Hysteretic Depinning and Dynamical Melting for Magnetically Interacting
Vortices in Disordered Layered Superconductors
C. J. Olson and C. Reichhardt
Department of Physics, University of California, Davis, California
95616
V. M. Vinokur
Materials Science Division, Argonne National Laboratory,
Argonne, Illinois 60439
(Received 2 July 2001; published 17 September 2001)
We examine the depinning transitions and the temperature versus
driving force phase diagram for magnetically interacting pancake vortices
in layered superconductors.
For strong disorder the initial depinning is plastic
followed by a sharp hysteretic transition to a 3D ordered state for increasing
driving force.
Our results are in good agreement with theoretical predictions
for driven anisotropic charge density wave systems.
We also show that a temperature induced peak
effect in the critical current
occurs due to the onset of plasticity between the layers.
A vortex lattice subject to quenched disorder offers an ideal system
for studying glassy dynamics governed by the competition between the
ordering forces (the vortex-vortex repulsion) and disorder [1].
One of the key issues of glassy dynamics is the nature of the
depinning transition separating a pinned state of the lattice at small
external drives from the sliding regime that occurs when
the external force exceeds a threshold depinning value. The early numerical
simulations revealed that in the system with strong
disorder plastic depinning occurs
with the vortices tearing past one
another and flowing in complex patterns [2].
The onset of plasticity at depinning was later confirmed in
experiment [3] as well as subsequent numerical studies
[4,5] and was demonstrated to
play a crucial role in
the pre-melting peak effect where the
critical current exhibits a striking increase
as a function of temperature or field [1,3] prior to
vortex lattice melting and
related phenomena.
As the applied driving force is further increased,
the effect of the quenched disorder is reduced and the vortex
lattice can show a remarkable reordering
transition at a driving force Fr [4].
At this transition a considerable portion of the vortex lattice
order is restored and the motion is highly coherent.
The reordering force Fr is expected to
diverge as the temperature T approaches
the vortex lattice melting temperature Tm [4].
Transport measurements provide
strong evidence for this scenario [3].
Additional studies on the
depinning and reordering transition and the nature of the
coherently moving phases have been
conducted theoretically [6],
numerically [7], and experimentally [8,9,10],
showing both
non-hysteretic continuous depinning transitions
and abrupt hysteretic V(I) curves.
The appearance of the hysteretic
portion of the V(I) curves depends on how the vortex
lattice was prepared: the initial current ramp shows a large
hysteretic critical current but subsequent ramps produce
only continuous non-hysteretic V(I) characteristics [9].
The underlying mechanisms of depinning and its interplay with the
reordering transition remain an open question. The issues to be
clarified include
a determination of the conditions under which
the depinning transition shows hysteretic or
switching responses, as well as an understanding
of the nature of these phenomena.
The mechanism by which a three-dimensional
lattice depins or reorders is also an open question,
as well as the issue of whether
the onset of plasticity coincides with a sudden jump in the
critical current (the peak effect).
Some of these issues were addressed in [11] in a context
of a
driven three-dimensional anisotropic charge density
wave system (CDW) with disorder. It was shown that if disorder is
sufficiently strong[12], the dependence of the lattice
velocity on the applied drive becomes two-valued or bistable:
at the same
applied force (close to the depinning force of the topologically
disordered phase) the lattice can either remain at rest or slide
depending on the history. This means that the depinning transition
shows switching behavior and hysteresis. The details of depinning are
determined by particulars of the system involved. For a layered
anisotropic CDW considered in [11] the depinning occurs
in two stages. The initial plastic depinning, where the decoupled
2D CDWs slide independently in each layer, is followed
by
a sharp transition or crossover at higher drives to a
faster coherently moving
3D solid CDW phase. Reversing the process reveals hysteresis:
pinning (immobilizing) of a coherent 3D CDW occurs at smaller
drives.
It was also shown that for
weak disorder there is only a
single non-hysteretic depinning
transition directly to the 3D coherently moving state.
The results of
[11] are very general, and the particular model used there
can be straightforwardly generalized
to other periodic media in quenched disorder.
