Transcript Hybrid models of magnetized discharge plasmas: fluid electrons + particle ions Gerjan Hagelaar

Hybrid models
of magnetized discharge plasmas:
fluid electrons + particle ions
Gerjan Hagelaar
Centre de Physique des Plasmas et de leurs Applications de Toulouse
Université Paul Sabatier, 118 route de Narbonne,
31062 Toulouse Cedex 9, France
Introduction
Magnetic fields used in low-pressure discharges:
 magnetron
 electron-cyclotron resonance (ECR)
 helicon
 Hall-effect thruster
 etc…
(magnetized discharges)
Magnetic field  complex physics
Insight from hybrid models
Plan
 Elementary physics
 Hybrid models
 Limits of hybrid models
 Illustrative model results:
- ECR reactor
- Hall thruster
- Galathea trap
Elementary effects of the magnetic field
 Cyclotron motion  confinement
 Perpendicular electric field  EB drift
 Collisions destroy magnetic confinement
electron
ion
cyclotron
frequency
c  e B
m
Larmor radius
electron
EB drift
(azimuthal)
collision
 L  v  / c
B
E
B
Typical conditions
plasma
pressure
plasma density
magnetic field
electron temperature
0.1 – 10 mTorr
1015 – 1019 m-3
0.001 – 0.1 T
2 – 20 eV
lengths
Debye length
electron Larmor radius
ion Larmor radius
mean free path
plasma size
10-5 – 10-3 m
10-4 – 0.01 m
0.02 – 5 m
0.01 – 1 m
0.02 – 1 m
frequencies
electron cyclotron
electron collision
3108 – 21010 s-1
3105 –108 s-1
Long mean free path
Electrons are magnetized  collisions + ionization
Ions have only few collisions
Magnetic field not influenced by plasma
Modelling
Low pressure  particle-in-cell (PIC):
 electron and ion trajectories
 space charge electric fields
K. A. Ashtiani et al, J. Appl. Phys. 78 (4), 2270-2278 (1995).
S. Kondo and K. Nanbu, J. Phys. D: Appl. Phys. 32, 1142-1152 (1999).
J. C. Adam et al, Phys. Plasmas 11 (1), 295-305 (2004).
Magnetized PIC models cumbersome:
 high plasma density  small time steps, small cells
 important 2D effects
 interest in simpler faster models
 describe electrons by collisional fluid equations
Electron fluid equations
 Electron conservation
n e
  e  S ionisation
source
t
flux
 Anisotropic flux
 e  n e    ( n e Te )
drift
 Mobility tensor
(classical theory)
diffusion
2
e /m e

  2
 //  2
2
  c
  c 2
collision
frequency
cyclotron frequency
perpendicular mobility << parallel mobility
Hybrid models
Non-quasineutral scheme:
 ion particles  ni
 electron fluid  ne
 Poisson  
no plasma oscillations
large time steps
 0  2   e ( ne  ni )
Quasineutral scheme:
 ion particles  ni = ne
 electron fluid  
(ne  (neTe))  i
no sheaths  large cells
(Ohm’s law)
R. K. Porteous et al, Plasma Sources Sci. Technol. 3, 25-39 (1994).
J. M. Fife, Ph. D. thesis, MIT, 1998.
G. J. M. Hagelaar et al, J. Appl. Phys. 91 (9), 5592-5598 (2002).
Limits of the electron equations
 Anomalous transport B  empirical parameters
e m e ( 2   c )     1 / 16 B
2
classical mobility
?
Bohm mobility
 Non-local effects //B: inertia, mirror confinement
But: flux //B limited by boundaries
 // n e  //    //  // ( n e Te )
drift
diffusion
  ( r )   * (  )  T e (  ) ln  n e ( r ) / n 0 
(Boltzmann)
potential = constant + diffusion term
Magnetic field lines approximately equipotential
Numerical problem
Extreme anisotropy  numerical errors tend to
destroy the magnetic confinement
Special precautions necessary (flux scheme)
0
10
insulator wall
(a)
-1
anode
a
uniform B

