# Variations of plasma potential in RF discharges with DC-grounded electrode

Paul Hiret, Patrik Tognina, E Faudot, Roland Steiner, Artem Dmitriev,

Laurent Marot, Ernst Meyer

► To cite this version:

Paul Hiret, Patrik Tognina, E Faudot, Roland Steiner, Artem Dmitriev, et al. Variations of plasma potential in RF discharges with DC-grounded electrode. Plasma Sources Science and Technology, 2024, 33 (7), pp.075019. 10.1088/1361-6595/ad6691. hal-05018134

HAL Id: hal-05018134

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# Variations of plasma potential in RF discharges with DC-grounded electrode

To cite this article: Paul Hiret et al 2024 *Plasma Sources Sci. Technol.* 33 075019

View the [article online](#) for updates and enhancements.

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# Variations of plasma potential in RF discharges with DC-grounded electrode

Paul Hiret<fnref n="1" />,* ID, Patrik Tognina<fnref n="1" />, Eric Faudot<fnref n="2" />, Roland Steiner<fnref n="1" />, Artem Dmitriev<fnref n="1" /> ID,
Laurent Marot<fnref n="1" /> ID and Ernst Meyer<fnref n="1" /> ID

E-mail: paul.hiret@unibas.ch

Received 19 December 2023, revised 24 June 2024

Accepted for publication 23 July 2024

Published 31 July 2024

<figure><img src="image_2.png" /></figure>

CrossMark

## Abstract

The self-bias voltage $V_{bias}$ and the plasma potential $V_p$ are the key parameters to control plasma-wall interactions in radio-frequency (RF) asymmetric plasma discharges. Knowing these two parameters allows to monitor the ion energy distribution on the electrode surface. However, in a direct current (DC) coupled plasma, the plasma potential increased to hundreds of volts while no more $V_{bias}$ developed on the driven electrode. In addition, the plasma potential is strongly impacted by the electrode-wall area ratio. Several analytical or semi-analytical models exist to explain this phenomenon and to approximate the plasma potential, knowing the electrode-wall area ratio considering capacitive sheaths or a combination of resistive and capacitive sheaths. The $V_p$ and several other plasma parameters were investigated experimentally with a Langmuir probe and a Retarding field energy analyser for different electrode/wall area ratios in low-temperature RF plasma. The validity of the models was studied for an extensive range of area ratios with different pressures and RF driving amplitude. Moreover, the presence of strong stochastic heating of the plasma for area ratios below 6 was evidenced by an increased ion flux and an increased electron temperature.

Supplementary material for this article is available online

Keywords: RF plasma, ion flux, plasma potential, DC-coupled plasma, notch-filter

## 1. Introduction

Controlling plasma-wall interactions in radio-frequency (RF) asymmetric plasma discharges is the main concern in driving plasma etching in the semiconductor industry [1]. Similar etching processes are foreseen to be used for removing contaminants on the first mirrors (FM) in optical diagnostics for the next generation of fusion reactors in order to recover

their initial optical efficiency [2, 3]. Such discharges are commonly driven with an electrode isolated from the ground via a blocking capacitor, which charges up negatively in asymmetric discharges leading to the formation of a self-bias on the substrate [1]. The self-bias voltage has an influence on the energy distribution of the ions on the surfaces as it contributes to the voltage drop into the sheath. In such asymmetric plasma discharges, the plasma potential ($V_p$) and the self-bias ($V_{bias}$) are then the main parameters to control the plasma-wall interactions. Knowing these two parameters allows for controlling the ion energy distribution on the electrode surface. The self-bias depends strongly on the area ratio between the driven electrode and the grounded wall. This dependency has been extensively studied for asymmetric capacitively-coupled plasma (CCP) [4–7].

