Transport in low-pressure plasmas:
From microelectronics to space propulsion

2019 Gaseous Electronics Conference Foundation Talk
College Station, Texas (not far from « Houston we have a problem »)Laboratoire de Physique des Plasmas

Pascal Chabert
Laboratoire de Physique des Plasmas
CNRS, Ecole Polytechnique, Palaiseau, FrancePlasmas in microelectronics++PlasmaCl+

P = 0.1-10 Pa

ClClx++

SiClySi• Plasma etching• Plasma depositioneClPlasma reactors

Radiofrequency Capacitive Discharges

Radiofrequency Inductive Discharges

Plasmas for space propulsion
TransferEarthLEO
GEOAll-electric platforms (Boeing 702 Small Platform, launch2015) - Orbit raising from LEO to GEO with EP
Miniaturization via the CubeSat technology
Orbit control not yet possible with EP
Requires downscaling to 1-100 W (order of µN)

Plasma Thrusters

Physics of low-pressure plasmas•

The physics of low-pressure (and low-temperature) plasmas
may be decomposed in two main phenomena:••

Electron heating (wave-particle interaction)
Plasma transport (or confinement)

Decaying fields~Ekz~
He- p  1 3 cm
Dielectricwindow•

Both include the physics of collisions

Agenda• This talk focuses on transport• First review the historical and “textbook” theories in thesimple case: the plasma slab or plasma column• Then challenge the assumptions: real discharges• Magnetized plasma column• The Hall thruster case• Transport instabilities and patterns

Transport in gas discharges
The low-pressure collisionless solution was studied by Tonks and Langmuir inthe late1920’s (L. Tonks and I. Langmuir. Physical Review, 1929)

f=0f(x)Positive ion generated
with zero velocity+acceleration

u(x,z)

dzzxx

Transport in gas discharges• The fully collisional solution was studied by Walter
Schottky, also in the 1920’s• A drift-diffusion theory that assumes that the plasma
density is zero at the wall and ignore sheaths
W. Schottky. Phys. Z., 25:635, 1924

The plasma-sheath transition
Problem identified by Langmuir: the plasma
equations are singular near the edge

Bohm in 1949 proposed that the plasma-sheathtransition occurs when the ion speed us is :

Fluid equations for plasma transport
Particle conservation equation

Momentum conservation equation

Commonly for a simple electropositive plasma:Isothermal approximation

Momentum conservation
The two limits in the pressure range
For electrons:

For ions in the steady-state collisionless (Tonks and Langmuir) limit:

For ions in the high-pressure (Schottky) limit:

Momentum conservationintermediate pressure regime
Valery Godyak showed in the 1980’s that a non linear friction force
should be used for ions in the intermediate pressure range:

Particle conservation
Particle conservation equation

Integrated over the discharge volume

Generalized h factor

In 1 D:

The h factor in 1D

Assumptions in the above (textbook) theories Isothermal electrons
 Constant neutral gas density Electropositive plasmas
 One-dimensional discharges

Isothermal electrons?
Strong heating near thesheath edge in CCPs

Constant neutral gas pressure?
Neutral depletion in high density plasmas:- Fruchtman , Makrinich , Chabert, Rax (2005)
Phys. Rev. Lett. 95 115002- Raimbault, Liard, Rax, Chabert, Fruchtman, Makrinich
Phys. Plasmas 14 (2007) 013503- Liard, Raimbault, Rax, Chabert,
J. Phys. D: Appl. Phys. 40 (2007) 5192–5195

New formula with neutral depletion:Raimbault and Chabert, PSST 18 (2009) 014017

Gas heating measurements in ICPs
Abada et al., J. Appl. Phys., 92 (2002) 4223

What happens with negative ions?
•Electronegative plasmas have a complexstructure•

Analytical solution of the plasma transport
is not easy (if possible at all)

•The generation and loss mechanisms ofnegative ions may have an important effecton the transport

Kouznetsov I, Lichtenberg A and Lieberman M J. Appl. Phys. 86 (1999 ) 4142
Franklin, J. Phys. D: Appl. Phys. 32 (1999) L71–L74.

