Moving Wigner Glasses and Smectics: Dynamics of Disordered Wigner Crystals
C. Reichhardt(1), C.J. Olson(1),
N. Grønbech-Jensen(2), and
Franco Nori(3)
(1) Department of Physics, (2) Department of Applied Science,
University of California, Davis, California 95616
(3)
Center for Theoretical Physics
and Physics Department, The University of Michigan, Ann Arbor, MI 48109-1120
(Received 25 July 2000)
We examine the dynamics of driven classical Wigner solids
interacting with quenched disorder from charged impurities.
For strong disorder, the initial motion is plastic, in
the form of crossing winding channels.
For increasing drive, there is a reordering
into a moving Wigner smectic with the electrons
moving in separate 1D channels. These different
dynamic phases can be related to the conduction noise
and I(V) curves.
For strong disorder, we show criticality in the
voltage onset just above depinning.
We obtain the dynamic phase diagram for driven Wigner
solids and demonstrate a finite threshold of force for
transverse sliding, recently observed experimentally.
Ordered arrays of charged particles
have been studied in the context of
colloidal suspensions,
atomic-ion Wigner crystals,
semiconductor heterostructures,
quantum computers,
astrophysics, biophysics, plasmas,
and arrays of metallic islands interconnected
by tunnel junctions [1].
A revival of general interest in charged
arrays has been fueled by the observation,
in 2D heterostructures, of
nonlinear I-V curves
exhibiting a threshold as a function of an
externally applied electric field
[2,3],
indicating the presence of a Wigner solid (WS) [4]
that has been pinned by disorder in the sample.
The depinning threshold
can
vary up to two orders of magnitude
in different samples [3].
Indeed, experiments [1]
based on transport
and photoluminescence
provide indirect evidence
for the existence
of the WS, and demonstrate
the very important role disorder plays
in the dynamics of the WS [5].
It is the purpose of this work to study how disorder
affects the transport properties of the driven WS.
The onset of broadband conduction noise has been interpreted
as a signature of the sliding of a defected WS
[2].
If the electrons retained their order and slid collectively,
narrow band noise resembling that in
sliding charge-density wave systems
should appear.
Simulations [6,7,8] on classical Wigner
crystals interacting with charged defects
indicate that a disorder-induced transition,
from a clean to a defected WS, can occur as
a function of increasing pinning strength.
For strong pinning, the initial depinning is plastic and
involves tearing of the electron crystal [6,7,8].
Many aspects of this transport have not been
systematically characterized, including the current-voltage
I(V) characteristics, conduction noise,
transverse threshold for sliding,
and the electron lattice structure
for varying applied drives.
Plastic depinning has been observed in the related system
of driven vortex lattices in disordered superconductors,
where it is associated with flux motion through
intricate river-like channels [9,10].
Defects in the vortex lattice strongly affect
the depinning thresholds and the voltage noise produced by the system.
For increasing drive the initially defected vortex lattice can
reorder to a moving lattice or moving smectic phase
[10,11,12,13,14].
In the fast moving lattice phase the vortex lattice regains
order in both transverse and longitudinal directions with
respect to the driving force, while in the smectic state
only order transverse to the driving direction appears.
It is unclear a priori whether the same
reordering transitions can occur for
long-range pinning, such as in
the WS interacting with charged
impurities. This is in contrast with the vortex
system, where pinning occurs only on a very
short length scale. Theoretical work on the reordering
transition also considered only short-range
(and often weak) pinning [10,11,12,13,14].
In the strong pinning limit, critical
behavior may occur near the depinning threshold,
leading to velocity-force relations
of the form v ∼ (F − FT)ξ.
As seen in transport in metallic dots,
ξ = 5/3 theoretically [15], and
ξ = 2.0 and 1.58 experimentally [16].
We use numerical simulations to study the dynamics
of a 2D electron system forming a classical WS in
the presence of charged impurities.
For strong pinning, the electron crystal is highly defected and
depins plastically, with certain electrons flowing in well-defined
channels while others remain immobile.
For increasing driving force, the initially disordered
electrons can partially reorder into a moving Wigner smectic
state where the electrons flow in 1D non-intersecting
channels. The reordering is accompanied by a
saturation in the dI/dV curves
and by a change from a broad-band to
a narrow-band voltage noise signature.
