# Conclusive Model-Fit Current-Voltage Characteristic Curves with Kink Effects
## 4 Discussions  
The as-measured data redrawn as characteristic curves can be fitted with the ones based on the modified conventional current-voltage formula. Even though it is quite engineering, such a fitting may be quite easy to be undertaken. The parameters are easily determined if the parameters are not required to keep the same. Instead, the trends or scales of some specific parameters always give some information [16; 17]. Moreover, the kink effects really exist, as referred to Figure 4a,b, for the 0.18-micron process, where the determined parameters for characteristic curves are listed in Table 1, as well as Figure 6a,b for the 0.09-micron process, where the determined parameters for characteristic curves are listed in Table 2. A kink effect model will efficiently help to work out the fitting, e.g., the sets, (Figures 4a,b, 5c,d, 6a,b and 7a,b), where the minimum delta (\(\delta\)) is highly reduced, as expected. Figure 5: TSMC 0.18 micron process (**a**) the strong correlation gives a straight line passing the origin as the kink is plotted against \(\mathrm{(V_{GS}-V_{th})}\) (**b**) \(\mathrm{k_{N}}\) containing mobility, \(\upmu\), is inversely proportional to \(\mathrm{(V_{GS}-V_{th})^{-1/3}}\). (**c**) \(\mathrm{I_{DS}-V_{DS}}\) characteristic curves and the corresponding fitting closely appear through elimination on kinks. (**d**) The differences between the fitting current and the measured current at various \(\mathrm{V_{DS}}\)’s are demonstrated through elimination on kinks. By comparing the FinFET 0.09-micron process with the TSMC 0.18-micron process, both share the same characteristics: one is for where the kinks are located and the other is for the electrical-field-associated mobility. As demonstrated in Figure 5a,c, kinks, \(\chi\)'s, \begin{table}
\begin{tabular}{c c c c c} \hline
**Gate Bias (V)** & \(\mathbf{k_{N}}\) **(A/V\({}^{2}\))** & \(\mathbf{V_{th,fit}}\) **(V)** & \(\boldsymbol{\lambda}\) **(I/V)** & **Kink (V)** \\ \hline \(\mathrm{V_{G}=0.25~{}V}\) & \(8.90\times 10^{-5}\) & \(-0.132\) & \(0.138\) & \(0.35\) \\ \(\mathrm{V_{G}=0.50~{}V}\) & \(7.40\times 10^{-5}\) & \(-0.140\) & \(0.120\) & \(0.58\) \\ \(\mathrm{V_{G}=0.75~{}V}\) & \(6.88\times 10^{-5}\) & \(-0.100\) & \(0.110\) & \(0.79\) \\ \(\mathrm{V_{G}=1.00~{}V}\) & \(6.60\times 10^{-5}\) & \(-0.020\) & \(0.106\) & \(0.89\) \\ \hline \end{tabular}
\end{table} 
Table 2: Transistors using 0.09 micron process technology with where kink is located. Figure 6: FinFET 0.090 micron process (**a**) \(\mathrm{I_{DS}}\)-\(\mathrm{V_{DS}}\) characteristic curves and the corresponding fitting without taking kink-effect factor into account. (**b**) The differences between the fitting current and the measured current at various \(\mathrm{V_{DS}}\)’s are demonstrated. Standard deviations: \(\delta_{0.25}=2.28\times 10^{-7}\), \(\delta_{0.5}=2.11\times 10^{-7}\), \(\delta_{0.75}=2.32\times 10^{-7}\), \(\delta_{1.0}=2.27\times 10^{-7}\). are almost located at \(V_{\rm{DS}}\), which is linearly dependent on \((V_{\rm{GS}}-V_{\rm{th}})\), meaning that there exists a strong correlation between the kink and the connection at the end of the triode region. Also, Figure 5b,d show that \(k_{\rm{N}}\) tends to be proportional to \((V_{\rm{GS}}-V_{\rm{th}})^{-1/3}\) and that the mobility (\(\mu\)) of the carriers is then understood to be due to delaying effects at the interface of the strongly inversed layer with the oxide, as shown in Equation (8): [15]  
\[\mu=\kappa(V_{GS}-V_{th})^{\frac{-1}{3}} \tag{8}\]  
As for \(\alpha\) and \(\beta\) in Equations (1) and (2), the maximum discrepancy between the fitting current and the measured current always helps to identify \(\alpha\), while the width of the kinks at a certain \(V_{\rm{GS}}\) determines what \(\beta\) value is more adequate. Both parameters are intriguing to explore profoundly in the near future [19]. Figure 7: _Cont._  
Nevertheless, a model might be useful, especially for circuit design, including digital and analog circuits.