The coupling effects between the bias power and the inductive power in the RF-biased inductively coupled plasma with synchronous control are investigated by measuring electron energy distribution function using a compensated Langmuir probe. With synchronous control, the inductive power and the bias power are driven at an identical phase and frequency. The experimental results show that the inductive power lowers the self-bias voltage, while the bias power changes the plasma density by introducing extra power absorption and dissipation. The bias power also enhances the electron beam confinement, leading to an increase in electron density at a low pressure. Furthermore, in the E and H mode transition, with the bias power increasing, the hysteresis power reduces, and the electron density jump decreases.

Low-temperature plasma has been widely used in semiconductor etching for its characteristic of anisotropic etching.1–6 Among various plasma generators, the inductively coupled plasma (ICP) has an advantage of high plasma density,7 while the capacitively coupled plasma provides high ion bombardment energy.8 Therefore, by applying a RF bias to a substrate immersed in ICP, high plasma density and high ion bombardment energy could be simultaneously achieved in a plasma generator, which is called ICP reactive-ion etcher or RF-biased ICP source.9–11 In a RF-biased ICP, the plasma density and the ion bombardment energy are separately controlled by the inductive power and the bias power, respectively. This separate control facilitates the application of plasma in semiconductor etching.

However, a fully separate control is difficult to achieve since strong coupling effect appears between the inductive power and the bias power. To specify, the bias power could affect the plasma density by acting like a capacitive discharge, and the inductive power could affect the ion bombardment energy by changing the DC bias in the sheath. Sobolewski et al.12 measured the electron density in a 13 MHz/13.56 MHz (13 MHz for the bias frequency, while 13.56 MHz for the inductive frequency) RF-biased ICP, and the results showed that thebias power increased the electron density at low inductive powers, while the opposite trend was observed at high inductive powers. Similar results were reported in 2 MHz/13.56 MHz,13 12.5 MHz/13.56 MHz14, and 27.12 MHz/13.56 MHz15–17 RF-biased ICPs. They pointed out that the bias power could increase the total absorbed power and cause extra power dissipation at the RF substrate, and the balance between them determines the increase or decrease in the plasma density.

Furthermore, since the drastic change in plasma density is a characteristic of the E and H mode transition in ICP, the bias power also significantly affects the process of mode transition. Zhang et al.18 investigated the mode transition in 12.5 MHz/13.56 MHz RF-biased ICP and observed that when the bias power is off, the mode transition is accompanied by a sudden jump in plasma density and hysteresis occurs between the E mode and the H mode. Conversely, when the bias power is on, the plasma density undergoes continuous change during mode transition with no hysteresis. They concluded that this can be explained by the enhanced power transfer efficiency with the bias power applied.

In the studies mentioned earlier, the frequency of the inductive power and the bias power are not the same, thus the phase difference between the two powers is not considered. The frequency difference results in a slow drift between the two powers, thus their phase difference covers the whole range from 0° to 360° periodically, i.e., they perform an average of discharges at all phase differences. However, when the inductive power and the bias power have an identical frequency, their phase difference is locked which will significantly affect the discharge state.19–22 As pointed out by Ahr et al.22 the phase difference between the inductive power and the bias power in 13.56 MHz/13.56 MHz RF-biased ICP greatly affects the electron beam confinement, particularly at low pressures. For small phase differences around 0°, a good quality of electron beam confinement is achieved, which could significantly enhance the electron–sheath interaction and the excitation rate. Therefore, for better application of the RF-biased ICP on semiconductor etching, addressing the coupling effect between the inductive power and the bias power with identical frequency and phase is of practical importance.

In this paper, we established an inductive discharge system with the synchronous RF bias, i.e., RF-biased ICP with synchronous control. In this system, the inductive power and the bias power are driven by the same signal but are amplified independently. Therefore, they have an identical frequency and phase but different amplitudes. The coupling effect is elaborated by studying the dependence of the inductive power on the self-bias voltage and the dependence of the bias power on the electron density. The influence of the bias power on the E and H mode transition is also discussed.

