C. Reichhardt and C.J. Olson Reichhardt
Center for Nonlinear Studies and Theoretical Division,
Los Alamos National Laboratory, Los Alamos,
New Mexico 87545, USA
(Received 7 May 2003;
published 8 October 2003)
We investigate the effects of finite temperature, dc pulse, and ac drives on the charge transport in metallic arrays using numerical simulations. For finite temperatures there is a finite conduction threshold which decreases linearly with temperature. Additionally we find a quadratic scaling of the current-voltage curves which is independent of temperature for finite thresholds. These results are in excellent agreement with recent experiments on 2D metallic dot arrays. We have also investigated the effects of an ac drive as well as a suddenly applied dc drive. With an ac drive the conduction threshold decreases for fixed frequency and increasing amplitude and saturates for fixed amplitude and increasing frequency. For sudden applied dc drives below threshold we observe a long time power-law conduction decay.
DOI: 10.1103/PhysRevB.68.165305 PACS number(s): 73.63.-b, 73.50.-h, 73.23.-b, 73.21.La
Recent experiments by Parthasarathy et al.9 have specifically sought to address the role of different types of array disorder on the current-voltage scaling. They considered varying the global structural order of the array by varying the amount of voids. For structurally ordered systems composed of a triangular monolayer of gold nanocrystals without voids, they observe a single power-law scaling with =2.25 in the current-voltage curves. In this case there is still charge disorder in the substrate and disorder in the interparticle couplings. For arrays where structural disorder is added, the current-voltage curves could not be fit by a single power law. Simulations by Reichhardt and Olson for structurally disordered arrays also produced similar behaviors. These results suggest that differences in structural disorder in the earlier experiments may be the cause of the differences in the observed exponents.
Less is known about how other types of disorder, such as thermal disorder, or perturbations, such as ac drives, would affect the scaling or the exponents. Recently Parthasarathy et al.14 have investigated the role of finite temperature on the current-voltage curves for ordered and structurally disordered gold nanoparticle arrays. They find for the ordered arrays that the threshold voltage decreases linearly with T while the scaling exponent is unaffected. For the disordered arrays the threshold also decreases linearly at low T. Decreases in the threshold with increasing temperature have also been observed in previous experiments on disordered arrays 12,13,15-17. In both the ordered and disordered arrays, for higher temperatures the threshold is lost and the nonlinear scaling of the I-V curve disappears and is replaced by linear behavior. Other experiments on ordered arrays have also found that temperature only weakly affects the shape of the I-V curves 6,9. Previous simulations have been limited to T = 0 1,4 or have considered only very small arrays17,18.
Another type of perturbation that has recently been considered in experiments on highly resistive samples is the application of a sudden dc drive19. The application of a dc voltage that is below the conduction threshold voltage produces a current response that shows a two stage decay. The first stage, shown in Fig. 4 of Ref. 19, is a rapid decay of the current at short time scales that does not fit to a power law. For longer times, however, the current shows a power-law decay with I(t)t-, with 0.1 < < 1.0, depending on the applied voltage and temperature.
A third perturbation which can also be applied to the system is an ac drive. To our knowledge, the effect of ac drives on the IV curves has not been investigated by simulation, nor has it been considered in experiments. It is not clear whether increasing ac amplitudes may cause the scaling of the IV curves to be lost, as the perturbation due to temperature did.
In this paper we use the RO model for charge transport through metal dot
arrays to consider the effect of finite temperature
and ac drives on the threshold behavior and the current-voltage scaling.
We also consider the case of sudden applied dc drives and examine the
conduction decay.
For this work we consider only 2D square arrays, and use system
sizes from 20 x 20 to 60 x 60. Our results are mainly presented
for system sizes of 50 x 50 which we previously found to
be adequately large to capture the essential physics.
The sample has periodic boundary conditions in
the x and y directions and contains Nc
mobile charges.
