Influence of external circuitry on CF₄ breakdown process in capacitively coupled plasma

Capacitively coupled plasma (CCP) tools are the most widely used tools for etching, deposition, and cleaning in the semiconductor industry.^1–3 Their simple structure has facilitated extensive research and widespread adoption.^4–7 To advance chip processing technology, it is essential to understand the discharge characteristics of CCP and effectively manage its discharge process.

In a CCP discharge system, an external matching network plays a critical role in optimizing the transfer of RF power from the generator to the plasma chamber.^8–10 This optimization ensures efficient power absorption and effective plasma generation. The primary function of the matching network is to match the impedance of the RF power source with that of the plasma load.^11,12 However, designing an optimal matching circuit for the CCP discharge is challenging because of the nonlinear properties of the plasma. The plasma impedance varies with power and plasma parameters, complicating consistent coupling with the matching circuit. Consequently, a well-designed matching circuit must consider various factors, including plasma type, gas pressure, electrode geometry, and frequency of the RF power source.¹³ The layout and components of the matching network are essential to achieve a successful CCP discharge. The matching circuit comprises passive components, such as capacitors, inductors, and resistors, which must be carefully selected and arranged for optimal power transfer and impedance matching.^11,14 Moreover, the matching circuit must handle high voltages and currents while maintaining stable operation, necessitating careful consideration of component materials and construction. In summary, achieving a well-designed matching circuit is crucial for efficient and stable CCP discharge.

Carbon tetrafluoride (⁠ $CF 4$⁠) gas has various applications in industrial semiconductor discharges. For example, during manufacturing processes, $CF 4$ or other fluorides and their mixtures are used to etch silicon¹⁵ and silicon dioxide.¹⁶ Electronegative gas discharges exhibit different plasma characteristics due to significant adsorption of electronegative gas molecules.¹⁷ The discharge properties of $CF 4$ vary with gas pressure.^18,19 At low gas pressure, the electron density is high, resulting in a positively charged discharge similar to that of argon gas, while at high gas pressure, the electron density is lower and the discharge demonstrates electronegative properties.

In recent decades, researchers have extensively studied the discharge of electronegative gases using models^7,17–24 and experiments.^25–28 In practical industrial applications, the presence of an external circuit outside the $CF 4$ discharge chamber significantly influences the plasma discharge process. While previous studies have focused mainly on the steady-state discharge process of $CF 4$⁠,¹⁹ the breakdown process of the discharge of $CF 4$ in the presence of external circuits has not been investigated so far. The external circuit can modify the distribution of the electric field within the chamber and alter the electron density and temperature of the plasma, thereby affecting the discharge process.^29,30 Therefore, a complete understanding of the properties of the circuit and their coupling to the discharge chamber is essential to fully grasp the impact of the external circuit on the discharge process.^10,29,31

Studying the electrical properties of the $CF 4$ discharge breakdown process and stabilized plasma when an external circuit is incorporated can provide valuable information on the behavior of plasma discharges in real-world industrial applications. This knowledge can be applied to improve the design and optimization of plasma processing systems for various applications, such as surface treatment, material synthesis, and environmental remediation. The aim of this paper is to investigate the electrical properties of the $CF 4$ discharge breakdown process and the stabilized plasma after integration of an external circuit, offering fresh insights into the interaction between the plasma and external circuit in industrial contexts.

This paper is organized as follows: In Sec. II, the physical model and numerical methods used in this study are presented. The simulation results are presented in Secs. III and IV. An overview of the general process and instantaneous avalanche breakdown as a special case of $CF 4$ under coupled external circuit conditions is provided in Sec. III. In Sec. IV, particular emphasis is given to the discussion of the effect of gas pressure and different matched capacitance values on gas breakdown. Finally, Sec. V presents a concise conclusion.

