**Kristin M. Abkemeier and David G. Grier**

**April 20, 1996
**

1/*f* noise in certain amorphous and polycrystalline materials may have an
origin in the intrinsic disorder in these systems.
Modeling such systems
as topologically disordered linear networks reveals how the range and scale
of the dynamical current redistribution due to isolated fluctuators depends
on the network's degree of disorder.
We introduce a ``disorder parameter'' which quantifies the degree of
topological disorder in simulated resistor networks. The magnitude of
conductance fluctuations in the networks caused by removal of single resistors
is found to scale with . Such bond breaking events are analogous to the
motion of diffusing defects in materials such as hydrogenated amorphous silicon
(a-Si:H). These results
extended to the scenario of multiple simultaneous bond breaking events
suggest how fluctuations on
all scales might occur especially in disordered systems.

Charge transport in many current-carrying systems is characterized by
conductance fluctuations as large as several percent of
the mean occurring over a wide range of time scales.
These conductance fluctuations usually are referred to as noise and
often are most pronounced in disordered and geometrically confined
systems such as amorphous and polycrystalline thin films [1, 2].
Despite considerable effort, the origins of many of the most common forms
of noise are not yet completely understood.
Their widespread appearance
in diverse materials, however, suggests that quite
general explanations might suffice for many cases and that reasonably
simple generic models could capture their essential physics.
This belief has led to extensive studies of transport in computationally
tractable network models in which the complex current paths
through the macroscopic medium are modeled as a
system of interconnected discrete circuit elements
[3, 4, 5, 6].
For the most part, such studies have centered on topologically ordered
arrays of bonds with disorder introduced through the choice of
coupling constants.
Recent studies, however, have demonstrated that disorder
in a network's underlying topology can influence
its mean transport properties [7].
The simulations described below explore
the relationship between topological disorder
and dynamical current redistribution in linear resistor network models and
provide insights into the
mechanisms by which topological disorder can enhance certain
systems' susceptibility to 1/*f* noise and the related phenomenon of
random telegraph switching noise (RTSN).

The traces in Fig. 1 are typical examples of 1/*f* noise
(Fig. 1(a))
and RTSN (Fig. 1(b))
measured in thin films of hydrogenated amorphous silicon (a-Si:H).
The power spectral density
of current fluctuations in signals such as Fig. 1(a)
falls off as a power law in the frequency,
, typically with an
exponent in the range .
For the trace in Fig. 1(c), .
This power law dependence is the defining characteristic of 1/*f* noise;
with a few exceptions [4, 5, 6],
the origin of this frequency dependence has been less investigated in
network simulations than the magnitude of the noise produced.
We will follow this path ourselves but suggest mechanisms by which the
processes we describe could yield realistic dynamics.
Most network models also have not been able to reproduce
the two-state step-like conductance fluctuations
which characterize RTSN signals such as that in Fig. 1(b).
The conductances in the two states
differ by roughly 0.02 percent of the mean and fluctuations as
large as 1 percent have been reported in comparable systems
[6, 8].
As depicted in Fig. 1(d), the power spectral
density for such two-state noise
is Lorentzian with a characteristic corner frequency
reflecting the dynamics of the microscopic fluctuations responsible
for the switching behavior [2, 1].

RTSN observed in transverse conductance measurements on a-Si:H devices has been attributed to charge trapping [9]. The small scale of the devices in these experiments was thought to render their conductance susceptible to fluctuations in the charge number via carrier trapping at a few sites. On the other hand, more recent observations of RTSN in coplanar conductance measurements on a-Si:H films at and above room temperature [8, 10, 11] are inconsistent with a charge trapping mechanism because of the relatively high temperatures of the experiments and the large number of electrons that would have to act in concert to account for the magnitude of the observed fluctuations. Parman et al. [10, 11] consequently conjectured that the conductance fluctuations in a-Si:H might instead be controlled by processes influencing a few critical current microfilaments with diameters comparable to the film thickness.

Intriguingly, the activation energy and typical rise time for RTSN in a-Si:H
are measured to be consistent with the activation energy and
hopping time for hydrogen diffusing at room temperature as measured by NMR
[8]. A plausible explanation for this observation is
that atomic hydrogen alters the defect
structure in a-Si:H by breaking Si-Si bonds [12]
and introduces deep carrier traps
which impede transport between neighboring atoms. Unlike the
carrier traps educed from the small-scale device measurements [9],
the electron-trapping defects in this picture shift spatially
according to the dynamics of the loosely bonded hydrogen atoms.
Hydrogen diffusion therefore leads to a time-dependent rearrangement of
local current paths in the Si matrix and thus to local conductance
fluctuations.
Hydrogen diffusion also has been implicated in noise generation in
metal films including Nb [13], Pd [14], and a-PdSi
[15] via its creation of interstitial defects.
The two-state fluctuations in relatively large area devices and their
apparent link to hydrogen motion have
led to the adoption of a-Si:H as a standard system for studying 1/*f*
noise and RTSN.
The question remains, however, as to how atomic scale conductance fluctuations
can lead to dramatic variations in a sample's global conductance,
particularly at room temperature.

