## 3 Results and Discussion
### Time Constants in Individual RTN  
As the extraction of multi-trap phenomena is difficult, as discussed in Section 3.2, we discuss the time constants and amplitude only in two-state RTN [59]. Figure 16 shows (a) \(\tau_{\mathrm{c}}\) and (b) \(\tau_{\mathrm{e}}\) as a function of \(\mathrm{I_{DS}}\), respectively. These data are measured at \(\mathrm{I_{DS}}\) of 0.1, 0.3, 1.0, 3.0, and 5.0 \(\mu\)A. The data in Figure 16 show how parameters from all two-level RTN can be extracted in common under four or five \(\mathrm{I_{DS}}\), and thus, the number of selected data points was 22. Figure 17a shows \(\tau_{\mathrm{c}}/\tau_{\mathrm{e}}\) and \(\mathrm{E_{T}}\)-\(\mathrm{E_{F}}\) as a function of \(\mathrm{I_{DS}}\) and Figure 17b band diagram of MOS structure for changing \(\mathrm{V_{GS}}\) (\(\mathrm{I_{DS}}\)). \(\tau_{\mathrm{c}}/\tau_{\mathrm{e}}\) and \(\mathrm{E_{T}}\)-\(\mathrm{E_{F}}\) are calculated from the data in Figure 16 and Equation (12), respectively. \(\tau_{\mathrm{c}}\) and \(\tau_{\mathrm{e}}\) decrease and increase with an increase in \(\mathrm{I_{DS}}\) (\(\mathrm{V_{GS}}\)), and the absolute slope of \(\tau_{\mathrm{c}}\) is significantly larger than that of \(\tau_{\mathrm{e}}\). Large trap energy (\(\mathrm{E_{T}}\)) decreases with an increase in \(\mathrm{V_{GS}}\) than that of channel electron (\(\mathrm{E_{C}}\): bottom energy of conduction band). Then, with increasing \(\mathrm{V_{GS}}\) (\(\mathrm{I_{DS}}\)), the energy barrier from the channel electron to the trap decreases and that from the trapped electron to the channel increases. As a result, the time to capture and time to emission decreases and increases, respectively, with increasing \(\mathrm{V_{GS}}\) (\(\mathrm{I_{DS}}\)). The transition probability depends on the energy barrier height between a trap and channel. \(\tau_{\mathrm{e}}\) depends only on the energy barrier because only one electron is captured in a trap. Meanwhile, \(\tau_{\mathrm{c}}\) depends not only on the energy barrier, but also on the number of electrons in a channel because the number of channel electrons increases as \(\mathrm{V_{GS}}\) (\(\mathrm{I_{DS}}\)) increases. Then, the dependency of \(\tau_{\mathrm{c}}\) on \(\mathrm{I_{DS}}\) is more significant than that of \(\tau_{\mathrm{e}}\). \(\mathrm{E_{T}}\)-\(\mathrm{E_{F}}\) in Figure 17 changed by approximately 175 mV during \(\mathrm{I_{DS}}\) (\(\mathrm{V_{GS}}\)) from 0.1 (0.53V) to 5.0 \(\mu\)A (0.75V). Based on these values, \(\mathrm{E_{T}}\)-\(\mathrm{E_{F}}\) changes by 0.18 V, whereas \(\mathrm{V_{GS}}\) changes by 0.22 V. The distance between the traps and the channel was 4.6 nm due to the gate oxide thickness of 5.7 nm. However, \(\tau_{\mathrm{c}}\) depends not only on the trap energy, but also on the number of channel electrons; thus, \(\mathrm{E_{T}}\)-\(\mathrm{E_{F}}\) values cannot be calculated using Equation (12). The distance is considered to be shorter than the calculated value. \(\tau_{\mathrm{e}}\) values for almost all samples monotonically increased with increasing \(\mathrm{I_{DS}}\). This suggests that the distance from the trap to the channel is shorter than that to the gate electrode. The distance between the trap and channel is shorter than 2.85 nm, which is the center of the gate oxide thickness. Figure 18a,b show the amplitude and transition frequency as a function of \(\mathrm{I_{DS}}\) for the same samples as those in Figures 16 and 17, respectively. For a sufficiently long measuring period, the transition frequency (TF) was calculated using the following equation. \[\mathrm{TF}\approx\frac{\mathrm{N_{e}}}{\tau_{\mathrm{e}}}\approx\frac{ \mathrm{N_{c}}}{\tau_{\mathrm{c}}} \tag{15}\]  
Figure 16: (a) \(\tau_{\mathrm{c}}\) and (b) \(\tau_{\mathrm{e}}\) as a function of \(\mathrm{I_{DS}}\). These data are measured at \(\mathrm{I_{DS}}\) of 0.1, 0.3, 1.0, 3.0, and 5.0 \(\mu\)A. The number of data points is 22. The amplitude decreases and the TF increases with an increase in \(I_{\text{DS}}\) for almost all samples.