Scale-invariant breathing oscillations are observed in similar magnetized discharges at different spatiotemporal scales via fully kinetic particle-in-cell simulations. With an increase in the similarity invariant B/p, i.e., the ratio of magnetic field to pressure, breathing oscillations are triggered, leading to an appreciable time-averaged potential fall outside the sheath. With the onset and development of breathing oscillations, the electron energization mechanism shifts from sheath energization to direct Ohmic heating in the ionization region due to the change in the potential fall inside and outside the cathode sheath. Based on the scale invariance of the Boltzmann equation and its collision term, the characteristics of breathing oscillations and the transition of the electron energization mechanism are confirmed to be scale-invariant under similar discharge conditions.
REFERENCES
1.
J. T.
Gudmundsson
, “
Physics and technology of magnetron sputtering discharges
,” Plasma Sources Sci. Technol.
29
, 113001
(2020
).2.
K.
Hara
, “
An overview of discharge plasma modeling for Hall effect thrusters
,” Plasma Sources Sci. Technol.
28
, 044001
(2019
).3.
A.
Simon
, “
Instability of a partially ionized plasma in crossed electric and magnetic fields
,” Phys. Fluids
6
, 382
(1963
).4.
F. C.
Hoh
, “
Instability of penning-type discharges
,” Phys. Fluids
6
, 1184
(1963
).5.
S.
Tsikata
and
T.
Minea
, “
Modulated electron cyclotron drift instability in a high-power pulsed magnetron discharge
,” Phys. Rev. Lett.
114
, 185001
(2015
).6.
O.
Koshkarov
,
A.
Smolyakov
,
Y.
Raitses
, and
I.
Kaganovich
, “
Self-organization, structures, and anomalous transport in turbulent partially magnetized plasmas with crossed electric and magnetic fields
,” Phys. Rev. Lett.
122
, 185001
(2019
).7.
J.-P.
Boeuf
and
B.
Chaudhury
, “
Rotating instability in low-temperature magnetized plasmas
,” Phys. Rev. Lett.
111
, 155005
(2013
).8.
J.-P.
Boeuf
and
M.
Takahashi
, “
Rotating spokes, ionization instability, and electron vortices in partially magnetized E × B plasmas
,” Phys. Rev. Lett.
124
, 185005
(2020
).9.
N.
Brenning
,
R. L.
Merlino
,
D.
Lundin
,
M. A.
Raadu
, and
U.
Helmersson
, “
Faster-than-bohm cross-B electron transport in strongly pulsed plasmas
,” Phys. Rev. Lett.
103
, 225003
(2009
).10.
A.
Anders
, “
Localized heating of electrons in ionization zones: Going beyond the penning-thornton paradigm in magnetron sputtering
,” Appl. Phys. Lett.
105
, 244104
(2014
).11.
K.
Hara
,
M. J.
Sekerak
,
I. D.
Boyd
, and
A. D.
Gallimore
, “
Mode transition of a Hall thruster discharge plasma
,” J. Appl. Phys.
115
, 203304
(2014
).12.
Y.
Yang
,
X.
Zhou
,
J. X.
Liu
, and
A.
Anders
, “
Evidence for breathing modes in direct current, pulsed, and high power impulse magnetron sputtering plasmas
,” Appl. Phys. Lett.
108
, 034101
(2016
).13.
L.
Wei
,
C.
Wang
,
C.
Zhang
, and
D.
Yu
, “
Effects of operating parameters on ionization distribution in Hall thrusters
,” Appl. Phys. Lett.
102
, 173505
(2013
).14.
A. A.
Rukhadze
,
N. N.
Sobolev
, and
V. V.
Sokovikov
, “
Similarity relations for low-temperature nonisothermal discharges
,” Sov. Phys. Usp.
34
, 827
(1991
).15.
Y.
Fu
,
B.
Zheng
,
P.
Zhang
,
Q. H.
Fan
,
J. P.
Verboncoeur
, and
X.
