Hybrid modeling

1
Hybrid modeling
Annemie Bogaerts Department of Chemistry,
University of Antwerp (UA), Belgium Annemie.Bogae
rts_at_ua.ac.be
College Computational Plasma Physics TU/e,
6/2/04
2
Contents
1. Introduction Glow discharge
applications 2. Overview of models for GD
plasma Advantage of hybrid models 3. Hybrid
modeling network for GD plasma Submodels
coupling 4. Particle-in-cell modeling for
magnetron 5. Modeling network for laser ablation
3
1. Introduction
Glow discharge e.g. cell
dimensions cm3 argon gas (p 1 Torr) V
1 kV, I mA
anode
cathode
negative glow
anode zone
cathode dark space
4
1. Introduction (continued)
Processes in the glow discharge
second. electron emission
anode
sputtering
cathode
5
1. Introduction (continued)

• Applications of glow discharges
• Analytical spectrometry (source for MS, OES)
• Semiconductor industry (deposition, etching)
• Materials technology (deposition)
• Lasers
• Light sources
• Flat plasma display panels
• Environmental, biomedical applications

Aim of our work Better understanding to improve
results ? Modeling
6
2. Overview of models for GD plasma A. Analytical
model
Principle Simple analytical formulas, valid
for specific range of conditions
Advantage Simple, fast
Disadvantage Approximation, limited validity
7
2. Overview of models for GD plasma B. Fluid model
Principle Moment equations of Boltzmann
equation Conservation of mass, momentum,
energy (or drift-diffusion approximation)
Advantage Simple, fast Coupling to Poisson
equation for self-consistent E-field
Disadvantage Approximation (therm. equilibrium)
8
2. Overview of models for GD plasma C.
Collisional-radiative model
Principle Fluid model Conservation equations
for excited species (balance of production loss
processes)
Advantage Simple, fast
Disadvantage /
9
2. Overview of models for GD plasma D. Boltzmann
model
Principle Full solution of Boltzmann transport
equation (terms with E-gain, E-loss)
Advantage Accounts for non-equilibrium behavior
Disadvantage Mathematically complex
10
2. Overview of models for GD plasma E. Monte
Carlo model

• Principle
• Treats particles on lowest microscopic level
• For every particle during successive
  time-steps
• trajectory by Newtons laws
• collisions by random numbers

Advantage Accurate (non-equilibrium behavior)
simple
Disadvantage Long calculation time not
self-consistent
11
2. Overview of models for GD plasma F.
Particle-in-cell / Monte Carlo model

• 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
12
Which model should I use ? Every model has
advantages disadvantages
Solution use combination of models !!
13
2. Overview of models for GD plasma G. 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

• Advantage
• Combines the advantages avoids the
  disadvantages
• Accurate self-consistent reduced calculation
  time

Disadvantage /
14
3. Hybrid modeling network for GD plasma
Combination of different models for various
species Species Model used Ar0 atoms
no model (assumed thermalized) or heat
conduction equation electrons MC for fast
electrons fluid for slow electrons Ar
ions fluid (with electrons Poisson)
MC in sheath Ar0f fast atoms MC in sheath
Ar excited atoms collisional-radiative
model Cu0 atoms thermalization after
sputtering MC Cu, Cu(), Cu
collisional-radiative model Cu ions MC in
sheath
15
A. Heat transfer equation for Ar atoms
Gas temperature f (power input in Ar gas)

• Power calculated in MC models
• elastic collisions of Ar ions, Ar0f atoms, Cu
  atoms,
• and electrons with Ar atoms
• reflection of Ar ions and Ar0f atoms at cell
  walls

? nAr p / (k Tg)
16
B. Monte Carlo model for fast electrons

• During successive time-steps Follow all
  electrons
• Trajectory by Newtons laws
• Probability of collision
• ? Compare with RN (0 - 1) If Prob lt RN gt no
  collision
• If Prob gt RN gt collision