The equation of motion for a charge i is
(1)
The mobile charges interact with a Coulomb term, U(r) = q/r. We employ a fast summation technique for computational efficiency to calculate the long-range Coulomb force20. The dc driving term is fdc=fdc which would arise from a dc applied voltage V. On each plaquette there is a threshold force fp, chosen from a Gaussian distribution, which prevents the charge from leaving the plaquette until fp is exceeded. This threshold originates from the energy required to add an electron to a dot with charge q. The electron is assumed to tunnel from one dot to the next when the electrostatic energy favors this motion 1. The threshold for dot j is Vthj=qj/Cj, where Cj is the capacitance of dot j. In experimental systems, this threshold can be altered due to cross capacitance between neighboring charged dots. In our model, Vth for a given dot is fixed regardless of the charge state of neighboring dots; however, due to the long-range Coulomb interactions between the charges, the effective threshold of the dot is altered by the presence of charges on neighboring dots. We measure the global charge flow or current Starting from zero we increase the dc drive in increments. When measuring I-V curves, we wait at each increment for 1500 simulation steps before taking data to avoid transient effects. We study transient effects separately in Sec. III of this paper. To explore finite temperature effects, we add a thermal force term fT which has the properties < fT(t) > = 0 and < fTi(t)fTj(t') > = 2kBT(t-t'). Here is a damping constant which we set equal to unity. The damping corresponds to dissipation produced by the motion of the charge. In Sec. IV of this paper, we also add a term representing an external ac drive, fac= A sin(t), which would arise from an applied ac voltage. Here A is the amplitude and is the frequency of the ac drive.
We first consider the case of different temperatures and zero ac drive. We normalize our temperature in units of Tth which is the temperature at which the threshold force for motion, fth, becomes zero. In Fig. 1(a) we plot the velocity-force curves (current-voltage curves) for a sample with fp=4.0 for increasing temperature, T/Tth= 0, 0.24, 0.61, and 0.95, indicating that the finite temperature driving threshold fth decreases with temperature. In Fig. 1(b) we show that the curves can be collapsed in the same manner as the experimental curves in Ref. 14, by linearly shifting the x axis an amount given by fshift(T)=fth(T)-fth(0). This collapse shows that the scaling exponent is independent of temperature. In Fig. 2, a log-log plot of V vs (fdc-fth)/fth for the curves in Fig. 1(b) illustrates a power law scaling with =2.00.15, in good agreement with the experimental values9,14. In addition, the thresholds are decreasing linearly with temperature, as indicated in Fig. 3. If the temperature is further increased above T/Tth=1, the threshold velocity disappears and the shape of the I-V curve begins to change. Thus above this temperature it is no longer possible to rescale the I-V curves by a simple shift of the x axis. This is also in agreement with experiment14.
In Fig. 3 we show the conduction threshold ft vs T for two systems that have different average disorder strength, fp=4.0 and fp=8.0. Both sets are normalized by Tth=1.0, the temperature at which the threshold reaches zero for the fp = 4.0 system (circles). Here the thresholds decrease linearly with temperature for all but the lowest temperatures, T/Tth < 0.1. For the sample with fp = 8.0 (squares), the overall thresholds are higher which is consistent with the increased average force to leave a plaquette. The linear decrease in the threshold is in agreement with experimental observations 14. We have tested several different methods for determining the threshold, such as using different finite velocity percentages ranging from 0.005 to 0.10, and find consistent linear decreases in the threshold with temperature.
The fact that the scaling of the I-V curves changes with increasing temperature only once a threshold temperature Tth has been exceeded can be understood with a simple physical picture of the channels of charge motion. It was shown previously at T=0 that, for low applied voltages, the charges move through riverlike patterns of channels inside the sample4. The exact pattern of the channels in a given sample is determined by the specific realization of disorder within that sample. Throughout the nonlinear segment of the I-V curve, charge motion is confined to a number of channels that increases as fdc is increased. Channels with the lowest barrier to motion open first, followed by channels with increasing barriers to motion. The order in which the channels open is fixed by the disorder realization. When temperature is applied, the barrier to motion in each channel is effectively lowered; however, the relative barrier heights of the channels are unchanged. Thus, at low but finite temperatures, the channels open in the same order as at T=0, but at reduced values of fdc. Therefore the shape of the I-V curve is merely shifted to lower values of fth, but not altered. In contrast, once the temperature is increased enough to completely eliminate the barrier to motion in some of the channels, all of these channels open immediately and the order in which the channels open is changed, which amounts to a change in the topography of the channel structure21. This alters the shape of the I-V curve and causes the scaling to be lost, in addition to producing fth=0.
To illustrate this picture, in Fig. 4 we show the flow pattern of the charges in our simulation at three different temperatures for fixed fdc=0.125. In Figs. 4(a) and (b), the temperatures T/Tth=0 and T/Tth=0.25 are well below the crossover temperature, and linear scaling of the I-V curves is still possible. Although the details of the smallest channels vary slightly, the overall pattern of the primary channels is the same in both panels. In contrast, Fig. 4(c) shows the channel structure at threshold, T/Tth=1. Here, although there is still inhomogeneous flow, the channel pattern seen at lower temperatures has been destroyed.