In this study, a non-ideal external circuit model is considered, as shown in Fig. 1. The RF power source is designed to generate a sine wave with a frequency of 27.12 MHz and an amplitude of 350 V, featuring an internal resistance of 50  $Ω$⁠. The L-shaped matching circuit serves to optimize the power delivered to the load. This circuit comprises two capacitors (⁠ $C 1$ and $C 2$⁠), a resistor (⁠ $R 2$⁠, fixed at 0.5  $Ω$⁠), and an inductor (⁠ $L$⁠, fixed at 0.5  $μ$H) connected in a specific configuration. The capacitor values can be adjusted during the discharge process to achieve the desired performance. The reactor functions to create and sustain plasma discharge, consisting of a parallel plate discharge gap (CCP), a stray capacitor (⁠ $C m$⁠, fixed at 200 pF), and a stray resistor (⁠ $R m$⁠, fixed at 0.5  $Ω$⁠). The CCP initiates and sustains the plasma discharge by generating a strong electric field between the parallel plates. The model assumes that the plasma discharge and the matching circuit are closely coupled near the supply electrode. The electrode used in the model exhibits symmetry and has a radius of 0.15 m.

Particle-in-cell/Monte Carlo Collision (PIC/MCC) methods are widely used to simulate plasma discharge processes in low pressure gases.^33–36 PIC/MCC simulations, based on first principles, resolve the motion of charged particles in the plasma and derive the corresponding electromagnetic fields.^33,37 In this work, a one-dimensional implicit simulation method is utilized to accomplish the simulation task. Compared to the explicit method, the implicit approach has a larger spatial step and a longer time step,^38–40 which is essential to simulate the extended evolutionary $CF 4$ breakdown process²³ in an external circuit.

The discharge gap is fixed at 3 cm and divided into 96 uniform grids. The spatial resolution is sufficiently fine to resolve the sheath, under the implicit code. The time step, $Δ$t, is set to $5.0× 10 − 11$ s. The initial density of the charged particle is established at $1× 10 8 m − 3$⁠, allowing the study of the whole breakdown process, including the avalanche. The simulation encompasses approximately 12 000 RF cycles to reach the final steady state, with each cycle lasting 16.67 ns (⁠ $T r f$⁠).

In this study, the equal weight method is employed for simulating charged particles. However, due to the significant disparity in particle density during $CF 4$⁠, an exceedingly low electron density results in a small number of macroelectrons, leading to substantial numerical noise that distorts the findings. To alleviate this numerical noise, the initial count of macroscopic particles in each grid is set to 1000, ensuring that thousands of macroelectrons remain in the discharge region as the discharge approaches steady state, effectively suppressing the numerical noise. Additionally, a simple particle merging algorithm, incorporating random particle deletion, is utilized throughout the simulation process. Specifically, if the average count of certain macroparticles in each grid exceeds 1500, one-fourth of each particle is randomly removed, and the weight of the remaining macroparticles is increased by the original one-fourth. During the breakdown process, particle merging is primarily observed in the pre-breakdown stage, where the electric field is dominated by the externally applied field, and the self-generated electric field is relatively weak. As a result, particle merging during this stage has almost no effect. In the post-breakdown stage, the electron density grows slowly, and with larger particle weights, particle merging can cause some minor effects.

Under radiofrequency power, electron-induced secondary electron emission will be the primary factor influencing the breakdown characteristics.^41,42 To enhance the accuracy of this simulation, we have considered an electron-induced secondary electron emission model based on the $SiO 2$ surface. In this model, the elastic reflection accounts for 0.03 of the high-energy region, while the inelastic backscattering accounts for 0.07.⁴³ 

Due to the complex nature of the reaction between the ions and the $CF 4$⁠, obtaining the collision cross section presents a challenge. Therefore, the standard MCC model proposed by Nanbu^44–46 is employed to address the ion- $CF 4$ collisions. This model approximates the collision cross section using the atomic or molecular radius and the reaction’s energy threshold. In this work, reference is made to the work of Wu²³ for the collisions between electrons and $CF 4$⁠, specifically focusing on one elastic scattering collision, four excitation collisions, two adsorption reactions, and one ionization-dissociation collision. The collision cross sections utilized in this study are obtained from LXCat.⁴⁷ 

