Recently, the cold source field-effect transistor (CSFET) has emerged as a promising solution to overcome Boltzmann tyranny in its ballistic regime, offering a steep-slope subthreshold swing (SS) of less than 60 mV/decade. However, challenges arise due to scattering, particularly from inelastic scattering, which can lead to significant degradation in SS through cold carrier thermalization. In this study, we delve into the theoretical investigation of the electronic excitation/relaxation dynamic process using the state-of-the-art nonadiabatic molecular dynamics (NAMD) method. The mixed quantum-classical NAMD proves to be a powerful tool for comprehensively analyzing cold carrier thermalization and transfer processes in semiconductor Si, as well as metallic silicides (NiSi2 and CoSi2). The approach of mixed quantum-classical NAMD takes into account both carrier decoherence and detailed balance, enabling the calculation of thermalization factors, relaxation times, scattering times, and scattering rates at various energy levels. The thermalization of carriers exhibits a gradual increase from low to high energy levels. Achieving partial thermalization from the ground state to reach the thermionic current window occurs within a sub-100 fs time scale. Full thermalization across the entire energy spectrum depends sensitively on the barrier height, with the scattering rate exponentially decreasing as the energy of the out-scattering state increases. Notably, the scattering rate of NiSi2 and CoSi2 is two orders of magnitude higher than that of Si, attributed to their higher density of states compared to Si. This study not only provides insights into material design for low-power tunnel field-effect transistors but also contributes valuable information for advancing CSFET in emerging technologies.