## 2 Device and Method  
The strained Si/SiGe MOSFET shown in Figure 1 demonstrates the propagation and amplification of space charge waves on the surface of the Si/SiGe inversion (100) layer, when the gate voltage surpasses the inversion threshold. To induce the phenomenon of negative differential conductance, a bias electric field of \(E_{0}>10\) kVcm\({}^{-1}\) is applied between the source and drain, where \(\mathrm{d}\
u/\mathrm{d}E<0\). At millimeter and submillimeter wavelengths, the input coupling element (input A\({}^{\prime}\)) excites space charge waves, which are subsequently amplified and collected at the output coupling element (output A) on the strained Si/SiGe layer. There, a thin layer of Si is grown on a relaxed Si\({}_{1-\chi}\)Ge\({}_{\mathrm{x}}\) buffer, the Si layer is forced to assume the larger lattice constant of the underlying Si\({}_{1-\chi}\)Ge\({}_{\mathrm{x}}\) substrate, see Figure 1. The Si layer is thus said to be under biaxial tensile strain where the amount of strain is given by the Ge content (denoted as x) in the substrate [19]. High quality strained Si/SiGe heterostructures can be fabricated by molecular beam epitaxy or by chemical vapor deposition [20; 21; 22; 23]. Figure 1: Schematic structure of a strained Si/SiGe MOSFET. A\({}^{\prime}\) and A are the input and output coupling element, respectively. In this system, high-field electron transport within the strained Si/SiGe MOS inversion layer is primarily governed by intervalley phonon scattering involving the sixfold degenerate valleys. To model this electron transport, we employ the quasi-hydrodynamic model in conjunction with [PERSON]'s equation. The strain-induced band ordering in the Si/SiGe structure allows for the occupation of only two valleys, with an effective transport mass \(m_{\text{t}}=0.19\ m_{0}\) along the channel and a quantization mass \(m_{\text{I}}=0.92\ m_{0}\) perpendicular to the channel. Consequently, we can describe high-field transport using a two-valley model. Under these conditions, electrons can acquire sufficient energy from the electric field to transition from the lower to the upper valley. We define \(n_{1}\) and \(n_{2}\) as the respective electron counts in bands 1 (lower) and 2 (upper). Thus, two sets of balance equations are required to model this system accurately. The conservation of particles in this uniform two-valley system is expressed through the following continuity equations, see Equations (1) and (2). \[\frac{\partial n_{1}}{\partial t}=-\frac{n_{1}}{\tau_{12}}+\frac{n_{2}}{\tau_ {21}} \tag{1}\]  
\[\frac{\partial n_{2}}{\partial t}=-\frac{n_{2}}{\tau_{21}}+\frac{n_{1}}{\tau_ {12}} \tag{2}\]  
The scattering time from valley \(i\) to \(j\) is denoted as \(\tau_{i\bar{\jmath}}\). For our analysis, we assume that electrons in valleys 1 and 2 are in equilibrium, meaning they have the same electron temperature. Although this assumption is not entirely accurate, it greatly simplifies the analysis process. In the two-valley system, momentum conservation is maintained through the following equations, see Equations (3) and (4),  
\[\frac{dp_{1}}{dt}=qE-\frac{p_{1}}{\tau_{m1}(T_{e})}-\frac{p_{1}}{\tau_{12}(T_{ e})} \tag{3}\]  
\[\frac{dp_{2}}{dt}=qE-\frac{p_{2}}{\tau_{m2}(T_{e})}-\frac{p_{2}}{\tau_{21}(T_{ e})} \tag{4}\]  
where \(p_{i}=m_{i}v_{i}\) is the momentum and \(\tau_{mi}\) is the momentum scattering rate in the \(i\) valley. Since we are assuming that valleys 1 and 2 are at equilibrium, we can rely on the simple one-valley energy conservation, see Equation (5)  
\[\frac{\partial}{\partial t}\left(n\frac{3}{2}k_{B}T_{e}\right)=nqv_{d}E-n\frac {3}{2}k_{B}\frac{T_{e}-T_{0}}{\tau_{E}}, \tag{5}\]  
where \(T_{e}\) is the electron temperature, \(T_{0}\) is the lattice temperature, \(\tau_{E}\) is the energy relaxation time, with \(n=n_{1}+n_{2}\) and \(J=qnv_{d}=q(n_{1}v_{1}+n_{2}v_{2})\)[18]. For our simulations, we need calculations of drift velocity and the average electron energy as functions of the electric field.