## 4 Results and Discussion  
Figure 3 shows the charges, length factor, and breakdown voltage for various values of N\({}_{\text{BPW}}\) ranging from 1 \(\times\) 10\({}^{17}\) cm\({}^{-3}\) to 3 \(\times\) 10\({}^{18}\) cm\({}^{-3}\). As N\({}_{\text{BPW}}\) increased, the charges reduced the electric field crowding at the trench edge by effective shielding in the BPW structure because the BPW charge gradually increased. This means that the blocking voltage and gate oxide reliability were improved. However, along with the increase in the N\({}_{\text{BPW}}\), the effective thickness of the drift layer was reduced. This was caused by the degradation of the length factor, where x\({}_{\text{j}}\) and W were increased and decreased, respectively. Therefore, the effect of the length factor reduces the breakdown voltage. In the case N\({}_{\text{BPW}}\) of 8 \(\times\) 10\({}^{17}\) cm\({}^{-3}\), the cross point between the effect of the charge and that of the length factor with various N\({}_{\text{BPW}}\) was determined using the analytical method. This indicated that the gate oxide field was below 4 MV/cm. The electric field distributions of the trench corner in B-B' for different N\({}_{\text{BPW}}\) are shown in Figure 4.  
Figure 3: (**a**) Charge density, length factor, and (**b**) the breakdown voltage of the TMOSFET according to BPW doping concentrations. When N\({}_{\rm BPW}\) was lower than 7 \(\times\) 10\({}^{17}\) cm\({}^{-3}\), this was insufficient for the shielding effect at the gate oxide, and the peak electric field of the oxide was still higher than 4 MV/cm. This caused the dielectric breakdown to occur before avalanche breakdown in the drift layer. For this reason, although an N\({}_{\rm BPW}\) of 7 \(\times\) 10\({}^{17}\) cm\({}^{-3}\) has the highest breakdown voltage during the simulation, it can reduce oxide reliability in the long term. When N\({}_{\rm BPW}\) is higher than 8 \(\times\) 10\({}^{17}\) cm\({}^{-3}\), the charges can effectively shield the oxide field, which is less than 4 MV/cm at the trench corner. However, the higher doping concentrations strongly influenced the degradation of the length factor, which caused the degradation of the blocking voltage. Therefore, in the case of N\({}_{\rm BPW}\) of 8 \(\times\) 10\({}^{17}\) cm\({}^{-3}\), we found that the highest breakdown voltage of 1345 V was extracted by the TCAD simulation results and the analytic method considering the oxide electric field and the effect of the length factor. Figure 5 shows the conduction band edge energy profiles at the poly-Si/oxide/BPW interface in the B-B' of the trench corner with various N\({}_{\rm BPW}\) values under breakdown conditions. As N\({}_{\rm BPW}\) increased, the conduction band energy of the BPW region increased. This indicates that the higher doping concentration is not fully depleted in the BPW region under the breakdown voltage, so that the gate oxide was protected from the crowding electric field. In the case of an N\({}_{\rm BPW}\) value of 1 \(\times\) 10\({}^{17}\) cm\({}^{-3}\), which was the lowest BPW concentration in the simulation conditions, the conduction band energy at the oxide region was pulled down. This means that a high electric field was applied at the edge of the trench gate oxide; consequently, a dielectric breakdown voltage was induced. Figure 6 shows the C\({}_{\rm gd}\) of the analytical modeling and numerical simulation as the Drain-source voltage (V\({}_{\rm ds}\)). The C\({}_{\rm gd}\) was extracted using a Gaussian distribution and a TCAD simulator for comparative analysis, when the N\({}_{\rm BPW}\) was 8 \(\times\) 10\({}^{17}\) cm\({}^{-3}\). The C\({}_{\rm gd}\) of the analytical modeling corresponds to that of the numerical simulation by approximately 86%. The analytical model was able to predict C\({}_{\rm gd}\) through the calculation of the BPW charge as the V\({}_{\rm ds}\), but all parasitic capacitances of the TMOSFET device were not taken into account for the simple calculation.