## 2 Impedance Spectroscopy
### Materials Modelling by IS
In the equivalent electrical circuit, \(R_{T1,2}\) represent the resistance of the transmission lines connecting the electrodes to the contact pads, \(C_{sub1,2}\) and \(R_{sub1,2}\) are the capacitance and the resistance between the electrodes and the substrate, respectively, \(R_{sol}\) is the solution resistance, \(C_{d1,2}\) represent the double-layer capacitance, \(Z_{W1,2}\) is a Warburg impedance for accounting for the phenomena at the interface between the two electrodes and the solution, while \(R_{d1,2}\) and \(C_{d1,2}\) represent the resistance and capacitance created on the surface of the polysilicon electrodes due to the formation of a thin silicon dioxide film on the electrodes' surface, respectively. Figure 10: Parallel association approach carried out by [PERSON] et al. [130] (**a**) Nyquist diagram of an IDE sensor; (**b**) Nyquist diagram of a three-dimensional sensor with two suspended electrodes; (**c**) electrical equivalent circuit model; (**d**) fitting of the Nyquist data of the IDE sensor obtained at 2 ppm DEHP concentration; (**e**) fitting of the Nyquist data of 3D sensor with two suspended electrodes obtained at 2 ppm DEHP concentration. [PERSON] et al. [131] also pursued a parallel association approach. In their study, an interdigitated array (IDA) microelectrode-based impedance immunosensor was developed for a new application of sensitive, specific, and rapid detection of the AI virus H5N1. The system was based on combining an IDA microelectrode with specific capture antibodies, red blood cell bio label, and impedance measurements. The collected impedance data were modelled with good superposition between the experimental data and the fits. Figure 11 shows the equivalent circuit in the presence of a redox probe and an example of the immunosensor electrochemical impedance response and respective fitted data. The circuit includes the ohmic resistance of the electrolyte solution, R\({}_{3}\), a Warburg impedance, \(Z_{W}\), which resulted from the diffusion of ions to the electrode interface from the bulk of the electrolyte, an electron transfer resistance, R\({}_{1}\) which resulted from the diffusion of ions to the electrode interface from the bulk of the electrolyte, an electron transfer resistance, R\({}_{1}\), and a double layer capacitance contribution modelled by a parallel association between a capacitor, C\({}_{1}\), and a branch composed by a second capacitor, C\({}_{2}\), in series with a resistor, R\({}_{2}\). In a work by [PERSON] et al. [132], undoped and niobium oxide-doped TiO\({}_{2}\):WO\({}_{3}\) MMOs were fabricated. The solid sensors were manufactured from TiO\({}_{2}\), WO\({}_{3}\), and Nb\({}_{2}\)O\({}_{5}\) powders; initially, TiO\({}_{2}\) and WO\({}_{3}\) powders were mixed in the molar volume ratio of 48.92:51.08%, respectively; the Nb\({}_{2}\)O\({}_{5}\) dopant was subsequently incorporated, in specific proportions of 4 and 6% wt%. Then, EleI8 was used to characterize the electrical response of the samples to variations in relative humidity, using an excitation signal with a maximum amplitude of 0.5 V, in the range of 1.5 kHz to 40 MHz, while varying the humidity concentration from 10 to 100%. In the proposed electrical equivalent model, using a parallel strategy of terms association, see Figure 12 (for which some examples of experimental data fitting are displayed), they identified several terms (in accordance with the known existent phenomena), contributing to the overall electrical response of the sensors.