l
 cathode
h c

insulator wall
normalized flux
10
standard method
-2
10
angle = /6
-3
10
/4
-4
10
analytical
(flux method)
aspect ratio 1/4
grid 80x20
/3
-5
electron flux in
the middle of
the channel
10
1
10
100
1000
Hall parameter
[cyclotron frequency] / [collision frequency]
Examples of model results
Non-quasineutral hybrid model  sheaths resolved
Fixed:
 Gaussian ionisation source
 uniform electron temperature (diffusion)
 electron collision frequency
Calculated:
 electron/ion densities
 electron/ion fluxes, currents
 self-consistent potential
Example I : Diffusion in ECR reactor
grounded wall 0 V
process
chamber
source chamber
grounded or insulator
insulator ionisation
wall
source
cylinder axis
ECR reactor with dielectric wall
radial position (m)
radial position (m)
no (pre)sheath !!
potential
0.2
0V
28 V
24 V
0.0
0.0
0.2
0.4
axial position (m)
electron density
0.2
14
0.6
0.8
11
-3
4x10 m
-3
4x10 m
0.0
0.0
0.2
0.4
0.6
0.8
axial position (m)
Magnetic confinement reduces loss to source wall
ECR reactor with grounded wall
radial position (m)
radial position (m)
normal (pre)sheath
potential
0.2
0V
12 V
16 V
0.0
0.0
0.2
0.4
0.6
0.8
axial position (m)
plasma density
& current lines
11
-3
0.2
current loop
4x10 m
13
-3
3x10 m
0.0
0.0
0.2
0.4
0.6
0.8
axial position (m)
Magnetic confinement shortcircuited by walls
A. Simon, Phys. Rev. 98 (2), 317-318 (1955).
Example II : Hall-effect thruster
cathode -300 V
ionisation
source
dielectric
gas
plasma
anode
0V
dielectric
cylinder axis
Hall-effect thruster
cathode sheath negligible
-300 V
potential
6
35 V
4
-100 V
-260 V
-230 V
2
radial position (cm)
radial position (cm)
6
electron
density
& current
lines
ion beam
4
2
15
17
-3
5x10 - 5x10 m log
0
2
4
axial position (cm)
6
0
2
4
axial position (cm)
acceleration region
Equipotential lines  magnetic field lines
Applied voltage penetrates in plasma bulk
6
trapped
low-energy
ions
Example III : semi-Galathea trap
ionisation source
gas
coil 0 V
source
cathode
-50 V
external
cathode
-300 V
plasma
grounded wall 0 V
dielectric wall
cylinder axis
A. I. Morozov and V. V. Savel’ev, Physics – Uspekhi 41 (11), 1049-1089 (1998).
Semi-Galathea trap
potential
radial position (cm)
6
-20 V
4
0V
70 % of ions
guided to exit
-25 V
0V
2
0
2
4
6
8
electron density
& position
current lines
axial
(cm)
6
radial position (cm)
negative
plasma
potential !
(inverted
presheath)
-50 V
electron current
from emissive
cathode to walls
4
15
17
-3
10 - 10 m log
2
0
2
4
6
8
axial position (cm)
Potential well reduces ion wall loss and guides ions to exit
Semi-Galathea trap without emission
potential
cathode sheath
radial position (cm)
6
-50 V
7V
-5 V
4
0V
2
0
radial position (cm)
6
2
4
electron density
axial position (cm)
6
8
6
8
4
15
17
-3
10 -10 m log
2
0
2
4
axial position (cm)
Potential well disappears because of cathode sheath
Conclusions
 In magnetized discharges, charged particle
transport and space charge fields are different
 This can be studied in 2D by hybrid models
 No predictive simulations, but insight in physical
principles