hybrid modeling annemie bogaerts department of chemistry, university of antwerp (ua), belgium...
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Hybrid modelingHybrid modeling
Annemie Bogaerts
Department of Chemistry, University of Antwerp (UA),
College “Computational Plasma Physics” TU/e, 6/2/04
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
Contents
-
V
+
cathode dark spacenegative glow anode zone
cathode anode
1. Introduction
Glow discharge:
e.g.: cell dimensions: cm3
argon gas (p ~ 1 Torr)V ~ 1 kV, I ~ mA
-
Ar0
Ar+e-
e- + Ar0 Ar* + e-
Ar+ + 2 e-
Ar+
M0 + e-
+ Ar*m
+ Ar+
Ar0 + e- + h
M+ , M*
Processes in the glow discharge:
anode
cathode
1. Introduction (continued)s
pu
tte
rin
g
second.electronemission
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
Principle:
Simple analytical formulas,
valid for specific range of conditions
2. Overview of models for GD plasmaA. Analytical model
Advantage: Simple, fast
Disadvantage: Approximation, limited validity
Principle:
Moment equations of Boltzmann equation:
Conservation of mass, momentum, energy
(or: drift-diffusion approximation)
2. Overview of models for GD plasmaB. Fluid model
Advantage:
Simple, fast
Coupling to Poisson equation for self-consistent E-field
Disadvantage:
Approximation (therm. equilibrium)
Principle:
~ Fluid model
Conservation equations for excited species
(balance of production + loss processes)
2. Overview of models for GD plasmaC. Collisional-radiative model
Advantage: Simple, fast
Disadvantage: /
Principle:
Full solution of Boltzmann transport equation
(terms with E-gain, E-loss)
2. Overview of models for GD plasmaD. Boltzmann model
Advantage: Accounts for non-equilibrium behavior
Disadvantage: Mathematically complex
Principle:
• Treats particles on lowest microscopic level
• For every particle – during successive time-steps:
* trajectory by Newton’s laws
* collisions by random numbers
2. Overview of models for GD plasmaE. Monte Carlo model
Advantage:
Accurate (non-equilibrium behavior) + simple
Disadvantage:
Long calculation time + not self-consistent
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)
2. Overview of models for GD plasmaF. Particle-in-cell / Monte Carlo model
Advantage: Accurate + self-consistent
Disadvantage: Even longer calculation time
Which model should I use ?
Every model has advantages +
disadvantages…
Solution: use combination
of models !!
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
2. Overview of models for GD plasmaG. Hybrid model
Advantage:
• Combines the advantages + avoids the disadvantages
• Accurate + self-consistent + reduced calculation time
Disadvantage: /
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
3. Hybrid modeling network for GD plasma
A. Heat transfer equation for Ar atoms
Gas temperature = f (power input in Ar gas):
P
r
Tr
rr
1
z
T g
2
g2
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)
During successive time-steps: Follow all electrons:
• Trajectory by Newton’s laws:
• Probability of collision:
Compare with RN (0 - 1): If Prob < RN => no collision If Prob > RN => collision
B. Monte Carlo model for fast electrons
tm
)sin(Eqvvt
m2
)sin(Eqtvyy
tm
)cos(Eqvvt
m2
)cos(Eqtvxx
tm
Eqvvt
m2
Eqtvzz
radyy
2rady0
radxx
2radx0
axzz
2axz0
00
00
00
))E(n(sexp1obPr collcoll
• 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) < Ethreshold (typically in NG, where E-field weak)
transfer to slow electron group
B. Monte Carlo model for fast electrons (cont.)
)RN1(81
RN21cos
Continuity equations for ions + electrons:
RAr+ and Re: from MC model
Transport equations for ions + electrons (drift-diffusion):
Poisson’s equation:
C. Fluid model for slow electrons + Ar+ ions
eee
ArArAr Rj
t
n|Rj
t
n
eeeeeArArArArArnDEnj|nDEnj
VE|0nne
V eAr0
2
Only in CDS (where strong E-field) !
During successive time-steps: Follow all ions + atoms:
• Trajectory by Newton’s laws
• Probability of collision, • Kind of collision, defined by RN’s• New energy + direction after coll.
