MONTE CARLO METHODS FOR ELECTRON TRANSPORT
Transcript of MONTE CARLO METHODS FOR ELECTRON TRANSPORT
MONTE CARLO METHODSFOR ELECTRON TRANSPORT
Mark J. KushnerUniversity of Illinois
Department of Electrical and Computer Engineering1406 W. Green St.
Urbana, IL 61801 USA217-244-5137 [email protected] http://uigelz.ece.uiuc.edu
May 2002
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MONTE CARLO METHODS FOR ELECTRON TRANSPORT
• The Monte Carlo (MC) method was developed during WWII for analysis of neutron moderation and transport.
• MC methods enable direct simulation of complex physical phenomena which may not be amenable to conventional PDE analysis.
• The method relies upon knowledge of probability functions for the phenomena of interest to statistically (randomly) select occurrences of events whose ensemble average is “the answer”.
• These methods are extensively used in simulating electron transport do obtain, for example, electron energy distributions.
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EXAMPLE: ELECTRON ENERGY DISTRIBUTION IN ICP
• Inductively Coupled Plasma: Ar, 10 mTorr, 6.78 MHz
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• EED at r= 4.5 cm vsDistance from Window
• Electric field (overlay) and ion density (max = 1.7 x 1011 cm-3)
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MONTE CARLO METHOD REFERENCES
• J. P. Boeuf and E. Marode, J. Phys. D 15, 2169 (1982)• G. L. Braglia, Physica 92C, 91 (1977)• S. R. Hunter, Aust. J. Phys. 30, 83 (1977)• S. Lin and J. Bardsley, J. Chem. Phys 66, 435 (1977)• S. Longo, Plasma Source Science Technol. 9, 468 (2000)• J. Lucas, Int. J. Electronics 32, 393 (1972)• J. Lucas and H. T. Saelee, J. Phys. D 8, 640 (1975)• K. Nanbu, Phys. Rev E 55, 4642 (1997)• M. Yousfi, A. Hennad and A. Alkaa, Phys Rev E 49, 3264 (1994)
• Computational Science and Engineering Projecthttp://csep1.phy.ornl.gov, http://csep1.phy.ornl.gov/CSEP/MC/MC.html
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BASICS OF THE MONTE CARLO METHOD: p(x)
• A physical phenomenon has a known probability distribution function p(x) which, for example, gives the probability of an event occurring at position x.
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p(x)
x
∫∞
=0
1dxxp )(
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BASICS OF THE MONTE CARLO METHOD: P(x)
• The cumulative probability distribution function P(x) is the likelihood that an event has occurred prior to x.
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1P00P
dxxpxPx
o
=∞=
= ∫)(,)(
')'()(
• Since p(x) is always positive, there is a 1-to-1 mapping of r=[0,1] onto P(x = 0 →x = ∞).
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p(x)
P(x
)x
P(x)
p(x)
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RANDOM USE OF P(x) TO REGENERATE p(x)
• By randomly choosing “product values” of P(x) (distributed [0,1]) and binning the occurrences of the argument x, we reproduce p(x).
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)(rPx 1−=
• The function which, given a random number r=[0,1], provides a randomly selected value of x is:
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P(x
)
x
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EXAMPLE: RANDOM P(x) TO REGENERATE p(x)
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],[)ln()(
)exp()(
)exp()(
10rr1xrPx
xx1xP
xx
x1xp
1
=−⋅∆−==
∆−−=
∆−
∆=
−
• WARNING!!! In practical problems, p(x) cannot be analytically integrated for P(x) and/or P(x) cannot be analytically inverted for P-1(x). These operations must be done numerically.
