Effective electrical resistivity in a square array of oriented square inclusions

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, μLED 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 Y V 1999 SIAM J. Appl. Math. 59, 1267-1287], 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 modelling, and mathematical analysis to generalize ρeff for an arbitrary inclusion size (0

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 μm-scale M4PP footprint.