## 3 Simulation Results and Discussion  
### SHE for Different Combinations of the Gate Oxide and Back Oxide Materials  
One of the main parameters of the SHE effect is the temperature in the middle of the channel. The resulting temperature at the center of the channel is defined by the rate of heat generation as well as the rate of heat dissipation through the oxide surrounding the channel. In this study, only heat dissipation is considered, assuming that heat generation is similar in all cases considered. The similar heat generation rate in all the considered cases is provided by choosing the appropriately applied voltages on the gate and drain electrodes. The rate of heat dissipation in the channel is defined by the transistor structure as well as the properties of the oxide materials surrounding the channel. Various considered structures are shown in Figure 2, and various combinations of gate oxide and back oxide materials are shown in Table 1.  
The simulation results of the lattice temperature dependence of the gate length at the center of the channel for case 1 (channel surface fully covered) and case 2 (channel surface partially covered) are shown in Figure 3. For all transistors, the temperature distribution along the channel was simulated at fixed (or constant) gate and drain voltages: Vg = 0.1 V and Vd = 0.4 V. As the resistance of long channel transistors is higher, their drain current is lower than the drain current of short channel transistors at the same applied voltages. Higher drain current in short channel transistors results in higher temperature in the middle of the channel compared to the temperature at the center of the channel in the case of long channel transistors. Our different numerical simulations have shown that the temperature at the center of the channel for short channel transistors is higher in both cases  
Figure 2: Structure of the simulated MoS\({}_{2}\) MOSFET. Two cases are considered: (**a**) the gate oxide length is equal to the MoS\({}_{2}\) channel length (first case); (**b**) the gate oxide length is equal to the gate length (second case). (i.e., cases 1 and 2) and for two specific combinations (i.e., combinations 1 and 4). Moreover, although the temperature depends on the length of the channel, it is important to note that it also depends on the structure and the oxide materials. These dependencies are discussed later in this section. The temperature dependence on the applied voltages Vg and Vd was also addressed in our previous work (see ref. [9]). With a gate length less than 40 nm, for combinations 1 and 4, the lattice temperature in the middle of the channel is slightly higher for transistors where the channel is covered with gate oxide only under the gate (second case) than for the transistors of the first case. This is related to the quite different thermal conductivity of gate oxide materials and air. In the first case, the transistor channel is fully covered with oxide (Figure 2a), while in the second case it is partially covered with air (Figure 2b). In the wide temperature range (between 90 K and 1473 K), the thermal conductivity of air has values between 0.0084 and 0.0915 W/(m K). These values of the thermal conductivity of air (even appropriate for 1473 K) are 2-3 orders of magnitude lower than the thermal conductivity of aluminum- and hafnium-oxides. Thermal conductivities of Al\({}_{2}\)O\({}_{3}\) and HfO\({}_{2}\), depending on the size and temperature, are in the ranges of 6-30 W/(m\(\cdot\)K) [21; 22; 23; 24] and 0.27-4.3 W/(m\(\cdot\)K) [25; 26], respectively. The effect of the thermal expansion of the channel is not taken into account in this simulation, because the coefficient of thermal expansion is too low (i.e., very small).