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p(x)
P(x)
x
p(x)
P(x) 2x =∆
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EXAMPLE: RANDOM P(x) TO REGENERATE p(x)
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• p(x) is reproduced within random statistical error (n-1/2=0.01).
ibins=100itrials=10000deltax=2xmax=10.dx=xmax/ibinsynorm=0.do i=1,itrialsr = random(iseed)x=-deltax*alog(1.-r)ibin=x/dxynorm=ynorm+dxy(ibin)=y(ibin)+1.
end dodo i=1,ibinsy(i)=y(i)/ynorm
end do
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p(x)regen p(x)
p(x)
x
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ELECTRON SCATTERING
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• An electron with energy ε collides with an atom with differential cross section
providing the likelihood of scattering into the solid angle centered on .
Note: Typically only the explicit dependence on polar angle θis considered. Scattering with azimuthal angle ϕ is usually assumed to be uniform.
( )φθεσ ,,
( )φθ ,
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DIFFERENTIAL SCATTERING
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• for real atomsand molecules can be quite complex (C2F6)
Christophorou, J. Chem. Phys.Ref. Data 27, 1, (1998)
• Accounting for forward scattering at higher energies (> 10s eV) is very important in simulating electron transport.
• Assuming Isotropic scattering in the polar direction yields:
( )θεσ ,
( ) ( ) ( ) ( )1r2d21P
0 oo
o −=== ∫ acos,''sin, θθθσσ
θσθσθ
1. Determine Eularian angles of 2. Rotate frame by so z-, x-axes align with3. Rotate by to yield direction of 4. Account for change in speed5. Rotate frame by to original orientation
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COLLISION DYNAMICS
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• To account for the change in velocity of an electron following a collision:
initialvr
( )φθ , finalvr
( )αβ ,
( )βα −− ,e
2initial
2
final m2vv /ε∆−=rr
( )αβ ,initialvr
initialvr
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COLLISION DYNAMICS
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• End result is the “scattering matrix” which transforms initial velocity to final velocity:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+−−+−+
⋅=
θαφθαφθβθαβφθαβφθβθαβφθαβ
coscoscossinsinsinsincoscossinsincossincossinsinsinsincossincoscossincoscos
finalfinal vv rr
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EXAMPLE: ELECTRON SWARM
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• A swarm of electrons drifts in a uniform electric field in a gashaving a constant elastic collision frequency and isotropic collisions. What is the average drift velocity?
• For constant collision frequency ν, the randomly selected time between collisions is:
• The change in energy in an elastic collision is
)( r1alog1t −⋅−=∆ν
( )( )⎟⎠⎞
⎜⎝⎛ −−=∆ θε cos1
Mm21 e
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EXAMPLE: PROGRAM DRIFT
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EXAMPLE: PROGRAM DRIFT
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EXAMPLE: PROGRAM DRIFT
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EXAMPLE: ELECTRON SWARM
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• Collision frequency = 1.048 x 109 s-1
• Drift distance = 20 cm• E/N (Electric field/gas number density) = 1-10 x 10-17 V-cm3• Electron particles=50-500 per E/N
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5 107
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50 Particles100 Particles500 Particles
DR
IFT
SPE
ED
(cm
/s)
E/N (10-17 V-cm2)
• Real atoms/molecules have many electron collision processes (elastic, vibrational excitation, electronic excitation, ionization) with separate differential cross sections.
• These processes can be statistically accounted for using MC techniques
C2F6
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MUTIPLE COLLISIONS
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Christophorou, J. Chem. Phys. Ref. Data 27, 1, (1998)
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MODEL CROSS SECTIONS
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ss S
ectio
n (1
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Electron Energy (eV)
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ElectronicExcitation
Ionization
• Compute collision frequencies for each process j having collision partner density Nj,
( ) ( ) ( ) ( )ενενεσεεν ∑=⎟⎟⎠
⎞⎜⎜⎝
⎛=
jjtotaljj
21
ej N
m2 ,
/
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5 109
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15 109
20 109
0 5 10 15 20 25 30 35 40
Col
lisio
n Fr
eque
ncy
(s-1
)
Electron Energy (eV)
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Electronic
Ionization
Total
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CUMULATIVE COLLISION PROBABILITY
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• Cumulative collision probability is sum of probability of experiencing “yours” and all “previous” collisions. (Note: Order of summation is not important.)
