Download Dear Editor - AIP FTP Server

Mapping of variable range hopping on non-universal and staircase percolation in
granular metal composites
Supplemental Online Material: a brief review of non-universal percolation as well as
some details of the experimental-data selection and their analysis
We present here a brief review of the general tunneling percolation theory in
terms of the non-universal percolation approach, as well as details of some of the side
issues involved in the selection of the experimental data that we considered and their
analysis.
We start by a brief review of the LNB model which provides the simple
common framework for the discussion of the critical behavior of the conductivity in
percolating systems [1,12]. In this widely accepted model it is assumed that the
network of local conductances is above the percolation threshold and that the
macroscopic, current carrying, backbone of the percolation cluster, can be envisioned
to be topologically equivalent to a cubic network of links that have an average
resistance Rχ. Taking the sample size to be L and the average link’s length to be χ,
we have on the average L/χ links that connect the opposite edges of the sample by
(L/χ)2 parallel links. This yields that the resistance of the cube sample RL = Rχ(L/χ).
The resistance of the link Rχ is just g-1L1 where g-1 is the local resistor value in the
system and L1 can be shown to be proportional to (x-xc)-, where  is an exponent that
is determined by the model of the structure of the link. For the simplest model that
considers only the singly connected bonds in the link it can be shown by a simple
1
argument that  =1 [1]. The more accurate accepted value of  for 3D systems is
quite close, 1.1 [1,12,24].
In percolation theory the average diameter of the finite size clusters gives the
correlation length of the system [1,12]. Since only finite size clusters of bonds can fit
into the "holes" of the above described network and since this length is the only
relevant length scale associated with the connectivity in the system, it was assumed
[1] (and confirmed latter) that the average length of the links χ is given by the
correlation length. On the other hand, the latter quantity (as in other phase transitions
[12]) is characterized by the critical behavior (x-xc)- where  is the corresponding
critical exponent (which has the value of 0.85 in 3D).
Recalling that the resistance of the of the cubic sample of size L is given by RL
= Rχ(L/χ) we have that the conductivity of the network σ  RL-1  g(x-xc)tu where tu
= + is the critical exponent of the conductivity. We emphasize here that all the
details of the network as a connected percolation network of conductors are
expressed then by tu.
The size of each local conductance (or resistor in the system) ℓ, as a detail of
the system is assumed in percolation models to be much smaller than the
"macroscopic" statistical quantities such as χ. In the near neighbor lattice-bond
percolation model ℓ is simply the lattice constant [12] while in the continuum it will
be of the order of the center to center distance between the particles [20], and in the
case of VRH it is expected to be the optimal tunneling range.
2
On the other hand, the details of the local conductance and thus the physical
mechanism that is responsible for them are given by g. In the case of a distribution of
the values of the local conductances one has to consider then the average <g-1>-1
instead of g [9]. The basic result that we apply by using non universal percolation
and transport by tunneling is the relation <g-1>-1 (x-xc)(2d/ξ-1). This relation can be
easily derived by assuming a one to one correspondence between the distribution of
the inter particle distance h(r)  exp[-(r-φ)/d)] and the values of the local tunneling
conductances that are given by g  exp[-2(r-φ)/ξ)] so that the normalized distribution
function of the local conductance values, f(g), is determined by f(g)dg  h(r)dr [10].
This yields that f(g) = (1-α)g-α [9] where α = 1-(ξ/2d) [10]. Considering, as relevant,
the local conductances in the system according to their g values from the one of the
largest value, gL, to the one of the smallest value, gS, there is, for x >xc, a critical
conductance with a value gc (such that gL >> gc > gS) that is determined by enabling
the system to be above the percolation threshold [9]. This value of gc is given then by
the condition x[gc∫gL f(g)dg] = xc. Since the average local resistance to be considered
is <g-1> ( gc∫gLg-(1+α)dg)) [9]) and since gL >> gc (the macroscopic conductivity is
determined by the smallest conductances that must be involved (i.e., the “bottle
necks” in the conducting network [9,25]), one finds that the average local
conductance that determines the macroscopic conductivity is given by <g-1>-1 (xxc)(2d/ξ-1) [10].
3
One must note that instead of considering <g-1>-1 where the average is over all
the conductors in the system [9], while keeping the (nearly, see above) universal
value of , one can use a more detailed model for the calculation of Rχ as a 1D
system. This approach appears to be useful in particular in the context of VRH [28].