17
B. Monte Carlo model for fast electrons (cont.)

• Kind of collision partial collision prob.
  compare with RN
• 0 1
• Pexcit Pioniz Pelast
• New energy direction scattering formulas
  RN
• Repeat until electrons (Ep Ek) lt Ethreshold
• (typically in NG, where E-field weak)
• ? transfer to slow electron group

18
C. Fluid model for slow electrons Ar ions
Continuity equations for ions
electrons RAr and Re from MC
model Transport equations for ions electrons
(drift-diffusion) Poissons equation
19
D. MC model for Ar ions Ar0f atoms

• Only in CDS (where strong E-field) !
• During successive time-steps Follow all ions
  atoms
• Trajectory by Newtons laws
• Probability of collision,
• Kind of collision, defined by RNs
• New energy direction after coll.
• Repeat this until
• Ar ions bombard cathode
• Ar0f atoms E lt Ethermal

20
E. Collisional-radiative model for Ar atoms
65 Ar levels (individual or group of levels) For
each level continuity equation (balance
equation) 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 gt
14.7 eV) Additional processes for 4s levels
Ar - Ar collisions -gt (associative)
ionization Penning ionization of sputtered
atoms three-body collisions with Ar atoms
radiation trapping of resonant radiation
21
Energy level scheme for Ar CR model
22
F. Cu0 sputtering, thermalization

• Sputtering
• EDF of Ar, Ar0, Cu (from MC models)
• Sputtering yield Y(E) empirical formula
• Result Jsput
• Thermalization Monte Carlo model
• cfr. electron Monte Carlo model
• Result FT
• FT Jsput source term for Cu0 in next model

23
G. Cu0, Cu, Cu, Cu Collisional-radiative
model

• 8 Cu0, 7 Cu levels, 1 Cu level
• For each level balance equation
• Production loss processes
• Cu0 FT Jsput
• electron, atom impact excitation, de-excitation
• electron, atom impact ionization, recombination
• radiative decay
• Penning ionization, asymmetric charge transfer
• Transport
• diffusion for Cu0, diffusion drift for Cu and
  Cu

24
Energy level scheme for Cu CR model
25
H. Cu ions in CDS Monte Carlo model

• cfr. Ar ion Monte Carlo model
• Cu ions important for sputtering !

26
I. Coupling of the models
27
(No Transcript)
28
Detailed coupling
Start Ar - e- fluid model

• Input arbitrary creation rates RAr, Re

• Output
• Electric field Eax, Erad
• Interface CDS NG dc (r)
• Ar ion flux entering CDS jAr, dc (r)
• Ar ion flux at cathode jAr,0 (r)

29
This output input in e-, Ar, Ar0f MC models

• Electric field (Eax, Erad) to calculate
  trajectories of
• electrons and ions (Newtons laws)
• Interface CDS NG Ar flux entering CDS
  jAr, dc (r)
• to define Ar ions starting in Ar MC model
• Ar ion flux at cathode jAr,0 (r) to define
  electron flux
• starting at cathode (secondary electron
  emission)
• je,0 (r) - ? jAr,0 (r)

30
Coupling between e-, Ar, Ar0f MC models

• Electron MC model (1)
• Input from fluid model
• Output electron imp. ionization rate
  creation of Ar ions

• Ar MC model (1)
• Input from fluid model e- MC model (creation
  of Ar)
• Output Creation of Ar0f (elastic
  collisions)
• Creation of e- (ionization collisions)

• Ar0f MC model (1)
• Input from Ar MC model
• Output ionization collisions creation of e-,
  Ar

31
Coupling between e-, Ar, Ar0f MC models (cont)

• Ar MC model (2)
• Extra input from Ar0f MC model (creation of
  Ar)
• Same output ( more creation of Ar0f and e-
  )

• Ar0f MC model (2)
• Extra input from Ar MC model
• Same output ( more creation of e-, Ar )

• Ar MC model (3)
• Ar0f MC model (3)

Etc until CVG reached (creation of e- constant)
32
Coupling between e-, Ar, Ar0f MC models (cont)

• Electron MC model (2)
• Input from fluid Ar, Ar0f MC model
  (creation of e-)
• Same output ( more creation of Ar ions)