( )( )
( )εν
ενε
total
j
1jij
jp∑
−==
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0 5 10 15 20 25 30 35 40
Cum
mul
ativ
e C
ollis
ion
Pro
babi
lity
Electron Energy (eV)
Elastic
Electronic
Ionization
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COLLISION SELECTION PROCESS
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• Choose time between collisions based on total collision frequency.
( ) )log( 1total
r11t −−
=∆εν
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0 5 10 15 20 25 30 35 40
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mul
ativ
e C
ollis
ion
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babi
lity
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Electronic
Ionization0.4
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0.58
( ) ( )εε j21j prp ≤<−
• The collision which occurs is that which satisfies
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NULL COLLISION FREQUENCY
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• The electron energy and collision frequency can change during the free flight between collisions.
• There is an ambiguity in choosing the time between collisions.
• The ambiguity is eliminated by the “null collision frequency” (NCF).
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n Fr
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Ionization
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FREEFLIGHT
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NULL COLLISION FREQUENCY
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• The NCF is a fictitious process used to make it appear that all energies have the same collision frequency.
( ) ( )[ ] ( )εεε totaltotalnull vvMaxv −=
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lisio
n Fr
eque
ncy
(s-1
)
Electron Energy (eV)
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Ionization
Total w/ Null
Null
Total w/o Null
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ulat
ive
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lisio
n P
roba
bilit
y
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IonizationNull
Null
• Collision frequencies with Null.
• Cumulative Probabilities with Null.
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COLLISION SELECTION PROCESS WITH NULL
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• Choose time between collisions based on maximum total collision frequency.
( )[ ] )log( 1total
r1Max
1t −−
=∆εν
( ) ( )εε j21j prp ≤<−
• The collision which occurs is that which satisfies.
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ulat
ive
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roba
bilit
y
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Electronic
IonizationNull
Null
0.34
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0.41
• If the null is chosen, disregard the collision. Allow the electron to proceed to the next free flight without changing its velocity.
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SPATIALLY VARYING COLLISION FREQUENCY
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• [e] and Cl2 densities in an ICP for etching (Ar/Cl2=80/20, 15 mTorr)
• In many of the systems of interest, the density of the collisionpartner depends on position and time.
• The choice of can be ambiguous.
( ) ( ) ( )txNm2tx jj
21
ej ,,,
/
εσεεν ⎟⎟⎠
⎞⎜⎜⎝
⎛=
( )εν total
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EXTENSION OF NULL METHOD TO ACCOUNT FOR N(x,t)
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• Sample time/space domain to determine .• Compute using .
( )[ ] ( )
( ) ( )
( )( )[ ]
( ) ( )MatrixScattering
rrFscattering
trajectorynextNulltxNMax
txNr
trajectorynextNullj
prpCollisiontavvtvxx
r1xMax
1tTimestep
541
j
j3
j21j
1total
•=•
→→>•
→=•
≤<•∆⋅+=∆⋅+=•
−−
=ƥ
−
−
,,
,,
:,
log,
:
φθ
εε
εν
( )],[ txNMax j
( )],[ txNMax j( ) ( ) ( )[ ] ( )ενενενεν nulltotaltotalj Max ,,,
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SAMPLING AND INTEGRATION METHODS
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• Electron distributions are obtained by sampling the particle trajectories; binning particles by energy, velocity, position toobtain .
• How you sample affects the distribution function you derive.• Integration ∆t should be less than: ∆tcol, fraction of 1/νrf,
fraction of or other constraining frequencies. • ∆t can be different for each particle. Particles can “diverge in
time” until they reach a time when they must be coincident. • Recommended sampling and integration strategy:
•Choose t(next collision) = t(last collision) + ∆tcol•Integrate using ∆t ≤ t- t(next collision) •Sample particles for every ∆t weighting the contribution by ∆t .•When reach t(next collision), collide and choose new ∆tcol
( ) ( )trvfortrf ,,,, rrrε
[ ] 1dxdE
Ev −
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EXAMPLE: ELECTRON ENERGY DISTRIBUTION
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• Compute electron energy distribution and rate coefficients for idealized cross sections.
