Noise and hysteresis in
charged stripe, checkerboard, and clump forming systems
C. Reichhardt, C.J. Olson Reichhardt, and A.R. Bishop
Center for Nonlinear Studies and
Theoretical Division,
Los Alamos National Laboratory, Los Alamos, New Mexico USA 87545
ABSTRACT
We numerically
examine noise fluctuations and hysteresis phenomena in charged systems
that form stripe, labyrinth or clump patterns. It is believed that
charge inhomogeneities of this type arise in two-dimensional (2D) quantum
hall systems
and in electron crystal structures in high temperature superconductors,
while related patterns appear in
manganites and type-I superconductors.
Recent noise and transport experiments in
two-dimensional electron gases
and high temperature superconducting samples revealed
both 1/fα noise signatures and hysteretic phenomena.
Using numerical simulations we show that 1/fα noise fluctuations and
hysteresis are generic features that occur in charge systems which
undergo a type of phase separation that results in stripes, clumps,
checkerboards, or other inhomogeneous patterns.
We find that these systems exhibit 1/fα fluctuations
with 1.2 < α < 1.8, rather than simple
1/f or 1/f2 fluctuations.
We also propose that the 2D metal insulator transition may be
associated with a clump electron glass phase
rather than a Wigner glass phase.
There is growing evidence that a wide variety
of condensed matter systems
intrinsically exhibit heterogeneous charge ordering in
the form of mesoscopic clump, labyrinth, stripe or checkerboard phases.
Theoretical and numerical simulations indicate that
such heterogeneities can arise in various systems including
cuprate superconductors
[1,2,3,4,5,6],
antiferromagnetic insulators [7], and
two-dimensional electron gas (2DEG) systems
[8,9].
Heterogeneous states are also believed to occur in
maganates and diluted magnetic semiconductors [10].
A natural question is what type of fluctuations or noise properties are
associated with such charge heterogeneities.
Does a charge stripe phase have distinct noise properties from a clump
phase?
Can noise properties be used to distinguish between uniform charge
phases and heterogeneous phases? Further, are the noise properties
connected to other types of transport or hysteresis properties?
In this paper we discuss these issues.
Recent transport experiments in 2DEGs
in the regime where a bubble phase is believed to occur
produced hysteretic current-voltage
curves [11].
Noise measurements in this same system have
revealed a remarkable number of phenomena ranging
from narrow band noise fluctuations to 1/fα noise fluctuations.
It has been difficult to interpret the meaning of these measurements, since
the frequency of the narrow band noise is much lower than
the expected washboard frequency
for a moving Wigner lattice or a moving clump crystal.
Transport experiments in underdoped samples of YBCO have
also revealed
hysteretic jumps in the current-voltage curves
at low temperatures as well as
non-Gaussian noise fluctuations
in the resistance curves [12].
As the temperature is increased,
the current-voltage curves become smooth and non-hysteretic.
These results have been interpreted as a signature
of the presence of some form of large scale heterogeneities
such as disordered fluctuating domains.
Other experiments on cuprate superconductors
in the non-superconducting region of the phase diagram
resulted in magnetic hysteresis and avalanchelike jumps
in the magnetization curves [13,14],
reminiscent of the Barkhausen noise that
occurs for domain wall depinning in
ferromagnets.
This magnetic hysteresis vanishes
at low and high doping and at high temperatures [14].
A simple model based on the random field Ising systems was proposed
that could produce Barkhausen noise signatures [15];
however, a large number of experimental features
were not addressed by this model,
including the disappearance of the noise at low and
high dopings, the temperature dependence,
and the noise spectral properties.
Nonlinear transport properties and non-Gaussian fluctuations have also been
reported in La2NiO4
[16]. In this system,
voltage jumps and hysteresis appear in the V(I) curves.
The time series of the resistance steps
∆R has an intermittent type behavior
along with power law features
in the noise distributions, P(∆R) = ∆R−1.3.
In most of these systems, some form of quenched disorder
from intrinsic defects in the sample is present, and it
can destroy any long range ordering in the patterns
[17].
Charged stripe forming systems have also been shown to exhibit
self-generating disordered glassy properties
[18,19,20].
