Plasma Modeling I: Modeling of plasmas Annemie Bogaerts · 1 superparticle = W real particles (W =...
Transcript of Plasma Modeling I: Modeling of plasmas Annemie Bogaerts · 1 superparticle = W real particles (W =...
Annemie Bogaerts
Department of Chemistry, University of Antwerp (UA),Research group PLASMANT
www.uantwerpen.be/plasmant
Plasma Modeling I: Modeling of plasmas
20th International School on “Low Temperature Plasma Physics”Bad Honnef, October 2, 2016
1. General overview of plasma models- Explanation of different models- Advantages and disadvantages
2. Examples of some models for various applications+ Typical results
Contents
1. General overview of plasma models
A. Analytical modelPrinciple: Simple analytical formulas, valid for specific range of conditionse.g.: Condition for self-sustaining discharge:
- +e- + A 2 e- + A+
L
= 1st Townsend coefficient (ionization) = 2nd Townsend coefficient (sec. e- emission)
A. Analytical model (cont)Electron multiplication in plasma, by electron impact ionisation: Balance equation: e
ed N Nd x
Solution: e e,0N N exp dx At cathode: Ne,0
At anode: e e,0N N exp L
Hence: for 1 electron at cathode: exp(L) electrons at anode Hence formed in plasma: exp(L) – 1
Secondary electron emission: Number of sec.electrons formed by these ions: * (exp(L) – 1)
Condition “self-sustaining discharge”: * (exp(L) – 1) = 11L ln 1
Number of electrons formed = number of ions formed
- +e- + A 2 e- + A+
L
A. Analytical model (cont)
Advantage: Simple, fast
Disadvantage: Approximation, limited validity
Principle: Rate equations (balance equations) for all species,based on production/loss by chemical reactions:
B. 0D chemical kinetics modelAlso called: Global model
ii i
B B
n Rprod Rlosst
Rprod / Rloss k n n (or :k n n n , or :An )
2 1 2 3 1 2 3 1
Electron impact reaction rates:From Boltzmann solver ( Look-up tables: k())+ Electron energy balance equation ( )
B. 0D chemical kinetics model (cont)
Advantage:Simple, fast (detailed chemistry)
Disadvantage:Approximation: No transport (assumes plasma = uniform)
However: based on gas flow velocity:can deduce spatial behavior from temporal behavior(equivalence batch reactor – plug flow reactor)
B. 0D chemical kinetics model (cont)
Example: Plasma jet
W. Van Gaens and A. Bogaerts, J. Phys. D: Appl. Phys. 46, 275201 (2013)
Plasma jet TU/e
(2 slm Ar gas feed,
6.5 W,
50% relat. air humidity)
B. 0D chemical kinetics model (cont)
Example: Plasma jetPlasma jet TU/e
(2 slm Ar gas feed,
6.5 W, 50% relat. air humidity)
• O, O3, O2(a), OH, H2O2
• NO, NO2, HNO2, HNO3
• O at longer distance
• O3 , O2(a) at longer distance
• OH dimerizes into H2O2 (~ 1.2 cm)
• Cluster formation important in
ion chemistry
• O2‐ and NO3
‐ : relatively low
(ppb) levels
W. Van Gaens and A. Bogaerts, J. Phys. D: Appl. Phys. 46, 275201 (2013)
B. 0D chemical kinetics model (cont)
Example: DBD reactor: filamentary character
Reactor length + gas flow residence time(correlation spatial – temporal behavior)
B. 0D chemical kinetics model (cont)
Example: DBD reactor: filamentary character
1 microdischarge pulse + afterglow- Te ~ 2-3 eV
(30 ns) - Ne ~ 1013 cm-3
Nanosecond pulsesInterpulse time ~µs
B. 0D chemical kinetics model (cont)
Example: DBD reactor: filamentary character5 consecutive pulses: species pass through several filaments(as function of distance, or function of residence time)
Principle: Moment equations of Boltzmann equation:Conservation of mass, momentum, energy
C. Fluid model: 1D or 2D
Continuity equations (for all species i):
ii i i
n (z,r) j (z,r) Rprod (z,r) Rloss (z,r)t
i i i i ij (z, r) n (z, r) E(z, r) D n (z, r)
Momentum equation: often drift-diffusion approximation: Transport equations (drift-diffusion for charged species; diffusion for neutrals):
Advantage: Simple, fast (detailed chemistry)Coupling to Poisson equation for self-consistent E-field
Disadvantage:Approximation (assumes species gain more or less same amount of energy from E-field as they lose by collisions