For weak disorder, the depinning is elastic
and a narrow-band noise signal appears
at all drives above depinning.
For strong disorder, where the depinning
is plastic, we find criticality in the
velocity force curves in agreement with
transport in metallic dots [15,16].
We map out the dynamic phase diagram as a function
of disorder strength and applied driving force.
Also, we find a finite transverse threshold
for sliding conduction, in agreement
with recent experiments [17].
Our results can also be tested in other systems,
cited in the introduction.
We conduct overdamped molecular dynamics (MD)
simulations using the model studied by Cha and Fertig
[6,7,8]. The energy of the system is:
U =
∑ i ≠ j
e2
|ri − rj|
−
∑ ij
e2
√
(|ri − r(p)j|2 + d2)
(1)
The first term is the electron-electron (Coulomb) repulsion and
the second term is the electron-impurity interaction,
where the impurities are positively charged defects out of plane.
Here,
ri is the location of electron i,
and rj(p) is the in-plane location
of a positive impurity located at
an out-of-plane distance d (measured
in units of a0, the average lattice constant of
the WS). The number (density) of electrons Ni
equals the number (density) of impurities Np, and the
disorder strength is varied by changing d.
Temperature is modeled by Langevin dynamics.
The long-range Coulomb interactions are evaluated with
a Lekner re-summation [18].
We neglect inertial effects from damping, originating from vibrations
induced by interactions with defect ions.
We have also considered (here and in [8]) many other cases
(e.g., ± charged impurities, Ni ≠ Np, etc.)
with consistent results.
The initial electron positions are obtained from simulated
annealing, in which we start from a high temperature
state where the electrons are molten and slowly
cool to a low temperature.
Once the electron configuration
has been initialized, the critical depinning force is determined
by applying a very slowly increasing uniform driving force
fd corresponding to an applied electric field.
After each drive increment, a transient of 104 MD steps is allowed before
collecting data, which we average over the next 104 time steps.
For each drive increment, we measure the
average electron velocity ( ∝ current)
in the direction of drive,
Vx = (1/Ni) ∑iNi∧x·vi.
The Vx versus fd curve corresponds to an I(V)
experimental curve and we will thus use the notation Vx=I and fd=V.
The depinning force fdc is defined
as the drive fd at which Vx reaches a value
of 0.01 that of the ohmic response.
We have studied system sizes Ni
from 64 to 800 and find similar behavior at all sizes.
Most of the results presented here are for systems with
Ni=256.
Figure 1: Electron positions (dots) and eastbound trajectories (lines),
for a sample with Ni/Np=1.0 and d=0.65.
fd/fdc = (a) 1.1, (b) 1.5, (c) 2.25, (d) 3.0, (e) 4.0, (f) 5.0.
When driven through a sample containing strong pinning,
the electron lattice undergoes a
gradual
reordering transition as the driving force is increased.
We illustrate such a reordering transition in Fig. 1
by plotting the
electron trajectories at increasing driving forces for
a sample containing strong disorder of d=0.65.
In Fig. 1(a) the onset of motion occurs through
the opening of a single winding channel, resembling
filamentary vortex flow [9](c). Electrons outside
of the channel remain pinned, and the overall
system is disordered. At fd/fdc = 1.5,
shown in Fig. 1(b),
several channels
have opened, some of which are interconnecting.
The
original channel in Fig. 1(a) has grown in
width, but regions of pinned electrons are still present.
Electrons moving past a pinned electron perturb it, causing
it to move (like a revolving door) in a circular orbit around
the center of the potential minima in which it is trapped.
Several of these electronturnstiles can be observed in Fig. 1(b).
In Fig. 1(c) for fd/fdc = 2.25
there are
regions where electrons do not flow, but none of
the electrons are permanently pinned. Some electrons
become temporaaly pinned before moving again.
If the trajectories are drawn for a sufficiently long time,
the electron flow appears everywhere in the sample, although
there are still preferred paths in which more electrons flow.