The presence of ordered or disordered heterogeneities should affect
the bulk transport, fluctuations,
and transient responses of the systems; however, little is known about
how the effect of heterogeneity on transport would differ from
that of homogeneous but disordered systems which form uniform
crystalline or partially crystalline phases.
In this paper, we examine a model of interacting particles
which form generic Wigner crystal,
clump, stripe, anticlump, and high density Wigner crystal phases
[21].
The particles move in two dimensions and interact via a
long-range Coulomb repulsion with an additional
short range attractive force. For low densities the particles are far apart and
form a Wigner crystal. In the presence of quenched disorder
the low density Wigner crystal phase
is strongly pinned and acts like a single particle system.
The transport noise fluctuations in this case
are of a 1/f2 form; however, the relative fluctuations
are very small since they arise from
single particle events.
Such 1/f2 noise is characteristic of single Brownian particles
or collections
of non-interacting single Brownian particles.
For the clump and bubble phases we observe
a significant enhancement of the noise power fluctuations
along with non-Gaussian noise features. The
1/fα noise fluctuations have 1.2 < α < 1.8.
In this regime the transport curves exhibit hysteretic behaviors.
We also find that
as the temperature increases, the heterogeneities become increasingly
smeared, which correlates with a drop in the noise power
and the whitening of the noise spectrum.
We argue that in the stripe and clump phases,
the system acts as a phase separated system. Complete
phase separation does not occur due to the Coulomb repulsion
between particles.
The hysteresis arises due to the
first order nature of the phase transition
from a uniform to a heterogeneous phase.
We also discuss the possibility that the recent noise
experiments in 2D metal-insulator systems
may actually be evidence for the existence of stripe or
clump like phases rather than a uniform Wigner glass. Such
a stripe or clump like phase would be
consistent with the recent proposals for charged microemulsions
by Kiveleson and Spivak [22].
We simulate a two-dimensional system
with periodic boundary conditions in the x and y directions.
Our system contains N particles in a system of
size Lx ×Ly where Lx=Ly=90.
We vary the particle density n=N/LxLy by changing
the number of particles N from 80 to 3000.
The particle-particle interaction between particles
at positions ri and rj is given by
fij=−∇U(rij)
^
r
ij
(1)
where rij=|ri−rj| and
∧rij=(ri−rj)/rij.
The interaction consists of a long-range
Coulomb repulsion combined with a short range exponential attraction
of screening length 1/κ:
U(r) = 1/r − Bexp(−κr).
(2)
In the absence of the attractive term (at B=0)
the particles form a triangular Wigner crystal.
With the attractive term (B > 0) there are three effective
regimes. If the particles
are far apart they interact only through Coulomb repulsion.
When rij is small,
the repulsive Coulomb term again dominates the interaction.
The attractive term is relevant at intermediate length scales.
We study these different regimes as a function of particle density.
The attractive term can also be modified by changing
the inverse screening length
κ and/or the parameter B; however, for this work we
keep these quantities fixed at
κ = 0.25 and B=0.29.
In the absence of quenched disorder, this system has been shown to
form crystalline phases at low and high densities and
stripes and clumps at intermediate
densities. The stripe and clump phases in the absence of quenched
disorder can have intrinsic disorder which arises from
the particle-particle interactions. Such
a system has been called a stripe glass [18].
In the presence of a symmetry
breaking field, the stripe phase can be partially ordered into a smectic phase.
Additionally, the bubble phases show a tendency to form a
Wigner bubble crystal phase.
In most materials there is some form of intrinsic disorder or
random pinning which has a tendency to disorder the system further.
In this work we model the random disorder as
Np
randomly placed attractive parabolic pins of
strength fp=2.5 and radius rp=0.125κ−1,
giving
fp=
Np ∑ k=1
fp
rp
rik Θ((rp−rik)/rp),
(3)
where Θ is the Heaviside step function and
the distance between a particle at ri and a pin at
rk is
rik=ri−rk.
The motion of an individual particle
i is evolved in time by the integration of the
following
overdamped equation of motion:
ηVi = fi =
N ∑ j ≠ i
fij + fp + fd + fT.
(4)
Here η = 1 is a phenomenological damping term.
