## 3 Results and Discussions
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **MOSFET** & **Planar TFET** & **DG TFET** & **EHBTFET** \\ \hline Average Vth or Average Vturn-on & \(-55.0\) mV & \(325.1\) mV & \(283.0\) mV & \(680.4\) mV \\ \(\sigma\)Vth or \(\sigma\)Vturn-on & \(14.0\) mV & \(20.1\) mV & \(14.8\) mV & \(4.0\) mV \\ Average SS & \(121.6\) mV/decade & \(64.3\) mV/decade & \(54.5\) mV/decade & \(4.6\) mV/decade \\ \(\sigma\)SS & \(0.5\) mV/decade & \(5.6\) mV/decade & \(4.7\) mV/decade & \(1.2\) mV/decade \\ Average Ion & \(7.11\times 10^{-4}\) A/um & \(1.08\times 10^{-9}\) A/um & \(3.26\times 10^{-9}\) A/um & \(4.33\times 10^{-10}\) A/um \\ \(\sigma\)Ion/Average Ion & \(0.122\) & \(0.348\) & \(0.263\) & \(0.155\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average and standard deviation of the turn-on voltage (Vturn-on), subthreshold swing (SS), and on-current (Ion) of the devices extracted from Figure 2.  
Figure 2: Transfer characteristics of the (**a**) MOSFET, (**b**) planar TFET, (**c**) DG TFET and (**d**) EHBTFET when the drain voltage (VD) is 0.5 V.  
distribution (var(\(\Phi_{\text{M}}\))) according to the number of grains (\(N\)) can be expressed as the following equation [55]:  
\[\text{var}(\Phi_{M})=\frac{1}{N}\left[\sum_{i=1}^{r}P_{i}\Phi_{i}^{2}-\left(\sum_ {i=1}^{r}P_{i}\Phi_{i}\right)^{2}\right] \tag{1}\]  
where \(\Phi_{i}\) and \(P_{i}\) represent the WF value of each grain and the probability of achieving the WF value. As can be seen from (1), var(\(\Phi_{M}\)) and the resulting WFV effects change when \(N\) changes. For example, as the grain size of the metal gate increases during the fabrication process (e.g., increase in the heat budget), the number of grains affecting the characteristics of the device decreases and the WFV effects increase. If the grain size is large enough that the overall gate area is filled by one metal grain, the V\({}_{\text{turn-on}}\) of 60% of devices is determined by a metal grain with a 4.6 eV WF and the V\({}_{\text{turn-on}}\) of the other 40% of devices is determined by a metal grain with a 4.4 eV WF. On the other hand, when the grain size of the metal gate is reduced during the fabrication process (e.g., by incorporating carbon into TiN), the WFV effects can be reduced [43]. If the grain size is continuously reduced and the metal gate reaches an almost amorphous state, it can be considered that there is no difference among the grain distributions of each device and the V\({}_{\text{turn-on}}\) values of all the devices converge to the average V\({}_{\text{turn-on}}\). As the grain size controlled by the process condition can affect the number of grains and the WFV effects, the device structure of TFETs can also affect the WFV effect if the structure can change the grain number, having effects on the V\({}_{\text{turn-on}}\) determination. In order to analyze the reason why the \(\sigma\)V\({}_{\text{turn-on}}\) of the TFETs is different, depending on the structures, the metal grain distributions and the energy band diagrams were compared for the lowest and highest V\({}_{\text{turn-on}}\) among the simulated results of planar TFETs with the randomly generated metal grain profiles. Figure 3a,b show the metal grain distributions of planar TFETs, with the highest value of V\({}_{\text{turn-on}}\) at 0.367 V and the lowest value at 0.289 V, respectively. In the gate region of Figure 3a,b, a red color represents metal grains with WF of 4.6 eV, while a blue color shows other grains with WF of 4.4 eV. Comparing Figure 3a,b, there is a clear difference between the two samples in terms of the distribution of metal grains adjacent to the source region. Figure 3a, with high V\({}_{\text{turn-on}}\), shows that most of the metal grains close to the source region have a WF of 4.6 eV, while Figure 3b, with low V\({}_{\text{turn-on}}\), shows that there are more metal grains with a WF of 4.4 eV around the source region.