Theory with negative ions
Solid lines fornumerical integration

Dashed lines:

Chabert, Plasma Sources Sci. Technol. 25 (2016) 025010
See also
Kim S, Lieberman M and Lichtenberg A J. Vac. Sci.Technol. A 24 (2006) 2025
Monahan and Turner, Plasma Sources Sci. Technol. 17 (2008) 045003

Geometry: 2D Effects
Non ambipolar diffusion due to 2D effects

See also Sternberg and Godyak PSST 20 (2011) 015018

Magnetized plasma column

Laboratoire de Physique des Plasmas

Effect of the magnetic field•

The motion parallel to magnetic field lines is not affected•

The transport perpendicular to magnetic field lines isstrongly reduced – but possible due to collisions•Standard theory predicts the following perpendiculartransport coefficients for electrons:

A. Fruchtman, G. Makrinich, and J. Ashkenazy. Two-dimensional equilibrium of a low temperaturemagnetized plasma. Plasma Sources Sci. Technol., 14(1):152, 2005.
N. Sternberg, V. Godyak, and D. Homan. Magnetic field effects on gas discharge plasmas, Phys.
Plasmas, 13(6):063511, June 2006.

•However, anomalous transport is observed
D. Bohm. The Characteristics Of Electrical Discharges In Magnetic Fields. R. K. Wakerling, A.
Guthrie, Mcgraw-hill Book Company, Inc., rst edition edition, 1949.

Effect of the magnetic field
Development of the diamagnetic
resistive drift instability

Azimuthal electric fieldDiamagnetic drift waves

Onset of the instability

Effective Hall parameter

Anomalous collision frequency

The effective collision
frequency is given by thefollowing formula at highmagnetic field:

The hm factor is boundedat high magnetic field (h0is the un-magnetized hfactor):

The h factor saturates at high B field

Dashed line from:
N. Sternberg, V. Godyak, and D. Homan. Magnetic field effects on gas discharge plasmas, Phys.
Plasmas, 13(6):063511, June 2006.

Anomalous transport inHall Thrusters

Laboratoire de Physique des Plasmas

Problems with Hall effect thrusters
 Difficulties to predict performance Role of wall materials (SEE): Insufficient [1,2] Anomalous electron transport along (Oz) Secondary electron emission (SEE)
due to e-/wall collisions
 Sheath instability[3,4] Electron drift instability
in the azimuthal direction

 Gradient driven fluid instability
Schematic picture of a HET[1] C. Boniface, L. Guarrigues, G.J.M. Hagelaar, J.P. Boeuf, D. Gawron, S. Mazouffre, Appl. Phys. Lett., 89, 161503 (2006)[2] N.B. Meezan, M.A. Cappelli, Phys. Rev., E66, 036401 (2002)[3] M.K. Scharfe, N. Gascon, M.A. Cappelli, E. Fernandez, Phys. Plasmas, 13, 083505 (2006)[4] A.W. Smith, M.A. Cappelli,Phys. Plasmas, 16, 073504 (2009)31

3D Dispersion relation for electrostatic
waves in magnetized plasmas
BrEzxyA. Ducrocq et al, Phys. Plasmas 13, 102111 (2006);T. Lafleur et al, Phys. Plasmas 23, 053503 (2016)zElectron Drift Instability (EDI)

J. Cavalier et al, Phys. Plasmas 20, 082107 (2013);

T. Lafleur et al, Phys. Plasmas 23, 053503 (2016)

Instability thresholds

Two EDIs:
1.
2.

