## 3 Simulation Results and Discussion
### SHE for Different Combinations of the Gate Oxide and Back Oxide Materials
According to [27], the coefficient of thermal expansion of the MoS\({}_{2}\) monolayer at 550 K is 6.5\(\cdot\)10\({}^{-3}\) 1/K, which means a channel expansion of only 1.95% at a change in the temperature of 300 K.  
For transistors with a channel length greater than 40 nm, the mid-channel temperature for the transistors of the first case became higher than for the transistors of the second case (Figure 3). This means that the impact of oxide materials on the establishment of the resulting temperature at the center of the channel becomes greater than the impact of the transistor structure (cases 1 and 2) for long channel transistors. In other words, for long-channel transistors, heat dissipation through oxides is much greater than in short-channel transistors, and heat dissipation through gas has no decisive role (i.e., is not so important) in establishing the resulting temperature in the channel. For long-channel transistors, the temperature tends to be the same for all types of transistor structures considered. In this case, it is observed that the heat generation rate plays a more sufficient role than the heat dissipation rate, since the resulting temperature is the same for both structures (case 1 and case 2) and for both combinations 1 and 4. From the temperature distribution along the channel for case 1 (fully covered channel top) and case 2 (partially covered channel top), it can be seen that with the same applied gate and drain voltages, the temperature variation when lengthening the gate, for the transistor with combination 1 is higher than for the transistor with combination 4 (Table 1) (Figures 4 and 5). Figure 3: Dependence of lattice temperature at channel center on gate length for 2D MoS\({}_{2}\) MOSFETs with Al\({}_{2}\)O\({}_{3}\) gate oxide (1, 2) and HfO\({}_{2}\) gate oxide (3, 4) with fully covered channel (2, 4) and with partially covered channel (only under the gate) (1, 3). The change in center channel temperature is 264 K and 192 K in case 1, 270 K and 194 K in case 2 for combinations 1 and 4, respectively. Apparently this is mainly related to the higher thermal conductivity of Al\({}_{2}\)O\({}_{3}\) compared to the thermal conductivity of HfO\({}_{2}\) (Table 2), which is used as gate oxide materials in combinations 1 and 4, respectively. Although the thermal conductivity of SiO\({}_{2}\) used as BOX material in combination 1 is lower than the thermal conductivity of HfO\({}_{2}\) used as BOX material in combination 4, it differs only by 1.5 times, while the thermal conductivity of gate oxide materials is about 5 times different (Table 2). The simulation results for a wider range of oxide material combinations show that SHE is gate length-sensitive for all combinations of gate and back oxide materials in the transistor (Figure 6). The SHE dependence on gate length has only been shown for transistors of case 1, since this dependence is the same for transistors of case 2. The sensitivity of SHE to gate length can be explained by the dependence of drain current on channel length, which in our analysis is based on gate length. Therefore, since the drain current defines the heat  
Figure 4: Temperature distribution along the channel for “case 1” transistors with gate oxide materials Al\({}_{2}\)O\({}_{3}\) and HfO\({}_{2}\), while BOX material is SiO\({}_{2}\) and HfO\({}_{2}\), respectively. The gate lengths are 20 nm (**a**) and 100 nm (b). Vg = 0.1 V, and Vd = 0.4 V.  
Figure 5: Temperature distribution along the channel for “case 2” transistors with gate oxide materials Al\({}_{2}\)O\({}_{3}\) and HfO\({}_{2}\), while BOX material is SiO\({}_{2}\) and HfO\({}_{2}\), respectively. The gate lengths are 20 nm (**a**) and 100 nm (**b**). Vg = 0.1 V, and Vd = 0.4 V.  
generation rate, a long channel (small drain current) reduces the heat generation rate.