Figure 1 shows the schematic diagram of the RF-biased ICP with synchronous control, equipped with a compensated Langmuir probe. The plasma is contained in a cylindrical chamber (300 mm in height and 300 mm in diameter). The distance between two electrodes is 10 cm. The top electrode includes a quartz dielectric plate with a thickness of 12 mm and a two-turn planar copper antenna with a maximum diameter of 200 mm and a minimum diameter of 160 mm. The bottom electrode is made of stainless steel with a diameter of 200 mm. To avoid sparking in the gaps between the electrode and the ground, the bottom electrode is connected via Teflon sheeting to the electrical ground. The top antenna and the bottom electrode are powered with two separate 13.56 MHz RF power supplies shared with the same crystal oscillator. As a result, the two power supplies output RF signal at identical phase and frequency but at different amplitudes, realizing inductive discharge with synchronous RF bias. The chamber is pumped with a turbomolecular pump and a mechanical pump to make sure the background pressure is below 5 × 10−3 Pa before discharges. Pure Ar (>99.999%) is introduced into the chamber via a mass flow controller. For a more pronounced influence of the synchronous control, the discharge pressure is set to be 0.4 Pa in majority of the measurements, except for some special situation where the pressure is 20 Pa.

FIG. 1.

Schematic diagram of the RF-biased ICP with synchronous control equipped with a compensated Langmuir probe.

FIG. 1.

Schematic diagram of the RF-biased ICP with synchronous control equipped with a compensated Langmuir probe.

Close modal

The self-bias voltage ( V bias) between the substrate and grounded electrode is measured with a voltmeter connected via 9 M Ω resistor and 10 mH inductor to the substrate. The compensated Langmuir probe settled at the center of chamber in the measurements is used to measure electron densities ( n e), effective electron temperatures ( T e), and electron energy probability functions (EEPFs), respectively. The probe tip is made of a cylindrical tungsten wire with a radius of 120  μm and a length of 10 mm and its holder is made of a 30 mm long quartz tube with an outside diameter of 200  μm. The radius of the probe tip and its holder are at least one order of magnitude smaller than the electron mean free path23–25 (about 2.2 mm at the highest discharge pressure of 20 Pa). Two groups of parallel LC filters are tuned to 13.56 and 27.12 MHz to diminish RF distortion. To further reduce RF distortion, a copper ring (300  μm in wire diameter and 30 mm in ring radius) is centered around the probe tip and parallelly connected with the measuring circuit via a 100 pF capacitor.

The EEPF f ( ε ) is obtained from the second derivative of the I–V characteristic curve,24–27,
f ε d 2 I P d U P 2 ,
(1)
where I P is the probe current, U P is the potential difference between probe and plasma potential, and ε = e U P is the electron energy. The electron energy distribution function F ε, n e, and T e are given in the following equations:24–27 
F ε = ε 1 2 f ε ,
(2)
n e = 0 ε max F ε d ε ,
(3)
T e = 2 3 n e 1 0 ε max ε F ε d ε .
(4)

Figure 2 shows the evolution of V bias with the bias power ( P bias) at different inductive power ( P ind). It is found that for all P ind, with increasing P bias, V bias experiences a slight drop initially and then keeps growing after that. Moreover, Fig. 2 also shows that increasing P ind causes a continuous decrease in V bias. These indicate that V bias is positively correlated with P bias but negatively correlated with P ind basically.

FIG. 2.

Evolution of V d c with bias power at different inductive powers.

FIG. 2.

Evolution of V d c with bias power at different inductive powers.

Close modal
The evolution of V bias with P ind could be understood by circuit analysis. In typical discharges, V bias is linearly correlated with V r f ( V bias = A s A g A s + A g V r f),8 where V r f is the amplitude of RF voltage applied to the plasma, and A s and A g are the area of substrate and ground, respectively. The equivalent circuit of RF-biased ICP system are shown in Fig. 3. C s h, R s h, R P, and L P are the sheath capacitance, the sheath resistance, the plasma resistance, and the plasma inductance, respectively. These parameters could be calculated with the following equations:8,28
C s h = 0.613 ε 0 A s s ,
(5)
s = 2 3 λ D 2 V 0 T e 3 4 ,
(6)
R s h = V 0 0.61 e n i μ B A s ,
(7)
R P = L σ d c ,
(8)
L P = R P ν m ,
(9)
where ε 0, s, λ D, V 0, μ B, L, σ d c, and ν m are the vacuum permittivity, the sheath thickness, the Debye length, the potential drop across the sheath ( V 0 V P + V bias), the Bohm velocity, the distance between the two electrodes, the plasma conductivity and the electron-atom collision frequency. The total absorbed power ( P a) by plasma could be estimated as follows:8,
P a = 1 2 I r f V r f cos φ ,
(10)
I r f = V r f R D 2 + X D 2 1 2 ,
(11)
φ = arctan X D R D ,
(12)
X D = I m 1 i ω C s h R s h + 1 + ω L P ,
(13)
R D = R e 1 i ω C s h R s h + 1 + R P ,
(14)
where X D, R D, φ, I r f , and ω are the total reactance, the total resistance, the phase difference between I r f and V r f, the amplitude of RF current flowing through plasma, and the bias power frequency, respectively. Ignoring the power consumption by matching circuit, the bias power is entirely absorbed by the plasma ( P a = P bias) and V r f, and V bias could be estimated as follows:8 
V bias V r f = R D 2 + X D 2 1 4 2 P bias / cos φ .
(15)
FIG. 3.