• Conditions:
• E/N: 100 x 10-17 V-cm2 (100 Td)• Drift distance: 3 cm (sample after
0.5 cm)• Number of Particles: 2000 0
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10
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0 5 10 15 20 25 30 35 40
Cro
ss S
ectio
n (1
0-16
cm2 )
Electron Energy (eV)
Elastic
ElectronicExcitation
Ionization
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EXAMPLE: ELECTRON ENERGY DISTRIBUTION
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• Sampling method• 1: Every ∆tcol• 2: Every ∆t (constant) < ∆tcol
• Rate Coefficients (cm3/s)•Elastic 1.1 x 10-7
•Electronic 2.0 x 10-9
•Ionization 9.9 x 10-14
• Number of Collisions:•Elastic 1.46 x 107
•Electronic 2.21 x 105
•Ionization 10•Null 5.93 x 107
• Lesson!!! Do NOT compute rate coefficients by counting collisions! Directly compute rate coefficients from EED.
( ) ( ) ( ) εεεσεε dm2trftrk 21
21
e
//
,,, ⎟⎟⎠
⎞⎜⎜⎝
⎛= ∫
rr
10-8
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10-5
10-4
10-3
10-2
10-1
100
0 5 10 15 20
F(e) Sampling 1F(e) Sampling 2
Ele
ctro
n E
nerg
y D
istri
butio
n (e
V-3
/2)
Energy (eV)
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EXAMPLE: ELECTRON ENERGY DISTRIBUTION
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• Required samplings are dictated by the tail of the EED. Rate coefficients for high threshold events are sensitive to the tail.
10-8
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10-6
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10-4
10-3
10-2
10-1
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0 5 10 15 20
50 Particles100 Particles500 Particles2000 Particles
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ctro
n E
nerg
y D
istri
butio
n (e
V-3
/2)
Energy (eV)
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10-5
10-4
10-3
12 13 14 15 16 17 18 19
50 Particles100 Particles500 Particles2000 Particles
Ele
ctro
n E
nerg
y D
istri
butio
n (e
V-3
/2)
Energy (eV)
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EFFICIENCY ISSUES
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• Create look-up tables where-ever possible (memory and lookups are cheap, computations are expensive).
• Minimize null-collisions by having sub-intervals of energy range with different .
• Be cognizant of “pipelining” opportunities. Perform array operations with stencils to include-exclude indices for particles which are added-removed due to attachment, losses to walls or ionization.
• Take advantage of cyclic conditions to bin particles by phase as opposed to time.
• NEVER hardwire anything!! Define all cross sections, densities from “outside.”
( )[ ]εν totalMax
ADVANCED TOPICS
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TYPICAL INDUCTIVELY COUPLED PLASMA FOR ETCHING
• Power is coupled into the plasma by both inductive and capacitive routes.
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WALK THROUGH: Ar/Cl2 PLASMA FOR p-Si ETCHING
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• The inductively coupled electromagnetic fields have a skin depth of 3-4 cm.
• Absorption of the fields produces power deposition in the plasma.
• Electric Field (max = 6.3 V/cm)
• Ar/Cl2 = 80/20• 20 mTorr• 1000 W ICP 2 MHz• 250 V bias, 2 MHz (260 W)
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Ar/Cl2 ICP: POWER AND ELECTRON TEMPERATURE
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• ICP Power heats electrons, capacitively coupled power dominantly accelerates ions.
• Ar/Cl2 = 80/20, 20 mTorr, 1000 W ICP 2 MHz,250 V bias, 2 MHz (260 W)
• Power Deposition (max = 0.9 W/cm3) • Electron Temperature (max = 5 eV)
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Ar/Cl2 ICP: IONIZATION
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• Ionization is produced by bulk electrons and sheath accelerated secondary electrons.