C. Fluid model (cont.)
eX X
eV n n n | E V
2
0
0
Electron energy balance equation:e
w w i e g ii
w e e e e
n j e j E k n Nt
j n E D n
5 53 3
Ions, neutrals: assumed Tg
Principle:Full solution of Boltzmann transport equation
(terms with E-gain, E-loss)
Advantage: Accounts for non-equilibrium behavior
jr vj j j el j inel j
ff (r, v, t) : v f a f C (f ) C (f )
t
D. Boltzmann model
Disadvantage: Mathematically complex Typically only used for electron behavior
(e.g., 2-term approximation): e.g., BOLSIG+
Principle: • Treats particles on lowest microscopic level• For every particle – during successive time-steps:
* trajectory by Newton’s laws:
* collisions by random numbers
E. Monte Carlo model
tm
)sin(Eqvvt
m2)sin(Eq
tvyy
tm
)cos(Eqvvt
m2)cos(Eq
tvxx
tmEq
vvtm2
Eqtvzz
radyy
2rady0
radxx
2radx0
axzz
2axz0
00
00
00
))E(n(sexp1obPr collcoll
Compare with RN (0 - 1): If Prob < RN => no collisionIf Prob > RN => collision
• Probability of collision:
• Kind of collision: partial collision prob. + compare with RN0 1
Pexcit Pioniz Pelast …
E. Monte Carlo model (cont)
2 RNcos 1 ; 2 RN1 8 (1 RN)
E. Monte Carlo model (cont)
- scattering angles (, ):
• New energy + direction: scattering formulas + RN, e.g.:- after ionization: Eprim = (E0 – Eion)*RN ; Esec = E0 – Eion – Eprim
or:
- after excitation: E = E - Eexcit
RNE d
E
ion diff
E
ion
prim
, ( , )
( )
00
0
sin( )cos( )sin( )sin( )
cos( )
cos( )cos( ) sin( ) sin( )cos( )cos( )sin( ) cos( ) sin( )sin( )
sin( ) cos( )
sin( )cos( )sin( )sin( )
cos( )
0 0 0 0 0
0 0 0 0 0
0 00x
- new angles (, ):
Advantage:Accurate (non-equilibrium behavior) + simple
Can be applied for all species
Disadvantage: Long calculation time + not self-consistentIn practice: used for electrons (in hybrid model)Or for species in sheath (detailed behavior, e.g., EDF)
E. Monte Carlo model (cont)
Principle: • Similar to MC model (Newtons’ laws, random numbers)
• But at every time-step: calculation of electric field
from positions of charged particles (Poisson equation)
Advantage: Accurate + self-consistent
Disadvantage: Even longer calculation time
F. Particle-in-cell / Monte Carlo model
Principle of PIC-MC (cycle):
Real plasma particles: replaced by superparticles1 superparticle = W real particles (W = weight, e.g. 2x107)
No
ΔtYes Postcoll.
velocities vi
’ vi
Δtelec = 1 ps
Δtion = 10Δtelec
Integration of equation of motion, moving
particles Fi vi
’ xi
Particle loss/gain at the boundaries (emission,
absorption)
Integration of Poisson’s equations on the grid
()i (E)i
Weighting (x,v)i ()I
(interpol. charges to grid)
Weighting (E,B)j Fi
(interpolate field to particles)
MCCollision
?
Which model should I use ?
Every model has advantages +
disadvantages…
Solution: use combination
of models !!