• Again Ar / Ar0f MC models
• Again e- MC model (3)

Etc until CVG reached total creation of Ar
ions and e- constant (typically after 2-3
iterations)
33
Coupling betw. e-, Ar, Ar0f MC models (summary)
e- MC model
Creation of Ar
Creation of Ar0f
Ar MC model
Ar0f MC model
Creation of Ar
Creation of e-
Creation of Ar
e- MC model
34
Coupling between MC models fluid model

• Global output of MC models
• Creation rates of Ar, electrons RAr, Re
• Input in Ar - e- fluid model (2)

• Output of Ar - e- fluid model ( same)
• Electric field Eax, Erad
• dc (r) and jAr, dc (r)
• Ar ion flux at cathode jAr,0 (r)
• Again input in MC models

Etc until CVG reached (E-field, jAr,0
constant) (typically after 5-10 iterations)
35
Coupling between MC fluid models (Summary)
Arbitrary creation of Ar , e-
e- - Ar fluid model

• E-field
• dc(r), jAr,dc(r)
• jAr,0 (r)

e-, Ar, Ar0f MC models
Creation of Ar , e-
No
CVG ?
Yes
36
Remark This is core of hybrid model

• Similar hybrid MC - fluid models for GD plasmas
• L.C. Pitchford and J.-P. Boeuf (Toulouse)
• Z. Donko (Budapest)

37
When Ar density ? constant, but calculated in
heat transfer model ? Additional model in loop
Arbitrary creation of Ar , e- constant nAr
Initial e- - Ar fluid model
E-field, dc(r), jAr,dc(r), jAr,0 (r)
e-, Ar, Ar0f MC models

• Creation of Ar , e- (? Fluid model)
• Power input in Ar gas (? Heat transf.model)

No
Ar heat transfer model
New nAr (function of position)
CVG ?
Yes
38
Coupling between MC fluid models strongest
coupling

• Hybrid MC fluid model determines
• Structure of GD (CDS, NG)
• Electric field distribution
• Ar ion and electron densities
• Electrical characteristics (I-V-p)

Other models more loosely coupled

• Other models
• Do not affect electrical structure of GD
• Are important for specific applications
• (e.g., spectrochemistry)

39
Results of the MC fluid models used as input
in the other models

• From e- MC model
• Electron impact ionization, excitation,
  de-excitation
• rates of Ar (? used as populating
  depopulating
• terms in Ar CR model)
• Electron impact ionization, excitation,
  de-excitation
• rates of Cu (? used as populating
  depopulating
• terms in Cu CR model)

40
Results of the MC fluid models used as input
in the other models (cont)

• From Ar and Ar0f MC models
• Ar and Ar0f impact ionization, excitation,
  de-excitation
• rates of Ar (? used as populating
  depopulating
• terms in Ar CR model)
• Ar and Ar0f flux energy distributions at
  cathode
• (? used to calculate flux of sputtered Cu atoms)

41
Results of the MC fluid models used as input
in the other models (cont)

• From Ar - e- fluid model
• densities of Ar ions and electrons
• (? used in some populating depopulating terms
• in Ar CR model and Cu CR model, i.e.,
  recombination)
• density of Ar ions (? used in Cu CR model,
• for the rate of asymmetric charge transfer
  ionization
• Ar Cu0 ? Ar0 Cu)
• E-field distribution (? used in Cu CR model,
• to calculate transport of Cu ions by migration)

42
Using all this input the Ar CR model 3 Cu
models (i.e., Cu MC thermalization model, Cu
CR model and Cu MC model) are solved in coupled
way

• (1) Ar CR model
• Input cfr. above, constant Cu atom density
• Output (among others) Ar metastable atom
  density
• (? used in Cu CR model, for the rate of
• Penning ionization Arm Cu0 ? Ar0 Cu
  e-)

43
Coupling of the 3 Cu models (i.e., Cu MC
thermalization model, Cu CR model and Cu MC
model)

• (1) MC model for Cu thermalization after
  sputtering
• Input Ar and Ar0f flux EDFs (from MC models)
• Ar density (constant or from heat
  transf.model)
• Output Thermalization profile
• (? used in Cu CR model, for initial
  distribution of
• Cu atoms (product FT Jsput))