A dc driving force
fd=fd∧x
which could arise from
an applied voltage in the case of charged particles
is applied in the x-direction.
The conduction is proportional to the
average particle velocity 〈Vx〉
in the driving direction,
〈Vx〉 =
N ∑ i = 1
vi·
^
x
.
(5)
The finite drive
is applied in small increments and averaged over many thousands of time steps
to avoid transient effects and ensure that the velocities have
reached a steady state.
Thermal fluctuations are
modeled as random Langevin kicks with the properties
〈fT(t)〉 = 0 and
〈fT(t)fT(t′)〉 = 2ηkBTδ(t−t′).
The initial particle positions are obtained by
simulated annealing.
Figure 1:
The particle positions (black dots) after annealing to fT = 0 for
density n = 0.1 and fp = 0.5.
Here a mostly ordered clump state forms.
Figure 2:
(a) The average velocity
〈Vx〉 of the particles versus time
for the system in Fig. 1 under an applied drive of fd = 4.5.
Here a washboard type of velocity oscillation occurs.
The x axis, in units of molecular dynamics (MD) time steps,
has been divided by a factor of 40.
(b) The power spectrum S(ω) of the time series in (a).
We first investigate ordered clump and stripe phases which form
when the quenched disorder is weak.
In Fig. 1 we show a image of a mostly ordered clump phase
at n=0.1 and fp=0.5.
Under an applied drive, the clumps depin elastically
with each clump keeping the same nearest neighbors.
Once the system is moving, the
average velocity has a characteristic oscillation as shown in Fig. 2(a)
at Fd = 4.5.
As Fd is
increased, the frequency of this oscillation increases.
Figure 2(b) illustrates the power spectrum
of the time series in Fig. 2(a),
determined from the velocity fluctuations δVx by
S(ω) =
⎢ ⎢
⌠ ⌡
δVx(t) e−iωtdt
⎢ ⎢
2
.
(6)
Figure 2(b) shows that
there is a characteristic peak in S(ω) which reflects the
main frequency of the velocity oscillation.
There are also some additional peaks
at higher harmonics of the main peak.
This oscillatory behavior of a moving elastic lattice
occurs at what is termed the
washboard frequency ωw, given by
ωw =
〈Vx〉
a
(7)
where a is the lattice constant of the
clump lattice.
Washboard frequencies have been observed
for moving charge density waves and ordered vortex lattices.
It is also possible for mode-locking effects to occur in these systems
when an additional ac external drive is applied.
The mode locking occurs when the frequency of the
ac drive ωac matches the frequency of the velocity oscillation,
ωac=ωw.
There is evidence for oscillatory conduction oscillations in 2DEG systems where
stripes or bubbles are believed to occur
[11]; however, simple analysis of the
frequency of the oscillations
in these experiments gives a much larger value of a
than would be consistent with the expected
lattice constant of the bubble phase.
We note that Fig. 2(b) shows the presence of
a variety of other
higher order peaks in S(ω) that are due to the fine structure in the
time series of Fig. 2(a). These higher order oscillations are due to harmonic
excitations within individual moving clumps.
Figure 1 indicates that there is a tendency for the particles to form
a distorted triangular lattice within the clumps.
The washboard frequency of the intraclump lattice is higher
than that of the clump lattice since the intraclump lattice spacing is
smaller.
From these results we conclude that ordered bubble or stripe phases produce
narrow band noise features rather than broad band noise.
Additionally,
the narrow band noise for composite objects such as bubbles and stripes
exhibits a second higher frequency narrow band oscillation due to structure
within the bubbles or stripes.
We have also considered the transport properties of
ordered stripe and clump states.
If the depinning is elastic, there is no hysteresis in the transport curves.
We next turn to the properties of strongly disordered stripe and clump phases.
Figure 3:
The particle positions (black dots) after annealing to fT = 0 for
fp=2.5 at
densities (a) n = 0.013, (b) n = 0.06, (c) n = 0.15, and
(d) n = 0.34.
If the disorder is weak, the stripe and clump phases behave
elastically; however, in the presence of strong quenched disorder the
system breaks into smaller pieces and becomes disordered.
In this limit the system behaves plastically.