Equivalent circuit of RF-biased ICP system.

FIG. 3.

Equivalent circuit of RF-biased ICP system.

Close modal

Comparison between the calculated and measured V bias with n e is shown in Fig. 4. The calculated results mostly agree with the measured results except for that at a low P ind. The reason for this discrepancy is not very clear.

FIG. 4.

Calculated V bias with n e in comparison with the measured V bias.

FIG. 4.

Calculated V bias with n e in comparison with the measured V bias.

Close modal

To trace the evolution of electron density and understand electron heating mechanism, effective EEPF measurements are conducted. Figures 5(a)–5(c) show the evolution of different types of distributions with P ind ranging from 0 W to 100 W. The pure capacitive discharges are set at three different atmospheres, low pressure and low power in Fig. 5(a), low pressure and high power in Fig. 5(b), and high pressure and low power in Fig. 5(c). With increasing P ind gradually, all the EEPFs finally evolve to Maxwellian distribution.

FIG. 5.

Influence of inductive power on EEPFs: (a) 0.4 Pa, P bias= 60 W, (b) 1.0 Pa, P bias= 400 W, and (c) 20.0 Pa, P bias = 100 W. (d)–(f) Influence of inductive power on n e and T e, respectively.

FIG. 5.

Influence of inductive power on EEPFs: (a) 0.4 Pa, P bias= 60 W, (b) 1.0 Pa, P bias= 400 W, and (c) 20.0 Pa, P bias = 100 W. (d)–(f) Influence of inductive power on n e and T e, respectively.

Close modal

According to the Eqs. (3) and (4), the calculated n e and T e are shown in Figs. 5(d)–5(f). It is found that a turn for n e and T e occurs as P ind reaches to be 40 W. After that, T e becomes either nearly constant or suddenly decreases, while n e experiences an exponential increase with inductive power due to E–H transition. The exponential increase in n e and thus enhancing e-e collision frequency ν e e ( n e / T e 3 / 2 ) is the main cause for EEPFs to be Maxwellianized in Figs. 5(a)–5(c).

The initial increase in T e in Figs. 5(d) and 5(e) with P ind is because the low temperature energy group confined in the dc ambipolar potential well can gain energy through collisionless heating in the skin layer formed by inductive coupling.29 While at a high pressure in Fig. 5(f), a sudden decrease taken by T e is observed, which may be a consequence of enhanced ν e e together with reduced plasma RF field E 0 J / n e.24 

The calculated n e with increasing P bias(0–150 W) at different P ind(40–120 W) are shown in Fig. 6. We can find that for low P ind, n e grows almost linearly with P bias. With P ind increasing to more than 60 W, n e experiences an initial abrupt increase and then a slow increase and even a slight decrease for P ind more than 120 W. Additionally, the turning point for the change of n e seems to shift toward right as shown by the dashed line in Fig. 6.

FIG. 6.

Evolution of electron density with bias power at different inductive powers.

FIG. 6.

Evolution of electron density with bias power at different inductive powers.

Close modal

The initial abrupt increase in n e found in our experiment is not observed in other RF-biased ICP without the synchronous control.13,14 One of the possible explanations is as follows. The difference between the RF-biased ICPs with and without the synchronous control is their ability of electron beam confinement.22 The electron beam interacts with the expanding sheath to gain energy to excite or ionize atoms. However, at pressures of less than a few Pa, the electron beam could not experience enough collision to consume the energy obtained from one sheath and will pass the bulk plasma to the opposite sheath. In such a situation, if the electron beam encounters the opposite collapsing sheath, they could overcome the sheath voltage and vanish at the electrode, while if the electron beam encounters the opposite expanding sheath, they could be almost confined in the bulk plasma and gain energy via the mechanism of bounce resonance heating (BRH).30 Without the synchronous control, the electron beam will eventually encounter the opposite collapsing sheath and vanish in a long period, while with the synchronous control, the electron beam always encounters the opposite expanding sheath, resulting in good electron beam confinement.

