Transcript No Slide Title

Hybrid modeling
Annemie Bogaerts
Department of Chemistry,
University of Antwerp (UA),
Belgium
[email protected]
College “Computational Plasma Physics”
TU/e, 6/2/04
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
1. Introduction
Glow discharge:
anode
cathode
-
+
V
negative glow
cathode dark space
anode zone
e.g.: cell dimensions: cm3
argon gas (p ~ 1 Torr)
V ~ 1 kV, I ~ mA
1. Introduction (continued)
Processes in the glow discharge:
Ar0
second.
electron
emission
Ar+
e- + Ar0
e-
Ar* +
Ar+ + 2
-
e-
Ar0 + e- + h
e-
anode
sputtering
Ar+
M0 + e+ Ar*m
+ Ar+
cathode
M+ ,
M*
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
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
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)
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: /
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
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 Newton’s laws
* collisions by random numbers
Advantage:
Accurate (non-equilibrium behavior) + simple
Disadvantage:
Long calculation time + not self-consistent
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
Which model should I use ?
Every model has advantages +
disadvantages…
Solution: use combination
of models !!
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: /
3. Hybrid modeling network for GD plasma
Combination of different models for various species
Species:
Ar0 atoms
electrons
Ar+ ions
Ar0f fast atoms
Ar* excited atoms
Cu0 atoms
Cu*, Cu+(*), Cu++
Cu+ ions
Model used:
no model (assumed thermalized)
or: heat conduction equation
MC for fast electrons
fluid for slow electrons
fluid (with electrons + Poisson)
MC in sheath
MC in sheath
collisional-radiative model
thermalization after sputtering: MC
collisional-radiative model
MC in sheath
A. Heat transfer equation for Ar atoms
Gas temperature = f (power input in Ar gas):
 2Tg
z 2
1   Tg 
P
 r
  