In this expression, $k e − i +$ is $3.95× 10 − 15 m 3/ s$⁠, $T e ( x g )$ represents the electron temperature at position $xg$⁠, and $T i + ( x g )$ denotes the ion temperature at position $xg$⁠. The reaction coefficient $k i − − i +$ is typically provided directly for the complexation of negative ions with positive ions. In the case of the $CF 4$ discharge process, the complexation coefficient of the negative and positive ions is $1× 10 − 13 m 3/ s$⁠.⁵⁰ During the execution of the recombination, in the particle data structure, the particles are typically traversed, starting from random positions. If any particle within the corresponding grid needs to participate in the recombination, it is removed from the particle data structure. It is essential to note that, in CCP discharges, because of the low degree of ionization, only a small fraction of the particles participate in the recombination during each time step. If the recombination is executed at every time step, traversing the entire particle array would significantly increase computational overhead. Therefore, it is common practice to perform the recombination at intervals of every 10s’ steps to reduce computational burden.

In addition to electrons, the simulation focuses on tracking three types of charged particles, namely $F −$⁠, $CF 3 +$⁠, and $CF 3 −$⁠, as they exhibit a higher density that determines the total charge density in the plasma. Simulations account for collisions between these charged particles and neutral $CF 4$ molecules, which play a crucial role in determining the plasma breakdown process.^20,51,52 Collisions involving other advanced dissociation ionization types and neutral dissociation are not considered due to their extremely low reaction cross sections and negligible impact on the evolution of plasma parameters during the breakdown process.^20,53 The primary concern is to analyze the evolution of $CF 4$ plasma parameters during the breakdown process and the feedback of the plasma on the external circuit.

In this study, when the value of the $C 1$ capacitor was set to 500 pF and the value of the $C 2$ capacitor was set to 260 pF, a lower electrode voltage was achieved. As illustrated in Fig. 2, electrons and $CF 3 +$ particles played a pivotal role during the breakdown phase. The electron density and electron temperature throughout this phase were analyzed, along with the field evolution parameters within the discharge gap, as shown in Fig. 3.

Before the breakdown (0–12  $μ s$⁠), the accelerating electric field could penetrate the entire discharge gap, directly accelerating the electrons. High-energy electrons oscillated in the discharge gap, ionized the neutral gas, or directly bombarded the electrode, inducing secondary electron emission. Under the combined effect of ionization and secondary electron emission, the electron density increased exponentially, ultimately leading to an electron avalanche. During this process, the density of charged particles within the discharge gap was negligible, exerting minimal impact on the external circuitry. The entire CCP could be regarded as a fixed-value capacitor.

Figure 2(c) shows that due to the presence of the matching circuit, it took 0.5  $μ$s for the voltage at both ends of the reaction cavity to reach a stable state. This process represented the capacitor charging process in the matching circuit. The capacitor voltage charging waveform was affected by the capacitor $C 2$ and the voltage amplitude was often amplified during this process, causing a voltage peak at both ends of the CCP. Under the influence of this voltage peak, electrons were accelerated to a very high energy level in the initial stage, resulting in intense ionization, and the electron density increased by two orders of magnitude in a short period of time. Consequently, the requirement for the breakdown voltage of the plasma was greatly reduced.

At 12  $μ$s, the electron density reached a value of $10 12 m − 3$⁠, leading to plasma formation. At this time, the Debye length of the plasma approached the gas spacing, and the shielding effect of the plasma on the electric field appeared. This caused rapid loss of electrons to the electrodes and the formation of a sheath layer. The plasma region maintained a high potential, while the sheath region maintained a high electric field. Thus, the plasma began a slower evolutionary process.

At the instant of breakdown, the electrical characteristics of the CCP underwent alterations as a consequence of the plasma’s shielding effect on the electric field. This led to an increase in the CCP’s equivalent capacitance and a decline in the voltage amplitude across the pole plate, from 85V to 73 V, owing to the feedback effect of the CCP on the matching circuit. Subsequently, the pole plate voltage experienced gradual changes in tandem with the plasma’s evolution and ultimately attained a stable value.

Due to the presence of energy storage devices within the matching circuit, the effects of charging and discharging during the initial phase of discharge might lead to elevated electrode voltage. In certain exceptional cases, a higher charging voltage amplitude could facilitate the plasma’s immediate attainment of a breakdown state and its eventual stabilization at a reduced pole plate voltage. When $C 1$ was set to 500 pF and $C 2$ was adjusted to 70 pF, the extremely high voltage caused the electrons in the plasma to undergo instantaneous avalanche breakdown.

