The third-generation semiconductor device known as High Electron Mobility Transistors (HEMT) has found extensive applications in high-frequency and high-speed electronic systems. Its widespread usage in critical technologies such as radio telescopes, satellite broadcast receivers, and cellular base stations has established HEMT as a foundational technology underpinning our information and communication society. This paper provides an in-depth exploration of these semiconductor advancements. Firstly, the paper utilizes the CMOS inverter as a representative example to elucidate the fundamental structure of Complementary Metal-Oxide-Semiconductor (CMOS) technology. Additionally, it employs Gallium Arsenide (GaAs) HEMT as an illustrative instance to expound upon the architecture of HEMT devices. Furthermore, the paper delves into the optimization of CMOS technology, focusing on topics such as Multi-Threshold CMOS and the impact of the Width/Length (W/L) ratio. These discussions shed light on ways to enhance the performance of CMOS-based components. Additionally, the paper explores strategies to optimize HEMT devices, including the introduction of carbon doping and the application of the Grey-Wolf optimization technique. These approaches are critical in achieving higher efficiency and performance in HEMT-based applications.