Repeat this until:• Ar+ ions bombard cathode• Ar0
f atoms: E < Ethermal
D. MC model for Ar+ ions + Ar0f atoms
E. Collisional-radiative model for Ar* atoms
lossprod2*Ar
2
*Ar*Ar
*Ar*Ar RR
z
nD
r
nr
rr
1D
t
n
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 > 14.7 eV)
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
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
2
3 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 8
9
11
(4p)
(4p’)
(3d) (5s)
(5p)
(5s’) (3d’)
(5p’)
12
13 16
18
14
17 15
19
(4d’)
(4d)
(6s’)
(6s) 20
21 22
23 (6p)
(7s)
(8s) (9s)
(7p) (8p) (9p)
(5d)
(6d) (7d)
(8d) (9d)
25
27 29 31
33 35 37 39 41 43 45 47 49
51
57 - 65
53 55
(7s’)
(8s’) (9s’)
(6p’)
(7p’)
(8p’)
(9p’)
(5d’)
(6d’) (7d’)
(8d’) (9d’)
24
26 28 30
32 34 36 38 40
42 44 46 48
50 52 54 56 - 64 Ar+ (3p5) 2P1/2
Ar+ (3p5) 2P3/2
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
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++
Cu,lossCu,prodCuCu RRJ.t
n
Energy level scheme for
Cu* CR model
E (eV)
0
1
5
4
3
2
15
11
10
9
8
7
6
28
22
21
20
19
18
17
16
1
5
4
3
2
8 7 6
11
10
9
Cu0 3d10 4s 2S1/2
3d9 4s2 2D5/2
3d9 4s2 2D3/2
3d10 4p 2P1/2
3d10 4p 2P3/2
3d9 4s 4p 4P,4D,4F
3d10 5p 2P1/2,3/2
3d10 5s 2S1/2
3d10 4d 2D3/2,5/2
3d9 4s 4p 2P,2D,2F
3d10 6p 3d10 6s 3d9 4s 4p 2P,2D,2F 3d10 5d
Cu+ 3d10 1S0
16
15
14
13
12 3d9 4p 3P2
3d9 4p 1P1
3d9 4p 3P1,0,3F,3D,1F3,
1D2
3d9 5s 3D1,2,3 , 1D2
3d9 5p 3d9 4d
3d8 4s 4p 5D,5G,5F
3d8 4s2 3F,1D,3P,1G
3d9 4s 3D1,2,3 3d9 4s 3D1,2,3
Cu++ 3d9
3d9 4s 1D2
H. Cu+ ions in CDS: Monte Carlo model
• cfr. Ar+ ion Monte Carlo model
• Cu+ ions: important for sputtering !
I. Coupling of the models
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)
Detailed coupling
• 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)
Start: Ar+ - e- fluid model:
• Input: arbitrary creation rates: RAr+, Re
• Electric field (Eax, Erad): to calculate trajectories of
electrons and ions (Newton’s 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)
This output = input in e-, Ar+, Ar0f MC models:
• Electron MC model (1)
Input: from fluid model
Output: electron imp. ionization rate = creation of Ar+ ions
• Ar0f MC model (1):
Input: from Ar+ MC model
Output: ionization collisions: creation of e-, Ar+
• 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)
Coupling between e-, Ar+, Ar0f MC models:
• Ar+ MC model (2):
Extra input: from Ar0f MC model (creation of Ar+)
Same output: ( = more creation of Ar0f and e- )
Coupling between e-, Ar+, Ar0f MC models (cont):
Etc… until CVG reached (creation of e- constant)
• Ar+ MC model (3)
• Ar0f MC model (3)
• …
• Ar0f MC model (2):
Extra input: from Ar+ MC model
Same output: (= more creation of e-, Ar+ )
• Electron MC model (2)
Input: from fluid + Ar+, Ar0f MC model (creation of e-)
Same output: (= more creation of Ar+ ions)
Coupling between e-, Ar+, Ar0f MC models (cont):
Etc… until CVG reached:
total creation of Ar+ ions and e- constant
(typically: after 2-3 iterations)
• Again: Ar+ / Ar0f MC models
• Again: e- MC model (3)
• …
Coupling betw. e-, Ar+, Ar0f MC models (summary):
e- MC model
Ar+ MC model Ar0f MC model
e- MC model
Creation of Ar+
Creation of Ar+
Creation of Ar0f
Creation of e- Creation of Ar+
• 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
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)
Etc… until CVG reached (E-field, jAr+,0 constant)
(typically after 5-10 iterations)
Coupling between MC + fluid models (Summary):
e- - Ar+ fluid model
e-, Ar+, Ar0f MC models
CVG ?
