# Numerical Analysis of the Breakdown Process of CF\({}_{3}\)I at Low Pressure
## 2 Models
### Physics Models
#### 2.1.3 External Circuit  
When studying the gas DC breakdown process, the external resistor plays an indispensable role, both in practical experiments and simulations.  
In practical experiments, the gas breakdown discharge occurrence leads to a significant increase in plasma density between the two polar plates, consequently causing a dramatic rise in conductivity. Without an external resistor in the circuit, there is a risk of damaging the experimental equipment.  
During simulations, the external resistance facilitates the attainment of steady-state discharge and enables more accurate determination of electrical characteristics. Prior to gas breakdown, the system comprising the polar plate and the CF\({}_{3}\)I gas inside it can be likened to a capacitor. As the gas undergoes breakdown, the system of plates and CF\({}_{3}\)I exhibits resistor-like behavior from a circuit perspective. At this juncture, the external resistor serves as a voltage divider. With increasing plasma current, according to [PERSON]'s theorem, the partial pressure on the outer resistor also rises, thereby reducing the voltage between the two pole plates, restraining the upward trend of current, and performing the dual function of voltage division and current limitation. Without the addition of an external resistor and direct application of supply voltage to the two plates for discharge simulation, the discharge may fail to reach a steady state due to the absence of a resistor to regulate the negative feedback of the voltage [20]. The simulation results can be adversely affected by either excessively large or small resistors: an excessively large resistor excessively divides voltage, resulting in insufficient voltage between the plates for gas penetration; an excessively small resistor fails to achieve the desired voltage division and current limitation effect. Additionally, it has been noted that when employing the PIC/MC model for DC breakdown simulation, the absence of external resistance in the model leads to inaccurate data such as current, thereby hindering accurate analysis of electrical characteristics [20].  
An appropriate external resistance value ensures that the external resistor shares approximately 20% of the voltage when the reaction reaches a steady state. Through the calculation of the equivalent resistance of the plasma and systematic exploration of external resistance parameters, we determined that setting the external resistance to 8 k\(\Omega\) is more optimal.  
The circuit diagram of the equipment simulated in this paper is depicted in Figure 2. The electrodes have a circular shape. Its diameter is 0.2 m. The distance between the electrodes is 2 cm. The pressure of CF\({}_{3}\)I gas between plates is maintained at 160 mTorr unless otherwise specified, a pressure low enough to induce the decomposition of CF\({}_{3}\)I gas.  
In practical scenarios, the circuit voltage does not instantaneously reach a peak; instead, it rises rapidly from 0 to a peak of \(-1000\) V. To simulate this phenomenon, a voltage function in the form of \(1000\left(e^{-\frac{t}{1\times 10^{-10}}}-e^{-t}\right)\) was employed. Plotting the function using appropriate software revealed that the rapid transition of voltage from 0 V to \(-1000\) V occurs within approximately 1 ns, consistent with the actual scenario.  
Following the approach advocated by [PERSON] et al. [32], the linkage between the external circuit and the plasma can be established and resolved. Subsequently, the PIC algorithm is employed to simulate the electrical parameters of each grid within the plasma over a unit of time. Solving [PERSON]'s equation enables the determination of the potential for the plasma region situated between the parallel plates.