• Ar/Cl2 = 80/20, 20 mTorr, 1000 W ICP 2 MHz,250 V bias, 2 MHz (260 W)
• Beam Ionization(max = 1.3 x 1014 cm-3s-1)
• Bulk Ionization(max = 5.4 x 1015 cm-3s-1)
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Ar/Cl2 ICP: POSITIVE ION DENSITY
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• Diffusion from the remote plasma source produces uniform ion densities at the substrate.
• Ar/Cl2 = 80/20, 20 mTorr, 1000 W ICP 2 MHz,250 V bias, 2 MHz (260 W)
• Positive Ion Density(max = 1.8 x 1011 cm-3)
MATCH BOX-COIL CIRCUIT MODEL
ELECTRO- MAGNETICS
FREQUENCY DOMAIN
ELECTRO-MAGNETICS
FDTD
MAGNETO- STATICS MODULE
ELECTRONMONTE CARLO
SIMULATION
ELECTRONBEAM MODULE
ELECTRON ENERGY
EQUATION
BOLTZMANN MODULE
NON-COLLISIONAL
HEATING
ON-THE-FLY FREQUENCY
DOMAIN EXTERNALCIRCUITMODULE
PLASMACHEMISTRY
MONTE CARLOSIMULATION
MESO-SCALEMODULE
SURFACECHEMISTRY
MODULE
CONTINUITY
MOMENTUM
ENERGY
SHEATH MODULE
LONG MEANFREE PATH
(MONTE CARLO)
SIMPLE CIRCUIT MODULE
POISSON ELECTRO- STATICS
AMBIPOLAR ELECTRO- STATICS
SPUTTER MODULE
E(r,θ,z,φ)
B(r,θ,z,φ)
B(r,z)
S(r,z,φ)
Te(r,z,φ)
µ(r,z,φ)
Es(r,z,φ) N(r,z)
σ(r,z)
V(rf),V(dc)
Φ(r,z,φ)
Φ(r,z,φ)
s(r,z)
Es(r,z,φ)
S(r,z,φ)
J(r,z,φ)ID(coils)
MONTE CARLO FEATUREPROFILE MODEL
IAD(r,z) IED(r,z)
VPEM: SENSORS, CONTROLLERS, ACTUATORS
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HYBRID PLASMA EQUIPMENT MODEL
SNLA_0102_39
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ELECTROMAGNETICS MODEL
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• The wave equation is solved in the frequency domain using sparsematrix techniques (2D,3D):
• Conductivities are tensor quantities (2D,3D):
( ) ( )t
JEtEEE
∂+⋅∂
+∂
∂=⎟⎟
⎠
⎞⎜⎜⎝
⎛∇⋅∇+⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅∇∇−
σεµµ
1 12
2
)))((exp()(),( rtirEtrE rrrrrϕω +−′=
( )m
e2
om
2z
2zrzr
zr22
rz
zrrz2r
2
22
mo
mnq
mqiEj
BBBBBBBBBBBBBBBBBBBBB
B
1q
m
νσνωασ
ααααααααα
αανσσ
θθ
θθθ
θθ
=+
=⋅=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
++−+−+++−+−++
⎟⎠⎞⎜
⎝⎛ +
=
,/
rv
r
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ELECTROMAGNETICS MODEL (cont.)
EIND_0502_11
• The electrostatic term in the wave equation is addressed using aperturbation to the electron density (2D).
• Conduction currents can be kinetically derived from the ElectronMonte Carlo Simulation to account for non-collisional effects (2D).
⎟⎠⎞
⎜⎝⎛ +⎟⎟
⎠
⎞⎜⎜⎝
⎛ ⋅⋅−∇=∆
∆==⋅∇ ω
τσ
εερ i
qEnnqE e
e 1/ ,
( ) ( ) ( )( )( ) ( ) ( ) ( )( )( )rtirvrqnrtirJtr veevorrrrrrrr φωφω +−=+= expexp,J e
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ELECTRON ENERGY TRANSPORT
where S(Te) = Power deposition from electric fieldsL(Te) = Electron power loss due to collisionsΦ = Electron fluxκ(Te) = Electron thermal conductivity tensorSEB = Power source source from beam electrons
• Power deposition has contributions from wave and electrostatic heating.
