## 4 Results and Discussion
### Mobility: A Kubo-Greenwood Study  
The aim of this section is to present a simulation study of the impact of the effective masses on the electron mobility in silicon nanowires as a function of their diameter and orientation, comparing the use of the bulk (m\({}_{bulk}\)) and the calibrated (m\({}_{eff}\)) effective masses. For this purpose, we chose a square shape NWT with an equivalent oxide thickness (EOT) of 0.8 nm and diameters ranging from 3 to 8 nm to describe the electronic behavior from a device with strong confinement impact (3 nm) to a near bulk-like device (8 nm) [18]. We studied the following intrinsic mobility limiting mechanisms: _(i)_ acoustic phonon scattering, with acoustic deformation potential 12 eV; _(ii)_ optical phonon scattering, with the fixed parameters for the different branches defined in Table 5; and _(iii)_ surface roughness scattering, with root mean square (\(\Delta_{RMS}\)) and correlation length 0.48 and 1.3 nm, respectively. In particular, we report the mobility for the total phonon (Ph) and the combined total phonon and surface roughness (Ph+SR) scattering effects with a sheet density of \(3\times 10^{12}\) cm\({}^{-2}\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Bulk [100]**} & \multicolumn{4}{c}{**Bulk [110]**} & \multicolumn{4}{c}{**Bulk [111]**} \\ \hline \multirow{2}{*}{**Valley**} & \(\mathbf{m_{y}}\) & \(\mathbf{m_{z}}\) & \(\mathbf{m_{x}}\) & \(\mathbf{m_{y}}\) & \(\mathbf{m_{z}}\) & \(\mathbf{m_{x}}\) & \(\mathbf{m_{y}}\) & \(\mathbf{m_{z}}\) & \(\mathbf{m_{x}}\) \\  & **[m\({}_{0}\)]** & **[m\({}_{0}\)]** & **[m\({}_{0}\)]** & **[m\({}_{0}\)]** & **[m\({}_{0}\)]** & **[m\({}_{0}\)]** & **[m\({}_{0}\)]** & **[m\({}_{0}\)]** & **[m\({}_{0}\)]** \\ \hline \(\Delta_{x}\) & 0.201 & 0.201 & 0.891 & 0.328 & 0.201 & 0.546 & 0.431 & 0.201 & 0.431 \\ \(\Delta_{y}\) & 0.891 & 0.201 & 0.201 & 0.328 & 0.201 & 0.546 & 0.431 & 0.201 & 0.431 \\ \(\Delta_{z}\) & 0.201 & 0.891 & 0.201 & 0.201 & 0.891 & 0.201 & 0.431 & 0.201 & 0.431 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Effective masses for Bulk silicon along different crystallographic directions. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Ellipse [100]**} & \multicolumn{4}{c}{**Ellipse [110]**} & \multicolumn{4}{c}{**Ellipse [111]**} \\ \hline
**a** & **b** & **Valley** & \(\mathbf{m_{y}}\) & \(\mathbf{m_{z}}\) & \(\mathbf{m_{x}}\) & \(\mathbf{m_{y}}\) & \(\mathbf{m_{z}}\) & \(\mathbf{m_{x}}\) & \(\mathbf{m_{y}}\) & \(\mathbf{m_{z}}\) & \(\mathbf{m_{x}}\) \\
**[nm]** & **[nm]** & **Valley** & \(\mathbf{[m_{0}]}\) & **[m\({}_{0}\)]** & **[m\({}_{0}\)]** & **[m\({}_{0}\)]** & **[m\({}_{0}\)]** & **[m\({}_{0}\)]** & **[m\({}_{0}\)]** & **[m\({}_{0}\)]** & **[m\({}_{0}\)]** \\ \hline \multirow{3}{*}{3} & \multirow{3}{*}{2.00} & \(\Delta_{x}\) & 0.383 & 0.238 & 0.986 & 0.469 & 0.251 & 0.574 & 0.972 & 0.292 & 0.570 \\  & & \(\Delta_{y}\) & 0.205 & 1.072 & 0.283 & 0.570 & 0.209 & 0.574 & 1.005 & 0.284 & 0.570 \\  & & \(\Delta_{z}\) & 0.683 & 0.287 &Figure 3 shows the Ph and Ph+SR electron mobilities considering both m\({}_{bulk}\) and the calibrated m\({}_{eff}\), which values are taken from Table 1, for different channel orientations: (a) [100]; (b) [110]; and (c) [111]. In general, as the transport effective masses were introduced in the scattering rate equations [12], their modification directly altered the mobility. Due to the relaxation time approach, the mobility for a particular scattering mechanism was inversely proportional to its rate and thus to the transport effective mass. As expected in Figure 3, the inclusion of SR significantly reduced the mobility because it dominated for very high sheet concentrations. Moreover, the lowest mobility was [111] devices due to the lower valley splitting energy and higher transport masses [11]. Then, different results can be highlighted when m\({}_{eff}\) was included for the different channel orientations.