-> “hybrid model”
Principle: Because some species = fluid-like, others = particle-like: Combination of above models, e.g.:• Monte Carlo for fast (non-equilibrium) species• Fluid model for slow (equilibrium) species + E-field• (transfer from fast to slow group: if (Ek + Ep) < threshold)
Advantage: • Combines the advantages + avoids the disadvantages• Accurate + self-consistent + reduced calculation time
Disadvantage: /
G. Hybrid model
H. Collisional-radiativemodel (for excited levels)
Example: Ar: 65 Ar levels (individual or group of levels)
Energy level scheme for Ar* CR model
E (eV) E (1000 cm-1) jc = 3/2 jc = 1/2
s p d f,… s’ p’ d’ f’,…
130
125
120
115
110
105
100
95
90
1
23 4
5
(4s)
(3p6)
(4s’)
11.5
16.0
15.5
15.0
14.5
14.0
13.5
13.0
12.5
12.0
0 0
6
7
10 89
11
(4p)
(4p’)
(3d)(5s)
(5p)
(5s’) (3d’)
(5p’)
12 13
16
18
1417
15
19
(4d’)
(4d)
(6s’)(6s) 20
2122
23(6p)
(7s)
(8s)(9s)
(7p)
(8p)
(9p)
(5d)
(6d)(7d)
(8d) (9d)
25 27 29
31 33 3537 39 41
4345 47 495
57 - 65
5355
(7s’)
(8s’)(9s’
(6p’)
(7p’)
(8p’)
(9p’)
(5d’)
(6d’)(7d’)
(8d’)(9d’)
24
26 28 3032 34 3638 40 4244 46
48 50
5254
56 - 64Ar+ (3p5) 2P1/2
Ar+ (3p5) 2P3/2
H. Collisional-radiative model (for excited levels)
lossprod2*Ar
2
*Ar*Ar
*Ar*Ar RR
zn
Dr
nr
rr1D
tn
For each level: continuity equation (balance equation):
Additional processes for 4s levels:* Ar* - Ar* collisions -> (associative) ionization* Penning ionization of sputtered atoms* three-body collisions with Ar atoms* radiation trapping of resonant radiation
Production + loss processes for each level:* electron, Ar ion + Ar atom impact excitation, de-excitation, ionization* electron-ion three-body recombination, radiative recombination* radiative decay* Hornbeck-Molnar associative ionization (for Ar* levels with E > 14.7 eV)
I. Model for plasma-surface interactions:Molecular Dynamics simulations
• Newton’s laws: Calc. of position + velocities of atoms, by interatomic interaction potentials
200
0
21 tatvrr
tavv
m/Fa
UF
Time-integrated trajectory of impacting C-atomon amorphous C substrate
I. Model for plasma-surface interactions:Molecular Dynamics simulations (cont)
• Interaction potential: empirical potential (parameters)
Potential energy surface for impacting H-atom on diamond {111} surface
• Binding energy: sum of binding energies between atoms i and j :
i ij
ijAijijRsystem )r(Vb)r(VU
)exp()()(
)exp()()(
ijijijijijijA
ijijijijijijR
rBrfrV
rArfrV
E.g.: C-films: Brenner interaction potential
• Repulsive (VR) and attractive (VA) terms:
• ‘Many-body’ chemistry: binding order function bij
bij = f(local binding topology)
I. Model for plasma-surface interactions:Molecular Dynamics simulations (cont)
• Semi-infinite surface: periodic ±{x,y} boundaries
• Time scale ~ ns (t ~ 0.1-1 fs)Length scale ~ nm
• Initialisation of the simulation:– Define substrate– Particle impacts
» Film growth: sequentially» Reaction mechanisms» Random orientation +
{x,y} position above surface– Bottom layers fixed– Heat dissipation by heat bath
I. Model for plasma-surface interactions:Molecular Dynamics simulations (cont)
Advantage: • Very accurate +
deterministic (self-consistent)
Disadvantage:Long calculation time…
I. Model for plasma-surface interactions:Molecular Dynamics simulations (cont)
Illustration of MD simulations for film deposition(diamond like carbon)
4-fold coordinated C-atom
3-fold coordinated C-atom