<figure><img src="image_3.png" /></figure>

Original Content from this work may be used under the terms of the [Creative Commons Attribution 4.0 licence](https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote n="1">¹Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland</footnote>

<footnote n="2">²Institut Jean Lamour, Université de Lorraine, 2 allée André Guinier, BP 50840, 54011 Nancy Cedex, France</footnote>

<footnote>* Author to whom any correspondence should be addressed.</footnote>
However, in the next generation of fusion reactors, FMs would also be exposed to high thermal loads and require active water-cooling. The grounded metallic water-cooling pipes would be in contact with the mirrors and cannot be physically decoupled from the mirrors leading to RF grounding. However, the water cooling pipes would be installed in the λ/4 scheme to decouple the RF from the ground [8, 9]. In addition, such DC-coupled antennae are foreseen to be used to increase the plasma temperature in fusion devices. Indeed, RF electromagnetic waves in the ion cyclotron range are absorbed by the plasma leading to an increase of the ion temperature [10]. In DC-coupled plasmas, the plasma potential increased to hundreds of volts while no more DC bias developed on the driven electrode as reported in our previous paper by Soni et al [9]. In such discharges, the sheath average potential drop is the same in front of the electrode and in front of the wall for non-resistive plasma, which is almost the case in low-pressure discharges (≤1 Pa). That leads to a homogeneous ion energy distribution on both surfaces (driven and grounded). In addition, the plasma potential is strongly impacted by the electrode-wall area ratio [11]. Several analytical or semi-analytical models exist to explain this phenomenon and to approximate the plasma potential knowing the electrode-wall area ratio considering capacitive sheaths [12] or a combination of resistive and capacitive sheaths [11, 13]. Köhler et al showed that considering a capacitive model the plasma potential depends only on the sheath capacitance ratio and the RF voltage amplitude [12]. The DC plasma potential increases with the wall-electrode area ratio to reach the RF voltage amplitude ($V_{RF}$) applied to the electrode when the wall area ($A_g$) is much larger than the powered area ($A_p$). Moreover, for constant ratio, $V_p$ was depending linearly on the $V_{RF}$. The analytical model developed by Aanesland et al used a combination of capacitive and resistive sheaths. This allows the DC current to flow from the plasma to the ground via the electrodes [11]. The plasma potential decreased from RF amplitude at area ratios $1 \le A_g/A_p \le 10$ to an asymptotic voltage of around 15.5 V for argon plasma with an electron temperature of 3 eV for $A_g/A_p > 100$. It was also shown that for a small area ratio, there was a linear dependency between the RF voltage amplitude and the plasma potential, whereas, for a large area ratio, $V_p$ was almost constant while $V_{RF}$ was increasing. Experimentally, they measured a plasma potential saturating while $V_{RF}$ was increasing for a large area ratio, which was consistent with their model predictions. Similarly, Faudot et al showed that the plasma potential was saturated for large area ratio while $V_{RF}$ was increasing [14]. They also compared the average RF sheath thickness at the DC-grounded electrode to different sheath models. The measured RF sheath thickness was at least three times higher than that calculated with the Child-Langmuir law in a low collisional helium plasma. Moreover, the Wang semi-analytical model [15] predicted similar RF sheath thickness magnitude. Recently, Faudot published a semi-analytical model considering both capacitive and resistive sheaths [16]. The ‘double saturated probe’ (DSP) model takes into account the electron saturation region, implying that the sheath can reverse as soon as the applied potential is strongly positive. The model showed similar results to Aanesland’s model in the case of a high

capacitive sheath. The plasma potential was decreasing while the area ratio increased for sufficiently large ratios. However, contrary to Aanesland’s model, for a low area ratio (lower than 5), the plasma potential increased with the area ratio from $V_{RF}/\pi$ to a maximum value close to $V_{RF}$. The particle-in-cell (PIC) simulations from Faudot et al [13] showed a wider collapsing region of the plasma potential for a large area ratio. However, for small area ratio ($A_g/A_p < 5$), experimental characterisation of the plasma potential magnitude, to our best knowledge, has not been studied compared to larger ratio as results of experimental plasma chambers with larger wall area than electrode area are commonly used.

Plasma potential dependency on the electrode-wall area ratio for DC-grounded electrodes has been studied theoretically through different analytical and semi-analytical models considering capacitive sheaths or a combination of resistive and capacitive sheaths however these models showed some discrepancy for several electrode-wall area ratio ranges. In this contribution, the experimental characterisation of the plasma potential magnitude in the case of DC-coupled plasma for an extensive range of area ratios is discussed.

