The configurations of the cavity and drift tube used in a 0.34 THz Extended Interaction Klystron (EIK) are theoretically studied. The results from particle-in-cell (PIC) simulations are presented and discussed. Based on the small signal theory, the coupling coefficient and beam conductance at the gaps were studied, leading to optimization of the multi-slot cavity. The physical analysis of electron movement was carried out to study the influence of electron drifting on device performance. PIC simulations were conducted with the results compared to analytical theory. Good agreement was achieved between analytical predictions and simulations, demonstrating the feasibility of the theoretical approach. The performances of an EIK under different conditions such as mismatching and self-oscillation indicate that an optimized structure can produce an output power of 143 W and a gain of 38.1 dB, demonstrating its potential to be a highly stable and reliable source of coherent sub-terahertz radiation.