## 3 Results and Discussion  
### Statistical Evaluation of RTN Characteristics  
The 1/f noise increases with downscaling of MOSFETs, as mentioned above, and RTN also increase with the downscaling [12; 36; 37]. Figure 6 shows the Gumbel plot of V\({}_{\text{RMS}}\) for the various MOSFET sizes [36; 37]. A large V\({}_{\text{RMS}}\) can be observed in small-size MOSFETs (L/W = 0.22/0.28, 0.22/0.3, 0.24/0.3 \(\upmu\)m). In this experiment, noise cannot be observed in large MOSFETs because the floor noise is relatively high at \(\sim\)2.5 mV. Figure 7a shows the Gumbel plot of V\({}_{\text{RMS}}\) for the various I\({}_{\text{DS}}\) varied from 0.13 to 12.7 \(\upmu\)A. The sizes of MOSFETs are L/W = 0.22/0.28 \(\upmu\)m and V\({}_{\text{BS}}\) = \(-\)1.0 V [39]. The data in (a) and (b) were measured by the frame and specific MOSFET measurement modes, respectively. The number of MOSFETs with large noise increases with decreasing I\({}_{\text{DS}}\), which is controlled by V\({}_{\text{gs}}\). This means that the event probability of large noise increases with decreasing V\({}_{\text{gs}}\) because the number of channel electrons decreases with decreasing V\({}_{\text{gs}}\), and then the effect of a trapped electron charge becomes large with decreasing number of channel electrons. The number of channel electrons also decreases with the shrinkage of transistor size shown in Figure 6, and then, the probability increases with decreasing channel size. Figure 7b shows the waveform of typical MOSFETs for I\({}_{\text{DS}}\) of 0.13, 0.38, and 1.3 \(\upmu\)A [39]. The time constants and amplitude are modulated by I\({}_{\text{DS}}\). With increasing I\({}_{\text{DS}}\) (V\({}_{\text{gs}}\)), amplitude and \(\tau_{\text{c}}\) decrease, whereas \(\tau_{\text{e}}\) slightly increases. An increase in V\({}_{\text{gs}}\) decreases E\({}_{\text{T}}\)-E\({}_{\text{F}}\), and then, the time to capture decreases as shown in Equation (11). The difference between the modulation of \(\tau_{\text{e}}\) and \(\tau_{\text{c}}\) is discussed later. The modulation of amplitude is caused by a decrease in the number of electrons, as discussed above. It is considered that decreasing the time to capture and increasing amplitude with decreasing V\({}_{\text{gs}}\) increases the event probability of large noise. Figure 8a shows the Gumbel plot of V\({}_{\text{RMS}}\) for the various back bias (V\({}_{\text{BS}}\)). V\({}_{\text{BS}}\) varied from \(-\)0.075 to \(-\)1.38 V, and Figure 8b shows the waveform of typical MOSFETs for V\({}_{\text{BS}}\) of 0.6 1.0 and 1.3 V [39]. The probability increases with the absolute value of V\({}_{\text{BS}}\) in (a). In this experiment, I\({}_{\text{DS}}\) was constant at 1.0 \(\upmu\)A, and this means that the number of electrons was almost the same for each V\({}_{\text{BS}}\). Increasing V\({}_{\text{BS}}\) caused channel percolation [46; 47; 48; 49], making the channel thickness narrow and percolated and increasing electron energy [50]. The probability is increased by channel percolation [46; 47], and the varying electron energy modulates the time constants. In MOSFETs with RTN, the amplitude does not increase with increasing V\({}_{\text{BS}}\) because the number of electrons is the same for each V\({}_{\text{BS}}\). This means that channel percolation increases the probability of RTN generation. Figure 8: (a) Gumbel plot of \(\rm V_{RMS}\) for various \(\rm V_{BS}\). \(\rm V_{BS}\) varied from 0.075 to 1.38 V, and (b) waveform of typical MOSFETs for \(\rm V_{BS}\) of 0.6 1.0 and 1.3 V. The sizes of MOSFETs are \(\rm L/W=0.22/0.28~{}\rm\upmu m\) and \(\rm I_{DS}=1.0~{}\rm\upmu A\). Figure 9 shows the Gumbel plot of the RTN amplitude for MOSFETs with varying channel doping [51]. The channel percolation is accelerated by increasing channel doping concentration [46; 47; 48]. This figure shows that the probability of the number of MOSFETs with large amplitude increases with doping concentration. RTN is increased by channel doping as well as doping the concentration near the source and drain regions.