If this picture correctly captures the behavior, it implies several experimentally testable features. First, the strength of the disorder will determine the persistence of the channel patterns. In samples with stronger disorder, the threshold temperature should increase, as in Fig. 3. For metallic dots, the disorder strength is determined by the inverse capacitance of each dot, which goes as C=4r. Thus, samples containing dots with smaller radii should show a higher threshold temperature. Secondly, the topography of the channels of charge flow is strongly correlated with the noise in the charge velocity, as has been demonstrated for the case of superconducting vortices21,22. If the structure of the channels changes above the threshold temperature, this should be observable as a change in the noise characteristics. Finally, in samples that contain voids, the channels are highly constrained by the voids themselves, and the channel pattern cannot be altered even by temperature. Thus, in heavily voided arrays, the linear scaling behavior of the I-V curves should hold to much higher temperatures than in similar void-free arrays.
For higher drives, as illustrated by
fdc/fth = 0.95 (Fig. 5, top),
V saturates to a finite average velocity after
an initial decay, even though fdc
In the experiments,
the decaying response persisted
for more than five orders of magnitude in time. In the simulations
we are limited by both the simulation time and
by finite size effects.
In the power-law regime at intermediate drives, the flow occurs in
decreasing numbers of channels
as previously seen in simulations.
In the transient experiments19
it was speculated that the
power law decays may arise due to the Coulomb interactions between dots.
We have also considered the case where there is no Coulomb interaction
between mobile charges, and find only exponential or stretched exponential
decays of the conduction. In addition, the channel structures are not
present,
indicating that the flow through interacting channels plays an important
role in the power-law decay of the response.
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FIG. 7.
(a) The threshold fth for the system in Fig. 1 normalized by
f0, the threshold at zero ac drive, vs ac amplitude for a fixed
ac frequency of =0.05.
(b) The threshold vs ac frequency for a fixed ac amplitude of A=0.2.
IV. AC EFFECTS
We next consider the case T = 0.0 for the sample that was used in
Fig. 1
and measure I-V characteristics for the sample
in the presence of a fixed frequency ac drive.
In Fig. 7(a) we plot fth/f0 as a function of ac
amplitude A for fixed frequency =0.05,
where f0 is the threshold for zero ac drive.
Here fth monotonically
decreases for increasing ac amplitude A,
but does not follow a simple functional form.
This decrease is reasonable since, during the positive cycle of the
ac drive, both the ac and dc drives combine to push the charge over
the barrier. We note
that f0=0.14, indicating that ac amplitudes that are considerably
higher than f0,
A > f0, still preserve a finite dc threshold
force fth.
We next fix the ac amplitude to A=0.2
and plot the dependence of fth on the frequency
in Fig. 7(b).
Here, the threshold increases with increasing ac frequency,
with fth saturating as
1/ to f0
at the highest frequencies. At
high frequencies, the mobile charge carriers do not have time to respond
to the ac drive.
We have also measured the scaling of the current-voltage curves, and find that
it is independent of both the ac amplitude and frequency.
The range of the scaling is, however, reduced by the ac drive.
V. SUMMARY
In summary, we have investigated transport in 2D metallic dot
arrays for finite temperature and ac drives.
For zero ac drive and varied temperature, we find a
finite temperature conductance threshold
which decreases linearly with temperature.
Additionally, the I-V curves obey power law scaling
with =2.0, which is independent of the
temperature below a threshold temperature.
These results are in excellent agreement with recent experiments.
For a sudden applied dc drive less than the threshold drive, we find a
two stage decay of the velocity response that shows first
a fast short time
decay that does not fit to a power law. This corresponds to
charge rearrangements less than a lattice constant.
For longer times
there is a slower long
time decay that is consistent with a power law where the flow consists
of correlated channels that gradually stop. For higher drives that are still
below the threshold, some of the channels can move across the entire
sample and become stabilized.
If the
long-range Coulomb interaction is removed we observe only a fast
exponential decay.
We have also studied the effect of superimposing an ac drive on the
dc drive and find that, for fixed frequency and increasing ac amplitude,
the threshold decreases.
Conversely, for fixed amplitude, the
threshold decreases for decreasing ac frequency.
The scaling of the current-voltage curves is independent of the ac amplitude
and frequency; however, the range of the scaling changes.
ACKNOWLEDGMENTS
We thank H. Jaeger, R. Parthasarathy, and C. Kurdak for useful
discussions.
This work was supported
by the U.S. Department of Energy under Contract No. W-7405-ENG-36.
Phys. Rev. B 68, 165305 (2003).
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Last Modified: 1/23/04