Note on particle balance in particle-in-cell/Monte Carlo model and its implications on the steady-state simulation

The steady-state criterion for plasma numerical simulations can be determined by the particle balance relation. In this study, we utilized a one-dimensional (1D) particle-in-cell/Monte Carlo model to investigate particle transport in a capacitively coupled plasma discharge, including particle density change, flow, generation, and loss. Our analysis revealed that the generation rate and loss rate are equivalent in both time and space, indicating a fine balance in the steady state of the discharge system. Additionally, we presented the spatio-temporal distribution and time-averaged particle transport term for electrons and ions to demonstrate how particles attain equilibrium at varying pressures. This validation method can be particularly useful in numerical simulations where determining steady state can be challenging. Our findings establish the correctness and reliability of the method.