Post on 22-Dec-2015
Computational Plasma Physics
Kinetic modelling: Part 1 & 2
W.J. Goedheer
FOM-Instituut voor PlasmafysicaNieuwegein, www.rijnh.nl
What are kinetic methods and when do they apply
Kinetic methods retain information on the velocity distribution(hydrodynamic/fluid methods first integrate over velocity space)
Needed when distribution is non-Maxwellian
Kinetic methods are to be preferred when “mfp> L” or “coll>T”
mfp and coll depend on densities and cross-sections
But what are L and/or T??
Examples from (plasma) physics?
Kinetic models: non-Maxwellian
Collisions electrons mainly with neutral species
Low degree of ionization
Effective cooling of parts of the energy distribution function
Counteracted by Coulomb collisions at high degree of ionization (>10%)
E
Variations in space and time
L
High T Low T
x
T
T
High Pow Low Pow
t
P
Boundary layersTransition layers
Transient phenomenaSwitching onModulation
Power modulated discharges
Modulate RF voltage (50MHz)with square wave (1 - 400 kHz)
Observation in experiments UU)optimum in deposition rate
RF
Modulated discharge in SiH4
Results from a PIC/MC calculation: Cooling and high energy tail
Examples of Ion Energy DF at grounded electrode
From Th. Bisschops, Thesis TU/e, 1987
Interaction between E-field and ion motion does not result in a shifted Maxwellian
Kinetic models: strong spatial variation
Very low pressures: L = size of vessel (applies for [e,i,n])
Space charge boundary layer: L = Debye length (applies for [e,i])
Micro-structures (etched trenches): L = size of structure (applies for [e,i,n])
Shocks: L = extension of shock (applies for [e,i,n])
There may be a difference between momentum loss and energy loss
Kinetic models: strong temporal variation
Microwave discharges / high frequency RF discharges (applies for [e,(i)])
Start-up of discharges (applies for [e,i])
There may again be a difference between momentum loss and energy loss
Methods based on direct solution ofthe Boltzmann equation
Tricks to solve BE: Use symmetry if present Expand f in some small parameter
A method especially suitable for electrons
Electrons have a low mass high momentum loss in collisions energy loss in inelastic collisions
Elastic scattering redistribution over a sphere in velocity spaceSmall deviation from isotropic f in the direction of the average velocity
Therefore: expansion in Legendre polynomials Pn(cos )with the angle between average and actual velocityf = f0(v) + f1(v)P1(cos ) + f2(v)P2(cos )+…….
Note: “Amplitudes” depend on absolute value velocityand vary in space and time
An example: f0+f1cos()
-4 -2 0 2 4
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-2
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f0=v*exp(-v)
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-0.08000-0.07000-0.06000-0.05000-0.04000-0.03000-0.02000-0.0100000.010000.020000.030000.040000.050000.060000.070000.08000
f1 cos() =0.1v*exp(-v/2)cos()
f0+f1cos()
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How to calculate the amplitudes fn
V
V+dv
Transport from Neighbouring volume
Shift in velocitydue to electric field(eEt/m)
Elastic collisons move electrons from outside in
Net change in t
How to calculate the amplitudes fn
Balance of number of particles in shell between v and v+dv in dxdydzTransport in real spaceTransport in velocity spaceEffect of collisions
How to calculate the amplitudes fn
Balance of momentum in shell between v and v+dv in dxdydzTransport in real spaceTransport in velocity spaceEffect of collisions
Note that f1 is a vector, directed along the average velocity (=0)
Cutting off at f1: The Lorentz approximation
For elastic collisions with atoms/molecules, with mass M:
Special case: homogeneous, steady state
Temperature gas is zeroConstant electric field, average velocity (f1) along E
Special case: homogeneous, steady state
Solution: Backward integration, tri-diagonal system …
Special solutions:
r=-1 ; s=2 : Maxwellr= 0 ; s=4 : Druyvesteyn
Druyvesteyn has less energeticelectrons
rm vq
Reduced electric field
Inelastic collisionsCouple parts of the distribution function that are far apart
Example: Excitation-electron looses excitation energy (a few to >10 eV)-electron is set back in velocity
Source proportional to vf0(v)NMexc(v)
Same holds for ionization: Energy new electrons to be specified
Inelastic collisions: two T distribution
