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Laboratoire de Physique des Plasmas
Global Model ofMicro-Hollow Cathode
Discharges
Pascal CHABERTLaboratoire de Physique des Plasmas, Ecole Polytechnique, Palaiseau FRANCE
Claudia LAZZARONI Laboratoire des Sciences des Procédés et des Matériaux, Université Paris 13, Villetaneuse FRANCE
Acknowledgments
• Micro-Hollow Cathode Discharges (MHCD’s):- Antoine Rousseau- Nader Sadeghi
100µm
Microdischarges
• History of microplasmas : plasma displays (since 60s..)– Microhollow cathode type (Schoenbach 1996) DC source
– Microarray (Eden 2001) AC source
– Microneedle (Stoffels 2002)rf source
– Micro APPJ (Schulz-von der Gathen 2008) rf source
• Applications in various fields:Surface treatment/ Light sources (excimer)/ Biomedicine
Global models
• Volume-averaged (0D) model: densities and temperatures are uniform in space
• Particle balance:
• Electron power balance:
Particle balance: low pressure• Particle losses at the walls. The term Pa contains
volume and wall losses. For the electron particle balance at low pressure, wall losses dominate and in one dimension:
• A plasma transport theory is required to relate the space-averaged electron density to the flux at the wall
Particle balance: issues in µdischarges
• In microdischarges, ionization is often non uniform. Classical low-pressure transport theories do not apply
• Fortunately in some instances volume losses dominate (recombination)
• However, evaluation of wall losses is a critical point for high-pressure discharges modeling
Other issues
• Properly evaluate the reaction rates:― Electron energy distribution function is
unknown― In microdischarges, Te is often strongly non uniform
in space, and may vary in time…
• Properly evaluate the electron power absorption:―Distinguish between electron and ion power (and
other power losses in the system)―Equivalent circuit analysis is often useful. I-V
characteristic of a device is a good starting point
So why one would use global models?
• They are easily solved, sometimes analytically. Therefore they provide scaling laws and general understanding of the physics involved
• They can be coupled to electrical engineering models to give a full description of a specific design
• They allow very complex chemistries to be studied extensively. Numerical solutions of global models with complex chemistries only take seconds
Laboratoire de Physique des Plasmas
Micro Hollow cathode Discharges (MHCD’s)
DC
• Gas : Argon
• Pressure : 30-200 Torr
• Excitation : DC
I-V Characteristics: mode jumps
0,0 0,4 0,8 1,2 1,6 2,0 2,4150
200
250
300
350
400
450
Vol
tage
(V)
current (mA)
0,6 Torr.cm
0,84 Torr.cm
1 Torr.cm
1,2 Torr.cm
self-pulsing
Aubert et al., PSST 16 (2007) 23–32
confined inside the micro-hole
normal: cathode expansion
Equivalent electrical circuit
P. Chabert, C. Lazzaroni and A. Rousseau, J. Appl. Phys. 108 (2010) 113307C. Lazzaroni and P. Chabert, Plasma Sources Sci. Technol. (2011) 20 055004
Hsu and Graves, J. Phys.D 36 (2003) 2898Aubert et al., PSST 16 (2007) 23
Non-linear plasma resistance
normal: cathode expansionabnormal:
confined in the hole
Rp controls the physics of the transition between the two stable regimes
0 1 20
100
200
300
400
500
Rp (k
)
Current (mA)
150 Torr
A3=0
Rp = switch
Stable and unstable regimes
1
2
3
13 Stable regimes
Self-pulsing regimeNormal regimeAbnormal regime
2
0 1 2160
240
320
Vol
tage
(V)
Current (mA)
equilibriumdId/dt=0
dV/dt=0equilibrium
Electrical signals/ Phase-space diagram
Electrical model of the MHCD
• Total absorbed power simply:
• What is the physical origin of Rp? What fraction of the total power goes into electrons?
Decomposition in two main regions
The resistance of the positive column is small
Makasheva et al, 28th ICPIG(2007) 10
Structure in the cathodic region
50 100 150 200 2500
20
40
60
80
100
120
Ar+ line maximum
Dis
tanc
e fro
m c
atho
de (µ
m)
P (Torr)
-0.2 -0.1 0.0 0.1 0.20
200
400
600
800
1000
Ar+ line
Ar line
200Torr
Inte
nsity
(arb
. u.)
r (mm)
Calculation of sheath thickness (d)
• One-dimensional cylindrical geometry:
• Ions/electrons fluxes: )(Ri
r0R
dsheath
bulk
)()( RR ie )()( RdR ii
Cathode
Sheath
)(Re
)( dRi
)( dRe
ee r )(. a
Sheath thickness vs pressure
50 100 150 200 250 3000
20
40
60
80
100
120
Ar+ line maximum
Dis
tanc
e fro
m c
atho
de (µ
m)
P (Torr)
Ar line maximum
dionizing
-decrease of the sheath size with the increase of pressure
-maxima of both emission lines located after the sheath edge
-same trends for the evolution of d than that of the maxima of the emission line
-the sheath edge coincide with the maxima of the ionic line
C. Lazzaroni, P. Chabert, A. Rousseau and N. Sadeghi, J. Phys. D: Appl. Phys. 43 (2010) 124008
Abnormal and self-pulsing regime
0 50 100 150 200
150
200
250
300
350
Vol
tage
(V)
t (µs)
0 40 80 120 160 2000
200
400
600
800
1000
1200
Vol
tage
(V)
P (Torr)
Allowed region for the cathodic expansiond>
R
Too low voltage
P and V high enough
• Electrons:
• Same for ions with:
Power absorbed in the cathode sheath
Power absorbed by electrons
• Fraction of power absorbed by electrons:
• Power absorbed in the volume under consideration:
Power balance
Power loss at the wall is negligible
Particle balance
Results in the self-pulsing regime
Experiments
Global model
0 2 4 60
100
200
300
400
500
600
700
n e (1019
m-3)
Time (µs)0 50 100 150
0
100
200
300
400
n e at
the
peak
(1019
m-3)
P (Torr)
- with excited species/2- without excited species*3
P=150 T
-ne(with exc states) ≈ 6 ne(without exc states) Importance of multi-step ionization
- ne(without exc states) = ne(exp)/3.7
- ne(with exc states) = 1.7 ne(exp)
Model with excited states in better agreement with experiment
Results in the self-pulsing regime
This has been observed experimentally:Aubert et al., Proceedings of ESCAMPIG, Lecce, Ed. M. Cacciatore, S. D. Benedictis, P. F. Ambrico, M. Rutigliano, ISBN 2-914771-38-X, Vol. 30G (2006).
0 2 4 6 8 100
30
60
90
120
150
0
30
60
90
120
150
nAr*(3P
2)
nAr*(3P1)
ne
ne (1
020 m
-3)
n ex
cite
d st
ates
(1019
m-3)
Time (µs)
P=150 Torr
Strong and short Te peak
e-impact
Multi-step ionization
Recombination
Electron temperature dynamics
C. Lazzaroni and P. ChabertJ. Appl. Phys. 111 (2012) 053305
Conclusions (1)
• Global models are very useful to understand the general behavior of plasma discharges
• They provide analytic solutions and scaling laws and allow fast numerical solutions of complex chemistry
• However, they rely on many assumptions and therefore need to be benchmarked by experiments or more sophisticated numerical simulations
Conclusions (2)
• The need for many simplifications and assumptions forces one to propose physical explanations, sometimes at the expense of being “rigorous”
• Global models do not explain the detailed and microscopic phenomena taking place in plasma discharges. they offer a representation of reality that becomes more accurate when more realistic fundamental assumptions are made