## 2 Materials and Methods
\begin{split}& H=\frac{\left|\overset{-}{J_{n}}\right|^{2}}{q\mu_{n}n} +\frac{\left|\overset{-}{J_{p}}\right|^{2}}{q\mu_{p}p}-T_{L}\left(\overset{ \rightarrow}{J_{n}}\
abla P_{n}\right)-T_{L}\left(\overset{\rightarrow}{J_{ p}}\
abla P_{p}\right)\\ &+q(R-G)\left[T_{L}\left[\frac{\partial\varphi_{n}}{\partial T} \right]_{n,p}-\varphi_{n}-\left[\frac{\partial\varphi_{p}}{\partial T}\right] _{n,p}+\varphi_{p}\right]\\ &-T_{L}\left[\left[\frac{\partial\varphi_{n}}{\partial T}\right] _{n,p}+P_{n}\right]divJ_{n}-T_{L}\left[\left[\frac{\partial\varphi_{p}}{ \partial T}\right]_{n,p}+P_{p}\right]divJ_{p}\end{split}  \tag{3}\]  
In the steady-state, the current divergence can be replaced by the net recombination. Equation (3) then simplifies to:  
\[H=\left[\frac{\left|\overset{-}{J_{n}}\right|^{2}}{q\mu_{n}n}+\frac{\left| \overset{-}{J_{p}}\right|^{2}}{q\mu_{p}p}\right]+q(R-G)\left[\varphi_{p}- \varphi_{n}+T_{L}\left[P_{p}-P_{n}\right]\right]-T_{L}\left(\overset{ \rightarrow}{J_{n}}\
abla P_{n}+\overset{\rightarrow}{J_{p}}\
abla P_{p}\right) \tag{4}\]  
Figure 3: (**a**) Interface trap density at the Si\({}_{3}\)N\({}_{4}\)/GaN cap interface, and (**b**) GaN buffer layer acceptor trap density doping profile with respect to depth. where \(\left[ \begin{matrix}\overset{\rightarrow}{I_{n}}\\ \overset{\rightarrow}{J_{n}}\end{matrix} +\frac{\overset{\rightarrow}{I_{p}}}{q \mu_{B}p}\right]\) is the Joule heating effect term, \(q(R-G)\left[\varphi_{p}-\varphi_{n}+T_{L}\left[P_{p}-P_{n}\right]\right]\) is the heating and cooling term considering recombination and generation, and \(-T_{L}(\overset{\rightarrow}{J_{n}}\
abla P_{n}+\overset{\rightarrow}{J_{p }}\
abla P_{p})\) accounts for the Peltier and Joule-Thomson effects. A simple and intuitive form of \(H\) that has been widely used in the past is:  
\[H=(\overset{\rightarrow}{J_{n}}+\overset{\rightarrow}{J_{p}})\cdot\overset{ \rightarrow}{E} \tag{5}\]  
where \(H\) is the generated heat, \(\overset{\rightarrow}{J_{n}}\) and \(\overset{\rightarrow}{J_{p}}\) are electron and hole current densities, respectively, and \(\overset{\rightarrow}{E}\) is the electric field. SHE can be estimated using either Equation (4) or (5) for steady-state calculations. By default, Equation (5) is used. We set the work function of the gate, source, and drain metals for Schottky and ohmic contacts, respectively. However, in the case of MIS-HEMT, the work function changes because of dielectric layers, even if the work function of the gate metal is fixed. This is due to the high-\(k\) effective work function model, which is expressed as:  
\[\Phi_{meff}=E_{highk}+S_{highk}\left(\Phi_{m}-E_{highk}\right) \tag{6}\]  
where \(\Phi_{meff}\) is the effective gate work function, \(E_{highk}\) is the high-\(k\) material charge neutrality, \(S_{highk}\) is a slope parameter of the high-\(k\) material, and \(\Phi_{m}\) is the metal work function in a vacuum (\(\Phi_{m}=3.73\) eV for a Schottky contact [30]). We considered auger recombination, Schockley-Read-Hall recombination, strain piezoelectric polarization, [PERSON]'s impact ionization model, self-heating, thermal conductivity, the Fermi-Dirac distribution function, Poole-Frenkel thermal emission, the Fowler-Nordheim tunneling model, and the high-\(k\) effective work function model for the physical calculations [31; 32; 33; 34]. Table 2 shows AlGaN and GaN material parameters. Figure 4 shows the thermal conductivities of GaN, Al\({}_{2}\)O\({}_{3}\), and Si\({}_{3}\)N\({}_{4}\) as a function of temperature. As temperature increases above 400 K, the thermal conductivity of Al\({}_{2}\)O\({}_{3}\) becomes lower than that for Si\({}_{3}\)N\({}_{4}\).