Current divisions and distributed Joule heating of two-dimensional grid microstructures

This paper presents current divisions and distributed Joule heating of two-dimensional (2D) grid microstructures. The current divisions on 2 × 2, 4 × 4, and n × n grid microstructures made of the same conductive beams are analyzed theoretically, and Kirchhoff’s voltage law (KVL) and Kirchhoff’s current law (KCL) are employed to determine the current division factors and directions under different voltage input cases. The equivalent resistances and Joule heating power are therefore derived. 2D 2 × 2 grid microstructures made of gold (60 nm in thickness) and those made of near-equiatomic NiTi (850 nm in thickness) for various independent voltage input cases are fabricated by electron-beam evaporation and co-sputtering, respectively. The equivalent resistances of these grid microstructures are measured by four-terminal resistance measurement at ambient conditions, which show a good agreement with the theoretical results. Further investigation on the electrical resistivities of evaporated gold layer (4.85 × 10^–8 Ω m) and co-sputtered NiTi (1.23 × 10^–5 Ω m) reveals that the influences of scale effect and fabrication process on the resistivity cannot be neglected. As such, it is found to be necessary to consider the materials’ resistivity in the fabricated grid microstructures before their electro-thermal analysis.