Wang
, “
Similarity of capacitive radio-frequency discharges in nonlocal regimes
,” Phys. Plasmas
27
, 113501
(2020
).16.
Y.
Fu
,
H.
Wang
, and
X.
Wang
, “
Similarity theory and scaling laws for low-temperature plasma discharges: A comprehensive review
,” Rev. Mod. Plasma Phys.
7
, 10
(2023
).17.
D.
Yang
,
H.
Wang
,
B.
Zheng
,
X.
Zou
,
X.
Wang
, and
Y.
Fu
, “
Scale-invariant resonance characteristics in magnetized capacitive radio frequency plasmas
,” Phys. Plasmas
30
, 063510
(2023
).18.
H.
Wang
,
D.
Yang
,
B.
Zheng
,
J. P.
Verboncoeur
, and
Y.
Fu
, “
Similarity-based scaling networks for capacitive radio frequency discharge plasmas
,” AIP Adv.
13
, 095001
(2023
).19.
B.
Zheng
,
Y.
Fu
,
K.
Wang
,
T.
Tran
,
T.
Schuelke
, and
Q. H.
Fan
, “
Comparison of 1D and 2D particle-in-cell simulations for DC magnetron sputtering discharges
,” Phys. Plasmas
28
, 014504
(2021
).20.
A.
Hecimovic
and
A.
von Keudell
, “
Spokes in high power impulse magnetron sputtering plasmas
,” J. Phys. D
51
, 453001
(2018
).21.
F.
Petronio
,
A.
Tavant
,
T.
Charoy
,
A. A.
Laguna
,
A.
Bourdon
, and
P.
Chabert
, “
Conditions of appearance and dynamics of the modified two-stream instability in E × B discharges
,” Phys. Plasmas
28
, 043504
(2021
).22.
B.
Zheng
,
K.
Wang
,
T.
Grotjohn
,
T.
Schuelke
, and
Q. H.
Fan
, “
Enhancement of Ohmic heating by Hall current in magnetized capacitively coupled discharges
,” Plasma Sources Sci. Technol.
28
, 09LT03
(2019
).23.
D.
Yang
,
Y.
Fu
,
B.
Zheng
,
H.
Wang
,
Q. H.
Fan
,
X.
Zou
,
X.
Wang
, and
J. P.
Verboncoeur
, “
Similarity properties in capacitive radio frequency plasmas with nonlinear collision processes
,” Plasma Sources Sci. Technol.
30
, 115009
(2021
).24.
Y.
Fu
,
B.
Zheng
,
D.-Q.
Wen
,
P.
Zhang
,
Q. H.
Fan
, and
J. P.
Verboncoeur
, “
Similarity law and frequency scaling in low-pressure capacitive radio frequency plasmas
,” Appl. Phys. Lett.
117
, 204101
(2020
).25.
J. P.
Sauppe
,
S.
Palaniyappan
,
B. J.
Tobias
,
J. L.
Kline
,
K. A.
Flippo
,
O. L.
Landen
,
D.
Shvarts
,
S. H.
Batha
,
P. A.
Bradley
,
E. N.
Loomis
,
N. N.
Vazirani
,
C. F.
Kawaguchi
,
L.
Kot
,
D. W.
Schmidt
,
T. H.
Day
,
A. B.
Zylstra
, and
E.
Malka
, “
Demonstration of scale-invariant Rayleigh-Taylor instability growth in laser-driven cylindrical implosion experiments
,” Phys. Rev. Lett.
124
, 185003
(2020
).26.
G. W.
Baxter
and
J. S.
Olafsen
, “
Experimental evidence for molecular chaos in granular gases
,” Phys. Rev. Lett.
99
, 028001
(2007
).27.
J. A.
Bittencourt
, Fundamentals of Plasma Physics
(
Springer Science & Business Media
, 2004
).28.
Y.
Fu
,
H.
Wang
,
B.
Zheng
,
P.
Zhang
,
Q. H.
Fan
,
X.
