## 3 Finite Element Analysis
### SiC-DBC Model Based on Homogenized Properties of Cu-IMC-Resin  
A 3D FE model was set up using ANSYS Workbench 18.2, which is shown in Figure 4. It consists of a bonding layer sandwiched between a SiC chip and DBC substrate. Homogenized mechanical properties estimated from the Cu-IMC-resin model based on 3D image reconstruction were used. These properties were considered as the average of the calculated results for loading along x-, y-, and z-axis directions. The other material properties are listed in Table 2. Dimensions of \(8\times 8\times 0.3\) mm,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{3}{c}{**Regions**} \\ \cline{2-4}  & **A** & **B** & **C** \\ \hline Cu-IMC-resin (PI) & Cu & IMC (Cu\({}_{6}\)Sn\({}_{5}\)) & Polyimide-resin \\ Cu-IMC-resin (Epoxy) & Cu & IMC (Cu\({}_{6}\)Sn\({}_{5}\)) & Epoxy-resin \\ Cu-IMC-air & Cu & IMC (Cu\({}_{6}\)Sn\({}_{5}\)) & Air \\ Cu-SAC-resin (PI) & Cu & SAC305 & Polyimide-resin \\ Cu-SAC-resin (Epoxy) & Cu & SAC305 & Epoxy-resin \\ Cu-SAC-air & Cu & SAC305 & Air \\ \hline \hline \end{tabular}
\end{table}
Table 1: FE analysis by 15-μm-sized model.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Young’s Modulus** & **Yield Strength** & **Ultimate Tensile Strength** & **CTE** & **Poisson’s Ratio** \\  & \(E\) (**GPa**) & _o_**\(\mathbf{\mathbf{\gamma}}\) (**MPa**) & _o_**\(\mathbf{\mathbf{UTS}}\) (**MPa**) & \(a\) (10\({}^{-6}\)/C)** & \(v\) \\ \hline SiC & 410 & – & – & 3.8 & 0.33 \\ \hline Si\({}_{3}\)N\({}_{4}\) & 314 & – & – & 3.0 & 0.33 \\ \hline Cu & 96 (0 \({}^{\circ}\)C) & 52 (0 \({}^{\circ}\)C) & 240 (0 \({}^{\circ}\)C) & 16.7 & 0.33 \\ \hline IMC (Cu\({}_{6}\)Sn\({}_{5}\)) & 102 & – & – & 18.3 & 0.31 \\ \hline \multirow{2}{*}{SAC305} & 46 (RT) & 45 (RT) & 58 (RT) & 21.2 (RT) & \\  & 35 (100 \({}^{\circ}\)C) & 15 (125 \({}^{\circ}\)C) & 23 (125 \({}^{\circ}\)C) & 22.0 (100 \({}^{\circ}\)C) & 0.36 \\ \cline{1-1}  & 4 (200 \({}^{\circ}\)C) & 3 (200 \({}^{\circ}\)C) & 4 (200 \({}^{\circ}\)C) & 22.5 (200 \({}^{\circ}\)C) & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Summary of selected physical properties used in this study; data is from [36; 37; 38; 39; 40; 41].  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Type of Resin**} & **Young’s Modulus** & **Yield Strength** & **Ultimate Tensile Strength** & **CTE** & **Poisson’s Ratio** \\  & \(E\) (**GPa**) & _o_\(\mathbf{\mathbf{\gamma}}\) (**MPa**) & _o_\(\mathbf{\mathbf{UTS}}\) (**MPa**) & \(a\) (10\({}^{-6}\)/C)** & \(v\) \\ \hline Polyimide-resin, used in this study & 0.120 (-40 \({}^{\circ}\)C) & 4.0 (-40 \({}^{\circ}\)C) & 16.4 (-40 \({}^{\circ}\)C) & & \\  & 0.021 (RT) & 1.0 (RT) & 4.0 (RT \({}^{\circ}\)C) & & \\  & 0.004 (100 \({}^{\circ}\)C) & 0.2 (100 \({}^{\circ}\)C) & 0.7 (100 \({}^{\circ}\)C) & & \\  & 0.001 (150 \({}^{\circ}\)C) & 0.1 (150 \({}^{\circ}\)C) & 0.2 (150 \({}^{\circ}\)C) & & \\ \hline Epoxy-resin & 2.0 & 30 & 70 & 38.0 & 0.37 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Physical properties of resin used in FE analysis.  
\(8\times 8\times 0.07\) mm, \(12\times 12\times 0.3\) mm, and \(12\times 12\times 0.6\) mm were chosen for the SiC chip, bonding layer (Cu-IMC-resin, IMC (Cu\({}_{6}\)Sn\({}_{5}\)), or SAC305), Cu electrode, and Si\({}_{3}\)N\({}_{4}\) substrate, respectively. Here, the 1/4 model was used for FE analysis considering the symmetry conditions. Quadratic hexahedral elements were used, where the number of nodes and elements in the model were 105,299 and 22,300, respectively. Macroscopic strain, which was used for the microscopic simulations to locally identify the crack initiation sites in the model, was calculated for ten steady-state cycles in which the temperature oscillated between \(-55\) and \(175\)\({}^{\circ}\)C. Thermal stress and strain were generated due to the CTE mismatch of the constituent materials during the temperature variation.