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 10 19 10 21 m -3 >> plasma density 10 16 10 18 m -3 Electron-neutral collisions with large mean free path (> source size)
Cyclotron motion 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 electron cyclotron frequency ion collision collision Larmor radius L v c B Hall parameter h c B
Drift Electric force causes drift in E B direction Similarly, any gradients causes drift across B (magnetic field, plasma density, temperature) electron EB drift Some drifts are macroscopic: they are not visible on individual particle trajectories Collisions with gas reduce drift velocity especially for ions collision E B 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: w 1 m mw w m w e ( nt ) ew B t n inertia collisions electric force pressure force magnetic force Drift-diffusion approximation: neglect inertia terms e 2 nw n ( nt) hb 2 n ( nt) h bn ( nt) b m (1 h ) electron flux nw n ( nt) mobility tensor driving force Hall parameter h c eb m b B B
Macroscopic transport equations (2) Mobility tensor parallel (large) // e m B electron perpendicular (very small) 1 h m eb 2 2 E collision EB drift & diamagnetic drift h 2 1 h 1 B EB drift Electron heat flux: 5 Q T etnt 2 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 Many sources heated by waves plasma transport by diffusion Without magnetic field: Transport current free ambipolar plasma potential Electron Maxwell-Boltzmann equilibrium electric force pressure gradient n ( nt) T constant Boltzmann relation n n exp 0 T 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
Example: plasma transport in ECR vessel 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 grounded wall 0 V source chamber grounded or insulator process chamber insulator wall ionisation source cylinder axis G. J. M. Hagelaar, Plasma Sources Sci. Technol. 16, S57-S66 (2007)
radial position (m) radial position (m) Vessel with dielectric walls no (pre)sheath!! 0.2 potential 0 V 28 V 24 V 0.2 0.4 0.6 0.8 0.2 electron density axial position (m) 4x10 11 m -3 4x10 14 m -3 0.2 0.4 0.6 0.8 axial position (m) Magnetic confinement reduces plasma losses to source wall
radial position (m) radial position (m) Conducting vessel: Simon effect normal (pre)sheath 0.2 potential 0 V 16 V 12 V 0.2 0.4 0.6 0.8 0.2 plasma density axial position (m) & current lines 3x10 13 m -3 4x10 11 m -3 0.2 0.4 0.6 0.8 axial position (m) 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!!
radial position (m) Example: dipolar plasma source Modular source for large-volume plasma creation Full fluid simulations of elementary source : 6 4 2 wall 2.45 GHz magnetic field lines magnet electron cyclotron resonance 87.5 mt 0 0 2 4 6 8 0.10 0.12 axial position (m) argon @ 1Pa, 300 K power 10 W cylinder axis 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)
radial position (m) radial position (m) radial position (m) radial position (m) 1D Electron Boltzmann equilibrium // B 6 5 4 3 ECR power density (log) wall 6 magnetic field lines 4 electron temperature uniform // B but varies B 2 2 1.5 antenna 1 magnet electron temperature (ev) 0 0 0 2 4 6 8 0.10 0.12 0 2 4 6 8 0.10 0.12 6 6 axial position (m) 3.5 3 axial position (m) 2.5 2 4 2 4 3.5 3 2.5 4 2 13 15 17 electron density (10 16 m -3 ) plasma potential (V) 0 0 0 2 4 6 8 0.10 0.12 0 2 4 6 8 0.10 0.12 axial position (m) axial position (m) plasma potential < classical value >> classical value p 5.7T e
EB discharges Some sources apply voltage across magnetic field Penetrates in plasma bulk due to low conductivity (< sheath cond.) e n ds I e ds e ( nt) ds i conductance voltage = current Heat electrons in plasma bulk, to sustain plasma Accelerate ions ion beam for propulsion and materials processing
Example: Hall thruster discharge cathode anode propellant B E electrons ions symmetry axis magnetic core coils channel ceramic walls 3 cm
r (cm) r (cm) Structure of Hall thruster discharge (time averaged) B-lines cathode electric potential electron energy 6 anode 5 4 300 80 V 20 5 ev 10 38 18 10 3 2 5 6 m -3 m -3 m -3 /s 5 4 10 20 10 18 10 18 10 24 10 23 10 22 3 2 0 1 2 3 4 5 6 x (cm) neutral density 17 10 10 16 0 1 2 3 4 5 6 x (cm) plasma density 0 1 2 3 4 5 6 x (cm) ionisation rate G. J. M. Hagelaar et al, J. Appl. Phys. 91, 5592-5598 (2002)
