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Modélisation de sources plasma froid magnétisé
Gerjan Hagelaar Groupe de Recherche Energétique, Plasma & Hors Equilibre (GREPHE)
Laboratoire Plasma et Conversion d’Énergie (LAPLACE) Université Paul Sabatier, CNRS
Toulouse, France
Magnetized low-temperature plasmas
Magnetron, helicon, ECR, Hall thruster, etc
Weak magnetic field < 0.1 T only electron cyclotron orbits
Magnetic field lines intercept walls transport losses
Neutral gas density 1019 – 1021 m-3 >> plasma density 1016 – 1018 m-3
Electron-neutral collisions with large mean free path (> source size)
Cyclotron motion
B
electron ion
Larmor radius
L
c
v
cyclotron
frequency
collision
Hall parameter ch
B
collision
Collisions with background gas destroy magnetic confinement Magnetized particles if both:
1) Larmor radius << plasma size 2) Hall parameter >> 1
Electrons magnetized, ions only sometimes/partially
Drift
E
EB drift
collision
electron
B
Electric force causes drift in E×B direction
Similarly, any gradients causes drift across B (magnetic field, plasma density, temperature)
Some drifts are macroscopic: they are not visible on individual particle trajectories
Collisions with gas reduce drift velocity especially for ions
F. F. Chen, Introduction to Plasma Physics and Controlled Fusion(Plenum, New York, 1984) J.-M. Rax, Physique des Plasmas (Dunod, Paris, 2005)
Macroscopic transport equations
Electron momentum equation:
1( )
ww w w w Bm m m e nT e
t ninertia pressure
force
magnetic force
electric force
collisions
( )n n nT w
Drift-diffusion approximation: neglect inertia terms
mobility tensor
driving force
c eBh
m
2
2( ) ( ) ( )
(1 )
en n nT h n nT h n nT
m h
w b b b
B
Bb
Hall parameter
electron flux
Macroscopic transport equations (2)
E
B
collision
electron Mobility tensor
2 21
m
h eB
2
1
1
h
Bh
/ /
e
m
EB drift
EB drift & diamagnetic drift
perpendicular (very small)
parallel (large)
Electron heat flux: 5
2T eTn T Q
thermal conductivity
tensor
Self-consistent magnetized plasma models
Particle-in-cell (PIC): trajectories of "super" particles tracked on grid and coupled with Poisson equation
No assumptions on distribution functions
Cumbersome due to: cyclotron orbits, high plasma density, 2D/3D needed for anisotropy
Fluid models: macroscopic equations for particle conservation, momentum & energy, coupled by quasineutrality or Poisson
Approximations/assumptions on distribution functions
Fast but computationally complex due to anisotropy
Hybrid models: combination of previous, e.g. fluid magnetized electrons + PIC non-magnetized ions
Plasma diffusion
Without magnetic field: Transport current free ambipolar plasma potential Electron Maxwell-Boltzmann equilibrium
0 expn nT
electric force
( )n nT
constantT
With magnetic field: Boltzmann equilibrium only // magnetic field lines Temperature gradient magnetic field lines Transport not current free, short-circuit wall currents Lower plasma potential
pressure gradient Boltzmann relation
Many sources heated by waves plasma transport by diffusion
Example: plasma transport in ECR vessel
grounded or insulator
ionisation
source
grounded wall 0 V
insulator
wallcylinder axis
source chamber
process chamber
Hybrid simulations: PIC ions + fluid electrons + Poisson equation Fixed: Gaussian ionisation source uniform electron temperature electron collision frequency
Calculated: electron/ion densities electron/ion fluxes, currents self-consistent potential
G. J. M. Hagelaar, Plasma Sources Sci. Technol. 16, S57-S66 (2007)
Vessel with dielectric walls
no (pre)sheath !!
