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