Noble gases have high first excitation energyFor lower energies only elastic energy losses: slow decay of f with vFor higher energies large energy losses: fast decay of f with vResulting distribution is characterised by two ”temperatures”
Eexc Eion
Inelastic collisions: ionization
Ui-du Ui+du Eion+2(Ui-du) Eion+2(Ui+du)
In ionization Eion is lostSuppose remaining energy equally dividedHow many electrons arrive between Ui-du and Ui+du
Ui-du < (U-Eion)/2 < Ui+duEion+2Ui-2du <U <Eion+2Ui+2du
So factor 2 from energy range + factor 2 from new electron:4f0(u)u1/2ion(u)
In steady state problems: new electrons neglected,Usually this has only a minor influence
An example, SiH4/H2, with inelastic collisions
EEDFs with 4eV av. Energy in SiH4/H2: non-Maxwellian
Some quantities (assuming f0 normalized)
Use of this approach in modelling
Local field approximation: Everything expressed in local E/N-fieldmobility and diffusion coefficientsreaction rates (ionization, excitation)average energy
Mean energy approximation:Use solution for various E/N-fields to construct table:(mobility, diffusion coefficient, rates)all as a function of the average energy(cf. table as function of temperature for Maxwellian f)Use fluid energy balance to obtain av.energy in simulation
Use of the mean energy approximation
Homogeneous gas of given composition, Nb1...bn
EEDF from Boltz.Eqn.Homogeneous electric field, constant in time
Mobility (e), Diffusion (De)
EEDF Average energy ( 1.5 kTe)
Reaction rates for processes Kproc (E,Nb1,Nb2,..Nbn)
Combine results in table for Kproc () , e(), De()
Modelling the electrons
+
Look-up table
One step further: time dependent E-field
Important characteristic times:
Loss of momentum: goes very fast f1 is in equilibrium with E-field
Loss of energy: Only fast in case of inelastic collision f0 can be out of equilibrium
Example: reaction of f0 in SiH4/H2
: E0cos(t): behavior depends on ratio and loss frequencies
Energy loss
Momentum loss
High frequency: smaller excursion f0
Collision frequency pressureTherefore: normalized to 1 Torr10% SiH4, 90% H2E=Emcos(t), f0 at Em, Em/2, 0, -Em/2 (1,2,3,4)
From Capitelli et al.: Pl. Chem. Plasma Proc. 8 (1988) 399-424
Time dependent, spatially inhomogeneous E field
Is possible in principle, but:More than 1 spatial dimension would take too much CPU timeReally steep gradients (sheaths) require fn with n>1
Solution: Monte Carlo methods Account in principle for all effects
Example: v*f0 in Nitrogen
E=3.6*104(x/L)5(10.8sin(t)), -L<x<L=2*80 MHz
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Position (128=0.04m)
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erg
y (1
28=64
eV
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y (1
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y (1
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Electron-electron collisions
Electrons efficiently share energy in elastic collisions
Collisions try to establish Maxwell distribution
More sophisticated operators conserve momentum and energy
Monte Carlo methods
Principle: Follow particles by - solving Newton’s equation of motion - including the effect of collisions - collision: an event that instantaneously changes the velocity
Note: The details of a collision are not modeled Only the differential cross section + effect on energy is used
Examples: Electrons in a homogeneous electric field Follow sufficient electrons for a sufficient time Obtain distribution over velocities etc. f0,f1
Positive ions in plasma boundary layer (ions have trouble loosing momentum)
Monte Carlo methods: Equation of motion
Leap-frog scheme
Alternative: Verlet schemeΔt))/2Δt(trΔt)(tr((t)v
t)O(Δt(t)ΔaΔt)(tr(t)r2Δt)(tr 42
Monte Carlo methods: B-field
Problem with Lorentz force: contains velocity, needed at time tSolution: take average
The new velocity at the right hand side can be eliminated by taking the
cross product of the equation with the vector
Monte Carlo methods: Boris for B-field
Equivalent scheme (J.P.Boris), (proof: substitution):
Monte Carlo methods: Collisions
Number of collisions: NMtot = 1/ per meter.
(x) = (0)*exp(- NMx) = (0)*exp(-x/)
dP(x)=fraction colliding in (x,x+dx)=exp(-x/)(1-exp(-dx/))=(dx/)exp(-x/)
P(x)=(1-exp (-x/))
Distance to next collision: Lcoll=-*ln(1-Rn) (Rn is random number,0<Rn<1)
Number of collisions: NMtot v= 1/ per second.