0.0 0.2 0.4 0.6 0.8
0.0
0.24x10
11 m
-3
axial position (m)
radia
l p
ositio
n (
m)
electron density
4x1014
m-3
0.0 0.2 0.4 0.6 0.8
0.0
0.2
24 V28 V
axial position (m)
radia
l positio
n (
m) potential
0 V
Magnetic confinement reduces plasma losses to source wall
Conducting vessel: Simon effect
0.0 0.2 0.4 0.6 0.8
0.0
0.2
12 V16 V
axial position (m)
radia
l positio
n (
m) potential
0 V
normal (pre)sheath
0.0 0.2 0.4 0.6 0.8
0.0
0.24x10
11 m
-3
axial position (m)
radia
l p
ositio
n (
m)
plasma density
& current lines
3x1013
m-3
current loop
A. Simon, Phys. Rev. 98 (2), 317-318 (1955)
Magnetic confinement shortcircuited by walls Non-ambipolar transport Walls affect plasma transport all over the volume!!
Example: dipolar plasma source
A. Lacoste et al, Plasma Sources Sci. Technol. 11, 407 (2002) A. Lacoste et al, Plasma Sources Sci. Technol. 18, 1015017 (2009) G. J. M. Hagelaar et al, J. Phys. D: Appl. Phys. 42, 194019 (2009)
Modular source for large-volume plasma creation
Full fluid simulations of elementary source :
0.00 0.02 0.04 0.06 0.08 0.10 0.12
0.00
0.02
0.04
0.06
2.45 GHz
electron
cyclotron
resonance
wall
magnet
magnetic field lines
87.5 mT
axial position (m)
radia
l positio
n (
m)
cylinder axis
argon @ 1Pa, 300 K power 10 W
1D Electron Boltzmann equilibrium // B
0.00 0.02 0.04 0.06 0.08 0.10 0.12
0.00
0.02
0.04
0.06
2.5
3.5
32
1.5
electron temperature (eV)
axial position (m)
radia
l positio
n (
m)
0.00 0.02 0.04 0.06 0.08 0.10 0.12
0.00
0.02
0.04
0.06
2.53
3.54
electron density (1016
m-3)
axial position (m)
radia
l positio
n (
m)
0.00 0.02 0.04 0.06 0.08 0.10 0.12
0.00
0.02
0.04
0.06
13
1517
plasma potential (V)
axial position (m)
radia
l posi
tion (
m)
electron temperature uniform // B but varies B
plasma potential < classical value
p 5.7 eT
>> classical value
0.00 0.02 0.04 0.06 0.08 0.10 0.12
0.00
0.01
0.02
0.03
0.04
0.05
0.06ECR power density
(log)
antenna
wall
magnet
magnetic field lines
axial position (m)
radia
l positio
n (
m)
EB discharges
Some sources apply voltage across magnetic field
Penetrates in plasma bulk due to low conductivity (< sheath cond.)
Heat electrons in plasma bulk, to sustain plasma
Accelerate ions ion beam for propulsion and materials processing
( )ie n ds I e ds e nT ds conductance voltage = current
ions
ceramic walls
symmetry axis
B
cathode
anode electrons
channel
E propellant
coils
magnetic core
3 cm
Example: Hall thruster discharge
electric potential B-lines
plasma density
electron energy
ionisation rate neutral density
2
3
4
5
6
r (c
m)
5
2080300
V
510 101838
eV
0 1 2 3 4 5 62
3
4
5
6
101810
20
m-3
x (cm)
r (c
m)
0 1 2 3 4 5 6
1016
1017
1018
m-3
x (cm)0 1 2 3 4 5 6
1022
1023
1024
m-3/s
x (cm)
Structure of Hall thruster discharge (time averaged)
cathode
anode
G. J. M. Hagelaar et al, J. Appl. Phys. 91, 5592-5598 (2002)
applied voltage
ions in phase space
potential + plasma density
Transit time oscillations in Hall thrusters
J. Bareilles et al, Phys. Plasmas 11, 3035-3046 (2004)
Example: End-Hall ion source