As depicted in Fig. 4(c), the magnitude of the charging voltage amplitude reached 350 V. Influenced by this voltage peak, electrons were accelerated to a considerable energy of 20 eV during the initial stage. These high-energy electrons subsequently collided vigorously with the neutral gas, engendering a robust ionization reaction within the plasma. This culminated in an avalanche breakdown of electrons and an exponential increase in the electron density. Consequently, the plasma formed immediately.

At approximately 0.5  $μ$s, the plasma sheath formed, generating a large electron current density, as shown in Fig. 5(c). However, when the capacitor charging process in the matching circuit ended, the voltage amplitude across the CCP rapidly converged and remained at a low level of approximately 50 V. The sudden change in voltage amplitude rendered the original plasma state unsustainable, and the plasma sheath began to oscillate intensely, resulting in a rapid loss of electrons. An intense evolutionary process occurred inside the plasma, and the electrons in the bulk region moved vigorously with the voltage change, producing a large electron current density. The energy of the electrons and the other particles also decreased, and the electron energy showed significant losses or even dropped as low as 0.1 eV.

Approximately 5  $μ$s after initiation, the plasma progressively stabilized and transitioned to a comparatively stable evolutionary phase. Within this phase, the electron energy gradually increased and the densities of the $CF 3 −$ and $F −$ ions increased accordingly. Due to adsorption and complex reactions leading to electron loss, the electron density in this system reaches only $10 14 m − 3$⁠.

The evolution of plasma parameters during the breakdown of the $CF 4$ gas under coupled external circuit conditions was presented in Sec. III. This section will explore two main sections. In the first subsection, the influence of gas pressure on gas discharge will be investigated, with a specific focus on the plasma evolution at different gas pressures and the variation of the equivalent capacitance of the CCP. The effect of matching capacitance on the breakdown process will be investigated in the second subsection by adjusting the capacitance value, accompanied by an analysis of two special cases.

The discharge mode of $CF 4$ gas can be changed by changing the pressure.^18,20,21 In the case of low pressure, electrons have a longer mean free path and lower collision frequency. This results in a positive discharge dominated by electrons and positive ions, which affect the electrical properties of the plasma. Conversely, elevated pressure conditions result in reduced mean free paths for electrons, thereby promoting increased collision frequencies and instigating modifications in the electron energy probability function (EEPF) of electrons. These disparities in EEPF significantly influence the electron attachment across distinct energy intervals, wherein enhanced electron attachment processes and electron-ion complexation phenomena precipitate the occurrence of discharges marked by negative charge. The electronegative mode is characterized by a higher ion density and a very low electron density, with charged particles dominated by negative and positive ions.

In the $CF 4$ discharge process, different particles dominate the plasma discharge, with varying masses and charges. Their microscopic motions have a significant impact on the electrical properties of the plasma. In positive discharges, the motion of positive ions is characterized by their significantly higher mass compared to that of electrons, resulting in slower velocities and lower sensitivity to the electric field. As a consequence, an ion sheath is formed around the electrode. In negative discharges, negative ions are mainly generated through the adsorption reaction with $CF 4$ molecules by electrons. As the sheath layer gradually stabilizes, the established electric field of the sheath restricts the movement of negative ions to the bulk region, preventing them from reaching the boundary. Therefore, even with extremely small adsorption cross sections, a small number of adsorption reactions can continuously increase the density of negative ions for an extended period after breakdown.^54,55 Therefore, the motion of the particles plays a crucial role in determining the electrical properties of plasma discharge in $CF 4$⁠.