Inspired by the current microfilament conjecture, simulations by Lust and Kakalios [5, 6, 8] of noise mediated by dynamical current redistribution on a square lattice in two dimensions achieve the necessary amplification by removing bonds at random until the remaining current-carrying pathways form a sparse cluster near percolation. The conducting backbone of a system-spanning cluster models a current microfilament; physical motivations for this scenario, such as the channeling of currents into microfilaments by disorder from long-ranged potential fluctuations created by charged defects, have been summarized by Parman et al. [11]. The percolation cluster geometrically confines currents so that a local change in conductance results in global-scale current redistribution and thus to large overall conductance changes. If the process of bond redistribution mimics the diffusion and trapping of hydrogen in a lattice, the dynamical behavior of the simulation qualitatively resembles that observed in measurements such as Fig. 1(b).

The natural disorder in amorphous and granular films hints at an alternate mechanism for noise amplification. Pioneering work by Priolo et al. [7] demonstrates that topological disorder can strongly modify a network's overall conductance. The observed dependence of global transport properties on the degree of disorder might be expected to reflect qualitative changes in the underlying local transport pathways. Such qualitative differences between ordered and disordered networks might in turn render the disordered networks more susceptible to large-scale current redistribution during bond breaking without the explicit introduction of current confinement.

We explore this possibility in a class of linear resistor networks with well-defined and realistic topological disorder. These networks are constructed from the Voronoi tessellations [16] of sets of points randomly displaced from triangular lattices. The Voronoi diagram for a set of points consists of polygons enclosing their nearest-neighborhoods, also known as their Wigner-Seitz cells. The interfaces between neighboring cells provide resistive pathways for the flow of currents, which we model as simple resistors connecting nearest neighbor nodes. Such cellular patterns arise naturally during Ostwald ripening of granular metal films, for example. In this case, the resistance between neighboring domains is appropriately modeled as being proportional to the length of the common interface. The analogy to a-Si:H is less direct, although we will argue below that the detailed nature of the topological disorder in a network is less important than the degree of disorder in determining susceptibility to noise processes.

Since we are interested in the
correlation between the local current arrangements and the global transport
properties of disordered thin films, it is important to
investigate a system in which the degree
of disorder is controllable and isotropic.
Each disordered network was generated from a regular triangular lattice of
approximately 5000
seed points inside a square of side length 66 lattice spacings.
These dimensions were
chosen as a compromise between convergence of the solutions and computational
manageability.
Disorder is introduced by displacing each point (*x*,*y*) by a random vector
, where lattice spacings for the data
presented here.
A Voronoi tessellation was performed on the resulting
array of points, yielding a network of
conductances *g* equal to the cell side lengths with connectivity determined
by the Delaunay triangulation [16] of the points.
One side of the square was set to a dimensionless
potential of 1 with the opposite side at ground.
The Kirchhoff
equation was then solved at each node to obtain the potential in each of the
cells within the sample.
From this, the global conductance *G* also
was calculated and normalized by the conductance across one of the cell
interfaces in the regular network, .

To quantify the degree of disorder in our networks, we borrow a result from the study of two-dimensional froths in which the distribution of side lengths in a random Voronoi tessellation is found [17] to satisfy

The first two moments of ,

where erf( *x* ) is the error function, can be computed for
a given realization of a Voronoi network and provide coupled
equations in the characteristic
side length *m* and its standard deviation .
Solving Eqs. (2) and (3) for *m* and
is equivalent to performing a nonlinear least squares fit to Eq. (1).
The success of Eq. (1) in describing our networks over the
full range of disorder investigated can be seen in Fig. 2.
The ratio

measures the spread of the distribution of cell side lengths (conductances) relative to the mean and therefore serves as a measure of the degree of disorder in the network.

As can be seen in Fig. 3,
the overall conductance *G* of the network depends non-monotonically
of the degree of disorder .
Qualitatively similar dependence was observed by Priolo et al. [7]
using as a measure of disorder, although does not
uniquely determine the degree of disorder.
The observation of a conductance minimum at intermediate
disorder, , suggests that similarly constituted
systems with differing microstructures may appear
indistinguishable on the basis of their mean transport properties.
Differences in the detailed distribution of current pathways, however,
might lead to substantially different responses to local
conductance perturbations
and thus differing susceptibilities to noise-inducing processes such
as the diffusion of hydrogen.
Such effects could explain, for example, why carbon film
resistors of equal value produced by different manufacturers
can have dramatically differing noise characteristics.
It is important to note that the dip in the conductance
as a function of disorder still appears for networks of equal-valued
resistors and thus arises principally from the topology of the network.
Selecting coupling constants based on proximity or boundary lengths
merely enhances the effect.

