Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2021 / Paper 2 / Section B / 19 / iii / a

A transparent metallic network must span the patterned electrode with many redundant current paths while obscuring little area. For random rods, number density times squared wire length is a useful percolation measure; greater aspect ratio generally reaches a connected network at lower coverage. Conductivity also depends on intrinsic metal resistivity, wire cross section, orientation, length distribution, wire–wire contact resistance and the number of junctions along each spanning path. Insulating capping layers can make nanowire junction resistance dominate even when the individual wires are excellent conductors.

Near the connectivity threshold, small density changes or wire breaks produce large resistance changes. For a narrow stripe, finite-size effects and orientation matter: a macroscopic sheet-average percolation result does not ensure a reliable 100 µm-wide path. Optical performance depends on coverage, metal extinction, wire diameter, aggregation, optical haze, substrate transmission and any binder. Increasing wire density lowers resistance but usually lowers transmission. Welding, pressing or suitable embedding can improve contacts with less additional optical coverage than simply adding metal.^[ref]

The stripe has $L/W=250$ squares, so its end-to-end resistance is $R=250R_{\square }$; the stated sheet-resistance limit requires a stripe resistance below 5 kΩ, not 20 Ω. The sketch below separates sparse connectivity, redundant paths and junction quality.