## 3 Results and Discussions
Figure 4 shows the energy band diagrams of these two extreme cases of planar TFETs when V\({}_{\text{D}}\) is 0.5 V and V\({}_{\text{G}}\) is 0.4 V. Due to the difference in the distribution of the metal grains analyzed above, the energy band bending between the source and the channel region becomes larger in the case of V\({}_{\text{turn-on}}\) = 0.289 V (red dash line) than in the case of V\({}_{\text{turn-on}}\) = 0.367 V (black solid line). As a result, when the same V\({}_{\text{G}}\) is applied, there is a difference in the tunneling width of the two cases and accordingly, the V\({}_{\text{th}}\) is also different. As described above, the planar TFET is relatively vulnerable to the WFV effect, as the V\({}_{\text{th}}\) of the planar TFET is only determined by the energy band of the channel adjacent to the source region, while the V\({}_{\text{th}}\) of the conventional MOSFET is determined by the entire  
Figure 3: Metal grain distributions of planar TFETs in extreme cases (red color: metal grains with a work-function (WF) of 4.6 eV, and blue color: metal grains with a WF of 4.4 eV) among the simulated samples. (**a**) Maximum V\({}_{\text{turn-on}}\) (0.367 V) case of planar TFETs. (**b**) Minimum V\({}_{\text{turn-on}}\) (0.289 V) case of planar TFETs. channel under the gate [33]. In the metal grain distributions shown in Figure 3a,b, planar TFETs have a \(\Delta\)V\({}_{\text{turn-on}}\) of 78 mV (0.367 V-0.289 V) between the two cases. On the other hand, the DG TFET and EHBTFET show a \(\Delta\)V\({}_{\text{turn-on}}\) of 65 mV (0.371-0.306 V) and 5 mV (0.685-0.679 V), respectively, when one of two gates has the same metal grain distribution, as shown in Figure 3a,b. As confirmed in the analysis of Figures 3 and 4, only seven metal grains near the source junction mainly affect the tunneling current and the V\({}_{\text{turn-on}}\) of the planar TFET. Meanwhile, in the case of DG TFETs, the addition of another gate on the opposite side doubles the number of metal grains (14 metal grains) that affect the tunneling current and reduces the effect of WFV. Unlike the planar TFET or DG TFET, in the on-state of the EHBTFET, electrons are collected under MG and holes are collected under SG. As the bias of MG increases, energy band bending and a tunneling current are generated in the channel between the two gates. The EHBTFET has a tunneling current at the channel between MG and SG, and all of the metal grains (70 metal grains) where MG and SG overlap are included in V\({}_{\text{turn-on}}\) determination, which greatly reduces the effect of WFV. This analysis can be confirmed indirectly by comparing the probabilities of the most extreme theoretical cases of the planar TFET, DG TFET and EHBTFET. As discussed above, the V\({}_{\text{turn-on}}\) of the planar TFET and DG TFET is controlled by the metal grains near the source. Therefore, the planar TFET has the maximum value of V\({}_{\text{turn-on}}\) when all WFs of seven metal grains adjacent to the source are 4.6 eV and the minimum V\({}_{\text{turn-on}}\) when they are all 4.4 eV; the probabilities are 2.8% (0.67) and 0.2% (0.47), respectively. Similarly, for the DG TFET, the probability of the maximum V\({}_{\text{turn-on}}\) is \(7.8\times 10^{-2}\)% (0.614) and the probability of the minimum V\({}_{\text{turn-on}}\) is \(2.7\times 10^{-4}\)% (0.414). In the case of the EHBTFET, as tunneling occurs from the valence band of the channel near SG to the conduction band of the channel near MG, the V\({}_{\text{turn-on}}\) reaches the maximum when all MG grains have a WF of 4.6 eV and all SG grains have a WF of 4.4 eV where MG and SG overlap, as shown in Figure 5a; its probability is \(2.0\times 10^{-20}\)% (0.635 \(\times\) 0.435). Additionally, in the opposite case, as shown in Figure 5b, the V\({}_{\text{turn-on}}\) reaches the minimum and its probability is equal to the probability of the maximum V\({}_{\text{turn-on}}\).