## 3 Results  
We explore the behavior of space charge waves in a strained Si/SiGe heterostructure at a temperature of 4.2 K, focusing on the range of frequencies and wave propagation characteristics. By applying an electric field bias parallel to the surface, in the direction of propagation, we achieve negative differential conductivity with E\({}_{0}\) > 10.3 kV/cm. Our analysis involves studying the dispersion equation D(\(\omega\),k) = 0, which establishes a relationship between the frequency (\(\omega\)) and longitudinal wave number (k). We consider cases where \(\omega\) is real and k takes the form of k\({}^{\prime}\) + ik\({}^{\prime\prime}\). A positive imaginary part of k\({}^{\prime\prime}\) indicates spatial increment or amplification, while a negative value signifies decrement or damping. Our findings, illustrated in Figure 3, demonstrate that amplification of space charge waves occurs for frequencies up to approximately 60 GHz at T = 1.3 K, 47 GHz at T = 4.2 K, and 40 GHz at T = 77 K. For more comprehensive insights at 77 K, refer to Ref. [11]. Notably, at 4.2 K, k\({}^{\prime\prime}\) becomes negative for frequencies above 47 GHz, indicating the possibility of space charge wave propagation without amplification. This observation suggests that a strained Si/SiGe heterostructure has the potential for amplifying space charge waves within a frequency range similar to that of an n-GaAs thin film, albeit at low temperatures. To carry out the simulations, we introduce an input signal, and it was applied in the input coupling element, see Equation (6). \[\vec{E}_{ext}=E_{o}sin(\omega t)exp\left(-\left(\frac{z-z_{1}}{z_{0}}\right)^{2 }-\left(\frac{y-y_{1}}{y_{0}}\right)^{2}\right) \tag{6}\]  
The output signal clearly demonstrates the amplification effect. In this scenario, a microwave input signal with a frequency of 10 GHz is applied to the input coupling element, resulting in the excitation of space charge waves in the 2D electron gas. These waves undergo amplification due to the presence of negative differential conductivity. We employ a stable implicit difference scheme for our analysis, utilizing the parameters provided in Table 1.  
Figure 3: Spatial increment \(k^{\prime\prime}(\omega)\) for tow-valley model for a strained Si/SiGe heterostructure at 77 K (blank), 4.2 K (blue) and 1.3 K (red). Figure 4 illustrates the typical output spectrum of the electromagnetic signal. The input carrier frequency is set at \(f\) = 15 GHz, with an input electric microwave signal amplitude of E\({}_{\rm m}\) = 25 V/cm. Despite a decreasing growth rate with increasing _rf_ frequency, our results show an amplification of 23 dB in this particular case. The input pulse duration is 2t\({}_{0}\) = 2.5 ns, with the maximum of the pulse occurring at t\({}_{1}\) = 2.5 ns. The amplified signal at the first harmonic of the input signal is evident, as well as the generation of harmonics resulting from the non-linear behavior of space charge waves. Specifically, we observe the second and third harmonics with frequencies of 20 GHz and 30 GHz, respectively. While it may be possible to obtain the fourth and fifth harmonics, these cannot be effectively amplified, as shown in Figure 3, and they are perceived as noise. In Figure 4b, we explore parametric amplification by introducing two input signals with frequencies \(f_{1}\) = 15 GHz and \(f_{2}\) = 20 GHz, respectively. The resulting output signals are displayed. Here, we can observe the output signals, including the second harmonics, as well as the combined effect of the two input signals. Figure 5 depicts the spatial distributions of the alternate components of the electric fields (E\({}^{-}\)z and E\({}^{-}\)y) and electron concentration \(\bar{n}\) at a specific time moment, \(t\) = 4 ns. Notably, the maximum variations occur in the output coupling element, and the spatial distribution in the E\({}^{-}\)z direction is larger than in the E\({}^{-}\)y direction. This difference arises due to the propagation of the space charge wave.