However, for the simplicity needed for the analysis of the experimental results, we
suggest that our non-universal percolation approach [10] enables to separate the
global network effect that is manifested by the value of tu, from the local
conductance effect that is manifested by 2d/ξ. We should further note that while in
the percolation-tunneling (or the simple hopping, or the VRH in the T→∞ limit)
approach one considers the critical number of near neighbors (2.8 in the dot like
particles [1]), in the VRH one considers farther neighbors in order to account for the
energetic requirement. Since the conductance unit length is the optimal tunneling
range we can associate this range with the basic building unit of the conducting
network (i.e., we can replace the above ℓ by R). In the systems that we study R is of
the order a few φ's and for the densities considered this length is much smaller than
the "macroscopic" statistical size of χ that must involve many more particles. That
such a "macroscopic" scale exists is suggested by the critical (Eq. (3)) behavior that is
exhibited in Figs. 2(a) and 3(a).
The system to which we applied our approach here is a GMC. The selection of
that particular GMC and the data available on it followed an extensive search of the
literature. We found there that t(T) values have been rarely derived for GMCs and
when derived the necessary pre-conditions (mentioned in our letter) for the
4
experimental validation of the VRH mechanism in GMCs (in particular the Eq. (1)
behavior) were not fulfilled. We concluded then that the best data from which the
t(T) dependence can be derived, and for which the preconditions are fulfilled, are the
data of Abeles et al [2,14] on W-Al2O3 GMCs. Correspondingly, we analyzed their
results by presenting their σ(T) data for several x values in terms of σ(x) dependences
for several T values.
As discussed in the manuscript the x-range relevant to the VRH behavior is the
dielectric regime that, as shown in Fig. 1, is the 0 ≤ x ≤ 0.34 range in those GMCs.
Using all the available σ(T) data points of Ref. 14 in this x-range we excluded,
however, in our derivation of the t values (by the applications of Eq. (3)), the σ(T)
data points associated with the smallest x (x = 0.1), that are available (or can be
extrapolated) from those data. This is since the inclusion of the latter data point
yielded, in the best fit of Eq. (3) to all the data points in the 0.1 ≤ x ≤ 0.34 interval,
non-physical negative xc, values. Avoiding then the σ(T) data points at x = 0.1 we
were left then with only five relevant experimental points (0.15 ≤ x ≤ 0.34) for our
analysis. However, it turned out that these five points were sufficient to get t values
with very small error bars (Δt < ±1) for all the temperature considered, whether x c ≠ 0
values were derived or xc = 0 values were assumed. The relevance of that to the
accuracy of our t(T) results that we derived is as follows.
The above considerations raise the more general question of the meaning of xc
in systems where the transport is by tunneling in cases where a lattice-like neighbors
arrangement does not exist [20].
In particular, when there is a monotonously
5
decreasing RDF and the transport is by tunneling (which is, in principle, long range
in spite of the short decay length), there should not be (classically) a finite
characteristic threshold value in the system and thus the xc = 0 value is the “natural”
threshold of such a system. The xc > 0 values that are derived from the best fittings,
when tunneling conduction is involved, are then cutoffs that result from experimental
or computational limitations. The a posteriori fact that with or without the xc = 0
approximation the difference that we got between the slopes of the linear t-1 vs 1/√T
dependence (Fig. 2(b) and Fig. 3(b)) is not too large, supports our main conclusions
regarding the order of the d/ξ values. On the other hand this difference can serve as
measure for the uncertainty of the results (noting that it is much larger, than the ±5
error bar that is associated with each of the slopes). We turn then to estimate the
accuracy of the d/ξ values by comparing the slope that we derived by the use of Eq.
(3) with the best fitted xc ≠ 0 value (i.e., 163√T), with the slope that we derived by
the use of xc = 0 (i.e., 120√T). The corresponding 120/163 ratio can be taken then as
suggesting that the slopes and thus the d/ξ values that we considered, have an
accuracy of 25%. In fact, the robustness of the linear dependence of t-1 on 1/√T
indicates that in practice all the information on the transport is indeed encoded in the
values of t. In turn, this shows that the results we obtained do not depend critically
on our choice of the xc = 0 value.
6