44
Coupling of the 3 Cu models (cont)

• (2) Cu CR model
• Input from MC fluid models (cfr. above)
• FTJsput from Cu thermalization MC model)
• Output (among others)
• Flux of Cu ions entering CDS from NG
• Creation rate of Cu ions in CDS
• (? both used in Cu MC model in CDS)

45
Coupling of the 3 Cu models (cont)

• (3) Cu MC model in CDS
• Input from Cu CR model (cfr. above)
• Output (among others)
• Cu ion flux EDF at cathode
• (? used to calculate flux of sputtered Cu
  atoms)

46
Coupling of the 3 Cu models (cont)

• With this extra output of Cu MC model (i.e., Cu
  EDF)
• Again calculation of
• Sputtering flux (empirical formula)
• MC model for Cu thermalization
• Cu CR model
• Cu MC model in CDS

Etc until cvg reached (i.e., when sputter flux
constant) (typically after 2 3 iterations)
47
Coupling to the Ar CR model
Output of the 3 Cu models (among others) Cu0
density (? used in Ar CR model, to calculate
rate of Penning ionization Cu0 Arm ? Cu
Ar0 e-)
Output of Ar CR model (Ar metastable atom
density) again used as input in the 3 Cu models
Etc until cvg reached (i.e., when Cu0 and Cu
density constant) (typically after 2 iterations)
48
Summary Coupling of Ar CR model and the 3 Cu
models
Output from Ar / e- MC fluid models
Cu sputtering flux (emp.formula) MC model for Cu
thermalization
Arm density
Ar CR model
FT Jsput
Cu CR model

• Cu flux entering CDS
• Cu creation rate in CDS

Cu0 density (coupling back until CVG)
Cu MC model in CDS
Cu EDF at cathode
49
Results of the Ar CR model 3 Cu models used as
input in the Ar/e- MC fluid models

• From Ar CR model
• Populations of Ar metastable other excited
  levels
• (? used to recalculate the e-, Ar and Ar0f
  impact
• ionization, excitation, de-excitation rates of
  Ar
• in the MC models)
• Some production loss terms of Ar excited
  levels
• (? can influence densities of Ar and e- in
  fluid model)

50
Results of the Ar CR model 3 Cu models used as
input in the Ar/e- MC fluid models

• From the 3 Cu models
• Cu atom density
• (? used to recalculate the e- impact
  ionization,
• excitation, de-excitation rate of Cu in MC
  model)
• Cu ion density
• (? can influence E-field distribution in fluid
  model)
• Ionization rates of Cu (EI, PI, aCT) (? can
  influence
• densities of Ar and e- in fluid model)