Transit time oscillations in Hall thrusters potential + plasma density applied voltage ions in phase space J. Bareilles et al, Phys. Plasmas 11, 3035-3046 (2004)
Example: End-Hall ion source back plate magnet gas injection anode B field cathode ion beam symmetry axis substrate http://www.intlvac.com/ argon gas @ 500 K, 0.5 mg/s, 8.4 mpa voltage 90 V, current 1A 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)
radial position (m) radial position (m) radial position (m) radial position (m) Example: End-Hall ion source voltage 90 V drops in front of anode 3 2 1 cylinder axis 3 0 1 2 3 3 2 1 potential (V) 90 anode 40 30 plasma density axial position (m) (10 18 m -3 ) log 10 magnetic field lines 20 0 3 10 cathode filament 0.3 1 2 1 ioniz. rate (10 24 m -3 s -1 ) log 1 ion trajectories 0 0 0 1 2 3 0 1 2 3 axial position (m) 3 2 1 electron temperature (ev) 12 6 0 0 1 2 3 1 0.1 10 4 2 axial position (m) 3 axial position (m) electrons heated in front of anode ions oscillate in potential well ion beam
Axisymmetric sources: closed drift Dipolar ECR source Hall thruster End-Hall source http://www.intlvac.com/ Magnetic drift closed loop no transport to the wall Magnetic confinement due to m 2 2 1 ( B) eb r z back plate magnet gas injection anode B field cathode ion beam symmetry axis
No axial symmetry: ITER negative-ion source Prototype source IPP Garching Simplified 2D Cartesian geometry B magnetic filter 1 mt Gaussian profile RF heating either or extraction electrode 20 V Simplified conditions hydrogen gas @ 300 K, 0.3 Pa simple chemistry, no negative ions rf power 30 kw, dc voltage 20 V 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)
position (m) position (m) position (m) position (m) Infinite drift: B 20 V 0.2 0.1 plasma density (10 18 m -3 ) B 0.2 electron flux 0.1 0.2 potential (V) 0.1 0.2 electron temp. (ev) 0.1-0.1-0.2 2.1 1.8 1.5 1.2-0.1-0.2 drift -0.1-0.2 30 Ie = 207.8 27 24 21-0.1-0.2 7 6 4 5 3 0.1 0.2 0.3 0.4 X Axis Title 0.1 0.2 0.3 0.4 position (m) 0.1 0.2 0.3 0.4 X Axis Title 0.1 0.2 0.3 0.4 X Axis Title Almost no electron transport across filter Electron temperature drop due to poor heat conduction Magnetic drift along infinite direction
Y Axis Title Y Axis Title Y Axis Title Bounded drift: B + walls 0.2 0.1-0.1-0.2 plasma density (10 18 m -3 ) B 1.8 1.5 1.2 0.2 electron flux 0.1-0.1-0.2 drift 0.2 potential (V) 0.1-0.1-0.2 Ie = 970.7 27 21 24 20 V 0.2 electron temp. (ev) 0.1-0.1-0.2 heat drift 7 6 5 4 0.1 0.2 0.3 0.4 X Axis Title 0.1 0.2 0.3 0.4 position (m) 0.1 0.2 0.3 0.4 X Axis Title 0.1 0.2 0.3 0.4 X Axis Title Hall effect: plasma polarization to re-direct drift across filter Oblique electron current Oblique heat flux Electron temperature drop less important
Y Axis Title position (m) position (m) Y Axis Title Closed drift: B + periodic BC 0.2 0.1-0.1 periodic BC plasma density (10 18 m -3 ) 2.1 1.8 1.5 B 0.2 electron flux 0.1-0.1 drift 0.2 potential (V) 0.1-0.1 Ie = 269.6 27 24 20 V 0.2 electron temp. (ev) 0.1-0.1 7 5 6 4 3 2-0.2-0.2-0.2 21-0.2 0.1 0.2 0.3 0.4 X Axis Title 0.1 0.2 0.3 0.4 position (m) 0.1 0.2 0.3 0.4 X Axis Title 0.1 0.2 0.3 0.4 X Axis Title Plasma nearly symmetric Drift does not cause transport across filter
electron current (A/m) Electron current across magnetic filter 1000 100 infinite closed drift bounded drift periodic 1/B 1/B 2 2 B 1 ( B) 2 1 ( B) 1 B m eb 2 2 10 0.4 1 2 B (mt) Closed/infinite drift transport governed by ~ 1/B 2 Bounded drift transport governed by ~ 1/B Bounded drift effect scales as anomalous transport = 1/16B (Bohm)
RF Hall effect? (S. Mazouffre, Orléans, France) RF bounded magnetic drift? RF coil Capacative mode Inductive mode
2D PIC MCC model (Periodic Boundary Conditions) Instabilities Electron Density 27 0.1 m (96 cells) Driver Expansion 0.7 10 14 m -3 0.32 m (224 cells) Filter B dt- 0.2 X 10 9 s. No negative ions. z Perpendicular Electric Field (JXB direction) y x 4 kw/m 40 kw/m 2 40 kw (scaling 5. 10 3 ) +/- 300 V/m drift instability in the JxB direction transport across the filter only slightly larger than in 1D GEC 2011, Salt Lake City, Utah. GREPHE Toulouse,France
Instabilities in magnetic filter Fluid model with inertia terms periodic BC periodic BC B 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
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 noncylindrical 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