## 2 Preparation and Measurements
### Fitting \(\rm{I_{DS}}\)-\(\rm{V_{DS}}\) and \(\rm{I_{DS}}\)-\(\rm{V_{GS}}\)
The value Vth = \(-\)0.150 V and the final minimum deviation 1.83 \(\times\) 10\({}^{-10}\) A\({}^{2}\) has the final minimum deviation as compared with the other two ways of parameter-fixing. Figure 1: W120L100 (**a**) Total deviation by summing over each deviation of (I\({}_{\rm DS}\),V\({}_{\rm DS}\)) versus various fixed Kn values obtaining the minimum deviation to determine the chosen Kn; (**b**) versus various fixed \(\lambda\) values obtaining the minimum deviation to determine the chosen \(\lambda\); and (**c**) versus various fixed Vth values obtaining the minimum deviation to determine the chosen Vth. As shown in Figure 3a-c, the transistor W120L240 has the final Kn = 1.13 \(\times\) 10\({}^{-4}\) A/V\({}^{2}\), \(\lambda\) = 0.130 (1/V), and Vth = \(-\)0.025 V. In addition, the corresponding minimum deviations are 3.70 \(\times\) 10\({}^{-11}\) A\({}^{2}\), 2.95 \(\times\) 10\({}^{-11}\) A\({}^{2}\), and 3.56 \(\times\) 10\({}^{-11}\) A\({}^{2}\), respectively. The value \(\lambda\) = 0.130  
Figure 2: W120L160 (**a**) Total deviation by summing over each deviation of (IDS,VDS) versus various fixed Kn values obtaining the minimum deviation to determine the chosen Kn; (**b**) versus various fixed \(\lambda\) values obtaining the minimum deviation to determine the chosen \(\lambda\); and (**c**) versus various fixed Vth values obtaining the minimum deviation to determine the chosen Vth. and the final minimum deviation 2.95 \(\times\) 10\({}^{-11}\) A\({}^{2}\) has the final minimum deviation as compared with the other two ways of parameter-fixing. Figure 3: W120L240 (**a**) Total deviation by summing over each deviation of (IDS,VDS) versus various fixed Kn values obtaining the minimum deviation to determine the chosen Kn; (**b**) versus various fixed \(\lambda\) values obtaining the minimum deviation to determine the chosen \(\lambda\); and (**c**) versus various fixed Vth values obtaining the minimum deviation to determine the chosen Vth. One intriguing observation is concerning negative Vth values. As shown in Figure 4a, Vth values plotted against channel length, surprisingly, is seen to be linear and the slope is determined to be 1.6071 (Volt/micron), as presented in Equation (4). \[Vth=Vth_{O}+1.6071L \tag{4}\]  
_where_  
\[Vth_{O}=-0.41071(Volt)\]  
This raises an interesting point: what if the channel length becomes larger, such as 500 nm which is 0.5 micron? The threshold voltage is found to be 0.25 by using conventional  
Figure 4: (**a**) The final determined Kn values are plotted against the inverse of different channel lengths, showing a straight line as expected. (**b**) The final determined lambda \(\mid\)(1/VA)\(\mid\) are plotted against the inverse of channel lengths, showing a straight line as expected. interpolation methods, as found in Figure 5b,c thus showing the channel-length-dependent line, which is encouraging and persuasive. Figure 1:0 The final determined threshold voltages are plotted against different channel lengths. (**a**) Vth values versus channel lengths produce a straight line; (**b**) IDS versus VGS on W120L500 giving a referencing Vth = 0.25 V. (**c**) Vth values versus channel lengths is drawn again with (500, 0.25) added.