In this study, we used the equation of the energy storage relation of a capacitor to compute the equivalent capacitance of the CCP. The capacitor’s energy was stored in the form of an electrostatic field, and the spatial electric field distribution was calculated for each moment using the PIC algorithm. Therefore, the equivalent capacitance of the CCP could be determined from the equation relating the electrode voltage to the electric field distribution.²⁹ 

Keeping $C 1$ at 500 pF and $C 2$ at 85 pF, the results of the $CF 4$ breakdown simulation in different discharge modes can be obtained by altering the neutral gas pressure. As shown in Fig. 6, the introduction of the initial charging voltage induces a variation in the electron temperature. Initially, the electric field directly accelerates the electrons; a larger free range results in a higher energy gain by the electrons. Consequently, the electron temperature is approximately 20 eV at 50 mTorr and about 10 eV at 200 mTorr. Although higher electron energy is observed at lower pressures, the ionization collision rate decreases as a result of gas rarefaction. This contributes to a more gradual increase in electron density at the initial moment. On the contrary, increased gas pressure leads to shorter mean free paths of electrons and more frequent ionization collisions, culminating in a more rapid growth rate of electron density under 200 mTorr conditions.

At 5  $μ$s, the plasma breaks down due to the dominance of electrons in the plasma at that time. Therefore, the thickness of the plasma sheath layer at different gas pressures is similar, resulting in similar electrical properties. The difference in equivalent capacitance is only 0.2 pF.

At 20  $μ$s, the attachment reaction in the plasma already dominates at higher gas pressures due to the increased electron density. Consequently, for the 200 mTorr case, the electron density exhibits a significantly slower rise after breakdown compared to the 50 mTorr case. This is replaced by a continuous increase in negative ion density, which gradually governs the discharge, resulting in the sheath layer gradually thinning and stabilizing. During this phase, there are notable disparities in the electrical properties of the plasma at different gas pressures, attributable to variations in particle density and plasma sheath layer thickness. As shown in Fig. 7, the thickness of the plasma sheath layer at high gas pressure is approximately 0.6 cm, with an equivalent capacitance of 66.4 pF, while at low gas pressure, the thickness is approximately 0.8 cm and the equivalent capacitance is 52.1 pF.

Between 20 and 440  $μ$s, in the high-pressure $CF 4$ plasma, the sheath layer gradually thins out as negative ions continue to accumulate in the bulk region.

Because of the low electron density, a strong drift electric field is generated in the bulk region, resulting in an elevated Ohmic heating rate within the bulk. Consequently, the average electron energy gradually increases. Furthermore, an increase in negative ion density directly leads to changes in the sheath, resulting in a gradual increase in equivalent capacitance.

In low-pressure $CF 4$ plasma, the larger average electron-free range limits the rate of ionization and attachment reactions, causing the growth rate of electron density to gradually slow down. As a result, the sheath layer gradually stabilizes. Furthermore, the negative ions are maintained at a lower density, resulting in minimal changes in the overall electrical properties. Eventually, both reach a stable value. As the plasma gradually stabilizes, at 50 mTorr, a more distinct dual-temperature distribution is observed. A wider sheath region leads to a longer high-energy tail, but the majority of electrons remain in the low-energy sheath. At 200 mTorr, after 5  $μ$s, the discharge progressively transitions to an electronegative state. Drift-bipolar field (D-F) heating in the body region results in higher electron temperatures, and the energy of the central portion of the EEPF exhibits a more pronounced raised shape, as shown in Fig. 8. This observation corresponds to the average energy evolution of the electrons depicted in Fig. 6.

Compared to the electropositive discharge mode, in a $CF 4$ plasma at high gas pressure, electrons are primarily concentrated in the lower energy range due to the strong influence of intense electron attachment processes.

This results in the rapid loss of high-energy electrons after breakdown. In $CF 4$ plasma at low pressure, a large number of electrons are in the very low energy range, while some electrons have high energy. This is because the average electron free length is larger, suppressing attachment reactions, and ionization still dominates the internal reactions of the plasma.