Time to next collision: Tcoll=-* ln(1-Rn)
Monte Carlo methods: CollisionsAnother approach is to work with the chanceto have a collision on vt: Pc=vt/
Ensure that vt<< to have no more than one collision per timestep Effect of collision just after advancing position or velocity introduces only small error
When there is a collision:
Determine which one: new random number
0.0 0.5 1.0 1.5 2.0 2.5 3.00.0
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0.2
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fracti
on
L/
no collision colliding once colliding twice colliding> twice
Monte Carlo methods: Null Collision
Problem: Mean free path is function of velocity Velocity changes over one mean free path
Solution: Add so-called null-collision to make v*tot independent of v Null-collision does nothing with velocity
Mean free path thus based on Max (v*tot)
Is rather time-consuming when v*tot peaks strongly
Monte Carlo methods: Null Collision
v
v*
v*2
v*1
v*3
v*tot
Max
v*0
1
1+2
1+2+3
1+2+3+..N
1+2+3+..N+ 0
’s normalized to maximum:Draw random number
Monte Carlo methods: Effect of collision
Determine effect on velocity vector
Retain velocity of centre of gravity
Select by random numbers two angles of rotation for relative velocity
Subtract energy loss from relative energy
Redistribute relative velocity over collision partners
Add velocity centre of gravity
Monte Carlo methods: Effect of collision
v1,v2 velocities in lab-frame prior to collision, w1,w2 in center of mass system
Monte Carlo methods: Effect of collision
A collision changes the size of the relative velocity if it is inelastic
A collision rotates the relative velocity
Two angles of rotation: and
usually has an isotropic distribution: =Rn*
has a non-isotropic distribution
Hard spheres:
Monte Carlo methods: Rotating the relative velocity
Step 1: construct a base of three unit vectors:
Step 2: draw the two angles
Step 3: construct new relative velocity
Step 4: construct new velocities in center of mass frame
Step 5: add center of mass velocity
Monte Carlo methods: ApplicabilityExamples where MC models can be used are:
- motion of electrons in a given electric field in a gas (mixture), see practicum- motion of positive ions through a RF sheath (given E(r,t))
Monte Carlo methods: Applicability
Main deficiency of Monte Carlo: not selfconsistent
- electric field depends on generated net electric charge distribution- current density depends on average velocities- following all electrons/ions is impossible
Way out: Particle-In-Cell plus Monte Carlo approach
Particle-In-Cell plus Monte Carlo: the basics
-Interactions between particle and background gas are dealt with only in collisions
-this means that PIC/MC is not! Molecular Dynamics
-each particle followed in MC represents many others: superparticle
-Note: each “superparticle” behaves as a single electron/ion
-Electric fields/currents are computed from the superparticle densities/velocities
-But: charge density is interpolated to a grid, so no “delta functions”
Particle-In-Cell plus Monte Carlo: Bi-linear interpolation
xi=ix xi+1=(i+1)x
xs, qs=eNs
i:=i+(xi+1-xs)qs/xi+1:=i+1+(xs-xi)qs/x
zi=iz
zi+1=(i+1)z
xj=jx xj+1=(j+1)x
ij:=ij+(zi+1-zs) (xj+1-xs) qs/(x z)
zs
xs
Particle-In-Cell plus Monte Carlo:Solution of Poisson equation
Boundary conditions on electrodes, symmetry, etc.
Electric field needed for acceleration of particle:(bi)linear interpolation, field known in between grid points
2
Particle-In-Cell plus Monte Carlo:Full cycle, one time step
Collisionnew v
Interpolatecharge to grid
Solve Poissonequation
Interpolate fieldto particle
Check lossat the walls
Move particlesF v x
Particle-In-Cell plus Monte Carlo:Problems
Main source of problems: Statistical fluctuations
Fluctuations in charge distribution: fluctuations in Eaverage is zero but average E2 is not numerical heating
Sheath regions contains only few electrons
Tail of energy distribution contains only few electronslarge fluctuations in ionization rate can occur
Particle-In-Cell plus Monte Carlo:Problems
Solutions:
-Take more particles (NB error as N-1/2 ) , parallel processing!
-Average over a long time
-Split superparticles in smaller particles when neededrequires a lot of bookkeeping, different weights!
