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