The slow increase or slight decrease in n e could be understood with a simple global model,8,11,13 in which the plasma density can be described as follows:
n 0 = P ind + P bias e u B A eff , g ε T , g + A eff , s ε T , s ,
(16)
where n 0, u B, A eff , g, A eff , s, ε T , g , and ε T , s are the plasma density, the Bohm velocity, the effective area for particle loss at the grounded wall and the substrate, and the total energy loss at the grounded wall and the substrate, respectively. From this equation, dependence of n 0 on P ind and P bias is not a simple linear relation, since ε T , s is basically positive relative to P bias under the collisionless regime. Initially, at low P ind, the increased P bias mainly contributes to the slow increase in n e due to the biased power absorption of electron through the oscillating sheath. With increase in P ind, P bias will reduce ne by increasing the ε T , s to be much higher than the ε T , g. This means that the ions vanishing at the biased electrode is of much higher energy than that vanishing at the grounded wall, resulting in extra power dissipation.

As is known, a typical characteristic of inductive discharges is the two distinct operation modes: the capacitive mode (E mode) and the inductive mode (H mode).18 At lower plasma densities, the discharges are primarily governed by the capacitive coupling, sustaining plasmas through the axial electric field. Conversely, at higher plasma densities, the discharges are dominated by the inductive coupling, where plasmas are maintained by the axial magnetic field and the azimuthally induced electric field. During the E and H mode transition, the discharge is always accompanied by an abrupt change in plasma density, and hysteresis occurs between the E mode and the H mode transition, particularly at high pressures.31 

In RF-biased ICP, mode transition exhibits different characteristics from a pure inductive discharge. Figure 7 illustrates the evolution of electron density during the E and H mode transition at different P bias. Figure 7(a) shows a pure inductive discharge, in which a distinct hysteresis loop is observed with a hysteresis power of approximately 4 W. With increasing P bias, shown in Figs. 7(b)–7(e), this hysteresis shrinks until it almost vanishes (around 2 W for P bias = 20 W, 1 W for P bias = 30 W, and almost no hysteresis for higher P bias). Here, we define η = K H / K trans to describe the electron density jump during the mode transition, where K H and K trans are the fitted slopes of electron density with respect to P ind in the H mode and E-to-H mode transition. For a sharper electron density jump, η is closer to zero. As shown in Fig. 7(f), η grows with P bias, which means that the electron density jump during the mode transition gradually becomes less sharp. In general, the effects of increasing bias power during the mode transition seem to reduce the hysteresis power and obscure the electron density jump.

FIG. 7.

Electron density evolution during mode transition under 20 Pa at different bias powers: (a) 0 W, (b) 20 W, (c) 30 W, (d) 40 W, (e) 60 W, and (f) is the calculated η for different bias powers.

FIG. 7.

Electron density evolution during mode transition under 20 Pa at different bias powers: (a) 0 W, (b) 20 W, (c) 30 W, (d) 40 W, (e) 60 W, and (f) is the calculated η for different bias powers.

Close modal

Taking into account the nonlinearity of both the absorbed and dissipated power,18,31–34 the energy balance equation8 can be used to explain the mode transition in inductive discharges. The nonlinearity of the absorbed power is primarily caused by the capacitive coupling of the inductive coil,34 while the multi-step ionization is considered to be a contributing factor to the nonlinearity of the dissipated power.18 

To account for the impact of the bias power on the capacitive coupling and the multi-step ionization, the calculated absorbed power including the capacitive coupling and the dissipated power including the multi-step ionization have been plotted in Figs. 8(a)–8(c) and 9(a)–9(c) (details are provided in the  Appendix). In each figure, the blue curve with two peaks represents the absorbed power, in which the left and right peaks are the capacitive coupling and the inductive coupling, respectively. The red curve represents the dissipated power, and the intersections of the above two curves indicate possible discharge states. Additionally, I denotes the discharge current in the coil. V c is the potential difference between the two ends of the inductive coil, and the contribution of the bias power on the capacitive coupling is expressed as an extra value of V c. ν 0 i and ν m i are the ionization rates from the metastable and the ground levels, respectively, and the impact of the bias power on promoting direct ionization is introduced as an increase in ν 0 i / ν m i. The impacts of the bias power on the capacitive power coupling and the ionization process are separately illustrated in Figs. 8 and 9.

