## 2 Modeling and Simulation Procedure  
Nowadays, there is no universal physical model that describes the spatial distribution and sizes of the single charge trapped at the interface or in the oxide defects in a MOSFET. Moreover, it is very interesting to implement these models in commercial TCAD programs. There are many geometrical sizes, which describe spatial sizes of an elementary component of solid states such as the Bohr radius, Van der Vaals radius or atomic radius. All of the above might be used as a size-related single defect physical model. We consider that the trap defect at the Si-SiO\({}_{2}\) interface or in the oxide is a broken connection between silicon atoms (or between silicon and oxygen atoms). In general, hydrogen atoms are introduced to recover the dangling bonds. However, because it (i.e., the recovery) is a statistical process, some broken bonds may not be recovered by hydrogen atoms. This case is very important in nanometer devices because the charged trap will have a significant impact on the device characteristics. In some cases, the trap in oxide may be represented by a vacancy V\({}^{0}\), which can capture carrier according to the reaction  
\[\mathrm{V}^{0}+\mathrm{e}^{-}=>\mathrm{V}^{-} \tag{1}\]  
where \(\mathrm{e}^{-}\) and \(\mathrm{V}^{-}\) denote the elementary charge and charged vacancy, respectively. Hence, the defect will represent a local charged area with some spatial shape. It is expected that the single trapped charge will be distributed in that local area. It can be assumed that the sizes of this area will be restricted by atomic sizes as well as by interatomic distances inherent to the considered materials. Interatomic distances at the interface, as well as in bulk, vary depending on defect's position and layer's thickness. Thus, the distance between silicon and oxygen atoms \(\mathrm{R}_{\text{Si-O}}\) varies in the range from 0.154 nm to 0.162 nm, and the distance between neighboring oxygen atoms \(\mathrm{R}_{\text{O-O}}\) ranges between 0.248 and 0.265 nm at changing SiO\({}_{2}\) thickness from 0.4 nm to bulk [17] (Figure 1). However, in [18], it is stated that \(\mathrm{R}_{\text{O-O  
of silicon and oxygen atoms is changed depending on the position. Therefore, the atomic distances indicated above, and, as a consequence, the interface defect sizes, can change in some range. When decreasing the transistor dimensions to decrease the tunneling current through the gate oxide, HfO\({}_{2}\) with a high dielectric constant and the appropriate equivalent thickness is mainly used instead of SiO\({}_{2}\). Therefore, it is also interesting to notice the characteristic interatomic distances in this material. Depending on orientation and number of atomic layers from 3 to 11, the distances between oxygen atoms vary from 0.254 up to 0.265 nm. In addition, the distances between Hf and oxygen atoms vary from 0.202 nm up to 0.248 nm [19] (see Figure 2). In the transition layer Si-HfO\({}_{2}\), the average distance between Si and Hf atoms is 0.260 nm [20]. In this transition layer, the break of the connections between silicon atoms as well as the oxygen vacancies is responsible for the arising of the interface defects [21]. Due to the facts indicated above, it is obvious that the interatomic distances and, as a consequence, the trap defect linear sizes are in accordance with the ranges indicated above. In some papers, the trap sizes are considered in the electron (or hole) capture cross-section point of view. For the considered materials, often, the range of the capture cross sections is taken from \(10^{-18}\) up to \(10^{-13}\) cm\({}^{2}\)[22; 23; 24]. This range of the cross section fits to the range of the linear sizes of defects from 1.8 nm up to 0.005 nm and it covers the range of possible defect sizes indicated above. It can be summarized that for the substrate and oxide materials used for MOSFET, the typical atomic linear sizes and interatomic distances are in the range between 0.05 and 0.55 nm.