r r  r 

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)
B. Monte Carlo model for fast electrons
During successive time-steps: Follow all electrons:
• Trajectory by Newton’s laws:
qE ax
 t 2
z  z 0  v z 0 t 
2m
qE ax
v z  v z0 
t
m
qE rad cos()
 t 2
x  x 0  v x 0 t 
2m
qE rad cos()
v x  v x0 
t
m
qE rad sin( )
 t 2
y  y 0  v y0 t 
2m
qE rad sin( )
v y  v y0 
t
m
• Probability of collision: Pr ob coll  1  exp   s(n coll (E)) 
 Compare with RN (0 - 1): If Prob < RN => no collision
If Prob > RN => collision
B. Monte Carlo model for fast electrons (cont.)
• Kind of collision: partial collision prob. + compare with RN
0
Pexcit Pioniz
1
Pelast
…
• New energy + direction: scattering formulas + RN:
2 RN
cos   1 
1  8  (1  RN )
Repeat until electrons: (Ep + Ek) < Ethreshold
(typically in NG, where E-field weak)
 transfer to slow electron group
C. Fluid model for slow electrons + Ar+ ions
Continuity equations for ions + electrons:
n Ar
t
   jAr  R Ar
|
n e
   je  R e
t
RAr+ and Re: from MC model
Transport equations for ions + electrons (drift-diffusion):
jAr  Ar n Ar E  D Ar n Ar
| je  e n e E  D e n e
e
Poisson’s equation:  V  n Ar  n e  0
0
2
| E   V
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 Newton’s laws
• Probability of collision,
• Kind of collision,
• New energy + direction after coll.
Repeat this until:
• Ar+ ions bombard cathode
• Ar0f atoms: E < Ethermal
defined by RN’s
E. Collisional-radiative model for Ar* atoms
65 Ar levels (individual or group of levels)
For each level: continuity equation (balance equation):
 n Ar*
 2 n Ar*
1    n Ar* 
 r
  D Ar*
 D Ar*
 R prod  R loss
2
t
r r   r 
z
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
E (eV)
E (1000 cm-1)
jc = 3/2
s
p
jc = 1/2
f,…
d
s’
p’
130
16.0
Ar+ (3p5) 2P3/2
Energy level scheme
for Ar* CR model
(9p)
(9s)
(8s)
(8p)
39
33
(7p)
(7s)
(9d)
45 (8d)
37
31
27
(6p)
43
(7d)
(6d)
(5d)
(5s)
13
12
(8p’)
(7p’)
(7s’)
38
32
36
120
(5p’)
(3d)
15
(4p’)
13.0
105
6
12.5
100
12.0
95
3
11.5
2
90
(3p6)
1
34
(5d’)
28
(4d’)
19
17
14
14.0
10
(4s)
(6d’)
24
110
7
40
14.5
(5s’)
0
0
(4s’)
46
(7d’)
30
21
13.5
(4p)
48
42
22
(6s’)
15.0
(9d’)
(8d’)
44
26
(6p’)
115
16
(9p’)
(9s’
)
(8s’)
(4d)
18
(5p)
125
50
29
25
20
15.5
35
23
(6s)
54
52
53
5
47 49 1
41
f’,…
Ar+ (3p5) 2P1/2
56 - 64
57 - 65
55
d’
5
4
11
9
8
(3d’)
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:
 n Cu
.J Cu  R prod,Cu  R loss,Cu
t
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++
E (eV)
28 16
Cu++ 3d9
3d9 5p
Energy level
scheme for
Cu* CR model
22
21 15
3d9 4d
3d8 4s 4p 5D,5G,5F
3d9 5s 3D1,2,3 , 1D2
20
19
18
3d8 4s2
14
17
16
13
3d9 4p 1P1
3d9 4p 3P1,0,3F,3D,1F3,1D2
12
3d9 4p 3P2
3
F,1D,3P,1G
15
11 11
10
10
3d9 4s 1D2
3d9 4s 3D1,2,3
9
8
7
3d10 6s
8
6
5
Cu+ 3d10
9
7
4
3
3d10 6p
3d10 5p 2P1/2,3/2
3d9 4s 4p
3d10 5s 2S1/2
6
5
3d10 4p 2P3/2
4
3d10 4p 2P1/2
1
S0
2
P,2D,2F
3d10 4d 2D3/2,5/2
3
1
2
1
3d 4s 4p
3d10 5d
2
0
9
9
3d 4s 4p
2
P,2D,2F
4
P,4D,4F
3d9 4s2 2D3/2
3d9 4s2 2D5/2
Cu0 3d10 4s 2S1/2
H. Cu+ ions in CDS: Monte Carlo model
• cfr. Ar+ ion Monte Carlo model
• Cu+ ions: important for sputtering !
I. Coupling
of the models
arbitrary RAr+ , Re,slow
Ar+/e- fluid model
Eax and Erad
dc(r), jAr+,dc(r), jAr+,0(r)
Re,ion,Ar
Ar+/Ar0f MC model
-
e MC model
Ri,ion,Ar and Ra,ion,Ar
new RAr+ and Re,slow
no
yes
convergence ?
-
nAr,met
Cu MC model
FT
Cu/Cu+ CR model
Ar* CR model
fCu+(0,E)
nCu
no
Rion,Cu , jCu+,dc(r)
Cu+ MC model
convergence ?
yes
final solution
e MC model: - Re,exc,Ar, Re,exc,met, Re,ion,met: for Ar*m model
- Re,ion,Cu : for Cu model
+
0
Ar /Ar 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
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
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)
This output = input in e-, Ar+, Ar0f MC models:
• 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)
