Size effects and the problem with percolation in nanostructured transparent conductors.

Much research is underway at present to develop nanostructured transparent conductors for use as electrodes. Transparent electrodes typically require high visible transmittances, T > 90%, and so must be very thin. We show that for most nanostructured films thin enough to display T > 90%, the conduction can be described by percolation theory. This means DC conductivities are lower than in bulk, giving correspondingly higher sheet resistances, R(s). To improve our understanding of the consequences of this, we develop a model which relates T to R(s) in the percolation regime. We define a percolative figure of merit, Π, for which high values result in high T and low R(s). High values of Π are achieved for high DC conductivity and low optical conductivity. In addition, the film thickness, t(min), where the DC conductivity first deviates from its bulk value and the percolation exponent, n, must both be as low as possible. We find that this model fits extremely well to much of the data in the literature. We demonstrate that t(min) scales linearly with the smallest dimension of the nanostructure in question (i.e., diameter for wires or thickness for flakes). This clearly confirms that low diameter nanowires or thin platelets are best for transparent conducting applications. We predict the properties of silver nanowire networks to improve as wire diameter is decreased. Networks of wires with D < 20 nm should display properties superior to the best ITO. We demonstrate the deficiencies of standard bulk theory and the importance of understanding percolation by measuring R(s) and T for networks of silver flakes. We measure the bulk ratio of DC to optical conductivity to be ∼35, suggesting R(s) = 100 Ω/◻ and T = 90% are attainable. However, the large flake thickness results in high t(min) and so low Π, resulting in actual values of T = 26% for R(s) = 100 Ω/◻. This makes this material completely unsuitable for transparent conductor applications.