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Title: 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 » 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.« less

Authors:
 [1];  [2];  [2];  [2]; ORCiD logo [3]
  1. Dalian Univ. of Technology, Dalian (China). School of Physics; Univ. of California, Berkeley, CA (United States). Dept. of Electrical Engineering and Computer Science
  2. Univ. of California, Berkeley, CA (United States). Dept. of Electrical Engineering and Computer Science
  3. 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},
issn = {1070-664X},
number = 8,
volume = 24,
place = {United States},
year = {2017},
month = {8}
}

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Cited by: 3 works
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Works referenced in this record:

On the physics of a large CCP discharge
journal, March 2016


Fast 2D fluid-analytical simulation of ion energy distributions and electromagnetic effects in multi-frequency capacitive discharges
journal, December 2014


Self-consistent simulation of very high frequency capacitively coupled plasmas
journal, May 2008


Effect of the resonant growth of harmonics on the electron density in capacitively coupled plasma
journal, November 2009


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
  • DOI: 10.1063/1.3406153

Sheath motion in a capacitively coupled radio frequency discharge
journal, January 1991

  • Wood, B. P.; Lieberman, M. A.; Lichtenberg, A. J.
  • IEEE Transactions on Plasma Science, Vol. 19, Issue 4
  • DOI: 10.1109/27.90327

Fast 2D hybrid fluid-analytical simulation of inductive/capacitive discharges
journal, April 2011


Modeling of Resonant Surface Wave Excitation in a Large CCP Reactor
journal, April 2017


Nonlinear electron resonance heating in capacitive radio frequency discharges
journal, April 2006

  • Mussenbrock, Thomas; Brinkmann, Ralf Peter
  • Applied Physics Letters, Vol. 88, Issue 15
  • DOI: 10.1063/1.2194824

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
  • DOI: 10.1063/1.1534918

Displacement current and multiple pulse effects in plasma source ion implantation
journal, May 1993

  • Wood, Blake P.
  • Journal of Applied Physics, Vol. 73, Issue 10
  • DOI: 10.1063/1.353841

Nonlinear plasma dynamics in capacitive radio frequency discharges
journal, April 2007

  • Mussenbrock, Thomas; Brinkmann, Ralf Peter
  • Plasma Sources Science and Technology, Vol. 16, Issue 2
  • DOI: 10.1088/0963-0252/16/2/022

Analytical solution for capacitive RF sheath
journal, January 1988

  • Lieberman, M. A.
  • IEEE Transactions on Plasma Science, Vol. 16, Issue 6
  • DOI: 10.1109/27.16552

Measurements and consequences of nonuniform radio frequency plasma potential due to surface asymmetry in large area radio frequency capacitive reactors
journal, July 2005

  • Sansonnens, L.; Strahm, B.; Derendinger, L.
  • Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films, Vol. 23, Issue 4
  • DOI: 10.1116/1.1868572

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
  • DOI: 10.1088/0963-0252/11/3A/331

Inductive heating and E to H transitions in high frequency capacitive discharges
journal, April 2006


The effects of nonlinear series resonance on Ohmic and stochastic heating in capacitive discharges
journal, June 2008

  • Lieberman, M. A.; Lichtenberg, A. J.; Kawamura, E.
  • Physics of Plasmas, Vol. 15, Issue 6
  • DOI: 10.1063/1.2928847

Nonlinear series resonance and standing waves in dual-frequency capacitive discharges
journal, November 2016


Self-excitation of the plasma series resonance in radio-frequency discharges: An analytical description
journal, December 2006

  • Czarnetzki, U.; Mussenbrock, T.; Brinkmann, R. P.
  • Physics of Plasmas, Vol. 13, Issue 12
  • DOI: 10.1063/1.2397043

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
  • DOI: 10.1063/1.1803608

Spatial and frequency dependence of plasma currents in a 300 mm capacitively coupled plasma reactor
journal, October 2006

  • Miller, Paul A.; Barnat, Edward V.; Hebner, Gregory A.
  • Plasma Sources Science and Technology, Vol. 15, Issue 4
  • DOI: 10.1088/0963-0252/15/4/036

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
  • DOI: 10.1063/1.3519515

Nonlinear standing wave excitation by series resonance-enhanced harmonics in low pressure capacitive discharges
journal, August 2015


Frequency dependent plasma characteristics in a capacitively coupled 300 mm wafer plasma processing chamber
journal, October 2006

  • Hebner, Gregory A.; Barnat, Edward V.; Miller, Paul A.
  • Plasma Sources Science and Technology, Vol. 15, Issue 4
  • DOI: 10.1088/0963-0252/15/4/035

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
  • DOI: 10.1088/0963-0252/15/3/002

