## 2 Modeling and Simulation Procedure
However, these above described sizes do not allow for arguing the defect's shape and/or assessing the spatial distribution of single trapped charge in the defect. The spatial distribution of the single charge in the trap is described by the wave function of the carrier in the defect. The wave function depends on the excitation condition of the defect. Obviously, in the basic non-excited state, the charge distribution can be modeled by a simple homogeneously charged sphere with the center allocated in the defect's center (see Figure 3, form (a)). For comparison, the idealized forms of the charge  
Figure 1: Model of the crystal lattices of (**a**) SiO\({}_{2}\) and (**b**) Si. Figure 2: Model of the Si-HfO\({}_{2}\) structure. distribution, which would be appropriate to the wave function of the excited condition, are considered too (Figure 3, forms (b) and (c)). In many papers studying the influence of a single charged trap on the characteristics of the MOSFET, the charged area is considered as a homogeneously charged simple cell [3; 8]. Therefore, for comparison, besides the forms indicated above, the single defect model in the form of a simple cell is also considered (see Figure 3, form (d)). Our analysis is mainly based on the McWhorter model of RTN and we have considered only the steady state, but not dynamic features [25; 26; 27]. We have considered the influence of the size and shape of the defect area on the amplitude of the RTN in a Junctionless MOSFET. For this purpose, the RTN amplitude dependence on the gate voltage overload (\(V_{g}-V_{th}\)) is simulated by trapping the single defect with different sizes and forms at the interface, as well as in the oxide layer. The spherical model of a single trapped charge is considered for the first time. It is therefore necessary to check whether this model is appropriate. In this context, we first compared the RTN simulation results induced by this model with the experimental results of a classic planar MOSFET, which were discussed in detail in [27]. The proposed physical model of the defect, and the transport model for the RTN simulation, are all calibrated by using experimental results. To calibrate the models, one considers the planar MOSFET structure with the same parameters as for the experimental sample [27], and for the model of the single defects, a sphere with radius \(r=0.13\) nm homogeneously charged by elementary charge (see Figure 3, form (a)) is used. Experiments carried out in [27] utilize silicon oxynitride (SiON) \(n\)-channel MOSFET devices with a physical dielectric thickness of 1.4 nm. The nominal channel widths are in the range 0.085 \(\upmu\)m to 1 \(\upmu\)m and the nominal channel lengths are between 0.055 \(\upmu\)m and 1 \(\upmu\)m. A full description of the RTN measurement apparatus and methods can be found in [27]. Indeed, source and gate electrodes are biased using battery-powered variable-voltage sources, while the substrate electrode is grounded for all measurements. The drain current is monitored by a low-noise current amplifier with 2 kHz bandwidth. The amplifier output is directly captured using a digital storage oscilloscope with a large memory depth (\(10^{7}\) Samples). The experimental measurements (obtained in the form of time series data) are further analyzed offline. The parameters of the simulated transistor are shown in Table 1. In the simulation, SiON is used for gate oxide; the same is done in the experimental sample. Figure 3: Models of a single charged defect with different forms of charge distribution. (**a**) is form of defect in basic state; (**b**,**c**) are forms of defect in excited state; (**d**) is simple cell form of the defect. The simulation and experimental results for a MOSFET with widths 90 and 130 nm are shown and compared in Figure 4. Regarding the RTN amplitude, it is seen that the simulation values are sufficiently close to the experimental ones when the gate overload lies in the range between \(-0.15\) and \(0.1\) V for both gate widths.