H. R. Kaufman, R. S. Robinson, and R. I. Seddon, J. Vac. Sci. Technol. A 5, 2081 (1987) N. Oudini et al, J. Appl. Phys. 109, 073310 (2011) G. J. M. Hagelaar et al, Plasma Phys. Control. Fusion 53, 124032 (2011)
magnet
gas injection
anode
symmetry axis
cathode
B field
ion beam
back plate
http://www.intlvac.com/
argon gas @ 500 K, 0.5 mg/s, 8.4 mPa
voltage 90 V, current 1A
substrate
Example: End-Hall ion source
0.00 0.01 0.02 0.03
0.00
0.01
0.02
0.03
ion
trajectories0.01
0.11
310
ioniz. rate
(1024
m-3s
-1)
log
axial position (m)
rad
ial p
ositio
n (
m)
0.00 0.01 0.02 0.03
0.00
0.01
0.02
0.03
0.3
1
310
plasma
density
(1018
m-3)
log
axial position (m)
rad
ial p
ositio
n (
m)
0.00 0.01 0.02 0.03
0.00
0.01
0.02
0.03
electron
temperature
(eV)
126
4 2
axial position (m)ra
dia
l p
ositio
n (
m)
voltage 90 V
drops in front
of anode
ions oscillate
in potential
well
electrons heated
in front of anode
ion beam
0.00 0.01 0.02 0.03
0.01
0.02
0.03
30
magnetic field lines
cylinder
axis
90
potential
(V)
cathode
filament
anode
0
40
20
10
axial position (m)
rad
ial p
ositio
n (
m)
Axisymmetric sources: closed drift
http://www.intlvac.com/
magnet
gas injection
anode
symmetry axis
cathode
B field
ion beam
back plate
Hall thruster End-Hall source Dipolar ECR source
Magnetic drift closed loop no transport to the wall
Magnetic confinement due to
2 21 ( )
m
B eB
r z
No axial symmetry: ITER negative-ion source
RF heating
extraction electrode 20 V
magnetic filter
B
1 mT Gaussian profile
either
Simplified conditions hydrogen gas @ 300 K, 0.3 Pa simple chemistry, no negative ions rf power 30 kW, dc voltage 20 V
or
E. Speth et al., Nucl. Fusion 46, S220 (2006) G. J. M. Hagelaar, J. P. Boeuf et al, Plasma Source Sci. Techn. 20, 015001 (2011) G. J. M. Hagelaar et al, Plasma Phys. Control. Fusion 53, 124032 (2011)
Prototype source IPP Garching Simplified 2D Cartesian geometry
Infinite drift: B
0.0 0.1 0.2 0.3 0.4
-0.2
-0.1
0.0
0.1
0.2
1.2
2.1
1.8
1.5
plasma
density
(1018
m-3)
X Axis Title
po
sitio
n (
m)
0.0 0.1 0.2 0.3 0.4
-0.2
-0.1
0.0
0.1
0.2 electron
temp. (eV)
3
7
5
46
X Axis Title
po
sitio
n (
m)
0.0 0.1 0.2 0.3 0.4
-0.2
-0.1
0.0
0.1
0.2 potential
(V)
27
21
30
24
X Axis Titlep
ositio
n (
m)
0.0 0.1 0.2 0.3 0.4
-0.2
-0.1
0.0
0.1
0.2 electron
flux
Ie = 207.8
position (m)
positio
n (
m)
Almost no electron transport across filter
Electron temperature drop due to poor heat conduction
Magnetic drift along infinite direction
B
drift
20 V
Bounded drift: B + walls
0.0 0.1 0.2 0.3 0.4
-0.2
-0.1
0.0
0.1
0.2 electron
temp. (eV)
75
46
X Axis Title
Y A
xis
Title
0.0 0.1 0.2 0.3 0.4
-0.2
-0.1
0.0
0.1
0.2
1.2
1.5
1.8
plasma
density
(1018
m-3)
X Axis Title
Y A
xis
Title
0.0 0.1 0.2 0.3 0.4
-0.2
-0.1
0.0
0.1
0.2 potential
(V)
27
24
21
X Axis TitleY
Axis
Title
0.0 0.1 0.2 0.3 0.4
-0.2
-0.1
0.0
0.1
0.2 electron
flux
Ie = 970.7
position (m)
Y A
xis
Title
Hall effect: plasma polarization to re-direct drift across filter
Oblique electron current
Oblique heat flux
Electron temperature drop less important
heat drift
B
drift
20 V
Closed drift: B + periodic BC