A detailed study of the character of the many-valued v−F dependence
near depinning was recently
carried out in [13] within the framework of a mean-field model
for a visco-elastically driven vortex configuration, where a dynamic
bistability and, accordingly, a hysteretic plastic depinning was
found to occur for sufficiently strong disorder.
Note that the layered CDW model
of [11] can be conveniently
applied directly to
highly anisotropic superconductors where
the magnetic interactions between pancake vortices
dominate [14].
In this system both a 2D regime, in which vortices in each layer move
independently from the other layers, and a 3D regime, where vortices in
different layers form coherently moving lines, should exist.
In this work we investigate the dynamics of magnetically interacting vortices
in a three-dimensional layered superconductor to numerically test the
predictions of
Ref. [11].
Our model is relevant to highly layered superconductors such
as BSCCO as well as artificially layered superconductors in which
the magnetic interactions dominate. Recent numerical simulations
with this model have found that for increasing applied driving force,
a transition from a decoupled plastically flowing state
to a coupled 3D coherently moving lattice occurs [15,16].
A sharp static 2D to 3D disorder induced transition also appears as a
function of field and disorder.
In this model,
the dynamic phase diagram as a function of driving
force and temperature, as well as hysteretic and
switching behavior in the current-voltage characteristics, have
not been studied until now.
We find that for strong disorder, the initial
depinning is 2D with the vortices flowing plastically and independently
in any one layer.
For increased drives we see a sharp transition to a coherently moving
3D vortex lattice
which manifests itself in an abrupt increase
in the vortex velocities.
We observe strong hysteresis when the driving current is cycled.
We have also investigated the driving versus temperature phase diagram.
We consider a three-dimensional layered superconductor in which
the vortices are modeled as repulsive
point particles confined to a layer with
an equal number of vortices per layer.
The overdamped equation of motion for vortex i is given by
fi = −
Nv ∑ j=1
∇iU(ρi,j,zi,j)+ fivp + fd + fT = ηvi ,
where Nv is the number of vortices, ρi,j and
zi,j are the distance
between pancakes i and j in cylindrical coordinates, and η = 1.
The system has periodic boundaries in-plane and open boundaries
in the z direction [17].
The magnetic energy between pancakes is [14,18]
U(ρi,j,0)=2dϵ0
⎛ ⎝
(1−
d
2λ
)ln
R
ρ
+
d
2λ
E1
⎞ ⎠
U(ρi,j,z)=−sm
d2ϵ0
λ
⎛ ⎝
exp(−z/λ)ln
R
ρ
− E2
⎞ ⎠
where
ϵ0 = Φ02/(4πλ)2,
λ is the London penetration depth,
R = 22.6 λ, the maximum radial distance,
E1 = ∫∞ρdρ′ exp(−ρ′/λ)/ρ′,
E2 = ∫∞ρdρ′ exp(−√{z2 + ρ′2}/λ)/ρ′,
d=0.005λ is the interlayer spacing,
and we set sm=2.0.
The uncorrelated pins are modeled by parabolic traps that are
randomly distributed in each layer. The vortex-pin interaction is given by
fivp = −∑k=1Np,L (fp/ξp)(ri − rk(p)) Θ((ξp − |ri − rk(p) |)/λ),
where the pin radius ξp=0.2λ, the
pinning force fp=0.02f0*, and f0*=ϵ0/λ.
We have simulated a 16λ×16λ system
with a vortex density of nv = 0.35/λ2 and a pin
density of np = 1.0/λ2 in each of L=8 layers. This
corresponds to Nv=80 vortices and Np=256 pins per layer, with
a total of 640 pancake vortices.
For our finite temperature simulations the temperature is
modeled as a Langevin kick fT where
< f(t)T > = 0 and < f(t)Tif(t)Tj > = 2ηkBTδij.
Figure 1: (a) Vortex velocities Vx versus fd for a system
with fp=0.02f0* at T=0
as the driving force is first increased from
fd = 0.0 to fd = 0.02 and then decreased back to fd = 0.