In Fig. 1(d) for fd/fdc = 3.0 the
electron flow occurs more uniformly across the
sample. In Fig. 1(e) at fd/fdc = 4.0
the electrons begin to flow predominantly in certain
non-crossing channels, although some electrons jump
from channel to channel. In Fig. 1(f)
for fd/fdc = 5.0 the electron
flow occurs in well defined non-
crossing channels,
which can
contain different numbers of electrons.
A similar channel motion exists for driven vortices
in disordered superconductors [12,13,14,10].
For samples containing very weak disorder, the pinned WS has
six-fold ordering and depins elastically, without generating
defects in the lattice. In this regime, the
electron crystal flows in 1D channels with each channel
containing the same number of electrons. Here, the
transverse wanderings of the electrons are
considerably reduced compared to the case of strong pinning.
Figure 2: Delaunay triangulation for electrons in a sample
with (a) d=0.65 (strong pinning) at fd/fdc=5.0; and
(b) d=1.76 (weak pinning) and fd/fdc=2.0.
The defects on the edges are an artifact of our triangulation
algorithm.
Large
circles indicate (5- or 7-fold) defects in the electron lattice.
To better illustrate the change in the amount of disorder in the
electron lattice,
we show in Fig. 2(a)
the Delaunay triangulation for the electrons
for the same drive as in Fig. 1(f).
Defects, in the form of 5-7 disclination pairs, appear with
Burgers vectors oriented perpendicular to the drive direction.
The computed structure factor (not shown) exhibits only two
prominent peaks for order in the direction transverse to the
drive, consistent with a moving Wigner smectic state.
In Fig. 2(b) we show the Delaunay
triangulation for the moving state in a weakly pinned sample
with d = 1.76 where the initial depinning is elastic.
Here the moving lattice is defect-free.
The structure factor in this case shows four longitudinal peaks
in addition to the two (more prominent) transverse peaks.
Much larger systems would be necessary to determine whether the
system is in a smectic state or in a moving Bragg glass [12]
in this particular case of weak disorder.
Figure 3:
(a) I(V) (dashed line) and dI/dV curves (solid line)
for a sample with d=0.65.
(b) Fraction P6 of sixfold coordinated electrons, as a function
of driving force fd. (c) Fraction Dtr of
transversely wandering electrons.
Inset to (c): Noise spectra for fd/fdc=1.5 in the
plastic flow regime showing broad-band noise (dashed line) and
for fd/fdc=4.0 in the smectic regime
showing a narrow-band signal.
(d)
Vx = (fd−fdc)ξ, in which ξ = 1.61 ±0.10 and
1.71. (e) Vx versus fd
for disorder strengths of d=0.5 and 0.65.
To connect the reordering sequence with a measure
that is readily accessible experimentally, we plot in
Fig. 3(a) the I(V) and dI/dV curves.
A peak occurs in dI/dV at fd/fdc ≈ 3, when
the electrons are undergoing very disordered plastic flow.
In Fig. 3(b) we plot the fraction P6 of
six-fold coordinated electrons as a function of drive.
A perfect triangular lattice would have P6 = 1.
For drives
fd/fdc < 2 the lattice is highly defected.
For fd/fdc > 3 (i.e., past the peak in dI/dV),
the order in the lattice begins to increase.
The value of P6 saturates near fd/fdc ≈ 6
which also coincides with the saturation
of the dI/dV curve. Thus the
experimentally observable
I(V) characteristic can be considered a good measure of
the degree of order and the nature
of the flow in the system.
To quantify the degree of plasticity of the
electron flow, we plot in Fig. 3(c) the
fraction of electrons Dtr that wandered a
distance of more than a0/2
in the direction transverse to the drive
during an interval of 8000 MD steps.
The peak in Dtr coincides with the peak in dI/dV.
Dtr then slowly declines until it saturates
at fd/fdc ≈ 6, indicating the gradual
formation of the non-intersecting channels as seen in
Fig. 1(f). The saturation in Dtr
coincides with the saturations in both P6 and dI/dV.
It is beyond the scope of this work to determine whether
the reordering transition is a true phase transition; however,
we do not observe any hysteresis in the measured quantities
through the reordering sequence.
The I(V) curve in Fig. 3(a) corresponds
to a highly irregular voltage signal as a function
of time when the electrons are in the plastic
flow regime (fd/fc = 1.5).