Arbitrary creation of Ar+ , e-
Creation of Ar+ , e-
• E-field
• dc(r), jAr+,dc(r)
• jAr+,0 (r)
Yes
No
Similar hybrid MC - fluid models for GD plasmas:
• L.C. Pitchford and J.-P. Boeuf (Toulouse)
• Z. Donko (Budapest)
Remark:
This is “core” of hybrid model
When Ar density ≠ constant, but calculated in heat transfer model Additional model in loop:
Initial e- - Ar+ fluid model
e-, Ar+, Ar0f MC models
CVG ?
Arbitrary creation of Ar+ , e- + constant nAr
• Creation of Ar+ , e- ( Fluid model)• Power input in Ar gas ( Heat transf.model)
E-field, dc(r), jAr+,dc(r), jAr+,0 (r)
Yes
No Ar heat transfer model
New nAr (function of position)
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: • Do not affect electrical structure of GD• Are important for specific applications (e.g., spectrochemistry)
Other models: more loosely coupled
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)
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)
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)
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: Ar*m + Cu0 Ar0 + Cu+ + e-)
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))
Coupling of the 3 Cu models (cont)
(2) Cu CR model:
• Input: * from MC + fluid models (cfr. above)
* FT*Jsput 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)
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)
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)
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 + Ar*m Cu+ + Ar0 + e-)
Etc… until cvg reached
(i.e., when Cu0 and Cu+ density constant)
(typically after 2 iterations)
Output of Ar CR model (Ar metastable atom density):
again used as input in the 3 Cu models
Summary: Coupling of Ar CR model and the 3 Cu models
Ar CR model
Cu sputtering flux (emp.formula)MC model for Cu thermalization
Output from Ar+ / e- MC + fluid models
• Cu+ flux entering CDS
• Cu+ creation rate in CDS
Ar*m density
Cu CR model
FT * Jsput
Cu+ MC model in CDS
Cu+ EDF at cathode
Cu0 density(coupling back
until CVG)
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)
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)
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
Summary
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)
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
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
Electrical characteristics:For given V,p,T: calculate current
Calculated: Measured:
Illustrations of some results
Densities:Argon metastable density (1000 V, 1 Torr, 1.8 mA):
Calculated: Measured (LIF):
z (cm )
r (c
m)
0E+000
1E+009
2E+009
3E+009
4E+009
5E+009
1E+010
3E+010
1E+011
5E+011
1E+012
c m - 3
0
0.5
1.0
1.5
2.0
-0.5
-1.0
-1.5
-2.00 0.5 1.0 1.5 2.0
z (cm )
r (c
m)
0E+000
1E+007
5E+007
1E+008
2E+008
5E+008
1E+009
2E+009
3E+009
4E+009
c m - 3
2.0
1.5
1.0
0.5
0
-0.5
-1.0
-1.5
-2.00 0.5 1.0 1.5 2.0
Densities:Sputtered tantalum atom density (1000 V, 1 Torr, 1.8 mA):
Calculated: Measured (LIF):
z (cm )
r (c
m)
0E+000
1E+010
5E+010
1E+011
2E+011
5E+011
1E+012
2E+012
3E+012
c m - 3
2.0
1.5
1.0
0.5
0
-0.5
-1.0
-1.5
-2.00 0.5 1.0 1.5 2.0
z (cm )
r (c
m)
0.0E+000
1.0E+010
1.0E+011
2.0E+011
4.0E+011
6.0E+011
8.0E+011
1.0E+012
1.3E+012
1.6E+012
c m - 3
2.0
1.5
1.0
0.5
0
-0.5
-1.0
-1.5
-2.00 0.5 1.0 1.5 2.0
Densities:Tantalum ion density (1000 V, 1 Torr, 1.8 mA):
Calculated: Measured (LIF):
z (cm )
r (m
)
0E+000
5E+008
1E+009
2E+009
4E+009
6E+009
8E+009
9E+009
c m - 3
2.0
1.5
1.0
0.5
0
-0.5
-1.0
-1.5
-2.00 0.5 1.0 1.5 2.0
z (cm )
r (c
m)
0E+000
1E+009
5E+009
1E+010
2E+010
3E+010
4E+010
5E+010
6E+010
8E+010
c m - 3
2.0
1.5
1.0
0.5
0
-0.5
-1.0
-1.5
-2.00 0.5 1.0 1.5 2.0
Energies:Electron energy distribution (1000 V, 0.56 Torr, 3 mA):
Energies:Argon ion energy distribution (1000 V, 0.56 Torr, 3 mA):
Calculated: Measured (MS):
Energies:Copper ion energy distribution (1000 V, 0.56 Torr, 3 mA):
Calculated: Measured (MS):
Information about sputtering at the cathode:Crater profile after 45 min. sputt. (1000 V, 0.56 Torr, 3 mA):
Calculated:
Measured:
0
50000
100000
150000
200000
250000
300000
350000
400000
300 400 500 600 700 800 900 1000
(nm)
Inte
ns
ity
(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.
