# Numerical Analysis of the Breakdown Process of CF\({}_{3}\)I at Low Pressure
## 2 Models
### Numerical Methods  
In recent decades, numerical methods for simulating the evolution of plasma breakdown processes have been widely explored, resulting in the development of two prominent algorithms: the explicit algorithm and the implicit algorithm. The latter has demonstrated distinct advantages over the former, providing accurate evolution parameters for the plasma breakdown process and expanding its applicability, which is unattainable for the explicit algorithm [33; 34; 35; 36]. Numerous studies have been conducted to enhance the implicit method [20; 21; 37; 38; 39].  
[PERSON] et al. successfully applied an improved implicit algorithm to numerically simulate the evolution of parameters in the electrical breakdown process of dual frequency (DF) capacitively coupled plasma (CCP) [20]. Similarly, [PERSON] et al. successfully utilized this algorithm to simulate SF\({}_{6}\) breakdown [21]. Consequently, this paper adopts the one-dimensional implicit \"evolution\" PIC/MCC algorithm.  
The initial plasma density is set to \(10^{10}\) m\({}^{-3}\). The value of the initial density does not significantly affect the breakdown process [40]. In the simulation of gas breakdown at low pressure, it is common practice to set the initial density of the plasma [20; 21]. An initial density of \(10^{10}\) m\({}^{-3}\), which is much lower than the plasma density after breakdown occurs, does not affect the nature of the breakdown. During the initialization phase, the density of positive ions is established at half of the electron density. This step ensures electrical neutrality and minimizes the effect of charge imbalance on the electric field. Particles reaching the boundary are absorbed by the electrodes, while secondary electrons are generated by ion bombardment of the electrodes, an essential factor in the growth of the electron number [20; 41]. The emission energy of secondary electrons is randomly distributed within the range of 0-5 eV, and their emission direction is also randomized [42]. The ion-induced secondary electron emission coefficient is assumed to be 0.2. The secondary electron emission coefficient and the energy of secondary electrons are not the focus of  
Figure 2: Diagram of external circuits.  
this study, but secondary electrons are essential in gas breakdown. Since the secondary electron emission coefficient for CF\({}_{3}\)I could not be found, a value of 0.2 was adopted based on studies of SF\({}_{6}\) by other researchers [21; 43]. The discharge gap between plates is divided into 97 grids, each initially populated with 2000 macroscopic particles. The number of grids and the spatial step of the simulation program are determined by considering equipment performance and simulation time factors to ensure accurate results. Theoretically, a smaller time step \(\Delta t\) results in a higher degree of decoupling between motion and collisions [21]. Given CF\({}_{3}\)I's strong electronegativity, a longer time is required for its discharge to reach a stable state. Setting \(\Delta t\) too small would extend the simulation time excessively. The time step must strike a balance that is sufficient for accuracy, but not exceeding the device's calculation performance. This paper carried out tests with various time steps (\(1\times 10^{-10}\) s, \(5\times 10^{-11}\) s, \(2.5\times 10^{-11}\) s, \(1\times 10^{-11}\) s, \(1\times 10^{-12}\) s) and found that until \(\Delta t\) increases to \(5\times 10^{-11}\) s, the accuracy of the calculation remains consistent. Therefore, the paper suggests that selecting \(\Delta t=5\times 10^{-11}\) s is the most appropriate.  
The diagnostic frequency is fixed at \(1\times 10^{8}\) Hz, corresponding to a diagnostic period of 10 ns, with each diagnostic period containing 200 time steps. This frequency setting ensures diagnostic results of a manageable size, facilitating image creation and result analysis. The duration of the full simulation spans 15,000 diagnostic cycles, totaling 150 \(\upmu\)s.