Conductivity of Nanowire Arrays under Random and Ordered Orientation Configurations

A computational model was developed to analyze electrical conductivity of random metal nanowire networks. It was demonstrated for the first time through use of this model that a performance gain in random metal nanowire networks can be achieved by slightly restricting nanowire orientation. It was furthermore shown that heavily ordered configurations do not outperform configurations with some degree of randomness; randomness in the case of metal nanowire orientations acts to increase conductivity.

orientation configurations are reported. In the first, a uniform distribution of orientations over the range (−θ, θ) with respect to a horizontal line is used. In the second, a distribution of orientations over the range [−θ] � [θ] is used, also with respect to a horizontal line. In each case θ is gradually decreased from 90° to 0°. Conductivity is measured both in directions parallel and perpendicular to alignment. It was found that a significant improvement in conductivity parallel to direction of alignment can be obtained by slightly restricting orientation of the uniform distribution. This improvement, however, comes at the expense of a larger drop in perpendicular conductivity. The general form of these results matches that demonstrated by Behnam et al. 13 for carbon nanotube films, although specific values differ. Surprisingly, it was found that the highly ordered second case is unable to outperform isotropic networks for any value of θ; this demonstrates that continuous orientation configurations with some degree of randomness are preferable to highly ordered configurations.
The computational model employed in this study is structured similarly to those previously demonstrated by Mutiso et al. 17 and Behnam et al. 13,14 . Random networks of nanowires are modeled by a network of randomly generated line segments. These nanowires are then assembled into a resistor circuit matrix using Kirchoff 's junction rule, where each length of wire and each junction between two wires is a resistor. A voltage arbitrarily set to 10 V is then applied from left to right across the network. Figure 1 shows two examples of these networks at different concentrations; fig. 1(a) is at a length concentration just above the critical concentration of percolation, C p , whereas fig. 1(b) is at a length concentration far above C p . The junction resistance is assumed to be a constant value R J ; the validity of this assumption was demonstrated by Mutiso et al. 17 . The resistance of the wires, R rod , is calculated using the resistivity of silver, and values for the length and diameter of the rods. The length of the nanowires is set to 35 μ m with diameter of 65 nm. This geometry is chosen based on silver nanowires which are commercially available from Seashell Technologies 18 .
A matrix equation is then set up and solved to find the voltage at each junction, which can be used to calculate conductivity of the network. Figure 2 is a 2-D color map of voltage at each node of an array with concentration much above critical percolation concentration (C ~ 5 C p ). It shows that voltage uniformly decreases across a high concentration network as expected. We include R rod in our calculations despite the fact that it is on the order of 10 −2 times smaller than R J so that the model may be applied later to similar situations where R rod may be comparable to R J . Figure 3 shows conductivity as a function of length concentration for isotropic samples. The conductivity, k, is the sheet conductivity of the samples. Since junction resistance (R J ) is dominant in this system, we define a normalized conductivity k N as: The length concentration, C, is defined as the number density of rods multiplied by the length of each rod. We define a normalized concentration C N as: where l is the length of each rod. We choose to normalize in this way because C N is independent of rod parameters. For example, the critical percolation concentration C p for rods in 2D space can be determined using the following formula 3 : Normalizing C p by multiplying by l thus yields a value of 5.71 which is constant for any set of rod parameters. This value is marked on Fig. 3. Since our manipulated variable is independent of rod parameters, it is necessary to conduct simulations only with one rod length. The inset diagram in fig. 3 is a log-log plot of normalized concentration as a function of C N − C N,p . This plot is a straight line, which suggests a power law dependence of conductivity on C N − C N,p , of the form: This equation (4) is the accepted general form of dependency of conductivity on concentration 19 fig. 3, as a function of θ, where orientation of the rods is restricted over (−θ, θ) from a horizontal line parallel to the boundary of the space. We refer to this horizontal line as L h . Fig. 4(b) shows an example of one such sample with θ = 45°. It is important to note that when orientation is restricted in this way, the networks are no longer rotationally symmetrical; conductivity measured in the same direction as L h is different from conductivity measured perpendicular to L h . In this sample, conductivity in the same direction as L h is the conductivity obtained by applying a potential difference from the left of the sample to the right (or vice-versa). Conductivity perpendicular to L h is the conductivity obtained by applying a potential difference from the top of the sample to the bottom (or vice-versa). All samples are generated at a normalized concentration of 29, with all other variables the same as in fig. 3. It can be seen that by reducing θ from 90°, the isotropic state, conductivity perpendicular to L h immediately begins to decrease. Conductivity in the same direction as L h , however, increases slightly before decreasing, reaching a maximum around θ = 60°. The improvement in conductivity over the isotropic state is an approximately 20% gain in conductivity. This is a significant improvement in performance for applications which only require current flow in one direction.
This pair of effects can be described as the result of two competing effects that result from decreasing θ. As θ is reduced, the number of intersections and number of parallel paths decreases, reducing conductivity. At the same time, the number of junctions the current must travel through decreases (same direction as L h ) or increases (perpendicular to L h ). This increases conductivity in the same direction as L h , and further decreases conductivity perpendicular to L h . In the same direction as L h , these effects are competing. This allows conductivity to initially increase before decreasing when the lack of parallel paths dominates. In the direction perpendicular to L h , both effects act to decrease conductivity, causing conductivity to immediately drop.  Fig. 5(b) shows an example of one such sample, with θ = 45°. Note that for any value of θ taken with respect to L h , the value 90° − θ is exactly equivalent when taken with respect to a line perpendicular to  L h . For the case where θ = 45°, as in fig. 5(b), it does not matter whether conductivity is measured along or perpendicular to L h . For any other value of θ with conductivity measured along L h , an equivalent network is obtained by taking 90° − θ and measuring conductivity perpendicular to L h.
Even though the configuration shown in fig. 5 is much more controlled than the restricted random distribution shown in fig. 4, no performance gain over an isotropic sample is achieved for any value of θ. θ = 30° is equivalent in conductivity to an isotropic sample; any other value of θ causes a decrease in conductivity. This yields the unexpected result that highly ordered orientation configurations do not yield the greatest increases in conductivity. In fact, randomness in this case acts to improve conductivity, and should not be removed.
In summary, metal nanowire networks show great potential for application in various forms of technology. Our computational model, which has proven itself accurate through its good fit with previously published data, has demonstrated quantitatively how different orientation configurations can impact conductivity of metal nanowire networks. Restriction of orientation can improve conductivity in a single direction by significant amounts; this could be relevant in a variety of technologies where current flow is only required in one direction. Surprisingly, heavily controlled orientation configurations do not exhibit superior conductivity; some degree of randomness in orientation in fact acts to improve conductivity of the networks.