Wang
, and
J. P.
Verboncoeur
, “
Generalizing similarity laws for radio-frequency discharge plasmas across nonlinear transition regimes
,” Phys. Rev. Appl.
16
, 054016
(2021
).29.
J. A.
Thornton
, “
Magnetron sputtering: Basic physics and application to cylindrical magnetrons
,” J. Vac. Sci. Technol.
15
, 171
(1978
).30.
M. A.
Lieberman
and
A. J.
Lichtenberg
, Principles of Plasma Discharges and Materials Processing
(
John Wiley & Sons
,
New Jersey
, 2005
).31.
C.
Huo
,
D.
Lundin
,
M. A.
Raadu
,
A.
Anders
,
J. T.
Gudmundsson
, and
N.
Brenning
, “
On sheath energization and Ohmic heating in sputtering magnetrons
,” Plasma Sources Sci. Technol.
22
, 045005
(2013
).32.
N.
Brenning
,
J. T.
Gudmundsson
,
D.
Lundin
,
T.
Minea
,
M. A.
Raadu
, and
U.
Helmersson
, “
The role of Ohmic heating in dc magnetron sputtering
,” Plasma Sources Sci. Technol.
25
, 065024
(2016
).33.
D.
Lundin
,
T.
Minea
, and
J. T.
Gudmundsson
, High Power Impulse Magnetron Sputtering: Fundamentals, Technologies, Challenges and Applications
(
Elsevier
, 2019
).34.
Z. A.
Brown
and
B. A.
Jorns
, “
Growth and saturation of the electron drift instability in a crossed field plasma
,” Phys. Rev. Lett.
130
, 115101
(2023
).35.
T. A.
van der Straaten
,
N. F.
Cramer
,
I. S.
Falconer
, and
B. W.
James
, “
The cylindrical dc magnetron discharge: I. Particle-in-cell simulation
,” J. Phys. D
31
, 177
(1998
).36.
S.
Hayakawa
and
K.
Wasa
, “
Discharges between coaxial cylinders in a magnetic field
,” J. Phys. Soc. Jpn.
20
, 1692
(1965
).37.
J. W.
Bradley
,
S.
Thompson
, and
Y. A.
Gonzalvo
, “
Measurement of the plasma potential in a magnetron discharge and the prediction of the electron drift speeds
,” Plasma Sources Sci. Technol.
10
, 490
(2001
).38.
R. P.
Brinkmann
, “
Beyond the step model: Approximate expressions for the field in the plasma boundary sheath
,” J. Appl. Phys.
102
, 093303
(2007
).39.
L.
Gu
and
M. A.
Lieberman
, “
Axial distribution of optical emission in a planar magnetron discharge
,” J. Vac. Sci. Technol., A
6
, 2960
(1988
).40.
A.
Marcovati
,
T.
Ito
, and
M. A.
Cappelli
, “
The dynamics of coherent modes of gradient drift instabilities in a small magnetron discharge plasma
,” J. Appl. Phys.
127
, 223301
(2020
).41.
D.
Levko
and
L. L.
Raja
, “
Magnetized direct current microdischarge I. Effect of the gas pressure
,” J. Appl. Phys.
121
, 093302
(2017
).42.
Y.
Fu
,
B.
Zheng
,
P.
Zhang
,
Q. H.
Fan
, and
J. P.
Verboncoeur
, “
Transition characteristics and electron kinetics in microhollow cathode discharges
,” J. Appl. Phys.
129
, 023302
(2021
).43.
B.
Zheng
,
Y.
Fu
,
K.
Wang
,
T.
Schuelke
, and
Q. H.
Fan
, “
Electron dynamics in radio frequency magnetron sputtering argon discharges with a dielectric target
,” Plasma Sources Sci. Technol.
30
, 035019
(2021
).44.
A.
Anders
and
Y.
Yang
, “
Direct observation of spoke evolution in magnetron sputtering
,” Appl. Phys. Lett.
111
, 064103
(2017
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
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