In Fig. 3 we illustrate
some of the representative phases that occur with increasing
density in the presence of strong quenched disorder.
In Fig. 3(a) at density n = 0.013,
the particles form a uniform phase of single charges.
In Fig. 3(b) at n = 0.06, the system
is comprised of heterogeneous arrangements of disordered clumps.
For higher density, such as n = 0.15 shown in Fig. 3(c), the system forms
a disordered labyrinth pattern. At very high densities
the system returns to a uniform phase
with considerable crystalline order as seen
in Fig. 3(d) for n = 0.34.
The heterogeneous phases occur for 0.05 < n < 0.34.
Figure 4:
The velocity 〈Vx〉 vs driving force fd curves for
a sample with fp=2.5 and
(a) density n = 0.15, fT = 0.1;
(b) n = 0.06, fT = 0.1;
(c) n = 0.15, fT = 1.3;
(d) n = 0.34, fT=0.1.
By examining the
velocity-force curves at fT = 0, we find smooth non-hysteretic
curves for densities where uniform phases occur,
while in the heterogeneous regions, 〈Vx〉(fd) shows
pronounced hysteresis. In Fig. 4(a) we plot the velocity vs applied
force curve for a heterogeneous system at n = 0.15 where the equilibrium
state is a labyrinth phase.
Here the curve shows two abrupt changes in slope,
and 〈Vx〉
on the decreasing sweep of fd is higher than on the increasing
sweep. If the drive is increased again the same ramp up curve is
followed.
In Fig. 4(b),
〈Vx〉(fd) is shown
for a system with n=0.06,
where the equilibrium system forms a clump phase. Here
a hysteretic region is also observed.
The current-voltage curve for n=0.34, plotted in
Fig. 4(d),
is smooth and has no hysteresis within our resolution.
This density n=0.34 corresponds to the uniform phase seen in
Fig. 3(d).
For the densities that exhibit hysteresis,
as the temperature is
increased the width of the hysteresis is reduced and
it vanishes completely at high temperatures.
In Fig. 4(c)
we plot the velocity-force curve for the system in
Fig. 4(a) for a temperature of fT=1.3
where the hysteresis has been lost and the
velocity force curve is smooth.
In general we find that uniform phases such as single electron bubble states and
the high density states do not exhibit any hysteresis in the transport curves
in two dimensions. This is also consistent with simulations for 2D vortex systems
in the presence of quenched disorder which
do not form stripe or clump phases. In 2D vortex systems there can be a reordering transition
at high drives or a transition from an ordered state to a disordered state as a function of
increasing pinning strength [23]; however,
these transitions are continuous or
are simply crossovers.
For 3D vortex systems there is a first order phase transition
between an ordered and a disordered phase and an associated
pronounced hysteresis occurs in the
transport properties [24]. Hysteresis is generally associated with
first order phase transitions and results when
one of the phases is effectively superheated or
supercooled. In the case of the clump and stripe
forming system that we study here,
the system resembles a phase separated system that has been frustrated.
In the strongly disordered regime,
a mixture of the equilibrium
clump or stripe phases and a uniform liquidlike phase
appears due to the quenched disorder.
If there is a first order phase transition between
these two phases, transport
measurements may bias one of the two phases
with a drive and cause an effective
superheating or supercooling between the phases. We note that
in the theory of Schmalian and Wolynes, it is argued that the
stripe glasses have a first order melting transition [18].
The appearance of hysteresis in
transport curves for effectively two-dimensional systems
then strongly suggests that such
systems have some form of heterogeneity that appears as a phase.
5. BROAD BAND NOISE SPECTRA IN DISORDERED CLUMP AND STRIPE PHASES
Figure 5:
Temperature fT vs density n phase diagram
for the sample with fp=2.5 shown in Fig. 4. The shaded region
indicates where hysteresis (H) is present
in the velocity-force curve. The nonhysteretic region is indicated by NH.
We next analyze the noise fluctuations in the different disordered phases.
We have performed a series of simulations for
different densities and temperatures. In
Fig. 5
we highlight the region of
temperature and density where the phases exhibit a finite
hysteresis in the velocity force curves.