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
  • DOI: 10.1063/1.1688334

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
  • DOI: 10.1088/0963-0252/11/3/310

Probe measurements of plasma potential nonuniformity due to edge asymmetry in large-area radio-frequency reactors: The telegraph effect
journal, June 2005

  • Howling, A. A.; Derendinger, L.; Sansonnens, L.
  • Journal of Applied Physics, Vol. 97, Issue 12
  • DOI: 10.1063/1.1940136

Electric fields in the sheath formed in a 300 mm, dual frequency capacitive argon discharge
journal, March 2007

  • Barnat, E. V.; Miller, P. A.; Hebner, G. A.
  • Plasma Sources Science and Technology, Vol. 16, Issue 2
  • DOI: 10.1088/0963-0252/16/2/016

Self-consistent electrodynamics of large-area high-frequency capacitive plasma discharge
journal, October 2010

  • Chen, Zhigang; Rauf, Shahid; Collins, Ken
  • Journal of Applied Physics, Vol. 108, Issue 7
  • DOI: 10.1063/1.3489950

2D fluid-analytical simulation of electromagnetic effects in low pressure, high frequency electronegative capacitive discharges
journal, April 2016


Improving plasma uniformity using lens-shaped electrodes in a large area very high frequency reactor
journal, May 2004

  • Schmidt, H.; Sansonnens, L.; Howling, A. A.
  • Journal of Applied Physics, Vol. 95, Issue 9
  • DOI: 10.1063/1.1690096

Modeling electromagnetic effects in capacitive discharges
journal, February 2008


Theoretical approach for plasma series resonance effect in geometrically symmetric dual radio frequency plasma
journal, February 2012

  • Bora, B.; Bhuyan, H.; Favre, M.
  • Applied Physics Letters, Vol. 100, Issue 9
  • DOI: 10.1063/1.3690870

Amorphous silicon deposition: Industrial and technical challenges
journal, July 1989


Modeling of dual frequency capacitively coupled plasma sources utilizing a full-wave Maxwell solver: I. Scaling with high frequency
journal, September 2010


Enhancement of Ohmic and Stochastic Heating by Resonance Effects in Capacitive Radio Frequency Discharges: A Theoretical Approach
journal, August 2008


Measurements of time average series resonance effect in capacitively coupled radio frequency discharge plasma
journal, October 2011

  • Bora, B.; Bhuyan, H.; Favre, M.
  • Physics of Plasmas, Vol. 18, Issue 10
  • DOI: 10.1063/1.3646317

Electromagnetic effects in high-frequency capacitive discharges used for plasma processing
journal, January 2007


Experimental investigations of the plasma radial uniformity in single and dual frequency capacitively coupled argon discharges
journal, December 2016

  • Zhao, Kai; Liu, Yong-Xin; Gao, Fei
  • Physics of Plasmas, Vol. 23, Issue 12
  • DOI: 10.1063/1.4971782

Comment on “Ion energy uniformity in high-frequency capacitive discharges” [Appl. Phys. Lett. 86, 021501 (2005)]
journal, August 2005

  • Howling, A. A.; Sansonnens, L.; Schmidt, H.
  • Applied Physics Letters, Vol. 87, Issue 7
  • DOI: 10.1063/1.2012528

Inductive Heating and E to H Transitions in Capacitive Discharges
journal, November 2005


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
  • DOI: 10.1063/1.1592617

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
  • DOI: 10.1063/1.4938204

A nonlinear global model of a dual frequency capacitive discharge
journal, August 2006

  • Mussenbrock, Thomas; Ziegler, Dennis; Peter Brinkmann, Ralf
  • Physics of Plasmas, Vol. 13, Issue 8
  • DOI: 10.1063/1.2244525

Simulations of electromagnetic effects in high-frequency capacitively coupled discharges using the Darwin approximation
journal, February 2013


Effect of electromagnetic waves and higher harmonics in capacitively coupled plasma phenomena
journal, November 2013


Nonlinearity of the radio‐frequency sheath
journal, April 1996

  • Klick, Michael
  • Journal of Applied Physics, Vol. 79, Issue 7
  • DOI: 10.1063/1.361392

A voltage uniformity study in large-area reactors for RF plasma deposition
journal, May 1997

  • Sansonnens, L.; Pletzer, A.; Magni, D.
  • Plasma Sources Science and Technology, Vol. 6, Issue 2
  • DOI: 10.1088/0963-0252/6/2/010

Relationship between center-peaked plasma density profiles and harmonic electromagnetic waves in very high frequency capacitively coupled plasma reactors
journal, January 2014

  • Sawada, Ikuo; Ventzek, Peter L. G.; Lane, Barton
  • Japanese Journal of Applied Physics, Vol. 53, Issue 3S2
  • DOI: 10.7567/JJAP.53.03DB01