Theoretical study of particle and energy balance equations in locally bounded plasmas

In this study, new particle and energy balance equations have been developed to predict the electron temperature and density in locally bounded plasmas. Classical particle and energy balance equations assume that all plasma within a reactor is completely confined only by the reactor walls. However, in industrial plasma reactors for semiconductor manufacturing, the plasma is partially confined by internal reactor structures. We predict the effect of the open boundary area ( A'_\mathrmL,\mathrme\mathrmf\mathrmf ) and ion escape velocity ( u_\mathrmi ) on electron temperature and density by developing new particle and energy balance equations. Theoretically, we found a low ion escape velocity ( u_\mathrmi / u_\mathrmB\approx 0.2 ) and high open boundary area ( A'_\mathrmL,\mathrme\mathrmf\mathrmf/A_\mathrmT,\mathrme\mathrmf\mathrmf\approx 0.6 ) to result in an approximately 38% increase in electron density and an 8% decrease in electron temperature compared to values in a fully bounded reactor. Additionally, we suggest that the velocity of ions passing through the open boundary should exceed \omega _\mathrmp\mathrmi\lambda _\mathrmD\mathrme under the condition E_0^2\ll \left(\varPhi /\lambda _\mathrmD\mathrme\right)^2 .