For high and low densities the transport curves are non-hysteretic at
fT = 0. As the density increases, the temperature at which
the hysteresis disappears
increases and reaches a maximum at n = 0.15, which corresponds to the
density at which labyrinth or stripe patterns appear.
As the density is further increased, the
hysteresis width decreases and disappears at n ≈ 0.34.
There is some asymmetry in the
hysteretic region, with the hysteresis extending further on the higher
density side of the peak. This is indicative of the importance of
collective particle interactions in producing the hysteresis; there are
more particles available to interact in the anti-clump state on the high
density side of the peak than in the clump state on the low density side.
In recent magnetization experiments
in doped cuprates, a similar dome-like
structure was observed where hysteresis is present as
a function of doping and temperature [14]. A direct comparison
to the magnetization experiment is difficult since our model does
not include magnetism; however,
hysteresis in magnetic materials or superconductors
can be modeled in general
as collections of particles or domain walls interacting
with quenched disorder and an external driving field, similar to the model
we are using.
It would be interesting to measure the hysteresis in the
transport characteristics of the experimental system
as a function of temperature and doping to see if a similar
dome under which hysteresis is present occurs, as found in our simulations.
We note that hysteresis is not seen in 2D systems with
quenched
disorder for
purely repulsive interactions such as vortices [25].
Hysteresis in transport can appear in 3D vortex systems when the sample breaks
up into two phases and these phases form a heterogeneous labyrinth
structure [26].
This suggests that hysteresis in transport is a sign of
large scale heterogeneities.
Figure 6:
(a) Time series of the velocity fluctuations for a system
at fT = 0.1 and n = 0.15. (b) The power spectra S(ω) for the
velocity fluctuations. Upper curve: n = 0.15, lower curve: n=0.3.
The upper solid line is a power law fit with α = 1.6,
and the lower solid line is a power law fit with α = 0.95.
(c) Noise power S0 for a fixed frequency range vs n.
(d) Power spectrum exponent α vs n.
We next examine the noise fluctuations at different densities and
temperatures.
We apply a constant drive fd
and measure the velocity fluctuations δVx(t) as a function of time,
as illustrated in the velocity trace curve of Fig. 6(a)
for a system with n = 0.15. From the velocity
fluctuations we can determine the power spectrum.
The noise power S0 is defined as the
average value of the power spectrum over a particular frequency octave.
In Fig. 6(b) we plot the power spectrum for n = 0.15, where
the labyrinth phase forms, as well as for n = 0.3 where the high density
uniform phase forms. Both measurements are performed
at fT = 0.1. In the labyrinth phase a 1/fα
noise spectrum occurs with α = 1.6,
while for n = 0.3, α = 1.0.
By performing a series of simulations, we can plot the noise power
and α for varied n and fT. In Fig. 6(c) we show the noise power
S0 as a function of n for fixed fT = 0.1.
In the hysteretic regions, α > 1,
while in the uniform regions, α ≤ 1.
The noise power is maximum near n = 0.15.
In Fig. 6(d) we plot α as a
function of density n, which also shows a maximum in the
hysteretic regime.
In the magnetization measurements of Ref. [14],
the power spectra for the Barkhausen type noise
were not measured. It
would be interesting
to examine the change of the noise fluctuations in
the hysteretic and non-hysteretic regimes.
Simulations of systems producing Barkhausen noise give
α = 1.77 [27].
The appearance of α > 1.0 and large noise power
in experiments and simulations has
been interpreted as evidence of glassy dynamics in
2D charged systems [28,23].
In contrast, noninteracting or single particle scenarios
for noise generation predict α ≤ 1.0
[29,30,31].
Figure 7:
(a) The exponent α vs fT obtained from the power spectrum for
a system with n = 0.15. (b) The corresponding noise power S0 vs
fT.
We next examine the noise at a fixed density and increasing
temperature as shown in Fig. 7 for
a system with n = 0.15. Here the noise power decreases with
increasing temperature and α also decreases, as seen in Fig. 7(a).
As the temperature increases, the heterogeneities
begin to melt and the system becomes more uniform.
In general, in the hysteric regions of the fT-n phase diagram,
α > 1. In Fig. 7(a) the hysteresis disappears in the
velocity force curves for
fT >~1.0.