2-fold coordinated C-atom
H-atom
5-fold coordinated C-atom
1-fold coordinated C-atom
Result: Calculated microscopic structure
of the film
4-fold coordinated C-atom
3-fold coordinated C-atom
2-fold coordinated C-atom
H-atom
5-fold coordinated C-atom
1-fold coordinated C-atom
Illustration of MD simulations for plasma catalysis
CH3 radicals on Ni catalyst surface
Illustration of MD simulations for carbon nanotube growth
Mechanism of cap formation of single walled carbon nanotubeon Ni nanoparticle
Illustration of MD simulations for plasma medicine
Breaking of peptidoglycan (~ bacterial cell wall damage) upon impact of O radicals
2. Examples of models for various applications
A. 0D chemical kinetics model: for DBD in CH4/CO2Greenhouse gas conversion (plasma chemistry)
B. PIC-MC model: for magnetron discharge
D. Hybrid model: for glow discharge with sputtering
C. MC model for electrons: for magnetron discharge
E. Hybrid model: for ICP etch reactor
A. 0D chemical kinetics model for DBD in CH4/CO2Greenhouse gas conversion
0D model (DBD in CH4/CO2 : greenhouse gas conversion)
Global warming ~ greenhouse gases conversion into value-added chemicals
Gas/vapor
plasma
Electrode
Electrode
Dielectric barrier
DBD reactor = very useful: Te >> Tgas
Enables reactions that would thermodynamically not occur at low Tgas
However: greenhouse gases = inert Classical processes: high T, p
Molecules Charged species Radicals
CH4
C2H6, C2H4, C2H2,C3H8, C3H6, C4H2
H2
O3, O2
CO2, CO
H2O, H2O2
CH2O, CH3OH, C2H5OH, CH3CHO, CH2CO, CH3OOH,
C2H5OOH
CH5+, CH4
+, CH3+,
CH2+, CH+, C+
C2H6+, C2H5
+, C2H4+,
C2H3+, C2H2
+, C2H+, C2+
H3+, H2
+, H+, H‐
O4+, O2
+, O+, O4‐, O3
‐, O2‐, O‐
CO2+, CO+
H3O+, H2O+, OH+, OH‐
Electrons
CH3, CH2, CH, C
C2H5, C2H3, C2H, C2, C3H7, C3H5
H
O
OH, HO2
CHO, CH2OH, CH3O, C2H5O, C2HO, CH3CO,
CH2CHO, CH3O2, C2H5O2
Plasma chemistry
75 species :
0D model (DBD in CH4/CO2 : greenhouse gas conversion)
Plasma chemistry
165 Electron-neutral collisions (ionization, excitation, dissociation,…)e.g.: e- + CH4 → e- + CH3 + H
1088 reactions taken into account:
50 Electron-ion recombination reactionse.g.: e- + O2
+ → O + O
873 Ion/neutral chemical reactions e.g.: CH3 + OH (+M) → CH3OH (+M)
CH4+ + CH4 → CH5
+ + CH3
0D model (DBD in CH4/CO2 : greenhouse gas conversion)
Calculated conversion, yields
Also calculated:Selectivities of formed products:At 50/50 CO2/CH4:‐ H2: 55%‐ CO: 48%‐ C2, C3: 6%, 30%‐ Formaldehyde: 13%‐ Methanol: 3% Future: plasma catalysis
CO2 + CH4 conversion + E‐efficiency, as f(SEI):(exp: Tu et al., Univ. Liverpool)
Trade‐off: conversion + E‐efficiency
0D model (DBD in CH4/CO2 : greenhouse gas conversion)
0D model (DBD in CH4/CO2 : greenhouse gas conversion)
Max. E‐efficiency: 15%(but low conversion…)
Computational optimization study (750 simulations):
(Effect of CO2/CH4 ratio, power, residence time, frequency)
Best results:‐ 84% conversion,‐ 8.5% E‐efficiency
Max + min obtained total
conversion + E‐efficiency:
DBD not competitive for industrial implementation Plasma catalysis to improve E‐efficiency + selectivity
Calculated conversion, yields
0D model (DBD in CH4/CO2 : greenhouse gas conversion)
Species densities: Different chemistry
CO2/CH4 : CxHy, CH2O, CH3CHO, H2/CO > 1
O2/CH4 : H2O, CH3OH, CH3OOH, CO2, H2/CO < 1
Comparison CO2/CH4 - CH4/O2
0D model (DBD in CH4/CO2 : greenhouse gas conversion)