FIG. 8.

(a)–(c) Schematics of absorbed and dissipated power considering capacitive coupling and multi-step ionization at different bias power. The effect of bias power on capacitive coupling is introduced. (d)–(f) Calculated electron density during mode transition from (a)–(c).

FIG. 8.

(a)–(c) Schematics of absorbed and dissipated power considering capacitive coupling and multi-step ionization at different bias power. The effect of bias power on capacitive coupling is introduced. (d)–(f) Calculated electron density during mode transition from (a)–(c).

Close modal
FIG. 9.

(a)–(c) Schematics of the absorbed and dissipated power considering capacitive coupling and multi-step ionization at different bias power. The effect of bias power on the ionization process is introduced. (d)–(f) are the calculated electron density during mode transition from (a)–(c).

FIG. 9.

(a)–(c) Schematics of the absorbed and dissipated power considering capacitive coupling and multi-step ionization at different bias power. The effect of bias power on the ionization process is introduced. (d)–(f) are the calculated electron density during mode transition from (a)–(c).

Close modal

In Fig. 8, the application of the bias power is represented as multiples of V c. For a pure inductive discharge, Fig. 8(a), within an increasing process of I from zero, n e grows until reaching point P2 and abruptly changes to P4, indicating the E-to-H transition. Then, it begins to reduce I until n e decreases to P3 and abruptly jumps to P1, signifying the H-to-E transition. This sequence creates a hysteresis loop represented by the points P1-P2-P4-P3-P1. In the absence of the bias power, Figs. 8(a) and 8(d), η is about 0.05, and the hysteresis power Δ I indicated by the discrepancy between the two critical values of I is about 0.053 A. When a weak bias power is introduced, Figs. 8(b) and 8(e), η increases to 0.08, and Δ I is reduced to 0.023 A. With a further increase in the bias power, Figs. 8(c) and 8(f), η reaches 0.54 and the hysteresis loop vanishes. These results agree with the results of our experiment.

Similarly in Fig. 9, with increasing bias power (represented by multiples of ν 0 i / ν m i), η increases from 0.05, 0.1 eventually to 0.63, meanwhile Δ I reduces from 0.053 to 0.043 and finally vanishes. However, the growing critical I for E-to-H transition, as shown in Fig. 9, disagrees with our experimental results; thus, we suppose that the enhancement in capacitive coupling with the bias power may be more important on mode transition for the RF-biased ICP.

In this paper, the coupling effects between the bias power and the inductive power in the RF-biased inductively coupled plasma with the synchronous control is investigated using a compensated Langmuir probe. In the RF-biased inductively coupled plasma, the bias power P bias and the inductive power P ind are expected to independently control the bias voltage V bias and the electron density n e, respectively. However, strong coupling effects are found that V bias is dependent on P ind and n e is also related to P bias. Additionally, P bias could considerably influence the E and H mode transition. Findings can be summarized in the following three aspects:

  • Regarding the dependence of V bias on P ind, V bias is lower at higher P ind. This can be explained by circuit analysis. Higher P ind provides higher n e, which can change the sheath impedance. As a result, the RF voltage ( V r f) supplied are accordingly varied. To specify, the RF voltage and V bias ( V r f) decreases with n e increasing.

  • P bias modifies the electron density. The reasons can be threefold. First, at low P bias, the synchronous control causes an abrupt increase in n e due to the enhancement of electron beam confinement. Second, P bias provides extra capacitive power absorption, which enhances the ionization near the sheath and thereby the increase of n e. Third, the ions are accelerated in the sheath and absorbed at the substrate. This process causes significant power dissipation, leading to the decrease of n e. The balance between the extra power absorbtion and the power dissipation determines whether n e will increase or decrease with P bias increasing.

