Effective medium theory for the electrical conductivity of random metallic nanowire networks

Interest in studying the conductive properties of networks made from randomly distributed nanowires is due to their numerous technological applications. Although the sheet resistance of such networks can be calculated directly, the calculations require many characteristics of the system (distributions of lengths, diameters and resistances of nanowires, distribution of junction resistance), the measurement of which is difficult. Furthermore, such calculations can hardly offer an analytical dependence of the sheet resistance on the basic physical parameters of the systems under consideration. Although various theoretical approaches offer such analytical dependencies, they are often based on more or less reasonable assumptions rather than rigorously proven statements. Here, we offer an approach based on Foster’s theorem to reveal a dependence of the sheet resistance of dense nanowire networks on the main parameters of such networks. This theorem offers an additional perspective on the effective medium theory and extends our insight. Since the application of Foster’s theorem is particularly effective for regular random resistor networks, we propose a method for regularizing resistor networks corresponding to random nanowire networks. We found an analytical dependence of the effective electrical conductivity on the main parameters of the nanowire network (reduced number density of nanowires, nanowire resistance, and resistance of contacts between nanowires).

💡 Research Summary

The paper presents a rigorous analytical framework for predicting the sheet resistance of dense random metallic nanowire networks (RNNs) by leveraging Foster’s theorem within an effective medium theory (EMT) context. Traditional EMT approaches rely heavily on symmetry assumptions and often require detailed knowledge of numerous system characteristics (length, diameter, and resistance distributions of nanowires, as well as junction resistance distributions), which are difficult to measure experimentally. To overcome these limitations, the authors first model nanowires as zero‑width conductive sticks of equal length l₀, randomly placed with uniformly distributed orientations and Poisson‑distributed contacts. Each stick has an intrinsic resistance R_w, while each junction between sticks carries a resistance R_j.

A key innovation is the regularization of the inherently irregular resistor network that results from the random placement of nanowires. By connecting the two ends of each nanowire with a zero‑conductance edge (effectively removing degree‑1 vertices), the network is transformed into a 3‑regular graph (all vertices have degree 3) without altering its overall electrical response. In the limit of very dense networks, the average number of contacts per wire is ⟨k⟩ = 2π n l₀², where n = N/A is the areal number density of sticks. Consequently, the total number of edges N_E = (3/2)⟨k⟩ N, and the edge conductance distribution f_G(g) consists of three contributions: (i) a delta peak at g = 1/R_j for junctions, (ii) a delta peak at g = 0 for the artificial zero‑conductance edges, and (iii) a continuous part arising from wire segments whose conductance is inversely proportional to their length. The segment lengths follow an exponential distribution with mean ⟨l⟩ = π l₀ n l₀², leading to a conductance PDF f_G(g) ∝ g⁻² exp(−C/g) where C = 2π n l₀² R_w.

Applying Foster’s theorem, which for a regular network of degree z states that ∫