Effective electrical resistivity in a square array of oriented square inclusions

Benny Guralnik*, Ole Hansen, Henrik Hartmann Henrichsen, José Manuel M Caridad, Wilson Wei, Mikkel Fougt Hansen, Peter Folmer Nielsen, Dirch Hjorth Petersen

*Corresponding author for this work

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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<c<1). We find that in the range 0.01≤c≤0.99, ρeff may be approximated (to within <0.3% error with respect to finite element simulations) by: [formula] 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 μm-scale M4PP footprint.

Original languageEnglish
Article number185706
Issue number18
Number of pages12
Publication statusPublished - 2021

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