To investigate mechanisms of dynamical current redistribution in a typical realization of a Voronoi network, we remove the single bond carrying the most current and solve the Kirchhoff equations again for the modified network. The overall conductance drops by over of the mean in the random case compared with for an ordered network. The overall increase in sensitivity to bond breaking with increasing disorder appears in Fig. 4. Note that the values for would be comparable to those seen in Fig. 1(c) if the simulation were scaled to the experimental sample's areal aspect ratio assuming uniform current density through its thickness. The error bars in Fig. 4 reflect the standard deviation of the mean of the responses obtained from 50 realizations at each value of . Not all networks of the same degree of disorder manifest the same sensitivity to the type of current redistribution mandated by our bond breaking protocol. Such variability from sample to sample also is common in experimental studies of noise processes in disordered films.

The system-wide effect of breaking the ``most important'' bond reflects local-scale rearrangements in the current pathways. To illustrate this, we calculate the absolute fractional change

in the current passing through the *i*-th bond at a distance
from the broken bond.
Averaging over angles
provides a measure of the range of current redistribution away from
the broken bond as a function of the lattice's degree of disorder.
To avoid the current-confining effects of the boundaries, we
considered only those realizations in which the broken bond was
no closer than 10 lattice spacings from any boundary.
The
results averaged over all relevant realizations for the ordered
case and four other
values of disorder appear in Fig. 5.
It is notable that
in the most disordered systems, breaking the high-current bond influences local
currents by at least up to twenty bond lengths away.
This suggests
that a local fluctuator in a disordered granular thin
film could be observed via conventional visualization techniques
as its influence
is felt over a region more than two orders of magnitude larger than the area of
the fluctuator itself.

While the magnitude of the local response to bond breaking increases with increasing disorder, its functional form does not change. To demonstrate this, we first assume that the disorder merely sets the magnitude of the current redistribution and does not affect its spatial dependence:

If we normalize so that , then
measures the role of
disorder in setting the magnitude of current rearrangements during
bond breaking.
As can be seen in Fig. 7, is on average
*30 times* larger for a disordered network than for
a triangular lattice.
Furthermore, the scaling collapse in Fig. 6 provides support
for the assumptions underlying Eq. (6).
The salient point of this observation and the central result of this
paper is that topological disorder *in itself* dramatically
affects the range and magnitude of current redistribution due to
bond breaking in a linear network without requiring any additional
assumptions regarding the distribution of current paths.

The magnitude and range of the response in local current rearrangements
to the breaking of a single important bond in a topologically disordered
network suggests how a process such as hydrogen diffusion can generate
1/*f* noise and RTSN.
Diffusion of a single broken bond in a topologically disordered lattice
will lead to conductance fluctuations reflecting not only the
dynamics of the diffusion but also the spatial correlations in the underlying
lattice.
Only a few bonds in the lattice are important enough that their
removal and replacement would be reflected in
dramatic conductance jumps such as those in
Fig. 1(b).
The persistence times of the two states would then reflect
the residence time of the random walker in the vicinity of that site.
Other sites would contribute small scale fluctuations whose
spectrum would reflect both random walker dynamics and lattice
topology.

The picture becomes more interesting and should be more applicable to real systems when multiple bond-breaking random walkers are considered. The long-range current redistributions observed in our simulations of disordered networks suggest that nearby walkers in a disordered system will not satisfy the statistical independence requirement of Cohn's theorem [19] to ensure linear superposition of their effects on overall conductance. Thus even a pair of random walkers can have a disproportionately larger cooperative effect on a system's transport properties than a single walker. The failure of superposition in such cases and its significance for noise processes recently was emphasized by Seidler et al. [20] and previously was pointed out by Garfunkel et al. [21] in the context of dynamical current redistribution. The statistics of multiple bond-breaking events in topologically disordered networks are beyond the scope of the present work and will be discussed elsewhere. Hydrogen diffusion in a-Si:H and activated defect motion in unhydrogenated a-Si also have been implicated [18] in irreversible microscopic structural rearrangements which could also change the distribution of local currents. These microstructural changes would appear in our scenario as a drift in the mean conductance of the system and also could affect the distribution of fluctuator strengths in the remaining network.

This study demonstrates that the effect of local fluctuators on thin film transport can depend dramatically on the film's degree of topological order. The introduction of in Eq. (4) as a quantitative measure of order in such systems makes possible the systematic study of the origins and ramifications of such dependence. The observed dependence of the range of current redistribution on the degree of disorder suggests a mechanism by which a fixed concentration of mobile defects yields more response in disordered systems without further assumptions about the geometry of local current paths. Generalizing the results from this work to include multiple mobile defects should provide new insights into the relationship between microscopic topology and the spectrum of noise produced by transport through thin films.

K.M.A. was supported by an AT&T Bell Laboratories Ph.D. Fellowship while conducting this research. D.G.G. acknowledges support from the David and Lucille Packard Foundation. This work was supported in part by the MRSEC Program of the National Science Foundation under Award Number DMR-9400379.

Sat Apr 20 09:23:47 CDT 1996