This corresponds
to the temperature
regime where the value of α drops to
α < 1. The noise power S0 decreases
exponentially with temperature, as shown in Fig. 7(b).
Single particle hopping models often predict an
increase in the noise power with temperature rather than the decrease
observed here [31]. This supports the importance of collective
particle interactions in determining the transport properties
and hysteresis.
6. DISCUSSION AND CONNECTIONS TO THE 2D METAL-INSULATOR TRANSITION
Our results show that the disordered phases do not exhibit hysteresis
in the transport curves and that the associated
noise spectra have a 1/fα
characteristic with α ≤ 1.0.
For the
disordered heterogeneous phases we find hysteresis and 1/fα noise characteristics with
1.2 < α < 1.8. We note that individual fluctuating Brownian particles
produce a 1/f2 noise signature.
We also found that the uniform phases generally
exhibit Gaussian noise fluctuations while the heterogeneous phases have
non-Gaussian noise fluctuations with long tails.
Comparing our results to results from some other systems, we note that
in simulations for a Wigner glass where there was no
attractive term in the particle interaction potential,
1/fα noise characteristics appeared [23];
however, in all cases α < 1.3,
consistent with the results we observe here for uniform phases.
Simulations of defect noise in single component
two-dimensional liquids also resulted in
1/fα noise with α ≤ 1.0 [32], while
2D vortex simulations
also gave 1/f noise spectra [23].
For particles on random substrates with
very weak interactions between the particles,
a 1/f2 noise signature occurs [33].
Based on these results,
we propose that for 2D systems, 1/fα noise with 1.2 < α < 1.8
is characteristic of strongly heterogeneous phases,
while uniform phases of interacting particles produce
1/f noise.
There have been recent noise studies of conduction fluctuations
for a 2D metal-insulator transition system [28].
In this system it is believed that the phases would most likely
be uniform rather than clumplike; however,
Spivak and Kivelson have recently proposed
that these 2D systems may behave more like charged
microemulsions than uniform phases [22].
Such a system
would strongly resemble the charge heterogeneous systems where there
is a mixture of two phases.
One phase is a liquid which corresponds to our uniform phase
and the second phase is solid. We predict that such phases will exhibit
1/fα noise signatures
with 1.2 < α < 1.8. Our previous simulations for
a uniform Wigner glass produced a 1/f noise signature
[23].
The noise experiments on the 2D metal insulator system [28]
find 1/f noise in the metallic state
as well as 1/fα noise near what they term the glassy state
with 1.0 < α < 1.9, although a
large portion of the data is consistent with an exponent of around
α = 1.75.
This suggests
that the 2D metal insulator transition may be better described by
a clump glass than a Wigner glass.
In summary, we have examined hysteresis and noise in a model system
that exhibits crystal, stripe, clump and high density uniform phases. In the
presence of weak quenched disorder the system exhibits elastic behavior
and there is no hysteresis in the transport curves. The elastic
phases do not show 1/fα noise characteristics but rather
narrow band noise signatures.
For strong disorder we find that at low and high
densities, the system forms non-heterogeneous phases that do not
exhibit hysteresis in the transport curves. Additionally, these
uniform phases
exhibit 1/fα noise with α ≤ 1.0.
At intermediate densities,
however,
the system forms heterogeneous clump and stripe phases which exhibit
hysteresis in the transport curves. As a function of temperature and
doping, we find a dome region where the hysteresis occurs. This
prediction can be tested in, e.g.,
transport measurements in high temperature
superconductors for varied dopings. The noise properties
in inhomogeneous and hysteretic regions
have a 1/fα noise spectrum characteristic with
1.2 < α < 1.8. We predict
the appearance of a peak in the noise power and α as
a function of particle density. Additionally both the noise power
and the exponent α in the heterogeneous regions
decrease as a function of temperature.
Finally, we argue that the recent experiments
on the two-dimensional metal insulator transition may be
indicative of the existence of a clumplike glass state
rather than a Wigner glass state.
This work was carried out under the auspices of the National Nuclear
Security Administration of the U.S. Department of Energy at Los
Alamos National Laboratory under Contract No.
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