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+
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)
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)
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 Ar+
Creation of e-
e- MC model
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)
Coupling between MC + fluid models (Summary):
Arbitrary creation of Ar+ , ee- - Ar+ fluid model
• E-field
• dc(r), jAr+,dc(r)
• jAr+,0 (r)
e-, Ar+, Ar0f MC models
No
Creation of Ar+ , eCVG ?
Yes
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)
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
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)
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-)
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)
Summary: Coupling of Ar CR model
and the 3 Cu models
Output from Ar+ / e- MC + fluid models
Ar*m density
Ar CR model
Cu sputtering flux (emp.formula)
MC model for Cu thermalization
FT * Jsput
Cu CR model
Cu0 density
(coupling back
until CVG)
• Cu+ flux entering CDS
• Cu+ creation rate in CDS
Cu+ MC model in CDS
Cu+ EDF at cathode
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
arbitrary RAr+ , Re,slow
Ar+/e- fluid model
Eax and Erad
dc(r), jAr+,dc(r), jAr+,0(r)
Re,ion,Ar
Ar+/Ar0f MC model
-
e MC model
Ri,ion,Ar and Ra,ion,Ar
new RAr+ and Re,slow
no
yes
convergence ?
-
nAr,met
Cu MC model
FT
+
Cu/Cu CR model
Ar* CR model
fCu+(0,E)
nCu
no
e MC model: - Re,exc,Ar, Re,exc,met, Re,ion,met: for Ar*m model
- Re,ion,Cu : for Cu model
+
0
Ar /Ar 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
+
Rion,Cu , jCu+,dc(r)
Cu MC model
convergence ?
yes
final solution
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
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
Illustrations of some results
Electrical characteristics:
For given V,p,T: calculate current
Calculated:
Measured:
Densities:
Argon metastable density (1000 V, 1 Torr, 1.8 mA):
Calculated:
Measured (LIF):
2.0
2.0
1.5
1.5
cm -3
cm-3
1.0
1.0
1E+012
4E+009
5E+011
0.5
3E+009
0.5
1E+011
2E+009
1E+010
0
5E+009
r (cm)
r (cm)
3E+010
1E+009
5E+008
0
2E+008
4E+009
3E+009
-0.5
1E+008
-0.5
5E+007
2E+009
1E+007
1E+009
-1.0
0E+000
-1.0
0E+000
-1.5
-1.5
-2.0
0
0.5
1.0
z (cm)
1.5
2.0
-2.0
0
0.5
1.0
z (cm)
1.5
2.0
Densities:
Sputtered tantalum atom density (1000 V, 1 Torr, 1.8 mA):
Calculated:
Measured (LIF):
2.0
2.0
1.5
1.5
cm -3
cm -3
1.0
1.0
1.6E+012
3E+012
2E+012
0.5
1.3E+012
0.5
1.0E+012
5E+011
0
2E+011
r (cm)
r (cm)
1E+012
8.0E+011
6.0E+011
0
4.0E+011
1E+011
-0.5
5E+010
2.0E+011
-0.5
1.0E+011
1E+010
-1.0
0E+000
0.0E+000
-1.5
-1.5
-2.0
1.0E+010
-1.0
-2.0
0
0.5
1.0
z (cm)
1.5
2.0
0
0.5
1.0
z (cm)
1.5
2.0
Densities:
Tantalum ion density (1000 V, 1 Torr, 1.8 mA):
Calculated:
Measured (LIF):
2.0
2.0
1.5
1.5
cm-3
1.0
cm-3
1.0
8E+010
9E+009
6E+010
8E+009
r (m)
6E+009
0
4E+009
0.5
r (cm)
0.5
5E+010
4E+010
3E+010
0
2E+010
2E+009
-0.5
1E+009
1E+010
-0.5
5E+009
5E+008
-1.0
0E+000
1E+009
-1.0
0E+000
-1.5
-1.5
-2.0
-2.0
0
0.5
1.0
z (cm)
1.5
2.0
0
0.5
1.0
z (cm)
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:
300
400
500
100000
50000
600
700
l (nm)
800
965.78
912.30
800
900
900
978.45
250000
922.45
700
978.45
965.78
922.46
862.29 852.14
866.79
826.45
840.82
912.3
763.51
842.47
3.E+15
852.14
842.46
600
840.82
350000
811.53
500
826.45
1.E+15
763.51
772.38
794.82
801.48
800.62
810.37
2.E+15
738.4
750.38
751.47
772.38 - 772.42
794.82
800.62 - 801.48
811.53
5.E+15
935.42
200000
750.39
400
696.54
706.72
2.E+15
751.47
300
603.39
604.49
3.E+15
866.79
150000
738.40
Measured:
727.29
5.E+14
415.86
420.07
434.52
Intensity (a.u.)
Ar(I) spectrum
Calculated:
696.54
706.72
Intensity (a.u.)
Optical emission intensities:
5.E+15
4.E+15
4.E+15
0.E+00
l (nm)
1000
400000
300000
0
1000
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
ye
ze




