Global Model of
Micro-Hollow Cathode Discharges
Laboratoire de Physique des Plasmas
Pascal CHABERT
Laboratoire 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
Microdischarges
• History of microplasmas : plasma displays (since 60s..)
– Microhollow cathode type (Schoenbach 1996)
DC source
100µm
– 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
Micro Hollow cathode Discharges
(MHCD’s)
Laboratoire de Physique des Plasmas
• Gas : Argon
DC
• Pressure : 30-200 Torr
• Excitation : DC
I-V Characteristics: mode jumps
450
400
0,6 Torr.cm
Voltage (V)
0,84 Torr.cm
confined inside
the micro-hole
350
1 Torr.cm
300
1,2 Torr.cm
250
200
150
0,0
0,4
self-pulsing
0,8
1,2
1,6
2,0
2,4
current (mA)
Aubert et al., PSST 16 (2007) 23–32
normal: cathode expansion
Equivalent electrical circuit
Hsu and Graves, J. Phys.D 36 (2003) 2898
Aubert et al., PSST 16 (2007) 23
P. Chabert, C. Lazzaroni and A. Rousseau,
J. Appl. Phys. 108 (2010) 113307
C. Lazzaroni and P. Chabert, Plasma
Sources Sci. Technol. (2011) 20 055004
Non-linear plasma resistance
500
150 Torr
Rp (k)
400
300
Rp = switch
200
100
0
abnormal:
confined in the hole
A3=0
0
1
Current (mA)
2
normal:
cathode expansion
Rp controls the physics of the transition between the two stable regimes
Stable and unstable regimes
320
1
3
2
Voltage (V)
dV/dt=0
equilibrium
240
dId/dt=0
160
0
1
equilibrium
2
Current (mA)
1
3
Abnormal regime
Stable regimes
Normal regime
2
Self-pulsing regime
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
Makasheva et al, 28th ICPIG(2007) 10
The resistance of the positive column is small
Structure in the cathodic region
1000
Intensity (arb. u.)
200Torr
800
600
Ar line
400
200
+
Ar line
Distance from cathode (µm)
0
-0.2
-0.1
0.0
0.1
0.2
r (mm)
120
100
+
Ar line maximum
80
60
40
20
0
50
100
150
P (Torr)
200
250
Calculation of sheath thickness (d)
sheath
d
R
• One-dimensional cylindrical geometry:
0
bulk
• Ions/electrons fluxes:
i ( R  d )
i ( R  d )   i ( R)
e ( R  d )
Sheath
i (R) Cathode
e (R)
e ( R)  i ( R)
 

.e  a (r)e
r
Sheath thickness vs pressure
Distance from cathode (µm)
120
-decrease of the sheath size with the
increase of pressure
100
Ar line maximum
-maxima of both emission lines
located after the sheath edge
80
60
-same trends for the evolution of d
than that of the maxima of the
+
Ar line maximum emission line
dionizing
40
20
0
50
100
150
200
250
300
-the sheath edge coincide with the
maxima of the ionic line
P (Torr)
C. Lazzaroni, P. Chabert, A. Rousseau and N. Sadeghi, J. Phys. D: Appl. Phys. 43 (2010) 124008
Abnormal and self-pulsing regime
Voltage (V)
350
300
250
200
150
1200
0
50
100
t (µs)
800
Allowed region for
the cathodic expansion
d>R
Voltage (V)
1000
600
400
P and V high enough
200
0
Too low voltage
0
40
80
120
P (Torr)
160
200
150
200
Power absorbed in the cathode sheath
• Electrons:
• Same for ions with:
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
P=150 T
Global model
-3
400
ne (10 m )
500
19
19
600
Experiments
300
200
100
0
0
2
4
6
Time (µs)
-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)
400
- with excited species/2
- without excited species*3
-3
700
ne at the peak (10 m )
Results in the self-pulsing regime
300
200
100
0
0
50
100
150
P (Torr)
Model with excited states in better
agreement with experiment
Results in the self-pulsing regime
P=150 Torr
Strong and
short Te peak
 e-impact
150
120
nAr ( P )
2
90
Recombination
ne
90
60
60
30
30
20
-3
* 3
ne (10 m )
Multi-step
ionization
120
19
-3
nexcited states (10 m )
150
nAr ( P )
* 3
0
1
0
2
4
6
8
0
10
Time (µs)
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).
Electron temperature dynamics
C. Lazzaroni and P. Chabert
J. 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