(b) Cz versus fd for the system in (a) showing the sharp,
hysteretic recoupling and decoupling transitions.
(c) Vx versus fd for a system with fp = 0.01f0*.
(d) Cz versus fd for the system in (c) showing
that there is a non-hysteretic 3D pinned to 3D moving transition.
In Fig. 1(a) we show the average vortex velocities in the
direction of drive, Vx= < vx > , as the applied driving force
fd is increased and then decreased,
and in Fig. 1(b) we show the corresponding c-axis correlation function,
Cz = 1 − 〈(|(ri,L − ri,L+1)|/(a0/2))Θ( a0/2 − |(ri,L − ri,L+1)|)〉, where
a0 is the vortex lattice constant.
At fd=0 the vortices
are pinned and uncorrelated in the z-direction as is indicated by
the fact that Cz has a low value.
The initial depinning, at fdp = 0.01f0*,
is 2D with the vortices remaining
uncorrelated in the z-direction. Near fd = 0.013
the vortex velocities sharply increase, coinciding with a sharp
transition to a coupled moving 3D state as seen in the jump up in
Cz. At higher drives
the vortex velocity linearly increases with increasing
fd. Upon reducing fd from this linear regime the system remains
in the 3D state for driving forces well below fd=0.013. At
fd=0.007
there is a sudden transition to a 2D pinned state as
seen in the drop in Vx and Cz. If the applied driving force is again
increased
the velocities and Cz will follow the same hysteresis loop.
In Fig. 1(c,d) we show Vx and Cz for the same system as in
Fig. 1(a,b) but with a lower disorder strength of fp = 0.01f0*.
Here the depinning transition is
non-hysteretic with a single
depinning threshold to the 3D coherently moving state as seen by the
fact that Cz = 1.0 at all times.
For a given coupling strength value sm,
there is a critical pinning force above which there is
a static 3D-2D transition. Hysteretic V(I) curves can only be
observed above, but close to, this transition.
The width
of the hysteresis in the V(I) curves is largest for
disorder strengths just above the static 2D-3D transition and gradually
narrows for increasing disorder strength.
Figure 2: Pancake vortex positions seen from the top of the sample,
for the system with fp=0.02f0 shown in Fig. 1(a,b).
Pancakes on a given layer are represented by circles with a fixed
radius; the radii increase from the top layer to the bottom. Vortices
that have aligned into vortex lines appear as thick circles.
(a) At fd=0.011f0*, in the 2D plastic flow regime.
(b) At fd = 0.015f0* in the coherently moving 3D phase.
The behavior we observe
is in excellent agreement with the theoretical predictions
of [11].
We note that unlike the
anisotropic charge-density wave system of [11],
where only inter-plane plasticity or slipping can occur,
in the vortex system in-plane plasticity is possible within each
layer.
The 2D-3D transition in the case of the vortices occurs by
simultaneous recoupling of the vortices between layers
and the reordering of the vortices in plane.
To illustrate this, in Fig. 2(a) we show a top
view of the vortices in the moving 2D regime,
indicating that the vortices are
uncorrelated from layer to layer and are disordered in plane.
In contrast, Fig. 2(b) shows that in the
moving 3D regime, the vortices are aligned
between layers and possess a triangular ordering.
Figure 3: The fd versus T/Tm dynamic phase diagram for a system
with fp=0.01f0*.
Filled circles: Depinning line. Open squares: Dynamic reordering line.
In Fig. 3 we show the driving force versus temperature phase diagram for
a system with fp=0.01f0*.
The temperature is plotted in units of Tm=0.00045, the temperature
at which the clean system undergoes melting in the form of
a single sharp transition from a 3D lattice to a 2D pancake gas.
As shown in Fig. 3, at T/Tm < 0.005 for low drives there is a
3D pinned phase where the vortices remain aligned. At these temperatures,
for fd/f0=0.001 there is a non-hysteretic depinning transition
directly to the coherently moving 3D state.
We find a static transition to the 2D state
at T/Tm = 0.005, which we label Tdc.
