Abstract
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, 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 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 μm-scale M4PP footprint.
Subject
Electrical and Electronic Engineering,Mechanical Engineering,Mechanics of Materials,General Materials Science,General Chemistry,Bioengineering
Cited by
3 articles.
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