## 2. Plasma models

### 2.1. Analytical models

Two analytical models depict the plasma potential evolution as a function of the electrode-wall area ratio. In the Köhler et al model, only capacitive sheaths are considered. In the case of a net current is possible through the powered electrode (i.e. DC-grounded electrode), the mean plasma potential $V_p$ is given by:

$$V_p = V_{bias} + V_{RF} \frac{C_g}{C_p + C_g} \quad (1)$$

where $C_g$ and $C_p$ are the time-averaged sheath capacitances at the grounded wall and the powered electrode, respectively [12]. If the powered electrode is DC-grounded, $V_{bias}$ is zero, and the ratio of capacitance is proportional to the area ratio, the equation reduces to:

$$V_p = V_{RF} \frac{A_g}{A_p + A_g} \quad (2)$$

In addition to the Köhler et al model, Aanesland’s formula considers plasma sheath to be a combination of capacitive and resistive sheath to calculate the plasma potential from RF potential $V_{RF}$ and surface ratio $A_g/A_p$

$$\cos^{-1}\left(\frac{V_p}{V_{RF}}\right) = \pi \left[ \frac{j_e}{j_i} - \exp\left(-\frac{eV_p}{k_B T_e}\right) \right] \frac{A_g}{A_p} \quad (3)$$

with $j_e/j_i = 0.61\sqrt{2\pi m_e/m_i}$ the electron to ion current density ratio, $m_i$ and $m_e$ are respectively the masses of an ion and of an electron, e is the charge of an electron, and $T_e$ the electron temperature. For further information, refer to [11].
## 2.2. Semi-analytical: DSP

The model simplifies by assuming Maxwellian velocity distributions for electrons and ions with constant temperature and a Boltzmann electron density in the sheath. Additionally, for the oscillations induced by the radio frequency (RF) field, the model uses the frequency $\omega_{RF} = 2\pi f$, where f is the RF frequency. This RF frequency is set to be higher than the ion plasma frequency, denoted as $\omega_{pi} = \sqrt{n_0e^2/m_i\epsilon_0}$ with $n_0$ the overall density of the plasma, and $\epsilon_0$ is the vacuum permittivity. By choosing $\omega_{RF} > \omega_{pi}$, the model effectively ignores the ions' response to the electric field. It disregards collisions in low-pressure sheaths, uses an electrostatic field solver, treats the plasma as a homogeneous resistive dipole, and models sheath conduction and displacement currents with a diode equation and a time-constant capacitance, respectively, while considering the plasma's contribution as an ohmic resistance and the external circuit's as another series resistance. In the case of the asymmetric electrode and grounded wall areas, each sheath drives different amounts of current proportional to their respective areas. However, current conservation is still verified. Applying this current conservation to this asymmetric DSP model with capacitive sheath described in figure 1, with $\phi_p = (eV_p/k_B T_e)$, $\phi_{RF} = (eV_{RF}/k_B T_e)$, and the hyperbolic tangent function for electron current yields:

$$c_p \frac{d\phi_{RF}}{dt} - (c_p + c_g) \frac{d\phi_p}{dt} = j_i (A_p + A_g) - \frac{j_e}{2} (A_p + A_g + A_p \tanh(1 - \phi_p + \phi_{RF}) + A_g \tanh(1 - \phi_p)) \quad (4)$$

where $c_p$ and $c_g$ are, respectively, the powered and the grounded sheath capacitance, depending on the sheath thickness, as follows:

$$c_{g,p} = \frac{\epsilon_0 \epsilon_{g,p}^s A_{g,p}}{\lambda_s} \quad (5)$$

where $\epsilon_0$ is the vacuum permittivity and $\epsilon_{g,p}^s$ is the sheath relative permittivity considered equal to 1 when $V_{RF} \gg T_e$ which is a case for these discharges. The sheath thickness $\lambda_s$ depends on the potential drop $V_s(t) = V_p(t) - V_{RF}(t)$ (on the grounded wall $V_{RF} = 0$) through the sheath as follows:

$$\lambda_s(t) = \alpha \lambda_{De} \left( \frac{V_s(t)}{T_e} \right)^{\frac{3}{4}} \quad (6)$$

where $\lambda_{De}$ the Debye length and $0 \le \alpha \le 3$ is the sheath model parameter [1, 13]. More details about sheath thickness parameters $\alpha$ are given in section 4.1. For further details about the DSP model, refer to [13, 16].

## 2.3. PIC

The MATLAB PIC code uses efficient algorithms for particle dynamics and a Fast Fourier Transform-based Poisson solver for large grids [13]. It has been validated for accuracy with xooopic [13, 17]. It prioritises computational efficiency without collision/ionisation modelling. More details about the PIC code are available in [13].