Comparison CO2/CH4 - CH4/O2
Different reaction pathways: Explains different product formation
CO2/CH4 : CxHy, CH2O
O2/CH4 : H2O, CH3OH, CH3OOH,
CO2
B. PIC-MC model: for magnetron discharge
Reactor geometry + Magnetic field
0Target (cathode)
Anodez
r
290 2 1Rotation axis
240
Brmax = 300 – 1100 G
Vext = -400 Vp = 4 - 30 mtorrArgon
PIC-MC model (magnetron discharge)
Electrons:
Calculated Electron and Ar+ Ion Density
Maximum ~ 1017 m-3
at r = 1.85 cm (maximum Br)
Ar+ ions:
PIC-MC model (magnetron discharge)
Fast Cu Atom Density and Thermal Cu atom density
Fast Cu0 atoms: Thermal Cu0 atoms:
Concentrated near cathodeMax ~ 1018 m-3
Broad distributionMax ~ 4x1017 m-3
PIC-MC model (magnetron discharge)
Erosion profile: Calculated - Measured
Good agreement
0 10 20 30r, [mm]
-1.2
-0.8
-0.4
0Nor
malised
ero
sion
pro
file
CalculatedMeasured
PIC-MC model (magnetron discharge)
C. MC model for electrons: dual magnetron dischargeMagnetic field
MC model for electrons (dual magnetron discharge)
Single electron trajectory
Closed field
Mirror field
Closed field
Mirror field
Calculated electron density – ionization rate
MC model for electrons (dual magnetron discharge)
Closed field:
Mirror field:
MC model for electrons (dual magnetron discharge)
Comparison MC vs PIC-MC model
MC = • faster (minutes to 1 hour vs. several weeks for PIC)• complex geometries possible
But: not self-consistent:- Input B needed (e.g., from
Gmsh (http://www.geuz.org/gmsh/)
& GetDP (http://www.geuz.org/getdp/)
- Input E needed (e.g., from PIC-MC model)
Combination of different models for various speciesSpecies: Model used:Ar0 atoms no model (assumed thermalized)
or: heat conduction equationelectrons MC for fast electrons
fluid for slow electronsAr+ ions fluid (with electrons + Poisson)
MC in sheath Ar0
f fast atoms MC in sheath Ar* excited atoms collisional-radiative modelCu0 atoms thermalization after sputtering: MC Cu*, Cu+(*), Cu++ collisional-radiative modelCu+ ions MC in sheath
D. Hybrid model for GD plasma with sputtering
Hybrid model (GD)
Application: Analysis of solid materials:Sample to be analyzed = cathode of GDSputtering → Excitation, ionization in plasma → OES, MS
Hybrid model (GD)
E.g.: Depth profiling: Concentr as f(depth)
Plasma
A
C
B depth
conc
A B C
Ar+/e- fluid model
e- MC model Ar+/Ar0f MC model
Ar* CR model
Cu MC model
Cu/Cu+ CR model
Cu+ MC model
arbitrary RAr+ , Re,slow
Eax and Erad dc(r), jAr+,dc(r), jAr+,0(r)
Re,ion,Ar
Ri,ion,Ar and Ra,ion,Ar
new RAr+ and Re,slow
convergence ? no yes
e- MC model: - Re,exc,Ar, Re,exc,met, Re,ion,met: for Ar*m model - Re,ion,Cu : for Cu model Ar+/Ar0
f MC model: - Ri,exc,Ar, Ra,exc,Ar : for Ar*m model - fAr+(0,E), fAr(0,E) : for Cu model Ar+/e- fluid model: - nAr+ , ne,slow : for Ar*m model - nAr+ , V : for Cu model
nAr,met FT
Rion,Cu , jCu+,dc(r) nCu Ar*m model: - nAr,met : for Re,exc,met and Re,ion,met
- prod, loss: for nAr+ , ne,slow Cu model: - nCu : for Re,ion,Cu - nCu+ : for E, V - asymm. CT : for nAr+ - ioniz.terms: for ne,slow
convergence ? no
yes
final solution
fCu+(0,E)
Coupling of the models:
“Modeling network”
Hybrid model (GD)
Hybrid model (GD)
Input data for the modeling network
• Electrical data:Voltage, pressure, (gas temperature) Electrical current = calculated self-consistently
• Reactor geometry:(e.g., cylinder: length, diameter)