30
922.
4593
5.42
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
ns
ity
(a
.u.)
415.
8642
0.07
434.
52
603.
39
604.
49
696.
5470
6.72 73
8.4
750.
3875
1.47
763.
5177
2.38
- 7
72.4
279
4.82
800.
62 -
801
.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:
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
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
y
nD
z
nDEnEn
1
1
y
nD
z
nDEnEn
1
1
eex
ee
yeex
zee
2
2x
ez
ee
eex
yee
zeex
2
2x
ey
very complicated
Fluid model: Effect of magnetic field (contd)
Therefore: Approximation: Effect of magnetic field in electron transport coefficients:
0 02 2 2 2
1 1 and D ,
1 1B B
eD qB m
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)
= electron gyro-frequency = average electron momentum transfer collision frequency
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 > 10 mTorr, B < 200 G)
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)
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
?
No
ΔtYes Postcoll.
velocities vi
’ vi
Δtelec = 1 ps
Δtion = 10Δtelec
PiC-MC model for magnetron discharge
5. Modeling network for Laser Ablation
laser
target
Heat conduction T melting evaporation
•Plume expansion• Laser absorption
by plasma
laser
target
Heat conduction T melting evaporation
Physical picture:
Typical operating conditions:
• Pulsed lasers (Nd:YAG 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)
Applications of Laser Ablation:
• Pulsed laser deposition
• Nanoparticle manufacturing
• Micromachining
• Surgery
• Spectrochemistry (MALDI, LIBS, LA-ICP-MS)
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
• 1D heat conduction equation:
• Laser intensity:
• Melting: data for molten Cu
• Vaporization
vapor density, velocity, temperature
= Input in next model
1. Target heating, melting, vaporization
)t,x(ICx
)t,x(T
Cxt
)t,x(T
pp
1xexp)t(I)t,x(I 0
Conservation equations:
(mass density, momentum, energy)
Godunov method (shock waves)
2. Expansion of evaporated plume
radlaserIB
22
2
I2
vpEv
x2
vE
t
vpxt
v
x
v
t
• Saha equations for ionization degree:
Ratio Cu+/Cu0:
Ratio Cu2+/Cu+:
• Conservation of matter:
• Conservation of charge:
• Internal energy density:
3. Plasma formation
Tk
IPexp
h
Tkm2
n
1
x
xx
Tk
IPexp
h
Tkm2
n
1
x
xx
2
2/3
2e
vap1i
2ie
1
2/3
2e
vap0
1ie
e2i1i
2i1i0
xx2x
1xxx
2i211i1e x)IPIP(xIPTk)x1(
2
3
mE
Inverse Bremsstrahlung:
Heating term in energy conservation eq.
4. Laser beam absorption in plasma
2i2
21i2
1
2/1
ee4
e36
ie,IB
0ene,IB
nZnZTkm3
2
mch3
ne4
Tk
chexp1
nnQTk
chexp1
1) Laser-target interaction: evaporation
Vapor dens, veloc, temp plume expansion
(boundary conditions for Euler equations)
5. Coupling of the different parts
2) Absorption of laser beam in plasma
• Energy gain in plume (term in Euler equation)
• Plasma shielding before reaching target
(Intensity at target << Laser intensity)
Therefore:
To describe entire picture of GD plasma (or LA):
Combination of different models desirable
General conclusion
Different plasma species (or aspects):
require different modeling approach
Hybrid model