• Kinetic (2D,3D): A Monte Carlo Simulation is used to derive including electron-electron collisions using electromagnetic fields from the EMM and electrostatic fields from the FKM.
SNLA_0102_41
( ) ( ) ( ) EBeeeeeee STTkTTLTStkTn +⎟⎠⎞
⎜⎝⎛ ∇⋅−Φ⋅∇−−=∂⎟
⎠⎞
⎜⎝⎛∂ κ
25/
23
( )trf ,, rε
• Continuum (2D,3D):
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PLASMA CHEMISTRY, TRANSPORT AND ELECTROSTATICS
• Continuity, momentum and energy equations are solved for each species (with jump conditions at boundaries) (2D,3D).
AVS01_ 05
• Implicit solution of Poisson’s equation (2D,3D):
( ) ( )⎟⎠
⎞⎜⎝
⎛⋅∇⋅∆+=∆+Φ∇⋅∇ ∑∑
iiqt-- i
iiis Nqtt φρε
r
iiii SN
tN
+⋅−∇= )v( r
∂∂
( ) ( ) ( ) ( ) iii
iiiiiii
i
ii BvEmNqvvNTkN
mtvN µ
∂∂
⋅∇−×++⋅∇−∇=rrrrr
r 1
( ) ijjijj
imm
j vvNNm
ji
νrr−−∑
+
( ) 222
2
)()U(UQ E
mqNNP
tN
ii
iiiiiiiii
ii
ωννε
∂ε∂
+=⋅∇+⋅∇+⋅∇+
∑∑ ±−+
++j
jBijjij
ijBijjiji
ijs
ii
ii TkRNNTTkRNNmm
mE
mqN 3)(32
2
ν
University of IllinoisOptical and Discharge Physics
FORCES ON ELECTRONS IN ICPs
EIND_0502_09
• Inductive electric field provides azimuthal acceleration; penetrates(1-3 cm)
• Electrostatic (capacitive); penetrates (100s µm to mm)
• Non-linear Lorentz Force rfBvFrr
×= θ
( )( ) 212
eeDDS en8kT10 πλλλ =≈ ,
( )( ) 212
eoe nem µδ =
Eθ
Es
BR, BZ
δ
λs
vθ
BR
vθ x BR
• Collisional heating:
• Anomalous skin effect:
• Electrons receive (positive) and deliver (negative) power from/to the E-field.
• E-field is non-monotonic.
University of IllinoisOptical and Discharge Physics
ANAMOLOUS SKIN EFFECT AND POWER DEPOSITION
EIND_0502_12
Ref: V. Godyak, “Electron Kinetics of Glow Discharges”
( ) ( )BvF
dtrdtrEttrrtrJe
skinmfp
rvr
rrrrrrr
×=
=
>
∫∫ ''','',,',),( σ
δλ
( ) ( )trEtrtrJeskinmfp ,,),(, rrrrrσδλ =<
University of IllinoisOptical and Discharge Physics
COLLISIONLESS TRANSPORT ELECTRIC FIELDS
• We capture these affects by kinetically deriving electron current.
• Eθ during the rf cycle exhibits extrema and nodes resulting from this non-collisional transport.
• “Sheets” of electrons with different phases provide current sources interfering or reinforcing the electric field for the next sheet.
• Axial transport results fromforces.
EIND_0502_13• Ar, 10 mTorr, 7 MHz, 100 W
rfBvrr
×
( ) ( )( )( ) ( )( )okk
kk
o
ttirvq
dAttirj
−
=⋅−
∑∫
ω
ω
exp
exprr
rr
ANIMATION SLIDE
University of IllinoisOptical and Discharge Physics
POWER DEPOSITION: POSITIVE AND NEGATIVE
• The end result is regions of positive and negative power deposition.