<figure><img src="image_4.png" /><figcaption>Figure 1. Electric equivalent circuit of the double saturated probe (DSP) model.</figcaption></figure>

## 3. Experimental setup

DC-coupled plasma potential measurements were carried out in a vacuum chamber with a background pressure of around $10^{-5}$ Pa. A height-variable cylindrical vessel was divided into two electrodes (one grounded (numbered 7 in figure 2) and the other driven (numbered 1 in figure 2)). In between these electrodes were added different copper rings (numbered from 2 to 6 in figure 2)), which were connected to the grounded electrode or the driven electrode through metallic balls to increase the area of the grounded part ($A_g$) and of the powered part ($A_p$). To modify the area ratio ($A_g/A_p$), rings could be added or removed, and their polarity could be changed. That setup allowed studying an extensive number of electrode wall area ratios ($A_g/A_p$) from 0.01 to 180. An example of two possible area ratios is given in figure 3, where the left area ratio is an example of the connection that could be done with all the rings drawn in the CAD model (figure 2). In this case, the area ratio was above 1. However, while changing the polarity of ring 5 and removing the 4, $A_g/A_p < 1$ was reached (right example in figure 3).

Argon (Ar) and helium (He) plasmas were generated with a 13.56 MHz RF generator (Huttinger Elektronik GmbH and Co KG, model PFG 300 RF) coupled with a matchbox (Huttinger Elektronik GmbH and Co KG, model PFM 1500A). The strong self-bias that develop naturally in asymmetric CCP plasma was suppressed by using a quarter-wavelength notch-filter (NF) made of an RG358-type coaxial cable short-circuited at one end while the other end was connected to the RF feed line using a T-connector. The cable length (3.64 m) was a quarter-wavelength of the RF frequency (13.56 MHz in our case), taking into account a reduction factor of the propagation speed in the cable compared to the vacuum.

The plasma potential, the electron density ($n_e$) and temperature ($T_e$) were probed using an RF-compensated ESPion Langmuir probe from Hiden Analytical Ltd with a tungsten tip of 10 mm long and 0.075 mm radius. The I-V characteristics were collected in the plasma bulk between the two electrodes. The typical error on $V_p$ measurements with ESPion Langmuir probe is $\pm 20\%$ [18]. A Semion retarding field energy analyser (RFEA) from Impedans Ltd [19] located on the grounded wall was measuring the ion current flux ($j_i$) and the ion energy ($E_i$)
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Experimental setup. The movable copper electrode parts are numbered from 1 to 7.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Example of electrode arrangement for two different area ratios. The electrode in red is driven, whereas the electrode in black is grounded. The numbers correspond to the movable copper parts numbered in figure 2.</figcaption></figure>

to the wall. From $E_i$ measured by the RFEA, the $V_p$ were evaluated, such as in plasma discharges, the energy of ions collected on the grounded wall is $E_i = eV_p$ where e is the elementary charge. The typical uncertainties for the measurements in

$j_i$ and $E_i$ as provided by Impedans Ltd are ±20% and ±2%. Finally, a voltage sensor was used to measure the RF amplitude of the driving voltage ($V_{RF}$) [9]. The error on $V_{RF}$ measurements was ±5 V.

# 4. Results and discussion

## 4.1. Sheath expansion and effective area ratio

Having a sheath thickness in the range of millimetres changes the volume of the plasma and affects both electrode and wall area. This sheath thickness for $\alpha = 1$ is deduced from the Child-Langmuir law and is valid for $V_{RF} > T_e$. While the Liebermann's $\alpha = 2.45$ and the simplified Wang model that $\alpha = 3$. Previous measurements of the sheath thickness in DC-coupled plasma determined that the Wang model was the closest to the experimental data for He plasma [14]. The Child-Langmuir law is valid only for semi-infinite and collisionless plasma and showed some inaccuracy for finite plasma and for plasma with electrode potential in the same order as the plasma potential [15]. The Wang model takes into account the possible collisionality of the plasma bulk and pre-sheath and shows that the sheath thickness is, on average, 3 times higher than the Child-Langmuir model [15]. In all models, the sheath is assumed to be free of collisions and ionisation. The effective area ratio, including the sheath thickness, has been calculated for $\alpha = 1$ and for $\alpha = 3$ and plotted as a function of the geometrical area ratio (i.e. the ratio of the material area) in figure 4. Taking into account the results from Wang and Wendt [15] and from Faudot et al [14], in the following of the manuscript, the area ratio $A_g/A_p$ corrected to take into account the sheath thickness with $\alpha = 3$ is used. $V_s(t)$ and so $\lambda_s(t)$ oscillated within an RF period. The time-varying effective area of the grounded wall and of the electrode was calculated over a few RF periods. The effective electrode-wall area ratio was then time-averaged. Due to the error on the plasma parameters measurements, the error on the calculation of the effective area ratio was about 25%.