0.0 0.1 0.2 0.3 0.4
-0.2
-0.1
0.0
0.1
0.2
2.1
1.5
1.8
plasma
density
(1018
m-3)
X Axis Title
Y A
xis
Title
0.0 0.1 0.2 0.3 0.4
-0.2
-0.1
0.0
0.1
0.2 electron
temp. (eV)
4
37 5
26
X Axis Title
Y A
xis
Title
0.0 0.1 0.2 0.3 0.4
-0.2
-0.1
0.0
0.1
0.2 potential
(V)
21
2724
X Axis Titlep
ositio
n (
m)
0.0 0.1 0.2 0.3 0.4
-0.2
-0.1
0.0
0.1
0.2 electron
flux
Ie = 269.6
position (m)
po
sitio
n (
m)
Plasma nearly symmetric
Drift does not cause transport across filter
B
periodic BC
drift
20 V
Electron current across magnetic filter
2 21 ( )
m
B eB
2
2
1
1 ( )
B
BB
Closed/infinite drift transport governed by ~ 1/B2
Bounded drift transport governed by ~ 1/B
Bounded drift effect scales as “anomalous” transport = 1/16B (Bohm)
0.4 1 210
100
1000
periodic
bounded drift
closed drift
infinite
1/B2
1/B
ele
ctr
on c
urr
ent (A
/m)
B (mT)
RF Hall effect? (S. Mazouffre, Orléans, France)
Capacative mode
Inductive mode
RF bounded magnetic drift?
RF coil
27
GREPHE GEC 2011, Salt Lake City, Utah.
Toulouse,France
2D PIC MCC model (Periodic Boundary Conditions)
Instabilities
dT- 0.2 X 109s.
No negative ions.
Electron Density
Perpendicular Electric Field (JXB direction)
x y
z
B
drift instability in the JxB direction
transport across the filter only slightly larger than
in 1D
+/- 300 V/m
0.7 1014 m-3
4 kW/m – 40 kW/m2 – 40 kW (scaling 5. 103 )
Filter
Expansion Driver
0.32 m (224 cells)
0.1 m
(96
cells)
Instabilities in magnetic filter
Magnetized plasma prone to instabilities, especially in plane B
Not only PIC simulations but also fluid models can describe instabilities, provided that inertia terms are retained
Open questions: Which instabilities arise in which conditions? Do instabilities destroy magnetic confinement? How do instabilities affect the particle distribution functions?
B B
periodic BC
periodic BC
Fluid model with inertia terms
METRIS: Magnetized Electron TRansport in Ion Sources
Project ANR - Jeunes Chercheurs, 09/2011 - 09/2014
Aim: improve fundamental understanding & modeling of magnetized low temperature plasma transport
Development of better, more robust, more realistic modeling methods (Gerjan Hagelaar, Romain Futtersack)
Dedicated experiments for model verification (Freddy Gaboriau, Laurent Liard, Romain Baude)
Collaboration with tokamak edge plasma specialists (Patrick Tamain, CEA Cadarache)
Conclusions
Modeling of magnetized low temperature plasmas is challenging due to strongly anisotropic transport and instabilities
Theories/methods from fusion plasma physics often not applicable due to different conditions: much lower B, field lines intercepting walls, collisions with neutral gas, …
Magnetized plasma diffusion depends on chamber walls and can be strongly non-ambipolar
Fundamental difference between magnetized plasma transport in non-cylindrical geometries vs cylindrical geometries: obstruction of drift by chamber walls causes 1/B transport and asymmetry
Magnetized plasma is prone to instabilities which can affect confinement
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