# Conclusive Model-Fit Current-Voltage Characteristic Curves with Kink Effects
## 3 Application  
The minimum delta (8) in Equation (3) can be used to determine the chosen parameters without considering kink effects, which are listed in Table 1[8]. In the table, the minimum  
Figure 3: **(a)** The subtraction values of \(\rm I_{DS}\)-\(\rm V_{DS}\) characteristic curves and the corresponding fitting without eliminating the Gaussian term into account. **(b)** with eliminating the Gaussian term into account. **(c)** The final corresponding fitting in Figure 3b.  
delta at different gate biases requires different \(\mathrm{k_{n}}\), lambda (\(\lambda\)), and threshold voltages. For one thing, the slope of lambda addressing the leakage current dominates a rough minimum delta. As referred to Figure 4, the fitting is not as good. The differences between the fitting current and the measured current at various \(\mathrm{V_{DS}}\)'s are demonstrated in Figure 4, where the peaks at different \(\mathrm{V_{GS}}\) are located at different \(\mathrm{V_{DS}}\), named \"kinks\" in Table 1. Surprisingly, the strong correlation gives a straight line passing the origin as the kink is plotted against (\(\mathrm{V_{GS}}-\mathrm{V_{th}}\)), as shown in Figure 5. That is to say, the kink is always proportional to (\(\mathrm{V_{GS}}-\mathrm{V_{th}}\)). In addition, \(\mathrm{k_{N}}\) containing mobility, \(\mu\), is proportional to (\(\mathrm{V_{GS}}-\mathrm{V_{th}}\))\({}^{-1/3}\), as shown in Figure 5.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Gate Bias (V)** & \(\mathrm{k_{N}}\) **(A/V\({}^{2}\))** & \(\mathrm{V_{th,fit}}\) **(V)** & \(\lambda\) **(I/V)** & **Kink (V)** \\ \hline \(\mathrm{V_{GS}}=1.00\) V & 0.2 & 0.625 & 0.134 & 0.35 \\ \(\mathrm{V_{GS}}=1.50\) V & 0.176 & 0.880 & 0.084 & 0.58 \\ \(\mathrm{V_{GS}}=2.00\) V & 0.156 & 1.14 & 0.048 & 0.80 \\ \(\mathrm{V_{GS}}=2.50\) V & 0.143 & 1.46 & 0.05 & 0.94 \\ \(\mathrm{V_{GS}}=3.00\) V & 0.128 & 1.78 & 0.05 & 1.06 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Transistors using 0.18 micron process technology with where kink is located.  
Figure 4: Shows the 0.18 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.  
Figure 5: _Cont._  
With taking kink effects into account, as shown in Figure 5, the deviation (delta) can be suppressed as low as \(10^{-7}\), and the fitting is improved a lot in Figure 5c,d. When turning off kink effects, as shown in Figure 5a,b, the fitting curves are always lifted up at \(\mathrm{V_{DS}}\sim(\mathrm{V_{GS}}-\mathrm{V_{th}})\) as compared to the as-measured data. To take care of the issue in which fitting curves are commonly lower than as-measured currents, the kink effects are thus considered. The solitary waves can be in thermal form, or maybe in phonons. The electron might be deflected because of collisions with phonons, and those collisions may cause the degradation of electrical performances. The item is thus introduced and subtracted, which is proportional to the exponential with the Gaussian form, as seen in Equations (1) and (2). In Figure 5c,d, the fitting is really encouraging. The enlarged figure in Figure 5d does enhance the fitting.