# Poster, presentation, PIC simulation of RF sheaths with non-adiabatic electrons

Léonel Tsowemoo Faabomve, Guillaume Urbanczyk, Stéphane Heuraux,
Jérôme Moritz

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Léonel Tsowemoo Faabomve, Guillaume Urbanczyk, Stéphane Heuraux, Jérôme Moritz. Poster, pre-sentation, PIC simulation of RF sheaths with non-adiabatic electrons. Topical Conference on Radio Frequency Power in Plasma, May 2025, Munich, Germany. hal-05322068

HAL Id: hal-05322068

https://hal.science/hal-05322068v1

Submitted on 20 Oct 2025

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## Introduction

* One of the main heating mechanism on future reactors such as ITER will be ICRF heating

* ICRF antenna are subjected to plasma particles accelerated in the RF sheath

* Heavy impurity production (W) by sputtering reduces the heating efficiency and causes disruption

* Need to study the sheath rectification phenomenon vs different plasma scenarios (grazing incidences)

### Simulation parameters

$$ \text{Hydrogen plasma / } V_{\text{RF}} = 100\text{V} $$

$$ T_e = T_i = 2\text{eV} \text{ & } n_e = 10^{17}\text{m}^{-3} $$

$$ L = 600\lambda_d \text{ & } B = 3T \text{ & } \theta = 1.25-90^{\circ} $$

<table><thead><tr><th>ω<sub>pi</sub>/nω<br>n<sub>0</sub>[m<sup>-3</sup>]</th><th>ω<sub>pi</sub>/0.1Ω<sub>i</sub></th><th>ω<sub>pi</sub>/0.5Ω<sub>i</sub></th><th>ω<sub>pi</sub>/Ω<sub>i</sub></th><th>ω<sub>pi</sub>/2Ω<sub>i</sub></th><th>ω<sub>pi</sub>/3Ω<sub>i</sub></th><th>ω<sub>pi</sub>/5Ω<sub>i</sub></th></tr></thead><tbody><tr><td>10<sup>17</sup></td><td>14.5</td><td>2.9</td><td>1.45</td><td>0.73</td><td>0.48</td><td>0.3</td></tr></tbody></table>

<figure><img src="image_5.png" /></figure>

## RF sheath properties

### Sheath with vs B incidence

<figure><img src="image_6.png" /></figure>

* Sheath width increases when magnetic incidence decreases (similar in DC [1])

* Transit time of electron $\tau_e$ within the sheath increases consequently

* Electron sheath heating is far more efficient at low incidence → both plasma potential and $T_e$ pikes when $\tau_e \approx \omega^{-1}$

### Electron heating vs θ

<figure><img src="image_7.png" /></figure>

* For $\omega_{pi} > \omega$, both ions and electrons react to instantaneous electric fields $V_{DC} \approx V_{RF}/\pi$

* For $\omega_{pe} > \omega > \omega_{pi}$, only electrons can screen the fields $V_{DC} \approx V_{RF}/2$

* With a tilted B, both $\omega_{pe}$ and $\omega_{pi}$ ↓ with, because the plasma density ↓ in the sheath

* Both particle dynamics depend on $\theta$ + transit time of electron can be too large compare to the RF time

### Typical expected behaviors

<figure><img src="image_8.png" /></figure>

### Grazing incidences issue

<figure><img src="image_9.png" /></figure>

* Regulating $T_e=2\text{eV}$ during the simulations by a collision operator leads to a different behavior

* At both grazing incidences and higher frequencies, electrons can not react anymore to the RF fields → floating potential + resonance at $1\omega_{ci}$

## Electron acceleration at grazing angles

Ion Matrix model [2],

$$ E_{sh} = \begin{cases} 0 & \text{if } z > s(t) \\ \frac{n_0 e}{\varepsilon_0} (z - s(t)) & \text{if } z \le s(t) \end{cases} \quad s(t) = s_m + s_m \sin(\omega t + \varphi) \quad \varphi \in [0, 2\pi] $$

* Electrons are injected in the sheath with a Maxwellian velocity distribution function at 2 eV for $V_x$ and $V_y$, and a 2 eV Maxwellian flux directed toward the wall for $V_z$

### Calculated electron energy at the sheath exit

<figure><img src="image_10.png" /></figure>

### Typical electrons trajectories

<figure><img src="image_11.png" /></figure>

* Sheath motion causes shadowing: more electrons experience an increasing field than a decreasing one → net energy gain.

* The shadowing effect is strongly enhanced at low magnetic field incidence. It is further amplified by the longer electron transit time in the strong electric field region compared to normal incidence, leading to a significant acceleration ('kick')

<figure><img src="image_12.png" /><figcaption>Ratio of electrons entering the sheath when it expands vs when it collapses</figcaption></figure>

## Conclusion

* At grazing incidence, assuming adiabatic electrons is not realistic, as their transit time through the sheath becomes comparable to or longer than the RF oscillation period

* Electron heating can be significant in such low-incidence scenarios, leading to a substantial increase in the plasma potential [3]

* Simulations with fixed electron temperature show that the plasma potential approaches the floating potential at high RF frequencies and grazing magnetic field incidence, thereby reducing impurity production in low-density plasmas ($n_e \approx 10^{17}\text{m}^{-3}$)

## Ions distribution function

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

At resonance, magnetized ions are in phase with the sheath electric field and are continuously accelerated within the sheath. At grazing angles (below 15° here), the energy they deliver to the wall exceeds 100 eV.

## References

[1] J. Moritz, S. Heuraux, E. Gravier et al. Phys. Plasmas 28, 083501 (2021)

[2] P. Chabopoulos, N. Braithwaite. Phys. Plasmas 3, 2511 (1996)

## Acknowledgement

This work was carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion), and partly supported by the french PIA project « Lorraine Université d'Excellence », reference « ANR-15-IDEX-04-LUE