Plasma Modeling I: Modeling of plasmas Annemie Bogaerts · 1 superparticle = W real particles (W = weight, e.g. 2x107) No Δt Yes Postcoll. velocities vi ’ v i Δtelec = 1 ps Δtion=

Annemie Bogaerts

Department of Chemistry, University of Antwerp (UA),Research group PLASMANT

[email protected]

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)


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)


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)


Example: DBD reactor: filamentary character

Reactor length + gas flow residence time(correlation spatial – temporal behavior)


Example: DBD reactor: filamentary character

1 microdischarge pulse + afterglow- Te ~ 2-3 eV

(30 ns) - Ne ~ 1013 cm-3

Nanosecond pulsesInterpulse time ~µs


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)


- 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)


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)


• 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


Advantage: • Very accurate +

deterministic (self-consistent)

Disadvantage:Long calculation time…


Illustration of MD simulations for film deposition(diamond like carbon)

4-fold coordinated C-atom



H-atom



Result: Calculated microscopic structure

of the film




H-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 :


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


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



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


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


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:


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


Erosion profile: Calculated - Measured

Good agreement

0 10 20 30r, [mm]

-1.2

-0.8

-0.4

0Nor

malised

ero

sion

pro

file

CalculatedMeasured


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


Closed field:

Mirror field:


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)


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


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


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)


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

Documents

Plasma Modeling I: Modeling of plasmas Annemie Bogaerts · 1 superparticle = W real particles (W = weight, e.g. 2x107) No Δt Yes Postcoll. velocities vi ’ v i Δtelec = 1 ps Δtion=