## 2 Materials and Methods
### Reference Staircase Microcapacitor Sample  
The electrical devices considered in this work are microsized metal oxide semiconductor (MOS) capacitors that have been widely studied by the metrology and research communities [38; 39; 40; 41]. The MOS capacitors are composed of circular gold electrodes evaporated on silicon dioxide (SiO\({}_{2}\)) deposited on a highly doped P-type silicon substrate of resistivity 0.01 \(\Omega\).cm. The SEM image of the reference kit is depicted in Figure 4. In order to vary the capacitance values, the diameter of the upper gold pad diameter varies from 1 to 4 \(\upmu\)m, and the SiO\({}_{2}\) thickness ranges from 50 to 300 nm, with about 80 nm steps. The reference kit is used for both calibration and verification. Prior to the measurements, the analytical derivation of the theoretical capacitances is considered. The impedance of the MOS structures measured at the tip apex of the probe is modeled by a series model consisting of an oxide capacitance \(C_{\text{ox}}\) and a depletion capacitance \(C_{\text{depl}}\). Both capacitances can be described by the parallel plate capacitor formalism. The resulting capacitance \(C_{\text{TOT}}\) is given by  
\[C_{\text{TOT}}=C_{\text{ox}}C_{\text{depl}}/\Big{(}C_{\text{ox}}+C_{\text{ depl}}\Big{)}. \tag{1}\]  
The capacitance \(C_{\text{ox}}\) is calculated from the areas of the gold pads \(A\) and the SiO\({}_{2}\) thicknesses \(d_{\text{ox}}\). \[C_{\text{ox}}=A\frac{\epsilon_{0}\epsilon_{\text{FiS0}_{2}}}{d_{\text{ox}}} \tag{2}\]  
The silicon dioxide is assumed to have a relative dielectric constant of \(\epsilon_{\text{FiS0}_{2}}=3.9\). The charge stored on the capacitor is distributed across a certain depth that adds the depletion series capacitance \(C_{\text{depl}}\) in series to \(C_{\text{ox}}\). The capacitance was estimated to be proportional to the area \(A\) of the metallic electrode and inversely proportional to the depleted zone depth \(d_{\text{depl}}\) according to  
\[C_{\text{depl}}=A\frac{\epsilon_{0}\epsilon_{\text{FiS}}}{d_{\text{depl}}} \tag{3}\]  
with \(d_{\text{depl}}=\sqrt{2\epsilon_{0}\epsilon_{\text{r}}\Psi/qN_{A}}\) and where \(\epsilon_{\text{r}\text{Si}}=12\) is the relative permittivity of the silicon bulk substrate, \(\Psi\) represents the interface band bending at the Si/SiO\({}_{2}\) interface and is set to 300 mV, \(q\) is the charge of the electron (1.6 \(\times\) 10\({}^{-19}\) C) and \(N_{A}\) is the doping level of the  
Figure 4: SEM image of the metal oxide semiconductor (MOS) microcapacitor reference kit. silicon bulk around \(5\times 10^{18}/\mathrm{cm}^{-3}\). Although the interface band bending at the interface is relatively unknown, due to the high doping, the depletion capacitance is higher than the oxide capacitance. Therefore, since the two capacitances are in series, an uncertainty on \(\Psi\) has a negligible effect on the total capacitance. The calibration procedure consists of determining a two-port error terms box to convert the measured complex reflection coefficient \(\Gamma_{\mathrm{M}}\) by the VNA into the complex reflection \(\Gamma\) at the apex tip. Then, the calibration model established can be used to determine the other capacitance values. The one-port vector calibration method model used to make the link between the reflection coefficient \(\Gamma_{\mathrm{M}}\) measured by the VNA and the reflection coefficient \(\Gamma\) is given by  
\[\Gamma_{\mathrm{M}}=e_{00}+\frac{e_{01}e_{10}\Gamma}{1-e_{11}\Gamma} \tag{4}\]  
The complex terms \(e_{00}\), \(e_{01}e_{10}\) and \(e_{11}\) correspond, respectively, to the directivity, source match and reflection tracking errors. These calibration parameters depend on the microwave path between the apex tip of the probe and VNA receivers.