Tures within this case present a smaller dielectric thickness when compared with the region in the electrodes. The geometrical condition d R (for a uniform field) is therefore satisfied, which validates the usage of Equation (three) inside the corresponding analytical calculations. For this, we regarded r,SiO2 with a relative uncertainty of 1 . Nevertheless, even if the impact on the fringing fields is modest for the case of typical samples’ structures, we nonetheless take into account it as a minor more correction term to the very first 2-Bromo-6-nitrophenol medchemexpress approximation expression in Equation (three). An analytical expression of this correction has been identified empirically and leads to an error term decrease than 20 for R/d 10 in a fantastic agreement using the numerical PX-478 Cancer calculation in the degree of 1 [32]. For the case on the high- samples studied here, the dimensions on the circular gold electrodes and dielectric layers’ thicknesses are described in detail in Section three.1.two with R/d 1, which tends to make the contribution of your fringing fields towards the measured capacitances higher. It’s therefore mandatory to consider a new analytical expression to right the initial approximation (uniform field) of parallel-plate capacitor CP . For this, we discovered the following expression: C = CP 1 1 where h(d, R) = 1 ln 1 h(d, R) , 3ln(r ) d R d , R (4)(five)and is definitely an adjustable parameter depending slightly on hpad , = 0.097 for hpad = 50 nm. For d/R ranging from 2 to ten, h(d,R) increases nearly linearly as a function of d/R with a slope weakly dependent on r in agreement with [34]. In case of d/R 1, this leads to a initial order approximation C = r 0 R, (six)( , ) = 1 ,(5)Nanomaterials 2021, 11,and ‘ is an adjustable parameter based slightly on hpad, ‘ = 0.097 for hpad = 50 nm. For d/R ranging from 2 to ten, h(d,R) increases practically linearly as a function of d/R using a slope weakly dependent on r in agreement with [34]. In case of d/R 1, this leads to a six of 19 1st order approximation = , (six)independent of your electrode separation as expected for capacitance of uncoupled circular independent of your electrode separation as expected for capacitance of uncoupled circular electrodes [35,36]. The capacitance calculation employing the relations (three) to (5) agrees with electrodes [35,36]. The capacitance calculation applying the relations (three) to (5) agrees with FEM FEM calculation in the amount of 3 for 0.2 d/R two.6 and to get a wide selection of r values, from calculation at the level of 3 for 0.2 d/R 2.6 and to get a wide array of r values, from 200 2001500, as shown in Figure 3. Additionally, the observed deviations weakly depend around the to to 1500, as shown in Figure 3. Additionally, the observed deviations weakly depend onr the r values, without exceeding 1 . Consequently, the FEM method will probably be preferred to values, with no exceeding 1 . Hence, the FEM approach will probably be preferred to analytical analyticalaones for capacitance calculation on high- on high- Nevertheless, Even so, the ones for precise a precise capacitance calculation samples. samples. the analytical analyticalwill be applied be evaluate the evaluate theofuncertainty of your capacitance process approach will to applied to uncertainty the capacitance calculation (by calculation (by propagating the uncertainties onand R)values d andestimate the uncertainty propagating the uncertainties on input values d input and after that to R) and then to estimate the uncertainty on the dielectric continuous determination. Theon the correction tocorrection around the dielectric continuous determination. The uncertainty unc.

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