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 . 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 . 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 . A plausible explanation for this observation is that atomic hydrogen alters the defect structure in a-Si:H by breaking Si-Si bonds  and introduces deep carrier traps which impede transport between neighboring atoms. Unlike the carrier traps educed from the small-scale device measurements , 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 , Pd , and a-PdSi  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. . 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.  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  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  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  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.  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  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.  and previously was pointed out by Garfunkel et al.  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  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.