## 3 Simulation Results and Discussion
### SHE for Different Combinations of the Gate Oxide and Back Oxide Materials
The reduction in temperature for long channel transistors and its convergence to the same value for all four oxide material combinations supports our assumption that for long channels the main mechanism responsible for the formation of the resulting temperature is the heat generation rate. The temperature dynamics (or behavior) shown in Figure 6 can be explained by the heat dissipation through the gate oxide and BOX. It is known that \"low thermal conductivity of oxide materials\" results in slower heat dissipation from the material and formation of a higher equilibrium channel temperature than in \"higher thermal conductivity of oxide materials\". Therefore, the thermal conductivity of the gate oxide and BOX materials is a very important parameter in SHE. Depending on the thickness, the thermal conductivity of HfO\({}_{2}\) films has values in the range of 0.27-4.3 \(W\)/(m\(\cdot\)K) at temperatures between 300 \({}^{\circ}\)C and 500 \({}^{\circ}\)C [26; 28]. The values of the thermal conductivity of SiO\({}_{2}\) are in the range between 0.5 W/(m\(\cdot\)K) and 2.5 W/(m\(\cdot\)K) [29; 30]. In our simulation, we used the thermal conductivity values shown in Table 2. We chose the thermal conductivities for HfO\({}_{2}\) and SiO\({}_{2}\) as average values from the indicated above ranges. Thermal conductivity for Al\({}_{2}\)O\({}_{3}\) considerably depends on the thickness and the temperature. It is varied in the range between 6 and 30 W/(m\(\cdot\)K) [21; 22; 23; 24] as mentioned above, in Section 3.1. The value 12 W/(m\(\cdot\)K) was chosen as the value corresponding to the nanoscale (less than 10 nm) oxide layer and for a temperature range of approximately between 100 and 300 \({}^{\circ}\)C.  
With gate lengths up to 100 nm, the mid-channel temperature is higher for transistors with combinations 1 and 2 than for transistors with combinations 3 and 4. Apparently, this is mainly related to a lower thermal conductivity of the BOX material in combinations 1 and 2 (which is SiO\({}_{2}\)) than the thermal conductivity of the BOX material in combinations 3 and 4 (which is HfO\({}_{2}\)). With the smallest gate length (e.g., a length of less than 30 nm) for combinations with the same BOX material, the temperature is higher for combinations with HfO\({}_{2}\) as the gate oxide material. With gate lengths between 30 nm and 100 nm, the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & **Thickness (nm)** & **Thermal Conductivity** \\  & **(nm)** & \(\frac{W}{K-m}\) \\ \hline SiO\({}_{2}\) & 80 & 1.4 \\ Al\({}_{2}\)O\({}_{3}\) & 2.385 & 12 \\ HfO\({}_{2}\) (gate oxide) & 5.641 & 2.3 \\ HfO\({}_{2}\) (Box) & 80 & 2.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Thermal conductivities used for simulation. Figure 6: SHE dependence on the gate length for different combinations of gate oxide and back oxide materials. temperature for combinations with the same BOX material is approximately the same and practically independent of the gate oxide material. With a gate length of 100 nm, the temperature is practically the same for all four combinations, which means that with long gate lengths, the resulting center channel temperature does not depend on the gate oxide and BOX materials.