= Solution
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.<Self-limited plasmonic welding of silver nanowire junctions by Erik C. Garnett and others>[Nature Materials 11 (2012), pp. 241–249, junction-localized welding mechanism]{ref}
The stripe has $L/W=250$ squares, so its end-to-end resistance is $R=250R_\square$; \b[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.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2021/paper-2-transparent-networks.png]
{title=Original network schematics distinguishing connectivity, optical coverage and wire-junction resistance}
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