SNLA_0102_19
• Ar, 10 mTorr, 7 MHz, 100 W
POSITIVE
NEGATIVE
University of IllinoisOptical and Discharge Physics
POWER DEPOSITION vs FREQUENCY
• The shorter skin depth at high frequency produces more layers ofnegative power deposition of larger magnitude.
SNLA_0102_32
• 6.7 MHz(5x10-5 – 1.4 W/cm3)
• 13.4 MHz(8x10-5 – 2.2 W/cm3)• Ar, 10 mTorr, 200 W
• Ref: Godyak, PRL (1997)
MAXMIN
University of IllinoisOptical and Discharge Physics
TIME DEPENDENCE OF EEDs: FOURIER ANALYSIS
• To obtain time dependent EEDs, Fourier transforms are performed “on-the-fly” in the Electron Monte Carlo Simulation.
• As electron trajectories are integrated, complex Fourier coefficients and weightings are incremented by.
EIND_0502_15
⎟⎟⎠
⎞⎜⎜⎝
⎛ΨΨ
=Ψ
= −
)],(Re[)],(Im[tan),(,|),(|),( 1
kin
kinkin
ik
kinkin r
rrW
rrA r
rr
rr
εεεφεε
( ) ( ) ,))(())((exp, 21
21 ∑∑ −∆±−∆±∆=∆Ψ ++
kjklkljii
jjjjkin rrrtintwr rrrr δεεεδωε
∑∑ −∆±∆=∆ ++k
jklkljj
jjk rrrtwW ))(( 21 rrrδ
• The Fourier coefficients are then obtained from:
University of IllinoisOptical and Discharge Physics
TIME DEPENDENCE OF EEDs: FOURIER ANALYSIS
• The time dependence of the nth harmonic of the EED is then reconstructed
∑=
=N
nn trftrf
0),,(),,( rr εε
[ ]),(sin ),(),,( rntrAtrf nnnrrr εφωεε +=
• …and the total time dependence of the electron distribution function is obtain from summation of the harmonics:
….where f0 is the time averaged distribution function.
EIND_0502_16
EXCITATION RATES: “ON THE FLY”
• In a similar manner, Fourier components of excitation rates can be obtained directly from the Electron MCS
• For the nth harmonic of the mth process,
• The resulting Fourier coefficients then reconstruct the time dependence of electron impact source functions.
( )∑∑ −∆±+→ ++k
jklklj
jjmjnmlnml rrrtinwkk ))((exp)( 21'' rrrδωυεσ
)Re()Im(tan
,)sin(][)(
nml
nml1nml
n
0nnmlnmlmllml
kk
tnkNetSm
−
=
=
+= ∑
φ
φω
University of IllinoisOptical and Discharge PhysicsEIND_0502_17
University of IllinoisOptical and Discharge Physics
ALGORITHM FOR E-E COLLISIONS
• The basis of the algorithm for e-e collisions is “particle-mesh”.
• Statistics on the EEDs are collected according to spatial location.
• A collision target is randomly selected from the EED at that location and a random direction is assigned for the target’s velocity.
• The relative speed between the electron and its target electron is used to determine the probability for an e-e collision
SNLA_0102_01
( ) ( ) 21, eeeeeee vvvtvvrnP rrr−=∆⋅⋅⋅= σ
( ) 22
1
22 4,ln14
vmqb
bbv
e
oo
o
Doee
πελπσ =⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+=
• If a collision occurs, classical collision dynamics determine the change in momentum of the electron.
• The consequences of e-e collisions on the targets are obtained by continuously updating the stored EEDs.
University of IllinoisOptical and Discharge Physics
ICP CELL FOR INVESTIGATION
• The experimental cell is an ICP reactor with a Faraday shield to minimize capacitive coupling.
MCSHORT_02_29
University of IllinoisOptical and Discharge Physics
TYPICAL CONDITIONS: Ar, 10 mTorr, 200 W, 7 MHz
• On axis peak in [e] occurs in spite of off-axis power deposition and off-axis peak in electron temperature.