As shown in Table I, due to the significantly higher electron energy in the prebreakdown stage compared to the threshold energy of adsorption reactions, the electron negative ion density ratios in the prebreakdown stage are much higher than in the postbreakdown stage. The appearance of the sheath causes rapid “cooling” of electrons in the bulk region, suppressing ionization reactions and intensifying adsorption reactions. This leads to a gradual increase in the negative ion density, replacing a portion of the electrons in the bulk region. This phenomenon becomes more pronounced at higher gas pressures. At 50 mTorr, there is a distinct electropositive behavior, at 100 mTorr, it exhibits weak electronegativity, and at pressures above 110 mTorr, it demonstrates strong electronegativity. As demonstrated in Table II, when $CF 4$ is subjected to electropositive discharge, the influence of gas pressure on the electrical characteristics of CCP is found to be limited, and the variation in equivalent capacitance, once plasma stability is reached, exhibits minimal dependence on gas pressure. On the other hand, when $CF 4$ experiences electronegative discharge, a more significant effect of gas pressure on the electrical characteristics of CCP is observed, leading to an increase in equivalent capacitance as plasma stability is achieved with increasing gas pressure.

Before breakdown occurs, the electric field on the electrode can directly accelerate the electrons, resulting in a higher electron energy. The presence of the matching circuit causes a voltage peak at 0–1  $μ$s due to the charging of the capacitor. The higher electric field signal instantaneously increases the electron energy, leading to an intense ionization reaction in the gap. This reaction significantly increases the initial electron and positive ion density, reducing the breakdown conditions and causing a substantial decrease in the voltage required to sustain plasma evolution. Typically, a high voltage at the initial moment is highly favorable for breakdown, as the breakdown voltage is often significantly higher than the discharge maintenance voltage.⁵⁶ 

The breakdown of plasma in the reaction chamber depends on the electrical signal at the supply electrode, which is determined by fixed conditions. Circuit simulations reveal that the value of the $C 2$ capacitance significantly affects the initial electrical signal in the matching circuit. For a given value of capacitance $C 1$⁠, inadequate or excessive capacitance $C 2$ can result in plasma breakdown failure. The nonlinear relationship between the $C 2$ and the electrode supply voltage complicates the selection of matching circuit parameters.

Figure 9(a) shows the breakdown of $CF 4$ plasma under varying $C 1$ and $C 2$ capacitance values in the matching circuit. In instances where $C 2$ is exceedingly large, it may be considered to be part of the wire due to its series connection with the reaction chamber circuit. Therefore, this study focuses mainly on the breakdown process for values of $C 2$ ranging from 0 to 400 pF. As shown in Fig. 9(a), a wider range of $C 2$ values is generally chosen when $C 1$ remains constant.

When $C 1$ remains constant, the variation in $C 2$ influences the voltage across the reaction chamber. Insufficient or excessive $C 2$ values can result in a voltage that is too low, ultimately causing the failure of $CF 4$ breakdown. Furthermore, alterations in the electrical characteristics of the plasma postbreakdown can induce substantial changes in the voltage across its CCP. Even minor modifications in $C 2$ within the range of 100–150 pF can cause dramatic voltage shifts. The simulation results for different capacitance values $C 2$ at a constant $C 1$ of 500 pF are presented in Fig. 10.

When $C 1$ is maintained at a constant value of 500 pF, the variation in $C 2$ exerts a direct influence on plasma breakdown. In successful breakdowns, disparate capacitance values $C 2$ significantly impact the breakdown time, while an appropriate capacitance value can substantially expedite the breakdown process and reduce the duration of the breakdown. In the absence of breakdown, various values of $C 2$ can similarly affect the plasma evolution process. For example, when $C 2$ is set to 400 pF, the electron density demonstrates an increasing trend from 0 to 0.5  $μ$s; however, after 0.5  $μ$s, the electron density within the plasma decreases rapidly and stabilizes at a density of $10 4 m − 3$ for an extended period, which differs markedly from the typical breakdown failure where the electron density descends directly.

Keeping $C 1$ at 500 pF, when $C 2$ exceeds 100 pF, the elevated capacitance charging voltage signal prompts the electron density to exhibit a growth trend during the initial phase, irrespective of whether the breakdown is successful or not. When $C 2$ is set to 50 pF, the electron density declines directly without any growth, which is attributed to the minimal fluctuation of the voltage signal throughout the capacitor charging phase. When $C 2$ is adjusted to 260 pF, the electron density increases to $10 9 m − 3$ in the initial phase; however, density growth subsequently decelerates due to reduction in voltage in the CCP. Following 12  $μ$s of evolution, the avalanche breakdown process occurs when the electron density reaches $10 12 m − 3$⁠, and its growth rate is further accelerated. In terms of electron energy, in the initial stage, the average electron energy, accelerated by the charging voltage, surges rapidly and remains at a high level until the breakdown takes place. After the breakdown, the electron energy declines swiftly as a result of the sheath’s shielding effect on the electric field. In the later evolutionary stages, the average electron energy increases as a result of the electric field heating in the bulk region.

