## 3 Effective Mass Calculation Methodology  
Effective masses are an integral part of all the NESS modules and here we discuss their extraction methodology. The transport effective mass is calculated from the curvature of the lowest subband in the band structure as [13]  
\[m_{x}=\hbar^{2}\left[\frac{\partial^{2}E(k_{x})}{\partial k_{x}^{2}}\right]^{-1} \tag{1}\]  
However, the calculation of the confinement effective masses is not as straightforward and is discussed below. In the parabolic effective mass approximation, the Schrodinger equation is given by:  
\[\left[\frac{\hbar^{2}}{2m_{y}}\frac{\partial^{2}}{\partial y^{2}}+\frac{\hbar ^{2}}{2m_{z}}\frac{\partial^{2}}{\partial z^{2}}+V(y,z;x_{i})\right]\psi_{n}(y,z;x_{i})=E_{n}\psi_{n}(y,z,x_{i}) \tag{2}\]  
where \(m_{y}\) and \(m_{z}\) are the effective masses along the \(y-\) and \(z-\) (confinement) directions, respectively. \(x_{i}\) is the \(i\)th cross-section plane of the device in the transport, \(x-\), direction and \(V(y,z;x_{i})\) is the total potential energy in the cross-section of the device at \(x_{i}\). By diagonalizing Equation (2), the subband energies \(E_{n}\) and their corresponding wave-functions \(\psi_{n}(y,z;x_{i})\) are computed. In addition, in Equation (2), \(\hbar\) is the reduced [PERSON]'s constant, \(n\) denotes the subband index, and the valley index has been suppressed for clarity. For calculating the effective masses, the electrostatic potential energy (the conduction band edge) is taken to be zero so as to be consistent with the band structure solver.  
To calculate the confinement effective masses (\(m_{y}\) and \(m_{z}\)), we have used the electronic band structure calculated with \(sp^{3}d^{5}s^{*}\) tight-binding method. These simulations were done using QuantumATK tool from Synopsys [14] with the parameter set specified in Ref. [15]. First, the quantization energy, \(E_{Q}\) (the increase in the subband energy due to geometrical confinement), is calculated by subtracting the conduction band edge of the bulk silicon (\(E_{\text{C}}^{\text{B}}\)), which is calculated from the tight binding method using the same set of parameters. Thus, for the subband energies from the parabolic effective mass approximation to match those from the tight binding formulation, the following relationship must be satisfied:  
\[E_{n}^{EPM}(m_{y},m_{z})-(E_{n}^{TB}-E_{\text{C}}^{B})=0 \tag{3}\]  
where \(E_{n}^{EPM}\) and \(E_{n}^{TB}\) are the \(n\)th subband energies from the parabolic effective mass approximation and the tight-binding method, respectively, and i = 1, 2. To compute the pair of confinement effective masses, we have considered the first two subbands for each valley. The corresponding system of  
Figure 1: (**a**) Flowchart of the simulator NESS including the modules herein used. (**b**) Schematic of a simulated nanowire device along with the doubly-degenerate \(\Delta\) valleys used in the simulations.  
equation can be then solved numerically using the Newton-Raphson method. The Jacobian of the system is given by  
\[J=\begin{bmatrix}\frac{\partial E_{1}^{EPM}(m_{y},m_{z})}{\partial m_{y}}&\frac {\partial E_{1}^{EPM}(m_{y},m_{z})}{\partial m_{z}}\\ \frac{\partial E_{1}^{EPM}(m_{y},m_{z})}{\partial m_{y}}&\frac{\partial E_{1}^ {EPM}(m_{y},m_{z})}{\partial m_{z}}\end{bmatrix}, \tag{4}\]  
and it is numerically evaluated by perturbing the effective masses. The initial guess for the effective masses \(m_{y}\) and \(m_{z}\) was calculated using the longitudinal (\(m_{l}\)) and transverse (\(m_{t}\)) effective masses of the valleys using appropriate rotations matrices for the different crystallographic orientations [16; 17].