## 4.2. Langmuir probe and RFEA measurements in Ar plasma discharges

Plasma potential measurements were carried out in the bulk of Ar discharges for different electrode-wall area ratios at a pressure of 1 Pa and a $V_{RF}$ of 50 V. There was a noticeable variation of the plasma potential, which was equal to ≈20%. For the area ratio of 1, the plasma potential measured with the Langmuir probe carried in the range of 35 V–45 V. The plasma potential measurements increased to 61 V, reaching the area ratio of 2.5, and then decreased to 4 V for $A_g/A_p = 180$ (figure 5, blue squares). RFEA measurements were consistent with the Langmuir probe (figure 5, yellow circles). On average, the dispersion of measurements for similar ratios with the RFEA was less compared to the LP; it was 8 V and 20 V, respectively. However, around the area ratio $A_g/A_p = 1$, the scatter of the results was similar (≈15 V). Moreover, different plasma regions were analysed with both diagnostics (figure 2). The plasma potential was homogeneous
<figure><img src="image_7.png" /><figcaption>**Figure 4.** Effective area ratio including sheath thickness as a function of the geometrical area ratio for Ar discharge at 1 Pa and with an RF amplitude of 50 V.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 5.** Plasma potential measured with the Langmuir probe and the RFEA in Ar 1 Pa plasma discharges with an RF amplitude of 50 V compared to the analytical model from Aanesland *et al* [11] and Köhler *et al* [12]. The semi-analytical model (DSP) and the particle-in-cell (PIC) simulation from Faudot *et al* [13] are also compared. The error were about ±10 V on Langmuir probe and about 2 V on RFEA measurements.</figcaption></figure>

in the whole plasma bulk. For the area ratio of 180, the high noise-to-signal ratio caused by low plasma density near the RFEA sensor made the data extraction unfeasible. To correlate our measurements to the models presented in the introduction, in the same figure 5, the calculations from Köhler's, Aanesland's and the DSP models are also plotted as a function of the effective area ratio. For Aanesland's and Köhler's

models, the maximum potential was the RF amplitude $V_{RF}$ (50 V), whereas, in the DSP model, the maximum potential was slightly lower than the RF amplitude. The general trend of experimental plasma potential measurements showed a collapsing of the plasma potential initiating in the range of an area ratio from roughly 16–40. The DSP model was maximum at the intersection point of Aanesland's and Köhler's models, close to the experimental one previously mentioned. Between the ratios 1 and 10, experimental measurements, PIC simulations and the DSP model predictions were comparable to Köhler's calculations, while for larger ratios ($A_g/A_p \ge 16$), the plasma potential magnitude collapsed, contrary to the Köhler capacitive model, which predicted a saturating plasma potential at the RF amplitude magnitude [20]. The PIC simulated a decrease in the plasma potential starting from an area ratio of 10. In contrast, Aanesland's and DSP calculations and the experimental plasma potential measurements showed an analogous decrease starting from an area ratio of 16. For $1 \le A_g/A_p \le 16$, the high plasma potential caused a high potential drop in the grounded sheath, leading to the collection of ions on the wall. On the driven electron sheath, the oscillation of the applied RF potential caused the potential drop in the sheath to be lower than the floating potential for a fraction of the RF period, allowing electrons to escape to the driven electrode [11]. When the area ratio increases, the electron collection area (driven electrode) becomes much smaller than the ion collection area (grounded wall), leading to a saturation of the electron current on the powered area [16]. Nonetheless, plasma neutrality requires the total current to be 0, which forces the plasma potential to increase the electron current to the grounded wall. The fall of the plasma potential at the area ratio of 16, measured experimentally and predicted by the DSP model (figure 5), resulted from the saturation of the electron current on the powered area. The ratio of the electron saturation current density ($j_e^s$) over the ion saturation current density ($j_i^s$) was measured experimentally using the Langmuir probe, as shown in figure S1. The plasma potential fell when $A_g/A_p = j_e^s/j_i^s$ as shown in figure S1 and predicted by Faudot *et al* [13].