MCSHORT_05_30
University of IllinoisOptical and Discharge Physics
TIME DEPENDENCE OF THE EED
• Time variation of the EED is mostly at higher energies where electrons are more collisional.
• Dynamics are dominantly in the electromagnetic skin depth where both collisional and non-linear Lorentz Forces) peak.
• The second harmonic dominates these dynamics.
SNLA_0102_10
• Ar, 10 mTorr, 100 W, 7 MHz, r = 4 cm ANIMATION SLIDE
University of IllinoisOptical and Discharge Physics
TIME DEPENDENCE OF THE EED: 2nd HARMONIC
• Electrons in skin depth quickly increase in energy and are “launched” into the bulk plasma.
• Undergoing collisions while traversing the reactor, they degrade in energy.
• Those surviving “climb” the opposite sheath, exchanging kinetic for potential energy.
• Several “pulses” are in transit simultaneously.
SNLA_0102_11
• Ar, 10 mTorr, 100 W, 7 MHz, r = 4 cm
• Amplitude of 2nd HarmonicANIMATION SLIDE
HARMONICS IN ICP
• To investigate harmonics an Ar/N2gas mixture was selected as having low and high threshold processes.
• e- + Ar → Ar+ + e- + e-, ∆ε = 16 eV
High threshold reactions capture modulation in the tail of the EED.
• e- + N2 → N2 (vib) + e-, ∆ε = 0.29 eV
Low threshold reactions capture modulation of the bulk of the EED.
• Base case conditions: • Pressure: 5 mTorr• Frequency: 13.56 MHz• Ar / N2: 90 / 10• Power : 650 W
0.18 2.7
Electron density (1011 cm-3)
University of IllinoisOptical and Discharge PhysicsSNLA_0102_25
SOURCES FUNCTION vs TIME: THRESHOLD
• Ionization of Ar6 x 1014 – 3 x 1016 cm-3s-1
MIN MAX
• Excitation of N2(v)1.4 x 1014 – 8 x 1015 cm-3s-1
• Ionization of Ar has more modulation than vibrational excitation of N2 due to modulation of the tail of the EED.
University of IllinoisOptical and Discharge PhysicsSNLA_0102_26
ANIMATION SLIDE
HARMONICS OF Ar IONIZATION: FREQUENCY
• At large ω, both υm/ω and 1/(υmω) are small, and so both collisional and NLF harmonics are small.
• At small ω, both υm/ω and 1/(υmω) are large. Both collisional and NLF contribute to harmonics.
0.0
0.1
0.2
0.3
0.4
0.5
0 10 20 30 40 50Frequency (MHz)
S2
S4
S1, S3S
n / S
0
• Harmonic Amplitude/Time Average• Ar/N2=90/10, 5 mTorr
University of IllinoisOptical and Discharge PhysicsSNLA_0102_27
HARMONICS OF Ar IONIZATION: PRESSURE
Pressure (mTorr)
0.0
0.1
0.2
0.3
0.4
0.5
0 10 20 30 40 50 60 70S
n / S
0
S2
S1
S3-S8
S4
• Harmonic Amplitude/Time Average
• Ar/N2=90/10, 13.56 MHz
• At large P, υm/ω is large and 1/(υmω) is small. Harmonics result from collisional (or linear) processes.
• At small P, υm/ω is small and 1/(υmω) are large. Harmonics likely result from NLF.
University of IllinoisOptical and Discharge PhysicsSNLA_0102_28
• 5 mTorr6 x 1014 – 3 x 1016 cm-3s-1
• 20 mTorr1.5 x 1014 – 1.7 x 1016 cm-3s-1
TIME DEPENDENCE OF Ar IONIZATION: PRESSURE
MAXMIN
• Although Brf may be nearly the same, at large P, vθ and mean-free-paths are smaller, leading to lower harmonic amplitudes.
University of IllinoisOptical and Discharge PhysicsSNLA_0102_29
ANIMATION SLIDE