T. Lafleur et al, Plasma Sources Sci. Technol. 26, 024008 (2017)

Azimuthal
Axial

Absolute vs convective instability

T.H. Stix, Waves in Plasmas (Springer-Verlag, 1992);

T. Lafleur et al, Plasma Sources Sci. Technol. 26, 024008 (2017)Electron-ion friction force
 Split f into equilibrium and perturbed parts. Equilibrium kinetic equation:

Multiply by mv and integrate over v

Electron-ion frictional drag force:2D z – q: PIC Simulations

Computed from
2D “z-q” PIC simulations: Average in time and over azimuthal direction. Axial momentum balance is: Azimuthal momentum balance is:

2D z – q: Azimuthal electric field2D z – q: Instability spectrum

 Spatio-temporal FFT spectrum matches that for the ion acoustic-type

dispersion relation of the EDI, and is similar to experiment**S. Tsikata et al, Phys. Plasmas 17, 112110 (2010);

Lazurenko et al, Phys. Plasmas 15, 034502 (2008)

2D z – q: Electron momentum balance Electron-ion friction force in PIC found from correlation function:

 This friction force completely dominates the azimuthal momentum balanceOther transport instabilities
and pattern formation

Laboratoire de Physique des Plasmas

0,5z = 37 cm

Courants de saturation ionique

0,00,5z = 34 cm

diminution fréquence

0,00,5z = 31 cm

0,00,3z = 28 cm

0,00,2z = 25 cm

0,00,2z = 22 cm

0,00,1zoom

0,0051015Temps (ms)2025z = 19 cm

30

Propagating double layers in expanding
electronegative plasmas
Propagating speed = 150 m/s

Z axis

HIGH
Vp26 cm- ++ -- + + - ++ 18 cm-LOW
Vp0 cmPlihon, Corr, Chabert, Raimbault, J. Appl. Phys. 98, 023306 (2005)

Courant
desaturation
Ion
saturation
ioniquecurrent

Transport instabilities
ooA complex dynamic develops as a function of the SF6 concentration C
Strong ion acoustic activity
Propagatingdouble layerIIIIIIIVV

13 < C < 14%

14 < C < 18%

VI

No D.L.
StaticD.L.
0 < C < 6%

6 < C < 8%

0ms 1ms 2ms

0ms 1ms 2ms

8 < C < 13%
0ms 1ms 2ms

0ms 1ms 2ms

Transition sequence
Tuszewski et al. Phys. Plasmas 10 539 (2003)
Plihon and Chabert, Phys. Plasmas 18, 082102 (2011)

0msC

10ms

20ms

C > 20 %0ms 1ms 2ms

Pattern formation in cylindrical ICPArancibia Monreal, Chabert, andGodyak
Phys. Plasmas 20 103504 (2013)
Song, Yang, Chabert and KushnerPhys. Plasmas 21 093512 (2014)
Desangle, Raimbault, Poyé, Chabertand Plihon, Phys Rev Lett (submitted)

Pattern formation in cylindrical ICP For non-Maxwellian distributions:J. H. Ingold, Phys. Rev. E 56, 5932 (1997)

 A linear stability analysis showsthat the system is unstable if :D. Mackey, L. Planti e, and M. Turner, Applied Mathematics Letters 18, 865 (2005)
Desangle, Raimbault, Poyé, Chabert and Plihon, Physical Review Letters (submitted)

Conclusion• Transport phenomena have been studied since the beginning of gas
discharge physics• However, there are still many unexplored or poorly understoodphenomena because of the complexity of low-temperature plasmas(complex gases, negative ions, gas heating, magnetized plasmas, etc..)• Theoretical, numerical simulations, and experimental efforts arenecessary.
• Dedicated experiments have to be built to validate the proposed theories• The physics of collisions and associated kinetic theory developments
have not been treated in this talk but are of fundamental importanceAcknowledgementsColleagues:
•••
•
•••
••••
•
•

Michael Lieberman
Allan Lichtenberg
Valery Godyak
Nick Braithwaite
Raoul FranklinTerry Sheridan
Miles Turner
Amnon Fruchtman
Mark Kushner
Scott Baalrud
Anne Bourdon
Jean-Luc Raimbault
Jean-Paul BoothStudents:••••••

Romain Lucken
Antoine Tavant
Thomas Charoy
Vivien Croes
Nicolas Plihon
Jaime Arancibia

Post-docs:•••••

Trevor Lafleur
Alejandro Alvarez Laguna
Roberto Martorelli
Victor Desangles
Cormac Corr