51
However Results of the Ar CR model 3 Cu
models do not really affect the calculations in
the Ar/e- MC fluid models
Hence Coupling back not really necessary at
typical operating conditions
52
Summary
53
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-consistently
54
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 analytical importance Crater
profiles, erosion rates at the cathode Optical
emission intensities Effect of cell geometry,
operating conditions
55
Illustrations of some results
Electrical characteristics For given V,p,T
calculate current Calculated Measured
56
Densities Argon metastable density (1000 V, 1
Torr, 1.8 mA) Calculated Measured (LIF)
57
Densities Sputtered tantalum atom density (1000
V, 1 Torr, 1.8 mA) Calculated Measured
(LIF)
58
Densities Tantalum ion density (1000 V, 1 Torr,
1.8 mA) Calculated Measured (LIF)
59
Energies Electron energy distribution (1000 V,
0.56 Torr, 3 mA)
60
Energies Argon ion energy distribution (1000 V,
0.56 Torr, 3 mA) Calculated Measured (MS)
61
Energies Copper ion energy distribution (1000 V,
0.56 Torr, 3 mA) Calculated Measured (MS)
62
Information about sputtering at the
cathode Crater profile after 45 min. sputt.
(1000 V, 0.56 Torr, 3 mA) Calculated
Measured
63
Optical emission intensities Ar(I)
spectrum Calculated Measured
64
4. Particle-in-cell / Monte Carlo model for
magnetron discharge
Hybrid MC fluid model for magnetron
discharge Very complicated (B) approximations
necessary Only suitable for certain B/n values
65
Fluid model Effect of magnetic field
Initial try Rigourous description for electron
flux Assumptions 1D magnetic field (Bx) Ar
ions do not feel magnetic field
? very complicated
66
Fluid model Effect of magnetic field (contd)
Therefore Approximation Effect of magnetic
field in electron transport coefficients
? electron gyro-frequency ? average electron
momentum transfer collision frequency
Note If ? based on cross sections ?/? 500
However, Bradley ?/? 7.7 ? 4.2 (max. 25) ?
We use ? as fitting parameter (physically
realistic results) ? This gives ?/? 15
(depending on position)
67
Limitations of the hybrid model
Strong B/n time of electron confinement ? ?
negative space charge, negative plasma potential
Only observed exper. at much higher B (3000 G)
Alternative Transport coefficients calculated
from Boltzmann equation (swarm
parameters) However only done for 1D constant B
? In practice Hybrid MC-fluid approach only
useful for certain B/n (e.g. p gt 10 mTorr, B lt
200 G)
68
PiC-MC model for magnetron discharge
Real plasma particles replaced by
superparticles 1 superparticle W real particles
(W weight, e.g. 2x107)
Integration of equation of motion, moving
particles Fi ?? vi ?? xi
Particle loss/gain at the boundaries (emission,
absorption)
?telec 1 ps ?tion 10?telec
Weighting (E,B)j ?? Fi (interpolate field to
particles)
MC Collision ?
?t
Weighting (x,v)i ?? (?)I (interpol. charges
to grid)
Integration of Poissons equations on the grid
(?)i ?? (E)i
69
5. Modeling network for Laser Ablation
Physical picture
70
Typical operating conditions

• Pulsed lasers (NdYAG or excimer)
• ns, ps, fs - laser pulse (10 ns fwhm)
• UV IR wavelengths (? 266 nm)
• Laser intensity 107-1010 W/cm2 (109 W/cm2)
• Laser beam diameter 100 ?m
• Target metals, non-metals (Cu)
• Expansion in vacuum 1 atm background gas (He,
  Ar, air)

71
Applications of Laser Ablation

• Pulsed laser deposition
• Nanoparticle manufacturing
• Micromachining
• Surgery
• Spectrochemistry (MALDI, LIBS, LA-ICP-MS)

72
Different processes to describe in a model

• Laser-solid interaction heating, melting,
  vaporization
• Evaporated plume expansion (in vacuum)
• Plasma formation in the plume
• Laser plasma interaction (plasma shielding)
• (Plume expansion in background gas
  interactions)
• (Nanoparticle formation in the plume)

? Hybrid modeling network
73
1. Target heating, melting, vaporization

• 1D heat conduction equation
• Laser intensity
• Melting data for molten Cu
• Vaporization
• ? vapor density, velocity, temperature
• Input in next model

74
2. Expansion of evaporated plume
Conservation equations (mass density, momentum,
energy) Godunov method (shock waves)
75
3. Plasma formation

• Saha equations for ionization degree
• Ratio Cu/Cu0
• Ratio Cu2/Cu
• Conservation of matter
• Conservation of charge
• Internal energy density

76
4. Laser beam absorption in plasma
Inverse Bremsstrahlung ? Heating term in
energy conservation eq.
77
5. Coupling of the different parts
1) Laser-target interaction evaporation ?
Vapor dens, veloc, temp ? plume expansion
(boundary conditions for Euler equations)

• 2) Absorption of laser beam in plasma ?
• Energy gain in plume (term in Euler equation)
• Plasma shielding before reaching target
• (Intensity at target ltlt Laser intensity)

78
General conclusion
Different plasma species (or aspects) require
different modeling approach
Therefore To describe entire picture of GD
plasma (or LA) Combination of different models
desirable
? Hybrid model