rj)/|ri-rj|.
A cut off is placed on K1(r/
)
after it reaches an extremely small value at
r=6
.
The pinning force is
Here,
is the step function,
rk(p) is the location of pinning site k,
fp is the maximum pinning force, and
=(ri-rk(p))/|ri-rk(p)|.
The Lorentz force is modeled as a uniform force fd.
All lengths, fields and forces are given in units of
,
/
, and
f0=
/8
,
respectively.
We focus on experimentally accessible parameters that are
close to those used in recent experiments [7].
Fixing the sample size at
36 x 36
,
with periodic boundary conditions, and the pinning radius
at rp=0.3
,
we examine an 18 x 18
square pinning array with Np=324, giving a pinning
density of np=0.25/
.
We slowly increase the driving force fd along the
horizontal symmetry axis (x axis) of the pinning lattice
and compute the average velocity in the x direction,
.
This quantity is proportional to a macroscopically measured
voltage-current V(I) curve.
In order to separate the different effects that each of the
pinning and system parameters have on the vortex dynamics, we
fix all the parameters and vary only one at a time.
In Fig. 1 we present a typical V(I) curve for
B >
as the driving force is linearly increased from 0 to
0.8f0 with fp=0.625f0,
=0.25
/
,
and B/
=1.062.
The V(I) curve exhibits several remarkable features which
clearly appear as discontinuous jumps and drops in
Vx. We label each of the features as regions I through V,
and approximate the fraction of flux lines which are mobile at a
specific driving force with:
=Vx/fd.
As fd is increased in region I, Vx is zero,
indicating that the vortex lattice is pinned. At
fd=0.146f0,
the onset of region II is marked by a finite Vx, caused by
the depinning of interstitial vortices. Here,
=0.06,
confirming that only the interstitial vortices are mobile
since the percentage of vortices above
is also
(B -
)/
=0.06.
Region III begins at
fd=0.406f0, where a very sharp jump up in Vx
is seen, along with an increase in the number of mobile
vortices to
=0.44.
The velocity fluctuations
Vx
are also much larger.
At fd=0.462f0, region IV appears with a
sudden drop in Vx, with
=0.23 and a
reduction in
Vx.
Finally, at fd=0.612f0, just under the pinning
force of each pinning site (fp=0.625f0), the
entire lattice becomes mobile with
=1, and the system enters region V.
To further characterize the dynamic phases, we have performed a number of hysteresis runs where the driving force is slowly increased and then decreased. In the inset of Fig. 1 we show a typical curve for the system with the same parameters as in Fig. 1. There is little hysteresis for the transition IV-V. However, there is a very strong hysteresis at the transitions II-III and III-IV that persists in larger system sizes and also for low values of the spatial disorder. The hysteresis and the sharp jumps suggest that the II-III and III-IV phase boundaries might be first order. Details will be presented elsewhere [11].
We use a series of snapshots of the vortex position and vortex flow paths for regions II through V to show explicitly that the features in the V(I) curve correspond to different plastic flow phases. Figure 2(a) shows the vortex trajectories in region II of the current-voltage curve from Fig. 1. It is clear that in region II only the interstitial vortices are mobile while the commensurate vortices remain pinned. The motion is confined to one-dimensional (1D) channels between the rows of pinning sites due to the square symmetry imposed by the pinned vortices. Such flow behavior for vortices in square pinning arrays has recently been experimentally observed [8].
Figure 2(b) illustrates that the vortex trajectories differ greatly from region III to II. The vortex lattice is now disordered, and the flow is no longer 1D but consists of channels that wind in both the x and y directions. Pin-to-pin motion also appears. Unlike the motion in region II, where only the interstitial vortices move and vortices at the pinning sites remain pinned, all the vortices in region III take part in the motion, with any one vortex moving for a time and then being temporarily trapped [12].
Another significant change in the vortex motion appears when
the system enters region IV, as seen in Fig. 2(c).
The vortex trajectories become more ordered and return to an
exclusively 1D flow, with the mobile vortices moving
along the pinning rows rather than between
the rows as in region II. Only certain rows are mobile,
and in these rows the additional vortices above
leave their positions between the pinning rows to create moving
incommensurate 1D structures along the pinning row.
An entire row does not move continuously, but instead a pulse
appears in which only four vortices, near the incommensurate
segment of the vortex row, are mobile.
As this pulse or discommensuration moves across the sample,
each vortex in the row is displaced by a single pinning
lattice constant a. This disturbance is thus crossing
the sample much more rapidly than the vortices themselves.
The vortex trajectories for region V, in which the entire lattice is moving, are shown in Fig. 2(d). Some portions of the vortex lattice have a distorted triangular order, although the incommensurabilities from region IV are still present. The flow remains strictly 1D and along the pinning rows as in region IV, except that now all the rows are mobile. Rows with an incommensurate number of vortices move faster than the commensurate rows. As fd is increased further, the density of incommensurabilities and the vortex lattice structure do not change, so the system is always undergoing plastic flow and a moving crystal is not formed.
The onset of these different phases is described with force
balance arguments that take into account the coupling of
the two different species of vortices, interstitial and
(pinned) commensurate.
In region II, while the commensurate vortices remain pinned,
the interstitial vortices begin flowing at a well defined
driving force and exert a force fc-ic on the pinned
commensurate vortices.
The total force on a commensurate vortex thus consists of
the driving force fd, pinning force
fp, and forces fc-c from commensurate and
fc-ic interstitial vortices. The commensurate vortex
will remain pinned as long the following inequality holds:
|fp| > |fd+fc-ic+fc-c|. (1)
Since here we are using a square pinning lattice, from symmetry
we have fc-c=0. If there are no incommensurate
vortices, fc-ic=0, and commensurate vortices depin at
|fd| = |fp|.
