Effective electrical resistivity in a square array of oriented square inclusions. (12th February 2021) - Guralnik, Benny; Hansen, Ole; Henrichsen, Henrik H; Caridad, José M; Wei, Wilson; Hansen, Mikkel F; Nielsen, Peter F; Petersen, Dirch H
Abstract: The continuing miniaturization of optoelectronic devices, alongside the rise of electromagnetic metamaterials, poses an ongoing challenge to nanofabrication. With the increasing impracticality of quality control at a single-feature (-device) resolution, there is an increasing demand for array-based metrologies, where compliance to specifications can be monitored via signals arising from a multitude of features (devices). To this end, a square grid with quadratic sub-features is amongst the more common designs in nanotechnology (e.g. nanofishnets, nanoholes, nanopyramids, μ LED arrays etc). The electrical resistivity of such a quadratic grid may be essential to its functionality; it can also be used to characterize the critical dimensions of the periodic features. While the problem of the effective electrical resistivity ρ eff of a thin sheet with resistivity ρ 1, hosting a doubly-periodic array of oriented square inclusions with resistivity ρ 2, has been treated before (Obnosov 1999 SIAM J. Appl. Math. 59 1267 – 87), a closed-form solution has been found for only one case, where the inclusion occupies c  = 1/4 of the unit cell. Here we combine first-principle approximations, numerical modeling, and mathematical analysis to generalize ρ eff for an arbitrary inclusion size (0 <  c  < 1). We find that in the range 0.01 ≤  c  ≤ 0.99, ρ eff may be approximated (to within <0.3% error with respect to finite element simulations) by: ρ e f f = ρ 1 α ( c ) − 1 − ρ 2 / ρ 1 1 + ρ 2 / ρ 1 α ( c ) + 1 − ρ 2 / ρ 1 1 + ρ 2 / ρ 1 c · α ( c ), α ( c ) = 1 + 0.9707 0.9193 + c 1 − c 2.1261 0.4671 . whereby at the limiting cases of c  → 0 and c  → 1, α approaches asymptotic values of α  = 2.039 and α  = 1/ c − 1, respectively. The applicability of the approximation to considerably more complex structures, such as recursively-nested inclusions and/or nonplanar topologies, is demonstrated and discussed. While certainly not limited to, the theory is examined from within the scope of micro four-point probe (M4PP) metrology, which currently lacks data reduction schemes for periodic materials whose cell is smaller than the typical μ m-scale M4PP footprint.