# Symmetry breaking in high frequency, symmetric capacitively coupled plasmas

## Abstract

Two radially propagating surface wave modes, “symmetric,” in which the upper and lower axial sheath fields ( *E _{z}*) are aligned, and “anti-symmetric,” in which they are opposed, can exist in capacitively coupled plasma (CCP) discharges. For a symmetric (equal electrode areas) CCP driven symmetrically, we expected to observe only the symmetric mode. Instead, we find that when the applied rf frequency

*f*is above or near an anti-symmetric spatial resonance, both modes can exist in combination and lead to unexpected non-symmetric equilibria. We use a fast 2D axisymmetric fluid-analytical code to study a symmetric CCP reactor at low pressure (7.5 mTorr argon) and low density (~3 × 10

^{15}m

^{-3}) in the frequency range of

*f*= 55 to 100 MHz which encompasses the first anti-symmetric spatial resonance frequency

*f*but is far below the first symmetric spatial resonance

_{a}*f*. For lower frequencies such that

_{s}*f*is well below

*f*, the symmetric CCP is in a stable symmetric equilibrium, as expected, but at higher frequencies such that

_{a>}*f*is near or greater than

*f*, a non-symmetric equilibrium appears which may be stable or unstable. In our study, we develop a nonlinear lumped circuit model of the symmetric CCP tomore »

_{a}- Authors:

- Univ. of California, Berkeley, CA (United States). Dept. of Electrical Engineering and Computer Sciences

- Publication Date:

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

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1540256

- Alternate Identifier(s):
- OSTI ID: 1472211

- Grant/Contract Number:
- [SC0001939]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- [ Journal Volume: 25; Journal Issue: 9]; 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; physics

### Citation Formats

```
Kawamura, E., Lieberman, M. A., and Lichtenberg, A. J. Symmetry breaking in high frequency, symmetric capacitively coupled plasmas. United States: N. p., 2018.
Web. doi:10.1063/1.5048947.
```

```
Kawamura, E., Lieberman, M. A., & Lichtenberg, A. J. Symmetry breaking in high frequency, symmetric capacitively coupled plasmas. United States. doi:10.1063/1.5048947.
```

```
Kawamura, E., Lieberman, M. A., and Lichtenberg, A. J. Mon .
"Symmetry breaking in high frequency, symmetric capacitively coupled plasmas". United States. doi:10.1063/1.5048947. https://www.osti.gov/servlets/purl/1540256.
```

```
@article{osti_1540256,
```

title = {Symmetry breaking in high frequency, symmetric capacitively coupled plasmas},

author = {Kawamura, E. and Lieberman, M. A. and Lichtenberg, A. J.},

abstractNote = {Two radially propagating surface wave modes, “symmetric,” in which the upper and lower axial sheath fields (Ez) are aligned, and “anti-symmetric,” in which they are opposed, can exist in capacitively coupled plasma (CCP) discharges. For a symmetric (equal electrode areas) CCP driven symmetrically, we expected to observe only the symmetric mode. Instead, we find that when the applied rf frequency f is above or near an anti-symmetric spatial resonance, both modes can exist in combination and lead to unexpected non-symmetric equilibria. We use a fast 2D axisymmetric fluid-analytical code to study a symmetric CCP reactor at low pressure (7.5 mTorr argon) and low density (~3 × 1015 m-3) in the frequency range of f = 55 to 100 MHz which encompasses the first anti-symmetric spatial resonance frequency fa but is far below the first symmetric spatial resonance fs. For lower frequencies such that f is well below fa>, the symmetric CCP is in a stable symmetric equilibrium, as expected, but at higher frequencies such that f is near or greater than fa, a non-symmetric equilibrium appears which may be stable or unstable. In our study, we develop a nonlinear lumped circuit model of the symmetric CCP to better understand these unexpected results, indicating that the proximity to the anti-symmetric spatial resonance allows self-exciting of the anti-symmetric mode even in a symmetric system. The circuit model results agree well with the fluid simulations. A linear stability analysis of the symmetric equilibrium describes a transition with increasing frequency from stable to unstable.},

doi = {10.1063/1.5048947},

journal = {Physics of Plasmas},

number = [9],

volume = [25],

place = {United States},

year = {2018},

month = {9}

}

Works referenced in this record:

##
Electromagnetic field nonuniformities in large area, high-frequency capacitive plasma reactors, including electrode asymmetry effects

journal, March 2006

- Sansonnens, L.; Howling, A. A.; Hollenstein, Ch
- Plasma Sources Science and Technology, Vol. 15, Issue 3

##
On the physics of a large CCP discharge

journal, March 2016

- Eremin, D.; Bienholz, S.; Szeremley, D.
- Plasma Sources Science and Technology, Vol. 25, Issue 2

##
Fast 2D fluid-analytical simulation of ion energy distributions and electromagnetic effects in multi-frequency capacitive discharges

journal, December 2014

- Kawamura, E.; Lieberman, M. A.; Graves, D. B.
- Plasma Sources Science and Technology, Vol. 23, Issue 6

##
Self-consistent nonlinear transmission line model of standing wave effects in a capacitive discharge

journal, May 2004

- Chabert, P.; Raimbault, J. L.; Rax, J. M.
- Physics of Plasmas, Vol. 11, Issue 5

##
Self-consistent simulation of very high frequency capacitively coupled plasmas

journal, May 2008

- Rauf, Shahid; Bera, Kallol; Collins, Ken
- Plasma Sources Science and Technology, Vol. 17, Issue 3

##
Standing wave and skin effects in large-area, high-frequency capacitive discharges

journal, June 2002

- Lieberman, M. A.; Booth, J. P.; Chabert, P.
- Plasma Sources Science and Technology, Vol. 11, Issue 3