$F −$ and $CF 3 −$ can be produced almost exclusively through the adsorption reactions of electrons with the $CF 4$ molecules. As a result, both the electron and negative ion density grow, although the latter grows at a slower rate because of the lower cross section of the attachment.

For $CF 3 +$ ions, ionization collisions are the only way to generate them. Thus, the $CF 3 +$ ion density follows a similar growth trend as the electron density. The higher capacitive charging voltage signal results in an increasing trend in $CF 3 +$ ion density during the initial stage. Due to the larger mass, the $CF 3 +$ ion loss rate is low, and the ion density decreases slowly even in the case of failure of the breakdown.

The capacitance value of $C 2$ not only induces alterations in voltage across the CCP but also affects the electrical characteristics of the plasma before and after breakdown. These modifications generate a feedback effect on the matching circuit, culminating in a change in the electrical signal, which, in turn, leads to a change in the voltage across the CCP, as shown in Fig. 11. When $C 2$ is set to 110 pF, the voltage across the CCP persistently increases to 450 V, surpassing the supply voltage after plasma breakdown, and the absorbed power of the CCP continuously increases until it stabilizes at a higher value. Conversely, when $C 2$ is adjusted to 120 pF, the voltage across the CCP initially increases but subsequently decreases following a breakdown, stabilizing at only half the maximum voltage, while the absorbed power of the CCP consistently decreases. The stable voltage and absorption probability across the CCP are lower than in the case with a capacitance value of 100 pF. When $C 2$ is set to 260 pF, plasma breakdown occurs at 12  $μ$s due to low voltage, and the voltage across the CCP progressively decreases until the breakdown occurs. The stable voltage is even lower than in the case of breakdown failure, and the presence of the matching circuit maintains the gas discharge at a reduced voltage. Following the breakdown, the supply current experiences varying degrees of reduction. When $C 2$ is adjusted to 110 pF, the supply current is markedly lower than the current at both ends of the CCP, signifying that the matching circuit protects the supply.

With a fixed $C 1$⁠, the voltage in the chamber exhibits a nonlinear variation as $C 2$ increases, as illustrated in Fig. 9(b). Initially, the voltage surges exponentially with the growth of the capacitance value but subsequently decreases exponentially upon reaching a specific threshold value. Around 100 pF, the voltage across the reaction chamber demonstrates substantial fluctuation. This variability is not conducive for simulation studies; therefore, in this article $C 1$ is maintained at 500 pF. When the voltage at both ends of the reaction chamber remains constant, typically two corresponding capacitance values for $C 2$ emerge. Furthermore, distinct capacitance values that match at an identical voltage can markly impact the breakdown process. Two of the most unique phenomena, corresponding to voltages of 87.3 and 83 V at the ends of the reaction chamber under vacuum conditions, have been selected for the analysis.

Keeping the gas pressure at 200 mTorr, keeping $C 1$ constant at 500 pF, and setting the voltage across the vacuum reaction chamber at 83 V, the corresponding capacitance values for $C 2$ are determined to be 80.7 and 263 pF. The simulation results are illustrated in Fig. 12.

When the value of $C 2$ is set to 80.7 pF, rapid electron loss occurs within the reaction chamber, leading to gas discharge failure. On the contrary, when $C 2$ is adjusted to 263 pF, a larger amplitude of the maximum voltage at the initial moment is observed, resulting in an initial electron density of $10 9 m − 3$⁠. This condition effectively suppresses electron loss and facilitates a gradual increase in particle density at reduced voltages. Gas breakdown is determined to occur at approximately 50  $μ$s, following which the absorbed power of the CCP experiences a gradual increase, which eventually stabilizes at a steady level.

