## 2 Materials and Methods  
This study analyzed and compared MIS-HEMT transfer I-V characteristics for different dielectric layer compositions and thicknesses, as shown in Figure 1. Figure 1a,b shows schematic views of a basic AlGaN/GaN-based HEMT (BG-HEMT) and MIS-HEMT, respectively. Figure 1c represents the composition and thickness of distinct dielectric layers A, B, and C. Table 1 summarizes the BG-HEMT layer information, where the nominal Al composition of the AlGaN layer was 26%. Figure 1: Basic AlGaN/GaN-based MIS-HEMT (**a**) without and (**b**) with a dielectric layer, and (**c**) dielectric layer composition and thickness for samples A, B, and C.  
We used two gate leakage current models to compare the BG-HEMT and MIS-HEMT structures' gate leakage currents. Figure 2a depicts the Poole-Frenkel thermal emission (PF) mechanism principle. Crystalline mismatches between layers when fabricating AlGaN/GaN HEMTs cause trap and continuum energy states in the barrier material. Electrons captured in trap energy states move to continuum energy states via high electric fields and temperatures, and current is generated by the voltage drop in the high reverse gate voltages. Figure 2b shows the Fowler-Nordheim tunneling (FN) mechanism. The barrier becomes thinner under high electric fields, allowing more electrons to penetrate the barrier and generate a current at high forward gate voltages [23; 24; 25]. The interface trap between the insulator and semiconductor is inevitable while growing epitaxial layers and causes the 2-DEG concentration and threshold voltage to become unstable. Therefore, it is essential to consider the interface trap density [26; 27]. To obtain more accurate data, we set the interface trap density at the dielectric layer/GaN cap interface and acceptor trap density in the GaN buffer layer. Figure 3a shows the interface trap density at the Si\({}_{3}\)N\({}_{4}\)/GaN cap interface = 2.1 \(\times\) 10\({}^{12}\) cm\({}^{-2}\), and Figure 3b shows acceptor trap density doping profile in GaN buffer layer = 2.09 \(\times\) 10\({}^{15}\) cm\({}^{-3}\) at the unintentionally doped (UID) GaN/GaN buffer interface [28]. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline \(\medriangle\)\(L_{SG}\) & 1.0 \(\upmu\)m \\ \(\medriangle\)\(L_{G}\) & 4.5 \(\upmu\)m \\ \(\medriangle\)\(L_{GD}\) & 9.5 \(\upmu\)m \\ GaN Cap & 2.0 nm \\ AlGaN & 21 nm \\ AlN & 1.0 nm \\ UID GaN Channel & 0.1 \(\upmu\)m \\ GaN Buffer & 1.8 \(\upmu\)m \\ \hline \end{tabular}
\end{table} 
Table 1: Basic AlGaN/GaN-based HEMT geometrical parameters. Figure 2: Mechanism of gate leakage current models: (**a**) Poole–Frenkel thermal emission, and (**b**) Fowler–Nordheim tunneling. Since the thermal conductivity of the material affects the heat transfer during operation of the device and depends on the temperature, the lattice heat flow based on the thermal conductivity model is essential to considering SHE [29]. The thermal conductivity model is given by:  
\[k(T_{L})=(TC.CONST)/\left(\frac{T_{L}}{300}\right)^{TC.NPOW} \tag{1}\]  
where \(TC.CONST\), \(T_{L}\), and \(TC.NPOW\) are the thermal conductivity constant at 300 K, local lattice temperature, and coefficient of temperature dependence of thermal conductivity, respectively. As shown in Table 2, thermal conductivity constants of GaN and AlGaN were applied. Then, the lattice heat flow model is applied by using the calculated thermal conductivity, and it can be expressed as  
\[C\frac{\partial T_{L}}{\partial t}=\
abla(k\
abla T_{L})+H \tag{2}\]  
where \(C,k,H\), and \(T_{L}\) are the heat capacitance per unit volume, thermal conductivity, heat generation, and local lattice temperature, respectively. \(k\) is calculated using Equation (1), and \(H\) is expressed as:  
\[