One can see that the DSP model also predicted that plasma potential was symmetrically centred at ratio 1. That symmetry was confirmed experimentally as shown in figure 5(Blue squares and yellow circles). DSP simulation presented in Faudot *et al* showed that for a symmetric discharge ($A_g/A_p = 1$), the DC current equals 0 while it is negative on the RF electrode for $A_g/A_p > 1$ and positive $A_g/A_p < 1$ [16]. Moreover, the current conservation has been verified, meaning that the current on the grounded wall equals the opposite of the powered electrode current. While the powered electrode was smaller enough than the grounded electrode, the ion current on the powered electrode saturated, implying the plasma potential to decrease to preserve plasma neutrality.

The transition area ratio starts now at $A_g/A_p = j_i/j_e$, which explains the symmetry in the logarithmic scale.

The simulation of the plasma potential oscillations for different area ratios showed that the sheath electrical regime changed from the capacitive regime for $A_g/A_p > 1$ to the resistive regime for $A_g/A_p < 1$ as shown in figure S2.
### 4.3. Electron temperature and electrode wall area ratio

In capacitive discharges, the two-electron heating mechanisms occur: ohmic (collisional) heating due to electron-neutral collisions (pressure dependant) and stochastic (collisionless) heating at the plasma edge due to momentum transfer from high-voltage oscillating sheaths (pressure independent) [21]. The total electron heating (i.e. the total contribution of both mechanisms) is proportional to $j_{RFF}^2/n_e$ with $j_{RF}$ the current density RF amplitude [22]. $j_{RF}$ was not measured however, the ion current flux $j_i$ on the grounded electrode was measured using RFEA. One can see that the plasma heating factor $j_i^2/n_e$ in figure 6 started to increase when $A_g/A_p$ was smaller than 6. Moreover, in DC-coupled RF plasma, two sheaths develop due to the high plasma potential. The first one in front of the grounded wall, which followed the oscillation of the $V_p(t)$ and the second in front of the driven electrode, which oscillated with $V_p(t) - V_{RF}(t)$. The oscillations of the driven sheath were at least 2 times larger than the ones on the grounded sheath because the RF amplitude $V_{RF}(t)$ was larger than the plasma potential RF amplitude [11]. This implies that stochastic heating occurs mostly in the driven electrode sheath. However, for high area ratios, the grounded wall sheath is much larger than the driven one, leading to a reduced plasma heating effect. For small area ratios, the driven sheath area is in the same order or larger than the grounded sheath, enhancing stochastic electron heating due to the high sheath oscillations. The proportion of the powered sheath oscillations over the total chamber volume $V$ is described by the powered sheath oscillations factor $\lambda_{sh}^{RF}A_{sh}/V$ with $\lambda_{sh}^{RF}$ the sheath thickness oscillations amplitude, and $A_{sh}$ the area of the sheath in contact with the plasma (figure 3). This powered sheath oscillations factor increased while the area ratio decreased, following the same trend as the plasma heating factor, as shown in figure 6. The volume of sheath oscillation increased to be higher than 50% of the total plasma volume for $A_g/A_p < 1$. The plasma temperature was measured to be constant around 4 eV for area ratios ranging from 2 to 150. Concerning the ratio below 2, temperature measurements ranged between 4 and 15 eV, indicating either a second population of electrons or a weaker RF compensation due to the smaller grounded area. Future measurements of the EEDF are planned, incorporating advanced compensation techniques to validate the temperature measurements for the area ratio below 2.

The measured $T_e$ was constant at 4 eV for ratio >6 and then increased exponentially to saturate at 16 eV for ratios <0.2 as shown in figure 6 following the same trend as the plasma heating factor.