  • The E and H mode transition is found to be significantly influenced by P bias based on two aspects, i.e., the hysteresis power reduces and the electron density jump decreases with P bias increasing. With P bias off, the hysteresis power is about 4 W, and the electron density jump is pronounced ( η 0.04). While as P bias gradually increases from 20 to 30 to 60 W, the hysteresis power decreases from 2 to 1 to 0 W. Simultaneously, the electron density becomes less sharp ( η 0.02 , 0.14 , and 0.66). By using a global model, the influence of P bias on the E and H mode transition can primarily be explained by the increased capacitive coupling rather than the weakened multi-step ionization.

This work was financially supported by the National Natural Science Foundation of China (Grant No. 12305221).

The authors have no conflicts to disclose.

Yi He: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Manting Lu: Investigation (supporting); Resources (equal); Visualization (supporting); Writing – review & editing (supporting). Xue Liu: Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Jiamin Huang: Methodology (supporting); Writing – review & editing (supporting). Jiawei Zhang: Investigation (supporting); Resources (supporting); Writing – review & editing (supporting). Xiaoping Ma: Visualization (supporting); Writing – review & editing (supporting). Lei Huang: Software (supporting); Writing – review & editing (supporting). Liang Xu: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Yu Xin: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal).

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

The total absorbed power consists of a capacitive branch and an inductive branch. In the capacitive branch, the field components are considered to be E = { E r , 0 , E z } and B = { 0 , B θ , 0 }, while that in the inductive branch are E = { 0 , E θ , 0 } and B = { B r , 0 , B z }. The appropriate Maxwell's equations are as follows:
· E = 0 , × E = i ω B , · B = 0 , × B = μ 0 J + i ω ϵ μ 0 E J = σ E , ,
(A1)
where ω, μ 0, ϵ, and σ are the discharge frequency, the vacuum permeability, the plasma or quartz dielectric constant, and the plasma or quartz conductivity, respectively. The boundary conditions for capacitive branch is that E r r , 0 = 0 , E z R , z = 0 and E r r , L + D = V c / R, where L, R, D , and V c are the height and radius of the chamber, the height of the quartz window and the potential difference between the two ends of the inductive coil, respectively. The boundary conditions for inductive branch is that E θ r , 0 = 0 , E θ R , z = 0 , B r r , L + D + B r r , L + D = μ 0 N I / R and E θ r , L + D + = E θ r , L + D , where N and I are the number of turns in the coil and the coil current.
The expressions for each component in a planar coil plasma reactor have been developed by El-Fayoumi et al.34,35 The total absorbed power ( P abs) is given by integrating the Poynting vector over the surface area of the interface between plasma and the dielectric plate,
P abs = P cap + P ind = 2 π μ 0 0 R r E r r , L B θ * r , L d r + 2 π μ 0 0 R r E θ r , L B r * r , L d r ,
(A2)
where L, R , and μ 0 are the height and radius of the chamber and the vacuum permeability. The dissipated power ( P diss) is calculated as follows:
P diss = e n e μ B A eff ε T ,
(A3)
where n e, μ B, ε T , and A eff are the electron density, the Bohm velocity, the total energy loss, and the effective area for particle loss, respectively. The total energy loss ε T consists of the mean kinetic energy loss per electron loss ε e( = 2 T e), the mean kinetic energy loss per ion loss ε i( = 5.2 T e),8 and the collisional energy loss ε c considering multi-step ionization. Intersections of the two curves are possilble working points, and the relationship between P abs and P diss for a stable working point is as follows:
P abs n e < P diss n e .
(A4)
The collisional energy loss ε c considering the multi-step ionization has been developed by Lee et al.36,37 At low electron density, ε c consists of a single-step ionization, excitation, and elastic collision and is dominated by the single-step ionization. While at high electron density, ε c is dominated by a multi-step ionization, thus is greatly reduced. The ratio of the ionization rates ν m i and ν 0 i from the metastable and ground levels is as follows:38,39
ν m i ν 0 i = n e k m 0 τ m n e k m 0 τ m + 1 g m g 0 ε 0 i ε m i 2 ,
(A5)
where k m 0, τ m, g m, g 0, ε m i, and ε 0 i are the rate constant for de-excitation of metastable levels, the characteristic time for the loss of excited species to the walls, the degeneracies, and the ionization energies from the ground and metastable levels, respectively. The collisional energy loss ε c is expressed as a combination of the collisional energy loss ignoring multi-step ionization ε c 0 and the collisional energy loss considering multi-step ionization ε c m,
ε c = ν m i ε c m + ν 0 i ε c 0 ν m i + ν 0 i .
(A6)
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