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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 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.
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.
This paper extends the applicability of the micro four-point probe technique from the sheet resistance measurements on large areas toward narrow (<20 nm) semiconducting nanostructures with an elongated fin geometry. Using this technology, it is shown that the sheet resistance of boron-implanted and laser-annealed silicon fins with widths ranging from 500 down to 20 nm rises as the width is reduced. Drift-diffusion simulations show that the observed increase can be partially explained by the carrier depletion induced by interface states at the fin sidewalls.
We derive exact, analytic expressions for the sensitivity of resistive and Hall measurements to local inhomogeneities in a specimen's material properties in the combined linear limit of a weak perturbation over an infinitesimal area in a small magnetic field. We apply these expressions both to four-point probe measurements on an infinite plane and to symmetric, circular van der Pauw discs, obtaining functions consistent with published results. These new expressions speed up calculation of the sensitivity for a specimen of arbitrary shape to little more than the solution of two Laplace equation boundary-value problems of the order of N 3 calculations, rather than N 2 problems of total order N 5 , and in a few cases produces an analytic expression for the sensitivity. These functions provide an intuitive, visual explanation of how, for example, measurements can predict the wrong carrier type in n-type ZnO. V C 2013 AIP Publishing LLC. [http://dx
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