## 1 Introduction  
In the past five decades, extensive research has been conducted on the amplification of space charge waves through the utilization of negative differential conductance (NDC) [1, 2, 3, 4, 5]. These devices find applications in a wide range of fields, such as amplifiers, oscillators, and frequency multipliers, particularly in the millimeter and sub-millimeter wave range. By harnessing the negative differential conductance phenomenon as a nonlinear medium, these devices can generate harmonics of input signals. The significance of these applications lies in their crucial role in high-frequency communication systems and photonics. Numerous materials, including GaAs, GaN, InP, and InN, exhibit the negative differential conductance phenomenon, when subjected to high electric fields at room temperature [6]. Unfortunately,most of these compound materials are incompatible with silicon technology. However, a strained Si/SiGe heterostructure offers compatibility with MOSFET technology and it demonstrates negative differential conductance as well, albeit only at extremely low temperatures--specifically at 77 K, 4.2 K, and 1.3 K [7; 8; 9]. In a previous study by [PERSON] et al. [10], a numerical investigation using simulations of space charge wave propagation in strained Si/SiGe heterostructures at 77 K was conducted. Additionally, two-dimensional (2D) simulations of space charge wave amplification in a strained Si/SiGe heterostructure at 77 K were presented in a separate study [11]. However, these articles solely focused on numerical simulations of space charge wave amplification at 77 K without considering calculations for temperatures at 4.2 K and without performing a parametric interaction analysis. The mentioned research works utilized electron mobility calculations at various temperatures as a function of the effective electric field, sourced from references [12; 13]. Nevertheless, it is worth noting that the negative differential conductance exhibits a larger dynamic range at temperatures of 4.2 K and 1.3 K, as revealed in [14]. Therefore, our research work aims to present 2D numerical simulations of space charge wave amplification in strained Si/SiGe heterostructures at very low temperatures--specifically at 4.2 K--where the negative differential conductance manifests. To accomplish this, we employed the hydrodynamic model to analyze electron transport at high fields in strained Si/SiGe heterostructures [15]. By utilizing the finite difference time domain (FDTD) method, we solved the hydrodynamic model in conjunction with the Poisson equation. Through this approach, we obtained the output spectrum of the input signal and calculated the variation of the electric field in both the Z and Y directions. The electric field was found to be greater in the Z direction, as the electrons predominantly move along this axis. Our results indicate the potential for developing cryogenic CMOS-based mm-wave amplifiers and frequency multipliers, presenting exciting possibilities for new satellite communication systems. Negative differential conductance (NDC) is a fascinating phenomenon observed in a wide range of electronic systems, wherein the current passing through the system decreases as the applied voltage is increased. This behavior defies the classical understanding of Ohmic conductance, where an increase in voltage typically leads to an increase in current. NDC represents a unique and intriguing characteristic, particularly notable in semiconductor devices. Essentially, NDC refers to a two-terminal component capable of amplification, converting DC power applied to its terminals into AC output power to amplify an AC signal applied to the same terminals. The discovery of NDC has paved the way for the development of novel electronic devices, including oscillators and amplifiers, with a particular focus on microwave frequencies. These devices play a critical role in communication, sensing, and computing systems by offering enhanced efficiency, speed, and sensitivity.