## 4 Conclusions  
This work presents a 2D numerical simulation aimed at studying the amplification of space charge waves utilizing negative differential conductance in a strained Si/SiGe heterostructure at a temperature of 4.2 K. The quasi-hydrodynamic model of electron transport is employed to investigate the amplification of space charge waves in this nonlinear medium, particularly focusing on the phenomenon of negative differential conductance. While negative differential conductance is commonly observed in GaAs thin films at room temperature, this study demonstrates its occurrence in strained Si/SiGe heterostructures even at very low temperatures. The results show that amplification and non-linear interaction of space charge waves in strained Si/SiGe heterostructure occur 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. Furthermore, the analysis of the electric field variation in the Z and Y directions indicates that the electric field along the Z direction surpasses that along the Y direction due to the movement of electrons in that specific direction, \(E^{-}z\) is nearly 50 times greater than \(E^{-}y\). It is worth noting that, although negative differential conductance can also be observed in strained Si/SiGe heterostructures at 1.3 K, it is crucial to acknowledge that the electron transport models differ significantly in this scenario. This emphasizes the need for further investigation into the underlying physical phenomena.  
Overall, these findings provide valuable insights into the amplification of space charge waves and the role of negative differential conductance in strained Si/SiGe heterostructures, emphasizing their potential applications in low-temperature electronic devices and expanding our understanding of electron transport in these systems. Further investigations and refinements of the electron transport models are necessary to uncover the underlying mechanisms and enhance our knowledge of this intriguing physical phenomenon.  
Conceptualization, A.G.-B. and E.A.G.-D; methodology, L.F.; software, N.N.; validation, S.M.-V., V.S. and S.Z.-G.; formal analysis, A.G.-B; investigation, E.A.G.-D.; resources, N.N.; data curation, A.G.-B.; writing--original draft preparation, A.G.-B.; writing--review and editing, S.Z.-G.; visualization, N.N.; supervision, E.A.G.-D.; project administration, V.S.; funding acquisition, L.F. All authors have read and agreed to the published version of the manuscript.  
Figure 5: The spatial distributions of the alternative part of the electric field component \(E^{-}z\) of the apace charge wave at 4.2 K (**a**); the component of electric field \(E^{-}y\) at 4.2 K (**b**); alternative part of the electron concentration \(\bar{n}\) at 4.2 K (**c**).  
**Funding:** This work was supported in part by CONACyT Mexico by Sabbatical fellowship at McMaster University and the PhD Scholarship with CVU number: 626570. Also, this work was supported by OEaD Agency at the Vienna University of Technology fellowship. [PERSON] gratefully acknowledges financial support through the Austrian Science Fund (FWF) grant P33609-N and the Austrian Federal Ministry of Labour and Economy, the National Foundation for Research, Technology and Development and the Christian Doppler Research Association.  
**Conflicts of Interest:** The authors declare no conflict of interest.