## 2 Device and Method
The two-valley model uses the single-electron-gas approximation and accounts for all intervalley transfer effects through the effective value for carrier velocity, energy, effective mass, and relaxation rates. We have chosen 0.25 eV as the energy separation between the lower and upper valleys. Depending on the strain, this energy separation factor may yield different values [23]. Figure 2a presents the computation of the steady-state drift velocity as a function of electric field at temperatures of 77 K, 4.2 K [9], and 1.3 K [7]. Beyond electric fields of 10 kV/cm, negative differential conductivity is observed at these temperatures. This behavior arises due to the transition of electrons from the twofold valley to the fourfold valley, where the effective mass is larger. At 300 K, the drift velocity reaches a notable saturation point of \(1\times 10^{7}\) cm/s at 10 kV/cm due to the low effective mass of electrons in the twofold valleys and reduced intervalley scattering of phonons between the twofold and fourfold valleys [24; 25]. Consequently, negative differential conductivity is absent. The presence of negative differential conductivity at temperatures of 77 K, 4.2 K, and 1.3 K ensures the amplification of space charge waves, similar to the behavior observed  
in n-GaAs thin films [1]. In Figure 2b, the average electron energy at 300 K, 77 K, and 4.2 K is plotted as a function of the electric field. The energy is measured relative to the bottom of the lowered valleys (valley pair 1 in the current model), and the sum of kinetic energy and potential energy (0 in the lower valley and \(\Delta\)E in the upper valleys) is averaged and plotted. At thermal equilibrium, the energy is \(3 kT/2=39\) meV at 300 K and increases monotonically with the field. The energy exhibits a clear dependence on \(\Delta\)E, although this dependency diminishes at fields larger than 100 kV/cm. In higher fields, electrons possess an energy greater than \(\Delta\)E, resulting in minimal differences for different \(\Delta\)Es. It should be noted that as \(\Delta\)E increases, impact ionization becomes more significant, as the band gap and threshold decrease with \(\Delta\)E. Figure 2b illustrates the energy-field characteristics at 77 K, comparing it with the same plot at 300 K. The electron energy is lower up to a field of 10 kV/cm, after which the order is reversed in higher fields, contributing to higher mobility at 77 K. However, an important consideration influencing performance is the temperature-dependent electron transport. Specifically, at ultra-low temperatures around 1.3 K, understanding the behavior of electron transport in strained Si/SiGe heterostructures becomes particularly important. At these temperatures, electron transport is primarily limited by tunneling through the SiGe barrier. However, at extremely low temperatures, electrons become strongly localized, resulting in a decrease in tunneling current. This phenomenon, known as Coulomb blockade, occurs due to the charging energy of electrons trapped in quantum dots, leading to increased resistance. Comprehending temperature-dependent electron transport in SiGe heterostructures at ultra-low temperatures is vital for designing and optimizing devices operating in this regime. Ongoing research in this area is expected to provide further insights into the underlying physics of electron transport in SiGe heterostructures and enhance the performance of future electronic devices. However, analyzing space charge waves in strained Si/SiGe heterostructures at 1.3 K remains an open topic, as additional important physical phenomena must be considered. The drift velocity at 77 K is constrained by phonon scattering, resulting in a decrease in electron mobility. As the temperature lowers to 4.2 K, the impact of phonon scattering diminishes, leading to an increase in the drift velocity. At ultra-low temperatures of 1.3 K, electron transport is predominantly limited by tunneling through the material, resulting in a substantial rise in the drift velocity with increasing electric field.