When interstitial vortices are
present, the term fc-ic causes commensurate
vortices to depin before |fd| = |fp|.
These vortices depin more vortices so that the number of mobile
vortices increases and region III appears. As long as
|fd| < |fp|, not all the vortices will be
mobile so that < 1.
For the parameters used in Fig. 1, the density of interstitial
vortices is sufficiently low that they do not interact
significantly with each other. In this case, we can solve
Eq. (1) for the transition from region II to region III to give
(2)
With the parameters for the onset of region III, Eq. (2) gives
fp=0.621f0, which is in very good agreement with
the value of fp=0.625f0 used in the simulation.
The appearance of 1D motion exactly along the rows of pins
in region IV might seem counterintuitive since for
B/ > 1 and fd=0, when the vortices are
not moving, an incommensurate vortex located along a pinning
row is unstable to perturbations in the y direction and will
fall into the interstitial area between rows.
For moving vortices, the situation is quite different since
the vortices spend part of their time in the pinning sites.
The pinning sites create a stabilizing force against perturbations
in the transverse direction, confining the motion along the
pinning rows. When the density of interstitial vortices is
low, the onset of region IV occurs when the driving force is
strong enough that interstitial vortices can depin commensurate
vortices from a distance a/2 in the longitudinal direction.
This distance, and especially the repulsion from the remaining
pinned vortices, allows the interstitial vortex to move towards
the just-vacated pin site. For sufficiently strong fd,
it will remain moving along the pinned rows
[11].
The transition from region III to region IV should occur at
(3)
where
r1=(a/+rp)/
.
With the parameters used in Fig. 1 at the onset of the 1D incommensurate
flow, Eq. (3) gives fp=0.624f0,
which is in very good agreement
with the numerical value for fp shown in Fig. 1.
To better characterize the flow behavior, we systematically vary
fp with the rest of the parameters fixed.
The resulting phase diagram in Fig. 3(a) outlines the onset of the
different dynamical phases. As fp is increased, region I
saturates at a value of
fd0.146f0. This occurs
because, although the pinning force fp of the pinning sites
is being increased, the vortex-vortex interactions which determine
the interstitial pinning force are not changed.
Region II only occurs when fp > 0.37f0, since for
fp < 0.37f0, Eq. (2) cannot be satisfied even for very
low fd, and as soon as the interstitial vortices move they
start to depin commensurate vortices.
The same argument applies for the onset of region IV, which
extends to even lower fp values. The II-III phase
boundary follows Eq. (2), which for high fd goes as
fp
fd,
in agreement with the phase diagram.
Similarly, the III-IV boundary follows Eq. (3), which is
also linear for large fd.
The onset of region V also goes as
fp
fd.
Next we vary the commensurability, from B/=0.75
to 1.7, producing the phase diagram in Fig. 3(b). Just
above B/
=1,
the five phases of Fig. 1 are present.
As B/
is increased, the disordered-flow
region III grows while the ordered-flow regions II and IV shrink. This
is expected since an increase in
B/
effectively
introduces more disorder via the addition of more interstitial vortices.
For B/
> 1.3, the flow becomes more disordered,
the II-III phase boundary becomes ill-defined, and the flow
in region V is no longer composed of 1D incommensurate flow
along the pinning rows but has a number of vortices flowing
between the pinning rows.
At B/
=1, the commensurate case, we find only
two phases: pinned and flowing, with the onset of flow
occurring at fd
fp.
The vortex flow for the
commensurate case is elastic since, unlike the case for
B/
> 1,
there are no discommensurations that cause certain rows to move faster. For
B/
< 1, where a
number of vacancies appear in the vortex lattice, we observe
a new vacancy flow phase, marked region VI. The depining
force for the onset of vacancy motion is considerably higher
than that for the onset of interstitial motion, in agreement
with experiments [8]. Flow in V' is like in V,
but now the faster-moving rows have vacancies.
Figure 3(c) shows a phase diagram in which the positions of
the pinning sites are gradually disordered by randomly
displacing them up to an amount r
from the ordered pinning lattice position. In terms of the pinning lattice
constant a, we consider the case
r=a/2 to be
a good approximation to a random pinning array. The
disordered region III grows and dominates the phase diagram
for disorder greater then
r=a/6,
so that only three phases occur, in agreement with other simulations of random
pinning arrays [2,4,5].
Figure 3(d) shows the phase diagram for increasing both vortex
nv and pin np density, determined by
.
Here, the ordered-flow phase IV grows while the disordered-flow
region III shrinks. Increasing
increases nv
and thus effectively decreases fp. As seen in Fig. 3(a),
this suppresses II and III, while favoring IV. Phase diagrams
for varying pinning radius rp, angle of drive, temperature,
and triangular pinning arrays will be presented and discussed
in detail elsewhere [11].
In conclusion, we have demonstrated several novel dynamical phases which are very distinct from those found in random pinning arrays. These phases are marked by pronounced changes in the V(I) curves and noise fluctuations which should be very accessible experimentally. We have directly related these features to the pronounced changes in the vortex lattice structure and flow pattern. These phases result from the coupling between two species of vortices. We have shown that certain phases show strong hysteresis while others do not. The dependence of these phases on various pinning and system parameters has been extensively studied and summarized in a series of phase diagrams [11]. We hope that these results encourage the experimental search of these new dynamic phases.
Computing services were provided by the University of Michigan Center for Parallel Computing, partially funded by NSF Grant No. CDA-92-14296.
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