For T > Tdc the depinning transition is 2D, and as the driving
force is further increased a dynamically induced reordering of
the vortices occurs.
As T is further increased above Tdc
the size of the 2D pinned phase is reduced, while the drive at
which the
2D plastic flow to 3D coherent flow transition occurs
diverges as T approaches Tm, in agreement with
the dynamical freezing model of Koshelev and Vinokur [4].
The thermally induced decoupling seen in Fig. 3
also coincides with a sharp increase in the critical current
or a "peak effect."
There has been considerable work on dimensional transitions in the
context of the peak effect in layered
superconductors. The static 3D-2D transition
and critical current increase in layered
superconductors have been previously studied as a function of interlayer
coupling and magnetic field [19,20].
Koshelev and Kes have proposed
theoretically that magnetically coupled vortices can show a
first order transition from a 3D to a 2D system
under certain conditions [21].
The sharpness in the 3D-2D transition we observe here suggests that
it is first order in nature.
The 3D-2D transition can be seen as a temperature induced peak
effect in which the combination of temperature and
quenched disorder bring about a decoupling transition where the
individual pancakes can adjust to the optimal pinning configurations.
The phase diagram in Fig. 3 is very similar to the one proposed
by Bhattacharya and Higgins [3]
in which there is an elastic depinning
regime for a certain temperature range, above which the peak effect and
the onset of plasticity simultaneously appear.
In our model the peak effect is caused not by the onset of plasticity
in the a − b plane but by the
onset of plasticity in the c-axis with the plasticity only
occurring for T > Tdc.
We find that the temperature Tdc moves further below Tm
as the disorder in the sample is increased.
Thus for large enough disorder the 2D phase extends all the way
to T = 0, and the dynamic phase diagram becomes the same as the
one proposed in in Ref. [4].
Figure 4:
Vx versus fd responses for varied T. Circles: temperatures below
the decoupling temperature Tdc: T/Tm=0.00027 and 0.0013.
Lines, right to left: T/Tm= 0.005, 0.014, 0.05, 0.11, 0.22,
0.5, 0.9.
In Fig. 4 we show the
Vx versus fd curves, which correspond to experimental V(I) curves,
for increasing T. For
T < Tdc there is a sharp elastic depinning transition
at which all the vortices start moving at once in the
3D state.
For T > Tdc the depinning response is more continuous since
the 2D depinning occurs inhomogeneously, with certain regions moving while
other regions remain pinned. As T is further increased in the 2D regime
the depinning transition falls at driving forces lower than that
at which the 3D depinning transition occurred.
Although for these higher temperatures the 2D pancakes depin
at a lower applied drive than the 3D state,
the overall velocity above depinning is still less than
that of a coherently moving 3D system at the same drive so that
a crossing of the IV curves occurs.
To summarize, we have numerically
investigated the dynamics of magnetically coupled
vortices in layered superconductors interacting with quenched
disorder.
For sufficiently strong disorder the depinning is 2D. Here the vortices
flow plastically in each layer and are uncorrelated from layer
to layer. For
higher drives there is a sharp hysteretic transition to a
coherently moving ordered
3D vortex lattice. For weak disorder there is a non-hysteretic
depinning transition from a pinned 3D to a moving 3D state. The two stage
2D-3D hysteretic depinning transition is in good agreement with the theoretical
predictions of Ref. [11].
We have also mapped out the dynamic phase diagram
as a function of driving force and temperature. At low temperatures we
observe non-hysteretic 3D pinned to 3D moving transitions. For
increased temperature there is a static 3D-2D transition which coincides
with a peak in the critical current as well as the onset of plasticity in
the c-axis. The drive at which a dynamically induced
2D-3D dynamic transition diverges as Tm is approached.
Our dynamic phase
diagram is very similar to that proposed in Ref. [3].
CJO and CR thank B. Janko for his kind hospitality at Argonne National
Laboratory. This work was supported
by CLC and CULAR (LANL/UC), by NSF-DMR-9985978, and
by Argonne National Laboratory through
U.S. Department of Energy, Office of Science under contract
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