# A nonlinear electromagnetics model of an asymmetrically-driven, low pressure capacitive discharge

## Abstract

It is well-known that standing waves having radially center-high voltage profiles exist in high frequency driven capacitive discharges. Capacitive sheaths can also nonlinearly excite driving frequency harmonics near the series resonance that can be spatially near-resonant, and therefore enhance the on-axis power deposition. The powered-electrode/plasma/grounded-electrode sandwich structure of an asymmetrically excited cylindrical discharge forms a three electrode system in which both z-symmetric and z-antisymmetric radially propagating wave modes can exist. We develop a nonlinear electromagnetics model for this system with radially- and time-varying sheath widths, incorporating both symmetric and antisymmetric modes, and the plasma skin effect. Waves generated in the electrostatic wave limit are also treated. The discharge is modeled as a uniform density bulk plasma with either homogeneous or Child law sheaths at the electrodes, incorporating their nonlinear voltage versus charge relations. The model includes a finite power source resistance and a self-consistent calculation of the dc bias voltages. The resulting set of nonlinear partial differential equations is solved numerically to determine the symmetric and antisymmetric mode amplitudes and the nonlinearly-excited radially-varying harmonics. Two examples driven at lower frequency (30 MHz) or at higher frequency (60 MHz) are examined. The first case displays the excited dominant ninth harmonic nearmore »

- Authors:

- Dalian Univ. of Technology, Dalian (China). School of Physics; Univ. of California, Berkeley, CA (United States). Dept. of Electrical Engineering and Computer Science
- Univ. of California, Berkeley, CA (United States). Dept. of Electrical Engineering and Computer Science
- Dalian Univ. of Technology, Dalian (China). School of Physics

- Publication Date:

- Research Org.:
- Univ. of Michigan, Ann Arbor, MI (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Fusion Energy Sciences (FES) (SC-24)

- OSTI Identifier:
- 1474308

- Alternate Identifier(s):
- OSTI ID: 1376737

- Grant/Contract Number:
- SC0001939

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 24; Journal Issue: 8; Journal ID: ISSN 1070-664X

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY

### Citation Formats

```
Wen, De-Qi, Kawamura, E., Lieberman, M. A., Lichtenberg, A. J., and Wang, You-Nian.
```*A nonlinear electromagnetics model of an asymmetrically-driven, low pressure capacitive discharge*. United States: N. p., 2017.
Web. doi:10.1063/1.4993798.

```
Wen, De-Qi, Kawamura, E., Lieberman, M. A., Lichtenberg, A. J., & Wang, You-Nian.
```*A nonlinear electromagnetics model of an asymmetrically-driven, low pressure capacitive discharge*. United States. doi:10.1063/1.4993798.

```
Wen, De-Qi, Kawamura, E., Lieberman, M. A., Lichtenberg, A. J., and Wang, You-Nian. Thu .
"A nonlinear electromagnetics model of an asymmetrically-driven, low pressure capacitive discharge". United States.
doi:10.1063/1.4993798. https://www.osti.gov/servlets/purl/1474308.
```

```
@article{osti_1474308,
```

title = {A nonlinear electromagnetics model of an asymmetrically-driven, low pressure capacitive discharge},

author = {Wen, De-Qi and Kawamura, E. and Lieberman, M. A. and Lichtenberg, A. J. and Wang, You-Nian},

abstractNote = {It is well-known that standing waves having radially center-high voltage profiles exist in high frequency driven capacitive discharges. Capacitive sheaths can also nonlinearly excite driving frequency harmonics near the series resonance that can be spatially near-resonant, and therefore enhance the on-axis power deposition. The powered-electrode/plasma/grounded-electrode sandwich structure of an asymmetrically excited cylindrical discharge forms a three electrode system in which both z-symmetric and z-antisymmetric radially propagating wave modes can exist. We develop a nonlinear electromagnetics model for this system with radially- and time-varying sheath widths, incorporating both symmetric and antisymmetric modes, and the plasma skin effect. Waves generated in the electrostatic wave limit are also treated. The discharge is modeled as a uniform density bulk plasma with either homogeneous or Child law sheaths at the electrodes, incorporating their nonlinear voltage versus charge relations. The model includes a finite power source resistance and a self-consistent calculation of the dc bias voltages. The resulting set of nonlinear partial differential equations is solved numerically to determine the symmetric and antisymmetric mode amplitudes and the nonlinearly-excited radially-varying harmonics. Two examples driven at lower frequency (30 MHz) or at higher frequency (60 MHz) are examined. The first case displays the excited dominant ninth harmonic near the series resonance frequency with the fundamental frequency having a weak standing wave. The higher frequency case shows a more obvious standing wave effect, enhanced by the nonlinear harmonics. Finally, the symmetric and antisymmetric modes are of similar amplitudes over the driving electrode, summing to give a larger bottom sheath.},

doi = {10.1063/1.4993798},

journal = {Physics of Plasmas},

number = 8,

volume = 24,

place = {United States},

year = {Thu Aug 24 00:00:00 EDT 2017},

month = {Thu Aug 24 00:00:00 EDT 2017}

}

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