## 3 Results and Discussions
Another measurement performed on the Cr/Al/Ti/Au-2 sample confirmed our analysis, in which Cr/Al and Ti/Au bilayers were evaporated separately with a process of opening the chamber door for 2 min between them (three-step method in Table 1), oxidation of the Ti also occurred as shown in Figure 4a. Sputtering all metal sublayers sequentially in the vacuum chamber may solve this problem. Nonetheless, in the process of photoresist stripping and cleaning, the sidewalls of Cu layer are prone to be oxidized and flake off, thus metal line width reduces (Figure 4b). Among four kinds of samples, Cr/Al/Ti/Au had clear interfaces with less problems of the binary phase and the oxidation mentioned above, as the Al\({}_{2}\)O\({}_{3}\) forming on the Al surface affords it a better corrosion resistance than Cu. Despite this, alloying between the adhesive layer and the conductor layer was observed in all samples. Measured \(Q\)-factors for these four kinds of inductors are presented in Figure 5a-d. Maximum value of \(Q\) curve (\(Q_{m}\)), usually referred to as one of the bench marks to evaluate the quality of inductors, was around 0.55 GHz for all inductors. \(Q_{m}\) was 6 for Cr/Cu/Cr whereas the smallest one was 5.2 for Cr/Cu/Ti/Au, \(Q_{m}\) of the Cr/Al/Ti/Au inductor was 5.6. The Al-based metal multilayer shows competitive ability in fabricating actual inductor on sapphire, as compared to Cu-based multilayers whose electrical conductivities degrade due to the oxidation or interdiffusion of the metals. Additionally, simulation results match well with the measured ones in spite of slight variations in \(Q_{m}\) and resonant frequencies that exist between them. Figure 4: (**a**) Segmental in-depth profile of Cr/Al/Ti/Au-2 sample; (**b**) Uncovering sidewalls of Cu layer are oxidized and flake off in Cr/Cu/Ti/Au inductor. Conventionally, three-element model ruling out the substrate-related elements on the basis of Pi model [16; 21] is often used for on-chip spiral inductors on sapphire (DUT in Figure 6b), in which \(L_{s}\) and \(R_{s}\) represent the series inductance and resistance, and capacitance \(C_{s}\) models the feed-through path between the spiral and the underpass metal line. In addition, the parasitic effect from GSG test pattern, which is used to be de-embedded relying on \"open\", \"through\" and \"short\" structures, was included into our model. Shunt admittances \(Y_{p}\) from the contact pads and the interconnects were modeled by capacitance \(C_{p}\) and resistance \(R_{p}\)[22; 23], while contact impedances \(Z_{c}\) between the probes and the pads at input and output ports were indicated by series \(L_{c}\)[24; 25], it cannot be ignored for its great influence on high frequency characteristic of the model. Except \(R_{s}\), the other elements are frequency-independent. Analytical formulas [21] considering skin effect in the conductor are used to evaluate \(R_{s}\) as follows:  
\[\delta=\sqrt{\frac{\rho}{\pi\cdot\mu\cdot f}} \tag{5}\]  
where \(\delta\) is the skin depth, \(\rho\), \(\mu\), and \(f\) represent the resistivity, permeability and frequency, respectively. \[t_{eff}=\delta\cdot\left(1-e^{-l_{me}/\delta}\right) \tag{6}\]  
\[R_{s}=\frac{\rho\cdot l}{w\cdot k\cdot t_{eff}} \tag{7}\]  
Figure 5: Comparisons between simulation and measurement results for the inductors fabricated by (**a**) Cr/Al/Ti/Au; (**b**) Cr/Cu/Cr; (**c**) Cr/Cu/Ni/Au and (**d**) Cr/Cu/Ti/Au, respectively. where \(t_{\textit{eff}}\) is defined as metal effective thickness, \(l\) and \(w\) represent the metal line length and width, respectively. A correction factor \(k\) is introduced in Equation (7). In our experiment, empirical value for \(k\) ranges from 0.6 to 0.7 depending on how well the model data fits with the experimental result. Figure 7 shows measured and modeled \(Y\)-parameters, \(L\) values and \(Q\)-factors for the inductors, respectively. Good agreement between them verifies the validity of our model and extraction scheme.