##
Effect of resonance in external radio-frequency circuit on very high frequency plasma discharge

journal, May 2010

- Rauf, Shahid; Chen, Zhigang; Collins, Ken
- Journal of Applied Physics, Vol. 107, Issue 9

##
2D fluid-analytical simulation of electromagnetic effects in low pressure, high frequency electronegative capacitive discharges

journal, April 2016

- Kawamura, E.; Lichtenberg, A. J.; Lieberman, M. A.
- Plasma Sources Science and Technology, Vol. 25, Issue 3

##
Modeling electromagnetic effects in capacitive discharges

journal, February 2008

- Lee, Insook; Graves, D. B.; Lieberman, M. A.
- Plasma Sources Science and Technology, Vol. 17, Issue 1

##
Experimental investigation of standing wave effect in dual-frequency capacitively coupled argon discharges: role of a low-frequency source

journal, May 2018

- Zhao, Kai; Liu, Yong-Xin; Kawamura, E.
- Plasma Sources Science and Technology, Vol. 27, Issue 5

##
Fast 2D hybrid fluid-analytical simulation of inductive/capacitive discharges

journal, April 2011

- Kawamura, E.; Graves, D. B.; Lieberman, M. A.
- Plasma Sources Science and Technology, Vol. 20, Issue 3

##
Modeling of dual frequency capacitively coupled plasma sources utilizing a full-wave Maxwell solver: II. Scaling with pressure, power and electronegativity

journal, September 2010

- Yang, Yang; Kushner, Mark J.
- Plasma Sources Science and Technology, Vol. 19, Issue 5

##
Modeling of dual frequency capacitively coupled plasma sources utilizing a full-wave Maxwell solver: I. Scaling with high frequency

journal, September 2010

- Yang, Yang; Kushner, Mark J.
- Plasma Sources Science and Technology, Vol. 19, Issue 5

##
Amorphous silicon deposition: Industrial and technical challenges

journal, July 1989

- Schmitt, J. P. M.
- Thin Solid Films, Vol. 174

##
Shaped electrode and lens for a uniform radio-frequency capacitive plasma

journal, January 2003

- Sansonnens, L.; Schmitt, J.
- Applied Physics Letters, Vol. 82, Issue 2

##
Suppression of the standing wave effect in high frequency capacitive discharges using a shaped electrode and a dielectric lens: Self-consistent approach

journal, August 2004

- Chabert, P.; Raimbault, J. -L.; Rax, J. -M.
- Physics of Plasmas, Vol. 11, Issue 8

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

journal, August 2017

- Wen, De-Qi; Kawamura, E.; Lieberman, M. A.
- Physics of Plasmas, Vol. 24, Issue 8

##
Analytical solution for capacitive RF sheath

journal, January 1988

- Lieberman, M. A.
- IEEE Transactions on Plasma Science, Vol. 16, Issue 6

##
Ion flux nonuniformities in large-area high-frequency capacitive discharges

journal, July 2003

- Perret, A.; Chabert, P.; Booth, J. -P.
- Applied Physics Letters, Vol. 83, Issue 2

##
Calculation of the electrode shape for suppression of the standing wave effect in large area rectangular capacitively coupled reactors

journal, March 2005

- Sansonnens, L.
- Journal of Applied Physics, Vol. 97, Issue 6

##
Linear electromagnetic excitation of an asymmetric low pressure capacitive discharge with unequal sheath widths

journal, January 2016

- Lieberman, M. A.; Lichtenberg, A. J.; Kawamura, E.
- Physics of Plasmas, Vol. 23, Issue 1

##
Effect of a dielectric layer on plasma uniformity in high frequency electronegative capacitive discharges

journal, September 2017

- Kawamura, Emi; Wen, De-Qi; Lieberman, Michael A.
- Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films, Vol. 35, Issue 5

##
Glow discharge processing in the liquid crystal display industry

journal, August 2002

- Schmitt, Jacques; Elyaakoubi, Mustapha; Sansonnens, Laurent
- Plasma Sources Science and Technology, Vol. 11, Issue 3A

##
Nonlinear series resonance and standing waves in dual-frequency capacitive discharges

journal, November 2016

- Wen, De-Qi; Kawamura, E.; Lieberman, M. A.
- Plasma Sources Science and Technology, Vol. 26, Issue 1

##
Simulations of electromagnetic effects in high-frequency capacitively coupled discharges using the Darwin approximation

journal, February 2013

- Eremin, Denis; Hemke, Torben; Brinkmann, Ralf Peter
- Journal of Physics D: Applied Physics, Vol. 46, Issue 8

##
Nonuniform radio-frequency plasma potential due to edge asymmetry in large-area radio-frequency reactors

journal, November 2004

- Howling, A. A.; Sansonnens, L.; Ballutaud, J.
- Journal of Applied Physics, Vol. 96, Issue 10

##
A Monte Carlo collision model for the particle-in-cell method: applications to argon and oxygen discharges

journal, May 1995

- Vahedi, V.; Surendra, M.
- Computer Physics Communications, Vol. 87, Issue 1-2

##
Comparison of electrostatic and electromagnetic simulations for very high frequency plasmas

journal, November 2010

- Zhang, Yu-Ru; Xu, Xiang; Zhao, Shu-Xia
- Physics of Plasmas, Vol. 17, Issue 11

##
Nonlinear standing wave excitation by series resonance-enhanced harmonics in low pressure capacitive discharges

journal, August 2015

- Lieberman, M. A.; Lichtenberg, A. J.; Kawamura, E.
- Plasma Sources Science and Technology, Vol. 24, Issue 5