• Gas (mixture)• Cross sections, rate coefficients, transport coefficients,…
All other quantities: calculated self-consistentlyHybrid model (GD)
Typical calculation results
General calculation results:* Electrical characteristics (current, voltage, pressure)* Electric field and potential distribution* Densities, fluxes, energies of the plasma species* Information about collisions in the plasma
Results of importance for applications(sputtering, GD-OES,…):* Crater profiles, erosion rates at the cathode* Optical emission intensities* Effect of cell geometry, operating conditions
Hybrid model (GD)
Calculated gas temperature:VG9000 cell(1000 V, 3 mA, 75 Pa):
0.0 0.2 0.4 0.6 0.8 1.0
z (cm)
-1.25
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
1.25
r (cm
)
Grimm-type cell(800 V, 50 mA, 400 Pa)
+ Corresponding Ar gas density:
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-0.4
-0.2
0.0
0.2
0.4
r (cm
)
(a)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
z (cm)
-0.4
-0.2
0.0
0.2
0.4
r (cm
)
(b)
Hybrid model (GD)
Potential distribution: (VG9000 cell1000 V, 75 Pa, 3 mA)
CDS: ca. 2 mm longNG: major part (9 V)
0.0 0.2 0.4 0.6 0.8 1.0
z (cm)
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
-0.0
0.2
0.4
0.6
0.8
1.0
1.2
r (c
m)
-1000-800-600-400-200-100024689
Volt
Hybrid model (GD)
Densities: (VG9000: 1000 V, 75 Pa, 3 mA):
0.0 0.2 0.4 0.6 0.8 1.0
z (cm)
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
-0.0
0.2
0.4
0.6
0.8
1.0
1.2
r (c
m)
0.0E+000
5.0E+005
1.0E+006
2.0E+006
5.0E+006
1.0E+007
1.5E+007
2.0E+007
2.5E+007
3.0E+007
3.5E+007
cm-3
Fast electrons:
0.0 0.2 0.4 0.6 0.8 1.0
z (cm)
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
-0.0
0.2
0.4
0.6
0.8
1.0
1.2r (
cm)
0E+000
1E+010
2E+010
5E+010
1E+011
2E+011
3E+011
4E+011
5E+011
cm-3
Ar+ ions ~ slow electrons:
Hybrid model (GD)
Energies:Electron energy distribution (1000 V, 75 Pa, 3 mA):
Hybrid model (GD)
Energies:Argon ion energy distribution (1000 V, 75 Pa, 3 mA):
Calculated: Measured (MS):
Hybrid model (GD)
Energies:Copper ion energy distribution (1000 V, 75 Pa, 3 mA):
Calculated: Measured (MS):
Hybrid model (GD)
Information about sputtering at the cathode:Crater profile after 45 min. sputt. (VG9000: 1000 V, 75 Pa, 3 mA):
Calculated:
Measured:
0.0 0.2 0.4 0.6 0.8 1.0
z (cm)
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
-0.0
0.2
0.4
0.6
0.8
1.0
1.2
r (cm
)
-1000-800-600-400-200-100024689
Volt
Crater edge effect due to anode front plate:
Hybrid model (GD)
Grimm cell
-1.5 -1 -0.5 0 0.5 1 1.5r (mm)
-4
-2
0
dept
h (
m) BerekendCalculated
-1.5 -1 -0.5 0 0.5 1 1.5r (mm)
-10
-5
0
dept
h (
m) GemetenMeasured
Craters more flatBecause equipot.lines // cathode
0 0.5 1
z (mm)
-1
-0.5
0
0.5
1
r (m
m)
-780
-600
-400
-200
-100
-20
0
5
8
V (Volts)
Hybrid model (GD)
0
50000
100000
150000
200000
250000
300000
350000
400000
300 400 500 600 700 800 900 1000
(nm)
Inte
nsity
(a.u
.)
696.
5470
6.72
727.
2973
8.40
750.
3975
1.47
763.
5177
2.38
794.
8280
0.62
801.
4881
0.37
811.
53
826.
4584
0.82
842.
4685
2.14
866.
79
912.
3092
2.45
935.
42
965.
7897
8.45
0.E+00
5.E+14
1.E+15
2.E+15
2.E+15
3.E+15
3.E+15
4.E+15
4.E+15
5.E+15
5.E+15
300 400 500 600 700 800 900 1000
(nm)
Inte
nsity
(a.u
.)
415.
8642
0.07
434.
52
603.
3960
4.49
696.
5470
6.72 73
8.4
750.
3875
1.47
763.
5177
2.38
- 77
2.42
794.
8280
0.62
- 80
1.48
811.
5382
6.45
840.
8284
2.47
852.
1486
2.29
866.
79
912.