### 4.4. Ar and He plasma discharges

The DSP model described our experimental measurements for Ar discharges more accurately. As mentioned in section 2.2, in the DSP model, the dependency of the sheath capacitance $c_{g,p}$ on the sheath thickness was included. Since the DSP model utilises $\lambda_s$, which can be calculated with different $\alpha$, three different values of $\alpha$ were used for comparison with our results in Ar, and He discharges for different area ratios. The pressure

<figure><img src="image_9.png" /><figcaption>**Figure 6.** Electron temperature measured as a function of the area ratio for Ar at 1 Pa and 10 Pa in semi-logarithmic scale for an RF amplitude of 50 V compared with the plasma heating factor $j_i^2/n_e$ and the powered sheath oscillation factor $\lambda_{sh}^{RF}A_{sh}/V$. An exponential fit of the electron temperature and its prediction bounds have also been plotted. The error on electron temperature measurements was about 20%.</figcaption></figure>

was 1 Pa and 10 Pa for Ar and 10 Pa for He. Measurements from both diagnostics (Langmuir probe and RFEA) were averaged and plotted in figure 7(and in linear scale figure S3). For Ar discharges, higher sheath model parameters led to a slightly lower plasma potential for area ratio from 0.04 to 30. Similarly, for He the plasma potential was reduced for the same range of area ratio while $\alpha$ increased. However, the choice of the sheath model parameter $\alpha$ implied more important differences for He than for Ar.

Figure 8 depicts the sheath model parameter impact on the plasma potential measurements for Ar at 1 Pa. The experimental measurements for an effective ratio considering $\alpha = 3$ were shrunk toward the ratio 1 compared to the one for an effective area ratio with $\alpha = 1$. For both $\alpha = 1$ and $\alpha = 3$, the experimental data and DSP model calculation showed good consistency. However one can see that in the case of $\alpha = 3$, the fall of the plasma potential occurred for the same area ratio as for the DSP model calculations, while for $\alpha = 1$, the experimental data fall for larger ratios. Nonetheless, the DSP model calculations showed a lower plasma potential than experimental measurements. Contrary to other models, the DSP model considered the plasma density in the calculation of sheath capacitances and sheath thickness [13]. A change in the density has a similar effect as a variation of $\alpha$, which led to a modification of the plasma potential, as shown in figure 7(a). DSP calculations were carried out with a density $n_e = 2 \ 10^{14} \ \text{m}^{-3}$, the average electron density measured experimentally. However the experimental density was not constant with the area ratio and varied from $3 \ 10^{13} \ \text{m}^{-3}$ to $10 \ 15 \ \text{m}^{-3}$, which can lead to a deviation of the plasma potential as high as 7 V. Moreover, the error in the measurements on $V_p$ and $V_{RF}$ could explain these differences in the plasma potential between the DSP model and the experimental results.

Experimental characterisation of the plasma parameters in He discharges was challenging, especially at 1 Pa where
<figure><img src="image_10.png" /><figcaption>**Figure 7.** LP and RFEA average plasma potential measured as a function of the area ratio for (a) Ar at different pressures and for (b) He in semi-logarithmic scale for an RF amplitude of 50 V. The error on plasma potential measurements was about 20%.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 8.** DSP model and experimental measurements of the plasma potential for different sheath model parameter as a function of the area ratio for Ar discharge at 1 Pa and with an RF amplitude of 50 V. The error on plasma potential measurements was about 20%.</figcaption></figure>

no parameters were accessible through our diagnostics. The trend of the experimental plasma potential at 10 Pa shown in figure 7(b) is similar to the DSP calculations. However, the deviation is stronger than for Ar discharges. That deviation can be attributed to a higher electron density variation and higher error in plasma potential measurements (figure S4). In He discharges, a negative plasma potential (≈ −5 V) has been measured for a geometrical area ratio of 520 (effective area ratio was not calculable due to low density).

As mentioned in the previous section, the potential col-
lapsed on both sides of the curves when the A_g/A_p = j_i/j_e

due to the current saturation. As $j_i/j_e = 0.61\sqrt{2\pi m_e/m_i}$, the quotient of the area ratio for which the plasma potential is collapsing for Ar by the one of He is theoretically equal to $\sqrt{m_{Ar}/m_{He}} \approx 3.3$. Experimentally, this ratio was found to be $\approx 2.6$. That confirmed that the current saturation was responsible for the collapsing of the plasma potential when the area ratio overcame $j_i/j_e$.

