## 4 Results and Discussion  
### Effective Mass Results  
Following the methodology discussed in Section 3 we calculated the effective masses for square, circular and elliptical nanowires. Figure 2 compares the parabolic band structures calculated with the effective masses and those calculated with the tight-binding method for square nanowires with a side of 5 nm. The comparison shows that the subband minima and the curvature of the lowest (dominant) subbands are accurately extracted using the methodology discussed earlier. For [111] direction, the non-parabolicity in the band structure results in the mismatch between the parabolic and tight-binding method at energies away from the subband minima.  
Tables 1-3 list the confinement (\(m_{y}\) and \(m_{z}\)) and transport (\(m_{x}\)) effective masses for square, circular and elliptical nanowires. For square and circular nanowires oriented along [100] direction, \(m_{y}\) of \(\Delta_{y}\) must be the same as \(m_{z}\) of \(\Delta_{z}\) and \(m_{z}\) of \(\Delta_{y}\) must be the same as \(m_{y}\) of \(\Delta_{z}\). This is because the structure is symmetric and hence the quantum confinement is the same along \"y\" and \"z\" directions. The small discrepancy in the effective masses arises due to the small difference in the subband minima obtained from the tight binding simulations. For square and circular nanowires oriented along [111] crystallographic direction, all the \(\Delta\) valleys are identical and hence their effective masses along \"y\" must also be identical and same argument also holds for the effective masses along \"z\". However, since the impact of \(m_{y}\) and \(m_{z}\) on the eigen energies is same the for circular and square nanowire (due to their symmetrical cross-section), their individual values can become interchanged depending on the initial guess. Thus, it is more appropriate that for these structures the pair of effective masses (\(m_{y}\) and \(m_{z}\)) must be the same for all three valleys for a nanowire with a particular cross-section. This phenomenon is observed in Tables 1 and 2. Small differences, in the pair of effective masses that was observed, are due to differences in the subband minima in the electronic band structure calculated.  
The transport effective mass, \(m_{x}\), reported in the tables are in good agreement with those calculated using the QuantumATK Tool. A clear trend in \(m_{x}\) is visible: as the cross-sectional area increases, the effective mass (\(m_{x}\)) tends to its bulk counterpart (Table 4). For example, in the case of  
Figure 2: Comparison of the electronic band structure calculated within the parabolic effective mass approximation and tight binding method for nanowires oriented along: (**a**) [100] direction; (**b**) [110] direction; and (**c**) [111] direction. Good agreement in terms of the subband minima and curvature at the minima are obtained for all the three orientations.  
square nanowire oriented along [100] direction, as the cross-section area increases, the \(m_{x}\) for the \(\Delta_{x}\) valley tends to \(m_{x}\) of \(\Delta_{x}\) value of the bulk silicon, which is the same as \(m_{l}\). The \(m_{x}\) for \(\Delta_{y}\) and \(\Delta_{z}\) also tend to their bulk counterparts which are same as \(m_{t}\). Similarly, for [110] and [111] orientations, the \(m_{x}\) (each \(\Delta\) valley) tends to its corresponding bulk values, which can be calculated by appropriate rotation of the coordinate system, as discussed in [13; 16; 17].  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Square [100]**} & \multicolumn{4}{c}{**Square [110]**} & \multicolumn{4}{c}{**Square [111]**} \\ \hline
**Side** & **Valley** & \(\mathbf{m_{y}}\) & \(\mathbf{m_{z}}\) & \(\mathbf{m_{x}}\) & \(\mathbf{m_{y}}\) & \(\mathbf{m_{z}}\)