392
2.46 96
5.78
978.
45
Optical emission intensities:Ar(I) spectrumCalculated:
Measured:
Hybrid model (GD)
Optical emission intensitiesMeasuredCalculated
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
2
4
6In
t. (a
.u.) ArI (750.3 nm)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
00.40.81.21.6
2
Int.
(a.u
.)
ArII (476.5 nm)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
01020304050
Int.
(a.u
.)
ArI (811.5 nm)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5z (cm)
02468
Int.
(a.u
.)
CuI (324.75 nm)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
00.020.040.060.08
Int.
(a.u
.) ArI (750.3 nm)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
00.0040.0080.0120.0160.02
Int.
(a.u
.)
ArI (811.5 nm)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
00.0040.0080.0120.016
Int.
(a.u
.)
ArII (476.5 nm)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5z (cm)
00.0010.0020.0030.0040.005
Int.
(a.u
.)
CuI (324.75 nm)
DC: 0.6 Torr 1.55 mA - 417 V 3.1 mA - 455 V
4.65 mA - 480 V6.2 mA - 495 V
7.75 mA - 505 V
Hybrid model (GD)
E. Hybrid model for ICP etch reactor
HPEM (M.Kushner, University of Michigan)
MaxwellequationsE, B fields
Monte Carlo(electrons)
ne, Te, EEDF, reaction rates
Fluid(heavy particles)
ni, Γi, Es
Monte Carlo(heavy particles)
IEDF, IADF
Surface modelingsurface
reactions,ΓRi
Monte Carlo(heavy particles)etch profileevolution
R (cm)
Z(c
m)
0 10 20 300
5
10
15
20
25
30Coil
Quartz window
Silicon substrate
Electrode Pump port
Inlet
Hybrid model (ICP etch reactor)
Calculated species densities (Cl2, e-)
• Cl2: Maximum near inlet, then depletion (chemical reactions)Fairly uniform density profile
• Electrons: Maximum in center/near coil (ioniz.degree ~ 10-4)
Hybrid model (ICP etch reactor)
Calculated fluxes + angul.distrib. to wafer
• Ion fluxes ± 1000x lower than neutral fluxes• Ions: narrow angular distribution (directed by E-field)• Neutrals: wide angular distribution
Position from center of the wafer (cm)
Flux
tosu
bstra
te(c
m-2
s-1)
0 3 6 9 12 151012
1013
1014
1015
1016
1017
1018
1019
Cl2+
O
O2
ArCl
Cl2
Cl+O2
+
O+
Ar+
Angle from normal on substrate (°)
Angl
edi
strib
utio
n(°
-1)
-90 -60 -30 0 30 60 900
0.02
0.04
0.06
0.08
Neutral angle distribution
Ion angle distribution
Hybrid model (ICP etch reactor)
Calculated energy distributions to wafer
• Ions: bimodal distribution (rf bias at substrate):Ions “feel” rf bias amplitude (timesheath < 1 rf cycle)
• Neutrals: Maxwellian distribution at low E
Energy (eV)0 100 200 300 400 500 600 700 800
0
0.005
0.01
0.015
0.02(a)
Ener
gydi
strib
utio
n(e
V-1)
0 0.2 0.4 0.6 0.8 1 1.20
1
2
3
4
5
6
7
8(b)
Ions Neutrals
Hybrid model (ICP etch reactor)
Etch profile calculationsInput:
- Fluxes, energy and angular distributions of bombarding species (from plasma model)
- Surface reaction probabilities (etch, oxid, sputt, redepos)of the various species (Cl(+), Cl2(+), O(+), O2
(+), Ar+)on the various surface sites (Si, SiCl,SiCl2, SiCl3, SiO, SiO2)
Hybrid model (ICP etch reactor)
SummaryPlasma modeling :Most appropriate model depends on application:- Fluid (1D or 2D) or (0D) chemical kinetics modeling:
* Detailed information on plasma chemistry* fast
- Particle-in-cell – Monte Carlo simulations:* Microscopic – non-equilibrium behavior
- Monte Carlo modeling:* Faster, but not self-consistent
- Hybrid Monte Carlo - fluid simulations:* Combination: combines the advantages + eliminates the drawbacks of individual models* Extra information (etch profiles, OES,...)
- Molecular dynamics simulations: Surface modeling