When the gas pressure is maintained at 200 mTorr, $C 1$ is kept constant at 500 pF, and the voltage across the vacuum reaction chamber is set to 87.3 V, the corresponding capacitance values for $C 2$ are found to be 82 and 250 pF. The results of the simulation are illustrated in Fig. 13.

Although the voltage at the end of the reaction chamber remains constant at 87.3 V, $C 2$ affects both the charging process of the matching circuit and the absorption of plasma power. As a result of the increased charging voltage, the electron density increases exponentially within the initial 0.5  $μ$s. With $C 2$ at 82 pF, voltage fluctuations (0–1  $μ$s) at the chamber ends before breakdown reduce the evolution time prior to breakdown. At breakdown onset, voltage change becomes negligible and the pole plate current surges, causing a larger plasma to absorb power, increasing plasma density after stabilization.

For $C 2$ equal to 260 pF, a larger initial peak voltage leads to more intense ionization, higher initial particle density, significantly lower breakdown conditions, and electron avalanche failure at 4  $μ$s. Changes in electrical properties in plasma formation create a feedback effect on the electrical signal of the matching circuit. After breakdown, the CCP capacitance increases, increasing the circuit capacitance containing the CCP. As shown in Fig. 13(b), the chamber voltage declines, and the reduced voltage and absorbed power cannot maintain a high plasma density, resulting in a lower density than in the 82 pF case.

The magnitude of the capacitance value $C 2$ significantly affects the charging process of the matching circuit. At the same voltage, different matching capacitance values can even affect the occurrence of gas breakdown.

In this study, the coupling issue between the PIC/MC code and the L-matching circuit was addressed using the IVM method. This approach facilitated the simulation of the $CF 4$ gas breakdown process, driven by an RF power supply. A detailed analysis of the plasma parameters and electrical signal evolution during this process provided several critical findings. The matching circuit was found to significantly impact the breakdown process by adjusting the voltage signal at both ends of the CCP. This effect is particularly prominent during the initial discharge stage, where a brief voltage oscillation across the CCP is noted, because of the capacitor’s charging process. The larger electrical signal that is charged accelerates electrons, leading to intense ionization collisions and subsequent electron accumulation. The strength of this charging voltage, influenced by changes in the capacitance values of different capacitors, was found to substantially alter the breakdown process.

Furthermore, a notable shift in the electrical properties of the plasma before and after the breakdown was found to generate a feedback effect on the matching circuit, altering its electrical signal. The discharge pattern of $CF 4$ gas, which changes at different gas pressures, was also found to influence the electrical properties of the plasma in the matching circuit, with lower pressures resulting in a thicker $CF 4$ plasma sheath and lower equivalent capacitance. Significantly, the matching circuit was found to mitigate the effect of nonlinearity of plasma on the RF power supply circuit, reducing the current amplitude of the power supply, which provides a protective effect on the power supply.

The observed influence of the matching circuit on the breakdown process implies that a more comprehensive examination of various circuit configurations could enhance plasma control. This finding is particularly relevant for applications such as etching and deposition in the semiconductor industry, where the precision of plasma property control is paramount. Moreover, the study sheds light on the impact of gas pressure on the electrical properties of the plasma, suggesting new avenues for fine-tuning these properties to achieve desired results. Further research is warranted to investigate the feedback effect of the plasma on the matching circuit. This exploration could yield innovative strategies for circuit design, potentially enabling the development of self-regulating circuits that automatically optimize plasma conditions.

This work was supported by the National Natural Science Foundation of China (NNSFC) (Nos. 12275095, 11975174, and 12011530142) and the Fundamental Research Funds for Central Universities (No. WUT: 2020IB023).

The authors have no conflicts to disclose.

Zhaoyu Chen: Data curation (lead); Investigation (lead); Writing – original draft (lead). Jingwen Xu: Methodology (lead); Validation (lead). Hongyu Wang: Resources (lead); Supervision (equal); Validation (lead). Hao Wu: Formal analysis (lead); Software (lead); Writing – original draft (equal). Wei Jiang: Funding acquisition (lead); Supervision (equal); Writing – review & editing (equal). Ya Zhang: Conceptualization (lead); Funding acquisition (lead); Project administration (lead); Supervision (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.