e
 x e n
1   x e e
e e
e


 n Ez   n Ey 
D
D
2 


z
x

1
2

e





x e e
 x e
1 
e e
e n

  n Ez 
 n Ey  D

D
2 

z

x

1
2

 very complicated
e

n 
y 

e

n 
y 

Fluid model: Effect of magnetic field (contd)
Therefore: Approximation:
Effect of magnetic field in electron transport coefficients:

B 0
1
1 
2
2
and D  D
B 0
1
1 
2
2
,   qB me
 = 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)
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)
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)
Δtelec = 1 ps
Δtion = 10Δtelec
Weighting
(E,B)j  Fi
MC
Collision
?
Δt
(interpolate field to
particles)
Weighting
Integration of Poisson’s
equations on the grid
()i  (E)i
(x,v)i  ()I
(interpol. charges
to grid)
No
Yes
Postcoll.
velocities
vi’  vi
5. Modeling network for Laser Ablation
Physical picture:
target
Heat conduction
 T 
 melting
 evaporation
laser
target
Heat conduction
 T 
 melting
 evaporation
laser
• Plume expansion
• Laser absorption
by plasma
Typical operating conditions:
• Pulsed lasers (Nd:YAG or excimer):
ns, ps, fs - laser pulse (10 ns fwhm)
• UV – IR wavelengths (l = 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
1. Target heating, melting, vaporization
• 1D heat conduction equation:
 T( x, t )
     T( x, t ) 


I ( x, t )



t
 x  Cp    x  Cp 


• Laser intensity: I( x, t )  I ( t ) exp   x  1  
0
• Melting: data for molten Cu
• Vaporization
 vapor density, velocity, temperature
= Input in next model
2. Expansion of evaporated plume
Conservation equations:
(mass density, momentum, energy)

  v 

t
x
  v 


p   v2
t
x


  
v 2 

   E    
t  
2 
x
 
p v 2 
  v  E     IB I laser   rad
 2 
 
Godunov method (shock waves)
3. Plasma formation
• Saha equations for ionization degree:
Ratio Cu+/Cu0:
Ratio
Cu2+/Cu+:
x e x i1
1  2  me k T 



x0
n vap 
h2

3/ 2
x e x i2
1  2  me k T 



2
x i1
n vap 
h

• Conservation of matter:
• Conservation of charge:
3/ 2
 IP1 

exp  
 kT
 IP2 

exp  
 kT
x 0  x i1  x i 2  1
x i1  2 x i 2  x e
• Internal energy density:
 3

 E   (1  x e ) k T  IP1 x i1  ( IP1  IP2 ) x i 2 
m 2

4. Laser beam absorption in plasma
Inverse Bremsstrahlung:
 IB,e n

 h c 
 Q n e n 0
 1  exp  
 l k T 

 IB,ei

 h c  4 e 6 l3 n e

 1  exp  
4
l
k
T
3
h
c
me



 2 


 3 me k T 
1/ 2
Z
 Heating term in energy conservation eq.
2
1
n i1  Z 2 n i 2
2

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 << Laser intensity)
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