## 2 Device Structure and Models  
The FFR and trench edge-termination structures were simulated using a Silvaco-TCAD two dimensional (2D) simulator. In these structures, with a total length of 30 \(\upmu\)m, the \(S\) and \(W\) parameters are the ring length and the distance between a ring and the next ring (or trench width at a trenched structure), respectively. The thickness and doping concentration of the drift region are \(t_{epi}\) and \(n_{epi}\), respectively, and \(n_{ring},\ n_{BPW}\) are the doping concentrations of the ring and bottom protection well (BPW), respectively, as indicated in Table 1. These parameters are considered by using a depletion width equation. \[W=\sqrt{\frac{2\varepsilon_{S}V_{bi}}{q}\bigg{(}\frac{n_{ring}+n_{epi}}{n_{ ring}n_{epi}}\bigg{)}} \tag{1}\]  
The depletion length is calculated about 0.9 \(\upmu\)m between the drift region and the ring region under a non-zero bias condition. The space and width parameters are varied to find a punch-through condition. This paper describes four types of edge-termination structures, as shown in Figure 1. The conventional FFR structure, Figure 1a, formed the periodically heavily doped rings at the surface. The trenched structures are filled with polysilicon or oxide implant Al at the bottom of the trench, as indicated in Figure 1b,c, respectively. Al implantation for BPW formation was simulated using Monte Carlo models by Silvaco-TCAD. This edge-termination combines the FFR with the trenched structure (Figure 1d). A conventional FFR five-ring structure and proposed trenched five-ring structures were designed. All edge-termination structures mentioned above were simulated by varying the \(S\) and \(W\) parameters. Numerical simulations were performed to analyze the breakdown voltage of these structures. This simulation of a reverse bias was considered by SiC ionization coefficients, which were used by [PERSON] [11]. \[\alpha_{n,p}=a_{n,p}\exp\left[-\frac{b_{n,p}}{E}\right] \tag{2}\]  
where \(E\) represents the value of the electric field, \(a_{n,p}\) are the electron and hole ionization fitting coefficient, and \(b_{n,p}\) are the electron and hole critical electric field. The magnitudes of the electron ionization parameters \(a_{n}\) and \(b_{n}\) used in these simulations are \(1.66\ \times\ 10^{6}\) cm\({}^{-1}\) and \(6.6\ \times\ 10^{6}\) Vcm\({}^{-1}\), respectively, and the values of the hole ionization parameters \(a_{p}\) and \(b_{p}\) used in these simulations are \(5.18\ \times\ 10^{6}\) cm\({}^{-1}\) and \(1.47\ \times\ 10^{6}\) Vcm\({}^{-1}\), respectively. \begin{table}
\begin{tabular}{c c} \hline
**Geometry Parameter** & **Value** \\ \hline Thickness and doping concentration of drift region, \(t_{epi}/n_{epi}\) & \(15\ \upmu\)m/\(4\times 10^{15}\) cm\({}^{-3}\) \\ Doping concentration of p+ ring, \(n_{ring}\) & \(1\ \times\ 10^{20}\) cm\({}^{-3}\) \\ Doping concentration of bottom protection well, \(n_{BPW}\) & \(1\ \times\ 10^{18}\) cm\({}^{-3}\) \\ Ring length, \(S\) & \(0.5\)\(\sim\)\(1.5\ \upmu\)m (\(\pm 0.2\ \upmu\)m) \\ A distance between rings, \(W\) & \(0.5\)\(\sim\)\(1.5\ \upmu\)m (\(\pm 0.2\ \upmu\)m) \\ Total \(S+W\) length & \(2.5\ \upmu\)m \\ \hline \end{tabular}
\end{table} 
Table 1: Geometry parameters and conditions for the edge-termination structures used in the simulations. The breakdown voltage for a cylindrical junction can be calculated by solving [PERSON]'s equation in cylindrical coordinates [1,3]. \[\frac{1}{r}\frac{d}{dr}\bigg{(}r\frac{dV}{dr}\bigg{)}=-\frac{1}{r}\frac{d}{dr}(rE )=-\frac{Q(r)}{\varepsilon_{s}} \tag{3}\]  
where the potential, \(V\), and electric field, \(E\), are a function of the junction radius.