4.5. Gas pressure effect on plasma potential

For Ar at 10 Pa, Vp showed a similar dependency versus the
area ratio than at 1 Pa (figure 7(a)). However, the plasma poten-
tial at 10 Pa was reduced for area ratios <80 compared to
1 Pa. This phenomenon was attributed to the plasma density,
which slightly increased at 10 Pa compared to 1 Pa. Indeed, as
discussed in the previous section, increasing the density can
lead to a decrease in the plasma potential. Moreover, the mean
free path was 8 mm and 0.8 mm at 1 Pa and 10 Pa, respect-
ively. That led to increased collisions, which induced a higher
plasma resistivity and, therefore, a higher potential drop in
the powered sheath. The plasma potential was then decreas-
ing toward the grounded sheath. That led to a lower plasma
potential in the centre of the plasma discharges compared to
1 Pa.

4.6. RF amplitude and plasma potential

The potential of Ar plasma discharges was measured at 1 Pa as
a function of the area ratio for 3 different RF amplitudes: 50 V
(orange), 100 V (blue), and 250 V (green) and is displayed
in figure 9(and in linear scale figure S5). In these 3 cases,
experimental data were consistent with the prediction from
the DSP model. However, at 250 V, the experimental data for
A<sub>g</sub>/A<sub>p</sub> > 30 diverge from the DSP model calculations, prob-
ably due to incorrect correction of the effective sheath area
A<sub>sh</sub>. Indeed, at high RF potential, the sheath size stayed large,
<figure><img src="image_12.png" /><figcaption>**Figure 9.** Plasma potential measurements in Ar 1 Pa discharges for RF amplitudes of 50 V, 100 V and 250 V compared with the DSP calculations. The error on plasma potential measurements was about 20%.</figcaption></figure>

even for a small electrode, so the minimum size of the RF elec-
trode A<sub>p</sub> tended to saturate, and then the area ratio saturated as
well.

**5. Conclusion and outlook**

The impact of the electrode/wall area ratio on the plasma potential, temperature, density and ion flux was investigated experimentally with a Langmuir probe and an RFEA. The high plasma potential obtained with a DC-grounded powered electrode was consistent with the results reported in [9]. The semi-analytical model presented by Faudot *et al* [13] showed very good agreement with the experimental results. It was validated experimentally that the plasma potential collapsed when the electrode/wall area ratio became larger than the electron-ion saturation current density ratio. This corresponds to the intersection point of analytical models presented by Köhler *et al* [20] and Aanesland *et al* [11]. The presence of strong stochastic heating of the plasma for A<sub>g</sub>/A<sub>p</sub> < 6 was evidenced by an increased ion flux and an increased electron temperature.

The DSP model showed better consistency for an RF sheath thickness roughly 3 times higher than the Child = Langmuir law (i.e. α = 3) in Ar plasmas as expected by Faudot *et al* [14]. The impact of the pressure was also studied, showing a similar trend to the DSP model despite the increase of the plasma resistivity due to collisions. Moreover, the change in the electrode potential RF amplitude did not affect the consistency of the DSP prediction of the experimental results.

The results presented in this manuscript are relevant for
applying plasma cleaning with DC-grounded electrodes in
the absence of a magnetic field. The plasma potential and

so the ion energy can be precisely chosen by modifying the
electrode-wall area ratio and the RF amplitude.

However, these potential structures developing in fusion
devices in front of ion cyclotron resonance heating antennas or
in diagnostics FMs cleaning devices call for further studies in a
magnetic field. The DSP model for magnetised plasma showed
similar behaviour to those in the unmagnetised case [13] and
requires experimental validation in magnetic fields. Moreover,
future measurements are planned regarding the measurements
of the EEDF using advanced RF compensation techniques.

**Data availability statement**

The data cannot be made publicly available upon publication
because they are not available in a format that is sufficiently
accessible or reusable by other researchers. The data that sup-
port the findings of this study are available upon reasonable
request from the authors.

**Acknowledgments**

This work has been funded by the Swiss National Science
Foundation under Agreement No 200021E_189448. This
work is supported by the French National Research Agency
(ANR) under project SHEAR ANR-19-CE30-033-01. This
work was supported by the Swiss State Secretariat for
Education, Research and Innovation (SERI) under Contract
Number 22.00424. The authors would like to thank the Swiss
Federal Office of Energy, the Swiss Nanoscience Institute
and the Federal Office for Education and Science for their
Financial support.

ORCID iDs

Paul Hiret https://orcid.org/0000-0002-2119-8648

Artem Dmitriev https://orcid.org/0000-0001-6790-167X

Laurent Marot https://orcid.org/0000-0002-1529-9362

Ernst Meyer https://orcid.org/0000-0001-6385-3412

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