Particle-In-Cell plus Monte Carlo:Stability
Plasmas have a natural frequency for charge fluctuations:
The (angular) Plasma Frequency:
And a natural length for shielding of charges:
The Debye Length:
Stability of PIC/MC requires:
Power modulated discharges
Modulate RF voltage (50MHz)with square wave (1 - 400 kHz)
Observation in experiments UU)optimum in deposition rate
Modulated discharges
Results from a PIC/MC calculation: Cooling and high energy tail
RF
Void
Crystal (21010 m-3)7.5 m radius
1-D Particle-In-Cell plus Monte Carlo Simulationof a dusty argon plasma
Capture cross section
Scattering:Coulomb, truncated at d
L/4L/8
w is energy electron/ion
Charging of the dust upon capture of ion/electron
The total charge is monitored on the gridpointsCharge of superparticle is added to nearest gridpointsDivision according to linear interpolationLocal dust charge is total charge divided by nr. of dust particlesThis number is density*dz*a2
For Monte Carlo the maximum v is computedNull-collision is used
Note the difference with the collisions with the uniformbackground gas: here we construct a grid-related probabilityof an event
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00
2.50x1015
5.00x1015
7.50x1015
1.00x1016
1.25x1016
1.50x1016
Ne
N+
Den
sity
(m
-3)
x/L
Simulation for Argon, 50MHz, 100mTorr, 70V, L=3cm
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00
1x1015
2x1015
3x1015
4x1015
5x1015
6x1015
NdQ
d/e
Ne
N+
Den
sity
(m
-3)
x/L
0 5 10 15 20 25 30 35 40100
101
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L/2 L/4 3L/16 L/8 L/16
EE
DF
(ar
b.u
n.)
Energy (eV)
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101
102
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108
L/2 L/4 L/8 L/16 L/32
EE
DF
(ar
b.u
n.)
Energy (eV)
dustfree with dust
Vd6V
Simulation for Argon, 50MHz, 100mTorr, 70V, L=3cm
dustfree with dust
0 5 10 15 20 25 30 35 40100
101
102
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109
L/2 L/8 L/16 L/32 0
IED
F (
arb
.un
.)
Energy (eV)
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102
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109
0 L/16 L/8 3L/16 L/4 L/2
IED
F (
arb
.un
.)
Energy (eV)
0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.0000.0
0.5
1.0
1.5
2.0
2.5
3.0
Av. El. Energy
Ion.Rate
3kT
e/2 (
eV),
Ion
.Rat
e (a
rb.u
n.)
x/L
0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.0000.0
0.5
1.0
1.5
2.0
2.5
3.0
Av. El. Energy Ion.Rate
3kT
e/2 (
eV),
Ion
.Rat
e (a
rb.u
n.)
x/L
Simulation for Argon, 50MHz, 100mTorr, 70V, L=3cmGeneration of internal space charge layers
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-1x1014
0
1x1014
2x1014
3x1014
Net
ch
arg
e / e
x/L
0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000-20000
-15000
-10000
-5000
0
5000
10000
15000
20000
Av.
Ele
ctri
c F
ield
(V
/m)
x/L
An internal sheath is formedinside the crystal
Ions are accelerated beforethey enter the crystal
This has consequences forthe charging + shielding
Particle-In-Cell plus Monte Carlo:What if superparticles collide?
Example: recombination between positive and negative ions
Procedure: number of recombinations in t: N+N-Krec t
corresponds to removal of corresponding superparticles randomly remove negative ion and nearest positive ion but: be careful if distribution is not homogeneous Again a grid-based probability can be used
A more sophisticated approach: Direct Simulation Monte Carlo
DSMC: Basics
Divide the geometry in cells
Each cell should contain enough testparticles
Newton’s equation: as before, but keep track of cell number
Collisions: choose pairs (in same cell!) and make them collide
Essential: the velocity distribution function is sum of -functions
DSMC: Choosing the pairs
Add null collision
Chance of collision of particle i with j is Pc=(Npp/Vcell)*Max(v)t
Average number of colliding pairs: n(n-1)* Pc/2
Select randomly n(n-1)* Pc/2 particle pairs (make sure no double selection)
See if there is no null collision, again with random number, comparingthe real chance for this collision (vr) with the maximum Max(v)
Perform the collision
DSMC: An example
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400
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1600
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3200
# p
art
icle
s
Energy (arb.un.)
25 50 75 100
Relaxation of a mono-energetic distribution to equilibrium20000 particles, hard sphere collisions, one cell contains all particles
Fast when possible, kinetic when needed:Hybrid models
Examples of hybrid models: Hydrodynamic ions and cold electrons Monte Carlo for fast electrons (tail EEDF)
Boltzmann electrons, Monte Carlo for ions
MHD model for plasma, Monte Carlo neutrals
Problems are due to coupling: iterations needed
B2-EIRENE for Magnum-psi
5 slm H2Th= 2eVTe=12 eV10^24 m^-2s^-1
T profile at inlet
Recycling at the target