The corresponding voltage noise spectrum
in the inset of Fig. 3(c)
shows that only
broad-band noise is present.
In contrast, at fd/fdc = 4.0,
in the moving smectic regime,
a roughly regular signal is obtained, and
narrow-band noise
appears, as shown in the inset
of Fig. 3(c).
In a system with d = 1.57 when the depinning is elastic only,
an even
more pronounced narrow band noise signal is observable.
In experiments, broad band noise has been observed above depinning
[2], but narrow band noise has not been seen.
Our results suggest that
plasticity may be playing an
important role in most experiments.
In Fig. 3(d,e) we examine the critical behavior above
the depinning threshold of Vx versus fd for disorder strengths
d = 0.5 and 0.65. In Fig. 3(d),
Vx ∼ [(fd−fdc)/fdc]ξ,
so that the curves are fit well by a power law with
ξ = 1.61 ±0.10 and 1.71 ±0.10, respectively. These values
agree well with the predicted value of ξ = 5/3 [15]
for electron flow through disordered arrays, and the
experimentally observed values of ξ = 1.58 and 2.0
[16].
A proposal [12] on the moving ordered phase
(for very weak and very short-range pinning) predicts
a barrier to transverse motion once channels similar
to those in Fig. 1(e) form.
As shown in the inset of Fig. 4, for a system in
the reordered phase we observe a
transverse depinning threshold that is about 1/6
the size of the longitudinal depinning threshold fdc.
In recent experiments, Perruchot et al. [17]
also find evidence for a transverse barrier that is
about 1/10 the size of the longitudinal threshold.
These thresholds are much larger than those observed in vortex matter
interacting with short-ranged pinning, where ratios of 1/100
are seen [14].
Figure 4: Dynamic phase diagram for the driven disordered Wigner solid.
Inset: Clear evidence for a finite transverse depinning threshold
for a system with d=0.9 at fd=0.16 in the reordered phase.
In Fig. 4 we present the dynamic phase diagram
as a function of disorder strength and driving force.
For fd = 0 we observe a similar static ordering
as in [6,7] where for strong disorder
(d < 1.4) the electron lattice is considerably defected,
the structure factor is a liquid-like ring,
and the depinning is plastic. We label this
static region the
disordered Wigner glass which depins into the plastic flow regime.
For weak disorder there
are few or no defects and the depinning transition is elastic.
We label this region the ordered Wigner glass.
For decreasing d, the pinned region grows while
a reordering transition to a
moving Wigner smectic state occurs for higher drive.
For fd=0 a hexatic phase may exist; however, much larger simulations
would be required to resolve this issue.
In summary, we have investigated the dynamics of an
electron solid interacting with charged disorder. We find that
for strong disorder the depinning transition is plastic with
electrons flowing in a network of winding channels. For
increasing drives, the electrons partially reorder and
flow in non-intersecting channels forming a moving Wigner smectic.
We show that the onset of these different phases can be
inferred from the transport characteristics.
In the plastic flow regime, the noise has broad-band
characteristics, while in the moving smectic or elastic
flow phase, a narrow band noise signal is observable.
We also find critical behavior at the onset of the plastic flow phase,
with critical exponents in
agreement with predictions for transport in
arrays of metallic dots [15,16].
We map out the
dynamic phase diagram as a function
of disorder and applied driving force.
We
obtain a finite threshold for transverse sliding,
in agreement with recent experiments [17].
We acknowledge J. Groth and A.M. MacDonald for very useful
discussions, and B. Janko for his kind hospitality.
This work was partially supported by DOE Office of Science No.
W-31-109-ENG-38, CLC and CULAR (LANL/UC), NSF DMR-9985978.
FN also acknowledges partial support from CSCS and MCTP at
the UM. This is preprint number MCTP-00-04.
K. Moon, R.T. Scalettar and G.T. Zimányi,
Phys. Rev. Lett. 77, 2778 (1996);
S. Ryu et al., ibid.77, 5114 (1996);
S. Spencer, H.J. Jensen, Phys. Rev. B 55, 8473 (1997);
C.J. Olson, C. Reichhardt, ibid61, R3811 (2000).
